Advanced Texts in PhysicsThis program of advanced texts covers a broad spectrum of topic,s which are ofcczOt=demerghgHtereMhphysiO.EaGbookproddesacompzGensive=dyet accessible introductioa to a âeld at the forelont of modern research. As such,Gese texts are intendedfor serlioz undernduate aadgraduate stnrlents atthe MSand PhD levek however, researc,h scientjsts seeklng an introduction to particalarareas of physics will also bemeât from the titles in tlzis collection.

SpringerBerïinHeidelbergN'ew YorkHong KongL ondonMilan17t4r:Tokyo

ONLINE LIBMRYPhysics nndAsaonomy

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Professor Dr- Friedhelm BechstedtFzieclt-iczz-ssr%a'llc Universitiit JenaInstim: ffr Festkörpertxeoz'ietmd Theoretiscàe OpdkMax-wien-?latz z

07745 Jena, Gerrnnnye-rnnll: beciLtêifto.physilc.uri-lYxde

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physics/ Medolm Becbstedt. p. cm- ,- tAdmaced texts izz physicsv ISSN z4Jp-zö74) Jndudes blbli-egtapbicalrieauces andindex- BBN 3-540-:0635-4 (acid-âee paper) z. Snrfnces (Php2cs) 1.Titlq.11.Series- QClD.4.S94 zool stso-4lz7-dcm zoo3oh'o6'1.t

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ISBN 3-540-00635-4 Springer-verlag Berl'in Heidelberg New York

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Friedhelm Bechstedt

Principlesof Surface Physics

WiG zo7 Figttres

L-

*-z Springero

To Andreas, Slzsanne, and UV

M the rivers raa iato the sea; yet the sea isnot h111: unto the place from whence the rivezscome, thithe,r they retlvn again.

M1 thina are full of hbour; mn.n rvmnot utterit: the eye is not satieed WI.II seeing, nor the ear

fLIIH with hfu.rlng.

Ecclezias'te(or, The Preadzea'l

Preface

J.n recent decades, surface atd intezface physics has become all increasiugly '

important subdiscipve withic the physics of condensed matte,r as well as a'ainterdisciplico feld between physics, c-z:ystallography, cltemsKtzy, biology,a'ad materials science. There are several driving forcu for the development ofthe âeld, among them semiconductor tenBnology, new materials, epitaxy andchemical catalysis. The (alectzlcal and optical properties of nanostructmesbased on diferen.t seMconductors aze governed by the interfaces or, at least,by the presence of ictezfaces. A microscopic kmdemtanding of the growthprocœses zequires the investigation of the surface processes at an atomiclevel. Elementary processes on surfaces, such as adsorptioa ar.d desorption,play a key role in the understanding of heterogeneo)ls catiysis.

During the course of the surface hwestigations, it has bee'n possible toobserve a dzomn.tic progzess i'a tile abttit.g to stad.y surfaces of materials ingeneri, aa.d on a microscopic scale in particlzlar. There are t'wo mai'n reasons

for this provess. Fcom the expem'mental poiat of view it is largely due tothe development am.d awdlability of aew types of powtkrhtl microscopes. Spec-taclzlar advances iu techniques suck ms srltnns'ng tl:nmeling microscopy nowallow us to observe individual atoms on sarfaces, and to follow tlb.e,ir patluswith a clarity qlnl'rnaginable a few years ago. bnrom the theoretical point ofview (or rathez the viewpoint of simulation) progress is related to the wideàvailability of computers and tEe c'lrnmn.tic inczeaze of their power. Todayjearly methodological developments suc,h as dezusity ftmctional theory allow afull Trnunttlm-mechanical treatment of electrons ia materiab. I'a the futm'e,compate,r experiments will be able to simtzlate the beavior of surfaces aadprocessE's on s'prTnces at the level of individual atomic corœ and their sm-

zotmding electrons with high accmacy and Temarlçable predic-tive power.TMS enormous progress in surface science has been documented i.c rnn.ny

excellent boolcs on surface structmw, surface pzocesses, theoretkal modelingof surfaces, a'ad surfaces acd interfaces of particular solids Like semiconduc-tors. Howevea', only very few books try to treat thc subject tn a lpnifed andcompreheasive way, This holds true in particulaz for the ex-persrnental audtheoretical methods tzsed in surface physics and, most of a11, fo< the princi-ples an.d coacepts. Hence 7 perceived the acsed for a book dealing wxith. smfacephysics at the level of aa advanced textbook. The n'l= here is to descdbe the

V1I1 Preface

ftmdaentals of the seld and to provide a lamework for the cliscussion ofsttrface phenomena in a siugle text. Examples of pazticulaz suzface.s of ma-teriaks suG as sczniconductoz's or metals aze only discussed as a means ofillus'tzating the fundamentals or pzinciples. Special theoretical or mcperimen-tal methods of suzface studies are mentioned but not described in detail.Particulaz attention ks paid to physical approaches that c-an be applied tothe discovery and dismlssioa of nove,l surface phenomena. Among them are

symmetzy azguments, energetics, driving forces and elements of gcometrickchauges, elementazy exdtationsl and other charadezistic propertics. Theseelements should lzelp to clazsify srface pzoblems and to facilitate their km-

derstanding. The only prior knowledge assumed is tmdewaduate phMcs andmathematics course material. Mp.imly textbook quantum meianics and ge-ometrical arguments az'e used to discuss and desczibe sarfaces and surfaceprocesses. Graduate-level topics such as second qxpxntization are avoideê.Whenevez macy-body argttments are needed, a bzief (more phenomenolog-ical) introduction is Tven. Greeu's ffmctions are intzoduced by using their'relationship to obsezvable quautities. The use of gzoap titeor.r is restricted togeometrical azguments and its notations. Feynmau diagrnml aze only shownto illustrate interactions between pazticlœ on s:Arfnces. An Hended mzbjxtiade,x will help s'tudeats and scientixs to use the book for reference and dmuiag their every-day scieutisc wozk. To keep formulas to a mnrageable length,thty are written in tke lamework of cgs uaits. In Hdttion, use is vnnzie of thefad that the energies of valence electrons are of the order of electron voltsand atomic distances are of the order of angstiozns.

The book is based oc lectmes given at the Humboldt-universttât z'u Berlinand the nieth4ch-snhlller-univemRât Jenn. and on s'tudeat sernsnars. I wouldlike to nrlmowledge maay discussions with colleagues around the world, Iakso thnunk my cozeagues azd students for their critical reading of paz'ts ofthe manuscript. Among others 1 am iadebted to R. Del Sole, N. Esser, J.Fnrkhrniiller, S. Glutsch, P. Kratze, J. Neugebauer, G. Onida, M. Ro1)l-fmg, A. Se.himdlrnnzyz, W.G. Sn%ml'dt, azzd J.-M. Wagner. The typing of them=uscdpt was achieved with competence snd inBnlte patience by my sem'e-

tanr Sylda Hofmnnn. Coordination and production of the book were tmder-taken by Petra 'mdber and Angeh Lahee 1om Sptinger Verlag.

. Jena, Ma.rc.h 2003 '

Friedhelm Sec/udeff

1.1 t%lodel S =es......................................... 11.1.1 S ace V=c B14IV ..........-.......m.......-... 11.1.2 The S ace ms a Physick Objed - . . . . . . . . . . . . . . - - . . 1

1.2 TvmpDimRnql'on/ Crys .-.-.-...............-......... 3 -

1.2- 1 Lattice Planes of BT7lk- Crys-tals . . . . . . - , . . . . . . . . . . . . 31.2.2 Om'ented Slabs ---..............--..............-. 81.2.3 Id S'arface. Pln.nn.r Pom- t Gzoups . . . . . . . . . . . . . - . , 111.2.4 R Smfaces: Reconstruction xnd Relaxation . . . . . . . . 161.2.5 Superlattices at S'urfa' ces . . . . . . . . . . . . . - . . . . . . . . . . . . (1.91.2.6 Xvood Notatl'oa ................................., 201.2.7 S etry t'Xs:l-qx-s6cat-on . . . . . . . .. . . . . - . . . . . . . . . . . . . . 25

1.3 Reciprocf Space ....................................... 291.3- 1 D1r- ect n.nd Rem-pro Lat-ti ces . . , . . . . - . . . , . . . . . . . . . 291.3-2 zBri'llo-l'lt-n Zones - - - . . . . . . . . . . . . - - . . . . . . - - . . . . . . . - - . 351.3-3 Projection of 317 Onto 2D Bm-llonain Zones . . - . - . . . . . . 37:..3.4 S etzy of Points aad Lm- es in *pro Space . . . 41

2. Thermod *cS . ,.............................. 452. 1 Km- etz-c Proceses nrd S aces m- Equilibrz-' Te'cl . . . . . . . . . . . - . . 452.2 Thermodynzmlc Relatio'nq for S aces . . - - . . . . . . . . . . . . . . . . 46

2,2.1 Thermody-aAmic Potentim.lA . . . . . . . . . - - - . . . . . . . . . L . , 462.2.2 S c.e Moda-qcatz'on of Thermodynxmic Potentin.lq . . . 48

' 2.2.3 S ace Tension a'nd STq'rface Stress . . . . . . . . . . . . . . - . . 492.3 E

-

-brixlm Sbnpe of Smnall als . . . . . . . . . . - . . . . . . . . - . . 512.3.1 à'nsqotropy of S ace Bnergy , . . . . . . - . . . . . . . . . . - . . . 512.2.2 Absolute Values for S ace Energie.s . . . . . . . . . . . . . . . . 552.7.3 'W:t1@ Clonntruction . . - . . . . . . . . - . . . . - - - . . . . . - . - . . . . 57

2.4 S ace Elergy n.m d lTorphology ...-...............-.... 592.4.1 Facett' and itoughe

'

...............--...... 59. 2.4.2 3D Versu 2D Gro<h...,...............,......... 60

2.4.3 Forraation of Qurttlm Dots ........-............, 632.5 Stoi 'oraetry Dependeace......,........................ 66

2.5-1 Thevodrp.m-lc Approvh .-.--.........-...- 66

X Contents

2-5-2 Appro -

ations for Smface Energies . . . . - - . - - - . . . . . - 672.5.3 Ch

'

Potenti .-............................ 69

2.5.5 St :1'- of A or tes ............................ 78

3. Bonaing nmd En etics 813.1 Orbitzs And Bondl-ng .-.-.-.-............-.............. 81

3.1-1 OnoElKtroa Picture -.-.-.-.-.-....-..-..-..-.-.. 813.1.2 Tl' Bua- ding Jkpproach .....-.....----...... 823.1-3 Atornl-c Oz-bitni!q imd Thelr- Tnteraction . . . . . . . . . . , . . .

. 85

3.1.4 Bonding Hybm'ds ................................. 893.1.5 Bonds And Band: ....................-........... 94

3.2 Danfling Bonds ........................................ 97' 3.2.1 Forrnatl'oa of D=g

'

Hybrids S'?3.2.2 Tn6uence o'n Electronic States . . . . . . . . . . . . - . - - . - - . . . 98

3.3 Total Energy an.d Atornl-c Forces . . . - - - . . . .-.

. . . . . . . . . . . . . . 1023.3.1 Basic Approvimatiomq . . . . . . . . . - . - . - . . . . . . . . . - . - . . . 1û23.3.2 Potential Enargy Suface Fmd Forces . . . . . . . . . . . . . . - . 1033.3.3 S iCe 171-6713 Fkl-on.

. . - . . . . . . . . - . - . . . . - . . . . . . . . . @ . . . . 1053.4 Quantitative Descn'ption of Structlzre Fmd Stabzh' 'ty' . . . . . . . . . 109

3.4.1 Densi'ty Rlnctiona.l Theory . . . . . . . . . . . . . . . . . . . . . . . . 1093.4.2 Bnzld-structure n.nd Tntezaction Coatm-butionK . . . . . . . . 1123.4.3 lklodeli-ng of Surfaces . - - - . . . . . - - - - - - - - - . . . . - . . . . . . . 114Bond Brenn-mg: Accompanpn' g Charge TrlmAfersn.nd .xtornic Displaceanemts . . . - . . . . . . . . - . . . . . . . . . . . . . . . . . . 1223.5.1 Charaderi qtie Chlmge m- Total Energy . . . . . . - - . . . . . 1223.5.2 Energy Gn.5n Due to Structtzral

a'nd ConGglzrational Ch es . - . - . - . . . . - - . . . . . . . . . . 1253.5-3 Energy Gn.l-n and Electron ''lnrnznqfe.,r . . . - . . . . . - . - . . . . . 129

4. econstructl*on Slements ...............,................. 1334.1 Recoctxcty'on acd Bonding ............................. 133

4.1.1 et 'c BôndS..,...................-............ 1334.1.2 Strong Ionsc Bonds .........,,,...-..............- 1354.1.3 1-x- Co ent and Iorsc Bon . . . . . . . - . . . . . . . . . . . 1364.1.4 P ' ciples of Semiconductor Surface Recomstlmction . . . 1384.1.5 Electron Co trc' g Tt es --......---....-.... 141

4.2.1 Z-)g-z -

of Cations and -ons . . . . . . . . . . . . . . 143

4.2.2 x-bond C '

-.....-...-.-.--.-.------..------ 1504.2.3

'

tZ *

.............................,.-... 1564.3 Dan- erS

---................................,.--...*.*.... 1584 . 3 . 1 S

-

c 1) ztnA- e r s . - - - . - . . . . . . . . . - . - . . . . . . . - . .

-

. . . . 1 5 84.3.2 etri c (Dil4-1 ers - . - - . - . - - - . - . . . - - . . . . - . . - - - . . . 1634-3.3 lRetero

*

ers . . . . . . . . . . . . . . . . . . . . . - . . . . . . . - . . - . - - 1.66

nt-nts 7(1

4.3.4 Bn'd '

Groum ..---............................ 1684.4 AdatomK xnd Adclustr ..-....-.-.-.--............-.,.. 170

4.4.1 bolated Adato=s.......................-........- 1704.4.2 Adatomq Accompnnnsed by Rest Atoms . . - . . . . . . . . . . - 1724.4.3 Adatomq Combined

with Other Recozustrudion Elements . . . . . . , . . . . . . . . . 176

4.4.5 Cpetrn.rners . . . . . . . . . . . . . . . . . . . . . , .. - . . . . . - . - . - . . . . - 183

Elementary Excs-tations 1:Sl-ngle Electronl-c Quasl-particles .'-

. . . - - . . . . . . . . . . , . - . , . . . - . 187 '

5.1 Electronq &ad So1%......,.............................. 187o- 1.1 Excitation acd Qua-sl-p icle Charader . . . . . . . . . . . . . l87a- 1.2 Sc

-

Ahlnn '

S tr py...............-.. 1885-1.3 Photo-mlssion Spectrosco

And Tnverse Photo-rrll-qm-on . . . . . . . . . . . . . . . . . . - - . . . . . 1945.1.4 Satelpltes .......................*......-.-....... 199

5.2 hlnmy-Body EFects ................................-.... 2O15.2.1 Quasip icle aatà'on..............,............. 2015.2.2 QAlxqip icle Sbsf'ts a'cd Spectral eights . . . . . . . . . . . 2035.2.3 Screening ENea,r S'urfaces . . . . . . . . . . . . . . . . . . . - . . . . . . . 2O7

5.3 Quasip icle S e States . . . . . . . . . - - . . . - . . . . . - . . . . . - . . 2115.3.1 S ace B=I'er.................................-. 2115.3.2 Chnracteristi c Energies - . - - . - . . . . - . . . . . . . . . . . . . . . . . 2 145-3-3 State Lof-iml l-zzation . . . . . . . . . . . . - . - . . - . . . . . . - . . . . - . . 2 175.3.4 Qusmip icle Bands ar.d Gaps . . . . . . . .. . . . . . . . . . . . . . 221

5.4 Strong Electron Correlation - . - . . . . - . . . . - . - . - . . . . . . . . . . . . . 2255.4.1 ïmnage Stat&...-----.....-....................... 2255.4.2 hlott-TTubbard Bxnœ...-....----...*............. 231

Element Excitations -.

Pnx-r nmd Collectezve Excitatl-onq . - . . . . . . . . . . . - . . . . - . - - . . . - - 2376.1 Probing S aces by Excitations . . . . - - . . - - . . . . , . . . . . . . . . . . 237

6.1- 1. Optical Spectroscopl-es - . - . - . - . . - . . . . . . . . . . . . . . . . . . 2376.1.2 L- t Propa tion in S aces . . . . . . . . . . . . . . . . . . . . . - 24O6.1.3 Eledron Enea'gy Losses - . - . . . . - - - . . . , . . . , . . . . . . . . . 2436.1.4 1?-,1../-f3 n'n Scattelnn- g . . , . . . . . . . . . . . - . . . . . . . . . . . . . . . . - 250

6.2 Electron-llole Paar- s: Excitons . . . - . . . . - . . - . - . . . . . . . . . . . . . . 2546.2.1 Polsm*zati on ction - . . . . . . . . . . . . . . . - . . . . . . . - . . . . 2546.2.2 Two-p icle llE:4.rn l-ltoniazl - . . - . . . - . . . - - . . . . .. . . . . . . . 2556.2.3 ExcitonR ........................................ 2586.2.4 Surface Exdton Bolnnd States . . . - . . . . . . . . . . . . . . . . - . 2626.2.5 S Glk-iodifled B'I1.k aitoxls . .

.-. . . . . . - . . . . . . . . . . 265

6.3 P1%monK.......-.......----------..................... 2696-3-1 Tntrabxnd Excitationq . . . . . . . . . . . . . . . . . . . , . . - . - . . - - 269

I Contents

6.3.2 PlnAma Osc:'1!atl'o=....................,,......... 27û6.6*6 6 2Cê Od1*6cationK ..........,.................. 27S

6.4 PhOLOnE...........-..-.-......v..........,.....-...... 2*f36.4.1 Hxrmont-c Lattice D nzni/w-q . . . - - . - . . - . . . . . . - . . . . . . 2736.4.2 S ace a'nd Blllk Nlodes . . . . . . . . . - . - - . . . . - . . . . . . . . 2766 . 4 - 3 ylei gh ave s . . . . . . . . . . . . - - . . . . . . . . . . . . . . - . . . . . 2 786.4,4 clnq-Kliewer Phonons . . . . . . . - . . . . . . . . . . . . . . . . . . . 2816.4.5 Tnfluence of Relaxatiozz zmd Reconmtmzction , . . . . - - . . . 285

6.5 Element citations for R uced Dl-znenqion . . . . . - - . . . . . 290

7.7.1

-

tl-c a'ad Ideal Surfaces . - . . . . . . . - '.

- . . - . . . . . . . . . . . . . . . 2S3U.2 P0Ul' t Defects .........................,............,... 294

7.2.1 V m'es ,......................,............... 2947.2.2 pun't'es ...--...........................---.... 3û0*/.2.3 Y-m-tes .............--..a.o......-.............. 302

7.3 Lu1' e Defeds'. Steps ..................................... 3037.3.1 Geoznet and 'NotatT-on . . . . . . . . - . . . . . . . . . . - . . . . - - . 3O37.3.2 Steps oa Si(10û) S aces - . . . - - - - . . . . . . . - . . . . . . - - . - 3067.3.3 Steps o:a Si(111) S s . . . . . . . . . . . . . . . . . . . . . . - . . . 310

7.4 X D : S *

V1 Y *............>....-....... 3127.4.1 Defed, Reconstrtzction Element or B Propc ? . . - 3127.4.2 Si on Si (111) 3x 3-.B . . . . . . . - . - . . . . . . . . . . . . - . - - - . 313

1. Symmetry

1.1 Vodel Smfaces

1.1.1 surface vez'sus Bulk

Evcry re,al soEd is bounded by surfaces. Nonetheless, the model of ac in6n7'tesolid which neglect,s the presence of sucfees works very welt in the casqof m=y physical properties. TNe remsoa is, Astly, that one usually dealswith properties, suc,h as traztsport, optical, magnetic, menhxnscal or thermazproperties, to wllic.h all the atoms of the solicl contribute more or 1- to thesame extent, and, seconiy, that there are many more atoms f.n the buik of asolid sample than at its surfRe, prokided t'he solid is of macrœcopic size. Irzthe case of a srdicon cube of 1 =3, for e-x=ple, one has 5 x 1022 bulk atolnsand 4 x 1015 surface atoms.

The surface atozcs are only visible in smface sensitive experimental tecy'

niques or by studying propertio or processes wif.ch are determined by sttrfaceatoms only. Among them ate phenomena like crystal growth, adsorption, ox-

. jdation, etching or catalysis. They cnmnot be described by the mode) of aa

l'nflnlte solid. However, there aze also eseds wlf.c,h are detnrmined by theinterplay of bu.lk and surface (or, more stridly spexlaang) the hterface). Fori'astance, the clmnne-l of the cnrrler transport in âeld-efect traaetors is dcu-termnened by the slxrfnœ (interface) staœ ws well ms tb.e bulk dopbzg. H one

of tlze ftrst theoretical approaees to the feld esect, Bardeen (1.1) applied theprtamise of charge neutrality at the sT:vhces/intcrfaces. TMs conditiön meaus

that in thermal equilibzium the surface bard bending adjusts in such a waythat the net charge in s'nrfnne states is balanced by a space nbxrge below tahesuxfve of the semlconductor fomning the maH paz't of 'the electzical device.

1.1.2 The Sttrface as a Physical Object

Under normAl conditions, i.e., atmospheric pressure an.d room temperatuke,the zeal smface Sf a solid is far removed from the ideal sylems desirable inphysical Gvestigations. A freshly prepared smface of a material Ls normally

vezy readive toward atoms a'ad. molecules în the enviro=ent. M kn'mds ofparticle adsorptioa - from strong chnmt'sorption to weak physisorption - giverise to an adlayer on the topmist atomic layers of the solid. One evxmple

:. Symmetry

is the l'rnrnediate formation of an extremely thin otde layer o:a a frebhzydeaved silicon crystal. Usually the chemicaz composition aud the geometrëcalstrudm-e of such a contmsnation adlayer are not well defned.

A.s az object of physical izwestigations a welydeMed soace kas to 5eprepared on a particular soEd, in a special preparâtion procces, under wemdeG'ned extez'nal conditions. Such a solid colzld be a czysinlli'ne material, asingle crystal or a mystplline filrn deposited by epitax'y in a well-controlledmy. A rathe,r cleaa slxrfar..ta of suG a crystnlline sygtem might also be preparedas an electrode surfaze in an elec-trrïz-ynn,l'ci cell, or a somiconductor snHnce

iu a reactor where vapor phase epîtaxy (V.PB) Ls performed . at s'tandardprvttre conditions and at elevated temperature. Howeverz the processes ofthe tmderlying methods and the results are l'athe.r comple,x acd dsocult tocharacterize. The simplest ways to prepare a sozd surface shotlld happea inaltrœgh vacuum (UHV1, i.e.) at ambient pressure lower than 10-S Pa (about10-10 to=l. There are essentially three ways to mauufacture cleaa suzfacestmder UHV conditiozzs: -

i. Cleamge of brittle materials in UHV. Of course, only snlrênres whicz are

cleavage planes of the czystal nnn be made in this way.ii. Treatmeat of imperfect and contnmsnated surface of arbitrary oriea-

tation by ion bombardment and thermal n.nnenling (YA), generally inseveral cycles. There are no lsrnstations io certain materials and to certain

tallographic orientations. '

1. Epitaxial growth of crystal layers (or overlayers) by mexns of emporationor molecular beam epitax'y (MRE).Obvioasly, a smooth and clean surface nn.nnot be realized i.c the ideal form,

bui rathe,r oxuly to some approxsmation. My real smface wi:tl exhibit irreg-l'lnr deviations from perfed smoothness and pttrity despite tEe care takenSn its preparation. A.zl illustration of such a surface is given in Fig. 1.1. J.nreality a surface consksts of a nltmbez of irregular portions of parallel sum

face lattice planes whic,h are displaced vertinn.lly by one or more lattice plaaesepratlons 'VM respec't to ev% other. Atomic steps occlzr at the botmd-

TERRACE xjxx STEP

ADATOMSTEP-ADATOM

VACANGY

Fip. 1.1. Jllustratîon of structural imperfeotioms of crystal sudaces. Atoms aï.dthekr elcctron shel)s are indicated by little cubu.

1.2 Two-Dimensionat Czystals 3

ade's of thesc httice-plane portions whic,h in this coatext are.called terraa.The steps may exhibit lrsnk-q. 1.n addition to teztaces, steps and ksnkq, otherstmmtmal irregularities may ocmzr wiziclz caa be substtmed under the temnisurface roughness'. Adatoms aud vacancies bdong to tMs categozy, as docomplexœ of thœe m'rnple ddects. ln the case of slzrfnrzxs of compotmd ezys-tals qaite often atoms of one of the contributiag elements are dcpleted morethn.n those of the other which ruults in an emrinbrnent of the latte,r and ina noa-stoiYometzy at the sllrfnre. The most signiâcaut fo= of chemicaldisttlrbauce of surfaces) whic,h applies both to compotmd aad elemental cnrs-tals, is the contaminxtioa by impuritics or adatoms of anothes spœies. Theimpurity atoms or adatoms may be situated at regular or nomegular sites of '

the surface lattice plane: at locations above and slightly below' it.

1.2 Two-Dimensional Cl-ys'tnlA

A complete Garadezization of a solid surface requires Howledge of not onlyatoms of Ewhat species' aze present but lwhere' they are. Just as in the bttk itLs not that the atoec coordinates as such are of pucb direct interest. Rather,besides the clmmical natme of the atoms their geometrical arrangememt gov-erns tke electronic; magnetich optical, and otNe.r propezties of surfaees.

1.2.1 Lattice Plnnes of BI:11z Crystals

A geometrical constzucdon which Ls of particular signifcxnce in describiugczystal surfaces is that of a lattia pltme. Ipattice plan.e,s are uslpnl'y deuotedby MCJer indicos (W) where h, k, î aie the iutege,r redpzoca) axis intezvalsgiven by the i'atersectioms of the lattice planes with the three crjrstallographicaxes. They have a simple men.nl'ng izz the czase of zectangalar crystal systems,e.g., the cubk system. The smbol (100), for example, denotes lattice plaauperpendicular to the cubic a-axis, (111) means lattice planes pezzlealc'tictllnr tothe body diagonal iu the ftrst odant of the cubic unit cell, aud (110) denotesthe lattice plan.e.s perpendiclzlar to the face diagonal izl the fzst (padran.tof the o-plane of the cubic lmii cell. Ustlnlly, the collection of such planœt'hat are eqttivale'nt by symmetry is labelH (à,k!J. 'lnhlzs (100J ld>nds forthe collection (100), (1.00)r (010)) (010), (001) aad (OOï), if these planes are

equivalent. The bar aotatlon ï ldicates the correspondiug negstive coeë-ciext. J.n the case of trigonal and hexagonal latiic%, four crys-tazlographic &es

are conside-red, tlzree bxstead of two perpendicular to the c-A='q. The lattiœplres are then cltazacterized by fom indices (h,kïI) instcad of three- The &stthree, However, are not independent of eack other. Iu fact h + k + ï = 0. Thefourth n='q (correspondbzg to t;e iude,x 2) is perpenclicular to the he-xagonatbasal platae. The Lhkélj aze sometimes termed Bvavais ï'rlJces.

A particular geometzical plane 'can also be characterized by its normaldiredion

4 ê. Symmetr,g

n = .;V1v5,1.J.n the ca-se of lattice planes it is convenien.t to relate it to a lincar combination

16, =

'jg gez + k% + !;a) (1.1)

of the primitive vectors b.f (/ = 1, 2, 3) of the reciprocal httice with theiuteger coescients h., k, a'n.d J. The vectoz's bj are directly related to theprlrnl'tive lattice vectors aé (ï = 1, 2, 3) by the relation

(1.2)(zç . bj = zmAj.'

Apart Fom the cmxe of priml'tive Bravais lattices) they are Hq'Ferent from thecrystallograpilic axes. M:yway, a lattiee plane can be charaderized by theMiller indices (i,kJ) anG hence, a normal parallc) to the vedor Ghkt = hbï +k% + Iba of the reciprocal lattice. Eowever, as a con<uence of rehlon (1.2)the Mllle,r indices depend on the particnllnr choice of the prîrnltive vectors oftba Bravais lattice.

Mûler irtdices are simplest to work with iu simple cubic (sc) Bravais lat-tices, since the redprocal lattice is also simple cubic and the M1'llmr inclicœare the coor8lnates of a vector nomr.al to the plaae irz the obdous Caztesiancoordinate system. As a general rule, face-centered cubic (fcc) and. body-centered cubic (bcc) Bravais lattices are de-scribed in terms of coaventiomnlmzbic cells, i.e., as sc lattice,s with bmsœ. Since any lattice plnme in a bcc or fcclattice is also a lattice plane in tv underlying sc lattice, the same elementazycubic ixdev'ng (hk!) ca'a be USE'CI to specify lattice planes. This agreem=tsimpves a variety of cociderations for a 1ot of matezials. M=y importantmeials consieing only of one element crystamze witlu'n the cubfc crystal sys-tem. AlRo many elemental and compolmd se-micoaductors or strongly ioniccompounds fo= diamond, zinc-blende, oz ronlrRalt crysta)s which also belongto the cubic cnrstal system.

The Miller indicœ of a plane have a geometrical interpretatîon in real

space. Therefore, a st'='lla,r convention is used to spect Oections ic thedirec't lattice, but to avosd conhiqion' <th the Miller t'adices (directioc i'athe reciprocal lattiee) square brackets aze used insiead of pazentheses. Forinsuuce, the body diagonal of a sz cubic lattice lîes iu the (111) diredion and,in general, the httice point haï +kcc -i-lca lies ia the directîon Fk( from the

. ozigin. J.n the ckbic case hàl deMes the norrnnl direction of the plane (/zkI).The collection of suc,h directions that are eqttivalent by symmetzy is labeled

(h,kI). This holds in pdndple also for non-cubic Bramis lsttices. However. ingemeral the direction hkJ) is not pezpendic'lln.r to the plane (?zk!).

The propert.y of the veetor Gskj = hbz +k;2 + Ibs of the reciprocal latticecan be proven chazactezizirtg the lattice plaues by all possible Bravais latticepoinà

s

Rï = 'T-lnlsc,s (1-3)H1

1.2 Twoeimensionaz Crystals

(a)*

* *

* *

*

00

OQ

OOO00

OOO00

OOO

OOOwc O O

OOO OOOJ .,,, o o o

oooooEiccl

jnrlO

grfj o(i r()l

OQOOO

00000

00000

000C400

O!..72:1

p1 07() 1 I:I

ng- 1-2. (a) Cubic Bravais lattices x, fcc, IXG (b) low-index plxnes (1K), (110),(111) in a sc cell; and (c) low-inde,x plaxtl.s resulting 1om cabic httce. Bravaislattice points are indicated as dots (+b) or sphmv (c).

O

6

Z

1(0 t' o 1 )

!1

(odolq

C

('î01'0)('1XO0) (011G)

p'I2o)

:214Xc

(-1 l'ooll 1 E()-I l'oj ----->

y.J

---- (21.:ib) Elol'o).-4

X

1. Sylnnàetry

Fig. 1-3- Charadezistic phnes in a àevxgonal Bravais lattice. Certain dlrvtionsh thss htlice are also indicated. The vectors œz, .2 (or aa), amd c r-'m be identifvwith the priml-tive Bravais httice vectors- The directiou (2E1Oq , (01E0! , and (0001qrepresez/ the Xexagonal Cartesian cooroate system.

with integezs as (ï = 1, 2, 3). The index I characterizes the n'nfsnl'te 6xrn51y of

paratle,l planes in a certatn ctistance from OC.IA otller. The plane l = 0, whichcontnsnR the zero poet, may afterwards be identised with the snrfn'ce of senzi-

l'n6nlte space. This is demonstrated in Pig. 1.2 for the low-index sttrfaces'of sc,

fcc, ard bcc Bra>is lattices (or monatomic metals czystalliziug witlnt'n tllese

stnztztavl. In the ca% of a hexagonal Bravaks httice such lattice planes az'e

indicated in Fig. 1.3. In pradice, it is oaly izl the descriptioa of non-cubicczystals that one mu.st remember that the MflTer indices are the coordsnntes

of the normn.l tu a system given by the viprocal lattice, ratxer tha.u tàedirect yttice. 'For tlzat remson, sometima for simplicit.y tzhe Vller lndices

(hàït) are also tused to nhnradezize the normal diredions as done in Fig. 1.3.

1.2 ToDimensional (717%* 7

à63);kt3b));qj,.''''''''''

''/ 'II

$ II111

Squlre (J$1 = IJ2IY = 90*

Y I

J l

1 I1

Rectangular

Iëh I XIJ: . 'r = 9D1

/2

#j' '

:

a'1 z

ê

J'

Hexagcnal IR1 = 1L1'f = 120*

%Y ' -

i ' E *<

! X : XY

$ ---arau-. qy! :: 1

centered redangular

1411 # lJa( , cas '/ = It7a(/(2 E-ct)

4

? zl

#'#'

l

Jh zl

l

J

l

oxkue 1,$1 .xrIa2Iy'Rr

Fsg. 1.4. The dve t'wo-dimensionnl Bravais latticœ. Besides pra-mstive llnt-t cells(dwshed linesl also a non-prirnitive ceE (dotted linos) ts shown.

Suc,h a l = 0 plane represents a.two-dimensional Bzavais lattice

2

R = rmaçï=1

wîth lz azd az az the prsrnstive basis vedors of tlzis lattice a'ad integer nn'=-

bers ru and mz. The three vectors 4z, 4a, zz fo= a, zlght-hand cooreate. rne possible îve t'qodimensional Bravais lattices of tNe four pla-

nar crystal systems are represented i'a bnig. 1.4. Apart from the redangalarcase, they are prlrnitive (p). Iu the c%tered (c) rect ar cue addition-ally the non-prlmitive ce2 is also iadicated. h pzactice one often uses thenon-priuzitive lattice for the convezzience of description. Sometimc, akso aon-

prlrnstive centered square me.shes are USGI in order to keep a cez'tain orientavtion of the tmit cell.

8 1. Symmetry

1.2.2 Oriented Slabs

Accoran'ng to (1.4) all lattice pla'aes in an a'rbitrar.g hxlfspace (! = 0, -l, -2, ...)or crystal (2 = 0, +1, éc2, ...) may be described by

.&; = J''l miö,z + ldst$=1

where J,a is a vedor complementing J,z and :,2 to form a set of (in general)non-prlrnstive lattice vectors 0,1, cz, tu of tlze three-dsnnfqrnqional (3D) bulklattice of the uzderlying m'ystal. The vedor Ja ca'a be deterrnined from theDiophantine equation 'Fz , Az = 1 with expressions (1.1) a'nd (1.3), as long as

the vectom &1, /2 satisf.y fk . & = 0 (1,2j. The choice of la is not :tnique, ofcotzrse, a'nd ary vector aj which dfezs fzom aa by a vector withT'n the latticeplane can also be used. We call 4a the stacking wedor because it deterrnî'neshow the chosen lattice pln.nes a-re stacked in the cozlsidered Bravais lattice.Foz t'wo Bravais lattices Table 1.1 shows a possible dloicc of thc vedors &:,&2, ar.d 5,a. The vectors :,: and ac span the lattice planes shown izt Figs. 1.2azld 1.3.

The selection of the titzee vectors J,z, 42, and ö.z shows that the primstivecell of a Bravais lattice may be ehosen as a parallelepiped vith one of its pairsof pazallel faees beiug parallel to a given lattice p7stne. This implîes that sacha lattice may be c'hnracterized as consistin.g of pa-ratlel lattice places whichare displaced with rœpect to enzq other as i'adicated in Fig. 1.5.

Table 1.1- Possible prsrnitive lattkc vectors of a plane and stanlclng vectoxs for cer-ttu'n plxne orieutations izz tke case of two Bravais lattices. Cubic (a:) and hexagonal(a, c) lattice constan'ts are used.

3D Bravais lattice Plane 217 Bravais lattice ë,z d,z az

fcc (111) hexagonal aa - tzz oa - a2 c.z

zzz = Ma (0? 1, 1)al = M.z (1, 0, 1) (110) Dreotangular fzl - c,z tz1 + c,c - az c,z

as = Ma ( 1, 1, 0)(100) nsquare c.a - ac /,1 ag

hpvngonal (0001) huagonal a; tutz1 = c(1, 0, 0)a2 = ! (-1, Vi, 0) (10ï0) Dzectangulaz 0,3 tzt

as = c(0, 0, 1)(1:tJO) Drectangular cu - c,z tzs c,1

1.2 rrwo-Dimensional Crystab

n

12

a1

Fig- 1-5. Construclion of a 3D Bravais lattice from its lattice planes.

Usually a crystal possesses an atomic basis with S atoras at the positions'ra (.s = 1, ..., S) ia the uzuit cell. k'a correspondence vith the lattîce planes,atomic phnes may be constructed. The lattice plane & caa be consideredto be occupied with atoms of speciœ 1, e.g., at rz = 0. The next atomicplane, ctisplaced by .2 <th respect to the ftrst one, is ocmzpied by atoznsof speciu 2, etc., a'ad the plane displaced by rs L5 cccupied by atoms oftype S. lt may happen that two or more atoms of the basis are located atthe same plane. In that case an atomic layer twhic,h is later identifed withaa ideat srface) consbts of 'two or more bmsis atoms. As a consequemce thepokacity of such a place clm be %ed according to the totaz charge. In costalswith partially ioaic bonds neutral, positively charged o:r negatively chargedatomic pla'nes arise. Tizis allows us to deA'ne the polarity of a surface or a

correspondicg halfspace. For two-atomic crystals with cations and anions,the equal or kmequal nllmbers of these ions ia a unit cell spltnned by thevectors al a'ad s,z characterize the polarit'y. The lattice plane Ao + rs mostdistatlt fcom & completes the constntction of a ezystal slab which in thevertical direction, encompasses exactly one prsrnstive unit cell. This slab iscalled a pvimibi'ce czwfcl slab. A lattice plane occupied by atoms is referredto aj a:a atomic Jcper. The second prlrnstive arystal slab again begins witha lattice plane occupied by atoms of species 1 an.d is displaced with respectto the zer0th. plane of the ftz'st sla,b by the stnrtku''ng veeor 5,a. A crystal can

iherefore be tEought of as consisting of successive crystal slabs situated one

above the other. A pile of xveral primitive slabs rxn #ve a new t'rnnAlationalsymmetry iu the direction of the norrnml zz. One o.n.lls it an émdudbl.e C*rI/SVtI)Jslab. For t'wo-atomic mzb. ic with zinc-blende or diamond stnlcttu'eand htlice constant tzn S'UG a slab Gmtains three for (111): t'wo for (110),ao two for (100) primitive crys'tal pylav 'G't,'N six for (111), two for (110),and fottr for (100) atomic hyezs. The corresponding st

'

vectors are

10 1.. Symmetry

(a)

(111)

w %* * * * * œ* %p

* 1 * % ** N FJ=

* o *

E1 -1 ()ll

Zinc blende

(11 1) (100) (1 10)@ IA @ IA @ IA

O 28 C) 28 C IB* 3A * 3A * M

o 4B o 48 O 28. 5Ao 68

(110)

1: 1

$ 4

$ 4$ $i $ IW

*1

F $ () T

Fsg. 1.6. (a) Top view of izzeducible mystal slabs with certain orientations 'n, forzino-bleade cgstals. The atoms i'a filserent layers a're indicated by cln'Fe-reut sizes.Dashed lines mdicate a possible 217 tmit cell. The size of the Slled and open circlesiadicates the layer bemeath the stlrfnne. It ks related to the layer index -J in. thelegend. The 6:1ing of the circles de-scribœ the cation or anion character of tEeconwpondjng atom. Mer (1.22.

c0(1) 1, 1), ao (l, 1, 0), ard co (1, 0, 0). For a t'w>atomic Eesxagonal cryl-tal wîthwurtzite strudum suc,h slabs contain four for (0001), tllre for (1120), aadfom for (1010) atmrni' c layezs. Projedions of suc,h irredudble s1,.:1:8s of two-atomic crystals aœe mœented in Pig. 1.6. Crys'tal examplœ from two Bravaissystezns are plottu: fcc with zinc blende (diamond) 'and ronGlt, hexagonalwith wttrtdte s'tmzdure. 'ne conwpondlng s'pve groups are FQ3m (F6I3m),Fem, aud .P6,= lming the internnisonal notation.

'

The location R@, J, zal, m2) of an iadividual at/om can be specfed by thenumbe,r I of the primitive crystal slab, the number s of the atomic sublatticeatd the hteger coordinates rru, t'nq of a point ia the 2D Bravais lattice as

1.2 Twfoimeasional Crystals

(b)

(1 11)

O o Q o

%'

%x *% '%%

o k o % Q& h

rpRt

t) D <> o

g1 -1 o'J

Rocksalt

(-100)

* œ +

1 11 I6 I

+ I œ 1 @1

t'

1 lI $1

r

&=o @ *

go 4 Q1

(111) (100) (110)@ IA @ IA @ IA

O 28 O IB O IB

@ 3A * M * Mo 4: o 28 o 28. 5Ao 6B

(110)

: Il

I l =I I S

.- &

F 1 01

Pig. 1.6. (b) Same ms Fig. 1.6a bat for rocksalt crystats.

c

Rls, J) mt, ??zcl = mk%. + Jë,:$ + n.c--cz

The complete set of atoMc sites in an irtGrtîte 3D crystal can be obteedby assigning all possible intege.r valztes 9om -x tzl +x for !, ml? zna aadall possible values s = 1, ..., S. T:e aormal direction rz, 'apoa whic,h theconstruction of the lattsce planes is based, is without snfluence on the sites.Any Goice of n yields the same curs-tal.

1.3.3 Ideal Snvgnces. Planar Poiut Groups

The above representation (1.6) of aa lnflnite czystal can irnmnediately be em-

ployed i.n desczibicg a C'I'yS:,aI with an ideal surfme. and normnl zz, i.e., abltlfkpace. Such a srtem may be gencrated 9om an in6rlste crystal by re-

moving azl atoznic layers above the suzface a'ad retaining those below. The

1. Symmetzy

(c)

(0001)

*. + * *

%% %* x% &.% 44%. %.

. o .. .. .. .. - -. . @ r'n&$v-Iw

e o *

Eal-i'o!

(1OT0)

1 4

I '* t , '

: 1l 1

r-&DCf' 't

F 1 o oél

(1 110)

WuAite

1 I

I I rD=-cOcL-Z

Ff 2 C-b!

(n001) (# #Jo) (-î0h-0)

@ IA @ IA * IA

O 28 O IB O CB. 3A @ M @ 2A* 48 o 28 (7 28

(4 3A 1 3A* 38 @ 3B

. 4Ao 4B

T'Cg- 1-6. (c) Sxrne ms Fig. 1.6a but for wurtdte crystals.

rernxining upperme atoxnic laye.r r' epresemts the s-urfnce or, at least, the up-perrnost atomii layer of the sazrfaee reon of the resulting bx.lfqpace. Eowmrmy atomic layers are counted to belong to the sllrfnce rcgion depends onthe method USCXII to ia<ate the system, e.g., on the penetration depthsof the exdting and/or detKted pnrticles.

Jn a 61rs-0 approe one may nv:me thsttthe atoms in the uppermcstatomic hyers, in pazticular in the topmost hyer, kcep the atomic posîtions ofthe im6nite cr-gsvtal. Such a czlnfgaration is usually tnmned an ideol dtzoce-Tlze atoms of a crystal having an ideat surface are thus located at the positions

1.2 Twccimensional Crystals 13

A(a, J, ml, rzal gfven by (1.6). However, only sites below the stuface place are

occupied.. Thue obey the relation

'h, .A(s, I,rk, zral = L + 9,6, . r. :< 0 (1.7)with ii- given iu expression (1.1). The snrfcce oz Ast atomic layer is obtaincdif the ief-t-haud side of tMs relatiou is taken to be zero. A possible solutionof (1.7) is l = 0 and 'rso = 0, so long as tke site of one atom sz of the atomicbasjs is identifecl with a BraYs lattke point. Tkus, the &st atomic laye.rcorresponds to the-particula,r lattice plane perpendictllar to the noz'mal A,

which goes throtzgll zea'o au.d. whox lattice points are occupied by basis atomsof species sz. There msy be other vedors ra beside rso which) although notbeing zero them-lvem have a zero projection a . rs. Tâem the bVs atomsof this species a are also located in the flrst atozaic layer. They are displRed'Mi,II respec't to the atoms of species sc by a veor ra paratlel to tb.e sttrfve.Suc,k mGtiple-spectes occupancy of a'a atomic layer occm's, for instance; inthe cmse of (110) snx4hces of zinc-blende-type . I11 this ca, two atoms- a catioa and = aaion - occtzr in each primitive uzlit cell of suc,h a 25 drjrstal.The (110) surface formq a non-polar face becaux of nlnnrge aeatralits wlzichis one of the reasons why the (110) pl=e represents the cleavage face ofzinc-blende crystals.

The resultiag hnlêqpace witk an ideal, bullc-terrnsnnn.ted surface (or even a

rea! stkrface as dîscmssed below) nvhlbits not only a 2D periodic'i'ty or, more

precisey' a 2D translatioaal symmetry with the prlrnitive basis vectors ë,1 andë,c but also apoint sgmmett'y. As a cozlsequence of the trxnqlational symmetzyaccording to the 2D BTavais lattice points R (1.4), physically equivalent spacepoints ca'a be zelated by

m/ = (sI.K :z)

= ê;c + R = z + A, (1.8)whea.e â denotes the transformation matrh characterizi!lg the element s. SuGpoints are displaced agxînKt ench other by a Bravais lattice vector .4t. Norotatioa or rededtion îs involved. This is indicated by the unit elemfmt aand the uait matrix J, respectlvely. A11 elements whic,h balong to a certaintranalaional prtlu, are abbreviated by (c1.K. However, ic addtion there can

be pokt group operations.tajo) wlzic,bu also relate physicAlly equivalent spacepoiats œ? azd œ. The point srametry dements tz are necessarily rotationsabou.t axes whic,h are parallel to the normal Tz, and rezedions at lines withiztthe surface or cell planes perpendicttlar to 'n.. Only n = 11 2, 3, 4, or 6-fo1drotation axœ perpendicalar to the snzrfKe may occur. Corrcxspondingly, thesmbol Jr sigoes a rotatioû around the surface norrnn.l directioa by theangle (360 mjnlo with zn. = 0, 1, ,.., zz - 1. rrhe mirror planes aa also normalto the surface. Jnversion centers) rnsrror planes acd rotation %e,s parallel tothe sllrfaze are aot allowed, since they refer to pohts outMde the surface. Thepbssible reEection lines m., z%, rzz, rza, md, ?pJ, p%, an.d zp,â aze specïedin Fig. 1.7.

1. Symmetzy

mx

rn 1

d'

I1 1 m, 21I 1

I1 11 )I1 I

1 tI 1

m, I YI 1 Xj I

1I 1I 1

1I p1I 1j 1i 4 rn#, 1 2

mldm'$ Fig. 1.7. Dmotation of re-

fection linœ.

By combl'nlmg the limsted nnrnhez of allowed symmetry operationsj one

obtalms 10 dilerent plane mïrzï gronps. In the ictmvnxtional system (Sczoen-=es system) they aze denoted by either n (Ck) or nm, nmm (Ckv). Thezmmeral s = 1, 2, 3) 4, 6 denotes rotations by Y aad the smbol m refers toreqections ia a mirror plane. The third smbol m indicates that a combina-tion of the meceding two operations generatœ a new mirroz plane- The 1Bpoint groups are 1 (Q), 2 (Ca)) m (Qv), z'm,zn (Qv), 3 (Cz), 3m (Gv), 4(C4), 4mm (C4r), 6 (C6), 6mm (Qv). They are geometrically represented i.aFig. 1.8.

The plaae Bravais lattices presented ia Fig, 1.4 also possess point symme-trie-s. Howevar, tlze possible mzzltipicities of a rotatîon spnmetry aads of pMe

'

latticœ are restrided to '?z = 2,4 aad 6. A lattiœ which only cxmtnsnq a Ffoldsymmetry a'ds is eithez an oblique lattice or a zectacgular one (independeutof the .p- or o-characterl. The poin.t voups of these lattkes aze 2 (tQ) a'n.d2mm (Cav), respectively. Quadratic lattices with a çfold symmetz'y nn'q pos-s%s fotzr reûection line,s whic,lz a're rotated through 45? with respect to eacizother. The point group of S'U?,h a lattice is theefore 4z?zm (Qv) . Eeagonallattices with a Gfold a'ds have six re:ection lices whic,h meet at an angle of800. Ia this c:se the point voup is 6mm (Gk). Sh'rnrnarizîng, there are thusfoar zlsFerent plane crystal svsfems - the obDque with the holohzdral min,t

gro'up 2, the rectangular wjth the ltoLoltedrat ptHzzt gro'up 2mm, the quadraticwith the hokohedral poïzzt gro'up 4mr?z, a'nd the hexagonal with the holohedraipofnf gronp 6mzru Thœe crystal systaq contain îve 217 Brawaâs lattices (cf.Fig. 1.4) .

.-

The low-kfill/r-indu sndhtvtq of fKe-centered cabic and body-cemteredcubic metal costals nvbîbit such lligh poiut-voup symmetries because oftheir stractuzal simplidty. As iadicated in Fig. 1.21 these sttrfates tend to

1.2 Twceimensioaal Czystals 15

t* m

--q(-- 2mm

Y-((j((( 4mm

L 3m

6 6mm

Fig- 1.8. Sclmrnntc reprto-tation of the 10 planè poiat groups.

have the highest degree of symmetzy and tEe smalHt uait cells. Exmples aze

fcctlll), fcc(110), aad fcc(100), which have tkreefold, twofold, aûd fourfoldrotational syxnmetrs r-edjvely. Othe.r A'mmplœ a're bcct4lllj bcc(110):and bcc(100), which have threefold., 'twofold, and fourfbld rotationi s.,-metry, respectively. Isolated (111) plaaes even possess a higker rotatiozM

(sidold.) symmetry. Of comse, all these surface,s akso have rnirror phaes inadditioa to the rotation axœ.

15 1. Smmetzy

1.2.4 Real Surfaces: Recozustruction aud Relnwtion

The 2D trn.nqlational symmetries of ideal surfaces and halfspaces M'KII bulkatoMc positions are aharactezszed by the primitive Bravais vectors az and a2.J.n addition to potat and line ddeds (cf. Fig. 1.1), on a real smface of a crystalthcre yre otb.er reasons that the msshlmption of an ideal surface is not validî:a gen' eral. Such a pictm'e does not Allly accotmt for the bonding behavior ofthe atoms in a crystal. Since the forcœ acting on atoms sila'xted beneath an

atomic pla'ae in an r'o6nstc crystal are partially due to tke atoms located abovethe plane, one c=, in general, expect that the forces azting on atoms i'a a

czystal witlz a surface shottld ri-lFe.r h'om tbose acti'ag i'a a,û im6mite cmtat. Thedeviation fzom the ln6n'lte case, however, dsrnl'ninhes with incremsiag diexnce

of the atoms 1om the surface. One nan thas assllme t'hat the forces actin.gon, and hence the posîtion of, atoms deep Mside the custal bulk are, to a

good approximathon, the same as those in au r'm6nr'te czystal. TMs is, Eowever,not true for atoms rauated near the sl:rfn.ce. The forcœ acting on tEem are

appreciably deerent, resulting in displacements SRLS, J, znt, mzl of atomicpositions RLs, J, mz, mz) (1.6) vith respect to those of the in6nite crystal.Constsquently, the equilibri:nrn conditions for surface atoms are moeed vithrespect to the sninste C.IFS-taI. One e'xmects altered atomic positioas

R'Ls, 1, nklrn'z) = A(,9, !, Tp't, Tn,cl + SRLS, 1) rnq, =2), (1.9)

vith

JA(s, 1, rzu, mzl -> 0 for l -h -x, (1.10)aud a smface atomic structure that usually does not agree with that of thebtfk. Thus a shprfsce îs not merely a tnlmcation of the balk of a crystal.

The distortion of the ideal bulk-like a'tomic conâguzation due to the exis-tence of a surface (more prcdsely, the non-existence of formerly neighboringatoms in the vamzttml depends oa thc boncling behndor of the materii con-

sidered.. In tetrahedrally bonded semiconductors, such as diamond) Si, Ge?Gn,Aq, 1aP, GaN, etc., strong directional bonds are present, The brtmldngof bonds due to the gcneratkon of the sarface is expecied to llav: dramaticeseds. Syste-ms with dangMg bonds shotlld be in general tmstable, siuce

' din ally lowers the tota' l enerr of tkte bxlRpace. Sometimes thisrebon g usu ,

process is acEompeed by briuging surface atoms closer together. Ozle ofthese me%smisms rtalltiztg in pairs of suzface atoms is schematically indi-cated ia Fig. 1.9m Howevc, suck a reazrangement exn nlltn yield roul surfacelayers, the stoiœometzy of whic,h is changed wjth reoed to the ideal sur-

fnr.e (see Fig. 1.9b). Lu b0th cmses, the 2D Bravais lattice of the sm'face iscizmlged. Suc.h perttzrbations detroying the translational symmetzy of theEditious idcal nurface are known as snrjace 'reztvgsfnsctï/=

'nere are general argttments for such symmeky-bremxlring atomic rear-

raagements. One is based oa the impossibility of âegenerate grotmd state,s

1.2 ToDjmezssional Czystals

Reconstrudion

a)

00 00 00QQQOOO000000000000

Relaation

c)

OOOOQO000000000000000000

Fig. 1.9. Schematic illustrations of atoznic rearrangememts tn the s'4rfnrte z'egion.(a) Pairhg retxlns-tzazdiiop (b) m$e row recons-ction; and (c) rllxvation of theuppermost atomic layer.

b)

O O OO O O C O OO O O O O OC) O Q () O O

of tbe system. Such a degeneracy zaay occtm foz iustance, for (111) surfacesof diamond-stmzctme czystats. In. tEeir case, on average there is oaly oaeelectron for eac'h clanglinpbond orbital pazallel to the (111) surface normal)althougk eac,h orbital can accomodate two electrons of opposite spius. Thesystem gwund state eAn thus be romlizM i'a numerous ways by plnning twoclectrons izl two orbitis, two iu one or aa equal distzibation ov< tEe danglingboads. However, the arrazgememt of these orbitals may rlsfer. According tothe well-known J'tzhn--Teller titeorem spontreoas spnmetry brenBing will oc-cur (1.31. The degeneracy is lifted. In the discussed (1T1) case this implie,sJxhn-Teller clisphcements of the smvfnre atoms which destroy their eqttivamlence. One may at leaast expect a so-called 2x1 reconstruction, in a sense thatis explained below.

1x1 stmple metals) instead one haa a gaA of quite delocalized electrons andcheznical boad.s which are faz less diredionat thr in sezaiconductors. Con-sequently, there are no preferred diredioms in the displaœments of atomswith the eazeption of that parallel to the sadace aormal vector itself. One

'

thtzs expects a displacement rnnsnly of the fa<daye,r atoms in a vehical di-rection vith respect to the sarface as indicated i'a Fig. l.0c. The 2D Bravaislattice and, hence, tNe 2D translational symmetzy remna'ns tmGaaged. Sucka imnqlatipnal-symmeiry-conxmdng change of the atomic stmcture is eAlleds'urjace r:lczctïm A special argument for simple metals is bmsed on the locinhsmge neutralitr. Ia the bulk the nearly $:26$e ealectrons are delocalized betweenthe cores mnking a'a ezectricatly neutral object. On a surface (see Fig. 1.10a),

18 1. Spnmetz'y

* *@ @

* *

(a) * ** @

* @* @

* @* @

@ *

(b) @ @* @

@ *@ *

Fig- 1.10- Sch=atic remœentation of a metal surface by cores (dots) and Wigner-Sdtz cells (hexagoms) before (a) and after (b) the surface relnvn.tion. Deformathonsof hrastgons indicate tàe redistdbution of the electfon density accompanying itssmearing out at the surface and the inward dksplacements of the cores.

this picture woutd lead to a rapidly varying electron density at the surface:azcording to the arrangement of the b4zlk atonzic sites. As i'acticated schemat-ically ilf Fig. 1.1Ob the surface eledrozdc charge temds to sm00th out, whichis only porible by vertical displacements of the ion cores. Simultaceously,the electrostatic repulsion of fzrst- and second-laye,r ioas is reduced, whichresalts i.n the izlwaird direction of the reln.vntion (contractîon). The efect isobsezwed for many low-inde,x metal sarfacœ.

Reconstructioas, for instaace that of the type incticated :in Fig. 1.9b9 rlmalso occm at metal surfaces. However, the relnvntion is not restricted to met-als but ca,n also ocmzr for surfaces of nonmetats. The cleaa cleaved GaAs(l10)surface is one well-lcnown e-xample in thîs respect (1.44. Due to the electrœstatic neutrality of the surface unit ceR with oae cation a'n.d one aoon (cf.Fig. 1.6), t'wo dangling bonds and. t'wo e'lectrons are available. Consequemtly,there is no need for Jn.hn-Teller displacemeats. Iastead: opposite verticaldisplacements, i.e., a surface bunkling accompanied by a,n electron trsmqferbetween the dangling bbnds, staboe the smface trnnqlational spnmetzylmowa from the truncated bullc.

Besidœ the argttments of the satuzation of dangling bonds or smoothnessof eledron distributions, another argttment is related to the zeduction ofthe electrostatic energy of systems with partially ionic bonds. Due to theirCoulomb character, the electrostatic i'ateraction of ions Ls of lonprange natkkrea'ad does not show a directional dependemce. Shce the only defned Oectionin the system is still the surface normal, again a tendexcy to a special type ofrelaxation occm's. One example for a class of corresponding crystals concernsArvBw semiconductors with paztiaz.y ioaic bonds and roclcsalt structure.Two diFereai types of sarfates are possible i'a these crystals, a pola,r type withexclusively A or B atoms, and a non-polar type with b0th A and B atoznsin the surface (see Fig. 1.6). J.n the casc of the non-polar (100) surfaces one

expects from considezations of the Madelung energy vertical disphcemeats ofthe surface atoms but nozte parallel to the s'tzrface place. 80th sublattices, Aand B, should relax inward but to dt'gerent degrees, because of their dlFeretltion sizes. TMs zœults in a so-called rumplin,g of tlze surface E1.5).

* * * @@ @ * @

* * @ *

(a) * * (b) @ ** @ * @

@ @ * ** @ @ *

Fig- 1.10- Sch=atic remœentation of a metal surface by cores (dots) and Wigner-Sdtz cells (hexagoms) before (a) and after (b) the surface relnvn.tion. Deformathozusof hrangons indicate tàe redistdbution of the elect/on density accompanying itssmea.iug out at the surface and the inward dksplacements of the cores.

this picture woutd lead to a rapidly varying electron density at the surface:azcording to the arrangement of the b4zlk atonzic sites. As i'acticated schemat-ically ilf Fig. 1.1Ob the surface eledrozdc charge temds to sm00th out, whichis only porible by vertical displacements of the ion cores. Simultaceously,the electrostatic repulsion of fzrst- and second-laye,r ioas is reduced, whichresalts i.n the izlwaird direction of the reln.vntion (contractîon). The efect isobsezwed for many low-inde,x metal sarfacœ.

Reconstructioas, for instaace that of the type incticated :in Fig. 1.9b9 rlmalso occm at metal surfaces. However, the relnvntion is not restricted to met-als but ca,n also ocmzr for surfaces of nonmetats. The cleaa cleaved GaAs(l10)surface is one well-lcnown e-xample in thîs respect (1.44. Due to the electrœstatic neutrality of the surface unit ceR with oae cation a'n.d one aoon (cf.Fig. 1.6), t'wo dangling bonds and. t'wo e'lectrons are available. Consequemtly,there is no need for Jn.hn-Teller displacemeats. Iastead: opposite verticaldisplacements, i.e., a surface bunkling accompanied by a,n electron trsmqferbetween the dangling bbnds, staboe the smface trnnqlational spnmetzylmowa from the truncated bullc.

Besidœ the argttments of the satuzation of dangling bonds or smoothnessof eledron distributions, another argttment is related to the zeduction ofthe electrostatic energy of systems with partially ionic bonds. Due to theirCoulomb character, the electrostatic i'ateraction of ions Ls of lonprange natkkrea'ad does not show a directional dependemce. Shce the only defned Oectionin the system is still the surface normal, again a tendexcy to a special type ofrelaxation occm's. One example for a class of corresponding crystals concernsArvBw semiconductors with paztiaz.y ioaic bonds and roclcsalt structure.Two diFereai types of sarfates are possible i'a these crystals, a pola,r type withexclusively A or B atoms, and a non-polar type with b0th A and B atoznsin the surface (see Fig. 1.6). J.n the casc of the non-polar (100) surfaces oneexpects from considezations of the Madelung energy vertical disphcemeats ofthe surface atoms but nozte parallel to the s'tzrface place. 80th sublattices, Aand B, should relax inward but to dt'gerent degrees, because of their dlFeretltion sizes. TMs zœults in a so-called rumplin,g of tlze surface E1.5).

1.2 woœimensional Costals 19

1.2.5 Saperlattices at Surfaces

J.Il accordance with the above dsscussions, the atomic displaœments &R(st 1,,rrzl, rrzzl i'a expression (1.9) may be divided into two clmsses wit?h regrd totheiz Gect on the tranqlational symmetr.y of tile stlrfime. V the trnoqlationalsymmetry Ls not asected, the displn-ments repr-nt a relxxxtion of the sm-

face (see Fig. 1.:1a). Thea &RLS, 1, mt, rul = L'rsk for all ml, ma. Ozuly thevecton 'ral of the atomic basis in the hnlfqpace belongin.g to tEe 2D Bravaislattice are altered. ln the case of suface reconstractton (see Fig. 1.1âb) eqtliv-alct atoms in diFerent llnit celks arc not all displaced in tAe ==e mnmne'r,

î.e., SRLa, J, rrzzl mg) depencls on zp,s and rzz. Both the atomic basis a'ad theBraYs lattice are nhn.nged.

Jn the cnse of a reconstrttcted sarface a new 2D Bravais httice with prlm-itîve basis 'vectoz's Ez aud &a occurs. Tzl th:is situation a perioclicity is prœentin the topmost atoMc layers wlliez ii HiFereat 9om the corresponding 2Dtrnanmlational symmetzy <th az a'ad 5.2 in bulk-like layers deep below the smu

face. ln other words, a surface lattice, nnlled a sayeriatticet Ls mlperimposed onthe substrate lattice widch evhsbits the basic periodidà. Consecuemtly, 't'wotranslational groups Ts (Garacterizing the uppermost surface layers by &zjQ) aud Tb (eaharactertziug bulk-like layez.s by 4z, 4z) Eave to be dkscussed.The trnnqlatîons wlzic,h trxntdbrm the m'ys'tal with sllrfMe (tà.e Ealfspacel intoitselfmust belong to b0th groups of traztslations, Ts and Tb. The trnnslationalgroup T of the w' hole c'orstal wîtlz smface is thus the intersedion

T = Ts fl Tb . (1.11)Alternatively, ore cau say that T is the largest common subgzoup of bothgroups Ts and Tb. There are two possiblities. Either T only consists of theidentity trazzslation, whic.h means thst the lattices dained by Q aud Tw azenon-commensnxte, or T contnsms more elemeats than just the idemtity, witie,hme= tlnn.t the soace (Q) axd bulk (Tb) aze comnzensurate- In the flrst =e,

the crystal with surface does not possv any htticotramslational symmetry.Reeations of non-commezssurate snlrfnres az'e more likely izl the case ofadsorption. In the second cmse, tke lattice assodated with T kq e-qlled a eo-

.<7 .do .47 a) b)Fig. 1.11- E'xamples for suzface relmtion (a) and sttrfve reconstruction (%) iu-duendng the ftrst aud second atoznk layers. A 2x2 reconstzuction Ls shown i.n (b).

1. Symmetz.y

incidence Jcfjice. 1$ in partictzlar: Ts is a subgroup of Tb, then T is equal toQ, i.e., the coincidemce httice is identicaz vith the httice of the sttrface. IfTs ks not a subgwup of Tyl then T cnnnot be equz $o Q acd is necesrw-trily a

proper sazbgroup of Ts, i.e., it is Krnxlle.r thn.n Ts.Iu general, the primstive vectors &z, Q of the lmrfaze Bravais lattice aad

those az, Ja of the corresponding blllk-lilce layers are related by

2=

= m a ,ai i;J ..) ,

j=k

i.e.: by a 2x2 matrh V with

s znzz rzzz 1 ya)= w (.'mzz M22

. '

'l4nls matrix Twm be xxpzvl to denote the sarfRe sazpelattice structure. It ro-

mzlts in the so-rxlled matrix notation. The matrix also allows a convenientclasslcation of the relation of two 2D lattices. There are tbree pezeentca-:

(1.12)

L When all matrix elements mv are integet's, the lattices of the sudaceregion and the bulk substrate aze simply related. The surface lattice Lscalled a simple saperlattice.

ii. Whea all matzix elements mq a're ratior./ ntunbers, the two lattice,s aze

rationally related. The surface Ls said to have a coincidcncc structtzre,and the s'aperstrudttre is referred to as comrnezzsuzate.

iii. Witen at lewst one matrix elemeztt ZZUJ is an irrational nlzmberj the t'wolattices are ixrationally related, and the supezstructme is termed inco-llerent or incomrnenlmrate.

Iu t'he Xst and second cœses, the combintd slprfnce layer and b411k mzbstœate ischaractedzed also by a Bravais lattice, azd the stkrface layer is in complete or

partial coinddence witlz the substrate. The, three cas/ (i), (ii), an.d (iii) capmore easily be Garactezized by the deterrnlnant of M, det V, Fhere det Mis atl i'ntege-r, rationv or irrational n=ber. Geometrically det .5.f relat% theareas of the primitive hlnnst cells spanned by the veetors Z51, Ea an.d az, lc. Inakny case the matzix V can be tzsed to c3aracterize a reconstructed surface.'I'he correspoadlug notation is called mcfzi'r notation-

1.2.6 Wood Notation

As an alternative to the matrixnotadon, the more trnmparent Wbod ràtltatf,onE1.6) is 'IZSGI in m=y ceases as a labnlsmg sche'me for the reconecucted stzrfaceaud the occumhg superstructare. The f1.rs1 step is t:e Garacterization of the(?zk!) crystanographc orientation of the substrate sudace (more preclely, the

pl=e) with the chemical composition S by SLhkl). Theze is a simple notation

1.2 Tw-Dimensional Czystals 21

for the reconstruction-induced superstmctlzres in terms of the ratkos of thelengths of the pm'rnl'tive lattice vectors of the t'wo 2D Braes lattices unde,rconsidezation. Such a smface Ls elnn.razterized by

J(&AJ)s (r3''l x 1*21) awo.a:I Iazl

Izl the notation (1-14), /ç ls either $ (for primltive) or tc' (for centered)accordicg to the way in whlcà the l'=a't cell of the mzrface Bravais lattice isdeoed. Wb.e'a the letter $p' is dropyed, the prirnl'tive notation is understoodimplicitly. The quantity tmxz;l = (14: l/1ë,l l x lQ 1/(5.2 1) indicates the ratios ofthe magnitudes of the tusuallyl prîmstive basis vedors of the surface latticea'n.d tEc bnll1r beneath One speaks about an (m x A) reconstruction. Thesymbol Rvl intludes the possibzity of a rotation (.R) of the tïnst cell of theoverlayer by p degeœ with respect to the n'nit cell of the substrate, i.e.,the aagle between zzl and @z. If e is zero, then Rpo is omitted Fom (1.14).Consequently, typical denotations could be

'

Slhkitm x r;, S(h,kl)c(m x a), and S(hA!)(m x n4RIV.Examples are plotted in Eig. 1.12. For a couple of evnmples of recomstractionsthe relation between the' Wood notatitm (1..14) and the matra'v notation (1.13)is givea in Table 1.2 (1.7).

Several additional remarks are necessaly First) sometimes the Wood no-tation ks not tmequlvocal. The lattice vectors iz = 'rrzal and Rz = rzë,z are

not nemqsrily primitive as o ' '

y assllmed in. tNe notatâon (1.14), ald inaddiiion to prlmstive (p) zeconstmzded sarface lattices, also centered (c) ona

S. >.c)

ng. 1.12. Three dr-fTerent tgpes of surface zeconstractiozus. (a) 1x% (b)(WxW)S30*, au.d (c) general =e. The Wood notation does not apply in thiscase; koweve'b the matrkx rotation do% with =zz = 5, mz2 = -1, mgz = 2, ma2 = 2.

22 1. Symmetry

Table 1.2. Wood and rnntrix notatitm of recomstructed sarfaoes of cubic and hexaponal crystals g1.7j.

ReconstructionIdeal suz'faœ Wood aotation Mntrix notation

r(1 )( 1)z'â1 X 1 (l' Z)j 1

fœ(100), pt2 x 1)A2 x 1 (0 î)bcc(1Q0), p(1 x 2)z%1 x 2 (1 .s0)

1 ldiamondtlœ), c(2 x 2)>(Và' x Vil.a4.5= (z Jzinc blendetloo) p(2 x 2)*2 x 2

'

C j)(2W x W)2450 (? 2)l lc(a x a) (0 z)p(z x 1)...1,1 x :L (1 0)j 1

fcctlll), p(2 x 1)=%c(2 x 2)*2 x l (m 0J1hc.ptt)Kll, p(2 x 2)*2 x 2 ( o)z

dinmondtlll), (./J x VJ'11?.:$00 (1 ))zinc blendegll), c(4 x 2) Cc J)graputetoool) lv'#' x ./flaa,rct=t./J/sl (( j)

p(1 x 1)*1 x 1, (1o D)lfcctllo), ,(2 x 1)*2 x 1 (j t)dwmondtllo), ,() x 2)i1 x 2 (j ()):zinc 'blendetsrto) c(2 x 2) () t)

p(1 x 7.) *:. x :, (à -0)1bcctllo) J,(û x 4*2 x 1 (-2 .)1 !p(2 x alec2 x 2 (0 p

are possible. TH c.FJ1 only take place, howeveer, for rectn.nglllar smface lat-tiœs- Thus, the mrvlifled aotation applies only to this cmï4ra, althouglt it is alsosometimes ased (formally incorredly) for sqwe httices. Ic the rect ar

case the notatlon ctl x m) descibes a iype of reconstruction witic.h is usuallynot covered by one of the notations W x 'rn) or (n?? x c//lswo. For squaxereconstzuc'ted lattices the ctzz x m) notatioa is Just a simple,r description ofa reconstzuction of type (rt' x r/)JM51. One example is shown in Fig. 1.13.

Second, aaother problem is related to the fact tlmt one alld the same re-

construction may be deflned in di%rent ways. The problem is a consequemceof tEe high poiut symmetzy of the crystat with aa ide.al surface. lf the latte,rhxq a square latticq an.d one of the two poiut syznmetry g'roups 4mm or 4,the directions of the two pvlrnl'tive lattice vectoz's aze symmetrica;y eqzziwalent. A surface reconstrudion which iz=eases the smface tlnit cell in the

1.2 Twceimensional Crystals 23

Ja :2

(y). --yky)s (y). ())t;k;-. l

O : . C)à O (91 ..' 1

1. (3. * Oe O

O1Q1O1Fig. 1.13. A surface supeatructua'e with the possible denotations c(2x2) and(WXW)X45O. '

direction of ë,1 by a factor n and i:a the diredioa of ë,c by a faetor mt isequivalent to an zrz x zz reconstruction. Azl evn=ple ks given izl Fig. 1.14. A.u.analogous statement holds for a'a ideaz sttrface, having a hezxagonal Jxtticeaad one of the p' oint groups emm, 6, 3m, or 3. J.n this case, three symmetd-cally eqaivalemt directiotus eist (see Fig. 1.14). If there is zzo physical reasonwhich malces one of the geomctrically diferent but symmetrically equivalentreconstructioas more likely than anotller, they will take place simultaaeouslyin dsFerent regions of the surface. As a result domahzs can be formed ofothezwise identical, but diFerently oriented, recozustructed unit cezs. Due tothe domain structme, the overalt trn.nKlational symmetzy of the surface isdestroyed. Strudaral imperfections of a more local natme occur wheare thebotmdn.rles of such dornxinn meet. In the cmse of the Si(111) surface, the 2x1qlnl't meshes may occar in one, t'wo or all three (211) diredions,' dependingon the cleamge conditions E1,8j.

Fig. 1.14. Symmdrically eqtzivaleni 2xl reconstructions of square and hexagonalideal latticu.

24

(11 1)a)

& YN. NN. %.N %.N %% N

r

l 1

1) -1 o'.l

1. Symmetry

Fk. 1.15. Two diferent 2x1 recoctructions of the (111) suzface of diamond-stmcttlre crystaks. (a) ideal mtdnne; (b) 2xl recons-tnzcted =z#v..e due to rthxl'u

formationi and (c) 2x1 reconstrudH mtrfnne due to an heqtlivalent bunlrMng ofm'rfnz,e atoms. Dots: nominal fust-hyer atozns; clple nomloxl second-layer atoms.

b)

z %-h. .# *

x.# %.

; %*

/ h.

< < , i'&> ; >'

Nw ê v-

z'

! Ih. /

t' w w ;

(:1 -1 0E1

c)., * N.

.: N## %

.f. ,' %ez %

- r=q v #. :N1% > v.& ., *

N + IW1& .,. 'f

i -

D 1 n:l

'PMrd, a certai'a recorsstruction denoted by an oression of the type

(1.14) ca,t be realized by rls/erent atomic coegurations. This is dezronstratedt'a Fip 1.15 for diferent arrangements of fzrst- and second-layer atoms oî a

diamond-like material (111) surface. The easiest way to reconstmzc't the ideals':rixz!e (Fig. 1.15a) is by bunkling the frrst-laye,r atoms (Fig. 1.15c). The daa-gling bonds paraDel to (11lJ become ineqzzivalent. ne accompanying deerentElJJ.M with electrons supports the tendency to an inert surface. However, the

' bonds can alqo be rebonded if they occur at atoms which are 62st-

zteaœest neighbors (from the bulk point of view). The sarface atoms may be

arranged in the form of c-lznsrtq lying nex't to eac,h othe,r (see a possible exam-

p1e ia Fig. 1.15b). A11 atoms of a càain are coupled togethe,r by 'r-bonds of

paralle: dangling orbitals. The gcceration of daqgling bonds at the formerlysecond-layer atoms icclude zemarlcable chaazges of the bonding topoloa be-

aeath. The atoms in ballt custals of the diamoad structme aze bonded in

sidbld zings. In a rr-bonded c'hain model I1.9J, however, fvefold azd sevenfoldHngs are formed.

Fourth, the Wood noiation e>n also be used for sllrfxvte overlayer stmlc-

1-%% dae to adsorbatœ. A periodk arrangement of adatoms or adsorbedmolemtles A.1qn givœ rise to a su c'turea which caa be classife accord-

ing to exprëssion (1.14). Eowever, one usqlnlly NICIS a ttRmn -pA to the Woodnotation. The chemical stoichiometzy of the atomic or molecular overlayer

is given by X, and T is the nllnnber of adspedes i.n tba overlaye.r unit cell.

For example, CO adsorbed molecularly on the Ni(100) susrface at a h'actionalsurface coveage of one 'Axlf formq an overlayer shown by the dots in Fig. 1,13.According to oression (1.14), in this case the suzface deaotation bcxcomes

Ni(1ûO)c(2x2)-CO or Ni(10O)(UfxVl)A45O-CO.

1.2 l'wo-Dimensional Crystals 25

1.2.7 Symmetry Classecation

In the commensurate case all the reconstracted sltrrn.ca with the tmderlyingbttlk halfspaces possess a translational symmetzy nyxracterized by the fourBramis classE's with group elements (slA). The corresponding 2D Bravaislattice tramsforms according to a certain planar point group (sce Fig. :.8),the so-called holohedral pott groap, with elememts (aI0J, The 2D crystal,the atoms in the surface and the bttlk below, transform according to a suN

group? its poict group. 'I'he combination of tNe translatiomal and point voupsmmetrie,s gives the spRe group. There are 17 plnmxr space groups. Latlicegwith a corr-onding spnmetry are shown in Fig.1.16.

It is evidemt that fur.h of the 10 'point p'otlps oi equivckleat directior.scombiued with the conuponding aodated lattice gives rise dizedly t,o aso-enlled symmorphic space group with ele-meats (tx(A). The spee groupsp1, p2l1, J1m1, pzrnm, p4, fmrn., p3, p3r11, #, and fmm orioate intids mn.nner. Since tke point groups of the rectangular crystal s'ys-tem are

each associatcd witk t'wo Bravais lattsces, prlmltive or ceatered, we 6nd twof:IVGe.r spae,e voups, elrp,l sad czmm. 1xï the ctkse of the point group 3,m,there exist t'wo dslerent possibilitiœ of positioning t'wo reNedion lines relativeto the hengonal lattice vectors, elther through the vertice of the equilateralhexagon of the Wigner-seitz cell as assumed in the case of p3mI, or suchthat they bisect its edges. J.c the latte,r c%e oae has, as the thirteenth s'pacegroup, the voup p31m. The point grou.p rfAmn.l'nR uncha'aged if in its spacegroup a glide refecvtion liue Ls substituted foz a.c ordinary re:ection liae. Onemusi therefore e-xarnine the 13 space groups alzeady established to determinewheGe.r the mzbstitutioa of a reNection line r?z by a glide reflec-tioa li'ae g (i.e.,a zefectioa irt rrù in conjtmdion with a trnrmlaf-ion 'v by half of thE shortetlattëce vector parallel to m) 1(.)* to a new space group. Oae easily 61.1.*that th.is is not the cmse for the hexagonal crys'tal system. Ia the quadkaticcrys'tal sys'te,m it is possible to s'ubstitazte a s'yste,m of glide reEectioa liaes forone but rot b0th. of the non-eqttivaleztt refedion lin.e systems. Tizis yieldsthe additional space goup pngrrt. The rematniag space groups .p1.gl (lom.y1m1) acd pzmn, p2.:.: (lom p2mrzl occtzr in 213 crystals with a prlmstive:rectangulaz Bravaks latuce. They contain elements of tahe form (aIA+'rJ with:i- ks a Factional latdc,e translation. The ceatered rectnmotlxr 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.

, ' . .

;',ïV Jnternational notation is used. Despite the fact that the atomic basks isElékvnded parallel to tite negative sttrface normk direction, we use a plane'' è' 8 Angular coordsnxte system with uzlit vectors ex, ev. The ozigin of the co-:?. ç '

'érdihate sys'tem Ls positioned on the rotation x='q if one ests. rne prlmitive', .bzsis vectoz's of the 2D Bravais lattice are az, Q. The second bar inclicat-

- * *' . *

'yh. . .

pt

@ @* @

* @ @* * @

y. .@* œ *

p211

@ ** v @

œ@ *@* ** o * .

* *@@ *

clrnl

@ * *. p . . * .* *

ol *** e *

@ * @@ o .

@ * œ * @*. **

2ggp

@ *.

'.*

1.*

plgl

+ @ *'*. '* * @* * @

@ @ e @ * .r* J* '*@ * *.

@ @ @ @ * *@ * @

p2mg

* @ *@ . * . *

@ * o @* * *

* @ ** @ . *

* * ** * *

* * *e * . @

@ @ *@ * *

* o @ .

* @

@ * ** * *

* @ *

* . * p

@ *

p3@ ** . * *

@ * @ ** @* @ ** @ @* * *

@ @. . *

e @ @* * *

* @* @e *@ @

*

* *@ *

p6

@@ *@* * * ** . * o *œ * e

* *.

**** **@*

@@ *@ *0* @ * @ * @* * @ *@ @ @ @ *

* **. *. * *..*

*@ @** @o *. œ *œ @*..* @.e

p6mm

Egg. 7..16. The 17 phnxr space groups represented ms parts of lat-tice,s satisfyingthe synzmetries of those space voups. Tîe sma:tl dots are at the corne> of the unitcells, or at their midpoints, for reference. One large dot is positioned at a.c arbitrarynon-synjmetrical lecation within the tmit cell, and khe other large dots are obuined1om thts one by appl>g a:ll the relevant syrnrnetry operatioc. ARe.r (1.7).

* @ @* * *

* * ** * @

* * œN'- . *

@

plml

@@ ** ***'o *:* @*w

@* @@ *@o. @@ +*

** 0* *@** * @**

p2nann

*@ ** @@.'o .'. @**

** *** .* *@ ** *. .. @ . .* @

. * * ** **

* œ @ m o

* o

Vmm

+ * @. @ . * . @

@ * @@ @ * . *

. @ @* * œ

* * * * @A . * *

@ * * *

* * * * @* *

@ * * * @. * @

* @ ** @ *

p4gm

* *@ ** . @* a @@ - *@ @

@ * ** * @* *

. @ * @@ * @* o @

@ *o @* @@ ** *

p31m

* @ @ @ * *@ @ * @ * @*. @. @

* @ @ @@ @ * * * *

* @ * @ * ** * * * * @* * * * * ** * * @ @ @

* * * * * @* * o * * *% @

* @@ * ** * @ @ * *

Mmm

* @ * @M . * * . @

* @ @*

@ * * @ * *@ * * * @

* e * *

* *@ * @ @

** **

p3m1

l.2 Twceimensional CUS'i?IS 27

Table 1.3. Symmetzy classifcation of 'twmdîmensional crystals (1.2, 1.10j.Crystal Bravais Point Space Symmetrysystem class group group elements

oblique DobDque 1 pl (cLA)2 c pall Le .a), (4Ix)rec.t- p-rect- r?z plrzl (s &, tznogxl

=rllpr

p1J1 (s1A), tznvr'r+Al, = = Ye. .

2mm pzmm (cIx), (JIIAJ, (NIAJ, (m.1AJpz''n'l,p (c1AJ, (4IAJ, '(mxl'r + A)', 'tmvrr + A),

v. = 1e. .. .

772:: (FIAJ, (JilA), frrul'r + AJ, (mvl'r + aJ,.

m .ve.+ ve.c-rect- m clzrzl (sIR), ('maIA)=>=

zrrzzrz czmm (slA), (m.1A), trrh/lal, (Jl IA)square p-squaze 4 .p4 (s)A), (J)1A), (JlgA), (JIIA)4nsm

4mm p4mm (slA)j (J11A), (J11A), (J11R), (mm1A),(Tr?'p EX), IMJIXJ, I?T4IAI

p4:%1 (cIJz), (J1EJQ), (J112Q), (d1l1Q),f-l'r + A), (,41,:1-, + &, (m:IA+,rJ,(m4IA + +), 'r = l(e. + ev)

hexa- p-kexa- 3 p3 (sIAJ, (JI(A), (JâIAJgonu gonal

6mm

r37%1 (c(A), ftQ1A)) (;JIA), frt>alal,(mzlA)) t=ilAl

6 ,31m (a1A), (J)1.R), (JlIA), trzvlAl,f'rrzz IAJ? t'm1 IAJ

,6 (c1A)? (tQ1A), (J3IA)) fJl(J0)IJâIA)., ((Q1A)

6mm p6mm (clA), (JàIA)? (dJLA), (#1A),(J1EA), (Jà1A), (m=IA), (mcIA),(m1IR), ('mXIAJI t'rru IA), (ml !R)

28 1. Symmetzy

Table 1.4. Space groups of ideal low-inde.x suzfaces of diamond-, zïc-blendeo andwurtzite-type crystals.

317 crystal Surfade Sp= groum

structure

Fit'st layer First First ïn6nite

two layers three hyez.s half space

diamon.d (111) p6mm lfzn,l p3m1 p3ml

(110) p2mg ,2r/ p2mg pzmg

(100) pkmm 2727p,rp, p2mm p2mm

zinc blende (111) pîmm p3m1 p3ml 3)3z?zl

(110) plzna plml p1zrz1 plzp,l

(100) p4mm pzmm 42mm p2mm .

wmtzite (0001) p6mm ,3>1 p3m1 p3rz1

(10ï1) pzmm ,27/,= plrp,l p1rrz1

(1120) plmm yzm'm plrnl plml

Space voups of reconstruced low-index surfaces of cliamond-kype cestals

Smface Model of reconstruciion Space groups

Fi:'S't layer First Tn6nlte

tw'o layem half space

(111)2x1 bunkqlng ,2mm plzn.l plml

'm-bonded ckain pzmg plzrtl plzr,l

gubonded buclded ckain plml y1'm1 plml

x-bonded molecule p2mm plml plml

(100)2x 1 symmetrtc 25=e.r p2m7p, p2mm g2mm

asymmetric dlmer plml plml plml

ing the reconstmzction ks dropped ic the followiqg. The oorrespondicg plaaelattice constaats aze tzz acd ca.

The general classiîcation of 2D cz-ys-tals izz Table 1.3 caa be used to elhn.rac-

terize the symmetry of mlrface of 1.ea1 . Fxxamples are given in Table1.4. Tllis table iudicates the space groups of low-index surfaces of tgpkals/micondactozs crystnl7n'zing in diamond? zinc-blende, or wurtzite structuzes.Ideal and reconstructed mlrface are considered and the reslzlting groups are

discussed for diferent nlzmbezs of atomic layers below the uppermost one.

Table 1.4 clealy shows that the reskfting space group depends on the czystal

1.3 Reciprocal Space 29

orientation, the nlzmbe,r of atomib laycs taken into accotmt, and the modelof reconstntction. That means, the space voup of a 217 Orstem depends on

all of the above-mentionH details.

1.3 Reciprocal Space

1.3.1 Direct and Reciprocal Lattices

A two-dîmensional solid smface is chazacterized by a 2D Bra,vais httice (cf.Table 1.3) 'ivith prirnitive basis vedors 41 and Ga. We use the vectors withonly one bar independent of whether or not a reconstruction ks pre-sOt. Iatermq of a rectangtzlar pla'aa.r Cartesslm coorrlirate system with l:mit vectorscx an.d ev, the bmsis vectozs read as

J,z = Azze. + Azzew,

5,2 = Azzew + Jbzev.

The determsmant det .j. of the 2x2 matrh

-4.1: Jkz./i =.,121 .d%2

gives the area W of the nlanl't cell of the Bravais lattice. 1.n fact A = 'n' , (&1x&z)= det ai with n as the surface normal.

A correspondsng reciprocal Ictâcc in Fourier space is mssociatecl with theBravais lattice in real space. TEe redprocat lattice, as we shall see throughoutthe book, is ex-tremely useful an.d pertinent in all dlfrraction methods, i.upwrticular in tlle case of low-energ,g eledron doaction (EEED). h.s in three-dimezssional space, the prt'mstive bmsis vectors 5z acd bl of the 2D reciprocallattlce are desned accordîng to the orthogonality relatior

% - 8J = 2rctjk (ï, # = 1, 2) . (1.18)With 'zz as the Alnit vector normal to the surface, solutions of the relatioa

'

(1.18) are

-. J.a x .p,bt = 27 jIë1 x :,21

n x a1% = 2x , . .

l41 )< 5,21 (1.19)

(1,16)

The leagtits of these vectors are 1b:1 = 2'V (c!.î sin (<, a2)J. The prl-mitîve. : basks vectors elm be used to constnzd the reciprocal lattice to a given 217.;'.: ' network. TV is Mlmmn.tizolly ryhown in Fig. 1.17. A general tnzmqlational:' vltor in rKiprocal spu is Tve,n by

è .

:'

.

gu = W + k82, (1.20)where h and k are iutegezs. The set of all vectors ghk #ves the reciprocal net.

1. Syznnzetry

SquareIattiœ

*(p2*

J ,- -* *

@ * o

''''''jk:)t!;yI

b: 1l1

# X

he=gonalIattice

ve

/#'

J'- - - -....4

*

X

bz

% 11

.. t XNx 1

->I

X X

**%J.1

- tbl t

1t11

1x. - - ..l

redangularlattice

ai oee' 1

. - - .- - 4 .

* * *

œnteredredangularIattice

X *N- x .R .......*

@ * e

* *

* @ *

lE l

7 ,111111

1 IX

2#

#2

1I

- lb1 1

l

X

obliquelatjce

%%

l

N'

Fig. 1.17. Direc't httice tleftl and correspondlng redprocal lattice (right). T/e217 Bravais latlices are presented.

1.3 Redprocal Space

kill (00): kul=kill

(01): k =k +gï1II iI1 () 1kikin

î

J2* . . @ *

R* e e @ e .

Fig. 4..18. DiAaction of an ixtddemt plaûe wave V:.IZ wave vector 1I. The mlrFnr.eSs reprœented by the correspoadùzg 2D Bravais lattice. Parallel momentum oomser-vation with any reciprocal lattice vedor .gsu create well-deoed 'llpvacted beams(hk).

'

The reciprocal lattice vedors have a direci physical mea'ning. I'a a dlfh'r-tion experiment, e.g., LEED, exrx dlfrrncted lvmm comwpoads to a reciprocallattice vector gsu =d, in fact, ea& such beam eAn be labeled by the watuesh and k as the btxnm (h,k). Thks is hdicated seematically i'n Fig. 1.18. Theangle of emugence of the ds*aded ben.ms is detemnimed by the corervatioulsw of the linear momentum parallel to the surface. The momentum of ind-

(ib) (00) (20) (40)Pig- 1.19. Bwazd construction for elastk scattering oa a 217 Bravais lattice. Ascaktering geomeW is considerH in which tzzq momentum conservation is 6'1G11edwith reprocal lattiœ vectors gtA parallel N) èz.

'

32

'Pable 1.5. Dh'H and reciprocal Jattices of twodimemsional czystals.

Brawais az , ,,z ura cen of s, , àa srioui.a

1. Symmetry

Z0n6

E2ey

vlc

ex0) ex Ds 2* (1 - *l'e )(a1x, oz. 1 oaoblique

lJa , 4t'v) a2* (0, 1)2v

1

*j5r:

2

wb,

p-rectaxs (c,,, 0) oxq Fz (1, 0)tralar (o, cc) 2:Js(0, :).- f:tF

class dired lattice

eyeyJa 6z

ex e:cu r e ct a.u - ( Mz

, - Mz ) F , ( 1 , - Ma

; )galar (s.z ea) 1 .4m(:, au; q2 ) 2 cc c'.t

2 bz

(c, 0) W 24 3E(1.0) e.x $XWC C

(O, c) L': (o, 1)az bz

ey. e

ex + z,,s :g . 1 ) e:(a, 0) a ( , :zjhexagonat!. ( 1 , Và') 2J ( 0 , w2 ) q

1.3 Reeiprocal Space 33

Fig. 1.20. LEED images of six dt'ferently prepared Gp.Aq(lO0) sufaces. Mterg1.15j. The stuface reconstmzction and the electron enerr are hcticated.

dent pazticlo is p = zki, where &. is the wave vedor. With ks ws the wave

vectdr of the diAacted particles the momentltnn conservation reads as

ksrl = ki r + guk. (1.21)After the c7''fFractive scatterhg the parallel component of the momentllm mxybe equal to that of the incident pazticle (e.g. an electron) bexm, i.e., gsk = 0.'Were is no relation betweau the componeats of ks and ky perpendicula,rto the surfacq because there is no trnanqlationx7 symmetzy in this direction.However, the particles stttdied in a difh-action experiment, e.g., the elcctromsin the LEED cwse, are elastically sottered. One therefore bn.q

Iksl = Ik:1, (1.22)

34

K1. Symmetry

3x3

.

-.-

..,

.

..y .,

K(W xW)œ30o

3x3

' tq..fj x 6'.W)a30oFig- 1-27.. Sjquence of LEXD patterns (with nlmost the same elecron enmgy $4$ 130e'Vl for the Sz-termlnated smface of 61LSiC(0O01). The lx1 bulk-terrnlnated phaseks stabHed by OH adsorption, whereby the following reconstmzcted surlaces restkltby E00 QC Annenlsng of the latter in Si-Sux (3x3 pbnAe) ibllowed by n.nnealing atabout 1000*c ((WxW)2R3r phwse) and at 11D0 IC ((6Wx6W).R3Ob pkase)(courtesy of J. Bqrnlurdt, U. Starke axzd K. Htain'z (University of Erlangenlj.

A solution of the two equatio> (1.21) and (1.22) always efsts for givenvectors kt and gsk. This is i'a contrmst to the caqe of scattering of particleshom blllr crystals with 3D tritnslational symmetzy Coherent scattering can

only occur if kk lies oa a Bragg reâection plnre. The solution of the aboveequations esm be readily cmied out using the Ewald construction shown inF'ig. 1.19. The points, at whick the vertical lines passing throur,h the rœipro-cal httice pohts ghk iatersect the sphere !ks) = Ik:J1 deterrnime the directionsi:a which dOaction mnxn'rna occm. There is exactly one mn.vl'm'r= for eachreciproci lattice vector. The reciprocaz s'arface lattice can tEus be read 1omthe di*action rnn.orima oc the LEED re>tration screen. The ralation be-

1.3 Redprocal Space 35

twee,n thq Oect lattice and the reciprocal lattice for the fve 2D BzaYs netsis shown expûcitly in Fig. 1.17. The dired relationqlnips between the directand reciprocat lattices of 213 systems are #ven in Table 1.5 i'a termq of a two-dirnertsional Cartezian coordinate system deMed by the vectors ez a'ad es.la more detaâl, the table relates the prlmltive basis vectoz's /z, J,a and 81, t)zto the Carteslxn vecto::s using the lattice constants cz, az (or cz = c2 = a) ofthe 2D nets. Moreover, the relatiomship between the Wignerw-seëtz cells of thedirect lattice (i.e,, the ltnst cell) and the reciprocal lattice (i,e., the Brillotns'nzone) is prœented. '

Typical LEED irnnges are preserzted for rectaagular (squaz'e) lattices iaFig. 1.20 and for hexagonal latticœ i!l Fig; 1.21. The bright spots correspond

'

to the reciprocal htt-lce of the ideal sltdhcez while 'i%e less bright spots are

related to tEe Mer reciprocal lattice of t'be r=nstracted smvfaae. One hasto mention that the construction in Fig. 1.19 Ls evnz!t only in the limit ofscatterixtg 1om a true 2D network of atoms. J.x1 a real electron dffrraztionex-perirnent: however, the primazy' electrons penetrate several atomic layersi'ato the solid. Therefore, the mea'a 9ee path of electrons deterrnlnes howthe third Eaue condition becomes more aad more tmportant. This lto-vs to a

modulation of the intensitic of the Bragg refections iu compnr'Kon with thecase of pme 2D scattering.

1.3.2 Brllloxal'n Zones

In trauslationAu7ly iuvadaut systems the wave vedor k dténes a set of tgoodlqlTnntlzm numbea's for each type of elementG excitatiom Iu the case of an

ordered surface of a czystal, such a wave vector, i, is zestrided to two di-meusions, i.e,j is pazallel to the surface. Wit%n'n a reduced zone sche-me it îsrestrided to a 25 Brillolli'n zone (BZ). The eatire 2D redprocal space can

be covered by the vectozs L- +J, where g ks a shtrfnno reciprox httice vector

(1.20). rne sndhce BZ is do6ned as the m'nxllest polygon in the 2D redprocal'

spaçe ïtuat'ed symmetrically with respect to a give,n latdce point (used ascoordinnte zero) and boamded by points k satisyng the equation

- 1 ak . g = -LgL .

2 (1.23)

The set of points deMed by (1.23) gives a straigb.t liae at a distance 141/2from tba zero point which bisects the connection to the n.e,x-t lattice poi'at .çat righ.t angles.

Since there are Eve dlfearen.t p7n.ne Bravais lattices aud, hence, dve dlsezentreciprocaz suzface lattice-s, there are also ûve HlFere'nt 2D or sarfnzwo Brillollr'nzonas. They are shown in Fig. 1.22. Their shapœ are the same as those of theWignez-seitz ce1 of the corresponding direct latiices (cf. Table 1.5), sincethe Bzavais type,s of the dizect and reciprocal surface lattîces always coindde.

1. SylaMletry

7 y

F

r px

7

Y7 Z R

a r Z'

>P .:.' D' x

(d)

yû z'' R z û

z z'

R' A'' îR i' R' x

@)

?' A ?'z ZR

s. p x z y

(c)

91 Z P2

h' E' QZh

P z v zA' z'

9.

(e)Fîg. 1.22- Brillolpi'n zones of the dve pla'ae lattices: (a) oblique) (b) peedaagalar,

(c) c-reda , (d) scpare, and (e) lmvltgonal. Symmetzy lines a'ad pohts aze alsoshown, and tlzeir notations are htroduced. The 217 Cartesian coordinate system tseosen so Gat the point symmetry operatîons $n Table 1.3 can be directly appzed.

Iu ng.1.22 m have labeled jome of the high-symmetry points of the Brillouiazones Msing lettea's X and r. The bar indicate.s such poht,s ia 2D Brillouhzones wherea points like X and r iudicate positions in the correswnding 317Brûlouin zzme of ln6nite 317 . We follow the convention of dmptiughigh symmetz'y points and linçs i'?/,Wtfe the BZ by Greek letters, e.g., r aad1, d, L'. Poiusts acd lines oa the bonndary of the BZ aa'e denoted by- Romanletters, e.g.) M and Z. 'I'he cente of the BZ ks always denote by r. ApartFom the hexagonal Bravais syste.m the Mgh-syznmetr.y lines parallel t,o theaxes of the 2D Cartesian coordinate system are indicated by the Greek letter1. Iu the hexagonal cmse, E or d is ttsed to hdicate a line from 2% to a corner

point of the hexagon or a midpoint on art edge. The primes on the Greek or

Romaa letters are uscd to allow azt indication of Xerent symmetries ia c.%e,s

where the point group of the 2D crystal is only a subgroup of the holohedralgroup of the Bravais lattice.

1.3 Reciprocal Space 37

Unforeately, the notation in the literatttre is not consistent. J.n the ori#-nal papers several mocls6cations aa.e wsed. WIJ.C,E of the asPere'nt points sho'aldbe indicated by a prime or not, is not exactly sxed.. For instauce, sometimesin papers about the cleavage face of zGo-blende crystals or the (110)1x1surface of group-W crystals, I'X is used to indicate the shope,r mds in theBZ, izz contrast to Fig. 1.22 where Ehn'q line is denoted by FX-/. The're aa'exlr.ry >amples where authors use the notatâon C tns'tead of X' (1.11j . In thecwse of the 2x 1 reconstructed (111) ar.d (100) sudaces of groamlv matezials,there is a tradition of following the notation of the squaœe lattice. Instead ofX acd V, the notation J' and 1% is used (1.12, 1.13j. In the latter case even

J and J' are lteraiuanged.. Sometimes one fmds a paper in whic.h the corner

of the BZ is denoted by &V and the midpoint of the edge of the rectangle by7 (1.14) .

1.3.3 Prolection of 3D Onto 2D Brilloxxx-m Zones

The fact that the 3D wave vector k from the BZ gives a set of Cgood' quaa-tttm numbers for elementary excitations ia an l'nilm'te crystal has several

con-uencc for the repr-ntation of the enerr specïrlxm of elementary e,x-

citatsons i.n a mygtal 'Zt,'N surface. 0n the (me hand, bulk eaxdtations shouldakso oc= in a semi-lnGnl'te hatfspace with surface. On the other bnrd, sucil asystem Ls o1y Gracterized by a 2D translational symmetry. Cozusequently,the elementaq excitations of the ftnite s'ystem ca'n only be elnnracterizmd bywave vectors k from the Brillouin zone belonging to the corresponding 2DBravis lattice. Ia order to lzse the Bloch-likn eigenvalu.o of a btto eleaneatarye-xcit,ation, the rehtiorship between the ekenvalue of the bulk crys'tal and thewave vector ha.s to be itered. To represent all allowed eigenstatesl umpnllythe component kjq of the 3D vector parallel to the surface cmn be Aed, whilethe perpendicular compoaent k.s has to be vazied. Generally spenving, thebulk eigeavalue must be assigned to a 2D wave vedor L ia the suzface BZinstead of a 3D wave vector k in the bulk BZ. For obvious reasons, suc,k arelauonship Ls called a projection of the Blocx-like eigenvalues ofsthe balkcystal; the 3D dispezsion relations, onto the suzfaz:e BZ.

, Withia = eprpûcit procedttre certain bulk directions and points of highsymmetr.g in the 3D Brillouiu zone are projected oato the 2D surface BZ. Fortbree 3D BraYs lattkes and some low-index gurfaces the relation Ls depictedic Figs. 1.23-1.25. Izz orde,r to illustrate the projection proceduze, ftrst theBrillotlin zonœ under considezxdon must be specised. The conwpondingbulk BZ is dezned by the Bragg reoedioa planes

1 ck . G = jqG , (1.24)

wkere G is a cer'taizz vector of the reciprocal lattice of the bulk crystal. Thesurface BZ mny. be calculated according to (1.23)- The prlrnstive vecvtors àz,1a- of tlle redprocal stlrfaee lattice neeed in derieg t%iq equation follow

38 1. Spnmetzy

fcc Iattice

(100)-.a= *

x

J

WU .

k WL* e

K J XY .- 1- A .

XA ' L.

''''''!k,4;;r, ''''''jj;(;:r,

K r - -

M KM K M K

KK

U WKlI

I?k 1 IJ XX l W'' 1l ' .e

: wK l 't Kt etl! * L#'

$'

àX W

Fig. 1.23. Relation betwee,n 2D Brillnuin zones of low-index RGMeS and the 3Dbulk BZ in the fcc cax After I1.16q.

(1 10)- MF

x- .k

: K.Lll

W 1W . :x , t

1. .'

kke' 1 )(K L 1

*W W

from (1.10) tzsing the prlrnstive vectors az, 82 of the dizect surface latticeconsidezed. Su& vectors are given lin Table 1.5 for all 2D srtems. Secozd,the bttl'k' BZ is projected onto the pla'n.e of the sadace BZ as indicated inFigs. 1.23- 1.25. We denote by kII the component of a wave vector k of thebtlk- BZ paraltel to the sarface. The bouad.at'y points of the pzojeded blllkBZ are located on straight li'aes detnvnîned by the t'wo equations

:1I = klI15l + k112à2,1 akjj . G = j.IGI . (1.25)

l.3 Rcciprocal Space 39

bcc laoice(100)

R $ EXû

P N p N1

1 :$ HH H wy--p ---

-j- .--.

2l Ii e

P N '

# 'J P: 'c

'g F l. r

F y

HP P

1

H :1 . 1x* . j N l .'

zr 1 Heee !

P N e '.

*N

PPH

(1 1 1 )-c

R ûR

PH 1 H

1 Nr: l 2

$ PIo I 1

I 'F' 1 XN ..K - -)*- - -

* ee * . Bxe K-* P

H *N H

P

Eig- 1-24. Relation between 2D Brillouin zonas of lowede.x sm'faces and the 35'bulk BZ izz the bcc case. Me.r g1.l6q.

In gemez'al, the projeded bulk BZ does not coincide 'with one surface BZ.It ks tlsllxlly larger tsee e.g. in Fig. 1.26 the tu-ample of the (100) surfMe ofazt fcc crystal) and one has to fold back the part of the projeded bulk BZ2not conteed m the surface BZ onto the latter one. Since these paz'ts of tlleprojected bnllc BZ agrœ with neighboring P,D Brizouin zones bezlonging toredprocal lattice vectors gtkjj )1 the foldiqg ts identical with a displacementby s(kgg). Consequently, all wave vectors k in tbe sizzface BZ are give.n by

k - k11 +s(à1I). (1.26)

40 1. Symmetry

hcp Iatjce

(Q005)

r-K

ë

H H

A HH 1l1 LL l

H I L Ftl

!1 .. e. ., * % x . 16

< F* ! =w.

--** ' M%-

j hx%-.

AJh1 l xw

K l M KlI

*A H

H L H

FY. 1.25. Relation beGeen the 2DBrtllouin zone of the (0001) stufacesan.d the 3D BZ of a hcp stmzcture.After (1-16q .

J.n tMs mnanner certaizz re/ons of the mnrface BZ are covered two or more

tHes by projected points of the brllk BZ.We elucidate the above general comsiderations, using as au e-xample the

(100) suzface of au fcc C.USIPZI. The 14 vectozs of the redprocal lattice de6nlngthe blll;r BZ according to (1.24) are G' = 21 (Aez + eu + cz) aud G =

@iMez,v,a. The prirnstive vectors of the redprocal surface lattice are 81 =

G .

Mlev - es) azzd 52 = 3X. (eg + ea). The four vectors de6mlmg the smface BZ%

are g = +b:, :1G. According to (1.25)* c

àlltte,v - ea)J+ :112(ev +e>)t7 = -1/12.4.,.

If G = Attex + ev + ea), for examplev is cahoscn, it follows that

3kllv =

u,aad

klll + kllz =.1,

(1.27)

(1.28)

(1.29)

1.3 Reciprocal Space 41

(a)l

(b)

F:g. 1.26. (a) Brillouin zone of a (100) suzface (shaded area) together with iheprojected bulk BZ of au fcc crystal- Projected critical poinlq of the 3D BZ areindicated along a (011) directioa. (b) BG BZ ïor comparlson.

if G = Mev. For the otb.er vectors of the 3D reciproe lattice, e'ither ssmilnmrelations resttlt or the associated Bragg redectioa planes are parallel to thepiaae of the surfRe BZ taad thus do not inte-rx'c't it). The eeuation of therelations of type (1.28) and' (1.29) reslilts in Fig. 1.26s. One notes that theeapcoidal are.as of the projected fcc BZ lying outside the surface BZ caa befolded ove,r the surface BZ by displacements along o=e of the httice vectors5z, -8:, Ja, -&. The little square i'a the center is part of the Bragg reNectionplane boundt'ng the b:llk BZ and associated witll a reciprocal lattice vectorG = +2E(2., 0, 0) perpendicular to the sudace. The mn.v'rnl:= (rn5n''nu,m) offlk.z (= kz) is thus +3X. Outside the Ettle squikrel k.z = k.u (k: j ) 'vades in a

smallez hterval âxed by the 313 BZ. 'I'he iaterval depends on the wave vedorkil-

1.3.4 Smmetry of Points and Lines in Redprocal Space

The spaual symmetry of a crysial with slldhce itas implimtions for the' possi-ble degree of degeneracy of elemento e'xdtatiozus with e'aergies zflgLkj. Thceigenvalces DgLk) as a fnlndion of the 2D wave vecRr k #ve the so-calleddispersion relation for the corresponding elemento excitation. The set ofindices Jz labels the remasning qtzantum numbers. Evnmples a're electzxm andhole e-xcitations with J2#s(i) as the surface energy barzs and g as the bandindecq sttrface phonomq with dispersion relations Xlkj of the vibrationalbranGes g, smface plasmons, etc.

The syatial symmetry results izl relations between the values f2s(i) fordxerent k values. The key for such conclusions are, in aaalog,g to the 't=6-nite bulk case, the irreducibie represotations of the space g'roup of the fvenczzztal with surface. This is based on the fact tllat the fzigenAlnctions be-lon#ng to a padiclzlar energy eigeavalue form s basis set of azt irzcdttdble

42 1. Symmetzy

represcntntion of thks group. Suc,h a repres%tation may be characterix'd bythe sta,r (V of the wyve voctor i and the irreducible repr-ntatiozts of thesmall Mzzf gnmp of k (1.10J.

A m'nnll poi'at group is a subgroup of the poim.t grqup of the crystal. Thepoiaçgroap elements â of such a subgroup trn.nsform k neither into itselfnori'ato a vector equivzem.t to k that difers fzozn k only by a Teéprocal htticevector g. The set of all HsfFe-rent aad nozseqlzivalent vectors (RA Ls cazed J' tarof i. At all points of the staz (V the eneror eigenvalues XLkj have thesame value. The s=n.ll point groups of Mgyh-symmetrjr points and lines i'n. theBZ are listed in Table 1.6 for the various space groups of BraYs lattices.

Table 1.6. Poizti rotzps of tHe high-symmetz'y points and liaes of the BZ (1.10j.The trredudble part of the BZ Ls indkated by the lzatched regon. .

a) Oblique lattice

Symmetz'y point Space voup

P1 17211

? '

: ;

J l 2

Heduciblepart of BZ

b) nrectaagular lattice

Symmetry point Space group

or line ylml plgl y2mm nzmg pzgg

2% m m 2mrs 2mm 2mm

gvwt srz rgv yyz

XXt 2mm 2mm 2mm

Z' m m m

V m m 2mm 2mm 2mm

ZZ' m m m m m

zl - - m zrs m

Irredvcibleêari Y BZ

1.3 Reckproc/ Space

3)àble 1.6. (contilued)c) csrectangular Lattice

Symmetzy point Space group

or line clml Omm.

P m 2mzp,ytvzs m

X 2m,m

Z m

Cz m

C m

,4 pz .

Irreduciblepazt of BZ

d) Square lattice

Symmetry point Space group

or Ltae J>4 p4mm p4mg

X 4 Qnm 4=,4 m Az

J 2r= 2mm

Z m m

X' 4mm 4rzr4

.D - m m

Irzedudblepart of BZ

44 :. Symmetry

Tbble 1.6. (conthued)e) He-mgortal lattice

Srmmetrz Space group .

points or lines ,3 p31m p3m1 .3% fmmJN 3 3m 3m 6 6mmzs',!;œ m m

4z-/' 3m 3aZZ' m m

##' m 2mm

AA' - - z?s - m

Heduc3blepat't of BZ

The Heducible repruentations of these point voups are gven in (1.10) forthe vazious space groups of a, corresponds'ng Bravais httice. TEe dimension ofthe Heducible representation deterrnsnes the degeneraey of an eigenvalue at a

given :. For the systtamK tmder consideration only Heducible reprœeutationswith fîsmensîons equal to 1 or 2 will appear.

2. Thermodynamics

2.1 Kinetic Processes and Surfaces in Equilibrium

The real surface of a solid tmder atmospheric pressure is ver.v diFerent homalt ideal system desirable in surface physics. Therefore, surfus as objKtsof physical studles are usually prepared in UlIV. Besides cluavage a'ad thecombination of jon bombardment and n.nn-b'ng, acother chssical methodto prepare a fresh, clean surface is thc euporation and condensation of an

overlayer on a substrate, e.g., within a:ct MBB procedure. 'ne resulting s'ur-face system, more predsely tke solid-vacu7nm interface, is no static system.Several Mnetic processes ocmtr to a,cez'tain extent depending oa the mzbstratetemperatm'e T.

A seledion of such processes is iudicated scornxticaxy in Fig. 2.1. Tlteyrepresent elemeatary events whic.h happen daring epiteal growth, e.g.,NIBE. Hbwcver, such processes also oc= at the substrate-vacuttm inter-

(a) deposition (aisorptson) orl a terrhaoe

(b) deposition on an island .

' q r

(c) on on a ce . .q . .

. .

. . . ! . I '

. .(..

(d) daorption . . .. . . . . . . .-. '

. '..

. . . . . . . z'

(e) cucleation . ,. w ... '. '

- . . '

. ' vh'

.5

J

., . . .. .. ... .-. .., .. ... .. .. . . ' ..'.,

, .. .. . . . . . ' '

.::4 7 ($ knterdlfrusioc (exchange)

.

' , .

' .

' .

' . .

''' . .

'''F

(g) atlcmont at an island

(h) de/chment from an is:and

(i) attachment at a step

F5g. 2.1. Schfemxtic representation of R'nziame-ntal atomic procasses occurriztg dur-ing epitaxizl vowth.

46 2. Thnrmodynamics

face, though with a reduced probability due to the lower tamperattlre audthe smxller number of atoms ia tike gas surrotmdiug the s:arlhce.' Ia any case,

the vacaum Ls not perfect, rather a reatgms occms. The atoms and moleculain the restgas interRt with the Rmrfaem. Adxrption and desorption of atomsare observed. Bœides the temperatme and t'hc cknmical nattzre of tlm atoms,the strength of 1%- proces- also depemds on the atomic sites on terrazes,at islands or near steps. Other proc- happ= on or in the surfaze, e.g.,dsfFnqion on terraces or along stem and jnterdlelnion due to exchange reo

tions. One possible conxqueace Ls nucleatiom Aaothea one is the atttementat stem or isLands giviag rise to a layer and/or isl=d growth.

The elementary ldnetic processes are also infuealced by the sudaze mor-

pkology ard geometry a,s the presence of tzenehes and steps which detmmnlnethe local suzface emergy. The spatial vadation of the surface energy deoesthe adsorption sites ard the difhcqion barriers. The clhn.razteristic bondhlgenea.a of an atom at a special site a'ad the enerr barriers betweea.such sitesdetèrrnsne the probabity for adsomtion, desorptiou and difhlqson of atomsat a given temperatuze.

In the thmrrnodpmrnic equilibrittm al1 surface processer proceed in t'woopposîte diredions at equal rates accoreg to the Sprinciple of detailed bal-a'ace'. Detailed balance means that the rate constacts for the forward di-rection, vç, a'nd. tke back-wa'rd direction, /b, of a process satLsfy the rehtion

vtjrs = exp(-1S/kBT), where AE is the energy dsFerence between theiaitial an.d ftnal states. Processes such as adsorption and desorption, decaya'ad formation of 11a41*, etc-, must obey the detailed balance. The equal-i'ty of forward azd baclcwazd rates in equilibrium is incompatible with thenet epitaxx growth of a surface overlayer. Crystal growth is clerly relate-dto non-equilibrium ldnetic processes. Yet the 'pzinciple of detiled balance' isstill 'h11671ed. However, closiag the shutteo of the emlsion cells, Le.t intenmpt-ir.g the molecular or atomic beamszua'nd holding the system at not too hightemperatuzes, a ratb.er s-tatic sttrfac'e' caa be prepared., which does not changethe principal strudme a,s is indicatcd, e.g.: by the conselwation of a cez'tatuLEED pattern. Such a surface can be considered to be in timmnodynnmt'cequilibri'tzm with the substrate and the surroundings, e.g., the restgas.

2.2 Thermodynnmic Relations for Surfaces

2.2.1 Thermodyzmmic Potentials

The eqttilibxium state of a one-component system consisting of N pxriucleat a %cd temperatmv T and pressure p is the one wit,h the mn'nimum Gibbs1ee enthalpy G(Ttp, N) (2.1),

G = F + p:r (2.1)where FLT, Jr A'),

2.2 Thermodynxmsc Relations for Sttrfacœ 47

(2.2)F = U - ST,

is the Eelmboltz Fee enera. It is related to the intelmal energy U =

tJ(S, J,r N) and the entropy S by a Legendre traasformation. The energyconservatioa law atd the relatioizship between heat and work n>.n be writtenia the fo=

KU = TdS - pdF + JzdN (2.3)for azt kTIIIiteSiIIIAI change of the internal energy. Variation of the numbe,r ofpmicles N Ls allowed due to pazticle Kxchaage with a reservoir characterizedby the cEemical potentiat Jz of the particle. For an isolated system with no

heat Gchange (0 = 0) a'n.d particle Gchange (dN = 0) the intcrnal energyis a constant at cozkstant volume tdy = 0). The corresponeg micascopicdistribution is the micronltnonical easemble of statistical met+n.nics.

The thermodynmp'c potential G (or F) can be used to deriw the thewmodynnmlc quantities of the considered system at consta'at temperatm'e T,paztide aumber N and pressur'e p (or volt'tme F). Tn6mitemnl changes ofthe three vadables give 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)1xl equillbriAlrn, F is a ml'nimllm wtth respec't to the Her variables at constantT, 7, and N, whcreas G is a nninirnllm at constant T', p, aud N. The cor-

responding microscopic distribution is the rltnonicaz ensemble. The Gemicalpotentia,l Jz ic (2.3), (2.4), or (2.5) is given by

.- (w'&) - (w'') - (N,x) r,a .

S,V :r'I5J

Under normal pzessm'e of abou.t 1 atmosphere, the dllerence betweea theHelmholtz 1ee energy F and the Gibbs free energy G,

J? = F - G = -pF) (2.7)is insîgniftcan.t for a bnlllc soDd or liquid. TMs holds'i.a pazticular for volllrne-induced changcs -pdF. 'nus, it is mlmcient to 'ttse F for most cases in solidstate physics. Thc difrerence (2.7) is Kramer's gzand potential J2 = .O(T, F) Jz)(2.22. Despite its smxl:ness, more precisely its vaaishing Muence on changesirt bulk systems includizlg phmse trnnsitions, the potettial is convemiOt to use

for system transformxtions that occur at a constrt temperature T9 vob:rneF, and chemical potential y. This may be of pazticular iuterest for tke surfaceregion of the estems under consideration. Together with tKe Gibbs-Duhem

48 2, Thermodyaamics

eqaation, SdT - Vdp+Nd;& = 0, inflnitesimal nhxnge of the variablœ result

dfz = -SdT -Jd7- Ndp. (2.8)In this cmse the thermodrmrnic properties of a syswte,m are governed by thegraadcanonical statistical operator. Comparison of (2.5) a'ad (2.8) indicatesa tqansforrnlttion law

J) t= .F' - /zN. (2.9)

This is a consequeace of the fact that the Gibbs free enthalpy varies lin-rlywit,h the zplmbe.r of particles (cf. (2.7) and (2.$),

G = P'N (2.10)vrith the proportionality factor /.: = gLT,p) for eaG homogeneous phaze (2.1j.

2.2.2 Sarface Modmcation of Thermodynnmlc Potentials

A stlrfnne of az'e,a A infaences not oaly the spectroscopic properties of a solidbut also its thermodynnaic pzoperties. Tn order to discass the Muence of ah'ee sllzûce of a hnlfqpacc, we follow Gibbsl idea of the dteqllsrnolarl dividingsmfaceh (2.2,2.3). This is illustrated in Fig.2.2 in terms of tke particle demsitjrn = NjV as a fnnction of the distaace normnl to the sltrfnce. It es grad-ually from its solid to its vapor vaâue. In Fig. 2.2 ihe vertical lines incticate apartition of khe total space into a bulk solid vobnrne Vi, a bulk mpor vol=e'Ui, an.d a volame W of the trxnRition region, the surface. The correspondkgdwlnm'tjes nz and 'pu characteriz,e the llnlform bulk phases, the l-nal'-in6nl'telsolid and its vapor with whic,h the solid coexists azd which occupies the othe,rhalfspace. The sllrfnre region, whose spatial eoctent is of atomic dimeasions(about 10 or less atomicylayers), is thus a strongly tnhomogeneous region sulu

rotmded by t'wo homogeneous phases, the solid and the vapor. Unforttmately,the partition in Fig. 2.2 is aot uitîque, siace the numbe,r of particles in eac,h

n 2 c Parucle density' ofg. v .

a onmcomponent system near

a surface arotmd z = 0.

2.2 Thermodynamic Relatioms for Surfaces

phase 1 or 2 depends on thc n=ber of particles Ns in the s'arface transition

regitm. The same holds foT the voblmes. TMS uncertatnty is of the same or-

der of martude as the surface eFect itself. Howeve-rj iu the framework of a

macroscopic theozy kere the titrqrrnodyztpml'œ, the partition is made unique

by applying the n.ataral conditions (2.1)ï,l = J4. + Vi ,

N = 'nzh + nzFz. (2.11)J.n compnriKon to the total nllmber of particles the nllrnber of partëcles Ns in

the surface region 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 azzd the free enthalpy % can be written as contributiozts lom phases1 and 2 plus a surface term. The ozigia of the surface t-rrn rltn be discussedtssing at least t'wo eqaivalemt vîews (2.1, 2.2, 2.41 . We follow tke derivation ofLandau aad Twifshitz (2.14 as we)l as that of Desjonqubes a'ad Smmjaard 12.2Jand consider Kramer's g'razd potential. Since iu thlrrnodpmmic equslibrillm

the pressare p is the same in the tvo homogeneous bulk phazes, at least fora plaue surface g2.l), eqaatioa (2.7) can thus be written in the fonn

J) = -p(M. + vi) + fls.

The surface contTibution J?s to the grand potGtial should be proportionaltothe surface area A

q = 'yA (2.13)( '

wîth y ms the'surface e-xce,ss density of .0.

2.2;3 Sttrface Tension a'ad Smface Stress

E Tlle proporttonality factor ''f in (2.13) can be identfed as the s'arjace czcexss

Lh-' cc energy mr zlrzif area or s'udace .free e'rterny for short (but imprecisely) .

$br a oae-component system the chemîcal potential p is equi in b0th phasestn thermodynxmlc eql'phbrinpnn. Witlz ihe total nttrnbe,z of particles accordiug

to (2.11), G = yN (2.40) is st2l A11611ed.. From our approximate descriptionof the tclividiug surface' with Ns = 0, one readily obtaius Gs = O for the

: sizrface contribution. Tlms, M'itll (2.13): I

Fs = f)s = JX (2.14)

holds. The surface e'xcess h'ee enezr 'y is sometsmes called the snrj'aco fensftm,

although this term is somewhat conhlsing despite the corresponding common

t'nst of measmement. Using the Glbbs-Duhem rehtion it cau be shown thatall thermodylmrnîc quactities of a s'arface can be expressed jn termK of J. The

50 2. Thermodynnm''cs

temperatme depezdence of ''f also allows the de6nition of the s'srfqce e'rcas,s '

entropy Ss (2.14, '

0 J2 s XSs = - = -.A .

OT &T,z1,/$ Aj/z .

The surface 9ee ene'rgy ,(2.14) contnalns two physically diFerent contribu-tioas. Au l'n6mitesimal change is given by

dl's = y ILA + A d'y.

The flrst term y dx represents the reversible work done to cllange the surface=e,a. by dA. This cllange may happen by incremsing the n'lmber of atoms ic'ihe surface at a Sxed averaged area per surface atom. lt ks therefore relatedto an ideally plastic deformation. Ic contrast, the srcond contribution A d'yis related to a,n ideally e'lastic defo=atiom The zu:rnber of suzface atorasremnsnq constant but their interatomic distances vazy. This is accompaniedby a vadation of the =e,a, per atom. Sach a stretched surface has a modifedsurface energy y-j-k, whie,h leads to azl additional contribution to the chaageof Fs .

The c'hacge of the smface area may be interpreted in a rnnnroscopic senseas the consequence of a (biaxial) strain ,ac'tjng on the smface atoms. Withthe components %j of the adual strain tensor azd assllrnl'ag linear elasticitythe work done i'a the surface region can be descibed as the c,hange of theelastic enerr

dFs = X Voudqj (2.17).i IJ'

wlth (7vJ as the components of the s'ur.fatte siress tensor (2.54. They are deoedas the spatiatly averaged deviations o'u = J dzko (z) - cz?j) of the local stress

(qjtz) from the stress âeld deep in the btlllc ct. Accortiimgly, consider a planenorrnnl to the sudace arld 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 ofintersection of the plnnae with the smface iu the Ftk. dtrection. The dementsof the strain ttmsor qJ are dqflned in direct analogy with the corrupondingbllllc quantities. With d.â = A E: dqç, comparison of expre-ssions (2.16) a'ad(2.17) leads to

*3i = '/V + ru . (2.18)witk

'

%g = ,w . (2.19)tlevLt

TMS Ls the venerable Shuttlewoz'th equation (2.6). Sinze in a liquid there isno resistaace to plutic deformn.tion, the second termq in (2.16) and (2.18)vanish aud the surface e'nergy and sudace stress become equ.at (2.5).

2.3 Eqailibdum Shape of Small Czystals 51

It1 the case of surfacœ of czystals this is usually not the case. The devia-tiozus vij play an important role for the reconstruction or relaxation of metazsmfaces (2.74. The stabitity of the 7x7 reconstruction of snrniconductor sur-

faces such as Si(111) sçems to be a consequence of stresses in cllfbrent atomiclayers (2.$. The secozd-rn.nk tensor '3' depends on the surface symmetry. If a

sllr'hce has tikreefold or lzigher rotational s-immetz'y, then the tensor becomesdiagonz with equal components, no' = 'ré%. So the smface stress (2.18) isisotropic in a surface plane. Anlsotropic smface stresses may have au Muenceon the step geneaatioa on surfaces. For instance, the 2x1 spnmetrsc tasy'm-metric) dimer reconstmtdion on Si(10O) has only ml'=or symmetry with thepoint group 2mm (m) (cf. Tables 1.3' azd 1.4) =dl Nence) the surface stress

teasor is anisotropic. Single-hyer s'teps (steps with a height of one atomic

layer) may occur witlz dlserent orientations with respect to the Hsmers. Theformatioa of b0th 2x 1 and 1x2 domaias (cf. Fîg. 1.14) o:a dsferent tezraces

is likely, since tEe sarface becomes isotropic 9om a macroscopic point ofview. For semiconductors smface stresses have beea direcvtly measured (2.9)or calculated (2.10-2.12).

'A liquid Ls chaœaderized by a vauishing resistance to a îow of atoms 9omthe bulk to the surface, a'ad 'dce versa. For solid surfaces one may thereforecoasider thc tenmr '? as a driving therrnodpmmsc force to move atoms h'omthe balk iato tlze stu'face layez. The signs of the diagonal elements deterrnsnethe strain stste of a surface. When 'lnrê > 0, then the smface teads to accw

mulate more atoms. When Trê < 0, the opposite tendency sho'ald occur. As a

consequcnce the surface atoms prefe,r smaller or lrger lattice spaciags than inthe bulk. This arzangement is accompaied by compressive or tensile stzrtinq.

In the cmse Tr'? < 0, a'aothe,r cozzsequence cokfd be a (srnzlletr.r-brenb'ng)smface reconstruction of the type schematicmlly iudicated in Fig. 1.9b. Stuwface rearrangements like atomic dislocations and elastic buckling of a sulfacecolzld also be possible. ln a metal surface atoms might be placed in un-

favorable bonding posltions with respect to tàe next laye,r of the substrate.Thereby, 1Trê1 should be large enough to compe=ate for this enerr expense.The important role of tke Hi/ereace be>een sllrfnzte'energy and surface s'tressfor the actual smface stracture is generally accepted. Howeve'r, there is a con-

tmveray about this dl'ference az a drivi'ag force for the recoas-truction of at

leazt metal surfaces, e.g., of the Aa(110) surface (2.11, 2.13, 2.14J.

2.3 Equilibrium Shape of Small Crystals

2.3.1 Anisotropy of Stu-face Enerr

The smface free e'nerg.r per llnit area 'y of a cez'tain crystal surface varies withits czystallograpkic orientation characterized by the surface pla'ae LhkLt or thesurface normal zz, i.e., 'y = 'y(h,kJ) or y = 7('n,) izl general. A plot of tMs energyversus orientation, a so-called W'IzlF plot (2.15!, plays azl importa'at role in

52 2. Tkermodynmics

ro 1(1 U n!

Ez-g. 2-3- A (ln) ce, whic.h -:s slightly mus- om-ented Fom the (01) s ace.

the theory of the eqllsFlbri:'m shape of crystals and morphological stabillt.y

(2.16,2.174. To illustrate this orientation dependence, a two-dimensional soltdwhich is mirw-a'gcted 1om the gO1) direction is shown in Fig.2.3 for a squazelattice with lattice comstant a. Its nomsnal (1n,) surface (1 >> 1) represents a'dcsnal (01) stzrface. It consists of a Mgh n'lmber of (01) ten'aces separatedby atomic steps of height c. With f 'w A as 'kke azlgle of orientation of (1p1agnsnqt the (01) orientation, the step density is given ms tan ela. If Js is the

emergy per step aud g(0) is the energy of a (01) face, the mndhce energy of a

(1n) surface is

taa 9'y(8) = cos 9 g(0) + ps . (2.20), c

The prefador cos 0 g'uarantee that the relative amount of the (01) terracesto the total surface =e,3, reduces with hcrewsing azzgle J. The htcaction ofsteps IhRA beea neglected in (2.20).

Asmlrnsng that tbe s'tep mode: also holds for large.r angles 8, tlze reslzltîngh'ndion .,7/) = g(0) I cos :1 + J.r-I siu 81 is plotted 'm Fig. 2.4 for angles # inthe fokzr quadzants of the plane. Suc,h a polar plot of àhe surface enerr cac

be dz'awn for 0e,1: place 'wit%x'n a 6n5te crystal. Even msmlma'ng the abovetrivial dependence of 'y on 3, it gives a WG plot of the surface energ.g (2.15J.

Y- > > - e '> >.

.- Nz' N

Zz .--w N/ N - !î ''k !$. ( j y;

N.hx . 0 ' e'

z N Xz ( N/

l X 1t ..ee lN /

N ZN A

N. ... -A

Fig. 2.4. Polar plot of the sar-face ezkergy for a vinlns'l s'urfRewith a 'mzsalignment augle 8. Thesimple energy exwession (2.Q0)has been USCXCI for several ratiosJ%/ h(0)4q = 1 lsolid linel, Aa

(dashed linel, and ) (dotted linel.

2.3 Equilibrsum Shape of Small Crystals

Polar (W:)1'FF) plots of the smface auerg.g are possible fol. arbitrazy plresin the crystal. lt is obvious that the variation of the surface energy withorientation is of great Lmport=ce for a a:lmber of pzoblezns related to sttrfaceinhomogeneities or surfaee morpholoor. Notice thst the surface energy inFig. 2.4, 'y(J), Ls a conthmous gmction but that it hms discontinuous derimtes'at angles limiting the quadrants, e.g., b = 0. J.n fact

-.2 - = z .'dù dê c:=+0 *=-0

That means, there is a cusp at 8 = 0. The increase of the acgle 9om : = 0iè large vazues is azcompazkied by an increaze of the s'tep density. A properixpression %r 'y(û) must hence i'nclude the iateraction between steps. Ia thisèase it has been sho'wn that 'y(p) lzas a cusp at eveo- angle wilic,h correspondsto a set of Msller indices whose ratios are rational Iplmbers (2.181. The shat'p-.

4npess of the cusp Ls a rapidly decreasing fnrnction of the index mw 1/a tseeYig. 2.3).

r rone exnmple, whiclz can be interpreted as a renlization of the modelEètsEctuqsed above: could be a surface of a diltmond-stzudme m'ystal, e.g., Si,àfiented between (001j an.d (111). As the sxmple orientation is tilted 1om.

j og 01) to g111j, i.e., û = 0 to 54.7 , the smface morpholor varies h'om (001).7 ''. 1 'j . ' .

:0012

1.. , ... r . . ()a,!). .t. .. .,.j. j.. (:

.. . '

è. t'I i 1 11a el-ls 1 17 -! 19 el.: $ -1.

' .: ..

. , -1 12 1 1 4 1 1S :1 1 12335 337

r3 r5

: : '

' . 55 12

:' (' :'

..

:' .' 7* .'

. :

''

E

''

'

1*

. .

.'

'

:' .' :' :' :' '' ''

.

.'

i: .

E'

( '

:' ''

.. .

.'

g

:'

'

''

'

5* :' .' .' i' '' '' .' ''

#lkkE 'a'à side view of a disrnond lattice betveen the (001) and (7.7.1) plsnes. Thep i''!7 ..... - - '

jkflsèçtions of the lattice li'aes represent projections of atomk positions onto the;E iï'd''' l%e The connections of the dots with tbe circle indicate the surface plpne;,E(. . .) ..p -

t@'' àtï' oied by (hk2). After (2.l9J.

54

(01 1)(111)

(11i)+(137)

(1 13)(114)

(001)

Thermodneœ

Pig. 2.6. Stereovaphic trirmrlè forcubic-czys'tal surfnrm ahîbiting thelow-index corner points (111), (011),and (001). In addition, one hig,h-index smface. is indicated in the i'a-tedor.

to (114) to (113) to (55 12) to (111) as has been mexqlpred for stli. con (2.19j.Possible imk-mmediate surfu orienvtions are shown in Rg.2.5 irt a (lï0)plane. The Mlller indices of these snxrfaces are indicated. Given a set of Mnmer

iatlices Lltkl), then tke polar aagle 8 can be obteed 1om tan 9 = Wvk.The sllrêxce orieatation r-qn atso 1ye VaHH Fom (oO1q in anothe,r direction

thnan (111). Arbikary orientations of high-index surfaces of cubic crystals are

visaalized iu a sterovaphic tzi=gle. The construction of the stereovaphic. tregle ks based on three-dsrnensional consîderatiozus. M crystallovaplkic

direc-tiozus cnn be given by poàt.s on a sphere. If one connects all points ofthe upper (north) half vith Ge south pole, eve.ry direction can be markedby the pokt of inteoection in the meridiaa plre. Tile aazest are'a of non-

eqdvalent points is givea by the s'tereopaplzic triangle plotted in Fig. 2.6.TEe three corners are givea by the three low-inde.x surfacw. The surfMeswhlch Iie on the co=ection lines be-een the low-index eoz'ae,r points are

formally lin a bulk-tmlncated view) composed out of the corner planes. Anyplane in the inteior of the trinmrle ks crmposed in the same sense out of thethree come,r place's. TMS shows that the compleYty of the diferent planesincreases 9om the corners to the connection lines to the interior.

The variation of the sllrfnne energy between the low-hdex surhces (1.11),(100) , (110), acd (311) may also be represented versus a polar angle 0 iu a

(110) plane as izl Fig. 2.7. Witk a surfaze normal E!l0q for $ = 0 one observ' e's

the variation 1om (ï12(, (I13q, (001)) (11:35, (1ï15, E1ï0), (1îl), (1IZ!, (00i1:gïllj to LI1J). Axvllmsng thn: the (110) and (001) surfae are b0th mirror

plauœ the measured values 'y(#) (2.29) ezm be represemted as a relative slrrênrza

energy verstus the polar angle as in Fig.2.7. Sn'ml'lnr Wlnlfrplots have akso beenex-tractH by other groups (2.21).

2.3 Equllibrium Shape of S=xll Czystals 55

:.20 glch4. (3.11) (a:1)

. W= ) ):.1.j =

r:l o G * wg * p

- a . pp r1.-1.0ve p s o : uw r Z :>- w o s au m = uN 5 an u'-N

': ns u ; z . u $f ': a& .

%-'' * t (lot)l k : e J (4(* I p>'r Z# % / %120

t t ! t(A11h (117.) (I:.'lJ (1111

:-95() 90 'I8O 167û ,36:

040)

Fig. 2.7. Slnrfnzte enera plot7L9j. The surface energy ratioshave been extracted by a reverseWtzl'm constnction fxm voids(2.20J .

2.3.2 Absolute Values for Surface Energies

The absolute 'value of the sarface 9ee euergy of solid materials is a fn'ndamen-tally importan.t energetic quantity which is needed for the tmderstandsng of alarge numbe,r of basic an.d applied pheaomemw suc,h as crystal growth, smfwzefafettklg, growth and stabllity of thln fzlms, the shape of sma:tl mystctesin a suppoded catalys't, and ma'ay generaz materials sdence applications.Dœpîte its wemrecogOed sigadcance) there are relatively little reliable pri-mazy data of ecperimental surface free energies because they are vezy diëcultto memsme.

k.

1u. contrast to ftzid interfaces, where the surface enerr or tension r-an

usually be obtaizzed quite emsily by 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'rnentalinformation cau be fotmd about J in the literatme. TMS is in parrticular truefor the Anssotropy of the surface energy'. Only a few ttvthnl'ques, such as zero-

camep (2.22, 2.23) aud cleavage technsques (2.244, have been USH repeatedly inthe past to obtain (pln.ntitative values for a, lirnited nlzmber of solids, mostlymetals. Irt other exmeriments the equilibrium shapes of voids izt crystals, suc,has Si (2.201, have been measmed. The void shape, to a good approximntioa a

tetmknldecahedron lsee also Fig. 2.10a), is related to the eqtlilibrblm czystalshape, and the surface Fee energy is extracted via the W111fF construction (seenext sedion) of such au equilibrittm shape. Results are lis'ted for Si sarfacesi'a Table 2.1.

Absolute surface 9ee eaea'gies e-q,n also be calrnllltted.. However, ârst-principles calculations are dlëc'alt for numerical and methodological rea-

sons as dkscussed in detail ia Chap. 3. The slab coogarations usually used

56 2. Thermodynamic.s

Table a.1- calculated slxrfnce energie.s a' (1 J/m2) of low-isdex surfacœ of fccsemiconductors cmtnlllzing in the diamond structure (C, S$ Ge) or zinc-blendestaxldttre (.fazAsl and of bcc (Mo, W) and fcc (,M, Au) metaks. Reconstraeted lre-laxcdl snrfxces lzave yyen s'tutliM for sezniconductmrs (metals). Iu the (311) columnthe value for tke (ï11) surface 1(211) Se'rGne) of TnAn Mo, W) Ls listed. ln t'hecompotmd caa the aaion cllemic.al potemtial is fLXI.d at Jzas = Jolk - 0.2 eV. Inthe cmse of Si, expe-ram' G't.'XI valtlt.s (2.204 are also given in. parectkeses.

Crys'tal (100) (110) (111) (311) Rderemce

C 5..r1 5-93 4.06 5.51 (2.24Si 1.41 (1-36) 1.70 (1.43) 1.36 (1.23) 1.40 (1.38) (2.25)Ge 1.00 1.17 1.01 0-99 (2.251I'nAK 0.75 0.66 0.67 0.78 (2-121Mo 3.34 2.92 3.24 3.11 (2-261W 4.64 4.01 4.45 4,18 (2.271A1 1.35 1.27 1.20 - (2-27JAu :.63 1.70 1.28 - (2-274

(2.12,2.25-2.27) possess two sarfaces. Izl the cmse of crystals consisting of onlyone element the use of symmetric slabs induces convergence problennq (2.25).111 the case of compotmds with partially ionic bonds, i'a the malorit.y of slaborientatiozss one buas to deal with t'wo inequivalent surfaces to A11611 the elec-trostatic neutratity coztditlon tc.f. Sec't. 3.4.3). Moreover, in th.is case the sur-

face free energies depend on the preparation.. conditioas or, more precisely, on

the chernical potentials of the constîtuents (cf. Table 2.1). Neverthelv, suckcalculations are =ow possible. Results for covaleat and ionlc semiconductozswith a'a fcc Bravais lattice as well ms for bcc and fcc metals are listed inTable 2.1. A rath.er compléte colledion of data for the llnrelaxed surfaces of60 metals Ls listed in (2.27). The absolute valuœ aze of the order of 1 J/m2.Eowever, the values vazy with the surface orientation az.d the bonding be-havior. For smicondudors a surface reconstrudioa may considerably logezthe smface free enera. .An intuitively reasonable, but rough estternxtion re-

latc the surface energy 'y of Ca material to the cohesive eaergy per bond?Le., apprnvfrnately half of the value given in Table 2.2. Together with an

area of about c2/2 per atom (a value whic,h is correct for (100) sttrfacœ of

ziuc-blende/diamond ) one Mds for econ 2.5 J/m2 withou.t '

reconstmzction into accm'nt. The absolute valuœ of the surface 9ee ener#esdepend on 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. Withdecreasing dis-tance of the eledrons rs = (3/4m) w (2.28) the sttrfRe eneiriucreases an.d vice versa- The behavior i'a the tow-densit'y limit rs k 4 fol-lo= tlle predictton of smface energies witlnsn the jel'tium model of a metalsurface (2.291.

2.3 Eq:lsllbrblm Shape of Sma'tl Castals 57

AI

1.2

DE 1.oR: >-> 0.8P Ga <:)

o6,t) o.6 In ol cd ca:

* Lieb g ug u

sr. w o.4 s't zn o@; = Ba(E;:I

o'E : n z Na

L Hg oK O oRb cs

g.g2 3 4 5 6

.rs @s)

1.4

t :E ' . .

q.:,dài:. a.B. Surface pee energy of snbonded metals (2.27) versus tke average distance'ij ii'ïnL' the electrons (2.282. The amuotropy of the ezzergio is negleoted. ortly values forrj'(zztlr,, (cubic or tetragona: cystats) or (0001) (hexagonal crystals) are plotted. fcc:'àkttirès, bcc: cdrcles, hcp: tHaugles, a'nd bct (body-centered tetragonal): diamonds.

'.,:;Lj . .. ' .. .. .

... :'''.; é''..' , '''r. 2:' .

(; ( ''( . :.y' .

.'

..; .

. . .

. .iE/'Lï:jrjj,,à) wujs constructiontjt' ; f'L. k .r. (. ;:'ry.! y.,:. : . . .. . .

.j( Lq(E ..k j . .:' .k .: .. 5 L ' ' ..'

' k 'àè''lhzajuotropy of the suxface f1.% eaea'gy determiles the equlibri:rrn shape)ik: f ': -. .

'

.ziti' 2111 ,costals at a particuur temperatme z, rrhe czystal is wsm:rned to:'

. : ( ''! ' .. .::..' ..

. . '1%'.l':J' f rnnnroscopic (or at least mesoscopic) size so that edge ar.d ap> egeds

..j.,,... ; yL ; r . ,.

k1.4 c%t keglected. According to expressioms (2.8) ar.d (2.13), the eqttilibriïlm9 t'$'f'! kt'û; shape (scs) at constaat temperature 7. witu fuxed voblrne >- and( '

Vkak'' 'èxl potential p, is determc'ned by the zairsvnxl excess suzface 9ee enero-; I.l '' ' ,. : i'' .. '

Liîz ukebhèct to ths surface a4. , .

i. 1. ttè.t);E,E... , .: ..)( E .lr '

.. E . ....

-- @'

-'

.

q'

q. '

E' :'

(

:' E'

. '

''

'E

''

:

.'

.

.,

.' '' '' .' .'

.

.' :' :':'

:.

.' :'

jiiyi u jjz (; gg;. #, , 'y('n,) .

.' .' r'

' .

.' ''

: .A(F)

58 2. Thermodynaaics

subject to thE constraint of âxed volttme kr = Yd'Z. The WIO theoremv'(z)

(2.22) (2.15, 2.16, 2.30) states that the ECS is not nec-arily that of tEemin:'mll= smface area. It may be a complex polyhedron witll the lowesttotal surface enerrr for a given volume. A minimal suzface only occurs for

a perfectly spherical Whtlfr plot, i.c., an. isotropic excess sarface free qnergpThe corzepponding ECS is a sphere. Tikis has been experimentally showa forwater dzoftets ia the absence of gravity. ln the case of czystals the variatjonof y with the normal .rt will produze, on eac,h surface elemènt dA, a foreeproportional to àyjon whie,h wi.ll tend to alter its direction a,t the same timeas ''/ tends to shtr'tuk 5ts area. Consequently, the ECS can no loager be a

. sphea'e.Figure 2.9 sehematically shows a res'alt i'n. two dlmensiorus. It indicates

that the WttlF plot c('n,) governs tke eqlTilsbrblrn shape ard the morphologi-

cal stability of a c'zystal (2.20j. In the case of real three-dimensional C'Iy'SC;aLSthe sîtuation ks more compncated. Besides the variation of 'y with the surface

normalh also the strength of the vaziation plays a role. This is demozustraîedin Fig. 2.10. The ECS is construded for silicon tytking mto account the surfaceenergies of fom' orientations, namely (0011, E011), (1131, and (111). In Fig.2.10a

and b the same eneagetic ordorlng of 'y(111) < 'y(100) < 'y(110) < 'y(113) butdl'ferent values (see Table 2.1) have been ased. The ftrst paramete,r set is

derived 1om meaaurements while the second one %nA been calculated by azl

ab initio method. Qaalitatively the shape,s kz thc t'wo âgtu'es (a) and (b)aze the same. However) the relative areaz of the cnrstal facets vaor with theabsolute nllmbe'rs of the various energles. Tn.king into accout only the two

lowest surface energies for the (111) a'n.d (100) orientations, the cazbic sym-metzy suggests that the W111fF shape is a regnzla,r ocwtalledron with (111) facets

Fig. 2.9. A polar plot of the stauface free energy for a 2D mystal(solid linel and the ECS basedon the W111fF construcion (dottedLinel. See also f1g7.u.% in (2.30J.

(a)

Fig. 2.10. Faquilibrîum shapes of a Si cryst.al based on the W411F consiructionlzs'tng (a) experlmental values or (b) calculated values. Fottr surface orientationsare consioered. The surface energies are taken from Table 2,1. The azems with theozieatation sequence (100), (311), (110), and (111) va,ry fmm blxztk- to white.

2.4 Surface Bneav and Morphology

(b)o .

. .

ykayyjo s .. .41% 'w. ''q , , ... . . . . .h ;.

t . J'ç

jh . .è '. . It

-

-g .. )k4',''',':L- 'è, <4k. ,. ,.j - ),.i . t b

&.g'% # 'ê .:x' . . . . . . . . x. . . .

v q i ! . (l's . . + .k . , . 1 . 'Il; ?* W r' ' .

. x :fk. y .i ; j. . ,... i . d bài. .

. . t)a Et... r

. ! ,,,.J k. jtqt.. qvk.lt.,' .v j;.. . .. ï. f'h l,gN.-K . 4 ; j.. qh; > ' . k

'

,

1.1. 'x$ %.!,p $ .

' --*t5 G$1q% ' :!jr%%!ujjy'J)j'jj<y.y)j) 2;.1'!.$M F-'. êb s . .r.., lk s. .-

'

59

tznlncated at each apœx by (100) planes perpendicttlar to the cube ames at thesn.rne distance from the oc'tahedrron center. The inclusion of the (113) eae'rgyêeastroys the octGedral shype discussed gequently for homopolar semicon-ductors (2.31). Oa tlze othe,r hand, the (110) facets in the tetrpkmsdeem.heclron

(Fig. 2.10a) disappear completely in Fig. 2.10b iudicating the l'mflaeace of thebsolute 'y values.

''

a

Czystaz pLanes that are part of a W1:1'fF constzuctitm are' tlmrrnodynlkrnn'-cally stable (2.301. Since all foar oriectations comsidered appear on thc ECSSto a cezdnin ex-tent în Fig. 2.10, the fo'tzr reconstmzcted or relaxed sïlrfn.ces

(111), (110), (100), azd (311) az'e stable surfaces i.c thts resped. Of co:tme,

the (110) areas in Fig. 2.10b are negligibly s=n.ll becaase of the high smface

eaergy 'J(110) give,n in Table 2.1. The inclusjon of the 16x2 leconstructionof t*e Si(110) surface should, however, lowe,r this vallp.

2.4 StuYace Enetgy and Morphology

2.4.1 Facctting and Roughening

ne vadation of the surface enerpr with the normxl caa already Muencethe shape of the surface of a semi-i'n:nite crystal as indicated by a possiblebuelclimg of the surface in Fig.2.11. This buclding happens on a mesoscopicleagth scale (Iarge,r thaa atoraic distances). Let tts compare the smface e'xcess

energy of a âat sudace linnlted by a plane A with its normal e = ()O a'ad. asucface with a s=n.ll polar buckling preserdng the ave-rage surface orientatioluThen

.

, (lAFa/ = 'y(8) d.A = a?(p) - (2.23)

cos 0A/ al

so 2. Thqrrnodynlmlcs

Fig. 2.11. Sml1.ll bunlêsng of a

sudace.

The azmtrnption of a, weak valatio:a of 'y 'with 0 yields aa exlmnqion of theintegrard up to second order,

d 1 d2'y'

= J(0)A + -2 dyl + 92 ,7(0) + z C1.A. (2.24)F, 'j d,dl p=o :=ox -4

V

The &st term gives the eleror of the Qat sarface. The second one vaaishes forsymmetry Teasons. The third tnrm #ves the enerr due tô surface bunkling.One concludes that for e/(0) + (d2'y/d82):so > () the fat surface is stable (or

W /d92)#=a < 0 the buckted stlrfaceat least metaztablc), whereas for 'y(0) + (d '-tis more si'able.

Izz a lzighly nansqotropic crystal with a strong variation of 'y with the smfaceorientation the buclclimg esed dlscussed above indicates that such a czystalwill ml'nlml'ze the surface energy for a given Itornsmal normal by the formationof facds. This jacetti'ng phenomenon waz Emt discussed by Herriug (2.16,2.301with the hèlp of a geomctzsc.al constntctiozu

Up to noo tempeaxture efec'ts have not been considea'ed. However: at :5-aite temperatures the discllmion must be supplemented 'by the Melusion ofemtropy esects. At low tempezature a stretched surface Ls :at on the micl'o-

scopic scale. Whea the temperatme incremses, thermal factuations appear:tlle surface is no longer Qat and may buclde. These thermal âuctuations are

related to the mean square deviatioas of the atomic positioas with respect tothe average posîtions. Their strengths depend o:a the actual temperature. Thefuduations may remsn'n Gnste or diverge. J.n the fzrst case one speaks about asm00th surface. J.n the second ll'rnstk one says that the smface is rough. Abovea criticat temperatare the sarface undergoes a roughœnén.g transition.

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 netgrowth. All elementar.y processe.s proceed in two opposite directioas accord-ing to tEe priaciple of detailed bal=ce. The czystal growth must be a non-

equilibriclm kinetic process. The xesulting macroscopk state of the systemdepends on the reaction paths in the covguration space ms tnclicated inFig. 2.1. Siace the result is ldnetiem,lly detmmnsned, tlle obtained state is notnecessarily the most s'table one. Neverthelcxss, pal'ts of the overall process may

2.4 Surface Ezlergy and Morphology 61

'be ldnetically forbsdden, whereas others may be in loci thermodynazrdc equi-librillc). Consequently, equilibrblm arrlments may be applied locally, eventhough the total growtiz process is a non-eqz:c'l'lbzlphrn process.

Exactly these ideas are used in modern Emt-priaciples studies of epitax-ia1 growth (2.32-2.341. The activation energies of the processes occlzmqng onthe growing surface, such as diffusion, nucleation, and attaeAment ot detach-ment of adatoms to existing islands and steps, a-rc ezxtracted from ab i'aitio:

total-energy calculations, The correspondiug atornic processes happe!l in thelength and time domain of 0.1-1 mm and femto- to picoseconds. Hence theèazclllntions for modeting atomistic aspects of growth have been hampered

: by thc need to bridge length an.d time scales by many orders of mn.gnitude.First-principles moleculaz studies, while being powerful tn the irwestigation

ipri)f individual events on a time scale shorter tlza'a 100 ps, a're not suitableYr accessing the time scales i'nvolved in epiteal growth, nor cnn they tackle

i,lq.r',iih: statistical interplay of the nttrnerous processes that are respoMible for tkeë.: ùtitcome of a growth expersrnent. Howcver, ldnetic Monté Carlo simulations' . T '

.

' .

g,tfger an emcien.t aad accurate way to cope with tMs dilctzltsJ ; ) .y . . .rz ) Vitead of following such atomisttc approaches one may consider the

krsàrii$th of an overlayer more pheuomenologically. ln the cmse of heterocpi-lsfiky, the overlayer grows on a substrate in one of thTee dilerent growthi:'illbdes as indicated in Fig.2.12. In tlle bwnk-va.n de.r Mezve mode (2.35q,:i>ibzizs or molecules grow layer-by-laye,r in asequential fashion. Jn contrastj theè:''V' yliher-weber mecûanism (2.36) leads to the fomnntion of indMdual threih-.EE..)L, .

irtéljliéùsional Lslands. The third mode is a combination of tke two - an initialsltkisdiiensional process creates afew monouyezs, the so-caned wetticg layez,;'': ' t. thèn udivi4ual islazïds form'ws more matarial is deposited. This process

' tttu it. i '

. . ' '' : .

' ' '.

r,.tkskàkuws ws t:e stranslu-xrastauov mode g2.aq. It migbz 'be induce: by sew'''''? E'sfacsors. (i) A cezta- lattice msmatcz betveen the substrate aud theij 7 tjkt ,

. Efi'' :v '

. . '

.

Jjrsktzlayer cnmnqt completely be accomodated in an elastic mauer as vithinj; . . q. . . 2 .j . . . ,

Es'p)j;t1k. J' ljhëkck-va,n der Merve mode. (ii) Alternatively, the dieezence in the czys-li ltjj ji 'lrE)i : (( jinzfzttzy or the orientation of the overlayer with resped to the subs-trate'

) ! ).i. (( 12 ' . .' ( ..

. .?(!!FEVd2'' tt's jucll a growth mode. As we will see below, the Strxnski-Krastanov@t'y: '''

gj. g s (jjuckat sr tue selpassembled growth of an azray of quantnnrn dots onPr)P . t !li (#.jtt' àitiutb.h',)ip?'k gtilryEfjuckssioc of the loc.al surface 1ee energy per t'nit area ey also azlows ajj . : . )j.. ' jj

'

jj .

. .j .. .E ' : ....L-.,.i- kk i .::.. E. .. .

!(:;di'' '

'àizin iti6n of the growth modes. One has to introduce three dsFerent energiœ,.t,.!11.' : ''a:ljj.!j jgu s:o energ.g of the sttbstrate-vamr:rn intezface, x, that of the.tf.t1(; ,

Ett -7j.j;';E ' . (tijjk jjg uyym yatesaae, yo , aud. tkat of the substrate-overlayer interfacc,k )kJ lav 9:(yï't)i#)) (; jiyl'''k.. ,l(t' tgkdjkg to the discussion in Sect. 2.7 the quantîty'. 'y ca,n also bek q,.j.:;ïj(q',L; - )tëi.. '.,;rj' 'kël.-- - .-rrr! ;..'-i!. ...

.k .

. :. 2. :

. . . . ,. ..

2

Jsiti bt, è884tks a force per Trnlt lengt. h of bolmdary. Disregakrztmg a possible.t ' j: . ' .: . 'J . . ' '.. .. ' ' ' .

tiltkitt/kllayvr, one may consider a. situation as showa ia Fig. 2.13, where theJfllfijtp) k' :,.. jj uuuu u assn,med to be deposited intla.e form of i'adividuil islands.#, . t .mlii! jzjgEEg(l rE.h.jp'r'yttu ujore substratej isln.ndj and vacuum toucz? the equilibrium ofs:'r.I ) . ; ë rj rj .! ..:.. . j'' ') .

''

j. ' iï. E . . .

. . .rltlb' J:kiy:'àè1ls zs1: ft ' ';.j:'.E: k. ' :: ''. .... . ;.:.(..k. ,. . . .. . .

62 2. Thmodynamics

Frank-van der Merwe Volmes-Weber Senski-Krastanov

6?2 mFk;

'b#' tqîh' kè' f '... s,)r <j)' ='.&5k

xj ;j . .,j ;qt . s : y.. 'xxt -.. . . . ; . . . .. ,, r:,:. 'V i 1V't4tl WqQ ' uts' 0..44 watj.%

9s'j)' %.>. .3'*. k3x lîL $? :û)1 '' ' k ! LfI''#xF ulép (x .' 1 Z Mjy 6% Iù'J 'i:!l 4 u.. j.t;. ...qo s<yu jj,e ..kà? gk.: .>..v n .r. jqp zjk. çç,. /jy j;v

5 it' .'(p (hîF ': h f N klu, a( K %69 t$%;à û '% . ' . . k c '. ajj E: :$ jk <j 'z . a jiyk jjz.. -16L b qql)r (': '' 19 1;1k 'i2h i; . ïY 11? ' ,

qrh'fl qjj). C;I'b 9p :. r ..Lg ' ' : . 1. b .' ;ï . ïplj. E.k ' I . : jk .. . . jgjjr .. L, jjjsj. . .. q kktgr q, .à;)R, 42j,/ .. tt . ... N> . y, tkjjs . . ajyg r,.;kj!. i h . 14 W) k%t .

'

;';i),?' 'CCj jllEh l(y''i Sjt'isjy t''k. .' 4)4 ypiiil L #hï$)j:) t j;. yr.g sk,tqkjjk/y. .. ..y.)t jtqkzdkti.iz , 34.'. .()jkL.,,kék n(+% ( -. ;'

j t ). (k:.. q;;(1j. llqtk..yvyjyk jj()s:. 'k;i. . , ijtyf.. j/- ?.rh 'îh. ... , .. ... i;j .,-!.A.,A . . ,:...1. . . . s j î.',. i :.: ;.'L ;.;k ; ..f. -.r ...@ ,, . ''.-;1qk,-;;. . ;. ...,.i..b,i ,, 41.-- ..jt. . ?,. . . ?..... kr- :, ., i ., , .-:1. . , ,1.g .s ... ..t .? . .. , - as .. i- ? ,3% ... .< , t.r -. .. :, ).j fk.!. .. ty . q . u.2y . q.è. . -st, .?f . , !

... .1).

Fig. 2.12. T rllmerent modes of epitxxial growth tseailezztatie--a.llyl for diler-ent coverages: below one monolayer, between one and two mono ezs, above t'womoztolayers. In tzhe 217 Fknmlc-van der Meme mode, layers of mxtfameAl grow on top of'

each othet'. In the Volmeer-Weber mode, separate 3D Is' hnds form on the substrate.1.n the Strsknmld-nutanov mode one or two monolayers (wetting laym') form fzzst,followed by m' dividual islands.

Js = Js/o + 7o cos $ (2.25)

with 4 as the angle between the overlayer-vacuum face and the substvate-vacuam hce. The two lsmiting growth modes, the Frank-vaa d. Merve mech-xniRm with / = 0 azd the Volmer-Weber mode with 4 > 0, can be charac-

Yo . . overlayer, : .. . .. . .. . 2 . ...

: :7 ., ' q ./' . ! :. z :' : : :. .t. '.L' . . .? J' 'j7 :: J f.''' t . .

' . ( .. L. :

' . .' ' : . .. .:.

' .'.. L.? : . j. . .. 7 . : .7 l .:l'diu/'r

,

', .: . ! .. .'.

: ., , p,jjujpr : . .

,., ,,' .. '.

, ,.y .Ey / t

,.:jr .

. j' g .. , g. .. ;

,. j : j . . .

. g ..

..

., .jjr jjji . ,

! . ;. . .( a.

.

. ..

jj,.. t.sy. jtggjy y . ; .y. g,: .jj j g...j j. ;., a.. . ty i.. . y .

.g . ;

:. .

, .g .y. y... .. ..

y

s st te4

Eig. 2.13. Faquil*lbrbtm of forces (schematically) at substrate, deposited islazïd(overlayer) and vactmm.

2.4 Surface Baergy aud Morp/ology 63

terized by ï'y f 0 or Ay > 0 with 17 = yo + wo - %. For the intermediateSiranski-Kraàtanov growth; A.f S O for the flrst atomic layers (wetthg layexand Arf > 0 for the islands. '

2.4.3 Formntion of Qv3Arttaarn Dots

For a long time, it was believed that the of 3D kslauds in b0th theVolmer-Webe.r mode and Strnmld-Krnstanov mode Ls rcompanied by plas-tic rehvxtion, for instancq by the formation of dislocalons n.ear the islandbase. In recent years, it 'hxq bee,a fotmd for severil heteroepitaMal systemsvith a lattice miqf.t 2: 2% that dislocation-hv (i.e., coheremt) islauds form'in Stransld-Krastacov growth on a wettiug layer. Suc.h systtarnq are Ge on Siwith 4% (2.382, Tn As on GAAK with 7% (2.39,2.40): and 1nP oa JAGa,P with 4%

' akisst. Even the system Ge on SiC with a norninally mueh high.e,r lattice mlp-

ft of aboat 23% (2.41) shows suc.h a growth mode. Thœe 1s1a11(1s reûeve mucphof the czisst-induced elastic enerr by ckaqging the implane lattice const=tin the island for layers away from the intezfacè. Tllese naztoscale coherentislands, which are often fouud to have a narrow size distribution a'ad to bearranged in a regular array (2.39,2.40j, are accompaGed by a spatial quan-tization of electrons aud holes i'n three dirnensions. Hence, the islands baaedon direct-semicondudor combinations suc,h as Tn ls/Gn.A.K aze promisicg foruse in quantum dot light-vitting diodes (LEDs) azd lmsers. lmages of suchqu-tum dots or nanoczystaltites, whicà have been obteed by sc-annîng ttm-neling Mcroscopy (STM), are shown in Figs.2.14 and 2.15 together with a

schvxtical represOtation of their facets. The examples are an Tn As pyrxrnldon GAAq(OO1.) (2.421 and a Ge nanoczystal on a 4H-SiC(0001) surface (2.43J .

(a) (b)

(% clT'

' ' r . .

' W ' W,A% - > : . ''k '

$ % % é. . . :: '&' : u ' '

ki ' e %. ' L xs;: 'k qY l ''ie '

''+ B'?: ' z--.w ' . ' ''-. .. m. V . .. '

. : . . . . val . ' ' w ' 'e'z t x# &. . . . l . z

g j . . '< ' . . .1 'u . . T < 6 . '* . . . . <;'' r.h # ' :' .

. .r @ .

'x@. , i. t ' '

t.y ? ' : ' . '' 1w .* h . I ',l r ' *s . . < NxW'à : ' ' '. ' ' '

4. @. . . s' :;

FAt . Z! ,, ' ' % 1 1) '.' '

c . . . : . jt.d ' $1 ' '' '' k;/ . . . . . . . ' '

Fig. 2.14. (a) Three-dimensional STM image of an T= Aq quantum dot grou ona Gn.An(001) smface. The oblique objects in tke foregrouad are due to As rlimers.

(b) Seematic reprœemtation of surface facets of the nauoczystallite shown in (a).However, the two representatiozts are ivgted agnsanqt each other. R'om (2.421.

64 2. Thermodpmmlcs

(a)

F'îg. 2.15. Ge nazlocustal on a 4&SiC(OO01) surface. (a) STM image, (b)scbematic representytion of tke dot (top view). After (2.43).

(451)(131 )

(541)(r54)(11 1 )

'

(145) (:.;,: .: )

(1 1 3)(514)

(415)

The mecxnfnimm of forznation of coherent island arrays iu ltigiuly lattlce-miRmatched Eetrroepitaxy is not fally tmderstoodu in partictzlar not the self-mssembly (2.441. From a thqrmal-eqplillbrillm picture, it is understood that theformation of 3D coherent 1s1a.n.c11 is energetically more favorable comparedto a aniformly stzn.ined 6lrn ms discussed above. However? even when theformation pf azl isla'ad array is not considered and interest is focused on azl

isolated 'nn.noçrystallite, a complete nmwer is still rnlssing. Suc,h a, strnsned,dislocationeee jsland vith a c'haradetris'tic extent of about 10 nnn representsa quant:'rn dot. Eledrons in suc,h systems containing several thousand atoms

tmdergo a spatial quantization, which 4'nfluences the electronic properties.The size, shape, and stabilit'y of such a quatzttnm dot grown on a substrateshould be governed by the energetics, although klnnetic infuences cnnnot beexcluded as, e.g., in the case of the nanocrystallite shown in Pig. 2.14.

The totaz-energy change due to the formation of a Iarge, isolated quantttmdot of the type showa in Fig. 2.16 can be characterized by three contrëbutions

(2.12). The enerr gna''n per tlnit voltlme due to islrding with resped to thesitustion of a homogeneous overlayer is

Zêlsland Zfsurf Zfrelax +

Zâlyyer= + ,

pr v' V' P' (2.26)

where lfusla.d is the total erergy gnln of an island with' vobnme J/: Zssurf isthe energy of the additional surface geaerated by island formation. Accordimgto exmresions (2.$6) and (2.17) it is give,n by

Zfsu'z.f = S '/ + 17 Yklfk A$ - 'ylayertoôlxo. (2.27)ï J,à

The stnrn over f rtms over all island facets jrnRtrained) with surface energies ''lazld areas Ai. The efect of the straiu on the facets is described by the second

2.4 Surface Energy and Morphology

term in (2.27). The tensors t'/ and Li characterize the islaxd smface stress a'n.dstrain aversged ove'r a facet. The esed of the covered wettiug laye,r suzfacewith area .A0 has to be substzacted. 'ytayerto- 0) is the smface phs iaterfaceenera of the wetthg layer with a nom'nnal thckmess 0- n (usually glvea i.nltnits of monolayers).

The diference in elaztic enera betveen a situatiox with an island aad ahomogeneously straiued 61rn witit the sn.me Arnotmt of material is given bythe second contribution ic exmression (2.26),

ïsrel,ux Rilrv. z. za) .= . . gqjm j ( o

7 &'

wEere feilmst is the elastc energy of the isiand a'c.d. ssm ks the elmstic energ.ydmnqity in a homogeneously, Al'nsforrnly strssned 61=, az obtained from elar,-tidty theory. The third. contribution to (2.26) descdbes the change i'a theformation enerr of the wetting layer as a restllt of its t

' '

g. lt is desncd

Iswer = (-.) - yP) lwy.tel - wyert6blq

with the combined surface and interface eaergy of the wetting layer, 'ylayerto),aud the are,a dezssitjr of iélands, zz. By de6mition., 'ylayerto) cocesponds to tNesurface ecergy of the substrate.

1zz order to understacd the energetics driviag the formation of Trt As islands'

on a Gn.AK substrate as a prototypici e-xnmple, total enengy gn.ins of the type

(2.26) are calculated for situatio' ns az shown schemxtically in Fig. 2.16. Thestudie,s begr with sqaare-based pymmids with (110) facets (2.45, 2.46) butwere latez refned by consideration of (distttrbed) hexagon-bmsed praxnidsidth (111) and (110) facets a'ad theiz trlpncation parallel to the (100) smface

(2.12,2.46,2.47j. Despite these actMties with increazing efort fuzther work isneeded in order to explna'rt the detailed facetting and the e-xad odentation of

. !

(001)

Lh-c'fz:1q1

lrtAs1o$ Wettin; layer(014 (ccj ) GaAs substrate

. : '

:., .: x joo)L. .(2

>nsg. 2.16- Square-based TnA.s pyramid with (101) facets over an 177 As wetting.

E' :'

, lbyer deposited on a GaAs(001) surface..:. , :

66 2. Tàermodpmmscs

the Tn As island sbowa in Fig. 2.14. Real dots are slkhtly fatter as calculatedand also possess high-index facets such ms (2.37) or even (2.5 11) (2.42j.

2.5 Stoichiometry Dependencé

2.5.1 Thermodynnmlc Approac.h

When only a single specie.s is presemt in the syste,m cha-ractezized by thesoEd-vapor eqailibrblm, we have seem in Sec't. 2.2 that in the fmrnrwork ofthe Gibbs çclividl'ng surface' it is always po%ible to choose tize position of the .

s:qriimce such tha.t Xs = û. Whea there aze several atomic or moleculaz spedes,this conditîon cnannot be reAlized simultaueotzsly for all species. lt is gezzerallychosen to satisà this condition for the major component (e.g. the atoms ofan oneucompontent substrate) but not for the rnsnnor one,s (e.g. atoms of a.c

adsorbate). For that reason we genernllze the result (2.14) of the Tdivids'ngsurface' to Ns # 0 also for a one-component s'ystem. The surface excess ofthe rand potemtial uOs, 'y, should not depend on spedal Goices and is givenby

1 1 1 '

'y = xfzs = 1(Fs - G) =

(;ï (Fs - p,Ns), (2.30)

where p. is the chemical potential. Fs and Gs are the surface e'xcess Helrnhoitzfree enerr and Gibbs free enth/py, respectively. .

For a mttlticomponent syste,m with species f = A, B, C, ..., the value of '?becomu (2.48, 2.491

1v = -x ss - )-) g : Nu ,

i

where /zç ks the ckernical potential of the component f and Nu denotes then'lrnbe.r of particles of that species i'a the sktrface region. The uzderlyiagsurface exce'ss thermodynnmsc potentials are

(2.31)

J?s = 'yz,

J's = JX + V/z:No, (2.32)ï

Ga = S ANU'J

for Iframer's gra'ad poteatial, the Helmholtz free energy, and the Gibbsfxee enthalpy, respectively. latrodlzc,ing the corresponding surface entropy Sa(2.15), ss=5lazly the surface e-xce-ss internal energy is

Us = Fs + TSs = TSs + V IsiIhS + qA. (2.32)'J

2.5 Stoiœometry Dependemce 67

This quaatity depends oa the variables satch as exc- eatropy &, surfacea'rea A, and the partide numbers Nu. Theqefore, oùe deduces immediatelythat

dgs = zdss + c dz + E la dms.f

Deereatiating (2.33) a'n.d identiyng the result with (2.34) yields

ssdz + xdg + E Nïsd;ss - o,

01*

(2.34)

(2.35)

1 1d'y = -

-x Ss cl?F -

-x S Nu dp i .i

(2.36)

Thks equation L6 called the Gibbs ndKorption equatioa (2.2, 2.501. No contri-bution corresponding to defomnatioas ks iacluded izz (2.34). ln the case of a

solid, one may therefore explicitly Gtrodace the esect of smface deformatioasaccording to tlle discussion in Sçct. 2.3.3 (2.505-

2.5.2 Approximatior for Surface Energies

The eqlTipibrillrn state of a surface ms a htnction of compobition ks deter-mined by minîmlzing the sttrfaae excess free enera y = .%lA (2.32). Theconwponds'ng contribution to the grazzd potential depeads on T, A, and p'x,/zs,.... Fzom expression (2.33) one derives

f2sIT, X, p'x, JIB, .--) = Fs(T, X: NKn, Nss, -..) - Jllz:,?1s, (2.37)j

'v

where the surface exce-ss 9ee energy Fs = Uv .- T& is a function of thenumbers Nxs, Nzs, .., in the sudace region.

In the caze of a solid system the fxee energ.y F in expressions of the type(2.37) n.= be replazed by the 96.e enthalpy G = F+pV. For novmnl pressme,càanges in the surface stoicometzy propordoual to AN. are accompaaiedby enezr variations pt.4zvkrlt=l with 1)2Om as a cyxraderistîc volume ofa'a atom of specie,s b ia a certain bonding coaNg=ation. Witâ a characteristicvalue of r2Om = 16 A3 azd normxl pressme of about p = 105 Pa one ftnd.s

enerr changes of about 0.01 meV pe.r aom. Heace, the direct Maeace of

pressure variations on the surface energy cau be neglected (2.511. .Xs m willsee later, tlle pressure may howeve.r play an importact role in the determlna-tion of the clzeakical potentials, at least describhg locally the situation aftezan epitaxy step.

In prindple, all contdbutions to the surfu excv part of the graadcauonical potentik (or surface ene-rgy, ifit Ls related to a certain stzrface area

68 2. Thermodynp.mics

X) Os in eopression (2.37) depend on temperatme, the internal ertergy, theeatropy and the chemical potentials. The term -T;% contairs the contribu-tion of the surf'ace foraation entropy to the surface eaergy'. lt is governed bythe lattice vibrations and, hence, its detailed itwestigation requires lmowledgeof tlb.e complete phonon spectra. However, a simplifed discussion is posssble,since (i) s5mllar contributionz appear in the chemical potentials and (ii) agsu'nonly chrmge,s have to be considered in peciple. A rough estimate of a three-fold coordsnated surface atom in compazlson to a fotlrfold coordinated bnl'k-atom yields Ss $:k$ 1.75 ks per atom in the case of GaAs (2.521. Ia contrmstto the pressure iuuence, this contribution cnannot be nveglected, not at room

temperature and not at a11 for highe.r growth temperatures. Another esti-rnnate for the metal oOde surface, Ruoc (110), gave a vibrational contributionto 'y less than 0.16 J/m2 i'a the temperature ruge up to T = 1000 K g2.51).However, compv'ng the stability of two surfaces vith. deerent compositions,oaly di/erences of suc.h entropy terms should appear whic,h are disregazdediu the following.

There is another' , .in m=y cases more impoztant argttment for the neglect

of the lattice contribution i'a the discussion of static smfaces. Wheereas theabsolute value of the iaternal energy Vs is essentially deterrnr'ned by the elec-trortic contribu'tion,tthe efect of the electrons on its temperature dependencecan widely be neglected. An apper liznit of the lattice contribution followsh'om the equipartition theorem. One fmds

t&(T,X, N,xs, Abs, ...) = ELNA.., Nss, ...) + 3/cars-llrïs

svith B beiag the total enera of the electrons and the non-vibrating ions at

zero temperature. ln the classical b'mlt fhe linear contribution 3:sT Eç Nuto (2.38) also governs tbe ettropy term TSs in the smface frec energy Fs.Therefore, a wide compensation of the lattice-dpmmicat contribtttioas to Fsis exmected. As a result, the fzree energy Fs in (2.37) can. nearly be replacedby the leads'ng contribution E (2.38) to the internxl enerr Z. Accordl'ng tothe disctussion of the emtropy tmmn, the contribution of tke lattice vibrationsto tlle internal eneror cxnnot be neglected inv all cmses, in particulr not at

growth temperatmes where the atoms staz't to move out fz'om the eqpxilibzixlmpositions. Neveztheless, in order to e-xtract the domn'nating physici processesstabilizing a certna''n surface phmse or to s'tudy a static suiface strudme, theentropy may be disregarded simultaneously in both the Us and TSs terzpand Fs i:a (2.37) Ls replaced by E.

The mnt'n contribution E izl (2.38) also contains the efect of the zero-point'eibrations. Within the Debye approvirnation their eFect is given by zàs/o

s

per elementau cell with eo a.s the Debye tempeaxture of the solid. Despitethe fac't tlwt the Debye temperatmes vazy for dlferemt materials (2.281 2.53),oni expects a compensation of the efects of the zero-potnt <brations on

tEe total enerzy azïd the chemical potentia'ls. The nnlmber of particles takenfrom tlle reservoir and occurrirtg on. the suzface (or vica versa) is the same.

2.5 Stoichiometry Dependence 69

P+ ? ? ' ' '+. .+ ;' i 5 !

: .. . ' 1 *. .F .

.-. ,., (p;j:?$;)k::!: ).., ... r ,1 1 ())

c(4x4)02(2.:4) 'v7(2x4) mlxed-dimor

(1 $ n1(2x6) . . t

Fig. 2.1Y.. Top view of (001) surfpce strtzctures of Gaks (a) and TnP (b), orderedaccording to the amoant of catio/ coverage. Empty (f2ed) drcles represeat cations

taaionsl. Posiiiozls in the uppermost t'wo atomic layers are indicaàed by lrge,rsymbok. The layer closest to the bulk contains catiorus.

Consequently) kt an exmlicit calculation usually E is replaced by the total

enerr of the electrons a'ad the classical repulsion enerr of the nucld (orcoresly and the apprordmate exprerssion (2.54)

fzst/zA, /zB, .--) = ELNX, .N'z, .-.) - )C P'GN. (2-39)

is used to discuss the relative stability of surface phases with dl'flbrent sto-

ieometrg and/or atoznic structme. The inde,x ;s' to the nllrnbezs N6s is

dropped.. Cakulations or discusssons are automatically restricted to the sttr-

face region with a 6nite nt:mber of atomic layers. The total-ene'rr calcula-

tiozzs have to be performed for several moctels of the smface structme with

varying nllrnbers Nh and NB. Suck models for relevant suzface strudm'es are

sho'wm for GaAs(001) ao.d 1aP(001) smfaces in Fig.2.17 (2.55).

k.s.3 chemical Poteatials

The chorn'lcal potential A = ihlpt T) (2.6) is defmed to be the derivative of

the Gibbs free enthMpy G for a g'iven phmse with respect to the nztmber of

pazticles of type ï, p.l = Lt3GjlelNàtpbvfqxauj and fzxed m:mbers jNjj. of other'particle,s apart from Nn. Since in eqlpllibrpzm the cornîcal. potentiaz /.4 of a

given spedes is the same in. all phases which are in contact, eac: /wJ cac beconsâdered as the free emtbn.lp.y per paz<cle in enzth reservoir for particles .of

70

Si substrate

.j.? :;>>.$. k Qijux. 1 Kjè.An' yyktiopjtobjwx.. .% /nk)gy .;).k.) . .. .$ k-jjuç T hjajfxl . .7 % o . ï a J& k .L o. c . ; E r r'L @' 4': t .; T '.% 2 ; ' .% ! ' (.. c l &' .. k . . ..u W. 'e k (. .

. ï '

w & , , r u?B. , Q iy o . . . 9 .2 F ,

t; 7 Tt: ' k .1 i. rk ' $2 ' ' hilk h . 'Tsh ï'. k4f. E ' ' k' .LL' EF ' 'k.Eë 'f é J çl L(l? ;r ..L;t '. '5; ' b''' 6 ç' '7 ' lh ' ' k $) '. ' f ' ' . ' t! f :'k';ï' ' ''rf' 'kl s: u; owV/kyuz i . ' W qg d w)

ikkkikh îikii'ik.ui/' 7 ;: hkl.hl i-l '-1 i-l sLzil'qs lkuz./dskik)' i'': ' 7 '' 'b.bLkê'Lt- A' ' -

H H . .. W

k? kJ. H w? f ' w'. .;4 H . .;c, k z !&' m .

hy .. , , H yy yyyj v. . xo qs

u jj y ; j , s yy yy yyel l H $.zZ H H 1+' ' ' 4

$ lg Gej.y ,çLq -$- ...? ,. hh-rs-hh-r--sh-s-

u-s-y,y/ AX

chemical vapor deposition (CVD) molecular beam epitaxy (MBE)

2. Thermodynamics

Pig. 2.18. Schematic representation of two diserent epstmdal tecAnl'ques, CACD audMBE, tzsed to prepaz'e a mrr'ace or to deposite au overhyer, here Ge on Si.

type ï (2.56) . The actual chemic,al poteatials depend on the surface prepaza- .

tion conditions or the kn'nd of epitaxy used to depostte a surface overhyer. Thelatter is schematscally indicated 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 exclude adsorption, rather we study thc preparation in-âueace on the eqlzilibdnlm smface of an AB compound ms a'a exnmple. Therelevant chemical potentials are p.x and #s. Eowever, i'a addition the clzemi-cal potential Vxkô of the corresponcllng crystal bulk also has to be take.a intoaccotmt: sirze the b'll1r solid Ls a reervoir whic,h e-q.n e,x e atoms withthe surface (2.541.

For condsmqed states, e.g., the GM.AS sudaces, bllllc Gadts, or blll1< Ga or

As, the Gibbs 96%, enthalpy per particle can be discussed in a s''mllar mpnner

a.s the surface f1.% Nnergy in the previot'ts section. The pv tmrrn is again com-

pletely negligible for pressures considered here. The temperattawdependentterms can be includecl in principle. Nevertheless, we igaore tite'm using sirn5lar

argaments ms in Sect. 2.5.4. There is at lert a partial cancellatiop with thetemperatme-dependen.t contributions to Fs. This is obviousj for mvltmplej forthe contributions of the lattice vlbrations to the irttea'nal energies, at lezustin the limst of the validit'y of the eqlzipeition theorem. Thus, for condensedphases, in a fa'st cousideration one ivores the explicit tempezatme depen-deace azd sets the c'hemical potentiat /.t,s of a specic ï = A, B equal to tketotal energy per atom calculated at T = O K.

Suc,h total ettergies cazt be caletzlsted by mexns of ab initio methodz(2.25) 2.52) 2.58) , S'abtracti'ag the energies of the correspondtng 9ee atoms,

2.5 StoiYometr.y Dependence

Table 2.2. Tkermodmrnl'cal data 1. Càeznical pntentials pthlk of elements ï =A, Bin ev/atom. Tkey are deined a-s negative cohesive energles, The cohesive energy5s the energy required to separate the element into neutrf aioms at T = O Kat atmospheric pressure. The expm' Tn'mental valuu (exp.) are from (2.31, 2.5/4. Thetkeoreticaz vazues (calc.) are cakulated b# means of ab Jnitio methods.

yulk-A

Blement Czystal structtzre exp. talc.

B rhombohedral 5.77 6.11 (2-58)A1 fcc 3.39 4.19 (2.581Ga ortûorhombic 2.81 3-58 (2.58jIn tetragonal 2-52 2.89 (2.58)C diamond 7.37 10.12 (2.25JSi d-txmond 4.63 5.94 (2.25)Ge diamond 3.85 5.20 (2.251N Na molecule 4.52

P white mod't6cation (cubk) 3.43

As rkombohedral 2.96 4,12 (2.52)

the chemical potentials p'%% '':k for the corstn.lline phases consisting of only one

spedes ï are given as negative cohesive energie. Calcalated values are listed

iu Table 2.2 aud compared with experimental value,s .j2.31, 2.57) . The calcu-

lated values are substaatîally larger than the mexastzredhones. TMs has mnsnly

two reasons. Firsth the use of total energies of free atoms calcmlated with spi'a

polazization would reduce the discrepancy. Howover, such a renormalization

of the energy zero does not in6uence the actual value J)s (2.39). The second

remson is due to the local deasity approvsmatson usually uzed for exchauge

and correlation izz the total-emergy expression (2.592. Improvaments of theexchange-correlatîoa qnearr exmressioa dramaticalty reduce the discepau-

des between theory and experiment 22.521. However, aLso these eseds due tothe local evhn.nge and corzelatioa used cancel eac,h other 'widely consideringJlFerences of expressions (2.39) for #)S. ln the case of the bulk group-lr e)-ements a ftzrther improvemen.t can be obtained by aoticing that sometimesthe bulk phase with the lowest energy is not considered as a reservoir. Forinstauce, the potmd state of the Ga metat is orthorhombic aud not fcc as as-

sumed in some calculations (2.5$. This lowers the euergy by an additional 0.2

ev/atom (2.541 . Slmilar remarlcs are valid for the chmrnical poteatiaks itbgkqïk of

bulk compound smrnscozductot.s given in Table 2.3.)The bttlk chemic.al poten-

tials of the crystnlllne phases of tke compotmd and the i'adividual elementsa're Telated by the heat of formatîon I.WAB of this compomld.:

bulk = bux + bulk

- zsAe (z.,o;#AB llx Jzs J ,

72 2. Thermodpmrn:'œ

Table 2.3. Thermochemâcal data I1. Chemical potentials yVV and heats of for-mation ZSrAB 5n ev/pair of AB compotmds crystnllszing in zinc-blende or wttrtzitestracture. The chemical potential is defned as negative cohesive energy. The heatof formation is given in (2.40). Values taken fwm experlrnental data (exp.) andtaken 9om ab 'imtio calculations (ca1c.) are listed.

t

-;LVk IJZ'/'BCoumpound ex'p. calc. ex'p. calc.

BN 13.36 (2.60) 18-95 (2.581 2-65 (2.58)Am 11.52 (2.60q 15.98 (2-5$ 3-30 (2.62) 3.28 (2.58)GaN 8.56 (2-60) 13.61 (2.5% 1.11 (2.62) 1-28 (2-58)JZaN 7.72 22.60) 1:-01. (2.5:1 - 0.38 (2.58)AlAs 7.56 (2.6û1 9-14 (2-52q 1-20 (2.63) 1.01 (2-52)GaP 7.12 (2.60; 0.91 (2.63)Gn.Aq 6.52 (2.60) . 8.26 (2.52) 0.94 (2.6!21, 0-74 (2.63) 0.66 (2.52)InP 9.96 (2.50) - 0,92 (2.63) -

SiC 12.68 (2-60) 16-65 (2.61) 0.72 :2.621 0.58 (2.61)

Tke comparison of the mcasmed and. calctzlated ZA/B values i'n Table 2.3shows very good agreememt. This fact corroborates the wide caucellationo? inaccuracies in the calclzlation if one considers quantitics suc,h as I,WA'Bor f?s wllich aze defned by 8sFerences (2.39) and (2.40). If the surface is i.'aequilibritlm with the bulk substrate, pairs of A aztd B atoms ean be exchangedwitiz the bulk, for whiclz the chemical potentixl is Jzbxtjk. The equilibrblmconftion (Gibbs phase rulel reads as u

b lkp,.'v + JZB = y.x% .

It1 principle: (2.41) may also be written in the fo= of a mass cctïtm Ja'tr, ifthe chemical potentials yx aud p.B aze rewritten as Apmctions of the partialpr%sures m aa.d ps an.d temperature T. The advantage of the eqln'libri:lmcondition (2.41) is that JIA and yz are linearly dependent. Hence, the fozow-ing considerations only have to be made for one element, e.g., A. This is ofparticular admntage i'a cases of mnrface prcparation., in whic,h molectzles sutzhas Nz am involvedu e.g., ia plasma-assisted MBE of groumlll nitrides.

Restridîng ourselves to trends in the stability of s'tzrfaces for rllFereutstoixometries, the dezimtion of prope,r chemical potentials p,x (or /zB) canbe avoided. Without detailed knowledge one cazl still set rigorous boundson the rarzge of ih values ms long as the surface system is izl eqTtilibrirpm. lnparticulaz, each /wJ mtsst satisfy /.q :K ;tt ulk since when pg = ybt u1k the gt)sl

phase condenses to fo= the elemental bpll'k phmse. Sometimes, the strlzdtzreof the bt:llr pbn.qe is not really clear. For instance, elemental phosphortuq showsa wid.e structural variel, the most cornmon allotropes betng wMte (cubic),

2.5 Stoickiometry Dependcnce 73

black (orthorllombic), violet (monoclirdc), and red (polmeric) phosphorusas well as some amorphous forms. Forttmately, their cohesive eaergies are rot

very di/erent. For that reason, in deriving the AHIB values in Table 2.3 theformatîon of TnP and GaP is considered fzom white phosphonzs, the solidhigh-temperatttre phase. However, the ase of a moiecular phmse insiead of asolid may sometimes cotusiderably change the results for the stability Oneexnmple is the alternatîve use of solid As (2.54) and Asc molecules (2.64, 2.65)discussing the stability of GKA.S sarfaces yrepared by means of MBB.

Tly variation of the surface preparatLon conditions may be representedby deviations of the actua: cheznical potentiats from these bulk values,

1/,/: = iq - phx u1k (ï = A, B) .

The combination of this defmition 'with (2.4û) and (2.41) gives

Ayx + zlpz = .-ZSfAB. (2.43)Since A-rich (B-ric,h) preparatiou conditions are flxed by Ayx = 0 tZï/zs =

0), one ûztds that thc surface ca'a be i'a eqp:slibziqlm with its surrounctings only

if the chemical potentials are within b0th upper and lower botmds

-s5.%*B < ztjs,j .,g O Li = A, B). (2.44)

Thus, the variation of each Jz4 is restric'ted to be in a rrge given by AHLA.B

below its respedive bulk value.Surnmarhhg this discussion one Mds that the preparation conditions of a

certai:a soace can be i'aduded in thermodrnmic corusiderations in a ratbereslegant mn.nner. Even withoat a detailed knowledge of the chemical potentials

gé as a A'nction of pressare and temperattlre, the reiations (2.44) allow one

to esstablîsh rrges for pç which are relevant for the deternûnation of thesuzfaee stzuctures Imder equilibrbTrn conditions. The values .-AHIB aud 0

defne ls=1ts on the allowable range in equilibri'am with all possible phascss.I'tl pazticular, the chemical potential for eac.h element cn.nnot bc above thatof the bulk elemeatal phxse. The cmse ztî/.t.à = O means that there is in general

bulk material present and the surface is in equilibritlm with the elementalcondertsed bulk phase. The case Ayé < 0 means that the bulk is not st>bleaind the surface is in eqtzilibrplm with the gaseous phmse. Both cases are

: schematically indicated in Fig. 2.18.According to the above discussiom the surface energy (2.39) can be writ-

tea for a'a AB compotmd ms a Annction of the variation A;h (2.42) of the'

cahemica,l potential of one specias, e.g.1 # = A. Hstead of (2.39) one hn.q.: J ,

f4(1/zA) = ELNg, Ns) - Jzx'ùb lkNs - abulktwk - Ns). - AjthLNx - #s). 4.45)' J.n the liznit of a monoelemental system, e.g., a groumW saiconductor or a

ssmple meta:, one observes A = B with the equilibrb'nn condition px =JIBzXirtstead of (2.41) . Consequently? the surface energy (2,39) changes into

74 2. Thermodynazniœ

q(/zA) = ELNA.) - JAAUAZVA (2.46)with Nx atoms in the sarface resom The surface free energies 'T = JA/X izlTable 2.1 kave been calculated usiug expression (2.46) or, in the case of 1nAs,asing (2.45) but 6='ng àyhz = -0.2 eV.

2.5.4 Phnqe Diagrnms

Ic order to fmd the surfaze with tlke low%t ettea'gy for a #ven c'hemical poten-tial g,x (or more strictly, given prepration conditions), one has to compareenergies f)s (2.45) detezrmlned for surface models with vazying smfacc stoi-clziometry and geometry but for a ftxed potential gx. Suc,h energies f)s derivedfor the surface structttres of GaAs(O01) and 1nP(001) in Fig. 2.16 are plottedin Fig. 2.19. The msninulm surface energy X corresponds to the most stablesarface phase for a giveu ehmmical potential or, more precisely, its deviationfrom the bulk value. Since these 'em.erar plots compare the energies of dsFerent

smface phases for a, g'iven c-h/rnical potential, one sometimœ denotes fgtzreéof the type presented in Fig. 2.19 already a,s phase diagmrns of surfaces. Thestabtlit.g of the surface phmse of a certain reconstruction ar.d stoiclliometzyis, however, not absolute. At fmite temperatures a ceztain surface phase withenergy f)s per (1x 1) tmit ce2 occurs with.a 6n5te probability

trp, x rzlf?s (j .G N ex (- yy (2.47)

For that reason the ene-rr Lm x zîlfls (measmecl with respect to that ofanoth.e,r smface pbnxe) is sometimes called the formation energy of the tznxzzlreconstructed suzface. Of course, in general this probablty is also Muencedby emtropy eects (2.36) whic,h may modify the stability of a cez'tain phaseat a given temperattzre.

I'n. contrast to condensed statœ, for gmseous phpses the efec't of tem-

perature T aad pressure p upon the chemical poteatiaks caacot be ignored.Accordiug to gas theory the c'hemical potentials Lh depend logarithrnl'callyupon .p azd T (2.5611 and the large mriatiozls in p.i crtn be used to controlthe state of condeased p'h>mes iu ecplilibdclm with the gas. Titey ace relatedto tNe change of tlle Gibbs 1ee enthalpy whem a pazticle is transfezred 1omthe ga.s phase into the condensed phase of the deposited 41m or surface layer

(cf., e.g., Fig. 2.18). U this trn,nqfe'r occurs exactly at the eqllilibritkm vaporpressure po:(T), then no emergy is needed. If, however, the particle of thef-th component ch=ges over h'om the vapor to the solid at a certain paztialpressme Jv, the h'ee enthalpy changes by

6.1.3* = /cBTI'n.IA/PO6I. (2.48)The spqm of the partial pressures Eç pi yiezds the total pressure p. Forideal gases the eqlnilibrblm 'vapor pressme poï = kBlnjkqie-#i/kBT is re-

lated to the vobnrne ls3 ï dfaAned by the thermal de Brogtie wavelemgth

2.5 Stoiclliometzy Dependence 75

As-dch . Ga-fch

(a) o.2

â.2:4.

S% (2xtfT. I0., (utogl y cy

B .e (2x4) (E mzed-dime'

' )2) -0.2 '

cmx4qq p'a o l

'#

-0.4-0.5 0

lgoo (eV)

(b)P-rth

0.2

q4.p

S* G2(2X4).t, 0

07a rhzssj ,

& ,& :m4) i

!; c(4x4) CSV muedsxmerY x.

@ -0.2tp Io 1

ct-0.4

-1 -0.5 0

ln-rich

Ahzln (eM

Eig. 2.19- Relative smhce energy fzs ?e.z. (1x1) llnlt ce,ll for various sttrface re-

constracions in Fig. 2.17 versms the catzon Gezaical p8teatial. (a) GaAs(001), (b)JnP(001) (2.55j. Dotted lines mark the approvlrnate auion- and cation-dc,h lirnp'ts ofthe thermodpmmscally allowed range of the deviations Ala,A. (A= Ga, 1n).

76 2. Thermodlmmlcs

,Ls: = R(2zr/JQksT)è with M6 as the partsde mmss au.d a thermal activa-tion factor (determined by the chemical potential lh of species ï in the solidphase) (2.662.

Expression (2.48) represents the chemical potentials of asomic or molec-ulaz spedes in a vapor izl the 1ow press'ure limit. J.n the cxse of matezf.aksconsisting ollly of one atomic species the index ï can be dropped. The suylimation of a pure sold at eqltslsbritrm is given by the condition that the .

dmmlcal potentials of the atoms in the solid and the mpor are equal. tuother words, b'y = 0 and p = po hold.. The zatio s = p/pc elm therefore becced s'upersatnratéon. Expression (2.48), b'y = kBTIIL s? characterizes thedriving thermodrmrn:'c force for the formatiozz of a thin 61m deposited 9oma'a ambient vapor pressme (cf. the seheme in Fig. 2.18). b'Jz is clearly zero inequilibzglm, is positive dttrimg condensation, acd negstive during subMationor evaporation.

Taking the vapor phaAe into accotmt, the condition for layer or islandg'rowth (2.25) has to be moeed. The characteristic quautity is Arf. = A'y -

c*kaT l.n s with zb''/' = Jo + 'ys/o - cs and c, as a certain constaat. .ây' K ()(,Ag* > 0) stauds for layer gzowth in the Frank-va,u der Merve mode tisla'adgrowth in the Volmer-Weber mode). The Garacteristic quantity .4J* forgrowth of a ceztaizb matcrial on a substrate is no longer a, constant materialparameter, 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'tzres and substrate temperatmes. In principle,the mse of these qppnrtities also allows the determination of the surface emrgy(2:39) and, hence) the stable surface ph%e for certain preparatioa conditions.Examples are given ia Fig.2.20 for the MBE preparation of GaA,s(O01) (2.67jand 1z1P(00:) (2.6$ surfaces. Htead of a partial pressure a beam equivalentpressure (BEP) js ttsed to account- for the presemce of molecttlar beams. lt isa pressttre whic,h is equivalent to the ftux of molecules or atoms impingingon the suzface. Together with the measurement of the s'arface reconstructionby refledion high-energy electron rllFraction IR.H%EDI at a given substratetemperatme, the beam eqaivalen.t presstzres allow the construction of BEP-Tp'hnxe diagra'ms. Jn tke cmse of GaAs(0O1) the BEP ratio for As molectzles audGa atoms is varied, wherems for ToP(001) tEe ûux of Pa molecules is varied inthe presence of an atmost zero ln ûttx. Iu the language of the theoretical p'hnsecliagrn.rnq in Fig. 2.19 an increasiug BEPASa/BEPCa ratio (increasing BEPPZûuxl as well as decreasing substrate temperatme T corresponds to a tendencytoward more As-rich (P-rich) preparation conditions. The opposite tendency,decreasing BEPAw/BEPOa (BEPP ) and incremsing temperature, describes?Ga-rich tln-z'ichl preparation condstioms. The meastkred phmse diarytrn!x inFig. 2.20 and the calctfated surface energies versus the cation chemical po-teatial i'a Fig. 2.19 show the same trends for the most stable reconstmzctions.For example, in the GaAs case the stable 4x2 reconstmzctions in Ga-rieh con-(2 i change over into a c(4x4)

reco' nstruction undez AS-HC,h preparationt ons

2.5 Stoichiometl'y Dependence

(a)

rr (0())800 700 600 500 400

1 () () ...'..- I 1 1 ( ( I 1 -'.,,'e - -- : : J : : '- : : : :..? ........ l - . . -

. -

. - s - . - - .-

....z''z' ' .- *-'' G (4 x4') : t ( IX 3) : :

ryr'z

! J M.: > = * I % .' K'. 'J ''. : ''. '.

*'

''V E -- I -

: ) : : '- : :'ZZ'Z''W

:5 h - - - - * - *

z'z J., . .-. , . . . . .

e''zz' m- .j- . . . - z'.-e : :

'. -.

:'''z'z 'é axm (zxr ) (zxs) . - - - .tu zz'.u. ( maz . . . .LD l E e'ez :,4 . . . .n ..- = z; % . . -

1 0 '. 4x1).(.1x1) ..-

:1: I 1 ( , : .rn / Az -û; - .

1.E V/; G - : .

----

(4u)!(3x1) -.-. ::xr ; > .- ..* W/ k 6 .

< -h z; s' r .--. zn zx .%v.1 .

1 I I < '''VAJ*'%*'T T'=rl-'71r4h1'û;m, ... ' ' '-

G a d ro p Iel:s :kN%.). a.q 6 pe'lgW.v -fn 4x2 4x rpliu' X1) a' 4xl -( .1 . ;:..' 1 %%%s%+ ' ' ' ' -

facetting

1.0 1.2 1.4 1.603 I T (K-1)'.1

(b) 10 V* c(2x8)4<

Xc(4x4)n:' (2x1)

2(M)

300 4O0T (oC7)

500

ng. 2-20. Phase diagrn.mn for GaAs(001) (a) (2.67) (copyright (2003), witk per-rninsion h'om Blsevser) and TnP(0O:) (b) (2.685 surfaces prekared by means of M3RB.The stable surface reconstmzctions are shown in beam eqmvalent presstae (BEP)-tempezature (T) ctiagrams. In the case of Ga,A.s the ratlo of the BEP for azsenicmoleculœ and BEP for gallium atoms is varieo, while for HP tke BBP of Pcmol-tles Ls varied at constant BEP for .1.zk atoms.

2. Thmrmodynnxnics

coaditions. In between, i.e., uader intermediate prepazation conditions, thereis a large re#on irt which 2x4 recocstructions are most stable.

2.5.5 Stability of Adsorbates

The theou developed to discuss the stability of smface phmses with varyingstoichiometry and reconstruction can also be used to discuss the stability ofadsorbate stractures depending on the preparation conditions. Accordiug tothe deftraîon of tlze surface e'nea'g.g (2.39) and the specfcations for a'a JkBcompotmd (2.40)-(2.45), ole Mds for the depositîon of a third element C,the adsorbate,

.f1 (zJu, zvc) - E (2çk, Nz, Nc) - gbxktxz - Jlusktxa - Ns)S

- pbculkyc - Alsxllqh - A's) - Aitclic (2.40)v'ith Nc ws the nnlmber of adatoms in the surface region.

In m=y experimentaz situations the c'hmrnt'cal potemtial of the adsorbateonly underlies the constraint gc S p/culk i.e., Agc S 0. Apc = 0 de-)

sczibes adatom-ric.h preparation conditions. ln the caae of ecporation inCIW, e-g., of M-BE, for instance, it may mea'n that the shutter of the efu-sion cell for the C atoms is open azld that so rnxny C atoms' are depositedthat they stazt to fo= clusters with a blllc-lilte crystal stractme. The om

ite 75=it Ayc -> -x menmq that practically no C atoms nrrsve o2 thepossttrfnce. Howeve'r, tlze interwat of the variation of Jzc has to be remn.rkablymodoed if s'table compounds AC, BC, or ARC (vith porssible adclitionalvariations of the stoickiometry) exist. One evmple could be tke adsorp-tion of C = Lî on a zinc-blende crystal Znse (A = Zn, B = Se). An' upperbotmd on the cxemical potential of the adsorbate is fouad by eloring tllevazious compounds that the adatom nltn form in its interactioa with thesys-tem. For Li, a possible tppe.r botmd on ;tu is of couzse imposed by Li

(bt17k) metal. Howevu, the most stringent constraint arises from the com-

pound Lizse; whie,h leads to the constraint jm the chernical potential of Li,b lk = 2 b!znc + Bjk - zxnaase 2 6gj2/:L2 + #se = Jilse Jzl,z p's f E ' '

For a certain smface structure or, more precisely, for a cez'tain adsorbaàestructure, the ntlmbe.rs Nx, NB, and Nc are fzxed in expression (2.49). Thus,f)s only depends on l/za aad zlpsc. Two adsorbate stmzctlzres are in ,eqlzi-librbtm for equal %. TMs condition allows the coastmzction of the phaseboudarie. ln a region of Zï/IA and Apn wbic,h Ls bouuded by such botmd-aries, the adsorbate phase with the lowest surface energy f)s Ls tlle most stableOne.

In order to iliustrate the stability of adsorbate structm'es on compolmdsuzfaces, the deposition of As atoms on TnP(l10), thc cleamge face of Iztls isstudied. Foz simplidty the translational symmetry is pstricted to the small-est lx 1 surface tmit cell. The fou.r favorable structmes (2.701 are presentedschematically i.a Fig. 2.21. They are related to four dilerent coverages @. A

S.5 Stoichiometnr DependMce

(a)

' :' . ly :

''

'( :' ' : :.J L. . :

'

(b)

(c)

coverage 0- = 1 conwponds to one monolayer of arsemic. 1c the case of a (110)cleavage face e = 1 3s related to t'wo adatoms per surface tmit cell. The cleanstuface with O = 0 is described by a reiaxed zig-zag cha,in strudttre. The &st

s'tep of art adsorbate with a coverage @ = àz represeat,s a'a exchange-reacted

geometzy The upplrmost phosphorus atoms are replaced by arsenic adatoms

resalting in an ï= AA monolayer. In the nex-t step of coverage, 0- = 1, an Asmonolaye,r occurs on top the ïnP(110) smface. Several structuzal models have

been suggested. The most stable one is a so-called eyïfczricl continzed zcper

stractur6 (ECLS) g2.702. The largest coverage @ = .j is represented by an

ECSL on top of a,n exchacge-reacted geometrsThe results for the surface ener#es of the fottr stradmes tmder coasider-

ation are slzrnrnarized in a m/z:n ,-Ayxs phase diagram with two triple points

i'n the allowed region in Fig. 2.22 g2.N. Despite the existemze of the stable

Fig. 2.21- Side views ofA.s/Tnp(11O)1x1 st:rmces withdl-lerertt As coverages e. (a)Buckled cl-n . InP(11O)1x l

(& = 0), (b) eacchauge-reactedgeometr.g (8 = 1z), (c) epitax-ial contûmed layer stntcture

Le = 1), -(d) ewxck=ge-reactedsurface 'wath a;a extra -&s oveaulayer Le = 2z ) . P (In, Asj atomsare denoted by full tempty,shaded) smbols (2.70j.

80 2. Thermodpmrnics

>

1S oange-readed 'ex=. exch .-

'< SUOCOreacted

-K + M L A s= .GZ -0.5 1* Ic clean 1nP(110) Ic. =

sudace 1% I2 I

tp .j.gr(2 ECLS

-1.0 M)-5 0.O

Chemical potential Agws (eV)Fig. 2.22. Pkaae diagmrn for the As/TnP(110)1x 1 sarface. The dashed lines enclosethe thermodynltrnlcally anowed regioa (2.70) .

Tn As compotmd? only an upper bound Ayxs = O is considered.. The occtu'-

rence of the clean 1nP(110) surface, the exchange-reacted geomet:y a wellordered As.monolayer with ECLS or the ECLS covered exehange-reacted sar-

face depends 4ensitively on the preparation conctitions. For a low amotmt ofazsenic in the rccipient bat P-ric,h conditions the clean TmP(110)1x 1 surfaceis stable. Uzzder 1- P-rich but more In-ric,h conditions there is a tendency forthe formation of the exchange-reacted geometr,y as lortg a,s As is presOt. Thiscorresponcls io tize fact titat dlnrs'ng the Annnltlsng procedare a pbosplloras d.e-pletion from TnP occms. Tn a wide raage of chemical potentials the exch=gereaction seems to be the preferred process. On2y for very As-rich and P-rkchconditions does the prepratiou of a,n ordered As monolayer seem to be pos-sible. However, because of the volatitity of phosphorus suc,h a structme maybe Hl'ëct'slt to prepare. Under extremely AS-HC,h conditions the formation ofatl BCLS monolaye,r on top of atl exchange-readed geometry is ecergetihallypreferred.

3. Bonding and Energetics

3.1 Orbitals and Bonding

3.1.1 One-Electron Pictare

Even though the sarfare of a crystal may appeaz ver.y sm00th at fzrst glance,exmezimental eddence shows that it is heterogeneous on a microscopic scale.On tllis length scale the thermodpmrnic treatment of smfaces in the previ-

ous clmpter Ls no longe,r secient and, hence, must be reHed by microscopicconsiderations. However, such studi% on a'a atomic scale havc to consider thebondâng between atoms in the surface layer and of sudace atoms wit: atopsbeneath in a butk-like eaviro=ent. The bonding behavior is governed by thevalence electrons, kusually s and p electrons of the outermost elecvtro/c shells.1.n the case of metals, e.g., transition metals, but also compotmd semicondqc-

tgrs, suc,k as GaN, semicore d electrons have akso to bc studied.Witht''n the single-yar' ticle picture, the electroGc states #@) with energies

s obey a one-electron Sceödinger equation

H#Lm) = c4(œ)with a Hnrnsltonian

:2H = - ,4. + 7(œ). (3.2)

2m

Et'a the shgle-electron Hn.rnsltonian (3.2) J?r(z) repnesents the total poteat#al

'energy of arz eledron (more strictly, a valence electron) with mass '??z in the, : ield due to the atonlic core,s and the othe,r (valence) electrons. This pot'en-

ùal energy (or brieoy poteatial) depends o:a the treatm>t of tile electron-. :

èlectrozz interaeion. In generi, besides the efect of the core and the HartreeSotential due to tite clmssical snteraction with the otlzer electrons, it coatxsnq

$ in contdbutions of exchange and correlatiou. The spin varisble of thecertaLelectron is not e'xgocitly considered here. Rather, we msmnrnç that it is in-

cluded in the space coordinate œ. The spinor character of tke wave hlnctions

i therefore also not e' xplicitly tikea into accotmt. ln cases where the spin: s

becomes importaat, if e.g., the spG-orbit hteraction strongly modifes theelectrozlic stmzeure, the spin variable and the spin quartttlm nllrnber will beexplicitly memtioned.

82 3. Bonding and Bnergetics

J.1l the case of an ordered aad comrnensttrate sttdace system the totalpoteutial in (3.2) obeys the trn.nqlatîonal symmetry of the halfspace with thesurface. Then

V'(œ + .&) = F@)

with R beiug a vector of the 2D Bravais lattce of the suzface. As > conse-

quence the eigenslnctions obey the Bloch theorem, '$Lm + A) = ebuR#Lœ),and an eigemstate ca.a be classiâed i:a terzns of a 217 wave vector k &omthe corresponasng 2D Brillouin zone (cf. Fig. 1.22) acd a b=d index v. ThescbrHnger-like equation (3.2) takes the fo=

'

JfW:@) = fy(i)#y:@) (3'4)

with 2D Bloc.h A'anctions #vx(z) a'ad corresponding Bloe,h bands zv(:) de-pendikg on the 217 wave vector i. One hms to mention that the theory pre-sented in (3.3) and (3.4) ks also valid for a system with 3D translationatsymmetzy and .wave vedors k from a 3D Brillolnl'n zone. ..

3-1.2 Tight-lBinding Approac,h

I.zl order to tmderstaud the bondmg behavior in the smface on a length scale ofthe order of neazest-neighbor distance-s, it is convenient to expand the' single-

. pazticle wave Alndions #@) in (3.1) izt termq of orbitals 4.(z) centered ataa atomic core at the origin of coordinates. The inde.x a labels the kiudof atom azd the atomic quantntm nthmbez's. Tizis approacll to the electronicstructure requires knowledge of the atomic positions A...

'

The solution ofthis strucvtmal problem Ls dkscussed i'a Sect. 3.3. Eere we adopt a locnlived-orbital basis set (4o(z - .&..)) and assn'rne that their locnlization centers

Rs, are lmowzu Withn'n the so-called linear combination of atomk orbitals

(1uCAO) method., the following equali'ty hole for an arbitrazy polyatomicsystem u'ade,r consideratioa

#(=) = V %i$=LT - .&)' (3.5)l,$

where in explicit calculazions atomic mMe fn'nctions, hybrids, or bonding andantibonding orbitals are IZSC.CI as /.(œ). ln fzrst-priuciple or semi-empiric,alelectronic stracttu'e methods the hmdions should be lmown a'ad may even

be adapted iu self-consistent c'ycles. In more or less e'mpirscal approaGes the=alytic fo= of the basis Gtmctions e-qn remain n''nk-nown. Mther, one intro-duces the matrix elemeats of the Hxrnsltonian H and the overlap integralsand tries to S'ad explicit expressions for these qu=tities.

With the a'asatz (3.5) the Schrödinger-like equation (3.1) tmnqforrrm into

an imfnite system of homogeneous algebraic equations. The expznqion coef-fcients restttt from tlze eigenvalue problem

3.1 Orbitals and Bond'mg 83

(3-6)I'''.2 (J2u(a., R,.) - sksabLltz' - .RJ)! ?pj = 0,b,j '

where

HzsLmt .7) = J d3tr/o' (œ - m'IHSb# - AJ)

are the matrh elements of the one-electron Hn.rniltonian S (3.2) of the s'ys-tem, and '

%sL.% - S.J) = j d.3z4.* (œ - m.I6LT - A#)

denote the intenatomic overlap i'ategraks. The intra-atomic iâtegrals are Kro-necker smbols because of the asgprned oz-tbonormnlbguation of the oae-cente,r

orbitals.I'n. ab izlitio calculations, the multicenter iutevals in (3.7) and (3.8) are

evaluated explidtly for a givea S and a given set of %Lm - .Rz+). Aparth'om possible self-conskstence requests suck a procedare requires heav,g nu-

merical calculations because, in general, the matrix elements do not convergerapiclly iu spsce. Therefore, m=y empideal and semi-empirical tecbnlqueshave been developed to reduce the tplrnerkal efort (3.1, 3.2). For eAample,

the extended Hfic'kel theory (EHT) (3.31 assllrnes that HC.bLm', .Rj) is pro-portiona) to S=SLK. - &) if i # j. Besides the enormous' reduction izl thecomputatioaal esoz't, empirical methods give muc,h insight iu'to the chnmi-

cal bonding processe involved an.d a deeper unde>tandi'ng of the trends .inproperties 9om oue system to azmther. A major development i.n thq historyof the empirical tight-binding method (ETBM) waswthe work by Slate.r aud

Koste,r (3.4j, who suggested a new and waiuable role foz the LCAO method,that of azl intezpohtion scheme. First, they suggested to treat the Hn.m51-tozzia'a matrix elements ILVL.%, .aj) (3.7) ms pazameters aztd to ft tkem tolœown tmeasttred or caklzlated) one-electron eaeates. Second: they invokeda theorem, ftrst proven by Löwdia (3.5), according to whic,h a,n orthonormalset of LCAO orbitals could be de6med rigorously suc,h that

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 trxnKformed into â. new set of orbitals whicit are

orthogonal to e'a.C'N othe,r aad preserve the atomic geometzyThe advantages of the empirical methods become obvious when consid-

ericg an isolated bond formed by two orbitals c = 1, 2 localized at neigbbor-ing .atoms. Examples codd be simple diatozaic molemzles, e.g., the hydxogenmoleculc H: with a pare colminnt bond a'nd a I;iI'I molecale with a, heterono-IJr (or simply, polcr) bond, for which, howevez, the considered orbitals are

restricted to the most importau.t s-like ones of the valence electroils. Withoutthe neglect of ovezlap (3.8), the dgenvalue problem.reads as

84 3, Bonding and Bnergetics

antibondingca

IH12l2 +(AE)2Ha

AE

AE1-11$

IH1zl2 +(A:)2

Eb

bonding

Eig- 3.1- Formation of bonding and antibonding molecule leveb from atomic level.

Szz - e Nlz - czsa :1= 0.S- - sS* Sa2 - e Qlc 12 -

Neglectiug the overlap of the t'wo orbitals forrnlng the bond, i.e,, S1c = û,

the eigenwlues sap = I ::i: glsza 12 + (zk)j are the eneates of the anti-

bonding an.d bonding orbital combinations, respectively. They are schematmically reprœented i'a Fig. 3.1. The abbreviations ë' = (fftz + Szz)/2 andzâe = (JA2-Szl)/2 defme the average of the two atomic ener#es ard one-halfof their dileaence, respectively. Tn the case of diferent atoms in the moleculewith 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 colmleni e'rJ-

ern 1Sz2 1, whic,h îs also nonzero in the lt'rnl't of equal atoms and orbitals.The polr energy determines to what extent the electron densit.y along thebond is deformed toward the anion. Iu. the point-nhnrge p6dure, zîc indicateshow many eledrons are traasferred betweO the catioa and anion along thebond. A quaatitative me=zre is given by the bond ,polczwzp (3.6)

Gp = .

Iffzzlz + (2!42(3.11)

(3.10)

It is related to the eigenveeors by IQ 12 = àz (1+ ap) ar.d 1?2 j2 = lz (), ::F: apl forthe bonding/anbibondzng statc. Ia the votmd (bondsng) state of the moleculeone Ods the rœtllt that the probability of an electron appering ou atom 1is (1 + (zp)/2 and the probability of Mdteng it on atom 2 is (1 - &p)/2. Thedipole of the bond is proportîomnl to (1ê1I2 - rJaJ2) = ap. Since ihe coYentenea'r is a, Alrctiozz of the distance d = lRz - Az I of the t'wo atoms, thepolarit.y aud the ezmrg.y splitting of the molectkle levels depend on d. Theecltsqsbrihtm bond length deq follows fz'om the condition of mypvlrnl:m enera

3.1 Orbitals acd Bonding 85

gain of the t'wo electrons due to bonding. The enero- (.FJ1z12 + (1s)2 shotzldbe a mprrim:lm. However, to perform sach an optlmlmation e-xplicitly, theoverlap interadion ($n general, a short-raage repulsive iateraction) has to betakeu into aecout (3.1,3.64.

I'n the case of trn.nslationally invariant systems of the type defned by (3.3)and (3.4) with atomic positions m' = .R + rz (Bravais lattice vector A audatomic basis vector .rv) the LCAO exnmnsion (3.5) can be speaifed as

'?Jb:(z) = 'h7 cvc,ililxt@l (3.12)c,ï

with Bloch sllrnq

1 s:.)z$:kf@) = J((2 e /.(z - A;)Gn

constmzcted fz'om the locatized basis ozbitals. The R-sllm exteads over Nelementa'cy cells of tbe hlndamemtal regkn of the 2D (or 3D, theayth wave

vector k) crystal. The Bloc.h bands ss,(k) at a.a azbitrazy poiut k az'e thenobtained by solving the eigenvalue problem

J-'-J gl:l'oçe''t:l - ,u(i)JoyJvq c,aj.(:) = 0,by.#

where the lbrnsltozsian matrix reads ms

1 -i:(a-.a,.).s (a. .%)Jfoïby (i) = y V c

. ab ,

AyR/

E E: 'but t:e overlap (3.9) of the orbit/ a localized at m. = A + Ju aad orbital b;.

itocnlized at Rj = R' + o is neglected.

j '

E.321.3 Atomic Orbltals aud Thelr Interactcon; ';

. .

.::.' :. :'!Eè 'Let us consider elamcnts A an.d B belonging to colllmns Nx=N aad Nz = (8-'':,X) (N = 1) 2, 3, 4) of the Periodic Table and formi'ng compolmds A.NBs-x.Vsually they 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-chloride structure. Sometimes the resultingLïKNBZ-N compound crystals belong to the hGagonal crystal system, e.g.,i ïhey possess the wurtzite structure with %ur atoms pe,r 'lmst celt. Th.e bondiug.

é'

Tbf such compozmds is governed by the N and (8 -N) valence electrons of theAo atoms of a cation-azzion pair. However, L'V-m compolmds, e.g., the lead

I'ubilts PbS) Pbse, and Pbrl'e: cau be descaibed i'n a &'=1'1n.r mxnne,r allowiz.g. Lïhe formation of a lone pair of electrons at the grottp-Rq atom.7 Tlle s and p atozaic orbitals (or, more strictly, the Löwdin orbitals) of the

!: : .'

oute,r electronic shell with orbital euergies cs, cp acd radial ftlnctions Ra (r),

86 3. Bondinz and Bnergetîcs

.Rp/) contxibute most to the clmmical bondiag. Usiug polq coordinxtes m =

rtsin $ cos p, siu 8 sinw, cos 0) and sphedcal hnmnonic,s ybmtp, w) the Arll wave

fllndions 4c(m) are umzally rephced by tlle,ir combinations

11J) = X04:, W)&(r) = .zw.&(r)'

-1 J' (p pj - 4&-z(p,p)) a,(r) = Z-J.w(z.;, .

1yu) = ( zz , 4x vWf

yi (9,w) +s-,(:,.,)q as(r) - -tX.w(r),yp) - w E z 4. p

(3.16)

t3 zlpa) = Fzûlû, W)&(r) = s-c&('r)'

where t:e corresponeg atom is asstaned to be sitaated as the origin of

tb.e coozdinate system. Suc,h orbitals are represeated in Fig. J.2. The new p

orbitals are directed along a Cartesia'a n='R.

la more complicatcd compouuds or even in mtztals, inzompletelf fktled d

and I shells may contribute to the cheraical bondi'ag. Jmportaat examples

itwolve t'ransition metal atozas. These belong to three'' series in the Peziodic

Table whic.h corrcspoud to the progresive 611n'ug of 3d (Ti to Ni), 4: (Za. to

Pd) and 5d (Hf to Pt) sbells. Real combinations of the d orbitals with a'a

enexgy ed and a radial part &(r) aœe

z

Pz

9S Py

Px

x

Fig. 3.2. Scàemlttic representation of s and p valemce orbitals.

3.t Orbîtals tmd Boadinz 87

1 5 Azltfwv) =

y,,.g a 2Q(r),T

15.:,IGz) =

s a&(r),r

15 zzIdam) = spadtr),

15 z2 - v2Idz2-#a) =

z .!Q(r),16zr r

15 3z2 - r2

Edaxu-ral = a .TQ(r).16c r

The atoznic orbitals givea ict (3.16) and (3.17) nnn be used to calculate thematzix elements .J.ILyIJk., Aj) (3.7), at lemst, to calcalate their smmetly andgeometzic dependemce. We illuswtrate suc,h a calculation of the Hxrniltozzianmatrix in a more general Famework. The following assumptions are made:

i. The basis set is restricted to one s orbital and the tkee p., pu, and pzozbitals of the mlence shell of each atom tsee Fig. 3.$.

5i. Orbitals on Werent atoms are assumed to be orthogonal so that (3.9) isflgfllled.

iii. Only the nearest-neighbor or second-nearest-neilbor iuteractionsS0:(.&, Ro'4 (3.7) are retainet with the additional asslfmption tha,t theHn=sltonia'a lnAA local cylindricaz symmetry around the n.='K couectinga pa,ir of interacti'ng atozns, so that J.Qs(A,., Rj) H IILSLR.. - Rj).

The adequacy of the tuse of s and p orbitals of the highest (partially) oc-cupied shell (i) seems to be quite clear from cizemical azgamertts, Le., thatonly valence electrons essKtially participate in chemical bonding. This is ob-vious for tetrahedrally coordinated hNBs-x compolmds but also for simplemetals such az: e.g.j A1. J.n addition) for trnmqition petals or transition-metalcompotmds the d orbitals (3.17) have to be taken iato accolmt. Of coul-se, allthese bmsis sets are incomplete. NevertheHs, electrordc stnzctmes axd totaleaergies cazz be deduced, at least, usi!zg the Harniltonian matrix elements asparameters to be ftted.. Foz iustance, the restdction to s a'ad p orbitals isqlzite accurate for the desciption of both valence a'ad. conduction bands ofs/miconductors. Howeverj baides the nearest-neighbor interaction (3.7) alsothe second-neazet-nejghbor inte-raction has to be taken into account, Onlyby the inclttsioa of suc.h matrix elements beyoad the fzst-nearut-neëghborinteraction, cau the Dlike condudion b=d msmima of the i'adired materi-als Si, Ge aud GaP be obtained near the X and L poiats of tNe Brilloqllnzone (3.8) . The improvement of tike condtzction bands ca,'n also be acllieved byan enlvgement of the b%is set wilbsn the fzstmearest-neighbor apprcfm'rna-tioa. For instance, the periplleral atomic states o-xn be approximated by e.n

excited s state (denoted e*) (3.9, 3.10). The additional s* statc couples with

88 3. Bonding and Bnergetks

the aatibonding p-like conduction states near X or L and pushes these statestowazd lower energy.

The neglect of overlap (ii) has often been viewed aa a sezious drawbac,kof the ETBM, since simple estimates fnd SabLm. - Aj) betweea atomic or-

'

bitals on nearest neighbors to be of the orde.r of 0.5. However, accordingto Löwdt'n (3.5)) in au.y case a'a ortkonormat set of basis Atnctioms may beconstructed without loss of local symmetry. The restrictson to frst- ar.d per-haps second-neazest-neighbor interacvtiozls (iii) in the Hamiltoaiaa (3.7) isalso aa mssllmption. A justiâcation cltn be given by viewing tlle JQ:(.R., .&j)

efecvtive average hteradions Jue to the Mterpolation scheme. The îttîngprocedure of known electronic bands or othe.r quazttitîes lduces the partialSnclusion of longer-ranged interactions izl the Xective quaatitites. The sm-metr.g restridiol in (iii) means that the intra-atomic matzix elements can bereplaced by one-center integrals wherems the interatomic rnntrix elements can

be treated as t'wo-cente.r integrals (2A.Witk localized wave faactions /atœ - Az.) of the type given in (3.16) or

(3.17) the intra-atomic (i m j) matrix elememts of the J'hmiltonian (3.7) can

be written with the symmetr.v restdction in the fo=

Hab (A4 , K' ) =

Xb eu + j d3z<* (z - Az.) (F'(œ) - Fk.tm - Az.)j /ts@ - .R;) ,

wlke're Fk. (2) is the one-electron potential of the corresponding free atom oc-

mzpyiug nomimally the site Rs.. The second term on the righvha'nd side of

(3,18) represents the eFect of the crystal âeld on a certnsn atomic level. Theinclusion of the crystal-âeld efect destroys the transjerability of an ETBMHnmlltonian to Tmknown atomic azrangcments. 1z1 the sense of the introduc-tion of a psetztb-Ehnnsltozf.an the crystat-feld shifis are usually negleded,

S'obll%:l NJ) = Jïc'ékôgo? (3-19)

and ceatnsn t'nsversal or cazmnical parxmeters ea are used as Kective orbitaleael'gies (3.1,3.$.

The interatomic matzix elements of tke Hlkrn'lltozlian, the hopping inte-graks .!.QSIAZ., Rîj , aze responsible for the spDtting of the discrete atomiclevels iuto molecule states (see (3.10)) or for their brooening into energybands ic trxnslationalty iavariant s'ystems (see (3.14)). They ca'a be a;)-pzovlrnn.ted by t'wo-center integrals. Tize distance dependence of the hop.ping mlttrix elements between orbitals at ril'fîerent sites can be obtainedby vaziotts methods (3.1, 3.2j. In the simple cubic case a l/dz-dependeneeLd = IAJ' - K' E: atomic distance, bond length in the nearest-neighbor caselcazt be derived (3.6). The geometric dependence of the Hamiltonian matrixelements is give,n by the Slater-Koster relations (3.41

3.1 Orbkta'ls and Bonding sg

Ilis (R., Rtt = Kag.,Ssp.(,!Q, .&j) = 'ruljstsptr,

EptxaLm', AJ) = '-rsarrlsptr'Hvop, (,&, RJ) = (TG1l ' npI rlvppcr + ('D=2. ' 'O-LIIV','

for the ss, sp, an.d pp iateractions. The hat on Zp. indicates that tiuis sp-intaraction is aigerent 9om k:sp. in the case of atoms of dslcrent chemicalnat'tuw at m and Rj. The direction of a pœ orbital, rzcz, is decomposed intotwo components, n.. = p..s + zurl, into a vector perpezzdiclzlar (.1.) au.d avector parallel (11) to the vector d = Jzj - R,. betveen the two interactiugatoms.

The Slatez-Ko<m parameters

Ka. = d3z4a@)JJ/a@ - J),

Kpcr = d3œ/s(œ)A/pI! @ - d),

Vpcr = Yœ/plt @)S/p(i@ - d),

Vmx = d3œ/p.z.WlN/p.l@ - d)az'e given in terms of p orbitals (3.16) parallel or pezpendicalar to the con-necting vccto'r d. The physical menmsng of the prazneters Ls itlustrated inng. 3.3. The zelatiozzslzips (3.20) and (3.21) are bven in the original paperof Slater and Koster (3.4) and in the book by Hazrison (3.6j not only for sand p states bat also for d orbitals. Geaeral formulas a'ad explicdt exwessjonsinvolving .J an.d g orbitals caa be fotmd in (3.112. The pazameteze Wscr, Wm.,'Uaw, 5fw, and 7/,,. depend on the atomic distance d. For typical nearest-neigûbor distanzes they have the signs Xse < 0, Kpv, Zp. > 0: $$w > 0,F's,rr < 0. J.n gerzeral, the absolute values of the c iuteractions lKwl, Iï%.I,1V;,.1, and I7pp, ( are of the same order of magnitude while IV's,pxl is smalle'r.Based on a comparison with the gee-eledron bandwidth ard with empidcaltight-biudjrtg parmeters, Harrison (3.12 deduced the following set of tmiversali'wo-ceuter interactions,

:2T'kv = rltzbtû s (3.22)M

with Tsu = -1.32, Taw = hspc = 1.42, Tppc = 2,22, and r/mg. = -0.63, whered is the bond length.

3.1-4 Bonding Hybrids

Ic solids with partially covaleat bonds the valence electrous are maimly 1o-cnllzed between atoms along their bond direction d. Therefore, it is more

90 3. Bonding and Energetics

atom A

P#

c interadion

,71 interadion

Eig. a.3. Iltustration of interatomic matrix elemenu for atoms & B coupled by s=dp orbitals. The line connecting A aud B Ls assamed to be t/e z-es. The matrixdement between two arrows denotes the interaction of the respectsve orbitaksk Thesign (+ or -) inclicates the odentation of = orbital.

3.l Orbitals aud Bondiag

advisable to use a basis set ill which.. the atomic orbitals are direded azongthe bond, bzstead of simple a and p orbithls (3.16): The new orbitals az'e linpa:rcombinations of s and p fltnctions and, hence, called spà hybrids. They #verise to probability distributions Scdsng an 'eledron that are pointed i.a thedirection of the p orbital entering the respedive hybzidr The resulting fottrftlnctions in the directions d) (J = 1, 2, 3,4) a're

l ' r za =/a,. (x) = 4.(11+. .y;( tAstr) + 31/ jopj j.j .%(r) .

The hybrids fulEll the orthonorrnn.lsty relation $, j = 1, 2, 3, 4)

y j dja.j .

1 + ASAJ = &g.(1 + ,k.)(1 + &) (dylldjl

If three bonding directions dj/ll, I (# = 1, 2, 3) are given., one is able todeterrnsne the four constaats kj describing the hybridization stage aad thefourth bonding directioa d</Id41 from the six orthonormality relations (3.24) .

The enerar of an electron in such a hybrid (3.23) Oected h'om the atom at

the origin in the direction j is deMed as the diagonxl matrix e'lement of theHnmsltonian H (3,19) formed with the hmctions (3.23)

1Cspn =

1 + y X + &f>l '

(3.24)

(3.23)

Iu tetrahedrally coorainated solids the fom bonding direcvtions are sm-metrjcal)y equivalent. They point Kxadly iu the ideal tetrahedron directions

.J.J .-g.cdz = (1, 1, 1, ), Jz =

4 (1) -1, -1)14.l.a .--EJdz = (-1, 1, -1)1 84 = (-1) -1, 1)4 4

with c,o a,s the cabic lattice constaat a.s shown i.n. Fig. 3.4a. The a'ngles betweentwo of them are eqllxl to 109*28: ard the orthonozmality relation (3.24)glves 1, = 3 (J = 1, 2, $ 4). The s and p atomic wave Alnctions (3.16) are

o h bridstrxmqformed into fom sp y1

l8p31) =

j. (L.s) + lpz) + 1pv) + lpzll ,l1'S1/2) =

é' (1-9) + 1pz.) - kpv) - kpall , (3.27)z

L.spS3) =

'j (Is) - 1.pz) + 1.p,) - l.pz)) ,

1I.s#4) =

.j (ls) - Lpm) - lpv) + Ipa)J .

They are sketGed izz Fig. 3.4a. For tetrahedrally coordinated compotmds thesp3 hybridseftnlflll the princéple ()/' 'rrltzlrlrlzcm o'verlap. Since tlle hybrids st

92 3. Bonding and Energetics

(b)

sp2 Sp1

.)4 '.:+ rrl -*.'. *d. :@: z ,ô;+:; +*.'*J.'JQr2e.'#;) **qJ. :. . @ .# * . d 4ia = 4 . - . + : v ' .

Fig. a-4. es of A hybn' ' titm: (a) fottr spD hybn' pom' tm' g tke corn

of ide tetr e on', (b) t 'ee 2 hybrids with angles of 120* to each. other 1-n

a plaae- (c) o sp hybrids polmtn'ng l-n. opposite -

ectiox)s ong the s e '

.

the neighboring atorcs point in the negative tetrvedron dîrediou -% thestronge-st overlap of two hybrids happens along the connecvtion line of twoatoms. The bondlng is mn.xn'mized when the exten.t to which the t'wo hybrjdorbitals ozt adjacent sites overlap spazially is mAvsmszed. The correspondinginteratomic interadson yz (see Fig. 3.5) becomc largez thxn the other in-teractions. The intra-atomic .and interatomic interadiohs in AB compolmdswith tetrahedral strttctuze (3.12),

3.1 Orbitals and Bonding 93

1 A A7zh =

ï (G + 3&, ) ,1 s s'nb = (cs + 3sp ) ,E1 x ..'h* = (G - o ) ,ï1 B B.nB = (ca - ep ) ,l

'-tz - k1

74 = -

4

Ysv - v'5>%.- Rvspc - 3/7,.)1

Mse, - v'àvrsw + .--srep. + lspv ,W1 1

% =

ï ltss. + .--ï4p. - Vijhp. + k-m. ,W1 1 1 1 4

= Ws. + -Kx. + '--vspv - jlip. + jlipcr ,'ys ï w yé1 1 1 1 8

= Wac + -Vlptr + -V. - ïïrppc - ï'f';zwr 176 k ygj xgéare explained in Fig. 3.5. All contributions to Jo possess the sn.rne sign a'ad.,

( hencej #ve rise to a negative Mteraction parameter with a lazge value ofabout Ja = -3.22/à2/(r12) according to (3.2$. The other nearest-neighbo:rZteractioc are 74 = ,% = -0.1852/('md2)j ys = -0.325,2/(md2), and ,./6 =

0.31h2/(md2). Their absolute values are indeed much smaller th= 1% 1.For other bondl'ng geometria tlze principle of madmllrn overlap is f7T1617ed

by hybz'ids with anotller character. J.n a thzeefold coord-lnated plm.nar bond-ing:coniguration sp% hybrids point to the atomic neighbors in the same plane.'Each atom is at the verte.x of three bonds at 1201 to eac,h other formiag a

hexagonal network as, e.g., in the bazal plane of graphite. ldentifying thisplane with the zy plane the directio:cs are dz1 1(1, 0, 0), Jaoi 1(-1, +W, 0)/2.l'hen ./1 = àc = zv = 2. The atomic wave Gnnctions (3.16) are trrmqformed1to (see Fig. 3.4b)

(3.28)

lsyp:) - 1

(Is) + v'ïlpzll ,Wc 1 ). 3

1.sp 2) = Is) - lp=) + glpv) ,VJ W1 1 3

ksp23) = 1s) - lpz) - glpv) .

vq Uf:: )T.he fotzcth flcnction is a pure p orbital (l4 -> x) in a dtrection 84 perpen-'

Edictzlar to the bonding plaae, in. the considered case the pz orbital.. k..'. Ia a ltuear cb.ain the bonding to the neighbors is governed by two oppo-sitely dâreded (d: = -da) hybdds with sp cahakracter (lz = la = 1). The

(3.29)

3. Bonding and Energetics

Eig. 3.5. Directed sp3 ùybrid orbitalson two adjacent tetrahedra ar.d the.irintra-atoznic aad Mteratoxnic (nex-t-nearest-ndghbor) interactions (3.28).

two other Alnovtiorus in the directioas perpendicutaz to the chain direcvuon

((%, dpa.Ldll are buik from p orbitals (,V = ,/4 -+ x). If the CI:a,Y is parazel

to the z-aads, the,n

11's.p1) =

.ya (1,s) + I.pg))' ,

1E.s'>2) = (ls) - lpxl'.l (3.30)

Gfor the two sp hybrids. They are plotted in Fig. 3.4c.

3.1.5 Bonds and B=ds

The consequences of the hybridization stage (Fig. 3.4) and tke interactioa cff

hybrids (Fig. 3.5) for tEe azlowed electronic states can be eazity demonstrated

for tetrahedrally coordsnated systems, e.g., covateat or partially ionic spmi-

cozductors c'oestn.lll'zing in the dinmond structurre or the zinc-blende strac=

ture. The lbmiltonian matrix Iïlàbllk) (3.15) follows with the fcc Bravais

lattice po>ts (A) and neighbors of an atom along the positive or negative

tetrAedron vectol's dj (3.26). Siuce t'wo atoms A and B are i:a one J'n:'t cell,

one ftnds an 8x8 matrix (3.7)

3-1 Orbitn.7q aad. Bon -

el o 0 0 Egagz (%) Eapn (k) Bvpgz (A) Espgh (1)c SA O () -.Esp:z(%) Aaagz (J4 .E'irs4l/o) Erugzqkjc o S'A c .-àspgzlkt .Ebvg4(%) ,EbzJz(J4 EnvgnLkj> .

o o o s) .-k.pg.iLkt .E)z.vgatpl scvgzllè z'==.vz(k).*l.,#2(b) -Xap>;(%) -Xop#1(A) -Xx,p9:(1) eBO 0 0 (1

Bspgqlkt Swzp''z (h) BzwgL (1) Ecug-o (D) 0 &Bp O 0

Bnpgllhs fwvp't (h) Em=g*z (â') fa,yv; (=) o 0 cpB c

.FQp,1(k) Exp.q.z (k) E=ugq (*) .%og*L (%) 0 0 0 cDP

(3.31)with the composite matrix elemeats of the ftrstrmearest-netghbor hteractions

(3.2:)zas = 4vuv,Eap = 4K,./7i2ap = -4!V./V% (3.32).pu = 4(k;p. + 27,.,.)/3,En = 4(vr,w - *9p=)/3.

The stlmq of the phase factors gjLkj (J = 1, 2, 3, 4), which depend on the 3DBloc,h wave vedor k, are dto-fned by

1 ipa, kud, + eipaa .j- (/Tea.j.çt(k) = .,.j ge + e ,

1 spaz :aaa -

eiaas - ppwj.ç2(k) =

'; Ee + e ,

:: ipaz - es/oax + eioaa - eipaij (:.a3)J3(k) =

'k Ee ,

1 l/od.z aaa - eipla + emaxj .:4(2) = - (e - e4

TMs LCAO I'Iackiltoaian matrix for the zinc-blende structure (3.31) ca'a

be easily diagonnlized for certnsm high-symmetry points and directions i'athe bz,lk fcc Brilloplin zone. For instaace, for the direction k = k(1, 0, 0)along a cubic nan's between the Mgh-symmetry poizzts F and X (see Fig. 1.25)one obtnsns in the lirnit of a dbmond-stract'are crystal (A = B) 'with latticecozustant tu (3.7)

c 2 2 -% *1c1/2/3/4(k) = ep :i: (f.2.cos2 1-J j + Qv sin (k < )j

for the two pg/z-lilte bands which are twofold degenerate, azzd

ss,e?,,,s(p) - ) (s- +sp + (s- + s-) cos (,-,-û )q '+'4

(3.34)

, a l1 jp - sa + (s..- sss) cos (k-% )j + .ss2p sin2 tk-t ): 4

for the Btr spz-like bands. Here the meaning of the *:':F notation is tha,t fo'arcombiuations of the two signs n.='.qe in (3.35) . They are + + +: + - +, - + -,

an.d - - - for the bands u = 5, 6, 7, 8.

3. Bonding and Bnergetics

>

>U1uXcUJ

Fig. 3.6. Ba'ad stmzctme of a sincon czystal witk.in the d:st-nearœt-nerghbor ap-provt'mation and mstng an ap3 basis set. The valence band mnr'm= (r1s) is taken1us zero energp Tke re#on of the Aamdamental gap ls shnzled.

There are m=y sets of frst-nearesfmneighbor tight-binding parameters.

Withsm the tplmsvezsal' moêel of Harrison g3.1! described in (3.22) one fndsSss = -7.28 eV, Bsp = 4.52 eV, Szz = 1.76 eV, az.d En = 5.24 ev for siliconwith a b177k bond length dqq = 2.35 A. Using sa = -14.79 ev and sp = -7.59ev the bands (3.34) an.d (3.35) are plotted in Fig. 3.6. The uppermost vatenceb=d varying Som J'cs, to ,X4 symmetry charader is twofold degenerate. To-gethe.r with the lowest conduction band vary'ing from Jlz to X1 it surroundsthe fl:ndn.rnezttaâ gap region. Wherems the valence bands are reasonably de-sczibed by tlle Ast-nearest-neighbor approvsrnation and the spn bmsis set:

the cpnduction bands show features it disagxeemeut with experimental ob-servations. Thls holds in particular for the position of the conduction b=dminirnttrn in the BZ, which is expeded to be situated at about 0.85TW. J.u-stead, 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 setor of tlle second-nearyst-neighbor interactions tke dispersion of the lowestcouduction band can be described corredly. '

2.2 Dangling Bonds 97

3.2 Dangling Bonds

3.2.1 Formation of Dangling Hybrids

There is a natlzral quasi-chemical view of the creation of a surface, as a,

process accompstnied by brexkeirg of interatomic bonds. Such a simpMedquasi-cimmlcal pictme can conveniently be discussed in the fmmework ofthe tight-binding appronn'h described izl Sec't. 3.1.2 =d, hemce, for homopo-1a.r and heteropolar semiconducztom with n.7=os% directional bonds.' I'a sue.hcrystals the eledronic structuze is deterrnîned by hybrid ozbstaks whidz are

directed toward eacz other and fo= bonding orbitals as sEown in Fig. 3.5for the t'wo hybrids coupled by %. When creating a surface with a certainorjcntation such bonds and bondir.g orbitaks are truncated. J.n the ideal cmse

(with bl:lk-terrninated atomic positions a'cd no electron redistributîon), thereappear hybrid orbitals which are directed out of the stuface ar.d remaiu ttzl-

bonded. These orbitals are to.lled danglLng hpérrld.v (Fequeutly, but less ap-propziately, also dangling bonds). For the discusslon of these dangling hybridsor bonds, prototypical systemq are the (111) a'ad. (100) surfaces of din.mond-type crystals. TMs is illustrated in Fig. 3.7 for silicon. The âgure shows the

(a)

.%py,;y. .,.-.-.'

,' .'

------.

'---------''''''''''''''

-----

.'

.-----...'..-'..................-......-------'-.-....'........................'./y(;, .. .y. -----,,,,,,,.,,,,,,,,,,,,,,,,,,,,,,,,,,,,,:z,

'--.,-.--..'....'-'' ..', ...-,'

.'

E.l

Fig. 3.7. Contour plot of the total electron density itz al (111) plazze at zight a'ngles: (a) b4'lk- siticon; (b) idealppacing of 0.10 1/A? is used.

E1 1 .( t

X# VV - C

,

-,-/ y j, ) ygjyy,

(110) plane intersectingSi(111) sttrface. A contour

98 3. Bondlng and Energetics

Tb - e?' le #'

2 1 e' 1e ' - # I

e 1 e# . 2 I

e * A 12 I.------mw- ------ e 1

: 1l ' I

t'

11 . tk ' 1.1 zI

&'#'

11 .p3 . ) 1I '

, . 11 F

.i. Ij .7. Ij 1

I II

1 2 '

j

e'

1 'I '

1 ' e1 el .*v.. t 4ç #'

, A

J2

Pig. 3-8- mustration of bozd cutting durin.z smface formation. sp3 hybrids in thefour bonds surrouztding an atom in a diamond- or zhc-blende-sirudme cryst/ are

shown: (a) (111) surface; (b) (100) sarface. The surface planœ are ùatched withinthe little cubes.

(b)

total electron denzity in a

shlr6nce.(1ï()) plane for a btzlk crystal and an ideal (111)

The cutting of bonds darizzg surfacedetai) in Fig. 3.8. It shows a little cube with edge lengkh co/2 that contains

the four bonds around one atom. For the (111) orientation iu a climond or

zinc-blende strudme, two types of surfaces az'e possible. One of these hws one

dangling orbital per surface atom while tile othe,r haz three daagting hybrids.

We conside,r here only the fzst type, the single dacgliag-bond smface, whic.k

is thQ natural cleavage plaae of diamoud-structme czystals. ln ma'ay cases it

seems to be the favorable surface, since only one bond hms to be truncated per1x1 surface uzzit cell. Bac,h smface atom hms one dangliug hybrid perpemdicu-

1ar to the surface and three bac,k bonds with atoms i.a the furst uuderlayer. In.

tEe second case, the formation of a triple daugling-bond surface woltld reqttire

tHe cuttmg of thzee bonds. Three dangling hybrids 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 diamondtllll suc,h triple êangling-bondm,rfnzves have indeed b*n dlrœulr.d?d i3.13!.

ln the (100) cax two bonds bave to be tmmcated dlxring the formntion of

the stxrfn.ce. Thts is reprxted in Fig. 3.8b. Each sarface atom has twolsp3

bulk-lilce dangling hybrids. It is bonded to atoms in the aext atoznic layerY. ytwo back boads.

formation is d=onstrated in more

3.2.2 soduence on Electronic States

The allowed eledxonic states of a, hzlfspace are ve,ry sensitive to the presenceof dangling azd bac,k bonds. This can be demonstrateê by tbe investigation

3.2 Dangling Eonds 99

of the c'hanges of the band structuze due' to the presence of an ideal sttr-

face. For two orientations the projected bulk bacd strtzctures are plotted izz

Fig. 3.9 for a, silicon crystal. Accordlng to the procedre described in Sect.1.3.3 the alloled ban.ds of a Si crystal are show.a for wave vectoz's vacying izlthe correponding surface Brillo':sn zone. Besides the Alndamental gap regiou

between the occupied valence ba'ads and. empty conduction bands, one n.lrxn

observes pockets in the projecvted bands ms a consequence of the forbiddearegions îzz the bulk b=d structme. M1 these regiozzs being forbidden for bulkstates allow a, clear identlcation of surfaceuderived states.

In spite of the faet that the ideal Si(111)1'x 1 surface with half-fttled da'n.-gling boads has never been observed, we discuss a Si imlfqpace vith sucb asurface for the purpose of illustratiom Tke llnit cell and the BZ are sbownin Fig. 1.6 and Fig. 1.22ej zespectively. The irreducible part of the BZ for the

space group p3m1 (Table 1.4) Ls indicated in Table 1.6e. The most promi-nent efect of the smface is the occuzrence of a band td' in the A'ndamental

gap of the projected band structme sllown in Fig. 3.9a. The correspondingelectronic states are rnninly due to the drgling sp3 hybrid orbitals as shownin Fig. 3.10. Only a small part of the wave hlncwtion ks loenllzeè i.n the back-

(a)

217 wave vedor

Si(1 1 1 )1x1

Fig. 3-9- Band stmzctures of the ideal, bulk-termsnated Si(111)1x1 (a) andSi(100)1x1 (b) surfaces. They are calculated by means of the empMcal tight-binrling method inclualng second-nearest-neighbor iuteraeions (3.141. Tile pro-jeded bulk band Gtructure (shaded rebozbs) aud the bar.ds of botmd surface statœ

(soDd lînes) of types $d' tbr' and %' are shown. Surfacc resonxnces and antizescs-1 ;

nances in the shaded retons are not plotted.

(b)10

Si(1O0)1x1

2D wave vedor

100 3. Bonding and Bnergetëcs

((l 1 1J

l YFig. 3.10. Contour plot of the wave-Gpnctionsquare for the td' state at the # point in t>e BZ ofthe Si(111)lxl surface (cZ Fig. 1.22). A (110) planetntersecting the (111) surface is shown. A contou.rspacing of 0.03 1/â3 is uged.

bond a'rea,. The dangling hybrid energetically resides between the bonding

(valence) a'ad antiborzdt-mg (coaduction) bands. Its energy ccT = npx (3.25) is

vet'y close to the top of the bl:lk rca, valence bands at T' vith energy sp -S=

(3.34). The interaction 'yc (3.28) svith the three hybrids in the back bonds

(see Fig. 3.5) shifi;s the dangling-bond band slightly to higher energies. Siucethe (111) surface atoms are secozd-neal'est neighbors (from the point of viewof the blllli-czystal), tlle interadion of the dacgling hybrids at an ideal surfaceis weak, =d, tims, the dispersion of the sttrface band td' is fairly smaz. TMsbazld pins the Ferms enerr. It is partially Oed, and therefore the ideal (111)surfaces of group-W crys-tals should have a metxlllc charactea'. Tn .adclitionto the dangling-bond-derived band i.n the fhindamental gap, the surface also

gjve,s rise to b=ds %' in pockets of the projected baad stracture arcund theK point in Fjg. 3.9a. They are mn.l'nly related to bonding a'aê antibondingcombimlntions of the t'wo hybrids fomning a b>ck bond. The bands db' relatedto the bondr'ng (antibonding) combinatioM occuz i:a the projected valence

(cozduction) bands. Since the Emt- azd secoad-nearest-nekghbor iateractionsof the bnnt- bonds with the environrnent are ed in compa'dson to thesituatson of bonds in the bu)k crystal, such an energetic splitting betweenblk7k' statœ ar.d ba'ek-bond stat% Ls understazdable.

1.n. the cmse of the (100) s'arface (cf. Fip3.9b) tEe suzface modiâcatioâof the electronic structme is more drmstic compared with that of the (111)smface. First, i:a the (100) cmsb two bonds have to be brokec per surface atom

(cf. Fig. 3.8). This resdts ia two dangling sp' hybrids. Sinee now these hybridsare localized at the sn.rne surface site, for symmetzy reasons they dehybridizetuto bridge-bond (tbr') orbitals which are pazalle: to the surface and danglinpbond (td') orbitals whic.h are pezwndiculaz to the surface plane (3.15j. As'aresalt, the orbital chazacter of the correpondsng smface states strongly difersfrom tha,t of the origiaal sp3 hybdds. A possible debybzidization of the tvo sp3

i otq

Eo1 14

3 3S7 Sp

.u - ..z. D e hyb n- d Qati O n

Bulk ldeal surface

Eig. 3.11- De-hybridizatioxp (schematic) of the surface sp3 dattgling hybrids on a

(â00) suzface.

hybrids into a p orbitaè acd an sp orbital is Mdicated in Fig. 3.11. Formxlly,with the de6nîtions (3.16) and (3.27) the zzew orbîtals can be written as

autisymmetric and symmetric linea,r combinations

1 :$ :$ 1lbr) =

xgj (ldp 1) - sp 2)) =

.a (1pv) + pa)) ,

1 a s 11d) =

v,j. ((.4:.'P 1.) + i'S.p 2)1 =

vg EIs) + I.pz)1 (3.36)

of the dangling hybzids, whea'e tke surface atom is asstlmed to be located 'at

tite origin. The two states with eaegies

cbr = Sapz -'- ,72 ,

d;d = hox + 'ya (3.37)aze no longer degenerate. The splitting is determn'ned by the interadsotl ma-

trix elements 'ya (3.28) of two sp3 hybrids at the sn.rne atom (see also Fig. 3.5).The strongest iuteradions be>een bridge-bond aad dangling-bond orbitalscemtered on dz'fbrent sites of the square lattice of s'arface atoms mxsnly happenvia the substrate and lead to a broadfming of the two levels sbr a'cd gd (3.37)i'nto surface bands in Fig. 3.9b. Since the smface atoms are second-nerest-neighbor atoms (from the poi'n.t of view of the blzlk crystal), the broaderiingcomes essentially from indirect iateractions between surface atoms located at

the same atomic row aloztg (0ï1). They are due to the coupling of tke surfaceorbitals (3.36) aad. the sp3 hybrid.s (3.27) of the srst underlayer atoms point-

ing along a bond. Nevertheless, the Id' band i'n Fig. 3.9b shows oruly a weak

dispersion because of the small efective rr-like interaction of the (d) orbita,ls

(3.36). The esective interaction of the lbr) orbitals is much stronger, iu par-tictz:ar in tke g()llJ dtredioa. TV corresponding baad therefore develops aquasi-one-dimensional character. Accordsngly, the dispersion of the (br' bandis strong along thc 13..11 and .1Re directions. The interaction of the bridge

102 3. Bondtng asd energetic,s

bonds along the chaia direction (0ï12 is of r type. The consequence is theweak dispersion of t:e (br; band ioag tke J'J an.d #J' directions.

Tke redaced intexaction of tize sp3 hybrids i'a the back bon.ds Zves rks:to states locnqized at fzrst- aud secoad-layer atozns (3.161. Their strongestiuteractioa happens in the (0î1q cha,i'a direction, i.e., parallel to tite ,high-symmetr.r lines /'J aad J'R. Consequently, the corresponding surface ban.dsappeaœ ia the # and J' pockets, ic particular izl the stomac,h gap arolmd #of the projec'ted bnxllc band stmzcture in Fig.3.9b.

3.3 Total Energy and Atomic Forces

3.3.1 Bmsic Approximatîozls

Jn Chap. 2 we fokmd that the smface free e'aergy and tàe thermodpmmicpotentiaks of a imlfqpace are domînated by the total energ.y E of the syste'm,

witicll is nearly the interni energy at zero temperatare. A sttrface systemcoasists of atoznic nuclei and electrons. The electrons are fisFerently boundto the auclei. Therefore, they c,an be divided into tightly bolmd core electronsaud valence electrons. Since the chemical bonding and, hence, the smface pro-cesses aze governed by the valenze electroits, it is usefut to regard a solid withshprfn.ce in terrns of a regular cozection of ion cores and valence eledrons. Forex-nmple, the free silicon atom with valence Z = 4 hms a'n. ion core of càarge't with the electzonâc coMguration 1s22,,22p6 acd a valence shell of chazge6 )

-4e with electronic covguration 3.623p2. Only in the case of atoms, withshaltow tt J, o:r g electrons does the picttzre need to be made morè compû-cated by the inclusion of seMcore states that should mostly also be treatedas Gence states. One example concerns the Ga3d eledrons, in particular in

the seznicouducting compound GaN. They are energetically close to the N2.sstates and, hence, conkibute partially to the chernical bonds. Apart fromthks compication, a hn.lRpaee caa be regarded as atl ensemble of ion cores in

equilibrpxrn positions (Rz.). and a valence electron density 'rz@) distributed

among dsFerent atomic cores. The spatial distribtttkon of the valence elec-trons is usually clmssled i'n. terrnq of metallic, ionic, covalent, molecular, or

hydrogen bonds. For the salce of brcvityt here we will refer to ion core.s aa

ions and valence electrons as eleerons.Anotker basic approximation coazerns the vibrations of the fons with

resped to their eqpnilibrbxrn sites (.Jk).. In Chap. 2 we have already learntthat the eiect of the vibratjng lattice on the s'arface free energy is s=''n.l! (atleast, 1ts vadation 9om one phase to acother) for not too high tempeaatures.

Oa thc other hand, i'a detemmlnlmg the eqailibrblm geometry of a syste,mwith Nx, NB, ... atoms of ldnd A, B, ... one lnnm to study many confgtzra-tions (Ao.J vith respect to the rnsnl'rnlzation of tEe toti energy B, i.e., thetotal 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 assumption that the dyaamksof the electrons and the ions can be decoupled so thatj vhatever the tlyzla'a!-ics of the ions are, thè eledrons are ia the eluronic ground state of theicstantaaeous geometrs This is the Born-oppenimlrner adiabatic approf-mation (3.17,3.1$ . It is usnlltlly a good apprnvlrnation because the eledroamass is much smaller t'hn.n that of the ioas. 'I'he electrons respozd idmostinstsataneotzsly to chauges in the positions of the ions. As a consequezmetb.e eledronic an.d ionic deree of leedom can be separated. More suctlyspensdng, the positions of the ions are parametea's; the total enerr depeencksparametrkally on the positions.

The Bora-oppenheimer approv'tmatîon is well jtzstifed for all stat'ic sur-

face problems. However, it zaay sometim. es breA do= for excitation phe-'cal tions. For e-xam' ple, irt rnnny clkfprnl'c.al reactions aanomena aad clmml reac

electron jumps from one enerr surface to anothe,r in a lon-adiabatîc way.The approzmation also f>.'11s for non-radiative trn.ngitions in solids where a;n

electron falls from a high-energjr satrfv.e to a low-enerr surfre not by emit-ting a photon but by emitting phonons. Electron transpol't (electron curreat,electroa ttlnneling) in surface regions mlty akso give rise to a Wolation of theBorn-oppfanbeimer approxn'rnation.

3.3.2 'Potential Enerr Surface and Forcesl

ta the iirnl't descibed above tEe total enerr E = ELNX, NB, ..:; (.R./), whenstudied' as a hlndion of the atomic coordmates (Rz.), is ofïen called.the p0-jentïc! energy dur/tzce (PES) in the 3 E:=A s,.. Jvk-dirnemsioaal. (atozaic) con-

fglzrazion spaze, because it defmœ the podezztkal enera laad=pe on whic,hthe atoms A, B, ... travel. Since oae hms assnlrned that the motion of thenuclei is not vezy fast, that the temperatme is very low, aad tlzat the elec-tronic states are at the grotmd state, the termq adiabatic poterlfùl s'ucface or

Bocrt-oppenheimer Jttr/cce are atso used.Usually a complete PES cxnnot be repr-nted graplzic-ally becatzse of

the many coordinates, However, important infovmxtion about the sttdacestmzcture azd energetics can be obtained when a test atom iq. displaced over

a surfaze 'M:,K a norninn.l geometzy Such a test atom may be a real adatomor aa atom of the same spedes as i'n the bulk. Then, the coordlmntes of thetest atom are âxed in the sudace plane. However, the normal distance ofthis atom and tEe coordinates of the surface atozns are allowed to relax. Oneobtains a spedal PES, that of a suzface with a test atom. Axl nvnmple of aresulting total enerr surfre is #ven in Fig. 3.12 for a Gn,AK(110)1x1 surfuaud a'a Sb test atom. The plm of the energ.y of the tmcovered stkdace andthe energy of the 1ee (Le., isolated) Sb atom is used as eneror zezo. For thatremson the negative total energy in Fig. 3.12,

Eaa = - lSttest atom + suzface) - ftclean sGace)- Elkee test atomll , (3.38)

:04 3. Bonding and Bnergetics

--12! . .$ ., ' ''

>

x - 3 . , (j -1 (jqap

3œc ,

F- œ -4 .

Iloïl

Tng. 3-1.2. Total energy smface of an Sb test atom on a GaAs(11O)1x1 smfaceplotted over an m.e.a of two lx 1 surface tmii cells as a three-asrnensional perspectiveview (lef1) and as a contour plot (right). The smface atoms are indicated by fIILIC.IIand empty cizdes. From (3.195.

Ga 4) As

t)-- t<t) ,

t:)(q)q)))))))()i2:---,'--represeats the adsocpiion epz6zrgp of the test atom. Tlb.e slTrihce atoVc struc-ture determl'nes tite shown PES. The most stvik-s'ng featttre in the plottedPES Ls the deep' channel wlf.ch Ls quite rectllinear and parallel to tlle (1ï0jdirection, i.e., the direction of the Ga-A.s zig-zag chnln. 1n. eac,h lx 1 surfacellnc't cell two eqtlivalent flat =ln5mn. occ'ar. They correspozd to a lonpbridge-bond position of the Sb tes't atom between the Ga and .ks atoms of ch'gcremtsurface chains. There are no isolated minima in gont of Ga or As danglingbonds.

Seaz'ching for optimal atomic structuzes (As.) i'n tEe cortsidered low-tempezature lr'rnit wotzld necessitate a calculation of the electronic grotmdstate for a given consgaration (.R:). The driving forces, sometimes calledHellrnxnn-yneynman forces (3.20, 3.2.1.j, are

Fi = -Va.& LNA., .Ns) ...; (a./) . (3.39)Besides the electronic cozltzibutions twlzic.h can be calculated by the Hellmxnn-Fe theorem (3.20: 3.211), the forces (3.39) also contain a repulsive con-

tribution due to the ion-ion iztteraction. For a given eomposition Nx, NB, ...

and a tven confguration f ,&J, the mltgllitude and the direcwtion of suc,h an

atomic force (3.29) #ve izzformation about how far a ceztain atom îs from aposition i'n a metutable or stable corïguratiom Tlze corresponding 'eqnlilib-rittm' atomic geometu Ls identïed by ellrnsnating a11 forces (3.39),

Fi = 0 (VRs.). (3.49)

2.3 Total Fnerc and Atomic Forces â05

Thus, the optimal smface atomic strudure correponds to a mlnimum of

the total energy. The resulting zainsnn:lrn does, in general, not necessazilyneed to be a global rnn'nirmlm. In order to ftnd it, usually several optimal

covgurations have to be studied and compared with respect to the resultingtotal-energy vatue.

THe PES azd the resulting forces are not oaly importaut to detemmine tEeequilibrhlm atomic geometry but also for the dpmrnlcs of surface systems.One dynnrnical esect is related to the vibratlonal properties. Their treatment

witln'n the hn.rrnortic approximation requires a liaeadzation of the forces ia thedisplacements of the atoms from the eq:nsl-lbrilpm positions. Other dynnmscalefects may be related to the time evolution of the atomic positions mltj.Suc,h an esect may be caused by heatîng up the system a'aê can lead to

a meltiug of the crys-tal. The perfommnmce of the corresponding molecalaodynxmscs simulabion rzeeds the k'ategration of Newton's equatîon of motion

(12Mi a m. @) = Fy (3.41)

dj

for eac,h core with atomic mass Mi. A Srst approae to the driving forces is

given by tlze expression (3.39). The classical descfption of the motion of theatomic cores is usually jlzstiûed. A simple criterion is based oa the thermalde Broglie wavelength lst = h( zx&-sTz5Q. It shouid be smaller than theatomic dist=ces ill the slprface (3.221 . At a tzpicat MBE growth temperatme

of 60O K one ftnds a value of about àsy = 0.2 A for Ga and As atoms. Iadeed.,the motion of the cores can be treated clmssically.

3.3.3 Sttrface Dettsion

The Born-oppfmlneimer total energy slprfn.ce is also importa'at for an usc.-

derstancling of all elementazy processes occurring duri:ag epitaxx growth

ms shown in Fig. 2.1. The desorption problem has akeady been reEectedirz the disucssion of Fig. 3.12. The adsorption energy Ezve (3.38) governsthe desorptioa process. The probabilit.y for the desorptioa of the test atom

at a givec temperatme T ks proportionat to the thermat activation factor'*= expl-fad/Wc'

Another importa'at process ivuencing growth is tke soace difhtsion ofatoms. Surface d'fhtqion can be coasidered as an isolated proccs for systems

with a high energy barrier for the penetration of au atom on the smfaceinto the bG. However, in many cases a sharp rlilereatiation botween sttrface=d. balk layers is decult. In the cmse of a metal with an I:nteconstructectsttrface the built-in of an adatom is energetfcally thnfavorable izï both the

upporrnost suzface layer and i.zl a 'bulk layer. The sëtuatson is ds/erent intlze case of recomstructed semiconduztor surfaces. The clzeznical bonds in the

surface layer rcay clsf'er 1om those in thc'bullc. Therefoze, a built-ia of an

adatom i'a the surface layer cnnnnot be excluded. NeglectMg suc,h penetration

1O6 3. Bondir.g and Energetics

efects, the PES determined for the movement of a test atom or adatomove,r the smface plxne may be 'LUSCXI:I to i'aterpret the elementary processescontributiug to the surface diRtniom lt is obvious that the local rninlma on

a PES should play a'n impolant role, since the dsfhnsi'ng atoms tzy to reachsuc,h a position. Chennisorption Eappcns at a minirmnm and the compbsites stem gains energpy

TEe results of a moleeular-dmnmical s'tudy of the atomic motioa basedon (3.41) ca'a directly be tzsed to descibe the process of surface difhlsîoa.The trajcetory Atesttfl of the te-st atom or adatom on the surface is relatedto the difhtqion. constant D by menz,q of the Eiastein relatioa. ln the cmse ofisotropic systems one has

'

ilztesttf) - J?.<.v(n)l2 a 4c)jEI uuz lsryh . ( .

z-yoc 41

Such a trajectozy is sGematically represented t'a Fig. 3.13. Most of the timethe te-st atom 'dbrates azotmd the boadimg sites, i.e., the mlnt'ma on the PES.Only seldom does the atom hop from one ml'nsmtnrn to another. However,these hops govern the surface dimlqion.

A.n elementary step in the Hifh:sion of a surface atom may be tmdemtoodizl terms of transition-state f/zeor.p (TST) (3.23) . The path in real space be-t'ween 'two sites on the surface corresponds to the reaction coordsnate. Theatom with its mlrrotmdings is the reacvtaat and the atom 54). its f-zzal positionafter an elementary process, a hop, is the reaction product. Proceeding be-

X

X

X X

X X

)M: ft (t)ttlst

X X X

Fig. 3.1a. Schernn.tic zepreserzatioz of tke tralectory of a test atom on a suzface.A satface with two eqtkivalent bondcng shtes (czosses) pe.r smface llnst cell is shown.

3.3 Total Energy azd Atomic Forces 107

treea the two sites the atozû usuany crosses a certaùl energy barrier Métha bawier height EB, whic,h tzlzarac-tfm'sro a tmnsition state as in the case o'f '

a càemical reaction. The bn.rrier energ,g Ls defmed ms the rlilerence of theiotal emergies of tXe t'ramsition state aud the lower minl'mtlm. Thereby, thetransition staie gives a saddle poiat at the PES. The trnrKitson-state theoryasstzmes that at eacà bonding site the test atom is iu local thermodpmmiceqnllibrblm with the heat baih of s'ys'te.m phonozss. Thea, the trxnsition ratebetwee.n two bonding sites c.an be calculated az a local tkermodynamic ea-

semble average (3.24, 3.252. The condiiion of the applicability of the TST is ahigh energy barrier EB :>.> kaT. Only in this'limd't does the particle stay longenough i.!l oae dfe, so that equilibration eltn occur.

The pzobability of such an elemeuta'cy hoppi'ag process ks given by theArrhenius (i.e., activation) behavior

(3.43)r = rcexpt-ls/ksTl,where ihe prefactor is related to the hoppi:ag rate'vithout thmrmal activation.It ks deterrnlned by the frequendes of ihe loc.al lattice vibrations, thdr interzzalene'rgy and entropy. Estîmates relate ro to dmaderistic phonon frequenciesof the system. Tjmical valu.e.s used are the Einste.in hwuency, t:e Debyeflwueaqy or ihe fxequency of the highest subsirate vibration (3.26,3.27j. Sucàa procedure gives a <ue of about ro = 8 TEz for Gaxs. It ygrees vith thosederived witlnln more sophisticated studies (3.281. For 6nite tem/eratures tkelattice vibl-ations may a.lso iniuence the bnm'er height EB via entropy efects.This becomH obvious if the bazriere Ss are computed as dlFerences of 9eeenerg-ies or eathalpies (see Sects. 2.5.2 and 2.5.3).

1n. the caae of semitxmdudor slxrlhces the sl:rfn.ce dn%tsion is highlyanisotropic. This behador is mnn'nly a consequence of the shape of the PES.This is atready obvious (cf. Fig. 3.12) in the case of a GaA.s(110)1x 1 surface,wlzich only shows suzface relna-xtion. TNe rn-ln'tmx. of the total energy surface,which correspond to long-bridge-bond positions of the Sb adatom, aze separrated in the (1ï0i direction by bzm'ers of Ssj j = 0.18 ev (in front of the 1%s

atoms) aud of Ssll = 0.35 W tin font of the Ga atoms) . Perpemdiculsr tothe cànnnel, Le., in the (0011 direction, the enea'r bazrier is caklzlated to beabout Eu = 1.2 eV. Altllough the numezical accuracy of sucit barrier en-

eates may be questionable, dae to direct and substrate-related hieractionsbetween Sb atoms in neighboring sapercells, qualitative coaclusions ezm bedrawn: The mobitity of a:latoms along the (1ï0! direction ezm be tmders'toodas they caa mm along the treac,h between the Ga-As zig-zag c>n,5nK while themotion along the (001) direction reqees that adatoms clîmb over tke zig-zagchxinq. In any cwse, the considerable variation between &1: aad EBS. shoidlead to a strong Ysotropy of the sYace dimnsion.

Smface rlllrtlqion also remarlrably lmfluence,s the homoepiteal p'owtk ofa material. This kolds in paztietzlar for the molecular beam epitaxy of com-

potmd semiconductors suc,h as Gn.Aq aloag a cubic a'ds. Depending on the

lû8 3. Bonding and Eneagetics

growth conditions, whethe,r they aze Ga-rich or As-rich, diferent reconstruc-tions appeaz'. Under moTe Ga-rjc-h prcpration conditions one of the most im-portant reconstructions is the GaAs(100)((4x2) smface (sce Fig. 2.17) whichhas been discovered recently (3.29). The potential enerar surfaces computedfor the adsorption of As a'ad Ga atoms (3.304 are plotted irt Fig. 3.14. Onefmds the adsorptgon behnuvior for ad-cations to be very dxereut from thatof adsorbed anions. The preferred bonding position for a Ga adatom (AC inFig. 3.14) is located in the tzenches, fourfold coordinated between the doubiyoccupied daagli'ag bonds of the Ast-layer As atoms. The calculated positiouis supported by a recent X-ray analysis (3.31), wizich givcs eWdence for a 19%occupation of AC sites by Ga adatozns. However, eve.n for extreme Ga-ric,h '

surface preparation couclztions the adsorption of a Ga atom in the AC posi-

(0 1 k

plïjFig. 3.14. Potential energy surface for the adsorption of As (>) and Ga (b) ontke Ga-rich GaAs(100))(4x2) surface. The contom spaaing is 0.15 eV. Bright(dark) regions indicate m''nlrnn. tmn='mal =d, hence, favorable adsorption po-sitiozzs (trxndtion-state regions). Open tfzlledl circles represent Ca (As) atonzs.Large'r smbols indicate topmost atoms. The rn''nn'rnn. AA and AC are described inthe text. From (3.304.

3.4 Quantitative Description of Structare and Stability 1û0

tion increases the total energy by 0.1 eV. This type of adsorption does not

therefore coutitate aa eqpll'll'briltm surface strucime. Thc most favorable po-sition of the As adatom (denoted .&A) corresponds to threefold coordinatioawith three empt.y Ga dangting bonds. The occttpation of the AA site inueasesthe total eaerg)r by 0.3 ev tmder Ga-rich preparation conditions. This eaezgyindicates a reduced metaztability of the As adsorption.

The PESS in Fig. 3.14 have consequences for the dlfhlsion charaderisticsof the adatoms. The dlmlsion is rather diFerent for Ga and As atoms. Gaatoms preferably migrate in trenGes along the (011) direction, where ehergybarriers of only 0.2 ev need to be overcome. The =lnsmpnrn energy barrierfor dl'mxsion along the (01ïj direction is 0.6 eV. The motion of As adatomsis somewhat less anisotropic. It preferably occurs along the L01î) direcdon.The miniml:rn eneror barzkitr in this direction is O.5 eV. lt is smaller than thebarrier of about O.7 ev in the (011J directiom The adatom difh3Kson dependsremarlcably on the surface reconstrudion considered. The previously acceptedJ2(4x2) surface strtzcture (cf. Fig. 2.17) gives rise to a completely dxerentlandscape of 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 calcu-lations deali'ag 'with the complicated coupled atozaic and electronic problem'mnllre use of deztsit.y Gtmctional theo' z'y (DFT) (3.34) and a certna'n approvima-

tion for the fhxcitazzgeucozvelation contribution to the total electron-electrontnteraction. Its standard descziptioa is bazed on tize local dezssit.g approfma-tion (LDA) (3.35) . However, geuern.lseations by adding gradient co=ectionsor by calculating the e-xad exchange using the DFT-LDA wave hlnetions aa'e

more hwuently used to accouat for the im%omogeneity of the electron gas.Due to its foz'mal and computational simplicitw as well a,s êue to its vezyimpressive successes in describi'ng grotmd-state propertie.s of manpparticlesystezns, the DFT-LDA has become the domsnn.nt approack for calcuhtingstmzctttral and electroaic properties of b'nllc solids and theîz surfaces. Withiathe two basic approvsmations (see Sect. 3.3.1) for the descdption of the en-

sembles of interactiag coz'es and dectrons) the total energy of a sttrface systemis divided into

ELNA, NB, ---; (.&,.)) = -E%n-:on((.Rz.)) + .Ee:ta, (A$)),

a s'Tnrn of the classicat ion-ion iateraction enerr

1 , zLzjel./ozz-iontta'll = -

2 Ia. - n.i Iï.?

11O 3. Bonding and Bnergetics

descibing the repulsion of the bare ions at the positions m or, more pre-cisely, of the cores vith the valence Z.t and the pure electronic contribution

Seltrz, (Rz.)) dependkg oa the electron density.The electronic contribution can be calculated witbln the DFT (3.34,3.35).

Tkis theozy is basicatly built on. the Hohenberg-Kobn theorem, whicah provesthat the em.ea'gy of a many-electron system izz the Jrotmd state c.n,m be obtainedfrom lmowledge of the corred electron density zz@). For a give'n ato' mic con-

âgttratiozz (A,.J, the quantity Seltn, (.R,.)) is a lznsque fllndional of r.t#. ltobeys a variationi pzinciple in the dtmqity. It acqtures a n75=14731:173 value whenthe eledroa densit'y is the corect ttmzel density. The extremum conditjon al-lows one to map the Mteracting Ne-electron proble,m onto the deterrnination

. of Ne single-pazticle orbitaks. This me--mq eac,h elcctron is movi'ag icdepen-dently of the other electroms, but it experience,s a,'n elective potemtial 5r(z)'whic,h emulates att the Mteractions with othe,r electrons. One obtaius thestation.ary Kohn-shnm equation (3.354

SkS#J @) = JJMC' (m),:,2

J:os = - Z= + 1'r@), (3.46)Jm

foz' enr'ln independe'at siugle-peicle state in analog.y to (3-1) but with theKohn-slmm Hn.rnsltonîn.n Nxs. Here %' (z) and s.f are wave Glnnctions a'aêeigenvalues, respectively, of norsinteracting âctitious single pazticle,s

'

suc,hthat the correct electron density is obtnlned axs

zztzl - J7'zjl#J@)l2, (3.47)j

wllere ni denotes the occupatioa nptmber of the eigenstate that is repre-sented by the one-pazticle wave fhpnction #j(z). I'a the cmse of a 2D trn.ns-iationat system, the e'igenvalue proble,m (3.46) formxlly replaces the one-

electron Scbröasnger equation (3.4) wit: a sjt of one-particle quantdtm nzlm-

bers J = ?A.The egective Kohn-shnm potential is deMed as -

F'(z) = Mo=@) + Vh(œ) + kkctœ). (3.48)

The extm'rnal poteatial Mon@) = E( 'tijonlœ - R.) is gemerated at space pointz by all Coulomb potentials '?J,Q@) due to tEe cores at m with charge Zq. lnthe case that oaly valence electrons are studied, these potentials 1(oa@l at As'have to be replaced by pseudopotentials (3.36). However, sach a stmple fo= of

Fiontz) is only valid for local pseadopotentials. For the non-local potentials'aded in the majority of modern applications, a geneaaDzation towa'rd non-

locat operators is reqaired. The Eartree potential

2 j d3a/ VtQU)F's @) = c ,. j:r - a: I

3.4 Quaniitative Deschption of Stmzcture azd StabiDty 111

is directly given by the electron demsity. The emxc-llaztgetxl-cola'eiationtc) coa-

tribution to the total potential follows as the Anmctîonal derivative of the XCcontribaiion to the totaz electroa energy SeT. In. general, thc XC contributionis ''nsrnown. lu the Famework of the comrnonly used LDA (3.354 one b.as

J dSam@)Exc(n(=)) and, hence,

d. hexclzzlg tWc @) = dn a=(.)

with exc (zz) as the axchange-correlation energy per eledron of a tmiformelectzon gas of densit.y rz.

The XC eue'rgy oxcln) has been calctzlated by several approaGes suc,h as

rnnany-body perturbation theoty (3.371 and quactttm Montc Carlo metkods

(3.381. For practical calcGtions, excln) is expressed as an acalytical 'htmction

of the electron density. Tiuts ca'a be illustrated by the LDA of the exc,haugeterm. Witbsx the X. approirnation (3.392,

9 c 3exctzz) = '-û'je -?z

K(3.51)

fozows vwitiz the parameter tz adjustabie in the intervat 2 S (z S 1, in3

orde,r to accotmt partially also for tEe electron correlaltiop.. Freqiently theparamdrization of Perdew and Ztmge,r (3.40) is tzsed for Exctzzl .

Many mtems such as magnetic transition-metal suzfaces, reconstmzcted

semiconductor surface with remnsnl'ng daugling boads or the dissociatedmolecules on a smface involve unpaired electrons oz molecllln.'r radicals and,th'tzs: require a spin-polarized method. 1.n this context it is the spin demsit.hrhTnctional theot'y witlnp'rl the local spin densit,g approvsmation (LSDA) (3.41).Usually the dependence of the correlation eae'rgy on the spin polarization is

replaced by the same htearpolation as fotmd for the e-xclzauge energy.For the majority of surface calculatiorus the LDA or LSDA gives a suf-

ûdent description of exchange an.d correlation in 'ltke gro'tmd s'tate. Surfacegeometries aad the nattzre of surface bonds a're reliably described. Eoweverlthere are also cases where the original nozslocality of the XC energy lzas to

be talcen i'ato account, at least paztially, alread.y for grolmd-state calcula-tioas. E'a a frst step, corrections related to the pactieat of tllc local elHron

density aze added to the XC energs and the geaeralized gradient approxima-

tion (GGA) Ls employed. Gradieat-corrected density Glmctionals have been'

suggeted by Perdew (3.42, 3.43), Beclce (3.z.t4), and others. For eqltz'librblmsurface geometries the GGA usually gives results sinnllar to the LDA. Oalythe tiny tmderestimation of tlle bond lenzths taad, hencel the overestMationof the bond energitws) is lifted an.d the bonds are weakened. For the descziption

of cl.tarnt'caz readions on a stzrface, in particttlr of tmnsition states, wherebren.ld'ng of old bonds and mn.lrs'ng of new bonds occur, the GGA appears to

be supedor to the LDA, i.e., the LDA often gives even (ptnlltatively incorrect

112 3.. Bonding an.B Energetiœ

resalts (3.45). Mothe,r example iu this respec't concerns the physisorptionof pazticles. Unfortunaiely, vau der Waals interactions near smfaces are notcorzedly described ic b0th LDA a'nd GGA. Moreovea, tEe DFT-LDA andDPT-GGA elmmot cozrectly describe ucited electronic states (d. Sect. 5.2).In Sec. 5.4 anotb.er deîcit concernicg the imageDke behavior of the single-pazticle poteutial will be discmssed.

3.4.2 Baud-structttre ard Interaction Contributions

The Kohn-en.m theory (3.46) atlows a splitting of the total ene-rg.r of theelHronic sylem ia the ground state (for a given atomic covguration (.&))into two physical contribuiions

.E'olt'z, (.R.)) = .ELt'n, t.R:)) - .àu(0l- (3-52)

T'he band-stmzcture energy

f'bst'z, (a.)) = J7zzjv (3.53)?

#ves the total eneror of Ge ron-hteracting system of electrons occapyingthe Kohn-shm states j. The sccond contzibution

'

j (pz ,.:(.) .j: v's(œ) + wd :xc (s) jEoeLn) z=k

=rs@l(3.54)

describes the e:ectron-electron icteraction whic,h acco=ts for double cotmt-icg (at letkst in the Hartree contribution (3.49) ill (3.48) but also paztially f.nthe XC contribution) în the one-electron energies sj (3.46).

The total energy of the suzface system, B, depenas'ng orz the particularatomic conâguradon IJQJ, was de'ned originally as the lrinetic energy ofvalence eledrons plus the potential emergies of electronvectron, eledron-ion, aztd ion-iot icteradions. From (3.z14) and (3.52) oae obtxsms

F(.Nk, NB, ..-; fAs'J) = Sbst'n,) (As'.).) + fost'n,: (Rz.)) (3.55)with the elearostatic enerr of the sudace system

.E'estn, (.&')) = lion-iontta.l) - .feetzzl. (3.56)This splitting of thc total energy is rather convenient for nlpmerîcal treatmentusing both a plane-wave bmqis set or a basis set of locldl'med hTnctiozus. Theelectrostatic energy is partimllarly impohant for the behsvior of surfacesof compoaads. Examples are the vdzious reconstrtzcvted GGs(100) s:ldxces.Their reconstructions aire remarkably driven by a temdency to mimlrnise theloztg-racge eledrostatic interactions (3.461.

3.4 Quantitative Description of structuze acd Stabitt

For ordered slTrfnces with a 2D trn.nqlational symmet:cy the moment'qm-

space formulation (2.47-3.40) caa. be apklied. The bazd-stracture ene'rgykeeps its representation (3.53) with J = vkt the baad iadex v an.cl the wave

vector L- i.zz the smfaee BZ. A vczy powerq:7 approac.h is the repeated-slabapproxqmation (cf. Sec't. 3.4.3): which allows one to use a'Et artiîcia,l 3D trans-

Eational symmetzy The electrostatic energ.y (3.56) is expanded into pla'n.ewsves defned by the vectoo of the correspondsng reciprocal lattice, G. Onefmds

1EosLn, (AJJ) = - fzat V'5,(G) (1 - Jcolyzlztt7l + VxctGl - ëxcllT

o

+ akx + 'Yswald (3.57)wsth the vobnrne perx atöm, flzt, aud the average valemce Irtrnber, X. Theovercotmting of the electron-electron interaction is govnrned by the Fouziertrn.nsfoz'ms of the electron densit'y, the Hartree potential, the XC potentialaud the XC enea'gy' per iectron. The constant

1 z$e2a: = )7 d3z kqostm) +

.%% : 1œ1

measmcbs the degree of repulsiveness of the ionic potentials or pseudopoten-tials averaged over the atomic basis in the unit cell. The Ewald eaergy

1 Z?e2= .Ekn-ozz((R;)) - J7 d.3œ * (3.59)v=ld 2.+: y (œl

is obtained by zemoving the divergence h'om the ion-ion Coulomb repulsion.tn the emse of a dinmond-stzucture czystal with t'wo identical atoms i.n the 'tmit

cell Jswala = -2.6936 Zzeljao holds with c,: as the lattkc constant g3.50j.The momentlnrn-space formn7sqm of the total energy allows a simpMed

representation of the Hellmg.nm-Feynman forces (3.39) (3.20, 3.21) adiug on

the atoms in the case of non-erlslsbrbtrn atomic coMgtlrations (a.). Thereare 'two rlilere'at somces contribttting to DmE. One is due to the explicitdependemce of the total energy on I.R.J and the other to the implicit depen-dence tbrough the solution %(z) of thc Kohn-slna.m equation (3.46). Thelatter contribation vanishes for a self-consistent solution of the Kolm-shn'rnequation, if the bac;is set msed to reprHent (#j@)) is complete (3.47, 3.51).Ihis can easily be show'a tusing the LDA, more exactly an XC potential (3.50)in an Xo-like form (3.61). The restridioa to tbe 'flrqt contribution corremspondsto ïhe spplication of the Hellrnn.nn-Feyrtmart theorem (3.20, 3.21) . However,the rcult corresponds akqo to the older force theore,m of Ehreafest (3.521. Us-ing the Kohn-sbnam equation (3.46) for the representation of tke eigenvalues;one fnds with (3.45) a rather simple expression for the forces

3. Bonch'ng and Rnergetics

, zszjcz (x; - Rt ).% =iA,' - Rs% 12 IA,' - Ri I3,/

- Re isa, 5-2 5-2 JgJ(G)goo(G)ciGAJ (1.60)J c

with 'J:oa(G) bebzg tke Fom'ie,r transform of the tonic potential situated at

.%. The 'smt term ca'a be evatuated usiug the Ewald plmmatioa method

(3.50, 3.53! . lt describes the force on the ion core at m. d'ae to the bare ionsat sîtes Rj. The second te= represents the force of the esedive electric Eeldindaced by the ionic (pseudolc>n.rge density on the (pseudoldemsity of theKence electrons or vice versa. Therefore, the atomic force Tk (3.60) #vesthe total Coulombic force excerted oa the ioa at a.. For a given covgaration

(.R,.) all quantities apppltrlng in (3.60) are knowa. Only the valence electron

(pseudoldensity llas to be calcalated self-consistently solviug (3.46).

3.4.3 ModelMg of Surfaces

Eve,n if the actual atomic geometzy (.Rs.) Ls lœowp one has to solve theKohn-sham equation (3.46) for a semi-ln6nste syste,m self-consisteatly. Sincethe respective uzkit cell is i'n6nitely long in the direction perpendicular to the

surface, it contzsnK sn6nitely many atozns. Thus auy standazd bulk band-stractm'e method leads smmediately to (x x c;c matrices that need to bediagorxlszed. Since that cauaot be achieved one resorts to either substitategeometries to simulate a sudace or to alteraative formal approachcs whic.h

. do not necessitate the diagonxh'zation of a Harniltonian of the type (3.7) or

(3.46). The model of a sernNin6nite soûd is mostly studied together with thesimple jelliclmn model for the material. In the jellium model the positive chaz'geof the atoznic nuclei is simply represemted by a llnifoz.m constxnt positivebackgmund imside the solid a'n.d zero outside an âp' propriately Gosea surfacePlane.

Maay, but by no men.nm all phenomena iu surface scieace are rclativelyshorwrange h ttature normal to the surface. Usually, the surface region cau

be restricted to a few atomic layers the nlnrnber of wilicll, howevez, has tobe tested carefully depending on the surface pheaomenon studled. The re-

striction e-qn be used to model the surface directly in space or by certainpemturbations.

lt is possible to choose geometric models wikich aze small enough tobe tradable by todayls electTordc stracture methods but aa yet still lazgeenough to be physically meaningful. Systems contnimimg of the order of 10@atoms per (repeated) l:'nl't cell can be treated a,t a frst-principles level usuallybased on DFT 'with today's progrnrnq and compater 'hltrdware. To deal di-rectly <th the geometz'y problem several methods have been developed, e.g.,the cluster method, the slab method and the supercell (or repeatedrslab)method. More formaz approaches need pertmbations that model tlze creation

3.4 Quantitative Description of Strueure aad Stability 115

of a surface region in a normaûy in6nite soDd. Fotzr of these metàods are

brie:y described below.

Slab methods. This method simulates solid suzfaces by studying relativelythin 61rnR of about 5 - 20 atomic layers embedded in a vacuqlm reton. De-

. pending on the electronic-structtzre metbod used either isolated slabs or pe-riodic repetitions of slabs ia the dâredion perpeadiculaz to the surface are

studied. In the centrosymmetric case t'h.e singale slabs are ckosea to be thic,kenough to approach b''lk-lilte behavior near tlze center of eac,h îlm. Iu noa-

syxnmetric cmses one slab side is pazsivated to simulate the tmderlyi'ag b:llk.An œxn.rnple is given in Fig. 3.15. The spacing of the slabs irt the nprmal &.-redton is taken to be large enough so that all artifcial interactions across the

1 b are m''nimszed. M' a reasonable n'lrnezic.alvacullrn region betweea two s a s

test of the 4.&m:41773 thinkness one may demand that the total one-electronpotential (3.48), averaged over the plane perpendiculaz to the surface nor-

mal, shows a plateau in the vacuntm region (cf. Fig.5.17 i'a Chap. 5). About10 - 20 ,l. are usually plmcient to f''l611 tisis reqeement.

The slab approvsmxtion waz alreaady used together with the ETBM forelectrooc-structt'tre calculations i'a 1969 (3.121 . The trblmplnn.nt advance cn.me

with the generalization to the repeated-slab approximatioa and its combi-nation with a pseudopotential tvbnique to describe the eledtroaic stmtc-

à The periodic arr=gement of the slabs (sV Fig. 3.16) recovers atature (3. 4).artfcial 3D periodicity. The new crystal represents a superlàttice with a lazge

(100)

(001)

Fig. 3.15. A slab consisting of staclcs of atomic laye!'s in the surface normal direc-tion. A slab modeling the (100) smface of a ziuc-blende mys'tal ks shown. The smallshaded spheres in the lower part indicate a possible passivatiol of the lower slabside.

tE01'l

>

116 3. Bondsng and Energetics

formal lattice period c

slab vacuum

Fig. 3.16- Periodk arrangement of slabs. A sitaation suitable for Si(l11) sudacesis sîown.

T:nlt cell consistin.g of a slab and the vacul'rn paa't. The formal lattice constantin norrnnl direction is givezz by the s'lm of the thicknesses of the slab a'ad thevacuTpnn region. For such a geometrjr, any three-Hsmensional batd-stmzcturemethod ca'a be used. The most common appToaches for the 317 band-structuzecalciations are the pseudopoteatial plxne wave (PPPW) method, the fh111-

potential linesrszed augmented pllme wsve (FLAPW) method., and the lin-eazized mnzGn-ti'a orbital (LMTO) method with its genera7lzation to F1'11 po-tentiaks. The 6mt t'wo methods use au expansion of the eigenhlnctions ofthe Kohn-shnmn problem (3.46) in terms of plane waves, at least in certaiu

space xegions, in partimzlar inbetween the core regions. Pradical applicationsof the combination of tZe repeated-slab method and 3D eledrovc structurecatcalations are lsmsted by thc n'trnber of atoms in the 3D supercell boatdedby the lattice constant c (Fig.3.).6) ard the surface 'nnl't ceE (Figs. 1.5, 1.6).Thus, a compromsqe needs to be foun.d betweea the sla,b thickness: the spacingbetwee'a the slabs, and computationn.l egoz't.

Izl the slab approac'nnation there az'e t'wo surfaces pe.r um't cell on oppo-site sides. These t'wo sides introduce severe problmmq in all cases, even forcentrosymmetric slabs (see e.g. Fig. 3.16) with eqtln.l non-pola.r surfaces ofcompotmds or physically eqlzivalent surfaces of simple metals and group-rk-semiconductors. In the lattèr case surface statœ, if they endst, appear in pairs.Because of the 6mite thiekness and sepvation of the slabs 'there is generallyan iuteraction between the tails of these smface states azld their ideally de-generate ener#es are split. The corresponding levels belong to symmetric andantisymmetzic combinatîons of surface states localized at dslerent slab sides.TMs situation is usect to decide on necessary slab thinlcnesses and also to

identià surface states i'a calculations. J.n the limn't of delocalized weak surfaceresonance.s the approvimxtion of nominterading surfaces brexkq dow.a and.states become b':lk-lilce with a qu=tization shift due to the dparticle i.n. abox' efect. The level spacGg due to this q'rnntization Gect is approvlmxtelykwersely propordonnq to the square pf the slab thinkncrss, au.d thus can be

slab vacuum slab

3.4 Q'lxntitzttve Description of Sttazd'tum and Stability 117

reduced by choosîng slabs of appropdate' thinLmess. Special care haa to betakea itt calollxting absolute surface energies by menm of centrosymmetricslabs 23.554. Mn'ny layers itave to be taken into accotmt and, because of thetwo physically eqaivazent soaces, the reosult of euemession (2.39) h.as to bedivided by 2.

Unlike slab systems with electrostatkally neutral atomic layers, slabsInimie4zmg polaz sbprfaces dâsplay a net chazge on evezy layez, and thus also onthe surface itself. A slab with an A and a B surface appears. 'lnypical exn.mplestx.e the slabs to model polaz (100) and (111) surfaces of ziuc-blende semkcozsdudors. In the ideal cmse one simultaneously observes a catiotsterroinatedsurface, (100) or (111), and a,n n.nn'on-terminxted sudace, (XO0) or (111). Thediserent polarities of the two surfaces will introduce a spurious eledxic ûeldin the vacullm region: whic,h will Wect the surface reconstrtzction. J.n ad-dition, inequivalent dangling-bond states associated with the two surfaceswould, artifcially, give rise to charge transfer from one surface to another,whic,h *11 prohibit converg:nce of self-consistent calcalations. The esed ofthe spurious electric Neld can be sappr%sed by applying a dipole correctionto the calculated electrostatic potential (3.564. However, one still has to dealwith the geometrg, the electrordc s'tructtzre, an.d the absolute smface energyof two completely deereat surfaces.

To oveercome such a situation, Kaxira,s et a1. (3.57) employed the dslcill-ful stab-ttonbnîque'. By putti'ag two identix slal)s together 'rith the cation-

E'j 1 1J

I

Vz'7. .'f. ///. Z'ZSZZSZZ'.X j j

Fig. 3.17. Schematk representatiozz of a sympetricSiC sla'b used for the simuhtion of a Si-temncnsttedSiC(111) surface.

ll8 3. Bonding aad Energetics

(anlon-lterrnsnxted surfRe fachg each other, the electric seld izt the mcaamrezion coutd be elsmt'nated. It does, however, imply that the t'wo centralatomic hyers consist of identical atoms, a'ad thus that the coeent boadsare bejng broken in between the twp slab halves. Heace, electronic stzatesdueto titks cemtral b%yer could Muence the iuteresting ene'rgy zacge of surfacestates. 1:a the case of the Si-terrnsnated SiC(111) surface (see Fig.3.l7) themethod wozks muc,k bette.r g3.58j. The artifdal C-C interface gives rise toelcdrozzic states far 1om the fandamentai gap of the zmc-blende SiC and,hence, the interesting suzfu staœ in this e'aergy region. Eowever, large s1a,1:zaze needed to avoid the (particle in a box' efec't izt OC.'E slab balf separatedby the artiâcial central bilayer. .

'

Considering the lont end of a slab to contaia the polar smface of in-tere, the back s'ttrhce rxn be suitably pnzqssvatM. Fozowing jh'is idea Shi-raishi (3.59) introduced an altemaxtive method based ozl saturation with fzac-tionally cbnrged pseudohydrogen as shown iu Fig. 3.18 for (100) stzrfaces ofX-V semiconductors. By czoosblg a hydrogezsbke pseudopotential with avalence nhmrge Z = 0.75 or 1.25, the dangling bonds ozt any (111) or (100)IRI-V surfve ztnn be fZIJM and convez't the back of the slab %to a perfect neu-

tral >-miczmâucti'ng suzface. Similar apprnnryes hold for IE-W compounds orgroup-rv crystals but vith pseudohydrogen of au appropriate Meace chazge.For group-N atoms tEe dnrgHg bonds 'c-q.n be satmated with tnte hydrogen,Z = 1. T'he bottom laye'rs of the pseudohydzogen-covered slab sides are kept

. Fozen duriug tite surface optl'mlzzations. They simuhte the bulk regions of thesnmsconductors under cozusidezation. The boading and antibonding states re-

(a)

As

Ga

0.75 charged pseudohydrogcn

(b)

As

Ga G

1.25 charged pseudohydrogen

E'ig. 3.18. Lower part of a GaAs(100) slab with satvration by lacdonally chargedhydrogen. ls=xll dots). 80th the lowe,r M-temnsnated (a) and Gatemnlnated (b)surfaces are shown.

3,4 Quantitative Description of Structuze an.d Stability 119

lated to the cationtanionl-pseudohydrogen bonds shodd be removed 9om thegap region. This haz to be checlced in detail. For instance, for cltun diamondslxrfn.ces this requirement is not valld for the CC-H antibonding states (3.55).rfhey appear in the Alndxnnentat gap zegion neaz the projected bulk conduc-tion bands.

'

After passivation of one shb side the geometry artd tike electronic state ofthe surface of i'aterest can be stuciied. Eowever, two problems remain. The t'woslab surfaces are 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 majority of applications the electronic single-paztide states àre we,ll separated euergetkallk. Thrts allows separate studiesfor the t'wo slab sides. Unfortunately, this is not possible for the total sudaceenergies. According to (2.39) oae calculates the s7lm of the absolute energiesof two diFerent surfaces. In order to divide the sll= i'ato the contributionsfrom the isolated uppe,r aad lower surfaces of a non-symmetric shb the eaerrdensity formalism of Chetty a'ad Maz'tin (3.60) ha,s to be used. Only in cases

of lucky coincidence can absolute surface energies be derived by combinatioaof results obtaiaed foz diferent shbs (3.551. The iaeqzzivalent surfaccss alsoizdude problezns in tlle calculation of other surface properties: e.g., the opticatpropelies of surfaces. Iu this cmse the contribution of the pseudohydrogen-covered slab side to the optical propeztiœ is separated by the Lutroduction ofa linear cutoF hïnc'tioa in the optical trnmqition opvator (3.614.Cluster methods. A sllrfnzte s'yste,m caa be modeled approxirn' ately using aseciently large cluster of atoms. In prindplej ezstiug methods of computa-tional qttanttlm chemiqtry or straightfozward generxlizations of sac,h methodscan be employed to deal with .suc,h clusters (3.62-3.64). A cluster-type ap-proac,h h,as the basic advaatage that ab initio calculations can be pedbrmedand total energies zwzm be rnsnsmszed with regard to atomic covgarationsfor s'ys-tems with not too rnymy atoms. DMculties, however, arise in studyingclusters of suocient siz,e to enable the ctiscrlmn'natîon between genuine local-ized sadace states and states whose loe.xllzatibn is merely a zesalt of the 'Fnitesize of the c'luster. In particalar, clusters whic,h enn be handled in practiced.o not give good bulk referoce energy value.s aud the resulttng ene'rg.y gapsare too large. Identifying the enerr position of a given surface featare on a,n

absolute scie may be problematic. The enlarged energy gap in small clustersresults Fom the (particle in a tktrcxe-dl'rnemsiolli box' behavior of the eleeronswhic,h are reQected from the cluster boundaries giving rise to standiug waves

insteG of true localized smface states.A second class of cltuster-type approaGes hms been developed to avoid the

shortcomings of 6nite-cluster methods: the smcalled efedive ield methods.The interaction of a Mnste duster with its real environment is represented byan esective îeld to replace the botmdades (3.651. The clttster-Bethe-latticemethod is the most comrnon method used i'a dexlr'ng with this embeddingproblem (3.65). lt has tmmed out to be a powergtll tecbrllque especially in the

12O 3. Bonding and. Bnergetics

case of amorphous matezisls, for instance Sioa, but has never been directlyused to model slTrflmes or, at least, substTates below surfaces.

Note that the remslnî'ng t'wo-dsrnezzsionat periodicity of surfaces or inter-faces is not exploited $a any of the possible cluster-type appxoaches, thusoaly iaformation of an integra: nature (in egect integrated over the sudaceBzillollsn zone) ca'a be obtnlned by these methods. Local dcnsities of states

in ternns of the Greea's 'hmction of the cluster can be calculated, but not thedispersion of the bands.

rrrarqfer-matrix method. More iadirect simulations of surfaces are f1-1.quently based on a representation by layer orbitals azd Green's Rtnctions.The layer orbita'b take advaatage of the 217 trn,nnlationn.l symmetry but varjrwith the Ipprnber of atomic layers beneath the surface (3.66, 3.6'4. The layer-orbital representation therafore combincxs the advartages of localized orbitals

(3.5) with those of Bloch snt'rnq (3.13). The Green's 'hzncvdon G@) solves theinhomogeueous equation (S - cS)G@) = 1 belonging to a homogeneousSfthrödinger-like equation of type (S - cS)# = 0 (3.4, 3.6, 3.46) vith thesiugle-particle Hstrnilioaian S and the overlap matrh S. Trn.nsfer matrices

relate diferent rnxtzix elements of Green's fundions to MC,II other! itt thelayer-orbital representation of surface calcttlations, they couple suc,h matrixelements for digerent atomiè layers. Using trnrsfer matrices of this type, theGzeen's Alncvtion of a semi-inGnite crystal can be calculated layer by layer.

The trn.nnfer-matzbc method avoids some of the problems of the slab methodand the cllzstir method by dealing directly with a semi-lr6nlte crystal. Detailsof the method hive been êescribed by Yndttrnsn and Falicov (3.682: where it isemployed for a model Hpmiltonian, and by Mele an.d Joannopoulos (3.691 whoapplied it in a study of the GaAs(110) surface. Bound states must be foundas poles of the Green's Alnction in the gap amd pocket regions, in contrast to

the more direc't amat-once mspect of the slab method. However, the transfer-matrix metlloct is not restricted to wemlocn' .lp'zz,ed. surface states. I'a principle,the method also gives an accarate description of resonances, antiresonances,an.d other features in the bar.d continua whateve,r their localization proper-ties may be. However, identifyin.g such feattzres involves subtracting the localdensities of states of the semi-snflnite atd sxtfmite solids wllic,h are large and

' nlmost equal qlllmtities. The trazmfer-matrix metbod has beea developed toattàck the elecvtronic-stracture problem of layered systems. lt caa akso beused to solve the surface geometry problem exadly. Its implementation îs,however, considerable more cTlrnbersome thaa, for e'xaample, a, slab method.

Scattering-theoretionl approach. A.n exad solation of the surface geom-etry problem cac also be obtained by anothe,r method which is bmsed on theorigi'aal Koster-slater idea (3.701 for localized pertmbations of mystalliae .

sohds. This idea was 6r!4t extended to ideal surfaces by Kouteclc.f (3.711. Themethod starts with an ''nBnite blllk' crystal, whose eigenstates can be rel-ative'ly emstly detfarmlned by exploiting the three-fqsrnension&l traaslationalsymmetzy azd then e-mploying these solutions to constract the bplk Gzeea's

3.4 Quantitative Descdpticn of Strudure and Stability

ltmction G0 (s). Finally free surfaccs are created by introducirg an sppropri-ate short-range pertlzrbation. The Greem's Atmction G0 (s) of the pefect crys-ta1 entirely contaiu the electronic structure of the unrelaxed surface. Tlms,the method hcorporate,s and retalns all the bllllc propeaties suc,h a,s bandcontinua, ban.d gaps, etc.) and the alterations resulthg from a cleavage canbe obtained Oectly 9om scattedng theozy based on G0 @) a'cd the cleavagepotential without subtracting large quantities. Both conceptttally a'acl com-putationalty tMs is a rns.jor advaatage compared to the othe,r methods, inwié.c,h one attempts to ft'td the blll'k as well ms the sudace-induced solutiorsdizectlyj from either a slab, a cluster, or asemi-im6mite solid. Sizzce the methodrequires the Green's Rtnction for the perfect in6nite solid., wllie,h can be folmdwith moderate efoz't by stnrnrning over (parts of) the bTtlk' BzilloTnsm zone, itis computationally Iess clrrnbersome thaa the transfer-matr.ix method.. Theimplementatîoa îs not as simple as in the case of the slab metbod., but onceimplemented and if thq surface geometr.g is lmown, it is more eëcient andreliable.

C) @ @ O

O * @ C)

O O #.* O C)Nl l

(a)

O @ @ C)

O @ @ C)

è C) * @ C).

'

( u -w cK)

(b):: Fig. 3-19. SGematic grapius showing the creation oftwin surfaces: (a) bond-cutting

hethod; (b) removal of atomic layers. The resulting uppnvost surface atoms are

j shaded. The arrows indicate (a) which interatoznic matrjx elements el'e set to bezero or (b) which intra-atomic matrbc elememts m'e shifted in eneror by the valueu. Mter (3.661.

l22 3. Bonding and Energetjcs

The Mvantagœ of the method have rettlted in a relatively wide appli-cation in the theory of relued or reconstructed suzface. Ia pazticulaz, Poll-mxnn and coworkers (3.6% have employed tlle scattering-theoretiY methodin dealing with ideal aud reconstzucted smfaces of smmiconductors of (lia-mond, 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'1t,0n1= H = SD + U jnto a bulk ezmtribution S? arzd ato divide tàe Hxml

pertmbatfon U simulatiug the presence of a sarface. Then, one has to solve aDyson equation G = GQA-GDUG. The b&t reprcentation of the pertmbationU Ls the layer-orbiial representation. lt allows eazy physical approaees for U,e.g., the bond-oming zp,efzW aud the layer-romovd rzzeszoé (3.66,3.72.. Thetèo approaches are illustrated in Fig.3.19. J.u tke 611% method (Fig.3.19a)the ftrstv and second-nearest-neighbor izzteraztiozts ia the bdk T-TmiltoGan

(3.7) decribhg interadioms between t'wo adjacent atomic planes are switGedo5. Jks a result two equivaleat surfaces are formatly cweatH. In the xcondcase (F$g.3.19b) aiomk layers aze forrnnlly (i.e,, with resped to the eaea'gy ofthe related electronic states) removed fxom the spRe by Klnl'fiing the atomiceigeaxalues (by a lazge value %) far from the eneror region of interest.

3.5 Bond Breaking: Accompan>g Charge Transfersand Atomic Displacements

3.5.1 Characteristic Chnnges ln Total Energy

In order to understand the efect of boneg of orbitals aad brealdng of bondsoa the total enerr (3.55) of slzrfax systmnnn, sevezal models have been de-veloped. l'a the force consta'at model of Chadi (3.734 the cùaage of the elec-trœtatic emergy (3.56) is daqnribed within th.e hxrrnoaic apprnvimation forthe geometry changes. The contribution of the geometrical n'hnanges is ex-

panded into powers of Factio:aal bond-lengtà chaqges %j. T:e vadation ofthe attmber of bonds dkNsond is desm-ibed by an additional erergy contribu-tion with respect to this nl:mber. The fradiom-kl bond-leagth changes of twonearest-nefghbor atoms at m and Hj are deMed as

qj = 1.R; - Rj I/Qq - 1. (3.61)In the presence of a perturbation, e.g.: a surface, tahe ralative nearest-neighbordisplacements and the brealdng of bonds nhn.nge the eledrostatic eaea'gy(3.56) formally accordlng to

zâfes = S(rAe< + tfaetl + WzNsond, (3.62)b<J

if orly one coatributioa to the second-order term, the raclial one, is hzcluded.Three-body terms such ms tkose proportional to 6çJEfa (J # k) a.re ignored for

3.5 Bond Brealdqg: Accompanytng Charge Transfea's 122

simplicity. Vaaderbilt aud Jcœmnopottlos (3.744 suggested to azcoant for thewiation of the number of bonds in the total eyergy by the tezm W/Nbond.The constant W has to be identiâed with the cohesive energy pe,r bond.The band-structttre enerr Eyn (3.53) is calculated for a given covgttration(.RJJ. The pazameters Uz and Ug foilow 1om the equiiibrium conition andthe dezaitioa of the isotkermaz bulk modulus B. From small perturbationsof the bulk bond lengïhs one derives

DmsUz = -

,0ê 2=o

:2as2% = 957 -

a (3.63)0 ? d.o

with F' ms the volume of the considered tmit cell and ê as tbe tensor (3.61) ofthe Fadional bond-len#h chauges.

Anothe,r interesting mod, e,l (3.66,3.754 Ls based on the ehxrge-self-consistent'tight-binding method. The basic idea of the charge-self-conskstent treatmentcan be easily illustrated formal)y asizlg a Hartree-llke (or a Hat-tree-Fock-

likel treatment of the occupation dependence of the total e-IIG'gieS and ofthe slgle-partide eigenvalues. Consideeg a free atom or 1ee ion with a

valence coMguration s'Hfp, accmding to the splitsng in (3.52) one has atotal energf

1Sltom Lslnyn.p ) = swy (tnj); ..- - ngjny ri.j2t--.a ,p *,i=s,p

with the single-pazticle energies

sv(.(zz:J) - c2 + ('z.f - 'zbôlr-&,f,J=ayp

here &Ds azd oo are the orbital ener#es i'n the couguratîon s'Zprz; trzoa +z4 =W

Z), i.e., the grotmd-state co ation of the neutral atom. Each c.0n551T>tion js characterized by certain occupation nl:mbers na and np. The energiesUé; (ï).i = s,p) can be idemtifed with egecwtive intra-atomic interactions of va-

lence electrons in the same or dsFerelt angulawmomentplm state. They maybe calculae as Coulomb intevals using the correpondsng wave Amctions.The expressions (3.64) a'ad (3.65) f1llfll1 the Jaaak theorem (3.761,

0 stozn nw ,zml . (a.66)c: = .B' (.s p&n%.

The eigenvalue eé equals the e'hn.nge of the total energy q'ith the clkange ofthe kccupation number zzç of the level ï = s,p.

Exw-ions l4smilnr to (3.64) can also be derived for tlke total eaerbes(3.55) of solids or solids with surface. A system for whiG this can exsily bedemonstrated is a tetralzedrally coordinated semiconductor, e.g., a group-N

(3.65)

124 3. Bonding azd Energeticg

rnxterial crystnllszing in the din.rnond. stnzcttzre. The most important contri-

butioa to the band-stracture eneror is due to the chernieal bonding gî.1).Thks foEows approvimately as the sl'rn of all bonding ener#es of the bondsof two sp3 hybrids pomtim.g to eac,h other along a bond direction. Withim themolecttle mode,l (3.:0) the bonding energy mny be written as

1sbond = -4 Lâs + 3%) - V'h + s'Ti

with fs and. G as the orbital enargies of the hybridized group.N atoras in the

covguration skpn. The Xect of the overlapp'mg of the t'wo hybrids) S H Slz,has been taken 'mto accotmt in the lowest llinearl orâer. A representatbnliuear ic the overlap integral has been Gosen introducing an eective matrixelemeat 5rh = (SzzNlz - J.G)/(1- S%). The energy W = -(Ws.- WIV. -

sfjxpc-svppcl/4 is the negative interactiozt energy (3.28) of t'wo ,sp3 hybrids.In a itting procedme the te= Skk is used to model also the repision ofthe cores. Near the eqllilsbriplrn positioas: Ji rxp 1/J2 holds for its vaziationwith the bond lezrh d (3.6â. Following the Hfickel theory f3.1p 3.32 the same

dependence S rxp 1/(J2 is valid for the overlap iategra). With the Cottlombiategral U = -Y (Jss + Guap + 9Uvp) of sp3 hybrids the total enerr (3.55) of

16a blhlk' covatent czystal consisting of N atoms is given by

ba:k(x,(a)) - N zs + so - 43-,:- (deq)2 i4J'J>h (deql' - su . (a.6s)d

The eqttilibrintm bond length doq has been iptroduced. t'a (3.68) fi and é'denote the corresponkding quNtities at the equilibrbnm distances.

The parametez's W and S fcoow 9om the conditions of mechecal equi-librbxm in the bnllk, vauislnn'ng bydrostatic pressure, and the relation of thesecond derivative of (3.68) with respec't to the volzzme to the btûk modulus

(3.63). As a result one fmds S = la . The dilerence Scoh = SMOm(s2p2) -

.&balktN, (4eq))/N deqnes the cohessve enerr per atom. lt is mainly deter-znined by the covalent bonding 'w Q and the sp promotion energy

csO - s0s

AScoh = 2Q - @p0 - s0s) -

(j. tr7s, - 2%p + Uppt .

The last term i.zl (3.69) describes the chaage in the intra-atomic electron-electron interacuon due to electron promotion lozrl an .:2.p2 confgaration to

the skp3 elec'tron confguration. The remaining fou.r parameters follow h'omthe ftrst four atomic ionization poteatials deuecl as ciiFerences of expxes-sions (3.64). The resulting tight-bincling paràmcters are lksted in Table 3.1for diamond, Si, and Ge.

3.5 Bond Breaking: Accompanying Charge Trausfers l26

Table 3.1. Tilht-bindtng pammeters es-timated from bulk eqllêibrittzn cônditionsard atomic ionzzation potezttials. M values are in eV. R'om E3.75).

Pammeter C Si Ge

71: 8.73 6.86 6.06

-Vks. 2.79 2.19 1.94

Vip. 3.67 2.89 2.55V'pp. 6.47 5.08 4.49

-kip,r 1.61 1.27 1.12-o0 4.71 4.05 3.86-cD 13.15 11.34 11.05.5

Uas 16.61 11.67 11.49

Usp 13.20 8.14 8.71Upp 13.11 8.19 8.05

3.5.2 Enerr Gain Due to Structuraland Confgurational Chuges

The total energ.g of a system of electrons and cores is esseatially detevninedby the caheznieal bondsng. Apart from plomotion azd changes in interactioresécts, the accompanying bond energy 'U'h is usually taken equal to tite cob.e-sive energ.y (3.69) , i.e., the heat of atomization, per nearest-neighbor bond.Conscquently the sarface energy .of a halfspace (2.39) is therefore given bythe excess envergy due to the broken bonds (see atso (3.62)) . Such a picturehdicates cozrectly tEat the msml'nulrn-enea'gjr surface of a covazent dixrnond-structme crystal is the (111) surface tsee Figs. 1.6 a'ad 1.15) whch is titecleavage surface and.an jmportant growth plane.

The simplicity of suc,h a boading picture makes it fmscinatmg. However,lt cxnnot dkectly give details of the smface behavior, such as sudace re-constnlction. Ncvertheless, one may also use it to de-scribe surface changes.The (111) surface of a homopolar seraiconductor with one dangling hybzidpe,r surface atom in the normal diredion allows aq easy Gltzstration of theeimplifed pictme. Using (2.39), (3.67), (3.68), and S = âz one f-nds a surfaceenezo- per atom (per 1xl tlmit cell) of

lwA = -5i

2

E q for the 'aaperturbed ideal sllrfnne. However, such a smface would be tmstable: .

, Ebecaatuse a single electron is placed in eac,h of the dangling hybrids despite itsEreadivity, Suc,h a surface represents a degenerate grotmd state. Neglectiag for

: a moment the eledron-electron interaction in neighboeg bybrids, one couldalso equally put t'wo electrons i'a some hybrids and leave others empty without

3. Bonding aud Energetics

changing the total energy. According to the Jnhn-Telle,z theorem (3.77,3.78) a

system having a degenezate Jrotmd state will spont=eously deform to lowerits symmetzy unless the degenez'acy is simply a spitt degeaeracy.

Iu this simple exmple the surface relntion doa not moclify ihe smfaceGymmetry. The plane of surface atoms tinds to be displaced slightly inwardor outward. TIIJS is accompanied by two efects: a rehybridization of sllrFnceatoms aad a variatioa of the bonds between the fzst aud second atomic lay-ecs. The reybridization mnc'nly Mueaces the dangMg-bcnd baad in Fig. 3.9.The sAlrfxce atom displaced outward (iaward) ends with an a-like @x-likeldangling hybrid, whereas the three equivatent hybrids in the bnnk bonds tendip become plike tspz-likel orbitals (cf. Fig.3.20). AII orbitaks are character-hed by the hybrid expression (3.23) with lz for the dangling hybrid and la =

la = A4 = A foz the hybrids in the back bonds. Without rehvxtion (v = 0),1, = .k = 3. In the case of an otztvard linwardl displacement of tke sudaceatoms by 'udeq <th -la S u < la (d. Fig. 3.21) ald the requiremeut that tkeeqttivalemt hybrids still poiat toward the atoms kz tEe second atomic layer, theaagle 8a between the ineqtzivalent hybt'ids incre.ases tdecreasesl accordiug tocos Q = -(1+3v)/W (1 - 'u)2 + 2(1 + w)2. The angle ou betweea the thzeeequiGent hybrids ks given by cos Iu = -(1+u)(1- 3tz)/((1-'u)2 +2(1+'u)2!.The ortkogonality of the hybrids (3.24) givœ the relatdons I-I-WWC,œ % = 0azld 1 + l cos % = 0. This results in hybrid energies (3,25) of the danglingbond ($db') and the back bonds (%') of

(1 + 3'$2edb = % - (zp - ês ) ,4

(1 + 1441 - 3a)as = âp - (% - ?a),4 (3.71)

(a) (b)

4v+Flg. 3-20. Schematic Mdication of the dehybridizatîon of atoms on a (111) surfacedisplace outvard (a) or inward (b). The UIX'XZJIW bybrid is shown wherems theûybrids in the back bonds are only indicated by straight lines.

3.5 Bond Brealdng: Accompanying Charge Transfezs

Fig- 3-21- Hybrids forming dangling bonds and back bonds in tke lx 1 cell of a(111) surface.

and izz a lenxgth of the bac,k bonds of d = deq (1 - u)2 + 2.(1 + tz)2/W.Izdeed the exwessioc (3.71) correctly desczibe the llrm'ting cmses. For a v=-

Ls'hlng atomic displacement the bulk struct'are aad the sf hybrid enectesare recovered. For au outward tinwardl displacement of 'tz = l (u = -j ) the

J$dangling hybrsd Jlms a pme s (p) character, while the hybnds izt the backbonds become p Lsp%j orbitals.

The raisaligne'at of the hybzids in the back bonds gîves rkse to (bond-bending' interactions whereas the variation of the length of the back bonds isrelated to 'bond-stretchmg' forces (3.61. Tile change in length of the three backbonds is seen to be udzgjh to lowest order in zI. The conopondiug change

ACo('u/3)2 vith the radia) force cozzstant Q. Thein enerr per bond is tims zatomic isplaceme/t misaligns the t'wo hybrids mxkn'ng up a back bond. Itatso reduces the hybrid covalent emergy aud thereby raises the e'nerg,y of enftbbonding electron. Iu lowest order the a'agle of rotation of the back bonds is

j'u. lf oaly a single surface atôm is moved, MC,II of the back bonds Ls zotatedby th:is angale while the oth.e.r three bonds at the atom in the second laye,rremain Sxed. The change in energy due to the change iu the angales betweeabonds meeting at eac,h bnr!k atom is 2C:12/3, to be added for each of thetla.ee back atoms. One must also add the angular ecergy i'n the three bondangles having apexes at the smface atoms. lt totals 4C:'u2. Consequently thetotal elwstic energy tper smface atom) amolmts to (C0/6 + 6Q)'u2 H jCu?.Tlze force constants r-q.n be related to the tight-binding prameters (3.22) or

3. Bonding and Energetscs

to the elaztic constants c11 and c1a of the cubic crystal, For silicon Q = 55.0eV, Cz = 3.2 eV, acd C = ,56.7 ev (3.6) hold appro-vsmxtely.

The described model for the deformed bac,k bonds aud their energy iscot suscient to describe the details of the i'award relaxation of a g'roup-IV(111)1x 1 sltrface (3.54, 3.55). Ic this ease the details of changes in theorbital hteraction have also to be takcn into account. Moreover: the elasticdeformation of the atomic layer beneath the surface gives rise to all increase i'athe surface enezgy: Nevertheless, simplifed consiêerations allow one to discussthe 2x 1 recoxxstractioa obsez-ved ota cleavcd Si(111) surfaces jn the fmrneworkof the b'ucklin,g Z/JIIIf:J lsee also Fig. 1.75). This wms sttggested originally byHanernn.n (3.791 and was the most cornmonly accepted reconstruction modeltmtil 1981. Todav it allows a didactic iscussion of the phenomena accom-

panying the pairwise buckling of atoms. Witbl'n the bunklimg model rows ofsurface atoms are, respectively, moved up a'ad down ms indjcated by Fig. 3.22.Accozdîng to (3.71) the bunk-ll'ng reconstruction is accompeed by a rehpbridization shown by the transformation of the hybrid characte.rs betweenFig. 3.22a and 3.22b. Consequently the d. aûgling orbital en.ergy will decreaseor iucrease for the raised ('tzzdoq) and lovered ('tzzdeq) atoms, respectively, pro-ducing a splitting of the dangl'mg bond surface state band in Fig. 3.9a. Whenthe amplitude of the bur-kling, (%: - tzzldeq, is sue,h that this removal of d.e-generac'y is large compazed to the initial dispersion of the surface state band,an absolute gap will occur inside this band, the lower bard being ûlled azdthe upper band. behg empty. The consequences are schematicatly indicatedi'a yqg. 3.22c. Negleding the changes in the electron-electron interadion thesudacc energy pe,r lx 1 htanit cell is given a-s

1 1ns = yï't + j. ).7 q(w)

i=1,2

jwith the contributions due o veztica.l atomic displacements

1 : a 1 g zf4(u) =

k g(-1) 611 - 911 j LG - êa) + j. tz .

The bunkling correction J'à is given by the emerc change of the danglingbond cdw (3.71) added to the elmstic enerr 3.Cw,Z. Tlle t'wo corrections

!5ztzql are plotted ia Fîg. 3.23. The energy mimmizatîon predicts that thesurface atom with the doubly occupied daugling hybrid is displaced rtlmost

until the hybrid is s-lilce, i.e., 'tzz = la . The predicted inward displacemeat

of the atom with the empty hzbrid is smaller. The msnimlTm occttrs at

'ttc = -êa (êp - J>)/(2z Vp - :s) + Q n$ -0.12.

3.5 Bond Brealdng: Accompanying Charge Traafeas 129

j(a)

(,) + (u,1Uz

(u11U2

Fig. 3.22- Efects of buaklîng of a stlrface) e-g-,of Si(111)2x1, on dangling kybl'ids and their oc-cupation. Norrnxl atomic clisplacements tsl audua are assttmed. (a) Unbuckled surface withhalf-fzlled 0p3 hybrids; (b) buclded surface withdehybridization; =8. (c) buclded surface withQlled s-like ar.d empty pa-like danglirg bonds.

3.5.3 Enerr GG azld Electron Transfer

Ia expression (3.72) a'act Fig. 3.22c complete electron trn.nmfer h'om the more p-lîke danghng hybrid iato the more s-lilte dargling hybrid dllm'ng the buclding2x l reconstruction ha-s been assnlmed. Tlzis wa.s oaly possible because theeEed of the electron-electron iateractioa izl tlle two daugling hybrids, 1 aud 2,has been neglected. In. order to discuss the efect of the întrabybrid electron-eledron interaction on the electron transfer, we restrict the considerations tothe two danglin.g orbitals in the 2x 1 tlmst cell of a (111) surface. lnstead ofsp3 hybrid energies there are the dangling bond energy cz - sds: of the raisedatom ar.d the enezgy E'c H aabz of the lowered atom. Asslnrni'ng an electrontrlmsfe.r b'n (0 K Jn K 1) and, hence; ocmlpation zrlrnbers nz/zàl = 1+Jn andn.a'(Jn) = 1 - Sn of the t'wo hybhds, it becomes obvious that tlze dangling-bond energies ss a're diFereni fzom the energiœ & bebnging to the slp3confgaration.

130 3. Bonding and Enezgetics

>Xm

c->xc)=(Dc

Fig. 3.23. Correcwtions fzztut) and f2zL%<) to tite surface enea.r of a &4111)2x 1

suzface due to imckltng. The atom vitll the completely Slled dangling hybrid Lsdisplaced outward by 'mdoq whereas the othe,r atom with the empt,g d

'

g bondi:a dksplaced inward By .-ltluldeq.

Quantitatively the GeC't of the electron-election interadion on the sur-

'face energy may be described ssmslazly to the behavior of eledrons in 16.e

atornR. Accordsmg to (3.64) au.d (3.65), for the staface energy one fmds aclzange duc to the electron trxnmfe,r Jn

1 ?éfzs = I''-.I m(J'n,)&:(éh) - y'rk (5n)&:=.1,2

1- rzf(0) sé(0) - j-zk?(O)rJ (3.74)

with the abbreviations

'?u(é'?z) = 1 - (-1)fJn:s:(Jzz) = êi - (-1)frJWa. (3.75)

For the sake of sMpllcit.y the orbital dependence of tlzo interactioa interals

Un = Uap = Upp = U has been aegledcd. The minlrnszation of the resulting.

e'aerr change (3.74)

Jfzs = -(fa - ê:)(h + U(J?z)2 (3.76)

gives au electron trnmqfer

1 éa - Jzôn = .ï rz

(3.77)

3 s Bord Br-pka'ng: Accompanying chacze Transfezs 131

It is proportion/ to the dxezence in the hybzidizatioa stages of the two&

dangli'n.g bonds. In the llm5t of a norrnltl U beha<or, U > 0j the repulsiveeledrou-electron iateraction hinders complete electron trnansfer. For the Siparameters in Table 3.1 one fnds b''rt Q$ 0.5, i.e., incomplete electron transfer.Coasequeatly the coatributious (3.73) to the lowering of the sudace energy

'

(3.72) are slightly overestimated.

(

4. Reconstruction Elements

4.1 Reconstruction and Bonding

4.1.1 Metallic Bonds

The most widely studied surface systems are those of metals. One importantaim of the eaharacterization of clean metal sttrfaces is to produce suitablesubstrates for adsorption studies. Sequences of low-inde,x surfaces have bee,nùwestigated such aas (111), (100)j and (110) of fcc metals (e.g. Cu, Ni, Pt) and

(11O)j (:.00), (111): a'nd (112) of bcc metals (e.g. W). I'a the early studies, formost cleaa metal surfaces the lateral pedodicity hms been found to be identicalto that of a parallel plane ic the blllk-. TMS conclusion was in gemeral based on

the observation that tlze LEED dOaction pattern did not contnrl'n any otherdsAactioc spots than those e-xpected from the Sdeal periodicit (4.1). lndeed.,the clean low-index (111) and (100) smfaces of Cu show dl'sradion patternswith hexagonal and square symmetries as expected accordsog to Fig. 1.2. Theclean low-inde,x (110), (100) and (211) surfaces of W yield patterns which are

also cbytracteristic of ideal lateral perioclicity.The conservation of the lx l tmmqlational symmetry for the above-

mentioned metal szzrfaces is accompn.nled by surface relration. A corre-

spondl'ng meeanism which is cbractezized by a smeazing out of the electrondistribution and a'a inward displacement of the eores in the topmost atomiclayer has been discT:.qqed in Sect. 1.2.4 (Fig. 1.10). The relative spacïg oftlle top layers is typically reduced by c.p to 12%. For e'xample, in the case

of the W(10O)1x1 surface a contraction of about 5% is fotmd for the top.most layer (4.2). The eledronic origin of the reluation o:a a trn.nKition-metalsmface sue,h ms the W(100)1x 1 seace may be unders-tood by considea'ingsimaltaueously the efects of bondsng of localized d electrons and delocalizedsp electrons. Ia the btllk of a transition metal, the bond formation driven by delectrons tends to deearease the interatomic distances while the free-elec'tron-like sp electrozzs minimize the,ir contribution to the total energy by expaadimgthe lattice. The balance between the two mec-haaisass leads to the eqltilib-rilrrn geometry. At a smface, the d-d bonding between surface azd subsurfaceatoms is emhnanced., i.e., the bond clistance is shortened, since the sp electronscan be pushed into the vacullm region above the surface. However, there are

exceptions. Akc;o outward relaxation may occtm The densest metal surfaccs

134 4. Reconstruction Blements

goc 1(I

Eo 1 QJ

Fig. 4.1. Possible lateral atomic displacements in the (01ï) direqtion of atoms on a

reconstructed w(100) smface. The open circles inclicate buût-like positions. Dashedlines desczibe a (WxW)JM5Q cell.

'

sometjmœ xlm show azz expausion of the tomlayer spazing. One mxxmple Lsthe Be(0001) stu'fruv (4.31. One reaszm seems to be relate to the strong di-rectional bonds formed in the Be bulk due to the hybzidization of : and. pbands.

'

Adsorbatocovered or contn.mimated meti smfaces often show beautifulzeconstracwtions. However, since a few years slTrGce reconstractions are alsolmown for cle,atk metal sttrface. For iustance, upon cooliug the W.(100) stuu

Lqc,e below room tamperature a p'hxqe transition 1om a lx 1 iuto a c(2 x 2)structure is observed (4.2). The reconstmzction is described by alternatiuglateral isplacements of W atozms abng a (110) diectioa to Vm zig-zagc,hn.''nm witk a (WxW)M5* geometry (Fig.4.1). The structural nlnm.nges can

be explalned by the lowering of the band-structtlre energy (3.53). Smfacerelamatioa barely reduces the dezzsity of states (DOS) for the surfaze layer at

the Ferml' euergy sv. The sur%ce recozzstruction is, however, accompaaied byopening of a sarface band gap arou'ad cp azld, hence, a sigvcan.t reductionof thq DOS in kbiq energy rauge.

Recently, reconstnzctions of clun (111) oriented Pt and Au sttrfazes (4.'4,4.51 have beea fotmd. However, the (1û0) surfaces of Ir, Pt and Au alsodisplay sarface recozlstradioa (4.6-4.8). The apparently widespread mvlntence

of surface reconstmzdions h.as evea raised the qucstion tslzolzld all sarfacesbe recoazstructed' (4.9!. It wms suggested that the dzivir.g force for a surface

h hanging the dertsit.g of surface atoms is the dsFerexce'bet-w'eenreconstruckio c

ihe surfaœ s'tzess nj (in tmits of forœ per unit length) (2.18) aad the sttrface

enerr J (in tmits of eaea'r pe,r tmit area) (2.13), Le., the strain derivativeof the smface en'

ergy qj = yks (2.19). Consequently a suzface would tend to

reconstruct towxrd a state in whickt the surfaze stres is equal to the s'Irfnne

enera. Whea the strain derivative of the surface enerr is positive, there is

4.1 Reconstractîon axd Bonding 135:

a tendency for the density of surface atoms to inœe-ue, and vke versa for aAegative value (cf. disctlssion in Sect. 2.2.3). The cxmclusion concerning the(:11-1v1)1g force for surfsce reconstruction is ssrnilar to results of other studies ofsurfacc densiscation g4.10,4.11q. However, in those stttdies it was shown thatthis driving force does not resalt in acy reconstruction. The reason for theobviously cligerezlt fmdings could be that the above criterion follows witbin

linea.r elasticit'y theory and? hence, applies only for s=n.ll tlnlibz'm strnainq whenthe surface Layer and bxlllr are strna'ned together. A reconstruction iuvolvingstrain:ing the surface layer alone necessarily changœ the surface-substratebonding: whic,h generally costs energy. The application of the above criterionto the case of the Au(110) surface should therefore fail (4.12, 4.13). Tikis

surface undergoes a missing row reconstruction (Fig. 1.9b) in which alternate

E1E0q rows of atoms are removed from the sudace. The removal of one halfof the smface layer of atoms cnunnot plausibly be discussed withz'n linear

elasticjt.y theozy. The missing row reconstraction involves very large es

ic the local environments of some atozzus, wlzich cn.nnot be descaibed witbln

the li'ne,a,r homework.

4.1.2 Strong Iozlic Bonds

Despkte tke completely diferent bottdGg behavîor ia ionic crystats with

strong ioaic bonds, the atomic reaazrangements on mnny surfaces are ratherweak aud keep the 1x1 trrslational symmetry. Surface relaxation or even

surface rttmpling (see Sect. 1.2.4) are favored a.s for rnnry metal surfacu

(4.14). Among such ionic systems are the lead salts PbS) Pbse, and PbTe, themetal ozde.s Mgo? Niol Zn0, A12oa, and Fealb, and s.lkxl:' halides, e.g., LiF.The lead salts, MgO, NiO, aud the n.1kn.17' halides, except for CsC1, CsBr, andCsl, wbic,h are most stable in the cesitfm chloride structure, crystnllize uzder '

normal conditions ic the sodillm chloride (rocksalt) stmzctme tseq Fig, 4.2).The other metal oxides favor tlze hexagonal crystal system. The a-F'ezo.s andosAlzoa coimpozmds czystn.llize iu the hexagonal com:ndl:rn structme whi)eZnO is a wuztzite materjal.

In contrmst to covale'at bonding the relatively strong ionic bondîng forcesact not only between nearest neighbors but also between atoms witich are

situated further away, because of the long-rrge natttre of the Coulomb i'a-

teraction ilwolved. Furtlmrmore, one hms to take into accotmt the Alrnost

directional indepexdence of ionic bondiag as a consequence of the hilercoordination tbxn in the rocksalt case (Fig. 4.$. As a result, mltlnpy smallstructmgl c'hxn. ges parallel to the sllrfytce normal appear. This holds in pawticular for nonpolar surfaces with an equal nplrnber of cations aud aaions inthe sttrface atomic layer. This is valid for tlle (100) faces of the sodip:m cblo-ride strudure and the (1110) and (l0ï0) facc of the wurtzite structure tseeFig. 1.6) . One important argtzment is based on the electrostatic enera (3.57)wjth a strong Ewald contzibution. For (100). roclcsalt faces both sublatticesrelax Snward but to dlFerent degrees, because of the dlFereztt ioak rnr155 of

â3$ 4. Recozetnzction Elements

r---ca.--/

Eoo-lll

Eo't oq

E1 onllFig. 4-2. Schematic representation of the zocksalt czystal strtzcture. A non-prlmltsve fcc Ilnlt cell, a cube with edge length ac, is shown.

x 'a

cations and n.nrozls. Tlzis results in the so-called rumpling of the sttrface (4.142.I'a the e-qme of polar faces, e.g., (0001) and (0001) of the wuztzite structme, one

expects larger rearrangements. This is iudeed the case for AW(0001) (4.15).However, this is not the case for the polsr (0001) faces of hexagonal comm-

cbnrn with -2LXO:$-MXO3- stacking (X = A1, Fe) in the (0001) direction. Theoxygen atoms fqrm hcp hyers, aztd the metal atoms ftli two-tbirds of the oc-

tahedral sites between these layers. 'T'wo Xaoa formttla tlrlts (10 atoms) are

izl tlze primitive rEombohedrat llnl't cell. J.n a description by a conventionihe-xagonal I:mit cell six Xatx formtzla llnits (30 atoms) occur. Usually lx 1trauslationf symmetr,g is observed (4.16-4.19). The phenomenon of surfacemtmplling does not seem to be restricted to ionic crystals. R7lrnpliug has abobeen observed for nozppnique Ti(10ï0) surfaces (4.20) and is even suggestedto contribute to the lonprange 16x2 reconstrtzction of the Si(110) smface

(4.21) .

4.1.3 Mixed Covctlent aud Ionic Bonds

Prototypical systcms with dizedional covalent or partially ionic nerest-aeighbor bonds are the tetrakedrally coordinated semicondudors cnrstal:iz-ing ill diFtrnond, zinc-blende or wurtzite structures (or even in othe,r hexagonalpolytres, e.g., 4.11 or 6H (4.221). Among them are the e'lemeatal semiconduc-toz.s C, Si, aad Ge aud the compouzd semiconductors SiC, III-V, II-W, andI-WI. The amotmt of ionic aharac'ter of a single bond can be successfullydescribed usiug diferent quaatities ard scales. One qurtity is the bond po-lxm'ky ap of Harrison (3.11); others are the eledronegativit (Ak/B) scateof Pallls'ng (4.23) or Phillips (4.241. The ionicity is related to the dference ofthe atomic electronegativitie-s Xx and ,X%. Meaawhile, sLightly revised Ns1-

4.1 Reconstrudion and Bonrlsng 137

Fig. 4.3. Valenceiectron density of cubic AlN along tEe (lllj bond direction (solidline). lt is decomposed into a s'ymmeiz'ic (dotted) aad an antoetdc (fazhed)pact. Drawn ushg results of (4.274.

ues of Pal:lsng's electronegativities ca'a be found ix the literature (4.252. Ascale based on ab initio calculations of the electron density (3.47) has beemrecently proposed by Garcia ard Coheu (4.26j. The ionicitie,s of this scale aze

computed az charge-msymmetr.v coedents g tn.king the squa're root of thcratio of the squares of the antisyrnmetric and symmetric parts of the elec-tron density htegrated over the vobnrne. These contributiorus are plotted i'a

Fig. 4.3 for zinc-blende A1N resaltin.g in g = 0.794.The bonds are illustrated seahe'mxtically ia Fîg. 4.4 for tetrvedrally co-

ordinated (i.e., .5p3 bonded) czystals. Each of the,m contnsrs two spin-pairedelectrons. When a soace Ls formed: some of these bonds 5dll be brokea, lead-ing to lmpasred electrons. As discussed in Sect. 3.2 tdangling bonds' a'ppear.They are foed with a fractiona,l zp:mber of elec'tro:as dependl'ng on the groupof the element A or B in the Periodic Table form'lng the A R compotmd. ThenTznnber of electrozts.amounts to Nx/4 (or Nz/4) for atoms A (B) from the

group NA LNz =.8 -XA) because of thc conserqation law, XA/4+.N's/4 = 2.Suc,h a btllk-truncated smface hms a high surface eaergy, and is therefore

vea.y lpnKtable. Hence, the atorns in the mIrface region relax 1om their blllkpositions 5n order to reduce the surface free energy by forming rew bondsoccupied by t'wo electro>. Reacldng a structure with a defced s'toichiometzywllich cxhibkts a local minirnl:rn of the stlrfacc free energy, implies that thechemical valemcies of the smface species (or at leazi of most of thcse species)are satisfed i'a the recoustructed geometrp Consequently, the observed ro-

constructious temd to be instfating. Surface bonds or d=gling bonds whoseenergies fall hto the bulk enerr gap, crossing the Pemni level, woald i'acreasethe sttrface energy (see Sect. 3.5).

Besides tlle geometzy also the sarfacc stoichiometu may vaty Sometimesit is useful to regard the surface region as a new clmrnlcal compound whose

l38 4. Reconstruction Elements

Fig. 4.4. Batl-and-stic,k model of a zinc-blende (cliamond) crystal. Open and 5l1ed.circles in the mzbe represen.t cations a'ad nmions, r%pedivezy. The 'bozzds are i'adi-cated by double ljnes (i.e., the sticlts).

stractm'e and composition may be evaluated theoretically by =5r,5=5,z111g thefree energy (cf. dkscussion in Sec't. 2.5). Multiple msnirna occyzr for most satr-

faces, difering i'a both composition and structare. Whicb. minimttrn is ac-

cesed e'xpersmentally depends upon how the surface is prepared. Thus, a

typicat compotmd sarface exhibits multiple structtzres depeadhg upon prepa-zatfon conditions as well as tlle ambient temperatme a'ad pressure.

4.1.4 Principles of Sezniconductor Sttrface Reconstruction

Surface reconstrudion aad the special case of surfacû relaxation fottow cez'tain

guidMg pzinciples discussed in detail by Duke (4.28,4.291. A surface producedeither by a cleaxrage process, a growth process or even repeated cycles ofsputtnrsng a'n.d n.nnenlsng 'tmderlies a

Basic Prirzcïple; The dtûr/ccc stmtctnre os,sezved '?All bn the l/.t(7c.#

Jree-enerlp structure kfrlcàfctzîî?/ acccssibln czàer the prqmration con-

tfftïonxs.

Apart from hozen non-equilibrblrn situations whic.h aœe also obseawable a'adsurfalce geometries detemnsnned by the growth kinetics, this prindple re:ectsthe fact shat a certain surface may be generated by a variety of processe'sand, therefore, corresponds to a local (but not necessarily global) =l'ns=plrn

of the fl'ee energy. J.n a sense somewut genernh'zed to the thermodyAmmt-cconsiderations in Sed. 2.3, we witl also call such a surface tstable' ia tlzefollowing.

The càearest e'xamples of the valiclity of the Bmsic PrGciple and tlle its

îuence of the preparatioa process are the (111) cleavage face of Si and

4.1 Reconstraction and Bonding 139

Ge. Low-temperatme or room-temperature deamge produces a 2x 1 struc-tlzre, wherems n.nnealing to Mgh temperatmes prodaces the Si(111)7x7 struc-ture (4.30, 4,31j and the Ge(111)c(2x8) surface (4.31-4.33), both of whic.hare stable upon subsequent cooling. The 2x1 structuze obtahed by low-temperature cleavage is a locally stable structme, but not the lowest fzee-

energy stracture whic,h is obtained for b0th smfaccxs by o.nnenling and subse-quent coollng. As we will see in the following sections, hir,h t-rnperattlres are

necessazy to overcome the eaergy barriers between deerent structuzw aadto generate the nAntoms iu the lowet free-euergy strudures.

There is an iuteresthg hypothv'A about the generation of the 2x1 re-

construction duiing cleavage (4.34!. Duzing eleavage in the /211) direction a

stanking fault is generated (see Fig.4.5). Its formation eaergy of about 0.02J/m2 (4.35,4.361 ts small compared to the Si(111) surface emergy in Tablez.l.A spedal bonding topole is suggeste. A double layer of six-member ringsis rephced by an alternate arraagement of âve- and sevea-mambez rings.Exactly such a ring stradure occurs in the topmost atomic hyers of a 2x 1reconstmcted salrfve of a dimnnd-structure mystal wiàht'n the widely ac-

cepted r-bonded CA:J.II model of Pandcsy 24.37).For electricatly tmclzarged systems, autocompensate,d vste'ms or systems

with no space Garge, the Basic Principle may be reduced to three maia'principlœ whic,h ex-plain the ddving forcG of recoxsstaatction (including the1 x 1 reconstruction, the reln.vxtion) of smmiconductor surfaces ia more detailand are consistent with a mlnsmtlm of the suzface free energyp.

Pcinciple 1: X sur/bce tends @0 rzinïzrlizc fâ,c nnmber of dtmglfp,pbonds by the jomation J.J n'e'tr ùo'ztrs. The rerrztzfnï'n,p dangkén.q 50.n,#,.:tend to ù6 satnraied.

(a)

7

I) 1 11

(2ï11

Ft. 4.5. Bond structuze of a (lllj-orlented Si czystal projeded onto a (01E) plxne:(a) ideal s't

-

, and (b) with a staœng hult. Afte,r (4.34).

14û 4. Reconstractiozl Elements

Reconstractions (or relaxations) temd either to saturate stzrface daztglingbon.ds via rehybridization or convert them into non-bonzlng eledronic stateswhic,h may be fzlled by a lone pair of electroms or be completely emp'y Promi-nent evnmples for the minimiv,atio'n of the nurnber of dangling bonds are theGe(111)c(2x 8), C(1.00)2x1, and 1l1-V(110) 1x1 s'urface. Jm. the Ge(111) case,six of the eight dangling bonds per c(2 x 8) cell are saturated by t'wo adatomsin Tzl positiozts (4.38, 4.391. This covezage results in a total nttrnber of fourdangling bonds. The four êangling bonds of the two C atoms per (100)2x 1ce2 are saturated by forrnlng a symmetric fqsrne,r L4.4O). The fottr electroms oc-

cupy the c atld x boncling levels of the dsrner. The cleavage (110) lx 1 surfaccxsof IH-V zinc-blende compounds indicates another menbn.nneKm of the passiva-tion of the two dangling bonds (4.412. There is a tendency for > transfer ofthree quaz'ters of au electron leaving the cation dangling bond empty and611ing that of the n'nioa with an electron pair.

Pziriciple 2: ./1 s'ttdace àontls to compnnsate càarges.

An important constrxint whic.h ls=l'ts the possible stoiclkiometzies of com-

pound semiconductor smfaces gs the reqttirement that no charge accumulatesat the surface (4.424. J.n the limit that sarface defects which accotmt for thecompensation of charge $n the space charge layer are negligible, tMs holds alsoon the length scale of the bond lengths. The relative stability of two surfacereconstructions 1511611:1g Principle 1 follows to a large extent from the min-imization of the electrostatic interactions i.n such a smface structure (4.43).The tendency to small electrostatic interactions may somettmes also drive aninterchange of stoms across an interface (4.u).

The 'vacancy reconstraction 24.45) of (111) snrfaces of IïI-V semicozduc-tors (see Fig. 4.6) may be considezed as a consequence of Principle 2. Theremoval of one cation generates arz equal nlpmbez of cations and anions i.ca 2x2 surface pnst cell, a,t least cotmtin.g their dangling bonds. hstead offolzr danglin.g bonds with a total of three electrons, one obsezves three catiortdangling bonds a'ad three Jmson dangli'ng bonds a,1e.1. formation of a. cationvacancy. The eledrostatic neutrality is locally g'uaratateed.. The sidbld ringsof atoms surrotmdl'ng the corners of the lnnit cell in Fig.4.6 conskst of alter-natmg threefold coordt'nated cation and n.nson atoms. The chains of attezingdangling bonds allow a slmslar electronic s'tzazcttlre as in tie (110)1x 1 =e.

ln agreement with Principle 1 the cation tanion.l dangling bonds are emptytflled with an electron pais.

Pclrzcfple 3: A semicondnctor mzr/acc tends to 5e izssulaténg (or .semb-nonducting).

In principle, all surface reconstructions f7,1.15111:),g Pricdple 1 and Principle2 evhibit a tendency to obey automatically Principle 3. Howevez, there arem=y cas%, in pazticular for quasi-pno-dimeasional surface structtlres, for

4.1 Reconstzuction and Bondir.g

Fig. 4.6. Top view of a Ga-termlnated GnAR(111)2x2 surface afier formatson of aGa vacancg. A possible 2x2 surface tlnl't cell ks drawn. Cations taazionsl ae indicatedby open tftlledl ecles.

whicb ad. ditional atomic relnvn,tion or recozustruction leads to a semiconduct-ing instead of a metn.llic eigenvalue spedmpm. For srteras with exterdedelectronic wave fllndions, metallic grotmê states do not occm in one dt'=en.-

sion (4.464. One clear example for the operation of Principle 3 on tetrahedrallycoordinated semicondk uctor s'trfaceur is the tilting of the dîmers on Si(100) andGe(100) (4.47-4.494. Tlb.e interadion of the dirners aloztg the rows in the (01ï1direction gives rise to a remarkable disperston of the %' and 'ir- bands. If thedimez's are untilted, tke 'zr and g'* bands associated with the clsmem over-

hp i.zï energy for certxsn 2D Bloch wave vectors. The chnsns of dt'mea's alongthe surface form a snrplmetal. According to the Peierls esecvt, semirnetals are

unstable in ttrael 1D. The l'esulting grouzd state is charadezized by Peierls-distoled (i.e., tilted) dimez's.

4.1.5 Electron Cotmting Rules.

Prindple 3 arzd the part conceratng the pusivation of the daagling bonds inPrinciple 1 aze automatically f7al6lled, if the studied surface stntd'tzre obeysan electron cotmting rule (ECR) (4.42, 4.50-4,522. Suc.h a mfe states that

bonding and non-bonding sttrface states that lie below the Fermi leve,l at

the surface must be ftlled, whereas the non-bonding aud antibonding states

whtc,h lie above the Fermi energy must be empty. This criterion is irt 'clearagreement with the condition of the insttlati'ng (semiconductlg) behavior ofthe surface, i.e., Prlnciple 3. Am. ECR cazl be applied directly to detmmnîme

142 4. Recozzstruction Elements

m4z

21)

Fig. 4-7. An acioreterminated (100)2xr?zsurface of a zhc-blende crys'tal with one

rnllsing dimer. Aniozzs (cations) are indi-cated as full (empty) circles. Hatched dan-gling bonds are ftlled.

allowable stkdaee compositions in agreemènt with Priaciple 2. Only in orderto deal with doped semicondudors must these rules be genernlszed to accotmt

for defec'ts at the surface which compemsate the space charge (4.531.The electron colmting is well tllustrated for the prototaical (100) sur-

faces of ziac-blemde semiconductozs (4.51,4.52j. The surface energy of thE'sesuzfacc is lowered after d''rnerlzation amd passivation o.f the daagling bondsin agreement wîth Prhcipie 1. More precisely, the dangling bonds are ftlledon surface axkions and are empt'y at surface cations. Only dsme'rs fo=ed solelyby catîorts or aaions (i.e., homorlirners) aad 2xm (or mx2) reconstrudions

are considereê tu the original fo= of such an ECR. The 12' LA a co>equenceof the Hsrnerization, while the zrz-fold periodici'ty is aamnrned to arise from

Lm - D) rnsqsi'ag dlmeas, leavhzg D fqirnep per qlnit cell where D :; m. Sue,ha reconstruciion is sbown 1. Fig. 4.7. J.n orde.r to determsne the relatioashipbetveen D and m, me cotmt the nlzmbe.r of electrons (4.51). tu the aaion-termsnated caae of Fig.4.7 ennln top cllrner requires six electrons, t'wo i'n the c.

clinner bond and two 5.u each of the remnlnlng dangliug boads forming .n' a'adz'* statœ. The total number of eledroc on the Hsmezized surface should beequal to that of the blllk-tlnlncated surface with a missi'ag aztion pair. Thisyields

6.D = 4Ja.D + 4.fcLm - D) .

He're fradions of electrons

Jc = NA/4'A = NB/4 (4.2)

4.2 ChJ4.lrlA

i'a cation or aaion daagling bonds llave been iutroduced (.& + Ja = 2). XAand NB = 8 - Nx are the nxlrnbers of walence eledrons of the atoms i'a the.A.B compound. In the case of cation-term4'nated (100)z?zx2 surfaces one hasa'a opposite ftllsng of tke daagling bonds resulting ia

2D + 8(r?z - D) = 4#cD + 4A(rr& - D). (4.3)Both equations (4.1) and (4.3) give the same res'alt for the n:nrnber of

aimel.s

1 NxD = zn.. (4.4)ï Nh -' 1

For cliamond-stracture crystaks with Nx = 4 no iateger solution caa be fotmd.This is in agreeaent with the fact that missing dimer reconstructioms are

zmt obsemred for C(100)j Si4100) and Ge(l00) sudaces. ln the ease of 111-V semiconductors (NA = 3) the rehtion D = 8a/4 is fltlflled for 2x4and 4x2 rcconstructions with three topmost dsmers. Hdeed, tNe resulting #reconstmzction (4.43,4.541 has been suggested to explain expnrsrnental Mclingsfor the As-ternal'nnnted GaAs(100)2x4 surface. Mfvnwhile, we Hlow that otherreconstmction models are more favorable tsee Fig. 2.19) aad that the idea ofthree topmost As Hsrnea.s has surdved in the rough.e,r 72 struct'are (see Fig.

2.17), where these dlrners are distributed over the flrst a'ad third atomic

layers (4.43,4.554. The 72 structure is more stable th= the .p recomstraction

because of îts lowe,r electrostatic e'aerr lpdaciple 2). '

For II-W semicondactom (4.4) holcls vith NA = 2 and D = m. No topdirner should be removed an.d the smallest possible tlnl't cells shottld be 2x 1

or 1x2. Hdeed, on ZnSe(100) and CdTe(100) surfaces 2x 1 a'n.d c(2 x 2) req-

constructions were obseznred (4.56-4.591. Larger reconstrudions such as theZn-t-rrnsnated ZnSe(100)4x2 one are described by combhed vacaccpdsmer

structares (4.60j. The condition (4.4) of a missing-dsmer teconstruction can

also not be htlAlled for I-WI compotmds LNx = 1). For instancc CuBr(100)surfaces show a c(2 x 2) LEED pattera (4.61). The surface structme is de-scribed by t'wo atozns and vacancies izz tbe top layer of a non-priraitive

c(2x2) lnnl't cell. The total nApnnbe,r of electroms iu the dangling bonds is2 . 2 . Je + 2 . 2 . A = 8. This nthmber seces to inslzre that fottr anion azldcation dangling bonds will be S'LIH and remain empts respedively. There-fore, a genvalized electron cotmting rule is fctlfllled in the s'pirit of Principle1.

4.2 Chains

4.2.1 Zig-zag Chains of Cations aud Anlons

Binary W-LV (SiC), EEI-V azd H-VI compotmd semiconductors, whic,h cry..;-taltize ia the cubic zinc-blende lattice, cleave along (110J planes. 1.zl the bulk

144 4. Reconstrudion Elemenis

E: i'oqk:k$1 t;l ()

gflo 11

JQ

22

: :: :: : Jcl .

: ' : 2: :1 : :1 .

;

: :: '

w :: :: 12 ;: ;

1 -....

)-1-

ds o

E1 1 ()!

. (josljFig. 4.8- Arrangement of atoms at relxved (110)1xl sutfaces of ziuc-blende semi-conductors. The zig-zag c'knsn!c of bonds in the Srst atomic layer are indicated bythiclc lhes. The dotted lines in the top view 5. ve a surface :lnlt cell. The mostimpozant pairamete,rs of the bord-rotatzon/bond-contration model - the rotationangle :zJ tke c'haizz-bunklsng amplittzde zds.k and the bond length ds witht'n the zig-.!! .

zag nhnzns - are show'n in the side view (lower partl. l'ntons: fzlled circles; cations:opem circles. Hatching indicates f1!5ng of dangling bonds with electrons.

of these semiconductozs (110) layers consist of plana,r zig-zag clln.7'nq of al-teznating cations a'n.d. anions along a (ï10) diredion (Fig. 1.6). Ic a blllk--ltlcecovguration eacbv soace atom b.as three nearest neighbors an.d one brokenor dangli'ng bond. Two neilboring atoms possess a bond in suc,h a (110)plxne parallel to a E1îî) or (E1R1 direction. The cation tanionl dangliug bondsarc partially Ved vith A = .&4 (A = 'V ) eledrons.

For (110) surfaces of the zinc-blemde compolmds the genezalized electroncotmtiug rule (Principle 1) can be satisâed without chn.nging the ) x 1 trausla-tional symmetry. The Gazacteristic chxnges of tlze atomic geometzy and thebonding behador with respect to the bttl'k termination e-qn be s'Immadzed bya combined sontf-rotcfion vetaantion model (4.62,4.63) an.d bond-coniraction

4.2 t7%s.lenq :45

reïcaation model (4.64J. In order to reduce the smface energy tBmsic Prin-

ciple) accordu to the bond-rotation relaxation model, at tba surface layerthe n.nson moves away Fom the b'nll- in favor of a p3 bondlng vith the threeneighbozing catioms restûting izl a local pyrnmldal geometry with bond anglestending to 90o (see Fig.3cz, 0a).

The cation iu a surface ''nit cell, on the othez hand., moves into the btllkin favor of au 4J2 bonding with the three neigbboring anlons restllti'ag i'aa tezzdenc.y for a loci pln.nltr geometzy with bozd angles dose io 1200 (seeFig. 3.20b). The corresponding bucldiag of the surface zipzag clmlns' is ae-

companied by d-like erson dangling bonds and p-like cation dangnng bonds.Therefore, in agreement with the dkscllrwqkon i.n Sect. 3.5.3 the n.nion danglingbond îs completely Xled 'witil eleetrons whtle the cation dangling bond re-mn.l'ns empty and, hencev Principlœ 2 and 3 are also f111611ed. Thjs sftuationLs indicated in the lower part of ng.4.8.

'nis picture is con6rrned by t'he measurements of fzlled aud empty sttrfn'ce

states by mp-emK of STM. The comhination of both state lmnges inclicates theformation of zipzag chairts of cations and n.nlons in the FI0) dtredion as

demonstrated in Fig.4.9 for the 1nP(110)1xl sudace (4.654. A czaracteristicfeatttre of the zîg-zag chp.5ns is their lateri extent lzrl in tbe Eû01) direction

(see also Fig.4.8, top view). Jm. the ideal brllkwterrninatect case) Zllj = co/4. Afurther low-ri'ng of the smfaee enerr (Principle 1) may be a consequence of ashozlening of the bonds in the zig-zag chains due to the attraswtive hteractionof catiotus and anions. This restfts in bond-contractfon relaxation. A spatial'represantation of the zipzag cllains is given i.u Fig. 4.10.

a . . . o k t..4qj . .j, %. . .. . ' ,. y, .:t . pj ,..o .. ljk .2 , . k .. xq ) y, .

. .

'. . 'o k-httv. ,vu

.1 k .';kp & ' :rfi ?# :::::/. . .. z .

' ' 'l k2$..k ; 1$. jrlt I . .

' .1 ' C 1)27. . ,: t 14 ki vt)'jr :,.*'! .

'à';/ kii'lë ' ' t't. :; l t m's .1 l k ' ' ' ïl' ' ' 1':h .- ' ! a!h 1 ;i.s iE t-? '

fi Iledd Z rl ; I 1 rl g b O n t

empdangling bond

Fjg. 4.9. Zig-oag chains of empty' and FIII?.II dapgling bonds at the rehxed1'nP(110)1x1 surface as measured by STM for bias wlth opposite signs (4.65q (coppright (2003), wit: permlnsion from Elseviez').

146 4. Reconstruction Elements

Fig. .4.7.0. Schematic indication of a relaxed GaP(11O)lx1 surface with the result-ing zig-zag eahsGoq.

Att key 3tructural paœameters of a relaxed (110) surface are showa inFig. 4.8. Tïe relative displacememt of the cation and acion dtmnes a vertic.alsllear ïza., the b'ankling amplitade, and a corresponding rotation angle Lo illthe (1ï0) plaue. The deviation of the length ds of a cation-anioa bond fzom

the bu)k bond length deq = VVo/4 characterizes the Coulomb interactionalong a zig-zag chain. The z'ig-zag character is governed by the deviation ofzlllj 9om its blllk 'value co/4 (Fig.4.8, top view). In addition, the cationsazd aaio:as in the subsurface layer show a cotmter-relaxation, i.e., cations

taaionsl move away h'om (toward) the bux albeit on a much smaûer scale.For coaveational III-V compounds the vertical separation betwee,n the smfacelayer and the subsurface layer becomes shorter than the (110) interplaaardistance ao/(2W) i'n the bulk.

Values for key parameters of the bond-rotation/bond-contration relax-ation are listed in Table 4.1 togethe.r with the bulk httice constants co andthe charge-azymmetry coeëdeats .ç of the b7llk- bonds. The values are take,ah'om ab initio caktzlations (4.41, 4.66, 4.67) a'n.d measarement of strttduralparameters LEED measurements (4.681. The key parameters indicate thatthe surface rele.xntion is strongly Mueaced by the interplay of covalent andionic 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

(antimozzides, arseaides, aad phosphide-s) ard II-W (ZnS) compotmds are

chazacterized by bond-rotation relaxation with a.n plmost bond-length con-

servation. The layer-tGt anrales ul vazy in a relatively small mterval of 25-320while the elnm'o tilt 11.u. amouats to 0.6-0.8 A. The (11O)1x 1 surfaces of thecompokmds with anions belongiag to the fzst row of the Periodic Table (zzi-trides, SiC) exhibit a mixed bosd-rotatîon/bond-xntrRtion rela=tion. The

4.2 Chstl'nq l47

Table 4.1. Charaderistic pazameters of bulk bonds @o, J) aud geometzy of(110)1x l surfaces tlz-u, az, G/doq - 1) for zinc-blende se 'mlconductoz's. The strac-t'aral parameters are taken from ab jnitio calculatioms (4.41, 4.66, 4.67) or LBEDmeaamements (4.68). The càarge-asymmetry coeëcient .q is taken from (4.26).

Compotud c,o (A) g lza. (A) u; (O) gs/deq - l (%) Re-f.

SiC 4.286 0.475 0-23 15-4 -5.7 (4.66)GVb 6.095 0-169 0.77 30 +0.4 /.68)Insb 6.475 D.230 0-78 28.8 +0.0 14-64Alàs 5.66 0.375 0.65 27.3 +0.D (4.68)GeA,s 5.559 0.316 0.67 30.2 -1.3 (4.411Tn AK 5.861 0.450 0.75 32.0 -1-2 (4.41)AJ.P 5.45 0.425 0.63 25.2 +2.9 (4.68jGaP 5.359 0.371 0.61 29.2 -1.9 (4.414ï:c.p 5-662 0.506 0.67 30.1 -1.8 (4.41)BN 3.60 0.484 0.18 15.7 -7.8 (4.67JAIN 4.34 0.794 0.17 11.6 -3.6 (4.671GKN 4-46 0-780 0.23 14.3 -5-3 (4-67)HN 4.97 0.853 0.24 13.1 -4.9 (4.67)ZnS 5.409 0.673 0-62 25 +2.9 , (4-6$

- 4.

surface neazwt-neighbor bond lengths are shortened by 4-8 % with ropec'tto the bulk bond lertgth, and muc,h srnxllez tilt angles tzg = 12-160 occlzr. Thebuckling ll.z of the surface clmz'ns is reduced to values of about O.2 i.

The key parameters are not independent of each other. Tn fac't

z v 8) i suw,Zlu- = (ds - %/, 2 8)i (1 - sias'l. (4.s)zzll - (ds - ao/

For that zeason we only discuss t'wo pazameters in terms of covalent andionic bondicg in Fig. 4.11. A.s can be seec tzl Fig.4.11a, there is a fairly goodlicear dependence between the bunldsng amplitude ïz-u and the length of thebonds ds in the sudace ahnsns, except maybe for BN. The gene'ral iêea thatthe bunkls'ng amplitude is Gversely proportional to the bonding strenkth iscon6rmed. This relation, howevea., is somewhat moeed by the ionicity ofthe bonds, the atomic sizes and electronic correlation efecwts. One esect ofthe ionic part of the chemical bondimg is demonstrated in Fig. 4.11b. Thestrength of the Cotllomb interaction along the zipzag cahnsns depen.ds on theegective atomic charges Zc/a = ZVA/B - 4(1 :F.t.?) of cations alld aztiorts (4.69j.Thetr diference Zc - Q = 8.: - (Nz - NAI governs the strength of thebond-lenlh contràdion in the cimsnq. There is an Mlrnost linear dependence.The exceptions BN, AIP, and ZnS are probably a consequence of the fact

148 4. Reconstruction Elements

1.0(a)

Q.8 jnxs oO 10I!1 !:.Rw GaV AMs

: o XP0-:< sap zns

= 0 4E

'

.;c

SICoo

0 2 GaN INK' C) o

BN xN

0.01-3 1-5 2-0 2.5 3.0

surpace bond length ds (A)

Dference En ionic charges g-(Na-NA#8

Fig- 4.11- GNrnical trends i.xv stractural parameters of relaxe (110)1x 1 surfacesof ziac-blende Mmiconductors; (a) nbnlu buekling versus mvrflw bond lemgtk; (Xbond-length contrn-ion versus Herence of esective ion c%nrges. Data are takenfrom flrst-prindples calculations 14.41,4.66,4.6% and LEED measurements (4.68).

4.2 Ckains 149

that deformations in a strongly bonded syste,m (in particdarly BN) cost toomuc/ elastic enerr or that the ab icit'iô ionidty scale skhtly overesb'mn.tœthe electron trxnsfer (e.g. in AlPI.

Besides the tendency to end with. completely ftlled or empty dangliugbonds (Prindple 1) or to m'lnirnsze the electrostatic interaction (Principle 2))also tite esed of Principle 3 (lowezing of the snrhce ene,rr by gap opeu-jngl ceaa be explored by studying the atomic rehxation of (110)1>t1 surfaces.Due to the natmal builçin civge trnrRfer beGeen neighbozin: cations azdanions even the ideally tmrml'nated, i.e., llmrelaxed, (11O)1x1 smface

'

of a

ec-blende material remnsns semiconducting (4.701. Two bauds appear ia thefntrtdamentat gap of the projected bllllc bard strucvtme. The lower band As isJmlon-derived az.d therefore Oed. The upper bazd Cz correspon.ds largely to

. X .

>-H

t::i .5 ' 3

BN A0 5

. .'M .. . .V F'

v pi '%' M. ' '''. ...

zy ..- c .w

.6 -. . c k.j/ ' >

'<.w ' ' ' w x .

AlN A,--w 50 - ..> .

. .

m -.-

<* N . . ev. '

..a v. ' > .-NSX .- .

=- . . ' Hn' ..-5>

rp . . xx .

Q) 5 n'=. ''' ''' '-= ''

c . !.

' ...

GaN AS0. ' ''7k. . . .. .

<. .

'ilkr v.. .

-5. mv #

. '

xaqw1!5; ... . l..rxcuuaz='

.. ..-z .

-!...,e ....

... -va'ze

.'.=. u . . .. '==..-.

' ' ' 'xu-.

. . . I n M w . .

: '% p ë M n M 5 5.0

w -. wtu=N=. v'Jv. ' ...5 . .

r X M X r

Fig. 4.12. Occupied (As)and empty (Ca) surface-state-related bands tthicksolid 1inœl of relued l11-N(110)1x1 surfaces wftilinthe projeded btz)k bands'tacture. R'om (4.67).

150 4. Reconstzuction Elements

empty cation d '

bonds. Dlxm'mg relaxatinn and hence the rehybridiza-tion of the dangling bonds, thae brds are displaced toward the bulk valencebands (a%) or bulk conduction bands (Q) aad ' e their orbital charac-ter. TMs situation is displayed in Fig.4.12 for the relred (110)1x1 smfacaof grouml'll aitxides. The ba'ad s'tates As correspond to mostly p-zlce azzioadangling orbitals, whereas the G baud gets a stronger spz-lilqe character. Therelaxation-hduced downward sïfts of the occupied As bands amouat to 0.09(BN)) 0.11 (A1N), 0.24 (GaN), 0.21 (1*) W at tke .j% point of tùe slnrLzteBZ (4.67). This gives rise to a relaxatiomizduced gair of band-structure en-

ergy (3.53). The corrcpoading shifts of the empty G bands are considerablylarget. They are 1.40 (BN), 1.03 (AlN), 1.10 (GaN), and 0.85 (TmN) ev buthave no dired inhuence on the tota) enerr. Howevez, as a consequence theCz band of JnN disappears from tEe Alndamental gap ia Fig.4.12.

4.2.2 Jr-bonded Chnlns

The (111) stzrfaces are the aleamge faces of diamond, Sij and Ge. An idealtrnncation of a buzc crystal perpenclickfaz to a (111) direction would resultin a surface with either one daagling bond (DB) or tltree d=gling bonds persurface-hyer atom depending on whether the trlmcation is made above orbetween one of tàe voup.l'v double 'layers steicked along the special (111jdizection în the bulk (see Fig. 4.5). Correspondingly, one distinguishes be-t'ween SDB and TDB (single and tziple daagling-bond) surfaces. The TDB=face is energeti-lly 1- favorable @n.n the SDB surface, siace the for'merhms three broken bonds while the latter ha,s only one. We therefore disctusstxe SDB surface 1st.

The ideal bonding topoloa of a gzoup-Nlllll surface is displayed izlFig.4.13a. The dangling bonds belong to atoms wllick are second-nearestneighbors iu the bulk. Qnherefore, they elmnot form bonds to gain emergyaccording to Principle 1. Reamugemeat of the atoms seems to be likely, atleast to eVplain the stable (C) or metmstable (Si, Ge) 2x1 rronstruction. Onepossibility is the formation of Evomember aud seven-member rings'in the stu'-face bilayer (see Fig. 4.13b) right panel). This may be induced directly duz'ingcleavage (4.A. Even a special Mnk mcschanlsm lln.q been mzggœted (4.71) .

Suc,h a ring strttctttre cau be nnhieved by displacements of atoms ia a (1ï0)plaae. Norlmlp and Cohen (4.72) have demonstrsted that the new conîjwration caa be obteed in a simple my fz'om the ideal structm'e by applpnga'low-energy shear distortion to the top double layer of atoms. Break-irtg tkebond betweea atoms 1 and 5 (Fig.4.13+ right pl.aell aad forming a Aew

bond betveen atoms 4 and 5 the new covguration in Fig. 4.13b appears.As a resttlt atoms l =d 2 are cow arzanged in the fzrst atomic layer, wbileatoms 3 aad 4 are in the second layer. ne interatomic dis%ances are closeto the bulk bond length, and ennh of the suzface atoms posseses a dangling '

bond. The dcct'ibed displaoamentss pr&erve the zmderlying bulk mirror re-

Nection symmetzy The threefold coordinated atoms 1 and 2 become nearest

4.2 Chzu-nK

Ej 1 f

E1 .1:1

2 42 3 1 3

$ W

(a) 5 6

8 7

2 r4 32 3

1 4(b) 5 6

z B 7

Fig. 4.13. Top and side view of a groumlvtllll sudace: (a) ideal lx 1 coafzg-mation; (b) rr-bonded chaiu model of the 2x l reconstruction. Large open circlesindicate ârst-layer atoms. ,

neighbozs. Their daagling bonds can satmate eac.h other via a rr-bonding-like mechanism accordiug to Pzinciple 1. The surface atoms 1 and 2 are eMzt'h

bonded to two other surface atoms a'nd form zipzag nhxins dtrected alongthe gI10) direction. The resulthg surface structure is itz complete agzeementwsth the reconstrtzcvtion model of Pandey (4.73). A perspedive view of suc,h a(111)2x1 sudace with zr-bonded zipzag chain.s on top and the new bondhgtopoloa is shown in Fig. 4.14.

The aew bonding geometr.y h.ms consequences for the surface electronicstzucture. To gain a cel'tain priacipal tmderstandicg we Vitzme that thesurface-state bands ocolrrlmg in the projected Alndamental gap can be de-scdbed by the interaction of the dangling bonds formhg the cizain and byneglecing their coupling to the bulk. Let sz aad ca be the orbital energiesof the dangling hybrids 4:(m - R - (k) (ï = 1, 2) on atoms 1 and 2. Thetwmdl'mensional Bravais lattice (1X of the (111)2x1 surface and the vectoz'sdi (i = 1, 2) of the atomic basks are iadicated ia Fig.4.15.

Wèthin a nearest-neighbor approximation of the tipt-bindîng description(3.7) one derivel an interatomic hlteraction matrix elemen.t Sta of orbitals1 and 2. Wit: kr = Sz2((d1 1) + Jzblida ()) one has a Fave-vector-dependentmatrix element of the Hn.rnsltonia.a (3.15), S:a (i) = 7S(i), with the gener-nls'zed geometric.al structure fador

4. Reconstrttdion Rlements

Fig. 4.14. Pezspective view of the A-bonded chain mode,l ol a group-Wtllllzx 1surface.

S (/1 ) = - kéeïLd'i

f=l,2

of the r-bonded chni'ns with X = Slatldz' 1)/2 atld 11 + Aa = 1. Dimerizationof the chlu'n bonds, dz # dz (dx' = Jtfz' I), and the bunk-ln'ng nrnplitude ztA =

(dl - dclz of the clminq are generally allowed (see Fig. 4.15). The infuence ofthe dimerizatiou on the absolute valaes of the hteraction between da'agtingbonds has been taken into accolmt by the jeneraWation of the structurefactor (4.6). The diagonitlsmmtion of the resultmg 2x2 problem (3.14) yieldsthe t'wo s'arface bands

1-1 1 2 a jg zs::lc (i) =

j. (sz + sa) + .k. (sz - s2) + 1RI 1S( ) I -

(4.6)

For tke (111)2x 1 surfaces an appropriate Caztesian cnordinate system is

zl l E11R , v1 I Eï101, a'n.d zg I (11lq . The two vectozs chnracterizicg the bonds of an

ideal cha,iu without Hbmerization and bucldiug a-re dïp = co(1, ui:vx 0)/2V&

E5 1 oq

E1 121

q%azW Fig. 4.15. Top view of the 'Jr-

bonded clml'nq on a (111)2x1surface. Bond dimerization or

chain tîlthg îs atlowed.

4 2 Chaizus

=1 -J 1r

- J.x r

ar

-1 (b)

r J K

Fig. 4.16. Schematic representatiox of the sudRe bands (4.7) withln tke gwbondedt+xin model: (a) tilted chains with ideal structme factor but tql'lYea'ent splittings ofthe dangling-bond energies.lal - s2 (/IVI = 0.0 (solid line), 0.1 (dashed line), and0.2 (dotted linel; (b) dimerized chltlns with dxerent interatomic matr!x el-entsAz and .:2 along the czain, ilz - Aa1 = O-0 (solid linel, 0.1 (dashed linel, aud 0.2(dotted line).

A s'trucime fRtor IS@)I = cost/cn/zWl rets. A chaia bu '

parallelto the z-aads does not change the structure fador. Accordiagly, tlze bands are

dispersiortless for k direct#ons parallel to (11Iqj i.e., along X,F and JX- tsee BZirz Fig. 1.22). Stzong b=d dispersion occurs along the chain Aection (ï10j,i.e., along P%ï azd ..FZ i'a the BZ. ne t'wo bards (4.7) are mb=aticallyshown iu Fig.4.16a for several d'l/erences of the daaglizzg-bond ener#es. Forideal rtbna'zus and identical dangling hybdds ihe surface system represents azezogap semiconduckm A d''Ference i:a the dnnrlinr-bond energies, however,opens a'a ene'rgy gap along J# aud, àence, makes the mtrfnre seznicondading.1n. any cmse, the formation of delocalized zr bonds and correspondi'ng zr and')r* bands (4.7) gives rise to azL enera ga,in accordi'ag to Principle 1. A gapopsmsng additionally lowezs the band-strudure eae-rg)r (3.53).

A gap opening along J# due to rllFeren.t danglinpbond enengies is relatedto a loweeg of the cb.ain symmetry iïz the sense of a Jahn-Teller distortion.One possibility is a bunlcling of the eyn.inn as slmwrt in Fig.4.17. The raised

(a) (b)

i At

Fig. 4.17. Side views of zr-bonded cakain struciuzes with positive (a) azld negative(b) burklsng of the Pandey chains.

-A it

154 4, Reoonatruction Elements

3 A

atom of the chaiu again tends to be p coordhated, wherems the loweredatom tends to axt spz coordl'nation. TMs bu ' i'adeed lowers the totalenergy. However, two rlieerent isomel's are possible (4.741. They dz'il'er oalywith respect to the sign of tke buclch'ng (Fig. 4.17), i.p., whether atom J. or2 (F'ig.4.13) is the raixd one. In the original Pandey geometry (4.37, 4.73j ,

there is no buckling at a11, ë.e.; the t'wo uppermost surface atoms in theseveremember rings are at the same heilt. Starting from thâs, one generatesthe t'wo isomers by titting these two atozns ' izl clockwise or anticlockwisedkectiom The two choices in Fig. 4.17 are not equivalent with respect tothe third atomic layer. The istandard' choice is the structure with pcksitivebunk-ling. However, the two isomers are nulrnost degOerate in enera. For

'

Ge(111)2x 1 the chain-leR tor chaizehigh) isomer $.74) semmq to be the moststable geometzy (4.75, 4.761, while for Si(11l)2x1 tEe eae'rr dlFearence tssmaller (4.76). For C(111)2x1 no buckling Ls foa'ad (4.76,4.77) because of theenezgy loss due to strain in the substtrface layezs.

For Si =d.-Ge the bu '

amplitudes az'e large, ztï = 0.2-0.3deq, whereasthe chain bond length dz = d2 is shozïer thnml tke bulk lengtk deq by a fac-tor of about 2W/3 (4.78j. The =ompanying gap openlng is 'isplayed ilzFîg. 4.18 for Si and Ge(111)2x 1 surfaces. The empty and Rled states form-ing the zr* aad rr bauds have deazly bpe.n identled in scazmiag ttmneliagspectra (4.79, 4.80:) whic3 also mclicate a negative bunlcllng for Gc. The ctis-persion of the ban.ds compa'res favorably witk data 9om angtzlar resolved

. photoemn'xsion spedroscopy IARPBSI and k-resolved inverse photoem:'xqionspectroscopy (KRIPES) :4.81-4.842. The cakulated absolute enerr positionand the separation between the two bacds show the tzsual DFT-LDA short-commings whick however, can be resolved by qllxqipartide band-structmecalculations (4.75,4.85,4.86).

Chnn'n tzttug is nmfavorable for diamondtllllzx 1 24.F, 4,784. However,thea'e is another possibility foz a suzface-state gap at the JR line by moclify-

7'' -16, ) $ ''lî%4 rt.- tr ?tk.:4. 4-:$) 't))h 'itrqot. . . ' ,, i4L;1.: L.y jf; . .)/-'h(($ k!.t). f' t . ., I . :.l , ' - f17 <. z ' q' i ttkt' .t xo

..y . . yt.,yy.x'.%.

4q7 '

x0t::::j )t. . l.t(.j(k), )).1--1

, ,: .)j..;t; . :p: I)yh,(t, vjk,z-jtj, .

. .1,;(12/ , . r >)j ,k :$s hj;,, . 4).yI.

$::1 /- lp ik.h :qj,- 78. . p q t4k-tj. .î; . ,) .;j; .

ë' rr r! '-:hk#;-?'ï.' $. #;. ' s

' '.: 't b p'y '

- 't,.-.qF: ' .

t'' - gylhlki . j:zs-tr. .. . - !i,, x-st-titgy,y:sslz.xp,,iygs-ktdqyt.-, .. , ?,),-h... k ,, - . .t .- .

''zthk'- Sttkas'k ,,..,jt.--,'-da.ù,œty--y..y.,- k,lo.,,. ..,,e.-.. -,,, j,..;.,t')',.4 . x.v.bu. s4-k ..,%.n,.s., $-, v-8 ' 5'f '

r J K J T'

7k//*..18 4;. . ...' Eh 1!1j). '

- ' sj , t&' ' . . kr Jiph $ . $rjà' -

k .,4r . :1 sxpX. à . tssyq.-. . l,, .f'' . .

- . ), .A. . y-,- :-q%% SIattjJ tj -Nç;l:'.ky f ,. . jk .. Fxqttt $ -t.

. yj'xlés /:/ ., . Ctjz. @@, s

xy .s-.61- ' ;i3b.. s N

tklr.&. '$(,fl,tt .?a.(... s , ,tk .. J.-j)a.t ,k'p

0s, :;1ï k - IJ .

s. z%à yy!: 1,. v ...

'.'j .. j o , . î.%'.%..%.. k' 'tk h 'kl

--1 ' . .. . x. lb'h 6. - y zo. j).'w+).. ,t-. qt . .,y )$. , .3.. ,v y s v

jtt'i.t, 7 t.' ' 'J. ' ' /.?? '.' ' ttq'' . 'fy .'

, uj . a: . . , ç, q .L..'X.. . sg . ,# y*s. . # As jzliu zv . slllàAsxlssttttllkNllh L. /j-2f' J K J r

piëlt.''l!cèf .. .

';?E@qil)'-. tCiii'''''.f ''

'

.'

':244!1/:--'* .

'lljiqi'.;.f . .(;.,. -.

,'

(21..... .. -. .. ,.

,.-rrjrjr'ljl;:j:rr

tt4 xzbtv' - h sër);. ) %yq kk y z ' '

k

- ' N %'.' ryjj. y.'.,.)y.yj ''vj x

' ' ' j . . : t . jr sj:y) . kt$- bbàvqqbn--bb..b '4k .',k;ru . '.

q$. - kN.i.!..>#;i.;(1t. ,'L -$:? , b.btt. hyk ,y,s . %a(. :--,..,k.f.., ,

:/..) ' '--Nw b' ''s $r.. u.s,y? xtstg- . avjjrt . r;. ); )y jf. -)j4, j!F h' t4. lqqtk

1:E2h yJ.-...s,sxs, krr.)gk)11 ?! , p)y.r.blbbb'''. k!!. -L .(..' p,lrfy 'kt, sll,'.?t:.ilr-?h$-',,-$/sîk$.p,A qîtl'- 'xï,%kï .:-..-'t-!(!:r-k .ii)

!t rlêîd lrt#hl -1F# ïi - '-?ir' dp,' t (It'/ lq'--îLtï*mn ..3.2.:3 lk . . s j))( .'t .

l i'sNrl '.')k;'.h7rt . '>7(1îzf5-? ' ; 4'$?; j)yt a? hy. j . ..r; . . . . . y.lyj j

. ., -. y.z .yyt, ,y,'L :41 ht7kt' ;t. if .1h.y%:;7' TA - '+')k:= 7%. J t: . jtu- .. , , .kci t'vxvvk.,.k. .,t-è '.k. f<. b zA. .y ';k..2 ' h

r J J' r'

Fig. 4.18. Band structures in DFT-LDA of tEe (111)2x1 sarfaces describt.d v'itb.izztbe rr-bonded chaizt model tchain-left isomez) (4.76j.

4.2 ClmlnK 155

ing the strudklre factor. This caa happen by a conjugation of the zr-bondedc'hains ms for hydrocarbons of the type ... =CH-CH=CH-CH=CH-.... Thisis accompanied by a dlmerization aud, hencet fliferent bond lengths dz and d,(see Fig. 4.15). The eFect c,a.rt be described by assuming Alp = j (1 + ,4.'t) irt

.(4.6). The geometrical strucwture factor becomes IS(i)I2 = cpszt%co/zWl +(1A)2 sinzfkvaojzsfk along ?.1 and LS(i)j = gzâà) along <W. The gap open-iug along J'# is given by 2I1AIfr. TMs egect is clearly demonstrated iaFig.4.16b.

For Si ard Ge no indication %r dimezization hms been obsen-ed. In thecmse of tlle C(111)2x 1 surface dirnezization is lmder discussion (4.871. Per-forms'ng grazing-in' cidence X-ray diAactîon studies Huism= et al. (4.88) favortilted chnsmq. Shim-sequenc'y genezation specwtroscopy experiments (4.8Sj e,x-

pla,in the observed selection rule,s by (I':G buckling. Apaz't from only one

reference (4.87j: converged Ast-principles calcalations (4.77, 4.00-4.92) fotmdalmost undsrnerhed a'ad unbuckled -C-C- chains on the diamond surfaceas a minimlnrn stzrface-free-enera- stnzctme. Bucklsng and dimerization ofthe clmsns resttlt i'a a deformntion of the tmderyng seven-member azzd fve-membe,r rings and, hencej increase the elastic energy whic.h reduces the enr

erg.g gain due to reconstmzctton. Ths eFect Ls strongest for diamond. Thewave înnctions resutting for the ideal x-bonded chai'n structtzre are shown inFig. 4.19. In qualitative agreement with the tight-binding estimates (Fjg. 4.16)the accompanying band structttre in Fig.4.18 resaltiug within DFT-LDAindicatœ a sernsrnetattic behavior. There is a need for fuzther studie,s in-

./* 1 U111

*

Or'i--la:l

1

I'ttz:l

(:r-;c2

1 (,1::1

Fig. 4.19. Plots of the square of the wave llnctîon of the 'C-artibonding and gwbonding surface states of C(111)2x1 for a wave vector 0.4 Jt%. The up#er panelsshow ihe projection onto the chain direction and the lower panels a top new of tàesurface c'hain. Lines of equal densit.g are separated by 0.075 boizr-'. Fmm (4.774.

4. Reconstruction Elements

cluding electronic exnhnange and correlation in a bette,r way (4.40) . On theother haad, the data poiats for occupied sttrface states fxom ARPES (4.93qa'nd t'wo-photon PES (4.94) have been shown (4.95J to be i.n good agreementvith the b=d dispeoion derived from DFT-LDA resutts for ideal x-bondedchains (4.771.

4.2.3 Seiwatz Chnlms

Qklite auothe,r 2x1 arrrgement of atoms on the (111) smface of diamond-struckt'are semiconductors mîght be expected if cleavage reslzlts in a tripledangling-bond (TDB) surface. I'n this case the cleavage occurs between twonarrowly spaced layez's (see Fig. 4.5) , izï coatrast to a single dangling-bond(SDB) smface, which sqpazates a widely spved bilayer at wlzic,h the bozldsaz'e orîented exactly i:ct the g111) direction. Ir. the TDB case three boae

E1 r rEl

(b) :.: ':t.l

12,11 n'.I

4E11Q

Fig. 4.20. Formation of Seiwatz cllairts of surface atoms resulting in a 2x1 recon-

struction of a TDB grolzp-lYtllll sarface: (a) ideal TDB surGne (side view); (b)Seiwatz ciminq (ssde '=ewl; and (c) Seiwatz chn.lnq (top 'eiew). .

4.: Ckalns I57

Eig. 4.21. Perspectivc view of the formation of rr-bonded zipzag chains on a tripledangling-bond (114) sudace of a diamond-stmzcure crystal.

are brokem per smface atom (Fig. 4.2Oa). Two of them might saturste whenneighbozing rows of the atoms are moved towards eac,h other tmtil the distanceof atoms of one aad the other row approaches the bullc bond length deq(Figs.4.20b and 4.20c). A perspective view of the resalthg stntctme is shownin Fig.4.21. Jt is consistent with the Seiwatz model of x-bonded c'haizus (4.964.A 2x1 reconstruction occms. The surface atoms form zig-zag czhnt'nq along the(ï10q direction. Two atoms cha,in are paz't of Ae-membered rmgs. Eac,h c'haiïtatom is threefold coorrlinxted. The remaining dn.ngling bonds form zc-bondede'hnsnq similar to that discussed for the Pandey chains on the SDB sttrface.They aze accompanied by 'zr-bondikg and x#-rtibondin.g suzface-state bandsin the ftmdmental gap of the projected bulk band strttcture (4.771 .

Until the discovet'y of the Pandey model the eazly Seiwatz cb.ain model wasused to explain the Si(111)2x1 surface, thougll it did aot completely describethe LEED fndings (4.97). Since 1981 a11 the properties of tke metastable Siand Ge(111)2x 1 sufaces are explained itt the gamework of the x-bondedckaiu model of Pandey. Only i'a the case of diamond are the TDB surfaceazld Seiwatz chnsns still tmder discuasion. Hydrogen ttarmimatîon may play animportant role, in pazticular fo< C'O-grow.a diamond samples. A tllzoe-stepmodel hms been suggested for the chenzisorptjon and desorption of hydro-gen at the C(111) TDB smface, which makes the creation of a correspondsngclean C($11)2x 1 stHace energetically possible (4.984. The outstarading role oîhydrogen is also obvious from the preparation of diamond 61mA svitlnin a CVDprocess. (WxW).R3OO saperstructures have been observed after heats.ag ttp

' (111) surface of polyczystnqll'ne Bl=s. They are related to TDB s-udaces forspnmetr,y reasons. However, STM ha.s also shown the coepdstence of 2x 1 do-maizss with IWXWIJBOO structure.s (4.9% . From the thezmodynamic point.of view the SDB C(111)2x ! surface is euergetically much more favorablethaa the TDB C(111)2x1 surface. Their surface energies dfer by 1.35 evper 1x1 sudace lmit cell (4.774. The TDB smface is not likely to occtlr upon

158 4. Recomstruction Blemexts

cleaxing diamoad bttlk material. For smface,s prepared by CVD growth andsubsequent Annen,ling the situation may be difFeyent, however.

4.3 Dsmers

4.3.1 Sylnlaetric Ilirners

The ideal; bulk-termlnated (100J surfaces of diamond- azd zinc-blende-

stzmcture crystals have a square ttnit cell, with one sarface atom (see Fig. 1.6).Thc seace atozns posseD two dangling .:.p3 hybrids. Eowever, suc,h an ideal

'

structure has not been obsezwed experimentally. Accordâng to the fmdings inSect. 3.2.2, on a (100) surface a dehybridization takes place yieldi:ag a bridg-

ing ppjz orbital ain.d. a dangling spx orbital (3.36) . AccordMg to Principlq 1 thearlmber of dangliag bonds elm be mimimszed by a pairiug mechanism, more

s'tdctly by the formation of dimers witk a bond length Q, as iaclicated ia

Fig. 4.22. Two atoms that aze secozd-neazest neilbors fxom a bulk poin.t of

view approaG each othe,r along (011). They form a cr bond emanatiug fromthe two bridog ozbita,ls Ibr) in between the dz'merized surface atoms an.d

a weaker zr-like bond emanatiug 9om the t'wo danglin.g ozbitals 1d) poHted

away from the dimer. In the orbital representation (3.18) of the sicgle-electron

Hmn'ltoxf.an the t'wo interatoxnic interactions are approvlmately givea byFjl = (brlslbr) = Fm. and Vk = (dlsid) = lz (Wa. +' V'ppxl, where the

matrix elements F'oys (3.21) have to be ta,ken at the dsrner bond length Q.Neglecti.ag deformations of t:e d=gling-bond orbitakq dnlrsng the dimemriza-

tion and interactions with tke surrotmding orbitals) the molecule Hnmiltonian

(3.7) e-n.n be written in the form

00

a'd - s W5Q ca - e

qrlth the intra-atomic matrix elements (3.37). TV resttltiug aatibonding aald

bonding Ievels are ec-jc = ssr + 1$61 1 and s.:r.fr = ca ::: lFk i.

!à ooli

E()r'IlFig. 4.22. Formatîon of 35me,.r boads along (011) on a (100) surface (schemxtically).

we-X'>. -=

x=>

4.3 Dimers l5S

Ga

9 tr

G .x. 616 o.e

V C.i g:%t. 3 .a cbr cur>x WP ;: c% :1 cd

-3 r

Fip 4.23. Molecule levels of a climer of eqcvalent surface Ga,, As or C atoms. Thesf' hybrid energy' of cazbon is taken as zero energy.

The correspondiug molecule levels of isolated Ga, A.s aud C dimers are

represented in Fig. 4.23. The orbital enertes n (Ga) = -11.55 eV) o(Ga) =-5.67 eV, aa(As) s= -18.92 eV: sp(As) = -8.98 eV, ss(C) m -19.38 eV,a'n.d cp(C) = -11.07 ev have been takea from the Solid State Table (4.1004.The interacticm mnnutrix elements (3.22) have been (nlculated with'dlrner bondlensh Q = 2.49 i for Ga on Gn.AK(1O0)2x4 E4.101), Q = 2.50 A for As on

GaAs(1O0)2x4 (4.102j, and Q = 1.37 .1. for C on C(100)2x1 (4.40,4.95j. Theenergetic ordrktg of the dsme'r levels i.n F1g.4.23 and the nlnrnber NA/s ofavazable electrons determsne the level occapation. 1n the crase of a carbondimer on a din.rnondtloolzx 1 surface with Nzvs = 4, each of the c and 'c'

bonding states is occupied with two electroas with spitsap ='d spia-down.Prindple 1 is also G'lfllled vith respect to the passivation of the dazlglingbonds. 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 additionalelectrop NB + 1 = 61 in aveement with the electron cotmting rule (see Sect.4.1.5). The sitaatîon js less cleaœ for Ga dsme'rs because of the high-lyingbridge-bond orbital energy sbr, The absolute level mlues however indicate atendeacy to donate electroas, at least oae, itt sprface As dimers resulting inNx - 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 obtnsned for goup.N matezials by an

arracgeme'nt of dsmers paralle,l to (011j in Fig. 4.22 (mght partl i'a linear c'hainsiong a g0ï1q d/ection. This dimer model was Srst proposed by Srllnller andFhrnqwoz'th (4.103) to explnsn 2x l reconstructions of (lool-oriented surfacesof silicon and germanblrn. It also chazactfm'mes the clean (100) surfaces ofdhmozd. For C(100)2x 1, neighboring surface atoms fo= double-bondeddsrners with a bond length of about 1.37 A (4.40, 4.95j, whiek Ls very closeto the doublmbond length of 1.34 i in, e.g., the Cam molecale. The C=Cdouble .boad is formed by the c- axd 's-like orbita,ls of the C dizner atozns

(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 projected Aln-

:60 4. Reconstruciion Elements

Fig. 4.24. Filled acd empty surface-statebands of C(l00)2x1 in the fandamental gapof the projected bulk baud structure (shadedarea) 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 dispersionof these bands is a consequence of the dimex interaction along (0ï1q, i.e.,along the dimer rows. Such an interaction V.S of two neioboring dangli'ag

qrbitals is smaller th= Vk because of the larger distance. Then approvimatelyVk = 7.1. (QVVc0)2 $4$ 0.37.1. is valid. According to (3.15) the correspondingtight-bindîr.g Hamiltonian is

. cd + ccost:aal'/..u eiEzfavr.k.#,r-,r(:) = -ixtza!z aa + acost:aalp.u (4.9)0 &

with @,z = co/W(O, lc 0) and dd = Qtl, 0, 0) and a new Cartesian coordinatesystem zl Ip11), :1 IE011), and zI1 (100q. 1ts diagonnlszation gives parallel one-

dimeasion.al baads c..y.(i) = cd .k. 2 cost/cvco/v/llvk ;:!z Vkl. They showstrong ctispersion alozg the J'X- and J'15 lines i.n the sttrface BZ (e,f. Fig. 1.22)<th an n.mplitude 47.1., but arc fat along f'J and ./1-.,7/. bbr Q > 0 (thesign of 'Q cxnnot be exmlained witilizz a simple tight-binding model, ratherQ has to be intezweted ay an egective matzix element :'nflaenced by the

enviromment) and l'Z.ul > 27.1. one observes quantitative agrecment with theresults of the DFT-LDA calculations iu F'1g.4,24, The indirect gap betxeenthe #+ bar.d along #T a'ad the 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 inFig.4.25. Their bonflimg or aatibonding nata're is clearly visible.

Symmetric dimez's have not only been obsezved on dixmondtlool surfaces,but also on (100) smfaces of zinc-blende materials. One exampie is thc Si-termiaated surface of cubic silicon carbide (SiC), whic,h besides the c(4x2),3x2, and 5x2 reconstructions also shows a 2x1 reconstruction (4.105). Thebonds are partially ionic. TMs is related to a nhnrge tranqfer from dlrner

Si atoms to C atorms in the second atomic layer, which malcu dsmer buck-ling ltnlskely. J.u addition, when % surface dlmers form, axlgular forces on the

4.3 Dimer: 161

ElKI Ct-----â's

1 p :() 1 1

Fig. 4.25. Sqlrnad wave fn'nctions for the occupied 'r state and the empt,y zrv stateat Ge poht K ir. lhp BZ. of the C(1Q0)2x 1 sudace (4.1041 . A pleme contnsm'ng t:edime.r and the surface normal is used. The contour s'pacing is 0.05 bohr-Y.

*

1) 0û1 .

) q>

: Eol $:1

secondrlayer C atoms are izwolved. They prevent a too strong atomic rear-rangement. Moreover, khe lsttsce const=t of SiC is about 20% smaxer thanthat of silicon. Consmuently, the Si sutface-laye.r atoms move only slightlytovctrds each other (with a dlmer leng'th of ybout da = 2.73 i (4.95, 4.102)ânstead of the atozaic distaucq co/'/f = 3.06 A on the ideal sudace). The di-s-taace of the two paired Si atoms is lazger thaa the typical Si-si bond lengthof abou.t 2.35 i. Therefore, one rnnnot really speak about the forrnntion ofSi sl:rêlme dMers in this case (4.95,4.1064.

The polar (100) surfaces of III-V compounds also tmhibit a dlmer recon-stzuction to hll6ll Peciple 1. For Hstance, As surface dimers are impolta'n.tbuilding bloclcs to e'xphln the various reconstractions of the GaAs(10O) smuface (cf. Fig. 2.17a). The surface dimqrs should be almost tmbuckled to avoidcaharging esects of the surface in contrmst to Prixciple 2. One important ex-ample Ls the 2 x4 reconstrudiop whic,h akso contains the basic unit of the

2

nj>&>PQ)cm0

Fig. 4.26. Surface-state bazlds (solid linœl for the GaAs(001)72(2x4) sarface ver-sus the projecwted bulk band stmcture (gray regfons). From DFT-LDA calcula-tions (4.1O2j ttsing pseudopotentio wltîcèt somewhat open tZe gap.

16t? 4. Reconstraction Elements

c(2x 8) reconstruction. Now, there is ageement (4.55) that the 72 stractme

(Fig. 2.17a) exeplnsns the 2x4 reconstmction appen.nlng kmde,r intermediateAs-ric,lz preparatiou conditions (see Figs. 2.1. 9a, and 2.20a). Under more As-Z'iC,h conditions tke c(4x4) structme is stabilized. For less Xs-ric,h conditiotusthe ('(4x2) structme (Figs. 2.17a a'ad 2.19a) :Ls stabilized. The top As dlmers

of the 72 structttre are @lrnost symmetric, i.e., the bu - ' mplitude is not

larger thn:n 0.02 .l. acd no twisting in the sudace plane occms (4.1û2j. TheHirner bond length of Q = 2.50 i is ver.y close to the bond dkstance of 2.51-t found for bllk- .&s where tEe atozns are threefold coordinated E4.107q.

The /2 structare ia Fig. 2.17a contnsnq two As dinners in each 2x4 mkitcell of the top layer. One pair of Ga atoms is removed from the secondatomic layer, aad aa additional hs dsrner is formed i'a the third layer. Tlle

p2 structm.e 6111611n the electron cotmthg rule (4.4) derived for a missing Asdlmer ia the toplayer in a somewEat generstlsaed sense. Origiaally, there azc 12As dangling bonds with A = 24 electrons and 4 Ga ctaagling bon.ds witlz Jc =24 electrozls. The total 18 electrons allow the complete occupation of the c, '7r,

and x* levelts of the As dimea's, whereas the Ga dangling bonds remaln empty.A semicozducting suzface results (4.102) ms shovn by the b=d stntctm'e in

o o C1 . . C2

@ Q oe Ct t7 o

V1 V2

y,-g. .

y'g. too (g;L- = . '

.Q'73 V4

So Vè % (ï* Q*>. Uca

Fig. 4.277 Cmntoklr plots of the squred wave Glnclions at X for surface states ofthe GaAs(100)72(2x4) stmzcttae as indicated in 57.4.26. The ccmtom spacing is10-3 bohr-3. A)l plots are drawn parallel to the surface normal. C1 is plotted alongthe x4 dizection, i.e., (011). C2, V2, and V4 are at a tomhyer d''me2. V1 and 5J3are shown at the third-layer d''rnea.. Filled (open) smbols iudicate As (Ga) atoms.

4.3 Dime'rs 163

Fig.4.26. TEe smface-state gap is la'rger than the btllk gap hdicating a, gainof batd-structme enera due to recomqtruction. The two lowest empty surfacestates C1 aud C2 as well p:ks the four highest S'II?'CI surface states V1, V2, V3,and V4 at the # poin.t are plotted irl Fig. 4.27. The highes't occupied statesV1 a'ad V2 are related to aatibonding rr' combinations at the topmost aadthird-layer As dl'mers. The states V3 aud V4 represOt bondiug combinations.The lowest empty state C1 is related to Ga daagliag bonds. Eowever) the nex'tone, C29 is mnsnly built by antsbondisg c* combinations at top As dsrners.For the GaXs(100)((4x2) s'tzrface the b=d stradttre and the djscvion ofthe orbital character of the surface bacds cau be fotmd in Ref. (4.*54.

4.3.2 Jksylnlnetric Ilitaers

In. contrat to C(1.00), for Si(100) a large nttrnber of reconstrudions hnA beeafouzd. The most impoztant ones are the c(4x2) a'nd p(2x2) reconstnzctionsobsezved at 1ow temperatmes, and the 2x 1 stzazdlzre at room tempezatme.The revezsible c(4x2) -F 2x 1 pitase trn.nKition takes place at a'rolmd 200 K.The Ge(100) surface shows siznilar behavior 24.28, 4.311. The large,r recon-

structions shoutd be possible for crystals vith covalent bonds weaker thautimse iu diamoad. J.n such sysorns the surface energy may be further loweredby allowing the d'lmers to buckle o'at of the sT:rf'nce plaae (4.47j or to Mstthe dime.r azs in the surface plane (4.7.081. The c-haractezistlc parameters are

zîb a'ad dk (see Fig. 4.28). The accompaayiug energy gain is larger than thecorrespoacling loss due to the elastic energy of deformatloa.

The driving force for suc,h additional atomic rearrangements is rehtedto Principle 3. For symmetric Si dinners the surfacomormat dangling-bondorbitals form a nearly degenerate pair of rr acd x* bands. A Jn.lnn-rfeller-likedistortîon is expected to open a surface gap between the 'r. and =. bam.ds(4.47, 4.9,$. Totat-energy calciations (4.95, 4.1091 indicate a bunldsng of the

:511 El Df!

- E01 f , :1 fFig. 4.28. Top and. ssde Wew of an msymmetric tî''=e.r geometzy Cbnractedsticlengths suc,h as the dimer bond lengtà Q: the bu#ing amplitude ïs parallel to(100), and the twisting nmplitude ztk parallel to (011) are indicattd.

du 4-àb

T

164 4. Reconstruction Plements

Fig. 4.29. S'urface electronic states itl the bttlk b=d gàp reglon of the Si andGe(100)2x 1 stracture within the asymmetric dime.r model. The DFT-LDA bandsare taken from (4.7$.

dirners but no twisting, i.e., lt = 0.'-1n accord=ce with the discuession inScc't. 3.5, one of the asmer atozns is pushed away from the surface, becorningmoTe pyrnmz'dal (p3-1ilcel bonded with a fdled s-lilce dangling bond Dup. Theother dimer' atom is pushed inward into a nearly planaz (spz-like bonded)cooguration. The accompanying pa-lilce daagling bon.d Ddonm donates itselectron. The restzltîng dimer ha,s a weake,r bond with a:a iucrease of the boadlength from Q = 2.20 i (symmetric Si di'rnea'l to Q = 2.25 k (asymmetric Sidimer). 1ts veztical buckqlng is ztlla = 0.6 A, malcing the tilt angle nearly 169.The xsymmetric dimer geometry results in an energy gsin of approxn'mately0.12 ev per dlrner (4.10% . F1:11re4.29 clearly shows the exmected band-gapopening between the surface bands bui)t mainly by Ddown or Dup orbitals.

The eHstence of azymmetric dimezs easily e-xplains lazger reconstrttctioassuc,h as c(4x2) snd p(2x2) as dlferent covgurations of left- aud right-tilteddime> (see Fig.4.30) . They give rise to small additional total-energy gains of0.05 ev per dimer hom knte-racwtions between naighboring dimers in a dsmer

d 0.003 ev per dimer â'om iuteraetions of nearest dimers in Vo aeigh-rOW an

boring dimer rows (4.1101. It turns out that the smface eneror is minimalif tbe Jimer-tilt direction altnrnates along and is perpenclictzlar to the dime,r

Tom. The resulting c(4x 2) reconstradion is indeed observed at 1ow tempemstures.

Despite ex-tensive expmrimental aad theoretical bwestigatioa, the natureof the reconstvction of the Si and Ge(10O) surfaces is still subject to debate.This holds in pazticular for the asymmetrjr of the Si dimers studied by me.ans

of STM. In early STM expe-riments the dirnezs appeared to be symhetric

4.3 Dimers 165

yX1

l'ï j j(2x2)P

l 11 d1 1.1 1

$'

1.p I

y'

11

y' ê'.y'

I

p'

II II 11 1

I 1I 1I 1I 1

Fig. 4-30- Surface unit cells of variotus reco>trudions of (100) sudaces. Theformxtîon of dimers is indicated by horlzontal lines. TEe lazge empty and shadedc'ircles indicate the inequivalent tomlayer atoms, e.g., the Cup' and sdown' atoms i'a

an aymmetric dimer. Dots represent second-layer atorns.

(4.111j. Tlzis experimental fmdsng, however, is not necessmrlly ilz coatradictiouto msyrnmetric dsrners. Studies of the dynsxnl'cs of the sttrface have shown

that the dlrnez's may Sip betweea their opposite tûtiug diredions (4.112,4.1132. At room temperabme, thia dimer Nipping happens on a t.l'me scale of10-10 -10-Ss. Most meaemements do not have the respective time resolution.

At room temperature, most dimers appear symmetric due to tb.ei:r dplnrnlcflipping motion, ar.d only dlme,rs close to defects are plrned in a buckledcoMgaration E4.114). The nlrrnbe,r of symmetric dz'mez's decreazes below 129 Kacld dirrlers buckle alternately witkin eac,h row, witit the formation of p(2x2)âztd c(4x2) domains correspon8ing to adjacent rows inu idezztkal or opposite

ozientations (4.114, 4.115J.

166 4. Reconstruction Elements

Thîs picture has recently been chatlenged. A series of low-temperatureSTM studies b.a's been reported that, while the c(4x2) structme is obsezwedbelow 120 K, farthe,r cooling below 20 K causc the dimers to again appearsymmetric. It llas been argtted (4.116) tha't. this is a drmmical phenomenoncaused by a loweeg of the potemtial energy barzie,r betxreen the t'wo backledcougurations, which allows the dl'rnez's to resllme the Eipdop motion czhar-acteristic of room temperatmz. Other authozz (4.117) claim that the observedsymmetz'ic dirners are static since their images d.o aot exhibit the noise associ-ated svitb. the Sipping motion obsezwed ita the same sample at 11O K. Howevcrlthere is also evidence for a sigocant imAuence of tlxnneling processes (4.118j.

To cla'rpy the low-temperatme bu '

behaWor 1om a theoretical poin.tof view, the subtle apects of electronic conelation near surfaces have tobe taken izzto consideratiom The syznmetric dimer with two g' dangling hy-brids, eac,h fllled with one electroa, represents a biradical state. Therefore,static cozbrelation (perhaps, beyozd LDA and includicg gradient corrections,see Sect. 3.4.1) must be properly talcen into accolmt. The clime,r bu '

isaccompeed by electron tzansfer. Thus, the net stabilit.y of the buckled con-fguration depen.ds on the devee to wilic,h tite repulsive Coulomb interaction(cf. Sect. 3.5) of the two electrons in the apper Si daazgling bonds is dynam-ically screened.. To accoaat for this dpmmical correlation esect, qttnantllmMonte Carlo studies have been perfo' rmed E4.119J. However, they qualita-ttveiy gave the same favorization of the bttcltled couguration as DFT-LDAor DFT-GGA calculatiou,

4.3.3 Hetetodkmers

On the GaAs(001) surface b0th. Ga and As dsrners f.t geometrically to theunderlyiag bulk. The covalen.t radii 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 mivedJlrner or heterodirne.r should ft muc,h bette.r to the bulk. Consequently, an

In-P e-xchange shoald be energeticluly favorable. TMs ide.a has been usedby Scbmsdt et al. (4.120, 4.1211 to explain the 2x4 reconstruction occlzrrlngi'a a large range of In-zic,'it prepration conditions tsee Fio. 2.19 and 2.20).Heterodirnezs akso appear to play a'n. important role i:a Imdeation a'ad growthon 1DSb(001) sttrfnce.s (4.122) as well as for submonolaye,r gzowtll of Ge on

Si(OO1) :4.1231.The suggested model for the 1nP(001)2x4 recortraction is shown in

Fig. 4.31. The sttdace is ternninnted by a complete In layer. ln tlle ideal case,eac,h 1'n. atom possesses t'wo dangling .6p3 hybl'ids along (111) and gïï1) with afraswtiom.al 6115ng Je = Qu

. The corresponding 2D square Bravais lattice is givenby a1 = .g-o.a (1, -1, 0) and s,c = e.z (1, 1, 0). A mived In-P dirner on a 2x4 celladds eigb.t clectrons to the available 12 electrons in the 16 Iu dazzgli'ag bondsaud 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 167

(110qI -' '

Eig. 4.31.. Top and side view of t;e rnlxed-dsrne,r model of the TnP(001)2x4 suz-face. Emqty tslledl cirole's represemt ln (P) atoxns. Large (srnM1) smbols indicatepositions m tàe &st aad second (thizd and fourth) atomic layers.

dangliug bonds at the I'n atoms being fa'r away from the heterodimer remainempty. Two electrons occupy the c' bonding orbital of J.n and P states ofthe buckled msved dimer. The remnsnn'mg two electrons Sl the e-lilte dattglingorbital of the P atom in the topmost heterodlrner. One may' conclude thatall reconstruction principles, Prvciple 1, 2, and 3, aqre A116l1ed. Consequently,this is also tzme for the electron counting rtzle in a geueralizect sense.

Total-emerg.y optsmizations witlnl'n a Mt-principles appronnb (4.120,4.121)com6rm the above considerations. The bond length Q = 2.57 i of the mivedfimer is close to, the sum of covalent radii 'ru + 'rp. TH dimer is bucldedwitiz an angle of j.7b. The ln atom is 0.43 i close,r to the substrate than theP atom. Ss=l'ln.r results are obtained for the mlxed-clime.r reconstractioa oftïe GaP(OO1)2x4 suzface (4.124) . The Ga-P dlmer is characterized by a bond

V1 V2

Fîg. 4.32. Contour plots of the squm'ed wave functions at R for the two highœtoccupied surface states of the mixed-dimer reconstrudion of ïnP(001)2x4. Theplots are clrawn in (001) planes 0.8 â below and 0.8 .â. above the uppemnost Patom (fzlled circle), respectively.

168 4. Recoustrudzio'a Elements

&'L

:

E1 10J

1nP GaP(2x4) Mixed Dimer (2x4) Mixed Dimer

ij)!!

. .- . i 7 ..' p:r ; ( 'E 'j . J ', t l l l )'

E: k L... î ,j: ? ... ,-' Ct.t,. y,;;r .... :

y y. : jj . ; .... !.. ,.,.,;j,,111121/ ::5.: 'q:.. .k ):.; .j . .).éq ,':LjL.fj4j.LL. ï ).,'c: è I : ;..

l <'%<'%ro jf8 k =v- t R

J(l1

. .'. L??(.j).: (' ?.jjî .:.( ': j

u ':i' .; n ::.!'ti. ; . ï),i ,.,...

. .k qq .q' 'i'!!.t ?..:. ; .L; ,; $!..- ;; k...@.- (). L, , ;ïi: .f j . . .

J1.66 nm 1.53 nm

Fig. 4.33. Filled state STM t'mnges calculated for Ir.P(001)2x4 and GaP(001)2x4in the mived-dsmer reconstruction using DFT-LDA (4.124J.

length Q = 2.36 i ('rca + rp = 2.32 i) and a bucldsng angle of 9.50. The Patom is 0.39 i above the Ga atom.

The resulting electroMc stractme is that of a, smmiconduding surface(Principle 3) (4.1202. The occupied surface states are below the mlencobandmaximlpnn in tlze b'Ilk. This fact infiicates-'-a gain of band-stmzcture energ.ydue to reconstrudion. A variety of surface-state bands occurs iu the pro-jechted G7ndn.mental gap close to the condaction-band edges. Most interestingaze the two uppermost occupîed szzrface states at # in Fig. 4.32. They areO.1 and 0.4 ev below the blllk- valence-band maxkrptm. One is related to thec-like bonds between the top In atom and the two cations below. The otherone represents the dangling bond of tlte top P atom. The lateraz positions ofthe resalting three msvlrna in the electron density due to thœe states forman isosceles triangle, ia excelien.t agreement with the corrugation measlzredby STM (4.121,4.124,4.125j. Theze is no need for an asslTmption of P trimerson top of the s'trfnne as ori '

y suggested to explain the STM images. TheSTM images obsezwed for negative bias ca,n be simulated assllrnlng a mixedHirner as demomqtratcd in Fig. 4.33. The single triangles observed consist of acenter struct'are (lhead') and two side stractmes (1e=7), the origin of whic.his shovm in Fig.4.32.

4.3.4 Bridging Groups

A nlnrnbe,r of stractm'al models has been suggested for tîe Gterrnsnnted po:1= sarface of cubic SiC, SiC(0O1) or actuazy SiC(0Oï) (4.95, 4.105, 4.1261.Because of the strong C=C double bonds (see Sect, 4.3.1) oae mlty expecta dimer reconstruction, in particular to explaia the obsezwed c(2x2) trax)s-lational symmetzy Jt may be desczibed by a staggered arraagemect of the

4.2 DMers :69

Fig. 4.34. Top 'dew of the C-term''nated SiC(001)c(2x2) surface: (a) staggereddimer geometzy; (b) staggemd bridog group arrangement. bqull (o?en) circles 1n-dicate C (Si) atoms. The squares of dotted lines zive tke surface lnn:t cells.

C dn'rnez's (se! Fig. 4.34a). Despite tlle ver,y dîffe-rent httice constants of SiC(deq = 1.88 A) aud dinmond tdeq = 1.54 i) (4.1001, C dimers with a C dou-ble bond a'nd a bond lezlgth of about Q = 1.35 .t$ (4.95,4.126j can be eazilyformed. However, because of the large SiC lattice constant a complete reac-

rangement of the top carbon layer is also possible az indicated i'a Fig. 4.34b.Instead of au azrangement of >C=C< dt'rners on squares of four s'ubsurface Siatoms, bridging Ca groups, -CHC-, in perpendictf.ar direction between t'woSi atoms are possible. The bond lengtil of thcse bridging C pairs is about 1.23i. This idca is consistent with the obseawation of triple-bolded C atoms inorganic chernsqtry. It seems to be in agreement with a LEED =alysis (4.127Jand spectroscopic EndA'ngs (4.1281. The formation of the anomalous bridge-bonded dimers is 5.11 agreement with the recortstnuction Principle 1 and theelectron counting rale. Because of the triple bonds, all surface cazbon atomsare fourfold coordinated. The second-layer Si atoms are atso paired in a direc-tion parallel to the bridging voups (see Fig. 4.34). Tkeir remniming danglingbozzds form a'a occupied bondlng orbstal.

From the point of view of the reconstradion-induced energy gnn'rl one can-

not favor the acetylenc-like bridge,s -CHC- versus the ethylene-like >C=C<dimers. The corzesponding values are aearly the same kdepçndveat of the de-tails of the ftrst-prindples calclllntiozus (4.95,4,1061 4.1264. However, there are

drxmatic dsferences i'n the resiting surface electronic strucvtme. Sttrprisingly,the staggered dimers give rise to a nazrow-semiconductor bazd structtuewith a.a e-xtremely small gap of about 0.033 ev (4.126). The gap between thesurface-state bands in the projected f'nndnmental gap is remarkably openedfor the triple-bonded Cc bridges abeady witl:dn the DFT-LDA. The hlnd.a-mental gap is n.lmost h'ee from sudace states. This fad is in good sgreementMdth photoemission measmements ac.d STM studies (4.128, 4.129) as weX as

the reconstnzdioa Principle 3.

17O 4. Reconstrueion Elements

4.4 Adatoms aud Adclusters

4.4.1 Lsolated Adatoms

Siugle danglicg-bond (111) suzfaces .of diamond-strttctm.e crystals, polar

(111) surfaccs of zinc-blende matea'ials as well a,s the (0û01) and. (0û01) sar-

faces of wultzite stntdures (see Fig. 1.6) possess one dangliug bond parallelto the surface normal at eack surface atom. The same situation holds not onlyfor the (llll-sarfaces of cubic 3C-SiC bat also for the (0001) and (000ï) sur-

faces of the hexago:aat polytypes 411 or 611 of SiC. ln the polytype case onlythe bond stacldng iu deeper subsurface bilayers is moeed (4.130q. The sur-

face atoms are second-neazest neighbors 9om the bulk point of view. There-fore, they wranldy interact =.d. thus cau only be satmated aaer remarkablereazzangements of the surface atomic geometry (see Sect. 4.2.2).

A minirnization of the number of dangling bonds (Peciple 1) e>m beïmbfeved easily, however, jtzst by adding a threefold coord-lnxted adatom,which co'ald be a,n atom chemlcatly identical with a blllk spedes or a sub-s'titutional atom. Sucll adatoms on (111) aud (0001) surfaces may occupytwo types of sites wlzie.h are illustrated i'a Fig.4.35. These geometries a're

distinguished ms hollow (Ha) aud atop sites (T4) depending oa whether thesubs-trate atom bebw the adatom is fotmd i'a the fout'th or secoad atomiclayer, respectively. i:a eacZ of the geometriez the adatom is in a threefoldt3l-symmetric site. J.nu the T4 case the adatom a'ad the second-laye,r atom bdow

' wemkly interact what gives neady a foulfoldtzl) coordiuatioa. I'a the geometrySs the adatom occmpies a substitutional site in the second atomic layer be-neath a T4 adatom of a bulk constitaent. lt possesses four neares't neighbors

Fig. 4.35. Adatozns on (111) surfaces of zinc-blende or diamond czystals i'a Ta,Ha, aud Ss sites. Adatoms are shaded. Top (a) aud side (b) 'dews are showa. Thecoorasnate systems are given in Fig. 4.l3a.

4.4 Adatoms attd Adclttstes's

but is also Huenced by a flfth (for that reason, index 5) atom :in the T4position. The illustration of the adsorption sites in F1g.4.35 is not only validfor (111) surfaœs. ln tke case of (tooly-oriented 4.11 and 611 polytres one

may expect tie same st '

of the bonds izt the appermost two bilayezs as

shown in the fgure.The Ss geometry Ls only pbservable for adatoms chnmle-q.lly diserent

9om the substrate atoms. Typical esxamples for the renalization of Ss are

B-induced (WxW).R30O recomstruc-tiozls on Si(111) surfaces (4.315 aad ozl

Si-termlrnted SiC(111) surfaces (4.131). J.n the pme silicon case the incor-poration may be considered as a'a exclnn.nge reaction. The adsorption ofother group-lll elements Ak Ga) a'ad Ia lcads also to the formatioa of

(WxW)-R30Q stractmu. Shce they are tzivalent, each group-/l atom sat-tzrates three dangling boncls of the group.N atoms and, hence, leads tosurface passimtiom The T4 site is favored. This pictme of adsozption alsoremains valid for constituent atoms. Total e'nergy studies of the gcoumWadsorption oa 1V(111) surfaces witbl'n the (Wx W).R3OO tmnKlational sm-metzy (4.130, 4.132-4.136) show that the T4 adsorbate site is mue-h favoredover the Hz site. Despite usicg diserent calculational approaches, there isagxeement for Si and Ge that adatom Vqorption in a T4 site is exothermkwith respect to the relaxed (111)1x 1 smface. This is in contrmst to diamond

(111) smfaces on whic,h adatoms are clearly energetically llnfavorable.On the other handj the Si and Ge(111) (W x W).R30O T4 adatom geome-

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 violatesPriaciple 3 and the electron couting rule. Moreover, r.% adatoms on (111)2x2cells shottld be energetically more favorable for Si a'ad. Ge (see Sect. 4.4.2).However: there is a mystery about the widmband-gap semiconductor SiC. Thesame type of, violations occm.s for the Si-termsnated SiC(111)(WxW)&0O

f the si-terrnsnated skc(1:1) (W3xW3)RV 0* surfaceF'ig. 4.36. Perspective view o

with si T4 adatoms. si (c) atoms are uicated by open (f:1led) circles.

E1 1 11

(1-:21E1 l'oq

172 4. Reconstruction Blements

>*

>P*

m

2.5

2.0

j X **>.. .**+Mw '** ' *œ v ... . *

wwus +.% >

1.00-50-0

-0.5

-1 .0 ..

l

-1.6 -. - ..- -

F M K F

Fig. 4.37. Ezectronic structure of the 3G.SiC(11))(WxW)&02 sttrface witht'nDFT-LDA (4.139). The projected bulk band structtlre is shown az the shaded region.The dotted line representas the Si danryling-bond-dedved band.

surface taad %r the corresponding (0001) ones of the 411 and 6H polytypes).Nevelheless, tEe L adatom geometry shown iu Fig. 4.36 gives a stable re-constructiou for not too Si-ric.h preparation conditions (4.130, 4.1364. TMSresult is iu agreement with othe.r total-enerq cakulations and e'xpezimentaldata meastlred for the (Wx W).R3OQ recdnstruction of 3&SiC(1l1), 6H-SiC(0001): an.d 4H-SiC(0001) s'urfaces (see also (4.95)).

The stkrfaze b=d structures of the Si(T4) adatom (Wx W)S30O ge-ometries of the Si-tcrminated 3C-SiC(111), 61FSiC(00û1) and 4H-KC(00O1)suzfaces possess a dangling-bond-related half-oed bar.d in the AJndamentalgsp, at least withim the DFT-EDA /4.137-4.139). One example is shown izlF1g.4.37 for the 3C,-SiC(l11) (W x WIA300 sudace. 55='2a,r ban.d stmzdaresare obteed in the cmse of the hexagonal polytype,s 611 and 4H. Only theprolected fttmdamental gap is videned by about 1 eV. The theoretical re-sults obtained witlnl'n DFT-LDA are in disaghree-memt with a comblation ofARPES (4.140) and KRIPES $.1414 inve-stigatiomq as well ms scnnml'ng t=-nelîng spectroscopy (4.:421. For the hexagonal polytrpes the expem'ments'indicate b0th a'n empty aad a ftlled danglinpbond b=d separated by a

tate g-ap of about 2 eV. The discreparcy is solveé by the mss:nrnmsurface-stion of a, Mott-Hubbazd insulator golmd s'tate of the SiC(Wx WIS3OQ sur-

faces (see Sed. 5.4.2) with a.c on-szte Coulomb Gteraction paramete,r U rz 2ev (4.137, 4.139, 4.1431. This is in agreement with Peciple 3.

4.4:2 Adatoms AccompYed by Rest Atoms

Although e,azh adatom reduces the dangling-bond density Vcording to Pri'n.-ciple 1, tbe (WxW)J?,30D Tecoastruction of group-lvtllll sufaces csmmotG11611 Priaciple 3 i'a tlze original sense. However, this principle fxltn be obeyed

4.4 Adatoms and Adclcsters

(b)

Fig. 4.38. Adztoms in T4 (a) and Ha (b) sites formsng 2x2 lattices on W(1:1)surfaces. The atoms are kdicated by circles (fzrst layer, adatom) cir dots (secondlayer). Adatoms are shaded, rest atoms aze hatched. Rectaurllar (Nmcagonal) cel'ls

are iadkated by thizz solid (dotted) lizïes.

considering a, 2x2 reconstlmcioruH tMs cmse one adatom occurs for eveay fot:rsarface atoms accomrnodating 75% of the broken bonds =d, conseqaently,

leaving aa adatom density correspondlng to .24 of tha original sarface. Thepm'rnl'tive cells have four times the azea of tke ttnrecozlstructed lx1 smface.There are two suc,h coverings, one with hexagonal syznmetl'y, the 2x2 stlr-

face, aud one with rectangular symmetry, the c(2x4) reconstractiom 80thcell types are hdicated iu Fig. 4.38. The hexagoaal cells an.d the T4 adatom

pocitions usually Tve zise to a lower surface enero'. The essential mechanispof gaining enwetrgjr is however indepeudeat of the cell shape. 1'a tke 2x2 case

the adatom bonds to three of the surface atoms, leaving one atom with a

dangling boudu knovn as the rest.atom. For Sî(111) and Ge(111) the adatomis displaced towaœd tlle btttlc, whereas the rest atom tends to be pobonded

and., hence, is displaced away from the bulk. Tlle dangling bond becomesmore p-like (&1ikel for the adatom (rest atom). TV rearraugement is ac-

companied by an electron tr=sfer from the adatom to the rest atom forming

Table d.2. Recoastmtction-induced enera gaizz in eV/lx 1 cell with respect tothe clearz relaxed smface for the 2x1 x-bonded chain modek 2x2 Hatom model

(hexagonal cell), the c(2x8) adatom model (4.38,4.39): and. the 717 dimer-adatom-stacking fault model (4.144,4.145) from a: izkitio dezuszty Glnctioual calculatioms.'

Surface 2x 1 2x2 c(2x8) 7x7

C(l11) 0.80 (4.76) -0.18 (4.76) -0.23 (4-76)0.83 (4.136) -0-10 (4.135)

Si(111) 0.24 (4.764 0.27 (4-76) 0.30 (4-7$0.28 (4.1341 0.27 (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

4. Reconstraction Elements;'

a lone pair in the daagiing-bond orbîtal of the rest atom. The resultiug sar-face geometzy fnllfllls Pzinciples 1 aud 3. A surface-state gap appears (4.134).The accompanying lowezing of the sl4rfnne smera comes in the range of valuescalclzlated for the x-bozded c'hn.in reconstrudioa (cf. Tab1e4.2).

Only for dlamoadtllll are adatoms less favorable because of tlle strongercarboa bonds in the bulk. Consequently, the 'r-bonded e'hain reconstructionrepreents the grouxd state of the C(111)2x 1 smface.

Otae caa gemerate other coverings of the adatom density iu ? such as c(2 x 8),

by decorating larger pm'mz'tive cells. Suc,h aa exmmple witlz T4 adatoms isshown in Fig.4.39. The two pairs of adatom and rest atom pe,r 'nn:'t cell ogernew degrees of freedom, which lower the total energy. On Ge(111)c(2x8),

. ab iaitio calculations on the simple adatom model have indicated that themsymmetzy of protrusions fotmd in the STM srnn.ges is mxsnly e'atzsed by the

:1 'tlll

p tol

Fig. 4-39. Atoznic structtu'e of a group-IV(111)c(2x8) surface (top view). Thesimple adatom model Ls represented. The atoms are indicated by cirdes t5.rst layer)or dots (second layer). Adatoms (on top) are shaded while rest atoms tin Erst Layer)are hatched. Possible 2x8 and c(2x8) zlnlt cells m'e demoted by dotted lines.

4.4 Adatoms and Adclustel's 175

Fig. 4.40. Sm'face b=d structures versus high-symmetry direc-tions in the 2D BZfor reconstruded (111)c(2x8) surfaces (simple adatom model) of C, Si, acd Ge.The shaded areas represemt the projected bullc band structures. From (4.76;.

bu '

between the t'wo rest atoms in the llznit cell (4.39j. The bualdlngmakes a 2x2 subnlnst more electron-ric,h thn.n tEe corresponds'ng c(2x4) sub-ltmt't ia the c(2x8) recozlstruction. Thks stabiGcs the c(2x8) trauslationalsymmetzy ove'r the 2x2 a'ad c(2 x 4) suzfaces according to Principle 2. Tllebunkllng of the rest atoms also happens i:a the c(2x8) orderiug on the Si(111)surface (4.76, 4.146). The chn.racteristic energy gn.s'ns in Table 4.2 clearly in-dicate an (even bette'r) stabilization of relaxed Si(111) and Ge(111) surfacesby adatoms in comparison wxith the formation of zr-bonded chalns. Tllis is izl

t1) '

dear contrut to dixrnondtl .

A'n idea about the energy gytlns due to a, c(2x8) adatom reconstnzctionis indicated in Fig. 4.40. The C(111)c(2x8) surface violates Prhciple 3. Incontrast, surface-state gaps al-e opened for Si a'nd Ge(111)c(2x8) . However,in the Ge càse the occupied surface states are much lower i'a enerr tha,nthe valence-band mnv'=:lm. Therefore, for Ge more baad-stmzcture energy isgained than i.n the Si cmse. More precisely, the adatom-induced elastic energy

(a) (b). G .

f'/e F qsl( 1 1 1 ) J' .a o . . . . . 0.4 CED co1- : ..

s:'s

. . . : . . .z (11

e

Fig. 4-41. Contpm plots of surface s'tates of tke adatom-covered Ge(111)c(2x8)sttrface in tke (110) plane contaMng a.u adatom aud a rest atPz-ty (a) state of tizelowest empty sarface band (Fig. 4.40) at a wave vector on thq CY line; (b) state ofthe higllest ocompied surface band. at a wave vector on the rF/ lice (Fiz.4.40) inthe smface BZ. The distance of the Ge atoms to the piane of the adatom and restatom is mclicated by the varying size of the dots. Fcom (4.78J.

176 4. Reconstrudion Elements

is obviously overcompensated more strongly by lowerirtg the band-stnzctureeaergy due to tEe redvuctioxz of the density of daagling bords'in the Ge case.

The orbitals beloaging to the lowe-st empt.g smface bacd and Mghest oc-cupied surface ban.d are plotted iu Fig. 4.41 for the Ge surface. Indeed, asdiscussed above for the 2x2 adatom situation, the wave ftlnction of the high.-est occupied band is mnr'nly localized at a rest atom (Fig.4.4lb).'The snme

holds for the localization at the adatom of the wave Atmction belongîng to thelowœt empt.y surface baud (Fîg. 4.41a). However, there are also contdbutîonsfrom the neighboring atoms.

4.4.3 Adatomm Combined with Other Reconstmzction Elements

The energy gains in Table 4.2 show that the grotmd state of the Sî(111) sur-face corresponds to a huge 7x7 reconstmzction. After long and controversialdisputes it is nowadays explained by a dsrneoadatom-stacldng fault (DAS)model with corner holes (4.144, 4.145) as represented izt Fig.4.42. Each 7x 7unit mesh contn.ins (i) a stanlrimg fault in one of its triangular subunits, (ii)a corner hole corresponding to one mlrwqiag atom i'a the second atomic layerand, hence, leaving a dvgling bond at its center atom ill-the third atomiclaye'r, (iii) aine dimers forrnsng domain walls abng the botmdar.g of one of itst'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 ofwlkic,h are not saturated by bonding to the adatoms. Consequently, 42 atoznsremah in the flrst atomic layea', among them the rest atoms, 48 atoms are iathe layer beneath, a'ad 12 adatoms i.n the top layer decorate a surface tmitcell.

Tlle DAS model also follows Pzinciple !. The 7x 7 surface mixtimizes itsdangling-bond densit.g by the formation of dimer-row dornnsn. watls whiczhare euergedcally favorable because of the relatively 1ow energies of stackinpfault au.d corner-hole formation. The decoration with 12 adatoms passimtes36 d=gling bonds in the ftrst atomic laye'r. 12 dangling bonds remain atthe adatoms, sbc more at the rest atoms, and one at the center atom ofthe corner hole. However; furthe,r dangling-bond saturation happens via thebunkling/charge-trxnsfcr mechaaism disccsed in Sect. 3.5 in accordance withPriuciple &. Danglittg bonds locsll-zed at rest atoms acd cqrner-hole atomscorssidera' b1y contzibute to occupied sudace bands jus't below the Ferml level.The contributioa of daugling bon.ds sîtuated at adatoms near corner holes(CoF, CoU) or the center of the llnz't mesh (CeF, CeU) are mu' c,lz smaller,independent of the occurrence in the fautted (F) or nlnfaulted (U) triaugle(4.78, 4.147). The empty surface bands mns'nly adse fzom daugling bonds ofadatoms. Thîs situatioû is indicated in Fig.4.43. In. gece-'ral the loe>lszation '

of the surface states is less cleaz. Partîal minring of dangling bonds may occur.Due to the reduced Csv point-groap symmetry the adatoms ar.d zest atomsare no longer equivalent. They HiFer with respec't to the position, Co or Ce,in a triaugular subllnst ard tlze occurrenvce of the stnrxing failt (4.148) (see

4.4 Adatoms and Adclusters

(a)

. * * ..

* *

. ' * @

@ * . @' * * * '

R. . Q .*' ..

* * e * *R G ..@ @ . . . @' *

* *' * * *R .

* * R* . *

é @. * * .

- * . .

110 . @ *

* *

* *

*

-2y(lq q

CoF CeF CeU CoUR R

(b)q1 1 f1' faulted > unfaulted

Fig. 4.42. Top view (a) and side plane vievr (b) of the dimer-nrlntom-stacldng fault(DAS) model of a Si(ll1)7x7 smface. J.n the top view (a) the shaded circles desipnate the adatoms. The circ'les 'with a letter R designzte tite rest atoms. Large opencîrcles designate triply Bonded atoms izz the ftrst atomic layer below the adatoms,whereas small open 'mrcles designate fourfold coordfnated atoms in tke lower partof the same bûayer. The dots designate atoms in the third and fourth atomic lay-et's beneath the adatoms. The lower panel (b) corresponds to a plaze view o: thenearest-neighbor bonding in a plxne normal to t:e s'arface coat 'aning tke long dî-agonal of the sttrface unit cen. A possible smface unit cell can be describid by 'lznes

(not shown) connecting the atoms with a dangling bond in the centers of the fottrcorner holes. Afte'r (4.14z1J.

Fig. 4.42b) . However, there are excellent expem'mental studiH using H''gerentmethods, e.g., STM (4.1491 or PES (4.:501, witiclk give information about thedangling-bond-relsted states. For tlznnelklg out of rnn.ny occupied smfacestates the STM image (see Fig. 4.44) shows twelve potrustons ia the smfacelTmst cell which caxl bc traced bnr,'t- to the adatoms. The corner holes aadthe regions between dimers ixï the domin walls aze also clearly visible inFig.4.44. .

Scanning tl:nneling spectröscopy (4.14% but also ARPES (see, e.g., (4.312)found a baxld whic,h extends up to the Ferrnl' level and pins it at 0.7 evabove the top of the valence bacds. The correspondsr,g state's are loem7szed atadatoms. Naively, the Si(111)7x 7 strttdm'e may not seem to satisfy the re-

1'J8 4. Reconstruction Blements

'

- . . QO . . . - .

'

-

'

: @ - . :' . . .: . . - ' . '

. - .. . .y .. . . . u < . .. u . . . . .

J5, () @ '

ô(a) k .. * .. i e : l : * . .

@@*@@*@@**@*@@ Q

œ@e*e*e*@*@*@@*@***@****@**œ

@ . . @ . . * @ . @ .

> . W . * . e :@ * * *; * * . . . A ' @p * -* *) * -> %

- Vœ *e . * , *

ê 4

@) ï . *, .. . e * . l : , : .

***@******@e*@

.@.*.*.*@@.*@*@*@****@@*@*@*

@ . o . e . . @ . . . @l . .

'. @ . @ :* . . *: . , * : @ . .* , o* .

p @ .@ @ .* m > ' F * @. ** . . . # % . . *

(c) i l , * .*. .. pe e .. .z% .* .% .. o.. . .

e@@@@@w *@*@e@@*@*@*@ * @@@@*B1.)1

> E11X)

Fig- 4-43- Contour plots of the sttrface states Pf the Si(111)7x7 surface. T:esquares of wave hlnctioms m'e represented in the (110) plane for tlzree surface bandsnear the Fermi level (for detKs see Fig. l in (4.76j). (a) Peially FLIIGI sudaceband in the gap, (b) au.d (c) ocyupied smface bands just below or above the 'bullcvalence-band maximl:rn. The (110) plxne contxl'nq the long diagonal of F1g.4.42 butis ceutered on a corne.r bole. Fkom (4.781.

construction pzinciples. The seemsngly metxllic c'hn.racter indicates residualtmsatisâed valencies at the surface, ia contradiction to the spirit of Prin-ciples 1 and 3. One possibility to solve the puzzle could be related to theoccuzzence of strong electron correlation (4.152J a-s obsenred in the cmAe ofSiC(111)/(0001)WxW and 3x3 smfaces (4.137,4.139) 4.:431. A scp.lr'ng ofthe Hubbard pamrneter & (cf. Sect. 3.5.1) vith the amo'ant of Si iu the sur-

face re#on may indicate a value of about a tenth of an eV. The accompanyin.gsurface-state gap shotzld be llardly nmmzqtzrable (see also Sect. 5.4.2).

The DAS model described for the 7x7 trnanslational symmetzy can begeneralized to (2n + 1) x (2n + 1) reconstractions with n = 1, 2: 3, 4, .... The '

corresponaing pnnc't cezs contns'n (27z+ 1)2 1x1 surface llmit cells and are dec-orated by zzlzz + 1) adatoms leaving 2a rest atoms. T*e resulting adatom

4.4 Adatoms and Adclusters 179

Fig. 4.o4d;. STM image of a Si(ll1)7x7 stzm

face (image ex-tentt 12x12 nm2) sxrnple bims;-3V). R-om (4.151) (copyrig'ht (2003), withpermsssion from Elsevier).

densit'y is reduced by 14 zo) ,) with respect to the derusit'y .14 in a c(2x8)structuxe. The domaia wxllg consisting of dimer rom with 3% Jsmers f:.ll 2/,1x1 tmit cells, the corner hole 6111 one, and %M,16 the sarface Ls faalted. A rel-atively simple attempt to tmdezstand the variotzs structures is that suggestedby Vanderbilt (4.1531. For Si this author esttmated the euergy costs per 4 x 1nlnst cell with rupec't to thearelaxed surface of the s'tnztlring fault, A.f = 0.06eV, of the domain wall, zl'tzl = -0.655 eV, and of the corne,r hole lc = 1.40eV. The enerr of the formation of a'a adatom is a = 0.28 eV (cf. also Table4.2). Thtus, the total energy gaiu per 1x1 smface region is

1 2n,11/2 + mc + ntx + 1)cAE :2n+1)x(2a+1) = 'j/.f + c .

(2a + 1)witiin the simple adatom model fotmd for Ge(111)c(2x8) the energy gainnmounts to

1ïfctcxs) = zc.

The comparison of the two energie,s (4.10) and (4.11) allows the eonstrttc-tioa of phase diagrams, e.g., 'using the relative star'zng-fatzlt formation enerrlj.fj6lj'trL aud the relative corner-hole fommyttion energy Ic/lzl'trl ms indepen-dent variables. Suc,h a phase diivam is shou in F1g.4.45 for a fLXe'd adatomeaezgy. lt exbi' bits a series of DAS stractmes tf I.J is s=n.ll, which have iIscreaskzg (2zz + 1) periodicity as lc increases. At lazge.r values of ztu, thestackizlg fault is n'rfn.vorable, and tiere ks a trxnsition to azl ordered adatomstracture, notioaally the c(2 x8) onc. .Xn inczease of the adatom formationenergy malces the lower (2a+ 1) x (2zz+1) recoastmzctions, e.g., 5x5, more far-vorable. The star indicates the stable Si(111)7x7 phmse with the parametersgivea in the text. If the cost of fomming the stanlrlng fatfts and corner holests 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 forma-tion of stacking fautts cos'ts znore energy in Ge thxn ia Si (4.154). ,A.s a result

18û

0.6

4. Recoustrudion Elements

O.5

,..S- 0.4R> a8)c(

.t1 0.3

Y.S 0.2(/)

.%,3x3 5x5 7x7 9x9

00 1 2 3 4

Corner hole Ac /IA wl

'Xg- 4-45. Phmse (liagram of a (111) surface with c(2x8) recorbstzudion (sknpleadatom model) an.d (2zJ + 1) x (2r; + 1) DAS reconstructiozls. The reduced for-mation energies of staddng faltlts =d corner holes are taken as variables. Thereduced adatom formation energy is fxed at c/1zl01 = 0.427. The star indicatesthe Si(11l)7x7 surface with the prameters listed in the text. n.

the c(2x8) yeconstruction is more likely tlnn.n for Si, thougk the enera isclose to that foy 7x7 (cf. Table 4.2) . However, the c(2x 8) reconstruction can

also be obsezared for quemched Si(111) surfaces (4.146, 4.155). On the othe,rhand, strnsmed Ge(111) overlayers growa on Si(111) substrate also show a7x7 reconstruction (4.156). 80th fac'ts indicate the outstandimg role of surfacestress 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

Si(111)7x7 suface, a few of more complicated adatom modeks' have beenproposed (see (4.157q), among the,m tEe milk-stool model (4.1584, the pyra-midat-cluste,r model g4.150), a'c.d the trimer model (4.160', 4.1611. However,subsequent studies of the Sî(111)7x7 suzface dismlssed these models. 'T%'me,rs

on group-W surfaces again extered the discussion after the obselwation of(WxW)R%O reconstructions on diamondtllll surfaces of crystallites inCvD-growa 61mq (4.991. One remson was that STM images have shown that2x l domnsnq coexist witE (WxW)A30O strudmes. '

For (WxW)S30ô reconstructions of the single dangling-boud (SDB) sur-

face, there eldsts no model that allows the half-ftlled dangllng boads to be-come nearest neighbou. One conceivable way is to reduze the density of thedangling bonds by adsorption of one addition.al carbon atom.pe,r thzee surface

4.4 Adatoms and Adclustezs

atoms ia a Ha or Tz site, still leaving one dangling bond per (WxW)A3OOtlnst ce2 tsee Sed. 4.4.1). Such adatoms are however energeticazy tlnfavor-able (4.135). Iit contrast to the SDB surfacej the triple dangling-bond (TDB)suface provides a nattzral way for the (111) smface to reconstruct by theformation of (Wx W)S300 l'nst cells without any additional adatom. The(WxW3)A30b reconstmction obset'ved ia STM foT dixmoacl curstxl7stes issuggeted to consist of trimer structures that are centered at a hollowll-ll-site position on the TDB mtrface. Such a stnzcture is showa in 1Rg.4.46.

The hollow-site position is characterized by the absence of atoms uitder-neath the trirner ic the second substrate layer. The atoms fomning the trsmer

are bonded with one bond to one earbon atom ia the top substrate layer. Theother three dangling bonds per atom pardcipate in a 'very s'trong bondingwithp'n the trjmcr. lt forms an isosceles triangle witk eqlllllbrip:nn bond lengtk1.39 .â. of the tvo equally spaced bonds. The elongated bond length equals1.52 i. The angle at the vertex of the triangle is 67O (4.77j. TEe distortionson the substrate are ex-treme'ly MmM,11. The odd apnmber of dangling bondssuggests a metallic charader of the stuface i.c contrndiction to Priadple 3.Neveztheless) tEe surface enera is lowe,r than that of the Seiwatz clzains tseeSed. 4.2.3). Three smface bands occttr i:a the projected Glndsrneatal gap.The wave llnction of the b=d pinning tbe Ferrn'' level i'a a midgap positiolconsists of dangling bonds at the three t='=e'r atoms. Til.ts b=d is very sat,kence, perhaps Gdicating the possibitit'y of a Mott-Hubbard metal-insutatortransition (4.136,4.138, 4.142) (see also Sed. 5.4.2).

Q

E'I 'f 01

l -

. Fig. 4.46. Top view of a hollow-site trimer reconstruction of the C(l1l)(WxW)A30O triple dauglirzg-bond surface. Carbon atoms in the uppermost sur-

face hyer are iuclicated as large open drclœ, whereas dots and small circles descdbeatoins in the two layers beneath.

182 4. Reconstrudâon Ele-ments

Other exnmples for trimer formation cotkld be polar suzfaces of compoundsemkoaductors. The formation of vacanc'iœ (4,45) on (111) safaces of 111-V semiconductors tsee Fig. 4.6) seenls not to occur at the As-teminated

GaAs(111) surface, of'ten referred to as Gau&stlïïl. Whtle the formation ofGa vacandes on GaAs(1I1) is exothermic, the formation of As vacaccieson Gn.AR(ïîï) is endothermic (4.1621 . Unll'ke the situation for Gn.Aq(111),for (>.A.s(1I1) a vaziety of dîferent reconstractions, s'ac-h xs WxW3, 2x2,3x3 and Ui-fxx/i-f, occuz's depemding upon the procerssing conditions (see(4.31)). The 2 x2 strucwtm-e is generally believed to be the As trimer strucvture

illtustrated in Fig. 4.47. Most of the evidence for this reconstntction modelis obtnsmed by STM (4.163, 4.164). Iu particlnlltr, the triangular potrusionsseen tu the STM images from the GauYs(ïîï)2x2 surface caa be explained bythe chernhorptiozz of A.s tz'imezs on the As-terml'nated stvface. Mnn.nwhile,the trimer model has received furthe.t support h'om a trausmisston eledronmicoscopy kwestjgation of Iusblïlïlzxz (4.164.

Witbi'n the tzimer mode.l an n.nsoa-terrnl'nated (ïïï) suçface is decoratedby a'a n.n5oa overlayer corresponding to a e = ) coverage. THee anions withâa

/J X,?

/ '>srz' s/ 's, .,

gj j.y/ N. %d' <.

..' 'y.J'.C %k

N z'%. /K y.. z

z'vn

..*

%, zz''AAy z

Fig. 4.47. Structural model (top 'view) for tke H1-V(!II)2x2 reconstruction withinthe Ant'on-trirner model and mss:tmlng the T4 site for the trimer position. Dotst *

second-laye,r catsoxxs, opem circles: Gmt-layer a'aions, ar.d hatched circles: n.nn'ons inthe top t '.n rn ers.

4.4 Adatoms and Adclttstezs 183

a 2 x2 uait cell (see Fig. 4.47) form a trimer that saturates tkee danglingbonds but leaving a rest atom with one dangling bond. The trsmer can occupya T4 or a Ha site. A R=n.ll dlFerence in the total-eneargy calcalations (4.163)might favor the T4 site. T'he aaioa-trîmer structure satMœ the pziaciplesof stamscozductor reconstrucvtion and thereby the electron cotmthg rule. Sixout of the- 15 valence eledrons 9om the anion tm'rne,r aze tzsed to bond the ''

trsmer atoms together and anoth.er sh aze IZSCSCI to ftll the thzee dangli'ngorbitals of the anions forming tke trimer. The remnsning three electrons Fomthe trime,r are transferred to ftll the dangli'ng bonds of the fou.r topmostsubstrate arkions. ê of a,a electron is transferred to form a loae pair in the4dangling bon.d of the res't atom. The othe,r 3x .?4 eledrons complet,e the threebonds between t'Z'IEn.e,T azd substrate. Since the daagling bonds of the restatom aad the tlzree trimer atoms a're occupied with lone pnsrq, the smfaceis passimted. It becomes semiconducting (PrGciple 3). The Bmsic Pedplein Sed. 4.1.4 inclicates that the considered recoastruction must be accessiblein an As-rich enviroaaynt, since the trimer is preided to be a minsmumfree-eaezgy strudure izz such a'a environment (4.163, 4.1661. Jndeed, thks isconqvrned exxpsrirnenully (4.163, 4.1641.

4.4.5 Qzetrnrners

Besides dimers or trimers JtTqn larger clusters of adatoms may occur on sur-

faces. SuG cluste,r coHgurations could be tetramers to e'xplain SiC(111)3 x 3or SiC(0001)3x3 surface strudures (4.136, 4.167) or reconstntdion elemeatson a Si(110)16x2 sttrface (4.168). Larger reconstrudion elements sach as peù' -

tagons 'or evem hesxeons mxy nbnmaztarize the 3x1/3x2 recomenlctions oflkigh-index snrfxtxas of Si axd Ge, such as (113) (4.16% 4.1704, or also theSi(110)l6x2 sutface (4.168, 4.1711. Under very Si-rich preparation condltiortsthe (111) surface of 3C-SiC as well aci the (0001) smface of 4H-SiC or 6H-SiC twhibit a 3x3 recoztstruction (4.167). Despite their slmllarities with theSi(11!) suzface a 3x3 DAS model contaiuing two adatoms pe,r unit ce2 withdxe-remt stack'ing orientation fn51R, lince only a siugle protrmsion is fotud e.x-pfm'rnemGlly (4.172) . Even a variant of the DAS model contain.ing one nzsntomcluster on a silicon adlayer is not in agre=ent with total-enerU mxMlms'za-

'

tioms (4.l3s) and LEED fndiugs (4.167J.A completely novel structure expln.s'ns tite 3x3 reconstructed Si-terrnsoated

SiC(111)/(0001) sarfaces. This reconstruction includtxs a twisted Si adlayezabove a bulk Si-terrnlnated SiC substrate with a Si tetramer adclttster on top.With e = 13/9 the Si coverage is large,r thaa ona monolayer. No stlmsrlngfattlts, Ht'mea's and corner holes appear. The central element of the reconstruc-tion ks a Si tetramer on top consisthg of a Si tm'mer aad a'a additional Siadatomj as shou în Fig.4.48.

Titis stmzcttzre widely satisûes the requirement of a fourfold coordlnxtionor of dangling-bond saturation (Principle 1). The adatom cluster (tetramer)saturates 'nîne of 27 dangliug bonds of the adlaym'. The Temnsnsng dangling

184 < Reconstmzdson Elements

E1 1 :(I(a) (b)

E1 -111

13-ljl= . a

'' . . ' .

. - ' #. :j' .- '*' '

) r> - ' t' ' - ' ' '' '

iA'.tc&'J '.i '.P '! Xm .fl#t ' w '# Q-' >' -.w L - ''MWQ..- .. ..%6 .zz .

' -'

TJ

F/g. 4.48. Perspective view (a) and top <ew (b) of a Si-rich Si-terminatedSiC(111)3x3 surface within the tetramer-adlayer model. Dots desigzza'te C atoms.Large open tfzlled, katched) circ'les designate Si atoms in the subMrate (tommost substrate layer, adlayer). The larger hatcked circies indicate Si atoms izï thetetrnrnez.

hybrids fo= a threefold coordination within the Vlayer. There is a tendencyfor 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,r atom formq a bond to the substrate. The adlaye.ratoms 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 tobe lmhybridized and bonded by p orbita!s as judged from its bond a'agles of90ö. Jn efect, this allows azl energetically favorable f-like drgling bond ororbital.

The fm'eence of this half-flled daugliug bond seeaus to contradid b0thPrinciple 1 and Principle 3. However, it haeeen shown that the more Si-ric,h3x7 reconstnlction forms a Mott-Hubbard insulator grou'nd state as the lessSi-rich (xCxVi).R30B rcsconstruction of the SiC(111)/(0O01) surfaces (4.13%(see Scct. .5.4.$. Only the Hubbard Mteradioa parameter U a 1 ev is mue,hsrnnller becattse of the larger mount of silicon on the surface. The fad thatthe Si-rich 3x3 reconstrucvtion satisûes the Basic Priuciple is iudicated iaFig, 4.49. This pbn.qe diagram b.as been constmzcted (4.13$ accordlng to therules describeê in Sect. 2.5.4. The tetrnmer-adlayer surface is predicted tobe the rnsnlmllm gee-energy structme ttnder Si-ric,h preparatioa conditions.This is con6rmed experimeatally since this stntct'are is accœsible in a Si-richenviromment (4.136,4.167,4.1721.

4.4 Adatoms and Adclusters 185Nx .

x N0.0N 5X1 clean xz'.y h x .9 S X* N 1./ N.

%î% x +u> xW.x N. A*A.& - (i - 2 N'4 - wx '%#' -w x

.o hhx p*% öio xwx tp xm x z o h.x 'h 4w. > xv N g%%. %.-' 'r x:j N N1* -().:$ 41 h% hv --'B' (Cz - Nx --* a) ,.c xcl : mz.- %

Jz> x .x ,9-: .4 h.x Kz) :.>. a x. xC'vo +qp %'h. hx* 0 6 > o h-4wr . . . +œ lz ''N.

o PT >t: Qlr xc ..>

=m -0.8 %o

Q'oz

,dho1 I

C-ricb Sl-richpbsf ' P.c

Fig. 4.49. Phase diavam of a Si-termlnated SiC(111) mzrface ms a function of thecàemical potential of the Si or C atolns. Relevant recozzstrudion models have beenseiected. The enerr of the rathe,r disordered 1x1 struct'are hms been lowered byazz emtropy term assuznirg Sa = 0-15 ks aud T = 1000 K, in orde.r to explain tllcsequence of pllase transîtions Wx W -> 1 x 1 -F 3 x 3 with increasir.g Si coverage.From (4.136) .

5. Elementary Excitations 1:Single Electronic Quasiparticles

5.1 Electrons and Holes

5.1.1 Excitation and Qllnqiparticle Chnracter

Surfaces are many-body syztems consisting of interacting cores and electrons

(Sec't. 3.3.1). In order to descrîbe rnxny propertiesj in pardctzlar, vound-state propez-tie, it is stnlcient to l'eplaze the system of interac'ting electronsby a system of independent particles (3.2). One exn.mple is the density Gpnc-

.tional theor.y (Sect. 3.4.1) witbs'n the local approvsmatioh for the exchangeaud correlation contributioa to the total eaergy (3.50). Using the KoM-Sham equation (3.46), the grotmd state of the electronic subsystem can bedescribed by Mdipendent (i.e., eiectively non-int. eractiag) particles, more

stridly eledrons moving in an eobcwtive single-particle poteltiat (3.48). An

Cindependent' electron i'a a Kohn-shnm state possesses a âxed single-particleenero- and a defmed probabilit'y distributioa of 0c111)g this eledzon in space.Eowever, exdtations of ac electronic system cnmnot correcvtly be describedby the independent Kohn-sbam particles in (3.46).

A 1ot of experbrnental studies are associated with spectroscopies a'ad,therefore: evitations of the eledronic (sublsystem. A.n elqdron may be addedto the system or an electron mn.y be taken away from the s'ystem and., hence,a h,ole is created. The e'xcited electron or the kole strongly Mteracts with the

many other electrons of the system. The electronic subestem is polarized andreacts with a redistribution of the e'lecvtroa deasity. Consequently, the enerrof such an electronic exdtation will diler from those for nozsiuteracting par-ticles. lt is renormmlszed with respect to energy azd to behavior i'a time:i.e., to the specwtral distribution. If azt excitation hnuq a, stnlciently long life-time, it however behaves like a particle. There-fore, it is called a cutkdpc,ràïcle,moze strictly a quasielecvtron or quasihole depending on the occupation of thecorresponalng single-particle state before excitation. The properties of thequasiparticle are better described by a spativy non-local spectralt-weight)hlndion thau by a'a eigenenezgy and a wave hlnction., Suc,h quaaipazticles are

adually observable in several surface-sensitive spectroscopies.

188 5. Blementary Bxdtations 1: Single Electronic Qttasiparticles

5.1.2 Scamling Tanneling Spectroscopy

The Xst scn.nning t'Tnnelizzg microscope was built in 1982 by Binnn'g aadRohrer (5.1!. The physical phenomenon at the ol'ign of this new inldrralment

is tEe brnneting of electrous through the vacuplm. J.n such a microscope ashar.p metallie tip is positioned at a distauce d (of the order of a few i)h'om the suzface of a conduding smple t'Fig. 5.1). l'a this way there is auoverlap bet'weea the electronic wave Rmctions of the tip and substrate. Avoltage F' is applied to the t'wo electrodes resulting in a tllmne.l curren.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 obtained by sc-nmning,i.e., by moving the tip over the lmrface, e.g., witlu'n conqunvcurreat mode.The condition of a constant tllnmel ctzrrent I.c ca'a be A116E.ed by vazying thedkstance d betwee,n the tip and the sample. The rœulthlg corrugation hlncoon

contnlns information about the sarface topography. A topograpidc 5V age oftke suzface is also obtained wit'hT'n constant-height mode, by memmnvlng the

zaartude of the ttxpneling m3rrent as tke tip is moved across the st-nce at a

fzxed distance d. Scxnnlz.g t:tmmfRll'ng microscopy (STM) Ls usaally performedic the constant-current operation mode.

More information aboat the eledronic s'tructare of the surface can beobtaiued by studying the dependen- of the STM signat on the siga andmagnitude of tlb.e tip-sample voltage. By varying the bims, stonmlng t':nmeo'agspectroscopy (STS) can be done. The sign of the voltage deterrnines whetheroccupied or empt,y states are studied, ms showa i:a Fig. 5.2 for a sernsconducvtorsnrfnme. For posjtive bias (a), ttmne,ling of electrons rA.n only occur fromoccupied tip-metal states inio empty suzface states or conduction-band statesi:a the substrate. lu the opposite case (b) with F < 0, elmstic ttnnnelingof electrons 1om the metal iato the semiconductor is not possible. . Only acurrent with oppYte siga is measurable. The measured 'ialnneling curzent I.roriginates from occapied sucface or valeace-band state,s in the semkonductor.

1

TipV

d

YSurlœ

d

Scan7

DiredionFig- 5-1. PrMciple of a scltnning tun-neling Mcroscope.

5.1 Eleckrons a'ad Holes 1%

(a)Energy Eneay

CBMx .s-1CBM x sF

eVVBM

y/ E, ,, a--- .-,ç ----------------------------------;,

Ffg- 5.2. Electronic band scheme along a sudace normal of a semiconductor sttrface(1eft) an.d a meta,l tfp trv,h' t) fbr oppos-ste values of the 'bias voltage in (a) and (b).The energles of the conduction-band rnlnlmum (CBM), valezlce-band mn.vlmum

(VBM) a'ad Ferml' level (ga) ak well as distributions of possible surface states areindicated for the unbhaaed seMconductor.

-1.y-. .eV

/ / , -,''>--u

By meastlri'ng the dependence of the curren.t I'j: on the applied voltage F,one can obtain an image of the energy distribution of the elec-trordc states iathe surface region.

Sizlce one is more interested i.a the general spectral behavior of the tl:nnelcurrent and not i'a its exact absolute value, a simpMed appzoac.h caa be used.At the surface-tip separations of interest in STM or STS, of the order of 4i or more, the surface-tip interaction is extremely weak. It is natllral theh'to calculate the tlmneling current using time-dependent pezttzrbatioa theozyThe result for an elementazy trlnnel process is given by Ferrxsts Golden Rttle.Axsplrning non-interacting electrons J.n the tip materii and the sxmple surfaceof the type chazac- terized e.g. ia (3.1) o' r (3.46), the probabûil per tmit timefor an Lsoenergetic tptnmel process across the barrier between a surface state

#s@) (SL?A) with energy ss = eu (i) and a tîp state #.r@)' with enerr s,.ris givezz by

2% ,- .:lWs =

-g- I'J.'Ts I 8 @'r + ek- - &s) )

where the trn.nnition matrix element

r'as - j d3z4,:t(z)>#s@)

i intToduccd for the taaaeling operator t. It may be issentially identiseds

M'ith the current densit'y operator (5.21. F' denotes the applied bias.First we consider a 'Fnste negative biks 1//, although tilis is relatively small

on the scale of the tip work fllnction or iomszation enerr of the sample. Thetqtal current for tpznneling from the sutface into the tip (7 < 0) is give:aby (5-34

J - ceyyltss) gi - ?.(cr)) wysTT,S

190 5. Elementary Excitations 1: Sinéle Electronic Queparticles

with J@) mq the Fermi dltzibutziom The factor 2 accolmts for the spin de-generacy. With (5.1) expression (5.3) can be rewzitten a.s

4rre +* aI'2 =

s L. dcltc) % -ulçe - eF')J 57 IT'z'sl J(c-cv'-e,r)J(s-ss), (5.4)T,S

using the properties of Dirac's élhlncwtion. Resuks sfrn-llar to (5.3) or (5.4) Eavebeen exploited by mrtny groups. However, genernlizations to obtaia an explicitformllln. for the cl:rrent density have also bœn usedv and real atoms of the tipand surface instead of model potentials kave been tékea into accout (5.41 . Amore or less e-xact way to determl'ne the tltnneling current vithin the single-particle picture is possible asing a Green's Alnction formalism aud solvir.gzplmerically the Lippmaa-seahwiuger equatiou (5.5). However, the zplmericalefort is large and, moreover, a detailed Mowledàe of the tip shape ks needed.

The theory of the tnlnnel currem.t (5.4) makes no distinction between sm-face and tip states. Howevez, ixl STM or STS this distiaction is cruciat. Ideally,one is interœted to relate a'a STM image directly to the stlrrlme propezties,wheremq in the above description the current mvolves a convolution of theelectronic spectra of surface and tip. nerefore, Tersof aud Hhmn.nn (5.6)proposed = approvsrnn.te way to elirninate the tiphpzoperties. They consid-ered au ideal tAlnneliag Ocroscope with a model for the tip that woald havethe highest possible resolutiom This goal is best aœeved assnprnlng that thetip is a rnnthematic,al point sotzrce (s-wave approvsrnation). 1ts potectial a'n.dwave Atndions are arbitrnHly locnll''zed. STM experiments obviously n.1m atatomic resolution. Therdbre, intuition suggeas that the tip must have atomicdimenm'ons as shown in Fig. 5-1. Iu oth.e.r words, the tip is not point-like on tshelength scate of the axperirnents. Nevelheless, the point-probe appzoxn'rnxtionleads to a reliable description of expersmental data, thougk it is not c prscrïcleaz why thks approvirnation holds. Witlnl'rt tlze polt-probe approximatioafor the tip at the positiou œt:p, the trnêndtion strength is given by

1T$s!2 :x: (#s(œt$p)12- (5-5)This approvsmxtion lfo-qrlm directly to the local electrozkic dtansity of state

(DOS) of the surface region of the snmple. Without cotmting the sp% degen-eracs $t reazitq

p(œ:i) = X--2 1#v/:(z)I2é@ - ::.(/2)). (5.6)vvk

This quautity represents the diagonal elements p@; c) = A@, tr: sj of a more

geuezal quantity, the single-pmicle spectratt-wtAirht) h'ndion A@, zT; e) ofthe electroic s'yste,m (5.7J. This hlnction also accouuts for the qua-siparticlecharacter of the' exdtations.

Comsidering the lirnl't of low temperatures witk J@)(1 - .f(s - cF)J =*

O(as> - c)O- (c - ey - sp) and introduchzg the giobat electronic DOS of thetip

5.1 Blectrons and Holes 191

sy) = EJ(s - cv),T

one falds for the voltage dependemce of the t'lnneling curzem.t

J,r = dsots - ekrlptœtyp; s).&F+W

Low temperattzres me-qn that the thermal energy kBT is small compared to$he position of cp with respect to VBM and CBM. 1.n tlze llmn't of smallapplied bias voltages or nearly constant DOS of the tip in the region of theFerzni energy, this expression becomes .

Je

JT = DLsvj / dsptœt:p; s).JFVW

A ssrnilitr result holds f6r a small positive voltage (7 > 0)el?+eV'

I.v (x DLep) dcptmup; c).dF

Eoression (5.9) shows that for Jzr < O (7 > 0) the tunneling cltrreat îsaccompaaied by the generation of hole (eledroas) ia the rebozl of the spmplesmface. Correspondingly, the tl:nneling clTrrint locnlly probes the pazt ofthe electronic dezusit.g of states of the surface that is occupied (unoccapied)for zero bitzs. Expression (5.9) is derived withsm the single-pacticle picture,The attractive Coulomb interacbtion of electrons (holes) in the tip with the

' holes (eledrons) generated j.n the surface re#on ha.s been neglected from the

vep- be ' '

. One argument justifyiug this neglect is the distance of tlb.eparticles, rother the dmmatic redudion of the eledzon-hole iateraction bythe metn.llsc screenlng in the tip reson.

The tlmneltug current (5.9) is proportional to the local eledronic densityof states (DOS) taken at the position of the tip. The local density of stata isintegrated over an energy intezval for F' > 0 (7 < 0) above (below) the Fermienergy, the length of wlzic,h is given by the applied voltage. Consequentlythe restzlting 'STM images depend on the siga artd the magnitude of theapplied voltage. Exadly thks dependence is damonstrated in Fig. 5.3 for Ga-ric,h GaP(001)2x4 surface reconstructions (5.81. A constrt-height mod.e witha tip-surface distance of about d = 4 â. is assïlmed for the simulatiom Toaccolmt paztially for the nqnideality of the tip, the specvtra are averaged over

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 theoryrequires timsn.mple distances larger thac the decay length of the wave hlnc-tions into tlle 'vacullm. Larger currents are modifed by multiple scatteringof tite eledrons. Mother problem for small distauces is related to the wave-

hlnction overlap of tip and sttrface. The resttlti'ag lbondsng' atld t=tibondsng'

192 5. Elcmertary ExcitatioM 1: Single Electronic Q'aasipartides

GaP (2x4)M'Ixed Dimer Top Ga Dimer

. . 1.. , t.

... j: , ,. .. y: j(. ? (:.:j s......) : ' . . . 1. . l ' .. . . .:- - t. àLt :

. j y '.

) . .t u. . . , r . , : ? .! ( . s .. . ; . ( 'p 't.? ..

. (tL ''. . . %% zk:'';.Jz .., )..i.)f ...t. :(.:. : /

. y: lk';..(

.' ë. .'r' ..!, .,

' (

'

' J .

. i

12 :

i

I'

, i

i

I 1

l. . )

. :

.... '.. j..v .t, .., '. ' ,. EI:1

i. !(

! l .

f

iG' . . ,.

' '.

+- L é xq 1 i' i. . ;1+-d

' ' 1.53 nm $ 53 nm7. .

17:. 1. û1

Fig. 5.3. Calctzlated STM im-agœ for two dimer geometries ofthe GaP(OO1)2x4 sklrface. In thecmse of the tomGamdlrner str'ucttlrethe P atom in the hetezodsrne.r ofthe mhed-dimer geometry (shownita ng. 2.17) Ls replaced by a Gaatom. The negative (yosijive) volt-agœ are measured =th respec't tothe CBM (RM). The ''brightnessindicates the magetude of the 10-c,al electronic DOS. From (5.8).

linear combinatîons may #ve rise to a complete change i'n the STM contrast

(d. the remarlcs in (5.9q). '''

The memsmement of the spatial variation of the local DOS for a givenvoltage is a powerfool tool for obtaining local stntctltrat 'Oormation aboutmetal or semiconductor mnrfaces. Moreover, the spectroscopic mode of STM,irt which the tlmneliag current f,r is recorded as a fhlmction of the appliedbias 7, gives direct information about the loci electronic strtzctttre of thesuzface. Asmtrnsng that the prefactor of (5.9) is nearly independent of thevoltage, diferentiating thks expression with rezpect to F yields the diferen-tial conductance, dfan/dvr (x: Dtsslptœtsp; se + e5rl. As a function of F' thisqt,n.ntit'y roughly reproduce.s the local densitjr of states of the surface in theneighborhood of the Fermi level. Density-of-states featares in I-V curves

appear as various lcin'k-s and b:tmps. They aze obscmed by the fact that the

$.1 Electrons and Holes 193

6 *

4% .* % . .r.r

. . 1.zs ..*. m a 2lb. *> Nà z, a ;.. a . . ,. ee sv<% 2 .- =..e . . z . hwxJ *-

w ' lw-k. ea jj * . x* @

'*'

. * *>

2 A*** & *

*0 T: Tr

..4 4 -2. -1 0 1 2 3 4Energy (eV)

Fig. 5.4. Relative conductance vezsus electron (hole) eaergy relative to the Fermienergy s'v measured by STM on a cleaved Si(lll)2x 1 sttdace. After (5.10).

tunneling cuzrent depends exponentially on tlb.e tip-sample separation and.in a norl'lnea,r mnrl'ne,r on the applied voltage. Most of this dependence can

be removed by computiug the ratio of diferential to total tTnnneling conduc-tance,

c'1.f 'r rrf 1 f F= = plztip, Ev + dV') dz#lttip, &).dy 'f/' cl'U'q ssicv

Eere only the restût for v' < O is givem J.n the opposite case 1/r > 0, tEe energyintegratioa hms to be Ganged accordsng to (5.9b). hdeed, (5.10) gives thelocal DOS of the surface at a hole (electron) energ.y whic,h is given by theapplied voltage. It :is normalized to the local DOS averaged oveer a,n enerorinterval of length elFl Tven by thks voltage.

Am example for the z'esttlt of such a procedttre is shown ia Fig. 5.4 for a

cleaved Si(111)2x 1 s'arface (5.10J . Within the region of the G'nndxrnental gapof bpllk Si two surface-state-related doublmpeak stnzctttres appear with peakenergies -1.1, -0.3, 0.2, and 1.2 ev with respect to the Fmrrns energy. ThespedTal featttres at higher or lower energies are due to bulk states. The fotzrcentral pen.lrK can be ex-plxined using the model of tûted em-bonded citains (seeSect. 4.2.$. The band structure (4.7) resulting witlnln a simple tight-bindingapproxsrnation gives a (normazzed) electronic density of states of zsbondedcàains as (é' = j 1s1 - sz 1 , sl + sg = 0) '

(5.11)The meastu'ed peak positions in Fig. 5.4 cac be identïed with the square-rootsingulHties in thjs DOS. This gives a surface-state gap of about rs) -s2 I = 0.5

194 5. Blementazy Excitatioms 1: Single Eledzonic Quasiparticlc

ev aud a.tè average band width of a rr* or rr band of about 0.9eV: i.e., Ikrl = 1.0W with F' bemg the averaged interatomk Mteractioa matrix rlement.

5.1.3 Photoemission Speeroscopy and Iuverse Photoeml'ssion'

If one bombards a sample surface with electrons or phètons, electroms and/orphotons will be emitted, whick have aa enerr spedmzm. Heuce, Hormationis obtained dkectly or indirec'tly about the jurface electrozzic states. Themost important and widely used experlrnental technlque to gain iaforma-tioa about occupied eledronic surface states is photoemlnqion spectroscopy

(or sometimes photoeledzon spectroscopy) (PES) (5.112. TNe solid surfaceis irradiated by monochromatic photons with energy ruo, holE's are gener-ated izt the sample, and the emitted electrons are analyzed witit respect totheir eetic enerr ck:u = :2:2/2m. When photons in the tûtraviolet (UV)spectral rauge are uset the teelnnique is cmlled UPS ('UV photerniAqioa spec-troscopy). Besid% the kiaetic eaezgy one may also use the enzission direction

kjk = (cos/sin8, sin/sin8, cosl) de-sczibed by the angle 9 with the suzfacenormal a'ad the angle / in the surface plaue to charactezize the geometry ofthe exp-rime'nt (Fig.5.5a). Vatyiag $ and/or / the method is then km own as

angle-resolved (AR) PES or UPS, ARPES or ARUPS.hwerse photonmsssion spectroscopy' ImESI can be regarded as a time-

reversed photoemission process (5.12J . It therefore probes the tmoccupiedsurface states. Ia this tenbnsque a be>.= of electrons with ene-rgy sktn and wave

vector k = àtcos ysin ù, ss.a/ siap, cos ;) is kcident on a smface (F)g. 5.5b).The electrons trn.nqmitted inside the solid decay to states with lower enerrthrough tlle Auge,r esect or by emitting photons,' which are detected. Thereare tvo operatbzg modes: either the energy &zJ of the detected photons is held

(a)/ ho

Ekin, k

T'sg. 5.5. schematic represenzation of a photoemission (a) and inverse photoemis-sion (b) process.

(b)Ekinl k

/

Z

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 spectranrn is taken as a funcwtioa of hu'. 1f, izïadditîon, oae takes advantage of the k-vector raolution, one ca,lls the methodk-resolved (KR.) DES, KR.TPES.

A rigordtls theoretical approach of azl elementar,g photoem7'Fmion (invezsephotomml'ssioa) process requires a 6111 q'aantllm-mechxnscal treatment of thecomplex coherent Hteraction processes starting with a photon (eledron) andEnishin.g with an electzon (photon) in the detector ms well as a hole (eledron)ia the smple. Titeoretical approaGes of this k'snd treat the pkotoemissionfor instaace a.s a one-step process (5.13-5.151. A more instracvtive approach isthe so-catled three-step model (5.16). The optical excitatioa of an electron ts.r.a 6149,1 state fk) and a hole (iL the izzitiat state ïi) can simply be desmibedagnin by Fe-rmils Golden Rule. Withln the dipole approvirnxtion aud theiadependent-pazticle approximatjon the trxmqition probability for the ftrststep is given by

2r i t k a j k) - aytjg) - spl .î&h(&) =

-..à- 1.,,/.2 ( ) ( @.f('

Withi'n the single-pazticle picture the pezturbation operator Xint the light-)

matter interactionj is given by the eledron velodty operator 'v = ihlV, a(- tiI1the cornmutator represeatation) oz' the m'omentamkoperator p li:a the case oflocal potentials i.u the single-particle Hxrniltonian Hj and ilze vector potentiazA of the iuddent electromagnetic wave as tin. cgs llnits) Xfnt '= --G.A .p withmcmatrix elementé

t e

J42 (:) - --A .p.s(:),mC

py:@) = d3œ#),@)p#ns@). (5.13)

The gauge of the electromagnetic âeld is here chose.n such that the scalarpotential vanishes. The vector potential .A is nearly spatially constant iu thelong-wavelength l'l=5t (in ITPS Ahe wavelength of rmiiation is still à > 10ûA). T:e J-fttndioa in (5.12) describes enerr. conservation dlpring a directoptical tmnsition from the Oed initial state sç(i) into the e'mpty ftnal statesJ(i) in the surface bands.

For photon energics lazger than the ionization energy of the system, f.nalstates with energies above tke vacullm enerzy svuc occm. In the caze of blllkstates) as a second stepj the e-xcited eledroms propagate to a certain extemtto the smface. 'This trn.nqpozt probabilsty depends also on the eaergy andwave vector. J.n aay case, for egLL-) > svac electrons may escape throughthe stkrface into the vacmtm with a probabiiit,y T/kin, i) as a thhd step.They appear in the vacuum with kinetic enerc slqia a'ad momentlnrn hk. Theescape processes are elmstic: i.e., dkia = sy(k) - emic holds. Together withtlze ettergy corzservation in (5.12), this equation results in the well-lcaownEinstei'a law of the photoeledric eF. ect (5.171 . Because of the 2D tr:mqlational

196 5. Elementary Excitations 1: Shgle Electronic QlxMipadâcles

symmetry of the system, the electron transmlAqion through the suz'face intothe vacqrlm requires momentl'rn conselwation in the generxliaed form kl g =k + g for the wave-vector component parallel to tbve surface, where g 'xs a

vedor of the 217 reciprocal lattice of the surface. On the other hand, thewave-vector component parazlel to the surface outside the crystal is relateclto the experimental pazmmeters clcis, 0, tj by kjj = zmskin/D,z sin ù ard )ç:g =

kgk (cos 4, sin/, O). Consequently, not only the energ.y of the initial state (Mthrespect to the vacutkrn level) i.a the surface b=d structme but also its wave

vector can be determineâ from the expem'rnental parameters FlaJ, skin, 3, and

/.In s , the three s'teps ylelcl an approvsmate expression for the pho-

toelectron current in the vacmlm or for the measured nlTmber N of photo-electrons emitted into a cone (with a small solid a'aglel pointing in a specfcdi'rection .

NLeysn,u), klI) cc EEltszn + svac - Fpx;l lJ2'y''lt::)l2t'kl kiys

x w4o.ét/l,sulu + sm-sc - r'zs)x Wyyti, cusa + cvacl ék!!,:+sT@kia, Y.

Here the diagomxl nantrix elements

k d3 q?W#. (z)./.t(œ tr'; s)# ,x(œ?)' (5 15)Avv' ( , s) = z .: 1 v .

of the single-electron spectral-weight function of the sl:rfn.ce system are takeuwith single-particle eigezzstates '(u@) (with quantzlm nllmbers Izil that havebeen calculated in a certnsn approvlrnation of non-interactiug particle-s, e.g.,withsn DFT-LDA (3.46). Thrus allows a genernlscation of (5.14) to interactiugelectrons ms indicated by tke replacement é'(c - cv(i)) --> Xvvti, s) of Dirac'sJ-fïtnctions in (5.14) by d-iagonaz matrix elememts (5.15) of the spedral-weighthlnction A(m, a/; s). Jn this my the idemtécation p@; s) > .A(œ, œ; sj of thelocal eiectzonic DOS (5.6) becomes obvious.

1.c the second spectral factor Jks izz (5.14) with v = #: izl which ortlyoccupied iztitial s'tates are considered, it is allowed that the exc-ited holes in-teract with the remairdng electrons. On the way to the Apacutlm the oatgoingphotoelectron, v = j', also interacts to a certnsn erent with the electroasin the surface acd barrie'r region and the remnsnlng holr (adiabatic approx-imation (5.18)). Howeverj this ezect should be weakened with increasing 1d-netic energy (suddezz approvlrnation), so that the replacement of the electronspectral hlmetion AJJ by a élAlnction is oRen a good approxn'mation. Thei'aterpretation of expersmeuts is mostly basecl on the sudden approxn'mation,expressing a photoelectro:a specwtrlprn i.zz terms of the one-pvticle spectralhlnctiozls of the iaitial s'tates, Aç(E-, s). However, in genezal, with the excita-tion interacting hole and electron quasiparticla occur (5.19).

5.l Eledrous and Hole,s 197

In the simpliîed expression (5.14) 'with tite factoeed representationA:(i, c - F?zslWyyt/1, s), the electron-hole interaction hms been neglected inagreement with its derivation witqhsm the origizlal picture of non-interactiugpartscsles. Since the e-xcitation energies r?w are muc,h larger th= the energyof the fl:ndnmemtal gap: the cledron a'nd hole excited i'a a photoemissioc

procv are energetically well separated. Their coapling in the surface shouldtherefore be smltll. This holds even more for the interaction of the oatgohgphotoalectron with the remainlng hole. Nevertheless, for not too large klnetic

enertes of the photoelectrons (adiabatie lc'rn-lt) intemsit'y vafatiozus of theplazmon satellite of the main photoelectron peaak versus energy have beenobserved (5.19) and have beem t'racect bnr!lc to the eseds of the electron-holeinteraeion, at least to vez'te.x correctious (5.20, 5.211.

A spectral Glnction (more stdctly a photoelectron spectznlm in the adi-abatic limjt) for a surface-state band is shovm in Fig. 5.6 for the occuyiedr-band of the C(111)2x1 surface (5.221. The enezgy Hssereace sv - Ev(k) isintezpreted a,s the binding eaergy of an electron with resped to the Femmilevel. The measurement with FlnJ = 50 ev has been performed unde.r sttrface-seasitive conditions. The measured azimutlnxl direction is fxed parallel to

(ï10). For 0 = 33.30 - 66.80 the wave vec-tor L therefore varies along the ?Rli'ne from its middle to a,a eqttivalent point in the neighboring surface BZ

Lg # 0). Despite the broadens'ng of the spectra, the peak positions shouldbe identiîed with surface 'bazd energies, here with those of the cr-band ofthe Pandey chnsn model. The observed dispersion is in qualitative agreemezztw'ith that of the calcltlated c-bands in Figs. 4.16 and 4.18. Near # the occu-

ied x-band comes closest to the Ferrnl' level whereaz in the directions towardr in the sxrne Bz aud. J' ic the adlacent Bz .a strong dispezsion of t:e x-

baud towa.d lower ezzergilas is observed. Thc measared value of cx(A) at lewst

K=c=

r=

w

>hM(DcQ)c

0 =

33.30

37.40

41 .60

45.60

49.70

54.00

58.2062.50

T( o66.8

5 4 3 2 1 0=EF -;2

Binding energy (eV)

Fig. 5.6- Angle-rœolved photoelec-tron spectra taken frop a:a AonealedC(111)2x1 surface. The photon en-

ergy Ls 50 ev at normal incidenceof the incorning light: and the mea-

sured =imuthal direction is (ï10q.F'rom (5.22).

198 5. Elementmy Excftatiorss 1: Single Electroaic Q'aimiparticlu

PES IPES3.5eVi;tk .-. ,k()!!.

ut qk . *û.i!. :,)zJixsiyqçs.; m . k!' . .k, % . . hx x

N. / ' I ' <*. . J

z : :Ga P k m-Z

.r ev ,

a& x T

- ; & J. ) tN..... z N . I .

x X - / %ql A :.

: / .

's .,(InP v

*.= .0eVc .

= .6 -. ' tv ..

- ,. & ;: . j.Q N ,J '

>-. . . /(1 .

we *

G%s i

>(D 2 4evc . >'

? 't .G) . j y ,

c 1 :

-... r .4: .

IDAS -:

nN 2.QeV

y x; .

:$

GYSb '

1. V:I -

z'

4 l* /'l n S b ' .1

Energy (eV)

Fig. 5-C. Combined photoemis-sion and inverse photoernlqqionspedra for the determsnation ofsurface band gaRs at the high-symmetry point X? ir the surfaceBZ of s'zx IH-V(110)1x,1 surfaces.The energy zero is given by theVBM. The photon eneries havebeen chosen to 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 sarface is stamiconductitlg, iucontrast to the DFT-LDA result i'a Fig. 4.18 tle:ît p=el).

A.n expression s7'=57ar to (5.14) may be derived for the tve-reversed IPESprocess. The corresfonding spedrnTrn is dorninated by the main peak in tlzespectral 'hlndion of the electrons in the empty ân.al state. Neglecting theiniuence of the trnnqition matzix elements and the veatex corrections, thespectral variation of KRIPES is governed by the empty i-vector-resolveddensity of states Ey AULL, c). 1n contrast: in the cmse of ARPES/ARUPS'the spectra aze governed by the occupied part of the Lvector-resolved dettsity

s.1 Blectrons amd Hole.s 199

of states of the smface s'tates, Eç .3ksti, $. Conseqaentiy, the combination ofARPES and KRDES allows one to detemnlne the complete i-vector-resolvedsmgle-paaâcle demsity of states of a sllrfnme slrstem.

An example is shown in Fig. 5,7, in wbic,h combiaûd photoemsAsion audinvezse photoèmlssion spectra (5.23) aœe presOted for the XT point iu thes'arface BZ in the enerr region of the ftlled anion-derived dxnglblg-bondband Jls and'the empty catiozl-derived Ca bard (see Sect. 4.2.1, Fig.4.12) forcleaved EFI-V semiconductor (J.10)1x1 slprfnrtes. For tkase surfaces the exper-iment has the aduntage that a cornmon energ)r reference for the two appliedspedroscopies coald be es-tablishfd. J.n all cases i'n. Fig.5.7, it is e'ddent thatthe anion-derived surface s'tate gives rise to a cler peak, whilè the cation-derived feature in some casœ is a rathe,r broad line with a large slope at lowerenergiesc The surface band gap for the (110)1x 1 surface of these materialsat the X' poin.t of the surface BrilloTlin zone is consideaxbly larger thn.n thefltndamental bttlk gap, bttt the hcrease 1om I'asb to Gap hms l'oughly thesame slope.

5.1.4 Satellites

The genern.lization of Dirac's élhTnctions for electron acd hole excitations in

(5.14) to shgle-quasiparticle spectral flmctions (5.15) allows a moce complexview of photoeelectron excitations beyond the trivial lifetime broadenimg of aJ-l:nnction (as, e.g., a porssible deascription of the spectra iu Fig. 5.6). ln thespitit of Sed. 5.1.1 such a geaeralization allows oae to account for egectsof the complete electron-elecvtron interacvtion at the leve,l of single-partideexcitations. The accompanying m=pbody eseds al'e best demonstrated forthe spectroscopy of strongly locltll'zed core states and a wider enera range.

As an exmple, the Sizp and Si2.s photoeledroa spedra of a Si mystalwith (111)7x7 surface are presented in Fig. 5.8. They are measured with a

photon energy of Fttzp = 1486.7 ev IAIAQ radiation) (5.24). Tlle large ki-netic energy mdicates the sudden lx'ml't. The escape depth of the outgomgphotoelectrons is varied by cblmrimg the escape direction from normat emis-sion (# = 00: mav'='lrn depth eqtln.l to the mean 9ee path Arafp of elecvtrons)to grarjing emsxqioa (ù = 80Q, smxll depth 'w àmfp cos 800). Iu. the flmt case,

the spectra are dornsnated by blllk losses. 1'a the second case, it is clear thatthe sltrface plays atl esserttial role for the PES. Beside the miu'n quasipamticle peak desmibing the core-hole excitation witkout losses, the positiou of

6 the core electron bindsng enera, -T P orwlliclt is usury used to d'R ne

pQP ith respect to a reference level, e.g., the vacul:rn level avac or the.-ggs j W

Fprmi level ek- (here: ss), the spedznlrn h.as an incoherent satellite structme.In dnbonded solids the satellite stractures are mainly due to shake-up ofsllrface aud blllk plmsmons. At the side of higher binr?îng energies, multiple

(n = 1, 2, 3, ...) losses by plasmons are Hdeed visible. The strongest satellitesare due to blllk pMmons at about ssizl .-nop (J = p, .s) with Fopp rts 16 eV. In

Si2s Si2pAop

Ot==71 pmtop.

= rM<- :3 pdhop-è I .

-i;5 j I bhœp- Icq) pzhrop-ecqs j,l j l-

8c0

I 111 fûs l

1 I

250 200 15D 100Binding energy (eV)

Fig. 5.8. Si2p and Si2s photoelectron spettra of a Si(11l)7x7 sarface meaured fora latge escape depth aad under surface-semdtive condîtions retative tr the Fetm!level. From (5.242.

the lower ctu'ves of Fig. 5.8 (memsmed tmder sarfMe-sensitive conditions) thebroG satnlllte featttres s110w shotûders at the position of the sarfce plxqmon

energy M?s F4$ Mzp/W. Figure 5.9 r'hows a similar photoelectron spedrum fora qirnple metal, Mg (4.29).

20

Gh Mg2se=c 1 5=

.d 5: 'G VV2?

%'-e 10A*Qc N%2 5

080 60 40 20 0

Binding energy relative to cî; (eV)Fig. 5.9. Mglp and Mg28 photoelectron spedra excited with AIAQ racliation (F?zd =

1436.7 e% relative to t:e binding energy cf Mg2s -sQaP = 49.8 eV. BG andsurfKe pbnmon losxs are ixdicated. Dom (5.25)-

() .J Mèllly-otlkly rllecb:

Suc.h satellite stzmctmes cm.nnot be observed in spectroscopies probing theenergy reg-ion arotmd the hpndnmental gap az in the cmse of STS. The satellftestructures i'a photoeelectron spectra of core electrons are due to two dferentmechazzisms. The main lossas originate &om Ax(c). TEey are a cozzsequemceof the polairization of 1he electroic system in the presence of the core holeand are probed by the photoeledzon before it leeaves the solid, This gives riseto intzinsic losses (5.18, 5.211 . On its my to the sttrface and csscape 9om thesystem the outgoing photoelectron itself pohrlze's the clcctronic system. Thismfvtbxnhm givœ rise to extrlnKîc losses. Between these two 1'.0% of lossesthere is quant:trn-mecllanical interference (5.20, 5.211. This hterference (thevez'text corrections beyond the single-pazticle approac.h in (5.14)) results in a

strong suppression of the satexite stntctures (5.20, 5.26), at lemst in the lsrn5tof 1ow klnetic energies, as compared to the predictions witbx'n the three-s'tepmode,l (5.162.

Neglecting extzinsic and surface losses the experimental spectra izt Figs. 5.8azld 5.9 may be represented only by the core-kole specvtral Alnction in the fo=

(J = s,pj (5.27q*

r pWaTzltc) = )((; c-7 J@ - sozy + zzhfggplj (5.16)V

a=0

apart 9om a broadenin.g of the main spectral li'ae and the sateltite daeto fmite lifteirne and plmsmon dispersion. The pcsition of the ma,in line,

QP z:ï is zenorrnn.li'zed by tEe same interactions that generate thegc; = fa2 + 2I,satellites, hence J9 = lcl/&azp. The center of gravit.y of the spectmnm (5.16)$s identical with the 'lnrenormnlsqed energ,g szl. Meanwlkile, rather completeshgle-particle Gcitation spectra have been calcttlate for semiconductoz.s audmetab (5.28-5.30q, at least for electrons ar.d holes excited in bulk states.

5.2 Many-Body Efects

s.2.1 Quasiparticle Equation

ln Sect. 5.1 we have seen that in important surface-secitive spectroscopiessucà as STS, PES, ard TPES the single-particle spedral-weight flmctdon

.A@, af; c) of the system or, at least, parts or matrix elements of it, are

probed. As a specetral hlnnctio:a it is a Herrnstiazt qurti'ty:

X*(a/ z; c) = .A(z, tr/;&).)

It 'F!11671i' importani stlnt z'alcws- The completeness relatsion ca'a be written inthe form

+x

dsaitœ, z'; c) = Jtœ - z'), (5.18)-X

202 5, Elementaz'y Extitations 1: Single Electronio Quasipmiclo

whic,h describes the couervation of the nllmber of skgle-particle states in-dependent of the treatment of the elecwtzon-electrozl interaction. The con-smation of the nltrnbe,'r of electrons gives the dectron densit,y (3.47) irê theform .

+œ '

n,@) = 2 J d&.f@)X@, œ; s) (5.19)e-X

with the Fermi functon fLej and the factor 2 cotmting foz' the sph degener-a'LF.

The spectral-weight function is directly related to a more genez'al function,the Greelfs 'hmcdon GLT, œ?; s) of tile system of ixé-rnftfaing eledrous. 'l-%ime>n be seen from the spectral repzœentation or Lehmn.nn representation ofthe Gzeen's ftmction,

+co .g.(m z; a;)G> m/1 f) = dE? 'h

a .et ..y jy ,

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's R'nction de-scibes the dylmrnp'cs of the No -> Ne + 1 e-xdtatiozzs, where Ne is the nlxrnbe,rof eledrons in the syste,m grotmd state (5,72. From its equation of motion withthe m=y-body Hstrniltoml'ym i'acludi:ag the f4117 longitudsnal eledron-eledroainttaraction, the Fourie.r trnnnnformation with respect to the time wiable leadsto the Dyson equatioa 25.31-5.33)

:2e + ip + zazrk - Mon@) - 5k@) &@, Z; &)

- d3a//.)J@, =??; s)J@M a/; &) = 6Lm - =p) (,5.21)1

with tke external potential of the ions Montœ) and the Hartree potentialV'H(z) (3.49). The self-energy ELœ, z/; s) accounts for the exchange (X) andcorrezatioa (C) efec'ts oa the single-particle level It represeats a non-local,complœx tnon-Hermitsxml, and Gergpdependent operator. Equation (5.21)or the correspoadhg homogeneoms oae is e-qlled the quasipmicle equation.The occarrence of tEe self-enersry J7 mctieates the better treatment of XCefects for excitations th= for iudependent particles in the electronic g'roundstate (3.46, 3.48).

' Due to the complicated self-consistent dependence of E' on G, solviag(5.21) is a dlec''ult task even for the simplest electronic model systems, e.g.,jellilxrn with s'zr-lhœ. Moreover, E Yntnsnn the bare Coulornh interactionn(m - :F) = eajjm - a?( to atl orders. The expn.nsion of 2 in the 677d: non-vazsisbl'ng orde,r with respect to 'o results in the Foc,k operator, the non-localexclnnnge icteraction. The ecpxmqion of E in ttarrnK of r Ls slowly convergent.

'

Anotb.ez approlmh is more promising. The electronic system kmdez considera-tion reacts to the pzesence of an excited electron or hole wsth a redistribution

5.2 Many-Body Efects 203

of the electron density i.e., a polarization of the electroaic system. For thatreason it is more conveGent to conside,r an epqmnqiozl of X i:a tnrmq of thedynmically screened Coulomb potential

T4r(œ, z'; rlaJ) = d3a//s-h (z œ&; w).sLm'' - m') (5.22)1

witll the spatially non-local and fzequency-dependent hwerse dielectric h:nc-

tion c-l (œ, œ/; aJ) of the system. The reducvtion of the iuteractioa potentialby screening makes it obdous that an expnnRion of E i'a g/m is more rapidlyconvergent.

The self-consistent dependence of 11' on G is provided by a closed set ofcoupled equations, the so-called system of fhlndamental equatioas (5.% 5.31Jwhich, ia the simplest approximation that iacludes dyzmmical polarization

processes, are decoupled 'by neglectiug the vezte,x corrections. !(n this case Ecan be wri'ttem as a convolution integral

i +x . +E(œ; œJ; c) =

s dec-oo G@, m'b e - &,J)W'(z, z'; &s), (5.23)

-K

where the vez'tex corrections have also been negiected in the calcqzlation ofc-1. This corresponds to a description of the screensng witbs'n the raadomphmse approvlrnntion IRPAI (5.7, 5.30, 5,31). Tlb.e resulting approvirnation

(5.23) is called the GW approdmation. Unfortunately, > strasghtfoNard tm-provement of the method is Himcult. A furthe,r ex-pn.nqio' n pf the self-energyi'a powers of the screemed interaction may yield tmpbysical resuks such as

negative spedral h:nctioxxs (5.33). I1l fact, the expxnsion itse,lf is only condi-tionally convergent due to the long-rauge natme of the Coulomb potential.S f there is no systema' tic way of cboosing wizic.lk diag'rams to sum to goo a,r

beyond the GW approxlmntion. The choice is usuatly dictated by physicaliutuition. Fortunately, already the GW approimation gives extremely goodreutts for quasipartide energies and spedral distributions.

'When W' in (5.23) is replaced by the bare Coulomb potential r, the rœult,-irg modiâed expressioa leads to the Fock operator for exelns.nge. The energyintegral of the single-particle Greeaps Alnctioa in (5.23) essentially gives theexchange demsi'ty, whicll is related to the of-cliagonal elrents of the spectralA'mction in contrrt to the eledron density (5.19). Consequently, the dfe'r-emce of the t'wo self-energies allows one to exctrad emsily the correlation part

of the shgle-particle self-enerr

$ -swo+gtw g? a.xo,y jw.ta, ttgg ss; - z,tz - a/)! ..S'C@, =/; s) = dFwJe ,2x

(5.24)

5.2.2 Quasiparticle Shifts and Spectral Weights

Apaz't from the inhomogene'ity the quasipacticle equation (5.21) is similarto the covesponding homogeneous Kohn-sh.am equation of density 'htnc-

2û4 5. Blementary Excitations 1: Single Blectrozkic Qaasiparticzas

tionnl theory (3.46). Only the XC self-energy operator X@j :r'; s) has to bereplaced by the Kolm.-shil.rn XC potential Fkc@ll@ - m/) (3.50). This sug-gests the reprexiation of the swtial dependence of all q',smtitie in tez'msof the oabonormallzed arad complete set of eigenAtnctioas (W:@)J of theKohn-shnm equation (3.46), including akso those of the empty statas, for a

surface system with 2D traaslationn.l symmetzy. Another reason is the wideavailability of the Kolln-slmm (KSI eigenfnndions. The majority of the op-t'rnsqations of the st:Hace structuzes and the cazculations of total en4rgies isbmsed on DFT i'a LDA (or GGA), The Kohn-shnm equation is always solvedusing these Rnnctions. TMS lp.1rl!; to the represeatation ..

Glm, tF;s) = X'.I 7-.7: t?vzzzli, s)#,:(=)#s!zx(œ'), (5-25)ukvl jp

as for all other quaatities, e.g., A@, œ'; s) and J7(z, m'; s).The og-diagonx! esements of Gwv' (/7, ej azzd rvrzzlij s) Mth. respect to the

b=d index are respousible for the fact that, in general, the quasiparticle

(QP) wave Ganctions are dieea'e'at from the Kohn-shnan ones. However, forexcitations near the h'ndxmental gap of b:pl'lr systems the near equivalence of

QP aad KS wave G:nctions has been sbown (:.34j. There is experieace thatthe equivaleme remains 'valid if the energetic ozdezing of the KS eigenvaluœsv(i) as well as of the QP energies CQ.,P (:), i.e., the posjtions oî the mainpeaks in the spectral A'nctions .âs,vti, c), is the snme. However, also in cases

of violation o' f tahis ordering, as in the case of the GaAs(110)1x 1 smface,the eect remai)ks small due to the small hybriOation of surf-related audb'ttk-related KS states (5.354. In generalj however, the QP wave ftlmctions cor-

responding to nearly degenerate states with diferent degrec of surface/bulkloemlszation may be h'nfluenced by a stronger hybridization, mnldrtg the QXwave h'nctions signlcazttly dfaent 9om the KS ones. Dxerences betweeaKS' and QP wave îtmctiozzs can be eoected in any noxsb::l'k system, i.e.,when there are resons of space where iEe electron densit.y goes to zero. 1:athœe zetons tlte LDA generaœ an XC potatial with iacorrect asmptoticbehavior (e.g., a rnsming k/r tail izl the caze X atoms).

Despite the asmnrnption of band diàgonnulîaation the resultizfg Dyson equa-tion can only be solved approvîmately via an iteration procedure, shce thepertatrbatlon itself Ls a, A:mctional of the Greelps fl'ndion. The dhgrstm-rnxtic representation of the self-energ.g contzibution iu the GW approv'ma-tion (5.23) Ls showu schematically iu Fig. 5.10. The solid electron line. rep-resen.ts the Gleen's Tnmction G, while the dnAhed line indicates the scrïvmedCoulomb interaction W. ln order to begia witù a Green's G'nction G0 (i,c),VV

the spectral fltnction of which is close to the true oneh a shl'ft 1v(k) of the .KS eigenvalue to

sQv P(:) = sv(i) + z;z(t-) (5.26)

is htroduced (5.36, 5.374, loding to

5.2 ManpBody Efec'ts 205

+ %.< h

+ xu' h

# hJ %

2 %#' *

#' %; N

k v'k' kv v

f'ig- s-10- Scllematic repraentation of the self-energy contributions v'itlzin theGW approvsmation. The eledron (hole) is represented byl a solid line, whereas thedashed line indicates the dynnrnscally screened Coulomb mteraction.

lG7v(:,s) -

s - sr(&) + iw(:)

with w(i) = +J (-J), J -> +0, for electrons tholesl, With (3.46) and therepresentation (5.25) for the Green's 'hlnnction and the seklenergy the Dysonequatiol (5.21) reads

Gu(:;s) = G2s(t-, s) (1 + pz'z,.(:, s) - zv(:)j ctv(&,s))with the diagonn) elements t'EvvLk, c) of the rtarnnlnsmg XC self-energy (irLthe GW approvsmxtion)

JJ7@, =/; c) = ELm, zJ; c) - J?kctœll@ - œ')

zzot included i'a the KS muation.Restridicg to the 'Ilrst non-vinishing order 'with rezpec't to the pertmu

bation /JCv(i, s) - z1.(ï,-)q , in whîc: the Green's fnnction G (5.25) in theself-enerr E (5.23) has to be replaced by Gz (5.27), ozle obtnsmq a ptmmaad

the right-bnnd side of the Dyson equation (5.28) being quadratic i:a GD.ozt

1n. order to avoid ata ucphysical double pole of the zesulting Gzeen's Atnctionat c = aQw P(i) in this terrn, one hms to set

zv(&) = Rc è'Euuv @; cQ.P(&)) . (5.30)

ht this way the nlnnlrnown quasiparticle sbiêk in (5.26) is now defmed by

(5.30) which. llas to be solved self-consistently. The restriction to the Green'sfntnction GQ (5.27) 'with the shift (5.30) cau be identfed with a Srst-orderpertmbatjozstheozy treatment of the pelurbstion 5.8 (5.29) beyond theDFT-LDA. Together vrith (5.23) azd (5.28), exwession (5.30) givez an ex-

plicit de6rition of the QP shkft, whic.h correc'ts tlle corrcxspondsng KS eigen-

1r. addition., îzl many practcal Gculations the energy dependence of theseY-energy matrkx elements is asstïrned to be linea,r arolmd c = cQvP(J&). Thjs

. gives a renormalizatioa ,t1v(:) '= a.(:)Re J&0.(i, s.(:)) where zv-l(i) =

(1 - t'iRefvovti, g)/&Ie=a.(:)1, 3,e., Avith a denominator of nearly the spectral

206 5. Elemeutary Excitations 1: Single Blectronic Q''nmiparticles

0.40>..Q 0.35 , xsc' Main QP peak

peakX' 0.30 Nzvtk) ,.1w gsgg2= 0-209 (ufetirnel-l 'w > <:e$ c lsc '

.Q-6 0.10a;z Plasmon satetliteOo 0 - 05&.@) g0-0

sîotkl <k) cp

Energy E; (arb- units)Fig. 5.11. Schematic representatioa of a quasiparticle spectral fntnction in theGW approvîmation for a smgle-pardcle Kxdtation below the Fermi levez (hole) . Forcompstriqon, tke spectral fanction of the non-interadin4Kohn-sha,m particle às alsoshown. The spectral weights of the main p'alm are Mdzcated.

weight of the mn.5'n QP peak in tEe spedral 'hloctioa (see Fig. 5.11). Detailsof the computations of the qumsiparticle shsfts for bl:lk aad surface s'ysteulsc--m be fouad ixl sever? redew articles (5.33, 5.38, 5.39).

Much in contras't to the QP shifk (5.30), the spectral Annction .k4.zz(i, s)of the Greea's fllnction (5.20) is strongly s'nBuenceê by the dmnmics of thescreensng. The flrst iteration of the Dyson equation (5.28) leads to

zs-(s,a) - g, + s' aessts,slj s (s -sr(:))

,,? p 1:i: v p x) -xJ=zJ2-(:, c)

s -ss (for electroas (+) or holes (-). The smbol P iudicates the Cauzhy principalvalue. Suc.h a spectrat Alnction tsee Fig.5.11) represeats a sharp peak at theQP enerar sQv P @) with a reduced spectral weight, 1 + wa Reycvti, s), andadditioaat broad stzudures x, Imfvovti, s) at shlled ener 'gzes (cf. the sateltestructme' in Fig. 5.11), whic.h may be lterpreted as replicas of the ma.in peakdue to additional losse's.

' The additîonal pealcs describe satellite stntdmes related to the shake-

u.p excitatious (5.7, 5.28) disctussed în Sect. 5.1.3. A set of spectral Alnctionscalculated for electron and hole excitations i:rl the lowEsst conduction bards or

valence bands of bulk Si is shown in F1.5.12 (5.29). For the hole exdtationsthe satellite strudmes are visible at large.r binding energies. This was alreadyshow.a for the core-hole spectral 'hxnction (5.16). Neglecting the band and '

plasmon dispersions the agreement of the entpressions (5.16) a'ad (5.31) is

obvious, at least in furs't orde.r with respect to dynxrnical QP eeds.

(5.31)

5.2 Many-Body Efects 207

io.2 k = (0-5,0-5,0.5) i

0 - E

0.2 k = (0.4,0-4.0-4) E

() '

lc.2 k = (0.3,0-3,0-3) j

. EF0 ic :2 k = (0.2,0.2,û.2) !

' i .

0 '

!0.2 : = (0-1 ,0.1,0-r ) ë

!0 l0.2 k = (0.0,0) @

. i0 t

0.2 k = (0.2,0.0) @E

Q IG.2 k = (0.4,0,0) E

i0

i0.2 k = (0.610,0) jo :

io.2 k = (0.8,04) j()

:

O-2 k = (1.Q,0,0) h0-10 -35 40 -25 Q0 -1 5 d0 -5 0 5

Energy s (eV) )

Eig- 5-12- The spectral-weight flmctions for electron aud hole quasigartides in bulkSi. The Bloch'wave vector varies iong ihe LT and T'x directioas. lt ls given in. uzzitsof z'Vco. The enerq zero ks fzxed at the ezterg of the topmost occupied DFT-LDAstate at T'. A'rrows mdicate tmdamped quampaztkle pulcq. They are J-fanctiozuswith a spectral weigkt smaûe,r than 1. h the lower J'X part the positions of thepeaks essenticy represent the band emergies descdbed in Fig. 3.6. From (5.29j.

5.2.3 Screensng Near Smfaces

Most important for tlle calclzlation of the GW self-energy J;' (5.23) is theknowledge of the drmmscally screened potential W' (5.22). TMs is deterrninedby khe inverse dielectric R'nctiop which js deMed by

d3œ''s(z, a?'; aJ)s-1(=& a/; (sl = J(= - œ'). (5-32)7

The correspondi:ag (longitudinal) dielectric 'h'nction

eLm, œ';fzJ) = J@ - m') - d3acwtœ - a//).P(z&, a/; tt)) (5.33)

îs rehted to the so-cnlled polazization flhncvtion or Heducible pohrizationpropagator, P, of the polarizable electronic sys'te,m (5.7, 5.33) . lt contnlns the

d ible cliagrxrnq of electron-hole paiz's excited virtually or physically asîzre uc

the response of the system to an externf pertttrbation.

208 6. Eltmentaor Excitations 1: Sixgle Blectronie Quastparticles

mydiuy vacuum',,'X',,'X c b q , *

yyyy?) l

The combiuation of the three equadons (5.22), (5.32), azd (5.33) attowsthe derivation of an inhomogeneotzs integral equation for the detnrmînn.tionof the dyaamîcally stzeened Coulomb potem.tial W' (5.7),

W'@, a/iMzl (5.34)= 't7(m - z') + d3œ'' d3œ''%(œ - x''')PLW'' œ'' U;jWLm'' a?; F=).'; 1 ,

Herej we are not interested in the details of the screent'ng re-sponse. Local-heldefects due to the atomic natlzre of rnxtter arc therefore neglected. Mahfy thespatial iohomogeneity due to the presence of the surface should be takext intoaccount. We consider a semi-imqnite solid such as indicated in Fig. 5.13. ThepolnHzable 'meclblrn occupies the negative lnnlfxpace, arzd the interface to thevacunlm is takd'n as the zv-plan.e with z = 0, in order to simplify the coa-siderations. The positive cores aa'e sîtuated witlnt'm the polarizable electroaics'ystem. Wit%sn the jellium model the positive jellillm edge $s displaced by z)mbelow z = 0 (5.40) . T.n the classical electrodyzmrnlcs z = 0 desne tite imageplane, whereas the plltne z = .-a:m nearly deMes the positions of the topmostatomic corem Sometimes, anothe,r notation with z = 0 for the plane of posijtive cores and z = zizn for the t'rnnge plnre is used- Exnmples a're Figs. 5.14a'ad 5.15.

The bnllr polvization propertiœ of cubic czys'tals are described by anisotropic wavmvectmr- and frmuency-dependent didecizic htmction s:(c,&7)with the 3D wave vector q or by a corresponding partlally Fourier-trn.nsformedpoln.rization hlnction .&(Q) z; fzJ) with a 217 wave vector Q. The polarizartion probtem Nms cylindrical symmetry. Therefore: one writees œ = Lp, z) andq = (Q, qz), where p and Q are vectozz parallel to the soace. For metals thesemi-t'n6mste Jezblrn, an electron gas conGaned by a'a l'n6rllte potential barrier?Ls a lequently used model q5.41). TMS approaG can be generalized for otherpolarizable electronic systems. A good s'tarting point is the mssllmption ofspecular qlectron re:edion at the surface (5.42). U the contHbutio:as of thequant'lm-mecalmnical interference terms to the polarization are neglected, thepolarization Alndion of the semi-in6aite system

PLQ, z, z?; fgJ) = &(-z)O- (-z') (2%(Q, z - z'; a7) + .!%tofz + z':u))J (5.35)

Fig. s.13. Iuterface betwee.tt thevacuum azd a polvhable medittm(schematically).

5.2 Many-Body Efects 209

caa emsily be exwessed by that of the inBnite bxxllz. J-n. tMc way the dielectricrespoMe of the halfspace Ls related to the respon- of the homogeneoua btTpt-system. TMs approvsmatîon (5.35) satioes the J-pïrn rale (5.42).

'I'he representatioa (5.35) allows a closed nanxlylical solutîon of the eqaa-tion (5.34) for the dyn.nnht-cally screened Coiomb potential. Oue obtnsnq(5.42)14r (# - #', z, V; M)

d2Q bQ(p-g) 2xd.2 , -vjx-arj .j. (y .x;e-o(a+z)j= J (2,zc)2e Q (t-76z)86''' ) qe

+ s g/tzlO- (-z')c(:, z'! wqe-Qz + eL-z4eLz')a.LQ, a;td)e-QJ') (5.36)+ 0- (-a)&(-z') gctQ, z = z';=) + aLQ, z + z'pa7l - saLQ, zqu?lato, z';w)j ),

where x = 2/(1 + a(Q, 0;t$). The quantity

2Q txl ccstozla(Q, ziuzl =

r j dçz qzsutr, w;

is dkectly related to the Fourier trnrqform of the screened potemtik i'c thebulk. In the limit of a local dielectric Aynction, c:(t2, :J) H sbtts1, one Mds 'Gt,hc(Q, z; :,J) = e-QI>p/sb@) the same screeaed potential W'(p - p', z, F; LJ) mqcalcalated using the method of clnnqical image charges (5.43,5.444.

Inter&tingly, the screeaed Coulomb hteraction of t'wo pazticles outsidqthe poladzable medbxm z, z' > 0, is !e,511 l'mfluenced by the pohrszation in thehalfspace, IEn the vacu.um xegëoa (z, zê > 0) the chauge L'E of thû XC self-eaergy wîth respect to the XC esects already i'acluded ict the XC potential(5.29) is dornsnated by the conelatioa part EC of the self-enerr (5.24),62 rks EC. Negseding the movcation of .DC due to the dyzmrnics in thescreening, asslnmlmg, e.g., cstqawl - &s(ç, 0), the corrdation xlf-ene'rgy XCis detemnined by the diFerence of tNe scree'ned and baze Coulomb potentîal(z, z/ > 0),

(5.37)

, , 2 =

, -<(z+z,) 1 - a(Q, 0: 0)N@, œ ; 0) - nlœ - m ) = -c dQJo(QIp- p l)eo 1 + J(Q, 0; 0)

(5.3b)with the zeroth-order Bessel fhlmction Jo. In the case of a dielectric constantsb(t2) e sb, (5.38) leads to -(sb - 1)/(ss + 1) x e2/ Lp - p')2 + (z + z/)2.. The accompanying change of the ener'gg of a point charge located at adistance Rz from the image pln.ne' z = 0 irt the vacqlllm ks #ven by (5.45)

e2 = 1 - a(Q, (); 0) npg. (5.w)Xcpoëattxg) = :F-g do eo l + t$(Q' % 0)

for dectrons (-) and holes (+). For large dist=ces from the image plnane,

(5.39) results in

2l0 5. Elementary Excitations 1: Single Rlectronic Quuiparticles

cs - 1 62..E'pco:attRzl = :!: . (5.4û)cs + 1 4az .

This Ls the classical image potential ei/-rar of a point charge at distance SaFom a poladzable hn.lfqpace elnxractezized by the static electronâc clieledricconstant sb. In the case of metals witi as -+ tx) the prefactor (cb -,'1)/@b + 1)in (5.40)4 the image chazge measmed in ltnlts of the elementary chazge, ap-pzoaches 1. The image-potential-like behavior of the self-emerg.g dl'seren.ce(5.29) îndicates that outside the electronic system mry-body esects, suchas exchaage azd correlation, cn.nnot correctly be described by the XC potemtial 7xc (;r) (3.50) of the Kolm-slnltrn tlleozy Because of its dependence on

the local electron densit'y, tEe XC potential slmws a rather abrupt variationnear the surface. This ks agaia an arlefad of t'he LDA or even the GGA.The exact DFT should #ve rise to the coz'rect behavior of tlle XC potential.Unfortunately) there is no ex.act XC enezgy fundioai that can be used in ex-

plicit computations. .An bzspection of the elecwtron self-energy E (5.23) ia theGW approvsrnation shows, howevez, that the electron correlation containedin E automaticazy produces an imagelike potentîal energp The Coulombcorrelation outside the surface is more long-range th= suggested by Fxctz).The exchange part still vanishes more rapidly for large distances from theimage plane (5.464. Exactly this behavior ks demonstrated in Fig.5.14 for a

simple metal surface. The key featum izl this fgtzre is that the local potenti?

0.0 .---.

o -0 1 .P'

>s ' m

ê

-0 2 /X .

= . zc #

c) Jt

O e.3 drs l

(.) :>< 1

.0.4 . 'IJII

-o 5 '

-1.00 -0.50 0-00 0.50 1.00

Dislnce z (%y)

Piy. 5-14. Exckange-correlation potential at a sHple metal surface calculated forjezrl!'n with am electron gaz paramete.r rs = 3.9318. Tke solid curve representsa local XC potential extracted from the GW self-enera (5.46J1 and the dottedcmwe is the correspohaing LDA potential. The dashed cun'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 measured iu uztits of the Fermi wavelength . :ê

Atp = 2a(4zr2/9)T7Z (here: ls = 12.9 aB). After (5.464.I

k'

. '

5.3 Qumsiparticle Surface States

correspon.diu.g to the GW self-enerr becomes imagelike outdde the surface,wherea.s the 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-ternal per-turbation is usually prevented by a surface barrier. The formatîon of a stlr-face barrie,r ca'a ewsily be ex-plnsned withsn the jellizlm model of a simplemetal (5.471. The positive ion cores are smeare out to a positive background.h the surface re#on it c-xn be modeled as a stemlike density (Fig. 5.15). The

' electronic wave Glmctions possess tn.!1K into the vamulm. The accompacyiugéxponential decay of the electron density is rœponsible for a redttcvtioa of theelectron concentration in the region of the positive b ouzd near to itsstep. lusid.e the crys'tal the electron density approache,s asymptotianlly thebltlk value tn the form of Fhedel oscitlations (5.484. .Xs a consdquence a netsmface dipole layer appears (see Fig. 5.15).

The fomnxtion of a smface dipole layer mtu'ns that the electrostatic poten-iial far in the vaclrnrn is higher than the mean electrostatic potential in thebulk. The mn.croscopic dipole and the accompanyin.g electrostatic potentialcu directly be obtxsned h'om the microsèopic iectrostatit potential 5Q@):ùr evnmple calculated withsn DFT-LDA (3.48). Jt is defmed by

e% zxO ê %

XM -- V X

. j.g ... xxxz j(U

> j +zx jr yX l:'*-R lg& I

a t: = 0.5 j;H7

Q) ï

'EP d; '7. %

(U %.'4n &

' c. .- X x

. 0 N-

.. -1 .0 -0.5 0 0.5 1.0

gjsjance Z / jF

Fig. 5.1.ô. Blectron densit.g proile (dashed liael and positve backgzound (solidnlikel at a jellium sueace. A metnllsc system with a 1ow electron density, rs = 5ûB)is considered. The Fezmi wavelength Ap = 2rs(4*2/9)1/3 Ls ttsed as the rthxracteristsc'length. After (5.47J .

212 5. Elementary Excitations 1: Sîngle Eledronic Quadparkicles

Vklœl = 5(on@l + kk@), (5.41)the local part of the ex-ternal potential Moatœ) and the Hartree potential5'k@l (3.49). As one is mnlnly interested in the macroscopic spatial depen-dence of the electrostatic potential in the surface normal direction z, it isconvenfen.t to perfozm a pln'nar average

1Ks(z) =

; dzdp7estœl,A

jtwhere z1. corresponds to the area of the surface tmit cezl. T e plan.ar av-eraged fnnction can therz be passed throagh a âlter of lemglh L in or-de,r to ex-trad tb.e macroscopic changes of the electrostatic poteatial (5.492,

Q HTIZ / t U all L is chosen to be tNe rnsn5rnum thic'k-Cesl.Zl = s Jx-zw dz 5%IZ ). Stt y'

ness of atomic layers 'glving clzazge aeutrality or, as a mncdrmlm, the extentof an irreducible cvystal slab tsee Sect. 1.2.2).

Examples for the averaged electrostatic potential are given in Fig. 5.16 forzinc-blende GaN(111)1x 1 with diFerent Gaor N overlayers and SiC(111)WxW surfaces with group-lff adsorbates. In the GaN case the surface stoichiom-etry is changed. The second eexample represents the Muence of dsFeront a&sorbates and adsorbate geometries tsee Fig.4.35). The graphs in Fig.5.16clearly show the ivuezzce of the compound, the surface oriemtation, the sto-iclliometry oz coverage, a'ad the geometzy on the actual surface bn.rrser. Thisholds ixt particula'r for the trNsition rebon itself. 0n the other hand) tlle efectof all these'details is weakened for the potential step .'A? = Zstxl -'Q(-x)

(a) (b)

Q ----

œAA

-5 ê*

l k.

+4(1 / #

> . a?

Q)we -$ 5K . ,w' l

fâ -2tl d @* t 1> I J

j ' '-'e.- In (T4)-:ts . 7 c.o (pl 't .-. n (.q )t ,

; (s )*30 1 ? --'--, $

$ --- B tsub.œ-35

41 :l.,: z': c,iDistance z (atomcc Iayor)

Pig. 5.16. Averaged electrostatic potentials of semiconâuctor sudases 'with difeer-ent stoickiometry (coverage) and/or diserezzt geometry. (llll-oriented sudaces ofthe zinc-blende polytype of GKN (a) and SiC (b) are chosen. The bulk valeace-bandmmvt'vnxrn ks taken as the energy zero in (a), whereas the vacuum levql desnes thezero in (b). From DPT-LDA calculations (5.50, 5.514.

5.3 Quasiparticle Smface States

Atomic bilayerFig. 5.17. Averaged total one-electron potential U(z) (dnmked tinel and averagedelecrostatic potential 5Q(z) (solid linel for a diamondtlllllx 1 surface. The sla'ba'ad 'vamrnrr, regbns are shown. From a DFT-LDA calmzlatîon (5.52).

which ks a dired coasequence of the macoscopic smface dipole. The de-tn5lq of the groupqll adsorption (element, atomic geometzy) have pradicallyno imfluence on the surface barrier. Additionnl overlayezs of Ga (N) on the

GaN(111)1x â sttrface) however, reduce (hcrease) the surface dipole by sev-

eral tenths of an ev (several eV). t'a addition, many-body efects, e.g., theimage potentii (5.40), modif.g thi absolute value of the surface barrier feltby an escaphg electron (see Fig. 5.14) but contribute practically nothing to

c'hanges of the surface dipole with geometry and stoickiometzyExch=ge a'ad correïa' tîon efects increase the surface barzier, ic pazticu!ar

within the DFT-LDA ms indicated in Fig. 5.17. There, besides the electrostaticpotentiat 5$s(z), also the total one-electron potential (3.48) averaged over

the surface plane (5.42) is plotted. 1n. the bltllr region it is much lower inenerr tha'a kk(z) as a consequence of the laœge attractive contribtttion ofe-xchange an.d corzelatiom I'n the b7llk region b0th potentials show oscilhtionsnormal to 1he sttrface. The wjdths of the oscillations are equal; only theirnmpDtudes vary sligEtly. lu this region the dxerence of the (macroscopicilly)averaged potentials is governed by e-xpremion (3.50) taken at the averageeledron derusity. Tllis pammeter is a blllk quantity. It is therefore smportantfor tke absolute energy position of the energy levels, but it doe not smflueace

tlve vadation oî the surface barrier wîth orientation and atomic covgttration

(5.481.

2 .

214 5. Elementary Bxcitations 1: Singte Blectronic Quasipartic!e,s

(a) (b)Energy

Evac

*

'E---' -'-EE);;-''',,,,,------''''''''-''''',-,,,,,,,,,,,,,,,-----,'--'''''''''

0 Z

Evac

X (p

CCBM l

&F ' '-

CVBM

0 Z

Fig. 5-7.8. Baud diagrams of a metal (a) and a rveml'coztductor (b) near a suzfaace,showlng the de6rition of the work hmction *, electron aëzkity xj aud ionizationen'era 1. A possible band bend-tng îzt tite semiconductor case is not visible on the

characteristsc length scale of a surface of a few monolayers.

5.3-2 Chntaceteristic Ene-rgîes .

The surface barrier can be characterized by characteristic ener#es suc,h Ets the

'workfzmction * of a metal or the Lonization ezzergr I au.d the eïectron clrzfàpx in the caze of a norrnetal. These qualtiti% can be measured by Narious

spectroscopies including PES/HVS. ln the case of a metal (Fig. 5.l8a)z the

energy of the highest occupied electronic 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 outsidethe curstal mizht be assumed to be zero, siace Ftzl and 1ts(# approe a

constant outside of the mystal, and the kz'netic energy of a $1.% eledron cazl be

made arbitrarily small. We call this level U(x) = Ws(x), the nacmum Jcpreôcvac, au.d usuaxy z'efer the absolate enerpr positkon of sGgle particle states to

emc. Therefore, the rnt'nirnlxrn enerr 1 required to zemove an electron 9om

the intedor of the czystal to a point outsid.e the crystal wéuld be

(5.z13)+ = cvac - sp.

Figure 5.16 icdicates that the position of fvac is ivuenced by thesurfaco barzier. Consequently, atso * may be influenced by sttrface relax-

ation/reconstraction and nziqorptiozt of other spedes. In the case of non-

metals, i'n particular for sezniconductozs, it may also depemd on the back-

gxolmd doping. Howeverl it is better to avoid a strong dependemce on tEe

doping level and to defne otb.e.r charactezistic ener#es .

X = fvac - FCBM)

1 = cvac- svsv. (5.44)

5.3 Qumsipartîcle Surface States 215

The clectron nmn7't.y x (photoelectron ionization energy J) is the energy of the'vacuptm level referred to the bottom of the conduction bands rcshc (top of thevateace bands svBM). Iu tMs way, the energies of the condudion-bazd rn1'n5-mplrn a'nd valence-baud mn.r'mthm are generally defmeê az quasiparticle values(5.26). Startin.g with the Kohn-sha.m valuesz QP corrediozss (5.30) have tobe added. The dferettce of the t'wo band-edge valuo, Sg = scsM - evnu,defmes the fundamental QP rnerr gap. lt exn be directly compared v-ithvasues derived 1om measured data for x a'ad J. Rmdamental gaps dezivedfrom optical mvuzements are in general somewhat s'mxlqer êue to e'xcitonicbinding eFects. The b=d line-u.p in Fig. 5.18b represents tlle situation for artxlds.qtic seznicönductor for which defects and fmite temperattzie a,re allowed.For ideal systems the introduction of a quantity * (lileren.t from I would notbe useful. Witbin the axact DFT of an ideal imsulator at zero temperatllre,the highest occupied Kohn-sham eigenvalue reproents the position of thechemjcal potential of the electrons (5.53).

J.n Fig. 5.19 the work hmdions for some typical metals az determlmed byvario'as experimental methods (including PES) are plotted vers!zs the averagedistance rs of the electrons (5.48). The small variation in the work hmction6ver tlsferent cnrstallographâc faôes Ls not shown. There is a cler itlcrease of +with i'acrewsiug electron dtanqî'ty (decremsing r,). This tread js well pronouncedfor nlkn.li and p.11m.15 eartk metals but 1% signifcant for group-x metals. Noclear trend ca,n be observed for the other elements, e.gk, noble metals. Their

4-s x (pysrx oF 4.o zjzxf is . x xo 4.x

3.5 ''v

g Nxt= x? s o N.

. N.uR x* xY x.o 2 s . wR '

o . o NN

xo o. x.. : 2.0 Nx

NNQ

2 3 4 5 6Eledron distance r.j (cs).; i'

;E : ..

)q Fig- 5.19. Memsm'ed work fklndioms * of selected metals verstzs average distance'(,E:/ of the electrozls in the system. Data are taken from (5.482. A11m.1't metals: opeu. . .2 .tilrcles, nll4n1l eartk metals; fllled drcles, noble metalm squares, groupqll elements:

siars, group-nr =d. -V dements: open tmnngles, trxnsition metals: 511M trianllesind group-lo metals: diamonds. The dashed line iuclicates a quadratic variatïonihf ô witk ra on average.

2l6

ZnSO

NN

Sh ''---

Znsee, . b'x-

v N6.5 '--..-

#; x.x

@$ x.

6 1nP zn-ry-s œ-reg x.* o x.w o: si GaM x.

55 -------1/1 N'ï ' + *s KR'

lx5 5 x oe-

xx K

4.5 VT

4 .

S.5 6.0 6.5Lattice constant co(A)

Elementary Excitatiozts 1: Single Blectronic Qumsipardcle.s

f'ig. 5.20. Ex-perimental ionization ener#es (opea syzjbols) and theoretical values

(flled symbols) calculated tusing a 61111 GW zpprovsmatlon (5.64) or an'dpprovlmaàeself-enerr (5.50,5.55) verst'ls cubic httice constant,s cm. The data collectzon is takenfrom (5.564. For zinc-blende seznicouductoz's only data for cleaved (11O)1x 1 sttrfacesare presented. In the ca-se of elemeatal group-l'v semiconductot's values for (100)2x1(trianrles) and (111)2x1 (squarc) are shown. The dashed lines indicate a lineazvariatzon of the experimental values witbln a compound class.

+ values vctry only izt the range 3,83 - 4.52 W with elecHzon gms pazametersra gs 2.12-3.0241. The tendency of a:a increase of * with the electroa densityLs in agreement with the simple pictttre of the fozmation of a surface ctipoleizl Fig. 5.15. 1ts strength should be proportional to the dezudt'y of particles.

Measmxd a'n.d calcalated ionization energies are pbtted in Fig. 5.20 verstts

the bulk httice constant. The ionizatkon energies I belonog to a certa.in

group of compouads IIVLIV, III-V, or I1-W) decrease with increasing latticeconstaut. TMs is rnna'nly a consequence of the l/oo-variation of the ppc and

ppr hteratomic matrix elements (3.22) whic,h donninate the valince-ban.dms.vp'm:xo sc4(0) (3.34). However, the ionicity of the bonds (see Table 4.1)a.lso plays a role. For group.W s-sconductors and III-V compounds theexperiment/ ard quasiparticle values for I agree excezettly. In the caqe ofthe II-W semiconductors the theoretic/ values tmderestimate the measuredvczlucs. The reason is not clear. Perhaps the QP shifts (5.30) are not correctlyctescribed by the applied oversimpMqd description (5.552.

Ionization energies a'ad electron Alnities are plotted in Fig. 5.21 for g'roumHI nitrides versus hlezremsing cation size. rfhe signs of these energies are chosenin such a my that the plotted levels represent tlze valence-baud mnvirna an' dcouductîon-band m5'n5rna. Their eaea'gy distance f - 'J' Tves the quasiparticle

gap Sg. The ioaizatioa eneyrgies depend weakly on the group-lll atom. This ,.

is in agreement with the fact that the valence-band ma-'dmum of strongly

5.3 Qumsiparticle Surface States 217

Pig. 5.21. Negative val'aes of the ionizatîon emeroe 1 (valeztce-baud mnvlrnum,

circles) and of the electron n.mn''ty x (conduction-band rnlnimtlm, squares) for cubicgroumTH nitrides. The level positions wah respect to the vammm leveâ have beendeterm':ned for (11O)lx1 surfacc in the framework of DFT-LDA (5.57j. They aresllifted by the relevant QP corrections (5.50) computed within a simpliûed mofel(5.58) .

ionic compounds is essentially detenml'ned by the anion. The electron xmnlty'varies dramatically with the cation size. Tlb.e variation mn.lnly reûects thehuge vadation of the fandamenti ettera gap along the series BN, A.IN, GaN,IZlN 25.591. The vatues for tlze elec' tron aEnity of AIN aad, i'a particular, BNare smalt but not negative. One may, however, eomect tbxt a negative eledronn.mnlty can be realized by the adsoation of suitable atozns.

5.3-3 State LocnliRation

The surface of a solid, brexlrsmg the thrce-dimensional peziodicit'y of the cvys-tal, 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 Brillotaimsone instead of the 3D vector k, represents good quautlnm nllmbez's (Sect.1.3.3), the actual surface stnzctme azd chemsqtzy mod.if.g the electronic stntc-ttzrc as indicated in Fig's. 5.16 ar.d 5.17 for the one-electron potential. lt isobvious that in the neighborhood of the surface the eledronic wave Alnctiomqare dferent 9om the Bloch waves of aa i'n6nite czystal. In the presencc of astaface the wave fttmctions of valence electrons are of two maia types:

i. The electronic ekgenvalues of the system with surface do aot correspondto Bloch ene,rgiœ of the l'n6nnste bulk crystsl. Thei.r enezgies lie iu theforbidden regfon of the. projected bulk b=d structure. The assodatedelectron states are accordingly locn.lszed at the pertmbatioa, the surface,

218 5. Elementary Exdtations 1: Single Electronic Quazipartides

(ét ) (th)

NuûWœX

1M .

e =w ..-**

7* . --*

> surface resonance --.- - c(:n

......... q..;:5u'o Stateœc >

Rbou surface

cta

wave vector E ----. coordlnate z ---->

Fig. 5,22. The tlzree type of eledron state of a trystal witk surface below thevacuum leve) (sckematically). Electrordc bands (solid lkinesl are skown (a) togetherwith the projec-ted butk b=d structure (hatched rebons). Re,al parts of the cmre-sponding wave ftmcdozus are also sketched versus the H'leance z &om the surface(b)-

and deOy exponemtially 1nt,0 the bulk tsee Fig.5.22). These sùates arecatled ùo'urd stöac.e sfxees and result i.n botmd surface bands when var./-ing the 2D wave vedor throughout the sudace BZ. Evnmple for realbozmd states are #vem ia Figs.3.l0, 4.18, 4.24, 4.29, aad 4.37. .

ii. The eigenvalues coindde with energies iu the allowed reson of the pro-jected bulk band structure. The amplitudes of the associated wave ftmo-tion are usually onhauced at the badace lsee Fig. 5.22) due to rœznaacebetveen surface azd bulk state. Mlvlng or Hybridizatioa of these statHis axowed for appropriate symmetries. TKe resulting states describe s'ur-Juee resoncr?,ce (sometimes antirosonanaj statœ. They decay into thebulk. The degree of the decay depeads on the actual state and n@n vazybetween tEe two extreme cases also showr in Pig. 5.22.

The criterion of the euea'gy overlap of states atlowed in the sezni-sn6nitecrystal and balk Bloch states is Tathe,r qttabtative. ne loexlizzktion of dectronstata near a surface is essentially êetnrmlned by the surface bamn'e,r axd it.smate.idng to the potential enezor of the electrons in thc bulk tsee Fig.'5.17).This e-'m be dronstrated in simple model Yculatjons. For a fzxed k m usea onedimensional repr-ntation to model the total potential F(z) felt byan electron. According t,o Flg. 5.17 tke potentîal deep inside the crystal isdescribed by a cosine variation 244 costzrrz/cl with lattice ,cozustant a azd

5.3 QuMiparticle Surface States 2l9

(b)

-a 0 z

V(z)Vco

2Vo

0-2tz -a o z

-2Vc

Ylg- 5.23. Model potential enera of an electron with Và > 0 in the presence ofa smface. wko diferent match>g positions z.o = O (a) a'ad zo = -c/2 (b) of theperiodic potential and the stepike bmier are cahosem.

amplitude 2W. 1ts average val'ae may be used as the zero of enerr. One rnnyimagine that the attradive ion cores are located at the minirnn. positio:as

.1 '

1 -2. ) if W > -1 or a = na (n = (), -1, -2, ...)z = (zz + zlc,o Ln = - , , ...

if Fh < 0. The surface around z = 0 îs modelled by matcht'ag aa abruptpotential step V'x = kr(x) (wlzici approaches the vacut'rn level cvacl atz = zo to the blllk potential. The resulting total potential is

V'x for z > zo7(z) = .216 costzrrz/c) foz z < zo (5.45)

lt Ls evident that the sig.a of 'Ui and the matnbfng position zo are extremely im->ortant for the locplization behaWor. T'wo emxn.rnplcs aze sketched in Fig. 5.23.W > 0 is asmtmed and the matchsng coaditgozt ls chosen to be p = 0 (a) orzo = -c/2 (b).

The motion of the cledron is described by a one-dinnensionaz Schrödingerequation of the type (3.1) or (3.4)

n,z d2

(- 2,,., dz2 + V'/)) #6Z) = fV7(I)'

The eigenvalue problem is solved within the nearly-lee-electron INFEI ap-proxsmation (5.48j. In the weak bondl'ng l:'ml't the eignnflmctions and eigen-Mues are oaly Gghtly flifkreat som. those of a 1ee electron îf the wavekector k (along the z-direction with -x(a < k % A'/a if restrided to the BZ)ls B a rvection pln.ne i.e., near a point k = ::Ex/a,, +3x/c,... . Ifnot near a r gg ,

however k is neaz suc,h a point, e.g.? 'Vc (as we vdll asmlme i:a the fozowing),iAz and ei(A-2x/c)a are xlmost degenerate. The truethe f:r% electron states estate is nearly give'n by their linear combination

#v.(z) - Aekkz + Beïlk-anlntz

Yhe rculting approximate eigenvalue problem reads aa

(5.47)

(5.46)

220 5. Elemextat'y Excitations 1: Single Electronic Qumsiparticles

21Vcl

Wave vedor k

F'ig. 5-24. The 'two lowest eaea'r bamds (solid lines) of a ID m'ys'tal in the NFEapprovlmation (schematically). The dashed lines indicate the free electron bands.

. :'

2 '

-4-.7:2 - svtk) 11 ,4am z = () (5.4s)Vi Kz

m ( k - REo ) 2 - s v ( k ) B

with Fo is the matzix elememt of the bulk potemtial with the cosine variation

(5.45) azd the t'wo h'ee-electron states (5.47) . This problam can be easilysolvect (5.60-5.621. Near the zone b. otmdar,g k = j+ t% the two bands take theform

a a urp, 2 x 2

acu ( 'j + q - ). (j ) + nz + ( v vol + 4a2 (-c ) . (:.49)

The degeneray is lifted and the free-electroa bards spht. There is an enerprgap of 2(W ( at k = E (s = 0) between the t'wo bands. The two bands are

shown in Fig. 5.24, The correspondiag eigenstates are

#cu.I+x(z)1 = 5,2 x 2

= Aeinz eioz + -v s:lc (-u, + s) -

zm (-c + zc) e-i', .

o(5.50)

The resulting banct-edge states foT n = O aud 11 > 0, 4:k.: (z) .x, cos (m.z) and- sbl (Iz), can be described by trigonometric sanctions.

,

ln 'contrwst to the ba,l'sr cmse, in tke presence of the surface altowed eigen-l z

states can occ.ar witbi.n the .hlndamental gap with energies s, -D=a.(I) - 1% l <2 2 ' '.

s < -#-zw,(I) + L5i L. They can be determiued by applying the methodof the tomnlea àarlé stmtctnre :5.6, 3, 5.64), Allowing complex wave vectors

5.3 Quasîparticle Smface States 22l

tz = -i(2 (0 < q < L'Zoj Pwq.s rr) ) i.e., k = ij - iq, the solutions (5.50) inside thecrystal z < za clescribe waves decayiag L'ato the czystat (z < O). The contin-ul:m of states (5.49) is sometimes also called the conthmlprn of drtucz gapdfafcd (ViGS) (5.561. They play an importaut role in the explanation of theelectronic properties of metal-semiconductor contacts by meial-çndnced .:c7)state.s (YGS) (5.561 5.641 . WMCJI of these states with energies (5.49) actuattyexist, depends on the botmdary conditions at the interface.

Ia the surface caze represented in Fig. 5.23, the wave Amctions outsidethe crystal (z > zc) must be exponentially decaying fxlmctioxts tc:1 < V'x)

2m,.$e. (z) = D exp -

a (1rx - sslz . (5.51)

The t'wo general solutions for z < za (5.50) azd for z > zo (5.51) have to bematched at z = zc. OT1c hms to assltrne a particular potential defmed by zcand W. Exn.rnples are plotted irz Fig. 5.23. The matching conditions requirefor the wave Rlnctions and for their dezivatives

o;!v/a,-sçlzol = '/u (.zn))d'lizzbmla-bq tf#a.+

= - . (5.52)dz =zo dz

a=z,Z

The two equations azow tlle detomniuation of two free parn.meters, the enerreigenvalue (5.49) or the wave vedor q and the ratio of the coeëcients A(D.

Extemded studies of the possible solutioms of the mat '

problem caa

be found in (5.62, 5.65j. For the kwo situations represented in Fig. 5.23 one

fmds:

i. If the surface is positioned smmetrically with respect to tke atoms

(Fig. 5.231, zo = 0) a botmd surface state exists for Fi > 0 but . notfor Và < 0. T*e bolmd surface state (ex-tsting for 'U8 > 0) is deziveâ fromthe analytical contizmation of the lower baad.

ii. The situation is revereed if the surface is not symmetrically located(Fig. 5.23b, zo = -c/2).'A boan.d surface state only exists if Fà < 0 andnot 7i > 0. This state is also derived fz'om tlze analytical continuation o' fthe lower band.

Anyway the trivial ex-numple described by the one-dimensional Schrödingercsquation (5.46) with the potentiat enerr izz Fig. 5.23 makes it obviotls thatsmface bolmd states, which decay into the blllk- and possess eigenMueswiths'n the Apndamental gap of the bulk, may exîs't, depending on the ac-

tual sttrface barrie-r.

5.3.4 Quasiparticle Bauds au'd Gaps

The grotmd-state calculations within the DFT-LDA (or DFT-GGA) allowfor an accuzate deterrnl'nation of many soace properties, in particular stuuface geometdes. For these geometdes the Kohn-shnam eigenvatues (3.46) are

222 5. E:ementat'y Excitations 1: Sinryle Electronic Quasipaaticles .

automatically detvmlned. However, there is no rigorous justlcation for theinterpretation of the Kohn-sham eigezwalues as single-particle excitation en-

ergies. A)l spectroscopies discussed,'sue,h as STS, PES and EPES, are related

to the removal or adclitkon of an electron. The cozaupondiug excitations are

desczibed by spectzal fnlnc'tions (5.15) the rnn5n quasiparticle peaks of whic-h

az.c Imt located at the KS energies &;z(i) (3.46). Ttather) the peakpositiotas de-fm.e quasipatrticle ener#es s,zQP(i) (5.26), whic,h are shifted by .X(i) agnsnqt

the KS values. These sllifts cnn be computed in a pertmbatîon-theor.g maze

ner (à.30). Mcanwhile, there eldst many calculations of such QP shifts for

semiconductor surfaces: including the detemmlnation of complete QP band

strudmvs (5.66-5.77).The Kohn-shltrn surfacestate energies clksagree with exparlmental obseo

vations since (i) bacd gaps bet'wee,n empty a'ad occupied surface-state band

energies are too smals (ii) the dispersion of the DFT-LDA smface bands istoo small i'n some cases, too Iarge in others, acd (iii) the placemen.t of occu-

pied surface-state energies is in some cases too high by 0.5 - 1.0 eV relativeto the bu.lk vcklence-band rnnixn'lrnlrn I'VBMI (5.66j. Thre'e missing pkysical

efects are crucial for the corzect' energ.g position of the QP states and, hence,

must be considered to remove the DFT-LDA failmes. First, the spatial non-

locality of the self-energy operator (5.23) is more sensitive to the localizationroperties of surface states tha'a tie only density-dependent XC potentiatp

(3.50) of the DFT-LDA. This requires a proper accoant of the nonlocality of

the Green's htnction G (5.20). This nonlocality leads to a modifed disper-sion of the quazipeicle enerr bands teoughout the surface BZ. Second,the inclusion of local ûe.l(ts due to the presenee of the surface i'a the bwerse

dielectric Annctîon s-1 (5.32) and, hence, in the screened lntezaction W' (5.22)is crucial for the QP approach, siace these local Vids describe the strongly

imbomogeneotzs screeniag (bulk-like versus vacullm, see expressioa (5.36) at

tlle surface). Third) a'a adequate treatment of the dyaamical efects în the

screenl'ng is more împortant than i'a the bulk cue. This %M.q to d.o with contri-

butions fzom botk bttlk and surface plasmons and the smaltez energy distanceof botmd s'arface states to the Ferm7' level. A11 these efects are importaat.

Fbcusing o:a only oae esed, e.g., the loewqlszation of surface states, ennnot

#ve a gegerally correct nanmwer for the QP shiAs (5.55).Qumsiparticle b=d s'tractures of intrinsic surface states i.n the prototypicat

elemental sernlconductors silicon and diamond are compared with DFT-LDA

electroaic structttres in Figs. 5.25 and 5.26. The 2xl reconstruded Si(I11)aad C(10O) sktrfacu are selected as evnmples (5.72,5.761. A buclded zr-bondedc'b.a.iu model with a positive bacldizk (Sect. 4.2.2) or a symmetric-dsmer mode:

(Sect. 4.3.2) is applied. In the cmse of Si(111)2x1, Fig. 5.25 als: shows bo'and

surface-state bands measmed by direc't and Yverse photoemission (5.78,5.794.1.c the case of C(100)2xl expevlmental data (5.22,5.80,5.81) are not izzcluded

izk the îg'ttre because of the presence of hydrogen axd contradidory Snd-iags. Figare 5.25 makes the priucipal situation obvio'as. Just as it happeas

5.3 Quasipartkle Surface State.s 223

2 a) 2 b)> .

-

yy.jg 1IIpgIj-6!2- -1 .. 1 r:, . ,> .

:) Enlil down zn Jp) down rJ K utyu>-o 0 uq, 0r D DupLu up

-1 -1

:

Fig. 5.25. Kokn-sitam (a) and quasiyarticle (b) band strtzcttue of the Si(111)2x1sudace. Thc hatched areas denote S1 bz'llc states. Fkom (5.764. Th: dots denoteexperimeptal data (5.78, 5.795. Dup and. Dcjowu are explnsned jn Sect. 4.3.2.

h: bltllr semiconductors, DFT-LDA is unable to provide ac accuzate descdp-tion of the ba'nd structure of these sarface êtates. Ozf.y the inclusion of a6111 masy-body treatment of the singvpazticle problam by usicg the GWapprovlmation (5.30) allows the reproduction of the expersnneutal electronicstmzctme. ïn fact, usually QP corrqdions may be even more important tbn.nin bulk semicondudors (5.821.

The igurcxs, Fig. 5.25 and Fig.5.26, iudicate general tendencies but alsospecialities for botmd sudace states of elemental semiconductoz's. Ln generakthe quasiparticle sblh:s of empt.y (f1led) smface states a,rç positive (negative).However, with resped to the bnl'lr VBM the result depeuds on the magnitudeof its negative QP shifi:. The empty surface states Ddo' wn and r* are shiftedtowa'rcls higher enerdes, while the relative shifts rwith respec't to the blll'lrVBMI of the occupied Dap an.â 'zr bands are small. The sign of the net sb7'f'hfor Dup (x) is positive (negative). E(n any cwse the indirect surface band gaps atJ ,-+ 0.57.8 azkd 0.25,/./1- --> X- are opened from O.4 ev to 0.7 eV (Si(111)2x1)

:

#ig. 5.26. Kolm-sha.m (a) and ql4asiparticle (6) band structtue of the C(100)2x 1suface. The shaded re#ons indicate the projected bulk band stracture. From (5.72).

224 5. Elementat'y Excitations 1: Single Electronic Quasiparticles. '. . : . ' ' . .

. , E . . :. ' .. j :.ù .7 . osj .. ... . . . . :: k : . l ' . . . . x' :.

' ' ' ..

'. a : .

' ç ' : : . ..

' ' . 5.. . .' .t b .J.(M . . I G # . . .

.'.. '.u.i:.J'p

'. .' i7 . '. 2... .'. .1 : . 1. '. ,' ; ' ': V:.k è a ; .

.T b(j; .i. :;: . .. .. t:l s .

.....: ..:... ' . . .. . , ' ;. .. ..r. ... : ;

, . .'.' J /?I (jj ..... .:.. Li. . ..

;... .x. ;:=... .. , ::. '... .

. ... .

.. .: ..z 4:91 . '

. . k . .. . . . . : ... > g . . : v' ;< * . '

2 ..' k . .

'.s 71.. . ..

. ., : k

'< e k.'. ,.V . .

' ' . : . .

. .

œ . :( '. . . :': '. . .

. %. .

>2.w ' : , , .k . qk ., ' , E . 2 . : .

r '

,. . p ' . . . :F . ' .. < . .

. . . . . ., : . .

mx !. . . . . . v . j> $ .5 ' cO 3

<'e C3>U).. 0 . 5(Dc . a . . x''k5 s A

' -5 al.u a à a-0.5 a ' 'a w . Go :

la : : ' FSLi e h : .: : : l . , ; M : ' : a :: . , . . .

-1.5F' X M X' F r' X M X r'

Fig. 5.27. Kohn-sham (a) and qumsîprticle (b) bard strucure of the IxP(110)1x 1surface. The slmded regbns indicate the projeded 61111: band structure. From (5.83).The tziaugles (5.842, squares (5.851, azzd circlœ (5.86J denote measured band energies.

or from 1.6 ev to 3.7 ev (C(100)2x 1). The openlngs of the indirect blllk gapsare somewhat larger (0.6 eV) or smaller (1.7 eV). The dispersion of the sudacebaacls is akso iHuenced by the quasipeicle character. The effed is rathersmall for Si(111)2x 1 (5.68) and the g'* band of C(100)2x 1. However, there is

d tion of the dispèzsion of about O.2 ev for the 'n- band in the diamonda re uc

fase. '

The situation is somewhat dlferemt for rnn.ny sTlrfacœ of compound semi-

conductors. As examples the KS and QP ba'ad structures are preseated forthe relaxed TnP(110)1x1 s'arface in Fig.5.27. Thks surface is nhnracterizedby almost resonant C& and Xs smface-state baxlds (see e.g. Fig.4.l2). TMS

energy overlap between the empty Cz b=d and. the bnllk- contdudion bandsis somewhat enforced wititin the quasipazticle pictme. Efectively the emptyCs surface-state band i.s slightly more shifted to Mgher ener#es than the blllrconduction-band edge. There is also a small downward shift of the energiesof the occupied surface states. Deospite the energy overlap the Cz and Jlssuzface states essentîally keep thejr localization at the smface. The orbitalchazacter of the occupied dangling-boyd states ié p-like (at least at X), a'ad itis localized on the smface anions. Thi tmoccupied danglinobord state is alsoprimarily p-like and is localized on the suzface cations. However, izt general ahybridha' tion of surface and blllk states due to the of-diagonal elements ofthe QP seklenera (5.23) cxnanot be excluded L5.351.

Characteristic QP gaps between surface states are Jzted Jn Tablc 5.1.They are compared wjth gaps obtained within the Kolm-s>nm theory and

gaps measured by combhmtion of PES and DES or STS. The small direct

gaps of botmd stlrface state within the flhndameutal gap öf the b:tlk ele- '

mental semiconductoz's a're considerably opened by the QP corrections, The

gap change could be 100% or more. In most caaes the zesulting QP gaps are

ia excellent agreement with measttred values. On average the discrepancy isabout 0.l eV or less. The situation is dsFerent for the more or less resonaat

5.4 Strong Bledron Correlation 225

Table 5.1. Almost direct surface-state gaks for selected semiconductor surfaces. AtrlmKition energy is characterized by the k point in the smface BZ. Three values(a1l in eV) are 'gwen: the allerence of the KS eigenvalues, the quasipaakicle gap andthe experimental value obtained 'by combination of ARPF.,S and R'THPBS results orfrom STS.

Surface '

, L point KS QP Expern'rnent

Si(111)2x l ./ 0.27 0.62 (5.6% 5.76J 0.75 (5.78, 5.79, 5.87)Ge(lll)2x l J.C' 0-38 0.66 (5+77J 0.61 (5-88-5.90)

0.54 (5-911Si(100)2x 1 0.2: 0.70 (5-731 O-9 (5.92)Si(l00)c(4x2) 0.39 0.87 (5.69) 0-9 (5.92)Ge(100)2x1 0.80 (5.74) O-9 (5.93, 5-94)

0.9 (5.95jGaAs(110)1xl X 1.8 2.7 (5.75) 2.4 (5.23)

X 1.9 2-9 (5-75) 3.l (5-231TJ 2.2 3.2 (5.75) 3.3 (5.23)k' 2.0 2.9 (5.75) 3.0 (5.23)

TnP(110)1x 1 P 1.8 2.5 (5.754 2.5 (5.231.% 2.0 . 2-8 (5-75) 2.9 (5-231V 2.2 3.l (5.75) 3.2 g5.231X-' 2.4 3.2 (5.75) 3.1 (5.23)

Cz aad As surface states on relaxed H1-V(110)1x 1 surfaces. The absolutevalues of the QP gap openings of abput l ev are much lazger than for thes'arfaces of elemental snmîconductors. Howevea', their relative contributionsto the totat QP gaps are much smalle.r than the KS gaps.

5.d Strong Electron Correlation

5.4.1 Image States

I'a classical electrostatic's, a,n elecwtron at a location z outside a polnrszablesystem induces a surfacc chazge axd in tllr.n experiences all attradive imagepotential whose aslrmptotic %nn for large z is given ms ,x? -e2/(4z) (5.40),where z = 0 is the image plane. This imagelike behavior ca,n be explainedclmssically by the polarization izducrd on the suzface regioa by axl ex-ternaleleçtron, and it has its analogous countezpart in the quantllrn-mechn.nscal XCpotential. Exactly tb.is has been disçussed in Sect. 5.2.3. 0n a microscopiclevel, the rearrangement of charge.s at the surface is due to long-range eorre-

latioa efects, wlkicll are absemt in the DFT-EDA because of tke e'xponential,

'

226 5. Elementary Excitations 17 Single Blectronic QRxqiparticles

rather than hwerse power-law decay of the DFT-LDA ex mcorzelationpotential outside the s'lrGce (5.401. For pazticles outside tlb.e slnrface in thevacua'm the real exnbnnge-correlation potential (as the self-energy (5.40))should have atl imagelike asmptotic behavior. It is likely tkat this behavsorhas ve,zy little l'nfuence i:a total-energy calculations, at lemst for bonded parti-cles, but it is cruciat in tuderstandi'ng the states of electrons at a certn''n dks-tance from 'the surface. The severe DFT-LDA failtzre iu this space region 1e*

' to a poor description of such smface ststes azd to aa absence of Smage states

bound by the ''rnnge potentii (5.961. The image behavior is also imfortaatfor a correct interpretation of surface-sensitive tèchnlque such ms low-energ.yelectron clsAactîon (LEED) (5.97J, srAnnlng tplnneli'ng microscopy (5.98,5.991,

' and inverse cr t'wo-photon photoemission experiments E5.100) 5.1O1q .

A.ta nvxrnple for the importance of such efects is the nvistence of additiozlalsurface states due to the image potential, the image-potentiz states or, inshort form, the image statc. Thqse statœ are not dedved in any way frombtrllr states or from the symmetzy-brealdng eFect of the surface. For theirdiscussion we study the surface of a simple metal with ap. l'n6nite electronicdietectric cozsstant ss -> x or even an elemental semiconductor with a Amlte

diezectrk constant sb. We know 'that the surfaee bltrrser of the system inthe grotmd state depeads on the geometric, clmrnîcal and bondlng detailsof the atoms in the sudace regba (Sect. 5.3.1). TnKid.e the crys'tal it shouldsmoothly matah with the czystal potentii. Outside the czystaâ the potentialbecomes constant with a value spon. Studying the localizatioa of crystal states

near the sudace (Sec't. 5.3.3) we have approximated the s'arface barrie,r bya stemlike potentii. Thks does not rnrnnsn valid for a'a eledron present or

exdted outside the crjrstal. The surface barrier must have the image-poteatialasmptotic fo= (5.40). Withim a '?e,ry crude model (5.1021 the total single-particle potential outside the crystaz (z > 0) clm therefore be described by

a .

cs - 1 eV/) = fvac -

+ 1 k-(' (5.53)ss

An electron placed at a distaace z tu front of a surface genezates an electdcâelcl. This ûeld leads to a rearrangement of the chaz'ge in the crystal in suc.ha way that the parallel component of the ûe'ld vaaishes a,t the surface, In thevacul:rn the electron an.d the image charge -@b - 1)/@b + 1) placed at -z

produce a'a eledric felct whic,h is perpendiclzlar to the surface in the wbolesmface plane. This results i:a the attradive interadion in (5.53) between theelectrozz aad Rs image charge. To avoid conhlsion the image plane ks identiEedin (5.53) with the plnane z = 0, i.e., approvimately Mrith the electroric surface.lt may be considered as the pl=e at a distance zsm from the last atomic hyer

(position of the cores) in the hallpace. The.n zim is approvlmately given byhalf 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 insidethe crystal for the energy of the electron below svac. The projected crystal

5 4 Strong Electron Correlation 227

Energy

//Eg

Gap

E u .%.24z

EF

'xx. XXXMz 0

Fig. 5.28. The potential energg of an electron in front of a metal surface. The metaloccuyying the z < 0 imlfkpace ::s assumed to possess a gap arozmd the vacuum levelwithm the empty'bands. The gap Ss characterized by tYe upper (U) and lower (L)band edges with eneries a-u aztd ss, respedively.

band stmzcture should have a gap or, at least, a pocket for an appropdateelectron momentttm paraMe: to the sudace. S'ach a situatipn is represented inFig. 5.28 for a metal. For energies e < svuc (but lazger th= the energy of thehighest allowed conduction states as below the vacultm level), tke electron(with appropriate parallel momentum) clmmot overcome the vaclrtrn barrierand becomes trapped in gont of the surface provided that there e-qn be sn

a'llowed (image) state. It can only exponentiaEy decay into the bulk. Neglect-iug also this penetration, the crystal is taken to be in6nstely zepulsive. Henee,the potential may be approtmated for z < 0 as

7(z) = x. (5.54)Outsicle the 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'nthe zp-plane e-an be taken into account in the ânmework of the Xective-mnssapproxsrnation (EMA) (5-48,5.103). The f:-% e'lectron rnnzqs m is replaced by an

eective rnxqs m., whic.h is asplmed to be positive am.d isotropic for simplicity.The motioa of the elechtron outside the crystal obeys a Schrödinger equationof the fo= (3.1). Then the wave Glnnction belonging to a s'tate with enerrj'.' .

f', e is of the form exp(ikq(p)#(z), #(a) being a solution oî the ID Schrödsnger

L'

. :: equation

:2 d2 hlkl

t-am dza + V/)1 #/) = E' - amtl /(Z)

228 5. Blementary Excitations 1: Single Blectzonic Quazipartiales

Insnite barrier

'Y>z0

Fig. 5.29- SGematic single-particle potential V'(z), (5.53) and (5.54), izl the vidn-ity of a surface. The accompan#ng hyclrogen-like enera levels (dotted lines) aadmve functions (dazh'ed lines) aa'e also shown.

witlz the poteatial enegg.g givea in (5,53) and (5.54). .'I'he energy c =.d. thewave vectof :, kjl = k + g, should lie in a Ipocket or gap of the projectedb1:7k band struâtttre. The Annctions #(z) obey the boundary conditîon

,11/) = O (5.56)For realistic sarface descriptions the condition (5.56) can be weakened. J.f the

energy and wave vector fall icto tke allowed zegion of the projected bpnl'lr bandstructure, the wave 'hlmction of an image state slzotzld 'fapidly decay hto thel))a1 1c.

The problem (5.55) is formally eqttivczlemt to that of the determa'nationf the radial part Ii (r) of the a orbitals of the hydrogen atom (5.1041.o

This becomes obviotts if one sets #(z) = rl(r) ir=z and replaces e by

@b - 1)/(sb + 1)c/2 (5.10 ,g5 5.106). Thus: the bound states of (5.55) with2k? /(2m*) are rendily obtltimed from those of theenergies sL < s < evzc + L j r

Rydberg series, the image-potential states. Thehydrogen atom. One gets aresulting levels and wave hmctions are iudicated in Fig. 5.29. More precisely,not only levels but two-dimensiomal bands s H sntkjl) ('n = 1, 2, ...) result:

52k2L

k'

s,z(kj1) = cvac + . + au,2r42

sb - 1 RHfa = '-

sb + l 16n2

with the Rydberg constant Jlu = meLjLzhlt of the hydrogen problem.

5.4 Strong Ezectron Correlation 229

For a given k;i = L + g the baud energies (5.57) must fall in a gapor pocket of the prolected btllk band stnlcture. For vzztishing momen.tltmk11 = 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 abovethe 'vacullm level are allowed. The forbidden region in the projected bandstmcture should be arotmd the 'vactstrn levei. Tlze localization of the imagestates varies rapidly with the quantllm Iglrnber 1z. The average dista,nces fromthe surface, (aw) = .(* dzzlk,ztzllz, possess the Gues (za) = 6n2cs(sb +

b t 3 k and inc' reases1)/@b - 1). The minzurnm distrce amounts to a ou

dramatically with n. lt is obvious that for stzr,h distance,s the treatment ofexchange and corzelation witlqt'm the DFT-LDA (3.59) or DFT-GGA fails.Only the msymptoticly cozrec't trestment of XC ia the form of the self-

energ.g (5.40) yields a correct description.In order to describe the Lmage states, XC elecwts, i:a particular the electron

correlation, have to be treated beyond the approvsmation schemes of DFT-LDA or DFT-GGA. Oruly XC self-enera esects (5.40) not included in thelocal XC potential (3.50) #ve rise to a correct description of the eledrons iathe vacunlrn dose to the smface/image plre. Therafore, one may interpretthe occurremce of image states a,s a consequence of esects of strong electroncozavlation.

We memtion that the inouenze of the trae surface barrier on the energiesof the imag-potentia,l states (5.57) can be talcen into account by replacingthe quantllm number 'n by (zz + :) witlz a quaatttm defec't 0 S ; K 0.5 (5.964.

(1h)Ni(001)

T X8

S26 s j jjjF&; 4 .

--

ps .e'

2 , B

P E R

Cu(0O1)P X

8

6

S 14 xx

h S2h

2

B0

F E X

Ag(O01)r R

8

6

S q *

4 CA7-S2

2 *<

BO

r k X

Fig. 5.30. Dispersion of smnge-state bands Sï and sudace-state bands Sc on the

(001) sudaces of Ni, Cu, and Ag altmg the syznpetq line rX. B labels denoteobsezved bulk interband trnnxitions. Hatched regnons mdicate the projected bulkb=d strudurœ. The ene'rgy scale is refet'red to the Fermi level ep. The vacuam

leve) is marked by a hozizontal arrow. From (5.1101.

230 5. Blementary Bxdtations 1: Single 'Electronic Quasipartîdes

The image potential does not exhibit the asjrmptotic 1/(4z) dependence i'athe vinsnit.y of the cr-gstal terrninxtion but, rather, exhl'bits a saturation iu itsz dependence. For mvltrnple, the behavior izz this vicizkity is better describedby (5.107: 5.108!

2

-2 z-fzl.,- t1 - e-N.''P (-l(z - zimll) for z > zun

7(z) = -

l+zexp r-aim for. z < zim

than by the potential (5.53)/(5.54). In (5.58) the parameters .A a'nd p are

%ed by flll6lling the continttity of the potentiat at z = zkm, and zim, W, aacll are three parametezs describing the dkstance of the image plre, the inner

potentialj a'ad the inverse distance ove,r which the image potential saturates,respectively.

Tke image states desczibed above az'e empty in the system grouzd state.Therefore, the/ detection needs a spectroscopy in which, iu a f11.s+ step, elec-trons are excited iato the image state,s or, in gealeral, i'ato unoccupied smfacestates. Such a spectroscopy could be the IPES or the two-plloton photoe-znission (2PPE) spectroscopy (5.109q. Figare 5.30 shou measmed surface-state bacds Sa togethe,r with image-state bands Sz for Ni(001), Ca(001)) andAg(001) surfaces along the symmett'y liae PX (5.110J. The 2D bands Sz are

vezy sensitive to cozztamination. J.n coxtrast, the image-state band Sz doesnot disappear aaer adsorption but only shifts izz energy. The brds Sï show

(5.58)

(b)

>O

w gu ( j gg jE ki n YG

x 1 01.6 nwx

; n .--

œb ---- n=3..-w-==. .------x. w * A:k...x

----...jyp n=:N

%N

1%Nx

1 2h Nx '

Qöa xN

x n=11.$

. . .4(t.i (;.ty

2PPE signal

Pig. 5.31. (a) Schematîc enerr diaram for the excitation steps in two-photonphotoernz'mion (2PPE). (b) Enmo-ze-solved ZPPE spectrtzm that records tke emit-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 231

Table 5.2. Bindsng ,energies ,-ers (in eV) an' d efective masses m* tin. ml of theMage states for various metal surfaces (5.1097

Surface -s1 -sz -as m-

Ag(100) 0.53 0.16 0.08 1.15

Ag(1l1) 0.77 0.23 l.3

Au(1ll) 0.bO - -

Cu(100) 0.57 0.18 - 0-9

Cu(111) 0.83 0.25 - 1.0

Pt(111) 0.55 0.15 - 1.0

Ni(100) 0-61 0-18 - 0-95

Ni(111) 0.E0 9.25 0.10 1.12

Co(OO01) 0.73 0.1'8 - -

'

Fe(ll0) 0.73 0.18 0.05 -

I .

öhe expeded parabolic depeademce on k;I (5.57). At kjj = 0: it albws thedetmrmination of the bindi'ng energy, .-ca, with respect to the vacuum level.

Witltsn 2PPE spectroscopy one ttltraviolet photon with e'atargy 1.,). exdtesau electron out of an occupied state below the Ferm! level cp into the image-potemtial state with the quant'zm zmmber rz (below cvacfor kII = 0). A secondphoton with energy huls ixl the ieared (1R) or visible spectral region exdtesthe electron to a'n enera above sxac (Fig. 5.31a). The eledron leaves thesurface, and its l-snetic energy is meastzred.. For normaz escape, ckis = Flso-l-caholds. The escaping electrons are recorded. As an e-xample a 2PPB spectrct'mis plotted versus the k-lnetsc energ.g in Fig. 5.31b. It îs measttred for a Cu(100)surface using tlle photon energies Xaa = 4.7 ev and Tun = 1.57 eV. TV ptu'k-positions allow the determination of the bindiag energiœ -sp, of the imagestates ms well as of tite e'Tecdve masses 'rn' for meazmements with AnitekI1. Corresponding valu.es are sllnnnnarized for several metal surfaces in Table5.2 E5.109). They indicate that the model calculations (5.57) give the correcttrends and correc't order of magzzitude. However a quantlam defed 0 % ; % 0.5is nececysazy to accotmt for the shape of the tru.e stMace barzier. Tùe paraboEcdispersion is con6rrned for the electrons in the image-state bands. However,thea'e are only small deviations of their esective masses from the leo-electronkslae.

à.4.c Mott-subbara Bazas

Xb Mtio tefinsques suc,h as DH-LDA or DFT-GGA have been employedV-tit g'rea,t success to descdbe the electronic stzazcture of weakly correlatedmaterials like semiconductors or simple metals and their surfaces. There was

.:

232 5. Rlementary Excitations 1: single Electro/c Quasiparticles

only a necessit.g to take into accotmt additional XC egects in the case ofexcitations. For more strongly correlated systems suc,h as d- and J-baud sys-tems, cuprates, etc.: on the other hand, the concepts behind available abiuitio tecHques are sometimes too lsrnited to correctly descibe the com-ple.x manpbody eFects. The question nrsqes whether surfaces with electronsi.n rather isolated dangling bonds cazl also show Xects of strong electroncorrelation beyond the esec'ts i'acluded in the standard technlques.

D=gling-bond-derived surface bands with partial occupation have beenpredfcted for seve'ral c-lean, reconstructed group-l'v surMes <th an odd atl=-ber of atoms in a complete layer of a surface tmit cell, suG as 7x7: 3x3,WxW or also 1x1 (see Sects. 4.4.1 and 4.4.3). 'I'heir reconstruction is gov-evrIe by adatoms or nzlxtom dusters. 'lqypicat examples are Si-t-mnsnatedSiC(1l1)/(0001)WxW and 3x3 stzrfazo (5.111-5.113). They possess one

dangling bond par slxvfn.ce Alnst CeIL Neglecting spin efects (for instance inthe Famework of the DFT-LDA), the correspondicg dangling-bond-relatedsmface barzd in the f'tmdamental gap is half-occupîed and, hence, pias theFmrrn'' levd. The dkspersion of this band is small, of the ozder of a tenth ofaa eV.

The metmc nature of the suzface band is in clear contrct to experiment.Photo/mirssion spectroscopy of the SiC(O0Ol)Wx W sarfaces (5.112, 5.114,5.115; shows one f:411y occupied sudace-state band which ks about 1 ev lowe,rtha,n the Fermi ievel at about 2 ev above the VBM. In the 3x3 cmse this baadis slightly s'hifhed toward Mgher energy (5.112). Fkrthermore, inkerse photoe-mission specvtroscopy (5.11,5-5.117) shows the e'xistence of an empty surface-state b=d at about 3 ev above tlle VBM for the hexagonal polytypes of SiC.I'n agreemerti with DFT-LDA c/ctzlations of the d=gling-bond-zelated band,the dispersion of b0th the occupied and the em'

pty stkrfàce-state ban.d in theprojecied hlndamental gap is fotmd to be small. The bandwidths amotmtto less th= 0.I ev (3x3) or 0.2-0.25 ev (WxW); The band. ml'olma andmn.vima occur in the WXW cmse at the same positions aa calculated forthe hamfdled dangling-bond band. The tmcertaintiG i'a the measurementsdo not allow suc,h axï identifcation in the 3x3 case. Sclmns'ng tnnneling spec-trœcopy on the 6H-SiC(000l)Wk W aad 3x3 surfaces co'nAm'n the evleeuceof a smfKe-state gap of about 2.0 ev or 1.2 ev (5.118, 5.11$. Interœtinglya gap free of Rmrface states around t'tte Fmn't level and a'û occapied marfxce-state b=d have also beea observed for specïcally prepared ùmSiC(00û1)1x1smfaces (5.120q.

TEe quœtion arise,s whether the opposmg electronic-strucvture results ofDFT-LDA and the eztpem'mental methods with respect to the band occupa-tion and, hence, the metallic or uonmetallic surface characte.r contradict thesurface reconstruction models, a Si tetramer on a twisted Si adlayer (3x3) '

or a, Tz-site Si adatom tWx W), which have been verled by various ex-

perimental aud toti-energy stttdies (Sed. 4.4). In tlle caze of the 3x3 andWx W surface transhtional symmetries of grolzp.l'v matezials one slwaye

5.4 Strong Eledron Corzelation 222

cuts an odd attmber of bonds FitAin one unit cell. Because of the fou.r valenceelecirozls per aiom all bonding states shoald be cömpletely Elled vith elec-trons, whereas more oz less noa-interacting d.anglîng bondz should be occu-;.'Lpied with only one eledron. As a consequence, the DFT-LDA b=d s-trlc'tllre

(cf. Fig.4.37) predict a metatlic behavior of the surface. The discrepancy withthe experlrnental Sndsngs suggests eects of strong eledron correlation be-yond the scope of the one-electron theozy at least that within the DFT-LDA.The eMremely small bandwidtbs of the measured s'urface bands suggest theimportance of stroag correlation efects on tke electronic struct'are in thesense of the Hubbazd model (5.121).

I'n order to include s'trong correhtion egeds on (zlectrons in Hmngling- '

bond states locaced at the top Si atoms of the addusters (3x3) or Tzl-siteSi nzntoms (Wx W) we consider a ono-band Hub. bard Hamiltonian 15.1221.lt consists of two parts. One descqibe the constdered band in a Mt-nearest-neîglbbor TB approximation (3.15), while the other term represents theCoalomb repulsion of electroas hl such band state,s but at (me httice slte. neHprni7tonia'a is therefore governed by two parameters. A hopping parametert of the type (3.7) i.n the tight-bincting pictu're descdbes the interaction of theS6 dangling bonds iu dferent sudace llnst cells. lt is 'asually ll'rnited to theinteradion of nearest neighbors. rne electron-electron iuteraction ks'limitedto the Coulomb integral, wllick Ls the largest one and not corzedly takenipto account withi:rt the local dœcription of the DFT-LDA. 'lahe parameterU (cf. also (3.64) and (3.65)) describ/ the esective Coalomb interaztion be-tween two electrons with opposite spin on thc same Si dxngling bond, whichis however e'mbedded kn a polarszable medb'm. Since the dangling hybridsare strongly locnlsmzed at the adatoo, they do not overlap sigaiftc=tly. Thehopping parameter Ià1 ks smatl compared to characteristic energies, e.g., the

'

Alndamental gap. The surface baud formed by the dangling hybrids is cottse-quently very îat, and its energy can nearly be takea to be a constant, equal7to the correspoading orbital eaerr (3.71). This flatnerss or nnmowness leadsto > strong electron correlation Garacterized by the on-site repulsion tmrrnew U in the efective Hmlltonhn. For axl iateraction parameter U large,r thanthe bandwidth of the dangling-bond baud, this co'rrelation efect beyond theDFT-LDA becomes jmportant. Lf a dangliug bond on a'a adatom Ls fdledwith two electrons of opposite spias, the Cotzlomb interaction betveen themcontributes to the electron energy aa indicated in (3.65).

A rough estimate of the Hubbard paramete.r t) follows from the orbitkproperties and the ezectronic poladzation hduced ia the vicinity of the dar.-gling bond by aa Mditional electron. The relation ?-/ = U/aee nearly hoidswith the atomic Coulomb integral U and an electronic dielectrsc constan.t&ce of the Gective medblm. n'om the tight-bindi'ag utimate of the cohesiveenergy m âerived a aralue U = 8.39 W for isolated Sî.s.p3 bybrids (see Ta-b1e 3.1). In the Rlid State Table of Hn.rrlmn one fmds the value U = 7.64ev (5.1231 5.1241. The Xective dielectric constant of a, sudace may be detem

234 5. Elementary Excitations 1; Sipgv Blectronic Quasipardcle

msned by the mean value iefs = lz @b + 1) of the bulk atd vacullm constrts.Fbr SîC with sb = 6.7 15.124) an esective hteraction parameter of roughlyV = 2 W is estimated. For more Si-zsc.h environments, e.g., for S# sutfaceswith ckk = 12j values of thc eFective Coulomb interaction slightly larger tlnxnL-t = 1 W m'e predidH. Tt is also 'pceble to estimnte the esedive interR-tion pxrxmeter C by mo-qznq of total-energy rligerences between dsFerent oc-mmations (zthn.rge s'tates) of the dangling-bond bands witbln the Fameworkof DH-LDA calculations. Using sue,h a ddta-self-consistemt feld (/SCF')method, 9 = M+) + f(-) - 2F(0) holds with .E(+), S(-), and S40) rep.resenting the ground-state energies (3.u) of a pœitively nlorged., negativelycharged, and netttral su' percell, respectively. The chazges have to be loo>15m.edat the dangling bond giving rise to the surface band of hteerests From th:D#T-LDA total-enea'gy diFerences E5.112j ozze œtimates a vatue U Q$ 2.1 W(V ss 1.û eV) for the Wx W (3x3) suzface. However, the calmklation of totalenergiu of charged supercells suf'ez's strongly from spurious electrêstatic in-teactions between the supercelks. Hence, the values reprment a rathe,r cnldeestimate. For the more Si-rielk 3x3 surface the value of U appzoaœes more

'

closely that of pure Si as a consequence of the rather lazge Si coverage of tbisstrttcture and, hence, incremsed screeaing.

The daagliug bon.ds are azranged in hexagonal latticu. For C > 0the cliagomnlszation of the corres/onding tight-binding l'Ixrnlltozkian Tvesa dangling-bozd band with a clispersion a(i)= 2t(1 + 2 c-ostrc,slqt wherethe parameter s describes the variation of the V vector iong .f'V, or

c(k)=2J(2cos((2r/3):) + cos((4r/3)sj) along the Z'V line (0 ; s K 1) in thecorrupozkdsng hexagona,l suzface BZ. A ît to the dispersion of the daugling-bond b=d calculated witbin DFT-LDA yields a hoppMg parameter t = 0.014ev (ï = 0.05 eV) for the 3x3 (WxW) stznzcture. ,

'

With electron correlation rw C the single-particle problem belonog tothe Hubbard Hxmiltotkiaa cn.nnot be solved exactly (5.121) . However) in theatomic llmit t << t-/ (5.1221 a'ad. a nearly equal distributiop of tbe electronsin the dangling bonds ove,r the spizz stattxs (i.e.) in the parmagnetic groa'n.dstate), the band with dispersion eLk) known from the uncouelated limst splitshto two bands

s+(#) - ) ts(:) + ?7 + /@) + t-;2) '

) . (5.59)

One obtains the approvimnte dispersion relauons 1z&(l2) and lzs(:) + t)' <ththe same spectral strength. As arnsequence of the strong correlation iu thenarrow emerpr b=G a gap rw U is opened. The smfves undergo a Motttrn.nm'tion 6om a metallic to an izlsulating state (5.125j. 80th Si-rich smfacereconstructions repr%ent a Mott-Hubbard insulator.

Thîs picture azïd the C and 1 values dezived Above explaâa the oerimen-tal observations and other calculatiozls. Thy gap between the two bands isdeMed by the eledron correlation eneror U. For the 6H-SiC(O0O1)W3x W

5.4 Strong Blectron Correlation 235

suzface Themlt'n et a1. (5.1161 have aligned thei.r k-resolved bwerse photoe-rnismion data to the occupied baacl in an enerpr distaace of about 2.3 eV. Apossible uncertainty pf tize alignment of about 0.2 - 0.3 ev may be asplmed.The resulting value U J:t$ 2 - 2.5 ev for the Wx W surface is in good agree-ment with the DFT-LDA estimate. Northrup and Neugebauer computed asomewhat smalle'z theoretici value of about 1.6 ev (5.111). Correpoadiugcombhed PES/DES data are not available for the 3x3 stmlctme. Anotherargumen.t for the validity of the strong c-orrelation pictme is the conservationof the ctispersion in the empty aud 6.lled. correlatiort bands and the reductionof the dispersion of the measured bands by about a factor 2 in compn.rsqonto the DFT-LDA restttt. ARUPS measmements fnd 0.2 - 0.25 ev (5.112)and. 0.2 ev (5.1141, where> the width of the uaoccupied band of about 0.35ev g5.116) seems to be slightly laœger 5.xl the W'kW cwse. On the other haztd,for the 3x3 s'arface, thfeozy fmds a measmable dispersion of 0.13 eV. TheARUPS value is srnnlle,r than the exptm'mental tmcertainties, i.e., smalle.rthan 0.1 eV.

There is another way to obtain the bazd structure givem in (5.59) from'.. an ab initio meareftelê approaœ (5.1132. lt Ls necessanr to accmately in-corporate tlle long-range cormlation a'acl screening efeds in the electronicself-energy operator (5.21). This can be done in a higltly rehable my by theGW appyovl'mation (5.23). However, as a basis for the GW calculation one

hms Srst to trea,t the sadace system withi'n the local spin density approf-mation (LSDA) (see Sec't. 3.4.1) to obtain the fltlly spin-pohrszed covgazra-tion. This already leads to a splitting of the former metatlic DFT-LDA bazd(Fig. 5.32, dashed linel izlto t'wo b=ds separated by a direc't DFT-LSDA gapof 0.6 ev for Uj'xué tsee Fig.5.32, dot-dashed 1i'aesl (5.1131. The reslzlt-

:'.

(..:. .

:( : Fig. 5.32. Quasipazticle band stractme (solîd linœ and hatched regions) of t:e 6H-ii'q jîc(0001)WxW stkrfve calculated ix a fully spin-polarized GW approximation..: '

; For comparison the dangling-bond-relatect bands in DFT-LDA (dmshed line) or izz'E' bFT-LSDA (dot-dmshed linesl are also shovn. From (5.113q.:;.

236 5. Elemeataur Excitations 1: Ssngle Electronk Qumsîparticles

ing QP ba'ads (Fig. 5.32, solid iines) a're fmther separated. Compared to thelower DFFIJSDA band, the ocmpied majority-spin baud. is s'hiffed down tolower e'aergies by 0.2 ev while the empty zninority-spin band is s'hlfted upto higher ettergies by 1.15 eV. 'I'his is accompeed by a sEght increase ofihe band widths. 'I'he mpzm direct gap betweu the two bands is iZLCZ'O'XSGIby 1.35 eV due to the QP corrections and amolmts to 1.95 eV. This value isitt good areement with the on-site interaction parameter r) of the Hubbardmodel.

'No remxrks are necvary. Fizst, at a real &C(0001) surface the s'pinconfguration may not be fltlly polazized. I'n fact, wîthin the DFT-LSDA cal-culation the totaz energy of the spin-poladzed smface is neazly the same ms

that of the urzpohm'zed one. lt eltn thus be expected tut the spin 1701=-hation, if Evorable at all, is easily brokcn by noazero temperatme oz otherpertmbations, so it senms likely that the real suzface is not spin polrized.Second: suc.h a metal-insulator trlmsqtion. due to strong correlation effecvtsand 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.s there are alsosome doubts tlla,t tùe Si(111)7x 7 swface tsee Set. 4.4.3) should be theonly true metallic surface of a semiconductor. The sîtuation is sirnn'lar tothe SiC(0001)Wx W a'ad 3x3 surfaces with their adatoms or adatom clus-ters. Only the average distance of the zemnsnsng half-fzlled dangling boncks atadatoms is somewhst larger than the distaace of the dangling hybfds in tEeSiC 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(SiC(0001)WxW) to 1 ev (SiC(0091)3x3) is enforced for the Si(111)7x7sttrface, rasultiyg izï an extremesy small Mott-Hubbard gap. Moièover, fogsmaner ratios J/rtl the gap opeeg îs reduced with respec't to the =lue Urelcvant in the atomic limit :5.1224. Probably cx'p' erimeatal tenlnnqques suchas PFXS elmnot rexlly omtzibute to resolve S'UC.IZ a very mnn.ll gap of the or-

der of 0.1 eV. At room temperatmm the styface should look metallic. Thenaturaz way cotzld be. to work at very 1ow temperattzres. Unforttmatezy, the7x7 surface shows a strong suface pbotovoltage shift. whic,h results iu an

tmdeftnH Fermi-level positioa in the eedra. TH lowers the preddon of thelow-t'emperatttre PES measurements (5.127, 5.128J . There is aaother compli-catjon for the theoretical and experimental studio of the Si(11l)7x 7 surface.Many bands appear kl the projected Alndamental gap (5.129). Their ene-rrspacing is small resulting i'a problems concerni'ag the peak identfcation inPES. Farthermore, a singlmband Hubbard Hailtonian cannot describe thereal situation of strong electron correlation efeds g5.130j.

'

6. Elementary Excitations II:Pair and Collective Excitations

$.

6.1 Probing Surfaces by Excitatiolas

6.1.1 Optical Spectroscopies

Optical ypectrdscopies a're emerging ms particalarly promising tools to probesurfaces, siace they allow for in sitn, non-destrueive ar.d realrtime monitor-i'ng tmder challenging conditions as may be encouutered, for instance, dzzringepiteal vowth. For epiteal growth by men.nm of clmrnl'cal zeadions, suchas, e.g., metal-6rgxnsc chemical mpor depositîon IMOCVDI, optical spectro-scopies provide the only possibitity for such monitoring. Other advantages are

that the materiaz damage an.d contxmsnation associated with charged particlebfoltrnq are avoided. Insulatozs can be studied without the proble,m of charg-icg eFects, and buried interfaces are accessible owing to the large penetrationdepth of the electromagnetic rnrliation. Optical tenbnsques ofer micron lat-eral spatial resolution a'ad femtosecond temporal resolution. Howeverj sincelight peaetration and wavelengt'h are mucbu larger than surface titiplcnersses(a few i), sucll tec>nique aze actually poorly snnRitive to surfaces. Someltrinlm' have to be employed in order to iucrease their smface sensitMty. Titeexpmrsrnem.tal progress k'a the characterization of s'arfaces using light hms beensplrnmarlzed in a couple of excellent feviews ar.d monograpbs (6.1-6.51. rfhco-retical considezationz can be fouud in review aztides by R. Del Sole (6.6, 6.*4.V d th of light in a solid, even ia the spectral range of MghestThe probkg epabsorption, is of the order of 10-500 n=. For a charactedstic smface layerof O.5 nm tbânk-ness, the relative surface contribution to the total opticalsignal only Jtnnotmts to 10 -10 . Several approaGes have been developedto improve the surface sensitMty. The bazic idea is to mexs'tzre dsFerence

signaks which enhauce tke surface contribution with rcspect to that of theblllk-. Fottr terthniques are comnnonly tlsed.

One is snöace (Jzfererùtïc,l reyectance (SDR) speciroscopy. It is based on

measuring the dsference ia reîectance due to chenkical modifeation of thesurface, for example, often the Vsorption of oxvgen or hydrogen. The per-centage dsFerence is related to the surface structare. However, to what exteatit is related to the c'lta-qn or to the c'hetmipsnrbed surface and. whether or notit is sensîtive to the spectrnnm of suzface states and/or to the atomk struc-ttzre of the smface, is in general dilcult to determîne (6.8). Figure 6.1 shows

238 6. Elementaz'y Excitatioas 11: Pair and Collective Bxcstations

*@ *

5 oc?-h *

o m *4

.5, **

W 3 *c @ @P *'œ 2 . @X *(n *

1 ; *œ

0 GWoXOOO 0 o O

6

0.4 0.5 0-6 0.7

Photon energy (eV)

Fig. 6.1- Diferemtial reûectazcespectra of a smgle-domainSi(:.11)2x1 smface for ligh.t po-laaqzqd along tlze tq- 11 (711) (opencizcles) aud v LI gOllq (dots) direc-tions. nom (6.91.

a spectrnlrn that docllrn' ents the breakthot'tgh of SDR spectroscopy becauseof the lzsc of polnHzed light on Si(111) samples with singllz-domai'a 2x1 re-

construction (6.9). A:a oxédized surface is tused as a reference. After oxygenchemîsorption, surface states are saturated a'ad therefore optical tmnqitionsacross them occtlr at higher energies, having the DR sigmql related only tothe suzface s'tate.s of the cle-an surface. Thc measured 10O % rmisotropy of the0.45 ev pemk yields s'trong evidence i'a favor of the Pandey cb.ain model (see4-2.2).

The memsurement of the relative reîectance diference for two orthogmnaz ligh.t polarhations (z and p) in the surface pla'ne is cnlled renectancecsùsfrop!/ (RA) syedroscoyy (R AS). Since the bulk of cubic materials Jtsoptically isotropic (at least, as loag as xni'rvotropies due to the photon wave

vedor are negligible), any RA observed for suck cnrsta.ls mus't be related tothe reduced spnmetzy of the surface or to another syznmetrsy-inre-q.kl'ng per-turbation, for >ample an electric seld. However: RXS is not restricted tosurfacœ of cubic czystals. For e-x=ple normxl-incideztce spedroscopy phkrat-le1 to the c-azs of a uuia'xial crystal (e.g.: wurtzite) can also be tzsed. Thegreat advantage of the ItA. ten'hnique compared to the SDR method is that itdoes not involve imdta6ned reference s'arfaces, so that it can indeed be used tomonitor C'O-based epitaxisl growth. The theoretical interpretation is alsosimpler in principle, since impoz'tant atzd poorly known iuformation) e.g., theatomic structmc of the refereace surface, is not necessary.

The t'wo othe,r methods are ''urlacs phoioabsovption (SPA) whic.h mea-

sures ambient-induced changes iu the npolarized reoectance at or near theBrewster azzgle, aud ellipsomeiry or spectral :JJ#UIIZ/JeJZ'.!/ (SE)1 whic,h meav

slzres the comple.x re:ectance ratio of s- and p-polp.rizeê light.

6.1 Probing Surfaces by Excitdtions 239

11 z

7 .1.#

*

y% '

X

ds! c=(f.b), syyltol

Eb(œ)

Fig. 6.2. Sckematic confguration of a rededance exwriment vith poladzed light.A three-layer system is aumed. Tie propagation clirection of tile light and tâedivisior of zts polnHMation directioa with respect to the surface are tdicated.

Theoretical descriptions of reâectaace experiments at surfaces with polm- -

ized light often start with the three-layer model (6.6,6.10) shown i.n Fig. 6.2.The solid consists of the butk aad of a surface layer with efedive tikiclolessds m'ach smalle,r tibsn thc wavelength ,j of the ûght. The bulk is mssalmedto have all isotzopic dieledric 'hlmctios cbtuJl. The surface is described by a

gequemzy-dependen.t dielectric tensor with eigenvectors parallel to z aad p.?1Ye corresponding comple,x diagonal elements are a=(aJ) and sw@). The

ttpper halfspace is the vacullm. For normal iccidence L$ = 00) the comple,x1tA. signal is given by the dslerence between refectance amplitudes at (z = 0Q

J .

and a = 900, 1-J. azd %, respedkvaly, (6.101I

zïiz zlrrids s..@) - evv (w) 6 1)=

y as (.; .y. ( .

W

with ,4f: = f:z - Dv and 'F = (irz + /p)/2. Genernlizations to a'a nrnbienthalfspace, a, non-norma,l ligb.t incicteuce, and an nniqotropic substrate cau bei fo' uzd izz g6.111.

The standard IRAS setup measkkres the polrm''zation state of the reâectedligsht, Le.) Re(.zA///) and 1m(z%//). Most eaverimentalists publish spectra:éf the real part of the relative variation of the reoection Amplitudes (6.1);.1te(1///). This quantity is formally related to the refectivities J?.z = I.i% l:

aud .% = 1/ (2 for the two polarizations. This situatiou is schematically! .s js uot directly measured.indicated in ng. 6.3, though the,ir relative dl erence

Thea

.4/ 1 Z.SRe =

-f- ï R (6-2)

6. Elemento Excitatiozts 1I: Pxir and Coûective Exdtations

Rx- Ry

y 9

X X

Fîg. 6.3. Schemafic reprœentation of the dxerential meamlring tec%nsqtze in areoectamce-auisotropy spectroscopy expersmeat. The rlîffbrent light poln.m'zation di-rectioms are indicated.

with M = & - % and 2t = (& + G)/2 ms the diFeremce and the meanvalue of the redectivities for orthogonat polnrszations. Herc the approxi-mate validity of the relation 2Re (,&%*) = I/z (2 + IFvI2 has been assqlmed.Second-orde.r tmrmq rw 1.4/12 are neglected. Stace tLe algeaence of two reîec-tivities is studied in (6.$, RAS is sometimes also termed rclccfcncz d@àr-ence spectroscopy (RDS) (6.7j. Accordsng to (6.1) the relative Gange of thepolnrlzation-dependent reiectivities Ls gven as

z.a zlu;ds s=(u,) - swta,l (6.a;= lm .R c &b(u?) - 1

The sensîtMt'y o? the reEectance esotropy to the smface reconstructîonard stoichiometz'y is demonstrated in Fig. 6.4 for GaAs(100) seaces (6.122-In Figs. 2.19 and 2.20 the reconstructions appenrlng in Fig. 6.4 are related tothe surface preparation conditioas. J.t1 Fig. 2.17 possible atomic structures aregiven for the indkated 2D transhtional symmetries.

6.1.2 Light Propagation in Surfaçes

1n. orcler to detammine reNectance and transmittaace of a'a electromagneticwave with Feque'ncwy a7 in a cr.ystal surface, one izws to solve Maxwell's equa-tions for tEe ezectrk displacement vector DLm, uJ), whic,h is related to theelectric Eeld ELT, a)) by the constitutive relation

D0:@, ta)) = J') d3a/staptœ, a/; w)Jy@/, (s).J7

Here t:s aud J label Cartesian coordlmates and eœp (œ, Wb (,J) is the microscopicdielectric Rnnction of the vacu4tm-crysta,l iutezface. Instead of the longitudinalfh:ncvtion in Sect. 5.2, here we use the generalization to a tensor in order to ac-cotmt for the transverse character of the light. The dielectric hlrction/tensor

6.1 Probing Suzfaces by Excitations

Photon energy (eV)f'ig. 6-4- Refectance antqotropy memsured for difrerent reconstnlctions of theGa&s(100) sllrfxce. The zr-tT/-la7ds is parallel to E0ï1) ((0111). The àorizontal linœmark the zero level of eacà spectrnlm. From (6.12q.

can be calculated for a'a arbitrazy system accordiag to linear-response the-

ory (6,131 . Actually the temsor càaracter can also be derived hom the space-dependent longitudiral response gtndioa in the l5rnl't of small photon wave

vectors by relating it to the projection of the gequezzey-dependent dielectrictensor onto the propagation vector.

J.n the case of light propagation ia bulk crysta'ls, thc constitative relation

(6.4) greatly simpMes. The electric displacement feld contnsmsi only 10%-wavelengtk components. Dgher Fourier components ic the total zaicoscopicelectzic ûeld cnn be neglected assttrnn'mg spstial homogeadt.y of the system.Thks results $n an @ -œ/l-dependence of the dieledric fnlnction. That m/-qanq,

one neglects the spatial dependence of the electron dersity induced by theatomic struM'are of the crystal, umzally referred to as local-feld Gects (6.14,6.15! . Nevezlheless, they play a role i:a m=y crystals, in particalar in thestatic lirnit (6.16). As an approvlmate resalt tbe dielectric respoase can bedccribed by a Fequency-dependeat tensor c.4(og) = J d3œ/stxygta - œ?; a;).In cubic crystals, because of the symmetrs the dielectric tensoz becomH a,

scaza,r q''p,ntit.g ssta7l witlz s.#(ttJ) = sbttbillap.

242 6. Elementary Excitatioas 11: Pair and Colledive Exdtations

In the ceaase of a surface: even when neglectiug local-âeld eseds due to theatomk stractme, the Aniqotropy, the spatial non-loelzi'ty a'ad iAomogeneityof the delectdc resporse, i.e., its z- an.d z/-dependeace, have to be takeainto accotmt. That makes the solution of lkfax-well's equations a non-tri'daltask. 'fhey are eexsily solvei however, for the rigorously simpved c,ax ofan abrupt interface between a snmûln6nite czystal occupying the halfspacez < 0 aad the vacuntm foz z > 0. Iû that case, the dielectric Ahmcvtioa takesthe fo=

c@; z) = #(z) + p(-z)cb@). (6.5)

The solution of the light-propagatioa equatsoms leads to the well-knowa Fkes-nel formulms of reâectivity (6.171. 0f comse, in thœe formulas the microscopic

. feata'res of the surface are lost, since the surface contriimtion i.s completelynegleded. One way to indude it ks to tzse th.e three-laye.r mode.l dvribed i.u.ng. 6.2 (6.10,6.11).

A more general (since microscopic) approvh was taken in 1979 by Bagchi,Barrera and Rajagopal (6.1S). They started 9om the jellblrn model for thehalfspace z < 0. The tmnncation of the bulk leads to a moocation of theoptical propertiœ in a mtrface region of thinlcness %. BaC.G on the assllmption.that da 'zs of the ozder of a few bllllc lattice const=ts or less aa.d tkus mtte,hsmaller than the light wavelength, % << lj the ligbt propagation eqaationswere solved. De,l Sole (6.19) genernllzed tids method to the case of largerAniqotropses ia real crystals a'ad obtained a'rl e-xpression for the pohm'zatjon-dependent stzrfM.n contlbution to the rezetaace ARJ/R for a given liglltpolazlation diredion c. The result for s-polxrized Eght aud normal incidencerods as (a = z, y)

zlJu x-zkszm t (1&.(x(ta)) ) .It c cbtazl - 1

A11 surface featmu are mrnbodied in the sudace respoase hmction (6.6)

(zM=$)) = dz dz' lsaolz, ,-/; a;) (6.7)

- dz'' dz'''As (z1 z'; tojez-l (,z?, ,z#; aJlnsaatz'', z'''T t.g)

with the non-local slprfkce contribution beyond (6,5)

lsrx#ta, z';fzJ) = eopLz, Z;tJ) - QpJ(z - .z/)c@; z)- (6.8)The inverse (pllmtit'y cz-z:tz, zJ; ?z) is defm.ed by ihe i'atega.al relation

&zl'e-ï (z, z''; fxlsaalz/', z'; w) = J(z - z').JJ

6.1 Probing Surfaces by Bxcitatîons 243

It was show.a by Del Sole and Fioriuo (6.201 that atotz, F, ; (.J) should be thenon-local macroscopic dielectzic tdosor of the solid-vactulrn interface account-ing for a'il macy-body aud iocal-âe'ld efects.

The occurrence of oF-diagonn.l tevms sœz of the dielectric tensor (a # z)makes tke calculatioa of the, surface respoztse (6.2) dz'mcult because the iuver-sion requires to obtssn sz-l, and because of tlze fourfoid iatevation. However:for special surface symmetries, such as for cleavage plaae.s of group-l'v sezai-conductors; these os-cliagonal tez'mq vaaish. Also a careful tuspection of theseterms for GaAs(110)1x 1 a'ad GaP(11O)1x1 surfaces (6.21) clid not iadicatea relevant contribution to the refedance. The os-diagonal eontributions a're

therefore usually neglected. The remaining fzst tnmn on the rigNt-lzaxtd sidef (6.:1 cata be evaluated bykephciug the seml'-'ln6nste crystal by a supercez:o

large enough to represent the vactrtm as well as the surface and bpllk regionsof the crystal tmde,r bwe-stigation (cf. discussion in Sect. 3.4.3). Provided that

(i) the slab is large euougll to pro'perly describe the surface region of the czys-tal, i.e., the surface as well ws s'urf'tïcfg-modc'6eê blllk wave flmdions, aad $)the oF-diagomnl tfarrns of the dielectric tensor are small compared to the diag-onal ones, a simple ex-pression for the surface contribution to the resectivitycan be deriveâ (6.6, 6.211. The sttrface response AAnctioa (zM..(a7)) in (6.6)has to be replaced by 4zrxhJ.(tJ) witit the half-slab polarizability

1 +* +K

xzlkstwtsgl = 'Yjs da dz' (ssokabtz, z'; aJ) - &Lz - z')) . (6-10)-X -X

The quantity in the square brackets Tapidly appzoaches zea'o for z and/orz' outside the slab. Therefore, the tmrofolâ iategral ove,r z aad z? convergeswithoat problems. Notice that xbtxt?gpl bn.q the flsmension of a ienwth, due tothe twofold ictegration. Iu explicit calculatioas it becomes proportiomn.l tothe think-ness % of the slab. The memsurable quautity, the n.nisotropy of therefecvdvit.g of light (6.3) vith two perpendicular pohrlzation dirèctions z andv iu the mnrface plane,

z.a lsrraa x:@) - xhgtxl= Im ,R c sbla7l - 1

:ks l'mnnediately obtained from the conwponding half-shb polarizabilities. Ina simi)ar way an eax-pression for the Hsferentîal re:ectivity c-an be derivediom (6.6). 1.I1 this caze one has to subtract the polnmizabity for a pmssivatedmtrênne, but for the sltrne polarization direction.

(6.11)

6.1.3 Electron Enerr LossesE: .('. é .

'

Anothe,r way to probe elementazy e'xcitatiou of a surface is the memsurementE i :.'': of the energy losses of incident electrons with point charge e aud velodty ,t7. 1'a' or near the stuface reglon the eledrons generate aa extermnl e-lectdc feld orj i'll

244 6. Elernentary Exdtatiors 11; Pair and Collective Bxdtations

Z

0

V

/sz.,yy-s. , , . , yyy

>xg. 6-5. Iuelastic scattering process (resectioa) of a low-energy eledzon withvelocity 't1 on a hn.lhmace (z < 0). Possible energy losses are due to the polarhablemedium v'itk a bulk dielectz'ic frnction sb/, (zJ). The electzon trajedory is quazi-elastic because of the smn7lrless of the energy transfcr.

the langaage of macroscopic electrodynamjcs, a dieledric displacement se-ld

(coride 'nng the most important longitudinaâ feld components and, hence,negaleaing retardation egec-ts) accordiug to

1 cr(z, tt = --V=r@ - 't7à) =

zja (m - mtc 1= - 't:

with w@) = e2/Iœ1 . To be surface sensitive a reoedion scatterlng geometzyand prsrnary energies m172/2 < 50 ev are used (Fig. 6.5) . The enerr isso small that the electrons penetrate only a few Mgstroms into the solid.Hbwever, these eledroas are accompanied by a lonprauge Codomb fteldwhic,h is screened by the electrons azld ions in the poln.m'zable halRpace. Thetot/ eledrsc Eeld E nxn be related to the sceened Coulomb potential W' in

(5,22) or (5.36) in a similar way as the D Eeld. to the Cotfomb potential r

in (6.12).The enerr losse.s of the eledrons pemetrating or approachtng the solid are

r#ated to the clynamics of the screening procœses. The total emergjr transfe,r

Q is give; by the c'hauge in the eaergy density of thc Coalomb âeld.j i.c refec-tfon geometry essentiatly by that outsid.e the polazizable halfspace bolmdedby the image place z = 0 (cf. Sect. 5.2.3): The totat (time-htegrated) energyt'rnnsfe.r 1om a'n inelastir-qlly reiected eledron to the solid occupying thehalfspacc z < O can be written az (6.22, 6.23)

:: ae/+-dt/daze(z)s(a,t)o(z,t), (6.1a)Q - s -Xî

where b icdicata the time dezivative. The totat etectric seld .E@, t) in or

near the hxlfxpace z < 0 kclades the efect of the pola6zable medbnrn.

6.1 Probing Smfaccs by Bxc-itations 245

Witbi'n the sxrne approvlrnntions whic,h have been useê foz the calculationof the displacement feld (6.12), tke total electzic ûeld ca'a be related to tkescreened Coulomb poteatial W' by (5.22, 5.34)

X@,:) = --lva j d3a/ j d/ctz, z?; f - Y)p(z?: f/). (6.14)

Since the dezusit.g of the scattered electron is simply g'iven by Dirac's &R:nction

plm, t) = 5Lm - .?J1)1 (6.15)one Mds

. +K1 , , ,X(m, 1) = --V. dt W'(œ,.?J'I ; t - t ).& -x

Without the polarizable medbxrn, i.e., for Wr@, =J; 1-Y) = r@ -a/)J(t-tJ),the eeressions (6.12) ar.d (6.16) become idemticàl.

The spatiaz symmetry of the scattea.ing problem and the tgme integrationin (6.13) sugges't the tlse of a Fomier representation of the tvo Nelds .f = D, Esirnt-lnr to that msed i:a Sect. 5.2.3,

d2Q +0*

.f@,t) = z

ciop cu e-i-zytq, z, ap) (6.1.7)(2.,4 -.

(6.16)

with p as a 217 vedor i.zl the zr-plazze. One obtes for the total energytrlmsfer

+x

Q = j.. dp,g j d2QF=#(Q,(a)

with the scattezing probabitit for the traasfe,r of the eaea'gjr &,d and a wave

vector Q parallel to the surface (image plane)

(6.18)

1 utkj*qjz.eLq,zzu,lnL-q,z, -u,)..ëLQ,u)? - -

,s,s,r c

The Fomier trsmqforms of the âelds caa be easily calcalated assllrnsrtg thatthe electron trajectorg m' = 'l:tl is ventially that of an elmstically scatteredelectron. The time t = 0 js takea to be the momem.t of reiection at the

surface/image plane (z = 0). Tlzeu, because of the smallaess of the energylosses (FâzJ << zp,12/2) it holds rztf < 0) = Iwl = -rz@ > 0) for tbe z-

compone'nt of the ve-locity. For z > 0 one :5.r.*

R y) erwq e-qa (6.ayD(Q,z,rs) - 2c(-i , , azj ,q (aJ - Qn4 + Q2 1

X(0'7'&J) =

1 + c,(Q,();a,) ss( vz + (u, - Q.)2jnk,u;)DçQ'z,u'1'

(6.19)

246 6. Elememtazy Bxcitations 1I: Pair and Collective Exdtations

where the quantity aLQ, 0; uJ) is defined ia expression (5.37), aud (-iiJ , 1)represents a 3D vedor.

The resulting scattering probability (6.19) is

cze,t?yX(Q)tzJ) =

2 c j(-u? - qn4a + :c.tg)2.V(Q?:zJ), (6'21)

x:

-1gLQ,u't = Im

a vvlaypjjro,) .

E1 + a(Q, 09 u7)) &s( Q + (t,J -

The probabilit.y (6.21) is charactem-zed by tsko factors. The resonance-typeprefactor is most important for graziag-incidence-like measmemen.t condi-

tions, lwl << :72 - rj. For these obsewatjon conditions, scattqring is most

like:y for f,l = Q'v, i.e., whea the electron velocity pazallel to the slzrfaceplsme equals the phmse velodt.y of the surface elementary exdtatiop e.g.) of a

surface plmsmon or pkonon that is created i'n. the inelastic scattezizlg process.The second factor é(Q,uJ) is the slzrface loss Aznction (6.24). The appear-

ing imaginary part indicates its relation to real ene'rr losses. This factordeterml'nes the spectral stractmw of the loss spectranm of the electrorbs mea-

s'tzred withszz a certnn'n electron energy loss spedzoscopy (EELS), tzsuaz.y izt its

high-resolutiox (NQ.) form, TOFRELS. TH is obdotss when the wave-vector

dependeuce of the bl:lk dielectric function rltn be zteglected, sb (q; (gJ) ss giy(tJ).Becatlse of a(Q, 0; cJ) = 1/sb((zJ) the sœcalled surface loss fnlnction becomes

.ç(Q,tz) = Im(-1/ E1 + sb(uJ)!J (6.251. Thtls, as ic the cmse of blplk scattering

processes x, Im (-1/cb@)) (see (6.23, 6.251), spectral strucvttzre are expectedfor energies &aJ for which LnwsLuq evh-lbits strong featmes. They are the rme

as in optical absorption, i.e.; electron-hole pairs created by intezband trare

sitions or trnmsverse optical phonozls. 1tt addition, signiâc-q,mt mnx-irna in the

enerr dependcnce of the scattering probability occm whe,n the condition

Reslattdl = -1 (6.22)

is A11611ed whi:e the dnrnping /xz lmpittzl Ls smatt. The solutions of (6.22) give

the eigeeequendes of the surface-related collective excitations of a polaziz-

able lnnlfspace, e.g., surface plasmozts and tpolarl phonons (see Sects. 6.3.3

and 6.4.4).The s'pectral featlzres of the surface eue-rgy loss% (6.21) measured in

reâectioa a're determined by the gequencndependent image-poteatial-lîke

screening of the externnl chazge outside (z > 0) tke polnr'zable halfspace.

T:e inelmstic scattezing of electrons witic.h pauetrate the m'ystal (z < 0) isdominated by the sceened Coulomb potential (5.36) ik side the polarizablehalfspace (z < 0). T*e correspoadûlg toti electdc Seld is governed by the

quantity a(Q, z - zû uJ) (5.37). In the ll'=l't of weak wave-vector dependence,

aLQ, z - z'b (,g) = e-QIz->?I/sb(uJ)j this te= generates the bulk loss fzlnction,xp Im (-1/cs@)).

The surface scatteakg menhn.nimm consfdered so faz is restricted to theimnge-potentiaz-mediated inieraction of electrons with a homogeneous poLzx-

6.l Probing Surfaces by Excitatious 247

izable hopace. The details of the atomic aud eledroic structuze of thesurface are not tnt-nn into account. In a rough approvimation eledron en-

crgy losses due to elementar,g excitations eo=ected diredly with the suzfaceitself can be treated xqgnming a three-layer system as in Fig. 6.2. .&n. addi-tional suzface layer with thîcuess ds and anl'qotropic dielectric teasor withdisgon.al elements cxtall La = m, y, # is nmnnned to l)e embeddH bet'weeuthe vacu.tzm and the bulk czystal with sb(w). The wave-vector dependence of

. t,hese diagoaal elements shotlld be reglected in the follo<ng discassion.Given the clielectric ftmctions of the sttrface layer atd the substzate: the

. loss fanction can be obtained as demonstra'ted above for the fwo-layer sys-E tem. The electdc âelds in the suface layer aud the vacuam are determsned

; by matrhing condit-ions for the transverse componeats of the total eledricâeld and the normal components of the clieledric dksplazement ûeld at the

: bulk/slxdhce4ayer and sndhce-layer/vamzum interfv>q. '

t%t thescattering occqzrs in the Jz-plan.e (Ql 1 l-axis1, one derives a loss fundion

(e,-) -z-(,+-.-1(e,.)),

wb-ic,h is sl'nnila.r to that of the two-layer system. The esedive dielectric gpnc-.'tion is given by (6.26)

(6.24)fp(al) cosll.g + f(tzJ) slnh sce.e(Q, aJ) = s@) 5.T, ss@) cosh s + c:(aJ) s

l With s(aJ) = c/v(aJ)c=(a;) a'n.d s = Q% a/o@l/cazttsl (7 = z, r). InEfhc Ts=l't of qrnx:l wave vectors, Q% --/ 0, (6.M) becomes ce.e(Q, œ) =k:ktwl + Qd. qs##(u') - *72: Lwljezz @)j . we note that tha loss stnction (6.23)ks Muemced by the norrnxl compone'nt suLwj of the sllrmce dieledric tensoz

E fwhic,h 1 asually not directly accessible to optiW ex-perimcnts.:'

'' 'r A.u rwnrnple of the irnrnediate n'niueace of the surface Ls given in Fig. 6.6. It;

:'

J . . .J.' 'sùows tbe electron distributios versus'

eneror Bsses for a Sî(12.1)2x1 surfaceylqand loss energies &,z smatler than the bulk indirect gap (6.27). In agzeememt. ;. . < x

E fwith the zr-bonded chain model tsee Sect. 4.2.2) a signlca'at Anlqotropy is

) pbserqd. Losses rnn.inly happem for trn.nsferred mue vectors Q parallel to

l':(:thq ?J diredion in tke suface BZ, whtle they are s=n.ll for wave vectors

t Q paroel to 'J'. The primary electron enerr m/2/2 Mueaces the peakl'Ft3 ositioa in ageement witlk the prefaztor 5.n (6.21). Anotb.e,z e'mrnple of the''lilue'ace of the surface Ysotropy but wîth Iosses above tke sandamentat gapks' shown in Eîg. 6.z. lt represents EEt,s spectra of the caA.s(001.)e(4 x 4)àgzface (see Fig. 2.17a) i.n the euergy region of the e'lectronic interbazd trn.nsi-'fions for trn.nsferred wave vectozs Q parallel (%10J) aud perpendicu.la.r (rf1û!)iö Ethe surface As airnezs (6.2$. The rehtive riigcrence spectmcrn gives addi-'ttonal -'nq-lgb.t into the aature of electro/c transitious-in the suzface relon.q'in' his holds ia partictllxr for the promiment derivative-loe stntd'ttre aroundùss enea'gies F= = 2 W wMc,h should be related to interbard traasitions on

248 6. Elemautary Excitzatiozls 1I: PaJZ aad Collective Rxdtations

Exchanged o. (â-$)0,04 0.Gs û.12 3-r6 02o

(a)

vw 2G0 40G 60O 800 1000

.a 023 0.05 0.07 0.()9 0.1ï= (b)L-

W

=9==

2*Q 29O 40G 600 800 $000.J

.G1 0-02 023 0.04 Q25

(c)

.e *-.x e' % x

z v *'

e % h.z - x

.. # %. - ..

203 400 600 800 1000

Energy loss ho (meV)

Fig- 6.6. ElKtron energy 1= fn:nctîon oftke Si(111)2x1 sarface versus tNe enerprloss Flt,l memsured for rlsFerent azicml. nlargles, Qr1!I10q (solid linel and QII g11k1(dashed linel. The azzgle of Ycidence is Sxedat .y('t/j (1112) = 700. ne pzimazy electroneae'rv mr2/2 = 2 ev (a), 5 ev (b), and 20ev (c) is varied. From (6.2:1.

the XJ line ia the surface BZ across electron states iuvolving As atoms ofthe second layer.

In tEe lsmt't of small wave vedors, Q% ..+ 0', the emerg.y loss fundionseparates into two con-butions

-1 :gslm s##@) - d@)/E'za(Y) (6.z5)J(Q,u8 = tm + a .

1 + cslss;l g + suta7ljIu the spedral range of loss enrgies Fla7 for which substrate emergies cau benegleded (1ms'b(az) = 0)7 the reEected electron can only trnneer enerr Fzu?azd momentum LQ to eacdtations in the smface layer. Thus

Q% z -1p(Q,*) =

z 1m(&/u(*)l + sb@)1m - (6-26)7 + cs@l! sz>@)

ln general, the second term in (6.25) is dornt'oated by losses i'a the sllrfnr;elayer. For metal substrstes and even on semiconductor sarfaces the factor

6.1 Probing Sarfaces by Excitations 24â

6(a)

5 J )Ojo.-. 4ldva -) y (; j'>5 3o

2

,.j , :;

1 '

0@)A

.-d 0.06VNm o

'& 0.0a

Io

l,..'.ru .4 0'n. J

-0.060 1 2 3 4 5 6 7

Energy loss Lek7Fig. 6.T- (a) Bnergy loss intemsity measmed for directions of the momectumtrxnKfer Q parallel and perpendicular to the dl'rneze of a MB&g'rOU As-cappedGGs(001)c(4x4) sn'vfpce. The inset shows the LEED patten acquired at 39 eV.(b) The relative Jeerence spectmnm. From (6.28) (copyzight (2003), with permsssionh'om Elsevier).

s2s@) exceeds 100. The mnsn stnlcture in the loss spectrnlm is therefore de-termined by gLQ, aJ) = Qdslm (-1/sza@)J, the tblnlk' loss fllnction of thethsm xnlsotropic suzfacc layer weighted by the sma)l prefactor Q%. A quali-tative explacation of tb.is e/ec't is givea in (6.254. In the ca-se of loss spedrafor the Si(111)2x 1 sudace iu Fig. 6.6, ezzLwj Ls real and nearly a constantas a Ahnction of ul (6.274. Consequently: in this case the meastzred anisotropyis dominated by the diference-s in the flmt te= cw Ilrz/?gttzJl for 7 = y an.clp = a. One has to mention that formtllas of the type (6.23) are also tlsedto cvaluate the electzon energ.y loss spectra of reconstructed semiconductorsarfaces (6.28,6.291 . The slab approvsrnation is appEed to calculatc the tensorof the slzrface clieledric R'nction so(aJ) Lp = z, y, z).

250 6. Elementary Excitations 1I: Pair aud Colledive Excitations

6.1.4 K-qmxn Scattering

Nonll'near opticsl methods are also succGetlly apphed to obtain iaforma-tion about sudaces. One ovnmple'is second-hnrmonic genezation (SHG). Forczystaks, suG as diamond-stractm'e czystals with vazkishing bulk coûtdbutioa,the SHG represents a powerhll tool for surface studies. Mother prominectexnmple ks the Raman efecf (6.302. lt iuvolves the inelastic scatteeg of pho-torus in tke visible or W raage by elemental'y exdtations of tite system: e.g.,surf'ace phonoas and plnmons. It therefore allows some additional inso' tinto the vibrational azd electronk properties of sudaces if its suzface snmq'l-

tivit'y is mxlnxnced. This tmhancemeat is substaatîal since izt general only amnn.ll nttmbe,r of photons is inelastially scattered ia a cel'tain solld angle.For snml'conductors the typical Qmmn.n elciezlc'y has been estimateê to beabout 10-6- 10-? jterad.c-ml-l (6.31J. In all inehstic light scattczing processeaea'gy ks trxnsêerred bet'wee,n aa incident photon with emergy Lulj aad mMe

vector g$ tmd the sample, resukGg 5.& a scattered photon of a dseerezlt enea'res and mme vedor qs. The aotmt of tmneerred energy corresponds to theeigecenerr Mu(Q) of an demeatazy excitation labeled by the izde.x x, e.g.,tàe phonon brrch) azd the wave vector Q. Eaergy conservation yields

fàtsli - tJsl = :l;#=v(Q). (6.27)The t+' sîgn stands for those Raman processcs in which a'n. elementanr axh

citation is generated. Tkese are called Stokes processes. The nnnnslnslntion ofan elementanr vdtation corresponds to the t-' sign. Thcy are referred toms anti-stokes proc-es. A correspor.ding R.q.rn= spectmlm is schematicallydrawn ia Fig. 6.8 versus the lequency shlfk (aq -&Js). Generation aad xnnihi-

lation of thealementary emxcitations depend ozt Mmperatlzre i;a a càaracinm'stic

way. For not too high temperatzzres NT L F?z,;v(Q) the Stokes scatttm''ng ismore tutertse. ID. analog.y to energy consezvatiop the qumsi-momot!zm conser-

ution 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 scattezed light versus lequency shl'ff (schemat-i:%--i)1)r) .

6.1 Probing Surfnces by Bxcitahons 2ë

0)o

.

(j:yk yzhk;à ,zz

j/rj-';j (j(,y;

Fig. 6.9- Sclnrnxtic representation of the mœt resonxnt onmphonon Klt-azz pr(ess. Stokes scatter.iag is assumed.9

. . *

. . . .1

'.'! parallel to the sudace and the 2D wave vedor Q of tha surface elementar' Hdtytion

'

l 'Q1 '

Lq? - 4alli = &Q- (6-21

iBecause of the smaltcess of tie transferred photon mm vectora practicall)11 elementar.y excitations with waye vedors sear the center r of the stup y

,f,1..e BZ are excited. Amquming tkat vksible light is used to eexcite the Ramaiqç>ttexing în a sample with rcfractive'index of about 3, the mnr'== transterred wave vector IQl is of the order of 10-6 cm-l. This vatue Ls about 1/101' J:6f the size of the BZ of an Dmeconstmlded smrface.E2:E ' A pronnl'ne'nt example is (resonance) Rn.rnan scatteeg by optical phonons:.rlthis scattea'c is mediated by the electroaic system. The photons hteracl .

. .tEvth the electrons and phoaons are created or xnnilln.ted via the electron.jj.(. ; . ' .

.: Xbnon icteradiota. Six elementary processes contribute to the so-called one) d crsayc'shoaon (Stokes) Rnmnn scattering. The mœt resonaat prouess is es

'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,llqFermi s Golden Rlzle (6.32)c,,v a

P@il a7sl =

-g. Xl lAsol&&l Ysll Jtruai - *x(0) - &,.zsl, (6.29

j' ' (

' '

Juotaq, u'sl (6-30.'

.)....) (o l/zi-eteq-azill z) (z 1.:-,(0,-(0))1 z-) (z.' 14-(.-,:.,-)1 0).(&,& - (Sz - -F70)! EAaps - (r.4w - .E%)1AJ'

(1n èontrast to the izdepemdent-particle picture used in (5.12), the excitartbn of the electzoaic system is heze described iu terms of electron-hole paisi2 tes 1d) with energîes EA. IO) denotes the bzitîal state of tbe scatteriata

E>rocess. It is usually identEed with the gzotmd state of the system, in witicl:/o electron-hole pair is e-xcited. Corresptmcllngk, the electron-photon iater.:'' . Jytut 4. (j d with tue siuglotrièfion Hn.mtlton (c, a7) has beem genern. 7ze compare

, oiitiic)e zepresentation (5.13) . The dependence on the polazization vector e i:, àE >m''..

l'Aplicitly indicated. The photon wave vector is mssnmed to be negligible. Th

252 6. Elementary Bxdtatiozzs 11: Pair and Collectlve Excktations

ligb.t scatterïg proceeds in three s'teps- In the 6rst step, the incident photouM;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 sec-

oud step, this eledron-hole pa,tr is scattered irto another iatermediate state

IT) by emittiag a phonon &zx(0) with vaaisAing wave vector Q ss 0 via theelectron-pbonon interaction Hn.rnsltonian X*. Iu the lmst step, the electron-hole pair ia JT) recombi'aes radiatively with emission of the scattered photonFpags with polarization es. The electrordc subsystem remn.lnq tmc-hanged afterthe onemhonon Rmmnxn process. Witln'ln the single-particle pfcture three sttr-

faze eledrouic bands are bwolved in azl elementary seatt/rimg process. EM%eledron-hole pair izl a:a intermediate state is azsoaiated with a conctuctionand a valence band. The electron-phonon interaction cac scatter the electrou

t1lolel in anothe,r cocduction (vezlence) band.The most prominent coupling mecltn.nistzns for bqllk semiconductors are

deformation-potential scattezing and pola,r Fröblic,h scattering (6.3% 6.321.80th. zone-cente,r trs.nwerse optîcal (TO) a'acl lon#turlimal optical (LO)phonozls show a deformxtion-potential eoupliag to the electrons. 1I1 systemswith pmially ionic boads the longitudinal phonons ca'a akso couple to theelectronic syste,m via the accompanying zong-razge electzic feld (Fzlblichm'vtbxnsRm). The strength of the Rm.=a,I1 Jtmplltude (6.30) depends on thescsttering geometnr, denoted a.s qitei, eslçs. The Rwtrnazl amplitude dependsnot only ozt the directiorus of the light polarization ez oz es but also on

the direcdon of the phonon displacemeut. For tEe (100) surface of a, zinc-

()01J

E0 1 O41

- ej qy

qs

es

Fig. 6-10. A possible pohrs'zation coHg-uration for backscattering at a (100) sur-

face. A.longitudinal phonon cnn be obsmwed.

6.1 Probing Surfaces by Excitations 253

blende czys'tal lo-symmetryl ' and a bnnlcscatteeg covguration, tw Ir (ï0()1and qs 11 (1OOj, the created/xommllated b',lk. optîcaal phonozzs are polaœizedaloz:g the k010)/(0011 direction (TO) or along the g100J directioa (LO). lnthis coHguration tke T0 phonons are not Rsrnazwactive. The zone-centerLO pbonon ca.n be excited via the deformation-potential meclmnsqm whec(cyvesz + escsv) # 0 holds for the light polarizations, e.g., whem ci 11 (010)and es IL joûlq or ei (1 es Ir E011). Suc,h a situation is showa in Fig. 6.10. Forsurfaces the symmetzy îs reduced.. A gene-ral discussiozz of the symmetqy ofthe so-called Rxrna'a tezssoz's (6.30) aad the correspondsng selection rules canbe found in articllas by Loudon (6.31, 6.332.

The 1bmazt seledion rules for bnztlrscattczing from a (110) surface of a

zino-blende crys'tal are completely dlfrereut. ln the blAl1- only TO phonozzscaa contribute 'via the deformn.tion-potential meclmnfqm for polnrssationslEipGz + figdsvl # 0 Or leiatGz - espl + csatc;-,c - eipl) # 0, for instatlcein a parallel covguration eë 1( es 11 (1ï0J or for perpendicala.r polarizationsci 11 (001) aad es Iy (1101. For that reason the strongcst peak in the Rnmn.nspectra of a relaxed IA(110)1x1 surface izl Fig. 6-11 is related to blll'lr zone-ceater TO phonons (6.344. However, the 'Lwo spedra for paralle,l and crossedpolarizations evhsbit signifca'at dilerenccys. This is due to the.ir surface sezl-sitivits Thc photon energies htz;â = 3.00 ev and Fttz?s = 2.96 ev of incident or

scattered ûght are in close re-sonancc qrktit the electronic sllrface b=d gaps at

* 6s :46 2,4270 a47R :5 . : :S i ; ! ;

S'm 2 i ! s* f : :c ;n 2 i io 5 ! i Eç-'l : : :, ë 2

cy ' : j . : (j jj (j

o E i J - l .â.: ë i 2 (1 10J(1 10)

w J ! : g :> : : , . :

...' - . ; : ' :* 3 ; '. i ':c: ! 'J ; : :o : i ; :

..-, , 2 : i :=- i x 2 e ..j. o-

i I sc(v ! po-jlg-l lcjE

'

i 30SXne

100 200 300 400

Raman shift (cm-1)Fig. 6.11. One-phonon maman spectra of the clean 1nP(110)1x1 surface observedîn the baclrzcatterlng geometry for parallel and crossed poln.rszation covgurations.The photon energy of the inmdemt light is huh = 3.00 eV. From (6.34J.

û54 6. Elemenàary Excitations I1: Pair and Colledive Exeitations

X and X/ in tke sarface BZ . (see Fig. 5.7 aad Table 5.1). Consequently, thegeneratkon and recombination of electron-kole paiz.s mediating the phononscattezing happen =n5nly in the locnlization reson of the contributing stu'-

face electronic states (see. Fig. 5.22). Rtrthermore, tke surface symmetry isBwered. The poiat group Ls reduced from Q to m (see Table 1.3). Apartfrom spectral featmes dae to multiphonon procves surface phouort peAs ofA' symmetry of 69) 146: 254, 270, and 347 c'm-1 appear (cf. (6.35-6.371 andthe discussion in Sec't. 6.4).

6.2 Electron-Hole Pairs: Excitons

6.2.1 Polarization nAnction

Tlb.e celtzal quaatity ia the calculatioa of a dieledric Ahnction or a polar-izabilit'y is the polarizatioa flnnction or irreducible poladzation propagator,

P, of the polarhable electroaic system. Accordimg to the dsénstion of the

(longitudinal) dieledric G3nction (5.33) and tite 3D Fourier trxnqformationuthe macroscopic dielectric tensor a.#(aJ) azd d-iezectz'ic pohrszability tsus-ceptibitity) x.p@) e.an be related to the corresponding longitttdl'nal quantity

depencllng on the vn.niqlnsmg photon wave vectoz in the direction # = g/1çI by

sçn, u,) = )7 tlceco.pta'lt.?p, (6.31)o,p

c(o@) = %n + 4rxcg@lwith

a d.a J -$q(x-=z).p t z 6 aa;:(4,t(J) = 1 - 55= 'ûLq4 d œ = e (zœ, œ z ; tz7) . ( -

q->0

The Fourier tzxnnform 'ïLqj = 4zrc2/(F')ç)2) (F' is the 'volllme of the sys-

teml of the ba're Coulomb potential v@) = c2/jm1 hms been i'atroduzed here.I'a contzast to the earlier notation of the polarization fanction in relation

(5.33), the four spatial argametats hdicate that the polsm'zation htndion

P(m, F; uJ) > ?Lxœ, a/z/; (gJ) 5.s: ia generaz, a four-point response 'h'nction i'aaccordrce with its rehtion to the No-particle Green's Al'ncwtioa (6.38-6.401.The horizontal bar oa P indicates that, in contrwst to the microscopic case

P, local-feld eseds are now explicitly included in the deterrnimxtion of thepolazization Glnctioa of the rnnmoscopic clieledrk h'nction, The combiua-tion of thc two representatiozzs (6.31) ar.d (6.32) allows the K-draction of thecomplete macroscopic dielectric teanqor. Consequently, the response of the

eledronie system to a trrmgveHe pertttrbation such as light cnn be describedby stat'tiug from the (locgitudfnnl) density response represeuted bz the po-lnn''zatîon fllnction X(a=, a/rt/i aJ).

Without Coulomb correlatioa, i.e., witht'n the independentmquasipmicleapprovirnation ttsed in Chap. 5,,it holds J'5 - Lo (6.22: 6.40) with

6.2 Eledron-Hole Px5m: Excitons 255

(6.33)

The replacement P = Ln z:l GG ks often also C'aIIH the random-pbxqe ap-provsznatîon (RPA). However, this is somewkat incorred (6.41j . Even when

J quasiparticle eEec'ts -are negteded, the Koha-sham Tnrniltonia,a (3.46) aadllence G contlin efects of both exchauge and correlation. Ic general, the'single-partic'le Green's 'A'nnctioas G for electrons and holes obey the quasipar-'ticle equation (5.21).

Thc Coulomb correlation of aa eledzon and a hole caa öe mcluded only ic'an. approximate rnn.nner. 'ne most important long-range mteraction of the

i les occllrring in P shoald be treated vithl'n the sceened laddeztwo pat't c

appTovimnAsoa (6.40, 6.421. This is based on the GW approvlrnation (5.23) forthe setf-energy of atl inclividual particle. The kernel of the most gemeral i'nte.gral equation foz P, the so-exlled Bethe-salpete,r equation (BSE), is given ms

a hlnctional dedmtive of the sel-enera with resped to the single-partîcleEGzeen's fllmction. Within the GW approvimation (5.23) the kernel is replaced''ijk the screened Coulomb interaction I'F (5.22)1 unde,r the additional assllmp-ïion that the derivative of the screened interadion W' with resped to G ca,n

.%' e'.neglected (6.22,6.40,6.43). A second short-raage contribation to the k/rnelpf the BSE for P can be traced to the possibilit,g of particle mccaha'age (6.4% or,

Eihen seamlhsng for tke mnnrtx%opic didectaic fnnction (6.3$, to a Coulombt(' .. ho . rpjte latterthteracthon g@) = rtmllsraugrt restrided to one ''n't ce,n (6.44:6.451.'àescribe,s locabûeld egects. For the pobrlqation flmction depending on only'.

é($ de gequenc'y a closeê BSE follows by negleding the dyunml-cs ita tlle screen-

CQ Le., replackv T,F (5.22) by the static one 'A'@, œ?; 0) taken at (z2 = 0..: .;

Ci'' he BSE then becomu (6,22, 6.42,6.46)J'5@zœc, œla(; ts) = Jmtœzaa, œ/aœ/zif,gl (6.34)

- j dza Jdz'a E.:?:@zœ2, œ'azaj u')Wr@3,z'a; 0) #(auz'3, œ'za/z:tz?l- 2.:0 @laa, traza; w) 'p @t? - a?a) P @'aa(, zla(;a81 .

tt he factor 2 incticates that the sulet ftmaion is coztsidered azd formany.

'. ry

th'i spin s':rnmation ha,s been cazried out.

E ( : ;' '

7d:2 2 'rwo-particle Hnmlltozgau'

:i.. r *

'ètti. '

''

f the C.w approvlmxuou suggets the diagonal approvlrnxtjon (5.25), e txse o

fud (5.27) for the siagle-particle Green's Aanction. Thks m/u.r'l that the wave

l'lttnctions are replved by thorye (#v,@)J 'without quaaiparticle e/ects, e.g.,Erttie' solutions of the Kohn-sha.m equation for the surface problem (3.46).E r ,

.

'Eltirt,h' ormoze the dîagonru Grœn's 'htnctîon cau 'be replaced by i'ts zvoth-( ... 'f.......'Vdèr (5.27) with respect to the satellite stmzcttzre. Only the eigenvalues..J (

' . . .Ejkiï shrfked by the quasiparticle corzections (5.26). A rigorotts proof of this

L.IJ. J' :' . .

+x (u ,

.!b@lœz,e1 œ'z;Y) = t?@l,a?z;f + *)J@'z,œz;s).2rrî-X

6. Elemectar.g Exciiations t1: Palr a,rd CoEeciive Exdiations

procedtzre cxnrot be Sven. Howevez', it caa be argued that tMs approximationjs cozzsisten.t with the neglect of the lequency dependence of T&r Lu the BSE(6.34). Jt has beev shown that the dyzmmic efects of W' in (6.34) laœgelycomptmqate the stzength of the satellite stntctures related to the frequeucydependence of the self-enerr operator (5.31) (6.47) .

Tke eomplete and orthonormzlszed set of zeroth-order quasiparticle wave

hlndions (%u(z)) allows a Bloc,h representation of the polazizabitity ic(6.31). For the diagonal elements with respec't to the light polarization direc-tions one dezives in the long-waveleugth limit

262:2 t /

(=) - - E X''2 (Ms-Q(:)pzc,J'v,(R )J'b(cw:,'n'c'& ; u')xpp vc z',5 c/,v',;'l

- M(,z,i (:)z4J*v, (k8')23* (ctJ:, 'c'c'k'q -ts)1 (6.35)with the Bloch matrix elemauts of the velodty operator 'tJ (6.411

tck8l'olvilMj,(:) - k) - evqvt .

sc( (6.36)

The fador 2 h'om spic sïlrnrnation has been talcen into account. ln the liznitof locaz shgle-particle potentiks (3.48) the matzix elements of the velocityoperator are identical to those cf the momentllm operator (5.13) apa,rt from afactoz given by the Fee-electroa myms. Jn the representation (6.35) only hllly

occupied valence bands ('n) or empty conduction bands (c) ard interbaudtrn.nsitions across tbem are considered. Jntraband trn.rqitio:as are discassedlate,r in the coctext of surface plasmons. Conseqpently, in (6.35) we sxl'rn over

pnl'm of electrons t'a condudion-baud states lck) acd holes in valence bandstates lwi) that are virtually or physically excited by photozss.

The Bloc,h representation of the macoscopic polarization ftmction

P(X1R, œiJ(iL;l (6.37)= F-1 J-.2 .P(1:A'z, l'ala; $b))#z. @1)#k (aa)#k (z'z)#.k(a()

A7.,Aj, àasàg

with the abbreviation l = vk obeys a BSE whose k-rnel is governed by them>trix elelents of the Coulemb interaction (6.49)

.F(ltI:, laàû)= - d3œ d.3z' /:k@)#àz(a/)k&r(z,z?; 0)#Aa@)#); (=')

- 2/,) @)#Az (œ)5(= - =')#Az@')#lj(œ')1 .

These do not on:y resonantly couple eledzon-hole pairs cw azld c''v' butcontain also antiresonant Mteradions cw and wfc? as we2 as non-particle-coaserving terms in whidz three conductiomband indices or three valence-

band izldice,s ocmm Usually only the 1m.q35mg resonaat terms are takem into

6.2 Electroa-Hole Pairs: Bxcitons 257

' accotmt (for a detaileê discussion' see (6.38) 6.43, 6.48, 6.40)). The omittedefects are only importaut in special casxses to appronr'h cxtreme accauvy.M exmmple is the calculation of plmsmon resoaances where the rnlxn'mg ofinterband trn.nsitions of both positive ar.d negatkve gequencies must be irscluded (6.50) .

Because of the vaaislasmg photon wave vector in (6.35) the :'mlnomogeneousBSE can be substantially simpved. Since, together with the siugle-particleGreen's Annc'tion (5.27), the Bloch representation (6.37) of the polarizationhlnctioa for inâepemdent quaziparticles (6.33) leads to azz energy denoMna-tor, the relev=t BSE c,an be transforrned i'nto a'n. iAomogcneous t'wo-parqcmleSclnrödsnger equation in this represeatatioa 26.4%

X-.2 'X-.I gzftcpi, c'''z/'k''j - &.(ùd + Lrnléccwévvz/lsszzj PLc''v''k'' z/c'#'; r,g)1c/d qzd?' tày

= -joc,Jvv,J/u, (6.3g)with

.FJ'tcw: c%'E') = gcQP(:) - cQP(E)j é'cczlvv/tkjp + a'(c.r:Ic'v?1') . (6.40), c 'lJ

A small dampicg r of the electron-hole pair c''t?i is introduced in (6.39). Thesolutions of (6.39) deterrns'ne the inter' baud poinrlrzability (6.3$). The He-rzni-tian quautity Hlcwv, c?rTV) (6.40) ca'a be interpreted rzs the Bloc,h represen-tation of a t'wo-partscle Hamiltooa'a dacribing the intertal interaction of arz

electron-hole pair (more precisely: quasielectron-vpnsihole pairl ar.d its inte'r-action with othe: pairs. Aftez real or virtual eaccitation of eledron-hole palrswith photons, the e-xcitud electrons and holes do not only interact with thestzrrotmris'ng remnsming mlemce eledrons resulting in the renornmlization tofwasipacticlœ, sv (&) -> gQ,zP(:) (cf. (5.26)); there Ls akso a direct intcractionIF (6.38) betwee,n the excited electrons a'ad holes. This contes the bng-range eledzon-hole attraction deterrnimed by the screened Coulomb interac-tion x' -W. The additional jmscreeaed) short-range iuteraction ,xz 2/ repre-sents a'a electron-hole e-xchacge (6.42,6.46). A diagrnmrnatic represenyation ofthe t'wo contributions is given in Fig. 6.12 (6.49;. For a vxnseîng electron-hole

(b)cT

j4LL4LLCvàck

vf :, v'J, cI'vP

Fig. 6-12- Seahernn.tic representation of (a) electron-hole attraction a,nd. (b)eleotron-hole mcchange. The screened jlnqcreened) Coulomb intezaction is indi-cated by a duhed (dotted) liue.

258 6. Elementanr Excitations 11: Pair aud Collective Excvitatiors

interadion .E H 0 ic (6.40) the polarization Alndioa in'(5.3S) follows to give

PL=k, v/c/i'; a7) = -%c'Xn'%k,( gcoc P(i) - avQP(:) - ?à(a) + irlj . rsjng tbjsqllxntity hz the stusceptibitity (6.35) one obtxlnq the well-kcown formu-la (6.514for the iaterband contribution to the dielectzic tensor i'a the Gdependent-quasiparticle approxnemation or, with the replacement sQ,, P (:) -> s;ztil, inthe independent-particle approYrnation (6.521.

6.2.3 Excitons

The inlnomogeneo'as BSE (6.39) can be trn.nRformed kto an e'igenvalue prob-lem involving the efective two-particle Hnmlltozzia'a (6.40)

5--2 )((g Hl=k, c'w'i')ad*'(#') = .Eu.4rtil (6.41)YTQ? LV

with eigeuvcklues EA and eigenvectors x4T(i). The iudex .t1 represents theset of a2 (plnamttl,'n aumbers of a.u i'ateracting electroa-hole pair in the b=dstates c,i and 'ck. Because of the attractivc electron-hole interaction bothbouad and scattering states are pos<ble. The botmd states are usually calledezdtons, but this term is also used more genezazly for Couzomb-cozrelatedeledron-hole pa'zs including those i'!z scattezing states.

Shce the eFecti've Hxmlltonian (6.40) is Hermitiaa, aa orthonormal andcomplete set of eigemvectors AJ (i) caa be derived. They satisfy the com-

pleteaerss relation

c..1)+ :ixczwz k' j. j' ,6. , .Az ( z ( ) = ccz vv klèz

The eigenvectom allow a spedral represOtation of the polazizstion flmdion

.AI'(:)A$'v'' (V)PLC'IIL 'u'ctk'; (,7) = - J-) (6.42), sz - s,(;.a + k.s;Z

with a small damping parn.rneter P. The diagonal dielectric susceptibility

(6.35) takes the trMal fo=2

282:,2x=(u') - v 57 J-2 MJv(:)-4$**(:)

z c,,z,:

1 1x + . (6.43)EA - Attd + ir) EA + hlul + ir)

The optical propez-ties are drmstically modsfled by excitoaic eseds. TMs holdsfor the oscillator strengtlls becamse the siagle-particle trxnKitiozsmatrix ele-ments Mcxh (i) are weighted by the eigenvedors WIJ (:). In addition, the jointdensity of states of the two particles is càauged as indicated by the new pair

6.Q Eleciron-Hole Pstlr.q: Bxdtons 259

: Fig. 6.13- Imag'inazy part of the hwudacpdependent dieledric fnnction ibr bulk Sii;it comparison with expezimental data (dotled liuel (6.542. The theoretical spectra:, 'have been calculated at rlsge-rent levels of the iuc-lusioa of many-particle eFeds::.(# iudependent-pYicle approach, (b) iudependent-quasiparticale appronc'h, and (c)gCouiomb-correlated quasipartkles. From (6.532.; t

lénergies Sz and theiz dependence on the quantlnrn ntlrnbea's A In. a s'pec-trtnmpve.r a wide eneror range a,s in Fig. 6.13c the ftrs't Xed, the coupliqg of (11.6-ferent clectron-hole paizs, Ls most important. Near a'a absorption edge, how-Eeker: the density-of-states efects, for exxmple addition.at bolmd-state penlcs,àr dorn:'na'atQ .) j ' .: The 'nBuenze of many-particle efec'ts on an optical spectmtm ks demon-strated in Fig. 6.13 for the optical absorption of a bllllr Si crystal (6.53) jn:ompazison 'wit: e-x-persrnental data (6.541. Three steps of approximstions areLfsed for the i'aclusion of mry-body eiects. For Coulomb-correlated electron-bole pairs (c) the fltll Hnmîltonian (6.40) js appDed- Witbin the indepOdent-Ruasiparticle approxlnnation (b) (6.411 the electron-hole interadioa S is ne-jlbcted. Iu the independent-particle appronrth (a) (6.41J the single-particleVergies i'a (6.40) are replaced by the Kohn-sham eigemwalues sv (k). TheEq. .

jiùasiparticle eFects sbsfi the spectrtlm to iligher ener#es iu accordauce withtîe openi'ng of the gaps aad trxnsition energies. Tke Coiomb iuteaaction: : .kikes rise to a signiâcant rMnqtribution of osczator and spectral strengths.:ie more or less corzect intensities and positions of the Eï and Eg pfa-q.kq'iV the absorption spectrnlrn of a Si crystal caxl only be de-scribed M'ith the: .

:'

.

.'

ikitdusioa of E. Due to the èarge nklmerical cost, calculations of optical spec-i 'tzar i'aclucllng quasiparticle shlf'fs .4.(2) and excitonic eseds .F have become:1' ..r. . .

'ljpéssible only reccntly for blllk semiconductors or instzlators with about tqrotiipms L'a the elementanr cell (6.55-6.572.;.,; The physici menmng of the pair exdtatioas defned by the pole,s of the: ( :. :

'

jbiàization A'ndion (6.42) as excitons nnqn emsfwy be tmderstood neglecting: :.:' . ,èiiue interactions with other pairs acd considering a dizect mode,l semiconduc-

.s 'C)l.

s. Elementazy Excitations II: Pair and Collective Excitations

tor with one conduction b=d c an.d one valence band 'v and a frequency- andwave-vector-indeperdeat screening constant a'b. The pair Hnmsltonia'a (6.40)then takes the form

1 t-FJtct,k, c'Wk') = scsôx''u, sku, (scQP(k) - svQP(k)j - -g(k - k ) . (6-44)

ck.

For b=ds with s'trong k-vedor dispersioa a 3D Fourier trn.nqformation of theelectron-hole pair eigenvectoo

1 a -ip..y(m;z47(k) = d xeW

yields the Schrödimger equation in real space

2p qe csQc (-iv.) - c. (-iv.) -

ssj-mj *7(œ) = Sz4JX'@)

for tîe internal motion of the considered electron-hole pair. The center-of-mass motion of this pair does not appear Lu (6.46) since vazaisbl'ng pho-ton wsve vectors have bee,n aamlrned. Therefore no momentltm trnnsfcr tothe electron-hole pair happems dltrc'ng the opticaz excitatiom The quasi-momenta of the electron and Eole are F&k azld -Dk, respectively. Witlninthe esective-mazs approfmation (EMA) (6.58) for the two bands, &Qc P(k) =X. + h2k2(2mc and svQP(1) = -mkklpmg with tke ecective ma.sses mc

and zp,v and tke separating quasiparticle etezgy gap X, the hydrogen-atom-like Scblfclinger equation for an telectron' with the reduceê mass

mr = merrtnjLmc + rzv) and a reduced tproton' c'hnmge of ejeb appears.Hydrogezslike bound states ./1 = nim (n, = 1, 2, 3...; 0 S ! f zz - 1;-! S m :; J) with pair emergies Saîm = X - Rqjn? below the gap en-

ergy eacist. For typical semiconductors, suc,h a.s Ga-ks, the e-xdtonic Rydberg

(6.46)

(a)

QOOOQQOn m -œ'e-*-- m l

(E) O zO C) U''. (:) (3g' N .

1111 -. 1111 11!1 1111 666ï%%- I:Iro7eooöo

p l K M / * h MOocoooo%

am =* v x x x x , . x

O C-hs O :) () ----0 Oé--. y ...'s u:1 -. .

O Q C) ' ' O O ()

Fig. 6.14. Sclmmxtic representation of 1he spatial extent of e-xdtons: (a) wealdybotmd Wnnnier-Mott exciton ia a semiconductor; (b) strongly correlated Frenkelexdton in an Sonio crystal. The dashed line fndicates the extmlt of the wa've ftmctionof $he inteamn.l motion in a fctitio:;s (here: 2D) czystal.

(b)

00100

O C) O Oi Om x m'C m

C) t7 C) O O

O O O O Ox = m m

C) O O O C)

6.2 Eledron.-Hole Pairs: Exdtons 261

constant .% = RHzr?a./tpzszsl is a factor of abottt 10-3 smazler than the Ry-dberg comqtant RH = 13.601 ev of tlte hydrogen atom. Consequently, theexdtonic Bohr radius câ = anesmjm.z is a factor of about 102 larger t'hn.n

as = 9.529 A of the hydrogen atom. The excitonic Rydberg cons-tant J?)jcorresponds to the bindicg energy of the e'xciton hl its grotmd state withquazztlzm zzttmber n = 1 and energy Szuc.

Such weakly botmd electroa-hole pna'm with srnxll biadiag ener#es and

large eledron-hole distances are kcown as Wxnna'er-Mott e-xcitons (6.59-6.621 . Their chaœacteristic spatii exte'nt is i'adicated in Fîg. 6.14a with respect

to the atomic stractme of the semlconductor. Another lfrnstiag case of e-xci-tons appears Ln ionic or molecular czystals. ïn these systems the k-ctkspersionof the band.s as well a.s the sc-reensmg aze geuerally weaker. The electron-holeGteraction is much stronger, and the electron and hole are tightly bovmd to

(a)

4œ

r.tc:D 3

=X 2== 1D-=& ..*--

X ****

& () . *

< F-a ..4 o 4 8. -12 $6

'hro- Eg (R))Fkg. 6.7.5. Optical absorption with (solid line) azd Mthout (dmshed line) electron-hole attraction in the WxnnleaMott l''m5t. (a) Schematic representation withoutlifettme broadening. The absorptiox by botmd pa,ir statœ is represented by verticalline. (b) Speotmlm lzz EMA. for parameters of the heayy-hole exciton of bttlk Ga.A.sv'ith JG = 4.7 mev and r = 0.25;z. (c) Spec'tr'am m the true two-dîmensionallimit with the parameters of ('b) but no modifcation due to the reduced smfacesceening. The pa,ir deztsit.y of states withotzt interaction shows a sqx'pte-root (3D)or stemûke (2D) variation with FIts - fg.

33Rrx ((;)c&ag 20u

c2 10=9o*r o .

<-8 ...4 0 4 3 12 IB

mo- Eg (R) )

26;

p.lu+ othe.r witbin the same or nearest-neighbor ttnc't cells. rnœe exdtons are

lœown as bnrenke,l exdto:as (6.61, 6.621. They are Rhematically indicated inEig. 6.14b. We mention that near the band edge E the absorptiou spedrum!can easily be calculated using (6.43) and the Wnnnnler-Mott appronmh (6.46).1n. the 3D case a hydrogen series occurs for photon energies &zg below Es,while a rather structmeless continu:nrn appeazs for F= > X i'a the regioa ofthe scattnrlng states (6.6:4. The drastic chamgœ of the optical absorption d'tzeto the electron-hole attradioa are hdicated i.n Figs. 6.15a and b.

.&n analytical solution for the absozption spectmlm can alrsn be fotmd inthe 2D >se (6.64) (see Fig. 6.15c). I'a the extreme two-dim-nsional llmn't of

(6.46), i.a whic,h th: pairs are located i!l a sheet at z = 0 (which may beidentifed with a surfacelj the pair e'xcitatioa enertes are (,/1 = rzzrl; n =

1, 2, ...; 0 S l'ml S zs - 1) (6.65)

Elementary lxcitations 1I: Pair and Colledivc Excitatiozus

2 2f 81, 1

E = Ssur - JLj . rz -om g a (jsarf

where tize gap enerr Egsk're and the dielectric constant ssurf relevaat to tzemlrfncm sheet are introduced. With csurf r? @b + 1)/2 Jk$ sb/2, exprvion

(6.47) predids a sigvezmt increase of the exdton binding energy by aboutfaztor 16 vith resped to the bulk val' ue .%. Consequeatly, i.n compm'qon'a

to the bulk situation stronger excitoaic efeds Ootlld appear for suzfaces.

6.2.4 Surface Exciton Botmd States

It should be possible to s'tudy botmd states of smface e'xcitons for the lowestlmrface-state Aansitiozss if these are energetie-qlly well separated from bulks'tates. This ts the cmse for the Si(111)2x 1 and Ge(111)2x1 stlrRztc (seeSect. 4.2.2 and F1.5.25). The sudace bands Ddown temptyl an.d Dup toccu-pied.) arising from tEe daïzgling-bond statœ of the buckled ûr-bonded clmsmqlie essentially kï the balk ftmdameat/ gap. The surface-iaduced optical ab-sorption has an onset energy Egsurî Ld. Table 5,1) that is smaller th= thei'adirec't bulk flmdamental gap. As a consequence a well-pronotmced peak isobserved ic the dsFerential zefedaace (see Fig. 6.1) for the altowed opticaltraztsitio:as with the transition operator paralle,l to the cimsns. A ssmllaz peak

J7is observed i'a Fig.6.16 in the N A s:1)4*:*,4-17= of t'he Si(111)2x l surface (6.66).rne broad peak possves a mnvsmltm at photon energiœ of about &gJ = 0.48eV. Its Mgll-energy tail becomes small close to the bulk iudired gap of about1.1 eV, Sl'mslar values of F= = 0.45 W azise fzom ch'fvential reoectMty spec-troscopy (6.9) and phototimmmxl deiection (6.67). On the other hand, 1om a :

4, :

combinatioa of dired and iuverse photoern:xqion spectroscopy, a surface gap E

of about 0.75 ev Ls deterrnl'ned tsee Table 5.1). Quuiparticle calcttlations give ::

'nimtlm dired gap of 0.69 W (6.68). The dxerence betveen the mn.='mn. :1a ml

in opticaz spectroscopies azld the deriyed gaps of about 9.2-0.3 ev can be in.- Jz.1

tezweted as an iudication of the large bindiûg energy of the surface exciton.The dtuation is slm51ar for the Ge(111)2x1 surfaze (6.69).

6.2 Electron-Hole P%1rR: Exdtons 26:

4%4.O . @*@

3.5*

en .

b 3.0 .

X @we *

2.5 @ch .

%1'&< *w 2.0œ * .

t *

1.5 **

*@*

1 O * **

* * *@ *

@ *

().61

().0().:$ ().:1 0.5 0.6 Q.7 ().19 0-9 1 .0

4.5

'

photon energy (eV):

. x'ig. 6.16. rlze reqectance xnisotropy spectmlm of a single-domain sitzzklzxl. :

sllrfmce ibr room temperatme aud norrnnl inddence. The two poln.rirzations aa'f:.: iosen paralle,l and perpemdicular to the zr-bonded nlnnsnq. 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, 6.68j by careftf tpnmeric,asttalctûations based on the solution of the BSE (6.39) and the t'wo-particlt:. .

Hnmsltozf.an (6.40) for the eledronic subsystea of the t'wo staface ban.dflàowzl i'a Fig. 5.25. For Si(11l)2x1 the results are presented in Fig. 6.17.'

.

!'

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: J ical transitîons for light polarizatîon paralle,l to the nbxims shows the dra.it . pt

) tlàtic ''nnuence of the bound exciton states. These generate the bz'oad ppm.Q- ir:rti' 'è apee-rentsaz resectMky spectmzm below the single-quwsiparticle absorptiortyèike Egs''rt = 0.69 eV. Above the st:rlhce QP gap, the dlferential re:ectivity

LtL'j :jj d juw,'..,. muc,h reduced.. Due to the electron-bole interaction, spectri att oscE:'E: iJr strengths are redistributed to smaller energies. Thc =n.6n reason is th

264 6. Elementary Excitations II: Pair amd Colledive Exciiations

oNo

>>

=OX

=œ=

=cœ=m

D

8(a)

eII chain6

*

4 * e x. z x

o l x* x

* * l so 7 --

x2 * -@ z x@ suf N'

* E N w# z g x.

z M'

0 .-*

(b)elchain

2

0 ' - - - -

0.0 0.2 0.4 0.6 0,8 1 .0 1.2

Photon energy (eV)Fig. 6.17. Deerential refectivity spectmlrn of the Si(11l)2x 1 sudace calculatedfor norrnnl inddence. The tclusion (nerslect) of the ezectron-hole interaction isldicated by solid (dashed) curves. The two lijht polarizatîons are chosen parallel(a) and perpendicular (b) to the zr-bonded chams. .&n artiûcial broadrasng of r?,r =0.05 ev Ls mcluded. The dots denote experimental data by Clniaradia et a1. (6.g;.From (6.684.

destrudive coupling of tmcorrelated pair oscillators c'ri Mritit diFerent wavevedors by the eigenvectors XJ (i) (cf. (6.43)). Bciow the seace QP gap, an4lrnber of discrete exeiton states are formed. The optical oscillatoz streng'this, however, nearly completely concecttateê in the lowest-energy exciton at0.43 eV.

The dominant sph-siuglet exciton at 0.43 ev possesses a,n exciton bintiimgeneror of 0.26 eV. This is more than one order of magnitude larger thaa thevalue of pbout 15 meV kl bulk Si. About a factor 4 may be due to the reduccdscreensng in the surface reglon with au esective dielectric constant of csus =

(sb +1)72 (cf. the ctiscussion ia Sect. 5.2.3 an.d (6.47)). The othe,r mairt reasonfor this icczeased bindlng is the spatii conBnement of b0th the eledron andthe hole at the stzrf'sce (see also (6.47)). For the direction perpendëcular tothc suface this is already dear hom the 217 model (6.47). Because of thelocalization of the surface s'ta'tes derived mns'nly 9om llze Ddown and Dgporbitals (see Fig.4.19 or 5.25), the b=ds get a partial 2D character with thestrongest dispersion parallel to the chains. TMs ID character may furtherMcrease the e-xciton bizlding.

6.2 Electron-Hole Pairs: Bxcitors

kqN----

hble'% 2

o @ : o cc,n so-2 .w .

X c o W

-6

-15 -10 -5 0 5 10 15

Distance from the hole (#Eig . 6.18. The electror-hole wave fht'nction of the lowest-energy exciton in realspace for a flxed hole position (irecated). For detn.17R see text. From (6.685.

A gencral visualization of the eledron-hole conelation at the sarface 2-lustrates the electron-hole wave hznction depending on b0th the electroncoordinate œe axzd the hole coordinate zh. It is plotted ia Fig. 6.18 for a flxedhole position œjz slightly above one of the ttp atoms in a r-bonded chatu,i.e., at a position whezc the amplitude of the Dup hole state twhicll con-

tributes 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 ftxedhole, m = tre - zh, in a (011) plane perpendic'ala,r to the Pandey nhn.l'n5k, can

therefore be interpreted pzs a visualization of the wave 'hlmction of thc inter-nnq motion desczibed by, (6.46). The Arnplitude of the electron is ve.r.g largeon the sxrne Pandey chna'n whe're the hole is located. Ou the neighboringPandey cAnsnq to the left and to the right, the ampûtude is mue,h weaker. Onthe second-aeighbor Pandey clzltlrtK, the xmplitude is already close to zero.

As a eonsequence of the qllnuqi-lD band structttre with strong dkspersioa inthe chain direction the exdton shows Frnnkel-like behavior izz the g2îîq clirec-tion and, hence, a large binrls'ng energy. The situation Ls dsFerent along thedmims (not shown in Fig. 6.18) . The probability distribution of fmdm' g theeledron in the (01Iq direction is more extended. The mean square distanvcein the closn direction of 40 J't may be interpreted as a'a indication of theWannier-Mott character of thc exdton in this direction.

6.2.5 Surface-Morln-Ged B111'k' Excitons

The interpretatîon of smface optical spedza for photon energies above tlb.ebutk hlndnrnentat gap is much more complicated. B111lc and surface opticaltrn.nRitions i'aterrnsv. Botmd states of one absorption edge occur izl the con-

tizmllm of scattering states of other absorption edges. Ic the presence of aresonaace iateraction mediatcd by E this mteraction may give rise $o a Fazolineshape near the bolmd states of e-xcitons (6.702. In orde,r to avoid conhtsion

266 6. Elementary Excitations F1: Pair and Collectâve Excitations

Fîg. 6.19. Uppermost atomic layers of t/e hyclrogen-covered Si(110)lx

wé conside,r a, model stuface, such as the hydrogetssaturated Si(1surface represented in Fig. 6.19. Because of the boncling to hydrqthe surface states are removed h'om the eaea'or region of the ftal bulk gap. The optical reQectance n.niKotropy (6.3) ca'a only

'

trnmqitioas between sttrfaco-modi6ed bulk states in the surface reg'idcussed in the fozowiug. According to ciculations tkis rezion is ret

less thaa 30 atoznic layers. The conwpoudiug experimental RA spFig. 6.20 (6.71) can easily be reproduced (6.72j. The spectmlm is ratsitive to the strtzc'tttral and clmrnicaz detnslq of the smface passiva,t

V 20.01 '

E 1

Experimentrzl<

Theory

2 3 4 5Photon energy (eV)

Fig. 6.20. Measua'ed 'R A spectmtm of the Si(ll0)1x 1-H sudace E6.71).pared with tbe spectrnlrn calctûated according to (6.11) and (6.39) forslab (6.721.

6.2 Electron-Hole Pn.5m: Bxdtozls 267

!It hms therefore become a calibration standard for RAS ,apparatus and a

tex-tbook evnmple for surface optical propertiœ (6.32j.The mcmsmed a'ad calculated spectra in Fig. 6.20 show two stzong posi-

tive RAS feattzres near the Eï and & blllk critical-point ener#es (6.321. Theàtrong polarization n.nl'sotropy of optical trsmqitions nea,r bl:lk-like miticalpohts Ls related to the symmetzy reduction to the relevant point grou.p m-

In detail it is mnu5nly due to the diserent deformations of tke 3D Bloclwlikewave hlnctions in the smface region for the z-direction ((1ï0)) and v-diredioa

(y001)). Amplitude, phasej oscl'llxtion width., a'ad decay into the vacunlm mayV i'nAuenced (cf. Fig. 5.22) by the tznlncation of the matezial and the orien-i'ation of the Si-si an.d Si-H bonds (d. Fig. 6.19). The botmdary conditions6f the blllk-like wave fllnctiorts at the surface depend on z aztd y. Tke Si-TIbonds lie in the pz-plre aud: henee, have a veater ln6uence ou the vztriar-tion of the hlnctions with &. As a qonsequemce, the trxnqition-matrix elaments

(6.36) for the excitation of one qumsipartide eledron-bole paiz a-s well a-s theEcotllomb interaction of electron-hole pairs (6.38) are modioed. This men.nq

Enot only the optical matrix elements Mm$ (i) but akso their tterference medi-, ated by the pair eigenvedors AI'(i) in (6.43) aze cilanged. The isotropy withresped to the z an.d y directions is destroyed. The strong opticz rmsqotropiesin Fig. 6.20 are directly related to the shb polarizabilities for light polnHza-iion parazle,l to (110) or (001q. Iu Fig. 6.21 their signlcant dependence on thejohvization direction is demonstrated for a lz-layer slab, in particu)ar ic thelz and E2 spectrat regiozus. Figttre 6.21 also indicate,s that the dip on tke

60

EcE4

>*'-

40r ...

. ------' a? %'

.tJ . 1'K.1 .' *

. Ix a. e *' *- K' J 1'%-- ; MM* %

? 4 *

... ,;, .i,k

'.O . w

(n. : N. nn .. w...

rn = VJ '. nw

r X .e xwE.--- # -w.... e. . J

O2 3 4 5 6

Photon energy (eV)i Fig. 6.21. Tmn.giuary pazt of the slab polnHzabili'ty (6:10) calculated from (6.31)bbr a lllayer slab. Tûe light po7xrization is pazalle'l to E1ï01 (solid linel or (Q01q('dotted linel. From (6.72).

268 6. Blementary Kxcitations H: Pair and Collective Bxcitations

low-energy side of the Ej peak in the RX spectrum (see Fig. 6.20) is moreralated to the Meshape of 1m(1/hb@) - 1;) as a consequeuce of exprassbn(6.11).

Because of the iazge slab witb. 24 atoms per supercell, t'wo conductionand two valence bands per atom a'ad l40 i-points in the surface BZ, about35O 000 pair s'tates are involved in a nl:merical calmtlation. To avoid the diag-onalization bottleneck (6.41), a novel time-evolution tenbnsque is applied tosolve the BSE (6.70j. I'n. partinnllnm, the remarkable arkisotropy near Eï is dueto e-xcitonic efects. This 5s demonstrated in Fig. 6.22. Hcludîrg the quass-particle corrections (5.30) the IRA. spedznlm is slniftcd to higher energies. Theshort-range electron-hole e'xchaage iateraction =d, hence, the locaz-âeld d-feds harfy in6uence the spectmnrn. The attractive electron-bole interadionin (6.38) strongly couples ezectron-hole patrs with quasiparticle energies nearthe b7:7k Eï and .F'z critical pomts. The constructive interference increasesthc strength of the Ek feature. The opposite eFect happens for tke Eg peak.Thas, the .E': peak remmîns almost l:n' Ahl'ffed wher- the excitonic E2 feamtm'e is. slightly shifted to lower energies. Coztsequently, one may concludethat e-xcitonic efects aze mop important near E1, at leut i'a the RAS ofSî($10)1x 1-H.

'

(0.01 Ez

Et

(d)k -

X (c)

(b) E2

- E1(a)-

2 3 4 5Photon energy (eV)

Eig. 6.22. RAS spedm'm of the Si(110)1x1-1I ptHnce cpzlcuhted for a lFlayezr slab:(a) independent-pattcle approvsrnxtion; (b) hdependett-quasiparticle approximamtion; (c) quaslpartîcle approxsrnxtion with electron-hole excAnnge/local-feld esects;and (d) fully Coulomycorrelated elecwtron and hole quasiparticles. &om (6.72J.

6.3 Plasmons 269

6.3 Plumons

6.3.1 Intrabaud Excitations

For the evesluation of (6.35) describing the optizal propemties of uonmetals,completely fzlled @) or empty (c) bauds have been azsnlmed. Iu metaks or

l'tighly doped semiconductors also paztiatly fzlled bands u appeaz. 1n. such

systems the sm'eenkzg is rather complete. Efects related to W' (5.22) are

negligible. 1lz a &st appronz'ln the local-feld esects .'w f should akso be n.e-

gle#ble, and the polarization Anmction P can be replaced by qh (6.33) witbl''n

the independent-quaskpartide approfmation, whic,h is eaentially identical

to the i'adependent-particle approv'rnation because of the GW approvima-

tion (,5.23) for the self-emergjr operator and the Almost vanishing screenedpotential W- in systems witlz fzee carziers. Combn'ning (5.25), (5.27), (6.32),aud (6.33), assllmsng low temperatres, a'n.d still consideriug vzmislnsng wave

vectors, the intratfand coatfbution to the dielectzic Aqncwtion (6.32) becomœ

u--(#,,s) - 1- 1...'2s(q) y'l * (&>' -&-*+t.')) - O ç''b- - f-t*) (6.48)f Lr')q->o

s svtk + g) - svlk) + 56u1 +

with the Fermi energ,g e's of the electronic s'ystem. A 3D b'xllr system is

consideaed as indicated by the replacement k -> k. The ktraband contzibu.-

tion to the dielectric glnctions of metals rtn.n also be calculate by ab initioeledzortic-stmctme methods (for copper see (6.734). In the limit of pa-rabolicbau.d dispezsion, ex-pression (6,48) describes for eadz partially fllled baad v

the watl-kaow.n Lindhard dieledric A:nctgon (6.74J.In the ll'=l't of small wave vectors (or large h'equencies) the intraband

contribution cau be approvlmately treated. With the elements of the ttmqor

of the inverse elective mmss

(6.49)

the intraband pazt (6.48) of the dielectrk flmdion takes the Drade fo= (6.75)

1 p2sv(k)gpzv-1(k)q., =

ks ok. ,k#

/p2 1 opsintrat*) (s; = 1 - jtx g (y(uJ + iF)o4

with the aaisotropic plmsma Feqttenc'y of the ealectronic system

2

yzlto = 8xv.6 :7 e (aF' - cv(k)) qzrz-v 1(k)q J? .P a

zsir

In the case of one partially SIIC'CI band u svith isotropic azld parabolic k-dispersiot characterized by the efedive rnnAs zn*, the plazma gequency be-

comes

(6.5û)

(6.51)

6. Elementary Excitatsoms 1I: Pair and Collective Exdtations

4rc2%aY =p m.

with the homogeneous electzou densit.y

2n =

p Eta yp - c.(k)) .

For 1ee electrons ia the jelliam model we have rn, = m-. Such a'a appcox-:imation allows tEe calckllation of tke intraband clielectric Alnction also foqnon-vcmishing wave vectors. One Snds a wave-vedomdependent plcma 1t.4 E

queucy wp E1 + 3cp/(5'rrJ*) Lq/iupllj (6.76j ,

(6.51)'

(6.53)

6.3.2 Pluma Oscillations

For vanishing damping the zesulting isotropic intraband clielectric A'nctio/'

('

sintrwtu,) (6.5c) uas a zero at the frequency so (6.52). TMs men'nq that an eletqtroa gas h:u an 'm6nttely lazge response to ields applied with this frlzqaency. k1.n. other words, there exist self-sustnsnceng chazge oscillations of the systeù; EThese are jas't tbe long-wavelemgth plasma osdllations of the eledzordc suV ï'

: , is tjsed . 1::)system. Consequentlyj the use of the terrn plasma frequencvy Jus . ,

eneral, zeros in the dielectric flmdion correspond to Gcited states of tVgsystem. They give the eigenenergies of (longitudsnxl) cozedive excitatiol.'In norrnxl metals the (anergy Fzto of thtase exdtations) the plmsmons at vanish:i'ng wave vedor, is typically severaè eV. The values for 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 gequendes aze muc.h smaller because of the srnnller carrierdensities involved and the presence of the fmite electronic interba'ad polarizhabili'ty (6.35), mpinly (cb - 1)/4x, at such small gequencies, which tscreed'Ethe plasma fawuency. Moreover, the modïcation of the eecvtive ban.d mais(6.52) has to be takezt i'ato consideration.

Physically the plasma oscillations correspond to solmdlik-e compzessionwaves in the electron gas. However, becatlse of the long-range natme of

' h illations their frequency doeqthe Coiomb potential, whic,h sustnanK t e osc ;

not approac,h zero at long wavelengths but approaches the 6nlte plmsma ::1-uency. BG plazma oscillations or bulk phsmozls may be excited icu meta.tsq

by flring Mgh-enea.gy electrons at tlnir fozs. Electrons i'ateract strongly witithe plasma modes a'ad the caharacterlstic eaergy loss is obseawable by stadyhgthe trsnqmitted elecqzon btoltrn (e,f. the discussion ill Sed. 6.1.3). Another pomsibility Ls the obsemtion of the satellite stzucvtttres in photoelecxtzon spectfi

(see Fig. 5.9) ,

6.3.3 Surface MoalGcations

The presemze of a smface morlmes the losses observed by EELS i'a a reâeditsgeometl'y (see (6.21)) or the satellite stzudares in PES (see Figs. 5.8 and 5:01

6.3 Phsmons 271

ty tnqncation a'ad image-potential eseds. One type of these modl'6cationsgives zise to plazma oscillations locxlized in the surface region arotmd z =

O (6.771. For the displacement of electrors in sach a sheet during plasmaioscillations the driving force is reduced in compn.rllon with the 3D cafe dueto the restriction of the density fuctuatiors to 2D space. This gives zise to a

fedttction of the phsma gequeccy. The eigenlequency of a surface plasmpn,als, follows from (6.22) using (6.50) a'ad neglectMg the interband contribution(6.35) to the dielectric înnction at this lequency as

ws = wp/v'it (6.54)Therefore, the energy of the surface plunfon at vazzis%ing wave vector Qappeal's to be a pzopert.y of the blllk'. Because of (6.52) it is diredly relatedto the bll'k electzon density zz. The wave-vecvtor dispersion of the surfaceljlasmon enerr is, howevea', completely diserent from that of the b771> (seeSect. 6.3.1). Jnstead of the eue'rgy increasing with the square of the wave

''qedor, a aegative stzrface-plasmon dispersion has been predicted (6.781. Moreprecisely, a linea.r dispersion relation a;s (1 - Ic1 IQr) occurs for small 2D wave( .kedozs.

> o .1= Q x 9.163 Av) .

=c)1-

S 2.6 meV

Q o c 1 6 A'@ '= . e

#'ae

O 1 2 t3 4

Energy Ioss (eV)fiik. 6.23. Electron energy loss spedza (measklred in the reèectioa geometzy) from> jiliclc K metal flm grown onto A1(lll) for tsvo diferent Mues of tke momezttzzmtransfer and primaz'y electron enezgy of 12 eV. From (6.79J.

272 6. Elementary Excitations J1) Paiz and Collective Excitations

2.7$

2.74,,-h %

> 2.72 Yd. m s

&* 2.70 Nf,rl s..1 N

x 2.68 %%

P NX 2.66 %= sku N

N2.64 %

2.62 A'A n

Ai

0.0 0.1 0.2 0-3

Q $1 )Eig. 6.24. Memstzred ctispersîon relation of the,surface plasmon as a fn'nctîon of IQ1for K metat The dmshed line corresponds to a tkeoretlcal prediction for the littearte= (6.781. From (6.794.

A soace-sensitive electzon energy loss spedrctm for au alka,li metal isshown in Fîg. 6.23 (6.79). The cozresportding measlzred dispersion relation of

the sarface plasmon is given in Fig. 6.24. There aze two discrepancies betveenexpeaiment and the An'rnplifying theozy mentioned above. The &st is that thevalu.e of als (6.54) deduced from Fig. 6.24 is 2.74 ev while the auticipated valuefor the density of the 11 metal would have been abou.t 3.0 eV. This djscrepancy

may be related to the fad that the bulk K metal is not an ideallellium. Band-structme efects have to be taken into accout. The second discrepancy thepositive dispezsion for large wave vectors, iudicates tkat Mgher-order termsia IQ1 become important. However, also the treatment of the liuear te= hasto be improved by the inclusion of exeange and correlatson esects (6.80).

We note that the observatîon of surface piasmons oa surfaces of dopedsemicondudors by N'RRELS is more diëcult than for metals (6.81, 6.82j. Adead layer of surfaee plssmons seems to exist. Moreover, their energies come

witidn the Tange of optical phonon eneargies and coupled sarface plasmon-phonon losses appear. Consequently, the sarface plmsmons in sezniconductorsère not sensitiv! to the details of the atomic geometzy atd eledroaic stmzcture

of the Srst atomic layers in the smface. An example for sach coapled modesis #ven in Fig. 6.25 for a cleaved InSb(110)1x1 smface with a bulk eledronconcentration of n = 1.5 x 1017 cm-S and a'n LO phoron entargjr of 24 mev.

The double-ppxk strudtu'e related to coupled suzface modes azd its mriationdue to the wave-vector-dependent prefactor (6.21) are deazly visîble.

6.4 Pkonons 273

(a) '

K

:3

Eo.' .

L (b)

0-90 -60 -30 O 30 60 90 120

Energy Ioss (meV)Fig. 6.25. EELS spectra measuzed ozz a deaved HSb(l10)1x 1 surface wiG t'wop 'nnrnary ener#es (a) 20 ev and (b) 5 eV. A doped 4:::.yst9 with a hiz,h electroncoscentration .p, = 1.5x 1017 cm-3 is studied. Fkom (6,82i.

6.4 Phonons

6.4.1 I'Iarmonic Lattice Dyzmmics

I'a order to dœcribe the lattice dyzomics n.ear surfaces, the same attemptsat modelicg should be used ms i'a the cmse of the electronic-stmzcztttre andtotal-energ.g studies in Sect. 3.4,3. Promlnem.t Rvnmnples are the slab andscatteriug-theoretical methods. The repeated-slab method or the Green'sfnrnction method togetb.er with a special mnrface ssnrllntio:a aze commonlyapplied (6.83!. The instantaueous position of an atomic core (atom, ion) isgiven by

-

. = m. + w (A) ,m

where R.t = R + 'rs (Bravais lattice vector R and atomic basis vector z%)gives the eqlliln'brinrrn position of the particle and w(A) the time-depemdentdisplacemeut. Tllese displacements a're the ceatral quantitîes of the latticedynanlics.

Withsm the hltrmonic approvimation the equatioz:s of motion of latticeparticles are

274 6. Elementazy Excitations 1I: Pair azzd Collective Excitations

d2 sj t ,Mk yg uJ.I.R t) = - J7 C.PLR, R lzb4 (S , t)

z?zo',J

witll the atoznic rnn.qses M:. In general: tNe hteratoznic force constaùtài ? 6 d a,s the second derivatives 02Ej@R%.=0.%p) of ià' e'Ci.;) (X, R ) are dm ne

total energy (3.$5) of the considered system iu agreement with the de6ri-tion of tke cvpilibz-blrn atomic structure (.Rs.) for vauishing forces (3.39)A nlamber of relations between the force constaats foltow from the beha'fuior of the totat eztea'gy a'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

(6:5q)

-'s 'i l ',i ?cutA, .&) = C;=çR, R) = - C%œ;?IR, Jt ).. zz4,:

The prlrne indicates that ltra-atomic terms are not taken into the s7lmmae

tioa.Because of the 2D trnmlational i'avariance of the s'ystem, the force ccfy!r

aiFeavnce (A - .Jr), C'U (A, RJ4 = JU (A - X$!)stants depend only on the .# œ;?Their tramqlational propezty implies that the norrnxl mode solutions to (6.5d)have the form of 2D Bloch waves

1 .J t(q(z?+.ï)-?.,,(e)tqu#(Jz,z) =

Jg )(T e (())e

where N denotes tke n'lmber of xlmit ctallq. J.n the cmse of bnlllc systmrns ràhd... . . ((:

. 'ain the repeated-slab apprornnation snrnliltr formu)a,s are valid witll 3D q >!ltities. Withi'n the repeated-slab appror'rnationj however, the modes stZ (Vz

. . '' ' .

pend maizsly on the 2D wave vector Q 5.n the surface plane since the dépetïudence on the third wavewector component is negligible for su/dently laq4slabs. Substitution of (6.58) i'ato (6.56) leads to the eigenvalue pToblem

JT-IZE/J (Q)? (Q) - dCQ)e' (Q) (6.(59)œp s# lna :

,f.J-3

with tlze Hermitian dynamçcal mcfrâ'r

.j 1 c'it x)e-w(a+r:-r,1D.,LQ) =

u syj 'X'I .p( .

: x

(6kt't) 1

In the general case of S atoms pe.'r supercelkthe eigenvalue problem (6.59): pps!2 1 2 3S) for s' at each point Q in the lst'afà4.dlj: ë35 solutions (as(Q) (/1 = , , .-., .. ...

Briliontsn zone, which can be interpreteê as the brauc'he.s of a multiokttd''hlmctiontsztol. Eaeh normalmode nQ with frequenc'y w(Q) and dgenviètbz!!ei (Q) desczibes a particular excitatioa of the vibratiug lattice of all atomd.:

&

A quaattlm of the lattice vibrations with energy hu)nLql is called a pltonon.

6.4 Phonons 275

Consequentlyj the relations expressed by the equatinns a7 = w(Q) aze lcowaV Itonorb dfapersïtm rdations.

.. There are many phenomenological models to determsne the force con-Eétants XJ (A, A/) in (6.56) (6.32j. Contrary to phenomenological models, abktitio caalcalations require an accmate ard parametergee Mowledge of themicroscopic zespoase to lozen-in lattice vibrations. The basic ide,a to allirst-prindples methods is to detemnine tbe inteaatomic force cortstants Niathe total enea'gy (3.55) of the s'ystem with gozen core .coordjnates. ln thelist decade, mnany theoretical advrces have beea made toward the appm'cation of these concepts to lattice dynamlcs. There are two cornmonly usedàpproazhes for this knd of caLculation: the direct lozen-phonoa method andfhe pemtmbattve approacz. These methods aze essemtially based on the den-

.'2àztr g:nctional theol'y presented in Sect. 3.4.1.

Withl''n the hnzen-pltonovt ûpprocch (6.85, 6.86) one considers the propa-cation of a phonon wave of a ftxed wave vector whic,h is comnnenstlrate with a,. :r q...l'

.

E( ,/etikproc,al lattice 'vector. Tbis causes the atomb to vibrate witiz a d.e6nite dis-7 ,.ylacement pattfacn. At a given moment the syste,m will cocespond to a new' E' al st dure corrcxsponding to the Aozea' vibrationi mode. For suc,h a',tjtllst rtz

jhonon mode the total energy and the atomic forces are calclzlated as a, fhnrdc-

ttickn of atomic displacemeats in a supercell appropriate for the new crystaldtrudare. Using the energy chfkre'ace between the distozled and tmdistortedEE:btructt'trE's or equivalently from the atomic forces iu the supercell geometzyibne construc'ts the force constant matrix. Meanwhile, this method has been'mâned to the placaz force constant method (6.87). Tite atomic planes per-E/çndicala.r to a given wave vector are regardeê a,s rigid bodies, and the latticei'2' nrnscs can be treated in the spirit of a linear ckta,in model.D

The most Mportant pexurbative approae comsiders the linea.r rezpozxse? the electronic system to the displacements. Based on the DFT it is calledis

'the density Jvrlcfïorli pectncbation theorl (DFPT) meih'od (6.88,6.891. For ajjveu 'lattice distortion the resulting chauges .zAn,(z) of tNe elecwtron densit'y(3.47) and 1v'(œ) of the total Kohn-siam potential (3.48) thus need to beqvaluated. These changes are Oectly related to the electroGc contributio:a toEtffe hn.rmonic force constrts. The ionic contribution ca'a be straightforwardlyi uated 9om the Ewald snlnnmation method (6.841.:'y: Polar czystals, suc.h as semicondudors with partially ionic bonds andktjzuc czystals, need addaional considerations. Irz stzclz systems, where dieerent::' .

iljrjes of atoms/ions are involved, the long-range r-haracter of the Coulomb.!.Yrcqs g'ives rîse to macroscopic electl'ic îelds for loagitudinal optical phonons''trt' the long-wavelength lz'mx't. As a consequence, the dynamical matrbc is aotE Valjrtic at q = 0 (ia 3D). There is an additional non-analytic contribution: Etd the dpormicaL matzix (6.60) . It %sm tke geaezal fo= (6.00)

( z- t - p h ( n- . z- ,* )CJ l a-!f k 3 -*7 = h.= %) JçD=o Lq -> 0) = 4cr: -

- - .

MLM.f V' q . êx - q(6.61)

276 6. Elementary Excitations 1I: Paiz and CoEective Excitatio>

k* denote the ielectzic and the efective dyzomicalThe tensors êx and ç ,

(Born) ion charge tensors, respectively. Thereby, the index x stands for thepurely static eledronic contribution to the screensng. The tensor êco ca'a becalculated tusing (6.J1) for vanishy'ng frequency ul = 0. J.'a the cubic llml'tacd without the lattice conebution, the scala,r soo corresponds to the btTlkcozlstant sb discussed above. A new abbreviation is 'ttsfafltl, since another &.-electric constrt so hcluding the static contribution of the lattice will late,r

X* d ê can also be obteed in the frame-be introduced. The qu=tities ç an x

work of the DFPT i'a termn of derivatives of the dielectdc polnm'zation feld ofthe s'ystem with resped to the dispacements a'ad the electdc âeld (6.88,6.90) .

However, at least for cubic systems it has been shown that the screenedBorn Garges Z;/ sx caa also be derived directly wlthsm a supercez calcu-lation (6-91, 6.92).

6.4.2 Surface and Bulk Mpdes

The poln.rszaticn vectors eix(Q) of the lattice displacements iu (6.59) have35 components, nxmely three cï .(Q) for each pazticle at 'rï izz the l'nst cell.

5They indicate how the atom ï vzbrates izl the paztioTlltr mode sQ. Since thedpmrnl'cal mlttrtx $s Herxnitian, the polarization vcdozs satisfy the orthonor-malit'y condition

5-2 et*.(Q)4-.(Q) - Jxxz (6-62)'lta

and the clomtre relation

X'le' (Q),f* (Q) - kiè'œp (6.63)nœ MJN

<th edx(Q) = eL.(-Qt and au(Q) = *s(-Q) . For ins'tance, the poTv'zationvedoz's of a givec mod.e are normalized to 'nn7't.y ove.r the whole thicAmess ofa slab. This enables as to determine, by izzspectîon of the variation of theei (Q) for that mode over the variation of the atomic positions vz parallelNG

to the sttrface normal withn'n the thieleness of the slab, the locxll'zation char-acte,r of the vibrational mode, i..e., whether it is a HJ: mode, a s'urfac.e modeor a m'ized (rc-s/zwzzcc.l mode. TMs is quite sre=l'ln.r to what we have lenrmtabout the zectroaic state of a surface system in Sect. 5.3.3, Consequotly,the identlcation of tkevdsferent mode Garacters r.a.zl also be based on thecompazqsozz of the uu(Q) of the surface system with the b:llk phonon dis-persioa relations projected onto the suzface BZ (see Sckct. 1.3.3). J.n the b:711ccase there are three tconstic p/ztmtm braaches and 3(S - 1) opticai Jl?lozltmbranches with S ms the Irlrnber of atoms i'a the prlrnstive n'n''t cell of tkecrystal. Mong high-symmetzy directions in the bphl'lr BZ, suc,h az the (100)and (111) directions in cabic c'nrstals, the phonons can be c-lassifed as trans-nerse or longitndino.t accorcling to whether their displacements o:r polarization

6.4 Phonons

vedors e,)' Lq) are perpendic'alaz or paratle,l to the direction of the 3D wave

vector q. The projection of the relations ulxLqj onto the surface BZ yieldsgap resoas, pockets or stomach gaps iu which sT:rfnne phonon modes tsu(Q)c,ac appear. Another possible lequeacy region for surface modes concerns

that above the mmcimllm frequeacy txmax allowed in the bulk cmse. Modeslocaliged a,t the smface appear izl thks region if lighter atoms are adsorbecl orforce constants aze increased by sulface reconstruction/relaxation. Projectedbulk phonon branches a'ad possible surface or resonrce modes are indicatedin Fig. 6.26.

We note that also the wave vector may Muence the locn.lîzation of aphonon mode at a surface. The locmuh'zation is geaerally stronger for shorterwavelen#hs. This holds in particular for so-called maccoscopéc surface modes(6.831 . The attenuation of the pazticle amplitude away from the sarface is insome way proportional to the reciprocal wavelength. Therefore, for long wave-

leagths these modes extend over consiêerable distance.s i'ato the czystal. Sinceat lon'g wavelengthz the atomic cnrstal stntctme (but not its anisotropy) islqnl'nqpordant, suc.h modes can be found jn thc framework of e'lutic a'ad di-eledric continut'lm theozy A.a example is the Rayleigh mode (see Sect. 6.4.3).The microscopic modes are chn.racterized by the fact that their penetrationdepth into the mys'tal extends over oaly a.few interplanaz distancc.e for allwavelengtlzs. Most sulface modes that are aucolmtered in practice are micro-scopic modes, whic,h are locn.lszed in the stttfaqe layezs.

t<Q) i!

*a !l

i

ii!E!E

0 l

0 QBz Q

Pig. 6.26. Schematic representation of tîe projeeed bulk phonon bzanches(hatched re#on) and smface phonon brazees (solid lines) or smface resonxncephonon branches (dashed linel. 'rlze boundary Qsz of the smdace BZ and the max-im= Wbrational frequenc,y aJ= i'a the 'bulk czys'tal are indicated.

278 6. Elementmy Excitations 11; Pair and Collective Excitazions

n - .

Qtransverse longitud-lnal shear horizonll

Fig. 6.27. Tite three mx'''n types of pollm'zations of surface phonons 'with res/èctlEto tke satttal plaue (Q, 'rz). ; :

;'

.

.' . ...

.. . .g

.' '' .' j' .'

k( .

:'

!

.'

.

.. . ':. j .

' . :1 '

''

('.. t tjkhz surface problems symmetry considerations aze fa.r 1= helpful in sim

: j ) '

plifying and clmssifyi'ag the solations of the eigenvalae problem (6.59) thV oi'bu)k problems. Nevertheless, some simpMcations due to the remnsnimg S#iV'E

al l is played by the 4J#ifOJ plaae devv' èmetry are still possible. A centr ro e.

.'

.

:' ''

'

''

1by the phonoa wave vectoz Q az.d the surface normal n (cf. Fig. 6.27). Tvi,

,1 referred to this plane. When it coktciè' t'êimode polarizations aze customn'm y. jj. j jj

'

wlth a re:edion plazle of tke slab, the dyzmrnlcal matrix can be reducedblocks. One contnsnR t'wo thirds of the modes, of wlkic,h the polarizafv/t'wo

are ellipse,s in the sagittal plane. These modes are labded SP (sa#ttal-/liùé' ). .. . ymodo. Titey may be classled into two typest transvez'se and longitudluak 'Vit is akso indicated in Fig. 6.27. The other bloek contnsnq the rernnlniug ozx

. -- '.'

,.

third of the phonon modes, which are lineearly pob.znzed normal to the sagittz> :

plane. These will be labeled shear-horizontal (SH) modes. Note that only theq!

SH modes have a well-defu'aed polarhation normal to the sagittal plane (si ti!Fi 6 27) whereas the so-called trn.nsvezse and longiturismnl modes are attti 17

f- - ) .( .r.Eatly coherent rnlx-tm'es of b0th polnvlzations, the name indâcating whie,h hl 5:

the lazgest relative amplitude.

6.4-3 Myleigh Waves

h f rrnnlism described in Sect. 6.4.1 glves complete iaformation about thètiq:T e o .. y . . .. ,.jg

lattice dyzmrnl'cs of a surface s'ystem. However, for ceztna'm frequeney aù2,),wave-vector regiorus it ks pozssible to apply certxirl approsnrnate descrtpttùriïythat give deepet physicaz insight. TMS holds in partichlln.r for rnnztroscolft'

:'

,

:' ;' :'

.

:'

vibrational modes. We lmow from practical exercises i'a solid state physics (i.èïe.g. (6.93j) that izl the lonpwaveleng'tk limt't the equations of motion (6.5à)l

.

' ' ' ' ' '

' *' '

for a linear chnnn eahauge over into the wave equations of an elutic contintplm.. .: ..: . ..7i

Witlnsn the frnmework of the linea'r theory of elasticii'y the equations of motiidof an elastic medlum are (6.94)

o, (-9.., (w, tjpazzralx, t) = )C ozpp

La,p = z, v, z), (6.6i''

6.4 Phonons 2$9

@ ere p is tEe mass density of the medium, .u@, tj is tite displacmment of thezzfèililll'n at tbe point m and time 1, aad &#t t) îs the stress tensor. The latte'r1k b Hooke's lawgiven y

J'') c.o.,o' -''tG' - J-.z eobofqeo, , (6.65)Cœp = Dzp' ,ue/ ,J3/ <k

khere the (tQ,.,,r) aze the elements of the elastiq sbfr-neôs tensor (at con-jtint macroscopic electric Eeld), and (etz.#) are the elements of the piezo-èidric charge tensor. The l'nnsuence of the macroscopic electric âeld X i'ali 'C. .

tfte medblm is take,n into account to allow aso the treatment of piezoeledrictikàteials The tensor of the pieaoelectric charge,s is symmetric Cfn the secondrkiji.of icdices. The elemeats of the tensor Coboorn' are symmetric j.nh a aad ;,1ik ;q?' aud p' a'ad ia the paizs an and a'p'. This makes it possible to exprersstrvm eqtzivalemtiy in a two-subscript' notation, Cœpœ,pr -> q.;, according tot11'è stbeme z:r = $, yy = 2, zz = 3, yz = zy = 4, zz = zz = 5, n = yz = 6,tkud to iutroduce the elements of symmetrized stress an.d stram tensors. Theelpstic mectb:rn Ls considered to occupy the lowe,r hnlRpace z < 0. The sur-fàce z = 0 ùs azsumed to be stresseee, i.e., cazlzxo = 0 (with cuz in thedkmmetrized form). The electric Nelds f1116l1 the botmdary conditions of elcc-tvàtatics.J ,,. ln order to simplif.g tbe considerations we study an Lsotvoptc medbnm4 k'i'h t the piezoelearic Kect. There are 'two iudeperdeat ehstic moduliou

tî:, cza, au.d cu with cu = c11 - czz. with Laml's mod.uli it = lz (c11 - cza)ttktd à = clc, (6.64) becomes (6.93,6.944

:2'tztm, t) = njl grad divuta, Lt - v:2 carl cuzlxztm, tj, (6.66)&2

.wrllre the longituainal and trxnwerse sound velocitie,s

',(a = L2g + à)/p,'nt = y/p (6.67)

1'e ltzoduzed. Equation (6.66) suggets splitthg up the displacament âeld''QL# 'tz! +w icto a turbulence-free contribution tcurlw = 0) and a source-free.7 .L, .

' '. .

.jz't ldivut = 0). In the btllk', tEe resulting dzflbrezttial wave eqttatiozls can beikved independently for b0th contrsbutioas, giving longituan'nal sotmd waveséï l '

, i;(jp, t) aad trxnsvezse (sheaz) sound waves 'tsutz: t) propagating with the: .. . ' (sotiid velodties /t& or vt. Eac,h of these soand waves obeys a wave equation

,2'tzvttm, t) = 'tf/taltrlt,l/ttm, t).(9$2

In the cmse of the elastic halfspace (z < 0) we only seek solutiozzs of the'Lqtpke equations (6.68) witic,h represent surface waves, i.e'., which are charac-Viiked by = ex-ponential decay iuto the crystal. With a wave vector Q in

i dùrface plaue (z = 0) one has

28û 6. Elementazy Fzxcitations 11: Pair and Colledive Fxcktatioz)s

5(Q=.-a7t) O2-bJ2/WpJ'tz:/tlz, t) = el/te e (6.69)for Q > Wqyt. The special fo= of the positive decay constant is a conse-

quence of the wave'equation. The polarization vectors F7r1611 the conditions

et . A-t = 0,

e) x AA = 0,

K%/ï = (Q:r1 Qv, -i Q2 - cJz/rtzs .

Wavcs of the type (6.69) aze called Rayleigh waves (6.952. Ffowevez, itsdividual waves (6.69) cnmmot fttlfll! the botmd.a'cy conditions of a stress-leestTrfnre. This is only possible by linear combitmtions ('t1: + 'tttl 'with a mixedlongitudlmal-transverse chxracter. Moreover, the bolmdar.y conditions cFmnotbe satisled for shear-hozizontal modes. ln a'a isotropic e-lmstic medî''m onlysagittal-plane E'ayleigh waves are. Vowed. The displacemem.t vector = of thcsurface wave lies iu the plaue spanned by Q and the smface norrnnl zz.

The botmdav conditions, cuzqmc = 0 (a = 1, &, # , lead to an eigenvalueequation. Slml'lar to the bulk case a linea'r dispezsion relation

uzawlol = rswkol,rRw = 'pt ' ï (6.71)

Ls obtained fov the Rayleir,h waves. The phase velocity luw is #ven by thesolution ( (0 < # < 1) of the polyaomial eqaation

2 126 4 a .-.% -.t-ï - 84 + 84 3 - 2 p

- 16 1 -

z = 0.

'tl 'r %J :

The quactity (' depends on the ratio of the sound velocities. The ratio (14/q)2VHe.S between 0 and 0.5 for the various mateziaks. Tllis corrœponds to a

xariatiozz of #' iu the interval 0.955 - 0.874. For Gaà.s a value ( = 0.92 isobtaiaed (6.974. J.n any case thc velodty of the Rsyleigh wave zxw is srnnller

tha'a the tzansvezse solmd velocià rt.. Thks Ss also tmze for cubk mystals withthree hdependent elastic constants (6.96j. Consequently, the Rayleigh modeshould be below the projeeed acoustic branches, at lemst for wave vectors inhigkssympetry directioms.

The rn5ved longitudical-t:ansverse chrader ks seen from the directionof the dîsplacements whic.h are paztially parallel acd pactially norm/ to thepropagatioa direction msslzmed to be parallel to the r-aMs. At the surface,z = 0, the polazization vector is given by

2 - ($2 2eaw = - 2 1 - $2, 0, -i$ . (6.73)

2 21 - ('pt/'t&) ('

The dispersion relation of a Rayleigh suzface phonon Ls shown 5.1 Fig. 6.28.TEe dispersion relation Ls measured for a clea,zz Ni(100) surface by men.nq

6.4 Phonons 281

. 5

150RS n4.E, %

80-> %ok

# 1Oo 8 3O

ö' u

@ a= u

g' a 2v:z u

Lc: 502 co a 1= uln-

0 '

()0.0 0.5 1.0

RX

k

>xOc

D=

<.

cOcC=œ

F XQ1Q (K )

Fiy. 6.28. Wave-vector clispezsiozz of the Rayleigh phonon on tke Ni(100) surface.It ls memsured by menx!q of EBLS usitg t*e primav emergiœ 180 W (squares) and322 ev tcirclesl. After (6.982,

of M-RMELS 26.9% . Near the bouad.azy of tile slzrênre BZ êeWatioas fromthe resdt of the continutlm occtzr as in the bullt case. The phnse velocityexhibits a wave-vector dispemion. For wave vedors at the zone botmdarythe localization of the Rayleigh phonon near the surface iacex'xse,s accordingto t. he decay constauts, e.g., Q 1 - ê2. J.n sach a Rayleigh wave only a fewatomic layers beneath the surface Wbrate. Consequently: Rayleigh waves withshort wavelogtkts may also be used to detect efects of bond changes aadreconstruction. One drastic example is the sharp (iip kl the surface phonondisperskon of W(110) whic,h appears 0e,r hydrogen satttration of the stlzface(6.99!.

6.4.4 Fuchs-Kliewer Phonons

The surface iHuence onw optical pbonons can also be studied i'a the long-wavelecgth tsud, hence, quui-contiauum) lirnst simn'lar to what we have)

learnt for acortic phonons. We consider the simple case where a bulk Eik-active czystal has tvo atoms in the uait cell with mmsses 3f+ aad M-. Sucha crystal could again be a polar smmicondudor or azt ionic c'rystal. The bonds

282 6. Elementary Exdtations II: Pair and Coiedive Excitations

should be partiatly ionic, ard the pair of atoms conwponds to cation and an-

lou witlz sm-'ned dynnrnscal ion charga e* and -c* with e. = ezijsfc (see(6.61)). Taking iuto accotmt only the efedive nexet-nearest-neighbor forceconstant .f, the equations of motion (6.56) for the disphcements 'tz.y and .tz-

read ms

:2M+ 'tt+ = -2.f('u+ - 'tz-) + z*E,

d@2d2

M- (uz

'tz- = -2f ('tz- - 'tz+) - e*B. (6.74)

I'a the considered loug-wavelength l''m'lt the displacements do not dependoa the position R of the Atnit cell. 1r. the continutlm limit, however: one

may discuss the dependence on the space coordinate z. Becaase of the ef-fectîve càarges of the cation aad anlon additional restoring e-lectric forcesoccuz. Tàe total electric feld E ac'ts at the position of the respective core.

It is therefore inBuenced by Bcal-âeld esects wizich wilk however, not bediscussed here (6.90, 6.93). Skce we are interested in lonpwaveleugth op-ticat phozmnsj there is only a need for stad>g the relative disphcement'u = '?z,+ - 'u- of the cation-anion pair, whic,h dbrate.s with the reduced mass

Mr = M+M-/ (M+ + M- ) . Then

d.2Mr z

'tz = -2.fu + 6*E. (6.75)dz

J.11 the ll'ml't of a harmoztic time dependence with frequency tzl one obtnl'nq

c* 1 .

n, = u g.E (6.76)V; savo - ts

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 c''tt ands

hencej a dielectric polnrization âeld

N.puaz = ..e w,.F (6.78*)

lat lat .s 6 zr8) cu%es jue latticèEBecause of (6.77) and tNe relation P = x (&J) , ( .

susceptibility a,s

+2N e 1

xhztfz;) = ; a . (6.79)V Mr ww - u'

In additiono atso the electrons contribute to the total polarization âeld by: e.l

'

L.PO = x E. Fbr frequencies below the absorption edge of the electrdiiv

6.4 Phonozus 283

Fystem the electroak polarhability xd = (:x - 1)/41 ex.n be replaced bythe static polarizability with ex ms the higisfreqttenc,y alectcotdc dielectricconstant. From the deAmition of the dielectric dksplacement îeld

D = E + 4zr,FDM + 4rr.Pd H LbLu)4E (6.80)the lcquenca--dependen.t blllk dselectric Alnction

Q - wzLto 'rosbtapl = cx 1 + z zu'To - LJ (6.81)

k derived for lequencie,s below the htndamental gap in the elèctronic bands'tmzctare. He're the (sceeaed) ionic plasma lequency

24re* N2 2

DJLO - û7TO =E'xafrF (6.82)

;i: fntroduced. This dm6nstion g'uarantees that the zero of the dielectz'ic flcnc-

jion, cbt(al = 0, eqz:xlsl the Sequency wLo = 1t?z,0(0) of the zone-center LOlshonon. With the statie dielectric constant sz = sb(0)j that also accotmtsf thc static lattice pobqrtzabilit'y i.e., &a > sx, the Eyddane-sn.nlnK-r-rellerPr ,

tèl>tion tu&o/u&olz = ca/cx holds (6.100).Since macroscopically the crysta,l i,s eledric-ally neutraz, one cau apply the

3àkkss hw '

dîvr = 0

or with (6.80) eqttivazently

skhtslltlivf = 0.

Nèglecting retardation efects, in addition the Maxwez's equatioa

curl E = O

(6-83)

(6-84)

(6.85)1i6l&. Equation (6.84) is f1116:1ed when eitheer dâvf = 0 oz sstoJl = 0. The, qt case ctivx = 0 implit's that tlze electric âeld E and with (6.76) alsoi1k4 relatkve displacemeze feld '?z. kave to be trxnmvezse, div'u = 0.. The di-Jt qqtric A'nction sb(uJ) h.as a resonxnce at ts = tga'o which is assocîated withtrxiivez'se lattice Wbrations. Thus) az'ro Ls catled the transverse rcsoncp'ce

''';.''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''d

, l ' . ,yyknertcy. In the second case, div E # 0 (but cuzl.F = 0) and sb(aJ) = 0,iiè :eledric Eeld E of the excited waves is longitadinal, Tke chvadetistîcièquenc'y is EO:E.II at w = fztoo the lonnitudinal rssonance Jrazuerzcr. Shcek'' w:r/ss(w), E ks cot necessxrlly zero even for D = 0. The displacement: . .

r' E'

.

iéïd.satisses cuzlvz = 0. -

.

;' .'

Fpr Fequencies tzx-o < ;(J < wuot the lattice polarization bnn simtztane-ottslf to An161!

284 6. Elementary Bxdtatîons F1: Pair and Colledive Bxcitations

div Jva,t = 0 cml-rfat = û (6.86)1 1

zesulting ia slmslar relations for the relative displacement feld 'tz. These con-

ditions have consequènces for possible displacement felds in a polarizablehaltspace (z < 0) with smface (z = û) . The felds ia such a system are cilar-azterized by a scalar potential /@) with E = -grad/. Because of (6.83) and

(5.85) the potemtial must Gal6ll a Laplace equation

Zm/@) = 0 (6.87)

a'ad. the standard bound.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 ILg-azds, whic,h decay into the vacuhtrn as well as into the crystal, are describedby the fozowing solutions of (6.87)

yuiozeom fozz < 0,/(œ) =

/(œ) = BeîQne-Q% forz > 0. (6.89)The 2D wave vector Q deterrnl'nu the attenuation of the electric îeld andthe displacement âeld into the blllk. The boundarjr conditions #ve a1. = Band the eigenvalue equation (6.22) for the dieledric smface exc-itatioas. With

(6.87) the eigenvalue equation, sb(uJ) = -1, leads to the gequency of the (inthe pre-sent approxlmation dispersioaless) Fizc/ls-fflïelccr phonon (6.1O1q

co + 1tz'.p'lc = u:sto , (6.90)

soo + l

whic,h indeed lies ill the intezvezl tppc?o < u: < Yco. The accompanying dis-placement feld '?z in the polarizable hnlfxpace Ls parallel to the vector ($ 0, 1).Therefore, the Fhchs-lcewe,r phonon also repreelts a sagittal-plre modve,but with a ftxed rnl'ved longitudinal-trnnsvease c'haactez. Since the trxns-fen'ed wave vector has no in-pla'n.e comporent, ushg the backscattering ge-ometa Fuchs-Kliewer phonons cazmot be excited ia a one-phonon Rxrnan

experiment.The Fuchs-newe,r phonons are nlnn.racteristic collective elementary e-x-

citations of the vibrating atoms in sarfaces of crystals with partially ioaicbonds. The formula (6.S0) derived for these surface phonons is generally valid.This ha.s been shown by inspection of the material parameters ugpx, tzro, cc,

a'ad scn for many compotmd smrniconductors (6.102j . Fttehs-Kliewer surfacephonons were flzst detected by HREELS with Zn.O surfaces (6.103) . Typi-cally tke Fkchs-cewer phonons gi've rise to stron.g (even mtlltiple) losses in

6.4 Phonons 285

Energy Ioss hro (meV)-J.0 -K O 20 40

0-:- aa:= 1D

XSCO.d%R

17orëp I xqa; -28: -!7c.S 1 lS

Q

-J.DO -2tt O 200 KC

Energy Ioss 'hro (cm-1)

Fig. 6.29. N-RFVLS speetmlm of a GaAs(110)1y l surface observed in the specalargeometry (.P poiut) for a prinzazy enerr 5 eV. Fkom (6.1O4q. '

the HRFELS spectra of polar crystals obseaved i'a spccular scattering geome-try. One example is presected in Fig. 6.29 (6.104q. The spectmlrn of a cleavedGaAs(:.1O)1x 1 stzrface using a semi-insulating substrate mainly shows theFuchs-lfliewer phonon at nso.x = 35.8 meV (289 cm-l). A second, extremelyweaker (by orders of magnitude) featuze in Fig. 6.29 appears at 21.1 mev (170cm-:) amd. is identïed a,s another surface phonon neaz 1-h.

6.4.5 Tmfluence of Relaxation and Reconstruction

Accordiag to the above discussions, the smface 'dbrations may be used toprobe the surface atomic structttre, since the restoring forces deviate â'omtheir bulk values. This holds for the micaoscopic modes but, in pzinciple,also for the macroscopic modes. For instaace, the dispersion of the Rayleighmode is Sn6uenced by the surface details for wave vectors aear the smface BZboundaaor. Soaces with an atomic s'tructure not too digerent from the bulkone are relaxed surfaces. The most intensîvely s'tudieâ relaxed smfaces arethe (110) cleasrage faces of EU-V compotmds crystallizing in the zinc-bltandestructme.

The (110) smface phonon dispersion of one of thE'se compotmds, Inpj isshown in Fig. 6.30. Calculated dispezzion relations (6.36j are compared withmeaslzred values (6.34, 6.1054. II).P îs considered to be a, prototypical material.Since the masse,s of the anion aad cation are vezy dxerent, a large gequemcygap between bulk optical ard acoustic branches exists that allows for szzrface-irduced gap modes. Tlle theozy, of course, predicts more brancahes of surfaee

286 6. Elementanr Excitdtions 11) Pair and Collective Excitatiozus

Wave vedor Q

Fîg. 6.30. Surface phonon deersion for a.rt 1nP(11:)1x1 surface obtnt'ned by dif-ferent methods. Ab initio D. FPT cakulations (6.36): solid lines, 'FFRIVLS E6.1O5!:squaro, au.d Rnrnan spedroscopy: dots (6.34J. The shnrled. areas desczibe the prœ

jected bulk phonor branches. From (6.341.

modes or surface resonxnce modes thn.n observed spectroscopically for difFer-

ent remsons, suc,h as scattering geometzies, matrix elements, selection rules,locnllzation, etc. The mayleigh mode appears as the lowest smface phononbraach, at least along tke PX' dtrection. For 11I-V(11O):tx 1 surfaces: typi-cmlly, it is found that upon sarface relaxation (see. Sed. 4.2.1) the Myleighwave modes at the zone bolmdary points R, X', aud a az'e shified upwardsby up to 2 meV (6.106j. Clnnzvtezîstsc feat'ares for sarface modcs and surface

resonance modes appear forïaptllollxl at lequencies of about 60/82 cm-l,

146 cm-l 254/270 cm-l, a'ad 347 cm-l tlsing the Rmman signature's (see)

Fig. 6.11). ln these frequenc.y regiozls more or less weaMy clisperskve phonon

branclzes are also observed by TTRRELS (6.1051 .

At the center of the surface BZ, lh as we2 as a2. ong ?X' (parallel to

g001)), the smrface vibzational modes c-q,n be claasifed according to tke trre-ducible representations of the poiu.t group m (or Qv or G) of the redangnllar

IE-V(110) smface llnst cell vith a rnirror plane that inclades the (0O1q direc-tion (see Tables 1.4 and 1.6). Accordingly, atoznic vibrations along (110), i.e.,along the III-V zig-zag abasn direction are represented as AM modes, aadvibrations perpendiculaz to the chain directionu are represented az AI. TheA'' modes represent shear-horizontal 'divations, whereas the A' modes are

6.4 Phonons 287

(transverse) sagittal-plane modes. Suc,h a clear dassfcation is not possiblealong the symmetzy directions PX (parallel to g1ï0J) and Pa (parallel to(1ï1)). The sagittal plane (Fig.6.27) deânect by the surface norrnnl and thephonon wave vedor is not a reoection plane. Consequently, along these &-redions modes show a rni'vklzre of sheaohozlzontal and sagittal polarizations,i.e., tbe atomic disphcements llave nonzero components in all Cartesian di-redions. The Rs.rnan feattu'es at 69-82, 146, 254, and 347 cm-l (Pig. 6.11)can be mssigned to smface phonon modes of A' symmetzy

The t'wo surface phonon modes at about 254 aud 270 cm-l (6.:?.44 arelocated in tEe gap between acoustjc atld optical bu?uk brarches. Their eigen-vectoo are irecated in Fîg. 6.31. They =M5n!y conwpond to nearly opposiugmotioas of the Ast-layer ar.d second-layer anions, with the most sîgnifcantcontributions from the furst layea. The chaœacte,r of the 17.n7,,= peak near 347çm-l is not complete'ly cleaz. A surface phonon mode at the 15 point, that ispredided at 353 cm-h (slightly above the LO phonon frequemcy) by DFPT

E calculations (6.36) aud at 349 cm-l by sezai-empirical calculations (6.37j,hms been assigned to matc,h the Fuchs-cewer pbonon with A' sym-petr-?.%. Fig. 6.82 the isplacement pattern of the corresponding surface F opti-,Vl phonon of GizAs(11O)1x1 at 286 cm-l (6.35q is showm For TnP(110)1x1é: XEELS however, yields a value of 342 cm-l for tize Rtchs-cewer pitonon1

Ebèiil.g in bewreec the TO and LO phonon energies of I'IIP (6.105J The chna'n. c

zitode of the relaxed (110)1x l s'arfaces for wave vectors along 1nX' possesses))/ symmetzy a'ad corresponds to the opposing motîon of the catiozzs and an-

ions in one atomic layer parazlel to the zig-zag-chna'n directions (1101. The dis-placeznemts ill the Ast-atomic layer are larger. This is shown for hN(11O)1x 1tàhd 1-h in Fig. 6.33 (6.107q. For the I'CN surface tMs mode lies i'a the uppe,r (L.a,If$î the bulk acoustic-optical gap at Hchautol = 55.6 mev. However, for othe,rVI-V compounds, it is also a gap mode, e.g., for A-lsb with *01t::$,:(0) = 35.0lev (6.10$.

r

E1 1 02

)

E1 1 0 E1 OE 0 :(I

Sng- 6.a1. Atoznic dksplacementy (eigewectors) on J.nP(llO)-1x1 whicz #ve rise toFîoizatkr:ed gap phonon modes at r. 'ne :engths of the arrows are taken from DFPTkalculatiozus (6.362. open t5J)edl smbols denote In (P) atozas.

288 6. Blememiary Bxcstations 11; PG and Collective Excitations

E1 1 0(1

%

E00 11

Fig- 6.32. Atomic displacements of the .? p/onon with energy 35.5 meV oq theGaAs(l1O)1x1 surface. The arrows give the Jtrnplitudes of the eigenvectors d(0).Filled drcles: nniorus, empty circles: cations. Fkom (6.351.

rrhe phonon modes measm'ed or c/calated for 1H-V(1$0)1x 1 surfacesshow rather clear clmrnlca,l trends. This is demonstrated in Fig. 6.34 for theFequendes of the Rmbq-Kliewe,r mode (Fjg. 6.32) acd the sttrfacolayer zig-zag clmt'n mode (Fig. 6.33). The xlmost linear vadation of Flau(0)c,c, the h'e-

quenc'y maltiplied by the blll1z lattice constant, versus T/ M: indicates thatthe efèctive force constant f of the type in (6.77) or J(&0 + 1)//x + 1) izl

Et ïOqlt>

Fig. 6-33. T:e qhizin phonon mode of A'' symmetry ozz a (110)1x 1 surface (topview). Here the In mode of IzzN(1:0)1x1 Ls shown. It 11% in the upper Bn.1f of thebulk acoustic-optical gap at ntzgcluuatol = 55.6 mev. After (6.10e(.

6.4 Phonons 28g

400

InN35o M

o:Fj; 300

AM&

X Alsb K (7

é' 253 X

mG Ga/ks:-) 200 MG Gasb

lnsb K

150 D

:a

1000.1 0.16 O.2 0.25 0.3

l/qo-r (a.u .)Fig. 6.34. Product of phozton enerr Nd lattice éonstant versas the reciprocalsquaze-root of the reduced mpA.q for two F'sn'vfnme modes of 1II-V(:10)1x1 surfaces.Fachs-Kliewer mode: ftlled sqaares; surface-layer zig-zag ckaiu modet open squares.From g6.107, 6.10$.

(6.90) varies nearly ms the reciprocal square of the lattice constant or blllkbond length.

The Si(111)2x1 surface can be considered a,s a prototypical sllrfnzte tostudy the l'nsuence of surface reconstruction. It is chvaderized by the for-mation of tilted zc-bonded ctlnnlnq iong tize Eï10J direction (see Sect. 4.2.2).Also this surface sbovrs a Rxyleigh mode. Along 13.1, i.e., along the chnsns,

there Ls excellent agreement for the dispersion of the Rayleigh mode betweenab initio theory (6.10% a'ud measuled results obtained from Ee-atom energyloss spectroscopy (6.110j. At the zone botmdarg the RW mode is YredlyOected by the reconstnzction. Tize Rayleigh mod.e at the J' point lies atapproxlrnately F/z.uxw(J) = 10 mev and is poladzed perpendicular to thc n'-

bonded chatus. The additional dispezsioeess resonant mode detected at 10meV aloag ?.1 g6.110j has not been veMed by the DFPT calculations (6.1094:but has beea reproduced in adiabatic bond-cZarge mode.l calcuhtions L6.111).

Ab izkitio latticedpmnnical calculations (6.109) 6.112) also predict severallocalizcd ar.d resonant modes in the energy range 45-68 mev arou'n.d the blllkoptical zone-ceater phonon with F/zs'ro = Flamo = 65 mev. The displacementpatterns of three characteristic .I9 modes being more or lesé resonant withblllk optical phonon bTanches a're shown in Fig. 6.35. These surface resonance

modes possess enertes of 57 meV (after improvement of the i-poin.t sxmpling

290 6. Elementanr Bxoitatioms II: Pair aud Collective Exdtations

D 1H

ET1 0)

l kk St!

E1 1 ZE1 1 12

I(@JI2

>xg. 6.35. Disphcement patterns of selected resonant surface pkonon modes at ?of the &(111)2x 1 sudace. ARe.T (6.112q.

$)

ita the calcalatioa) (D), 55 meV (r) and 50 mev (J??). However, they ap'çstrongiy localized i'a the St'st atomic layers. The D mode corresponds t3.longitudinal-opticat vibrations along the chnsns in the &st atomic layer. TVtwo other modœ 11 au.d J& also contain contributiozss from the secoad atomiçlayer. These modes are essentiaxy polarized pazallel to the surface normk.

. jyliyvith a smaz (121) component perpendiclzlar to the c-halms. The D mode wita'a encrgy of 57 meV (6.112) or 59 meV (6.111) is accompanied by a lafgè:polxrization âeld. lt may therefore be identled with the strongly dipol/:

taive mode detec'ted by HR.EELS 5.zt the optical-phonon zegion at 56 meVK

(6.7134 or 57.5 meV (6.114q.

6.5 Elementary' Excitations for Reduced Dimension

'Fhe mnmpbody theory of intenading particles, espedally eledrons, in solidàpredic'ts the existence of mazty dementary exdtations. Because of the,ir mtl itaal interactiou or their inte-raction with other particles they are renorrnniz

ized and therefore named qltn.qipal-ticles; whiclk may show a one-to-one corzr

respondence witk non-interacting electrons tsee Sect. 5.$. In the case 6fmetals the correspondjng physics can be described within the Femni-liquv

% (6 761. The properties of electroms become more and more exotiè&PPrORn .

. . ;L1as oae progrcsses from the three-dpmensional world into lower dnrnenskonà.In a two-zirnertsionat eledron gms one a'tready obsez'ves sttrprising phenons:ena) suc,h a,s fractional nbxrge aad statkstscs ia the regime of the gactiona?quaatltm Ha.IJ eFect. The corelated motion of electrons and magnetic vof.tices generates these uzzasual phenomena (6.115,6.116). Efects sternming frozithe jncreased eledron corelation aear smfaces have bee,n Ocussed in Sedi:5.4 and 6.2. Predîeions for a one-dsrnensional electron gas are even morrexotic. This applies ic particular to one-dn'rnensional metallic clmimq. Suçfqlow-dimensional systems nxbibit a variety of novel physical phenomena) suci

6.5 Ezemerzary Exdtations for Reduced Dimension 291

J: charge-density waves (CDWs), Peierks irtstabilities, or the formation ofrlcm-Fezrfrti-liqdd-De grotmd states (6.761 6.117-6.1212.

Quasi-onew-fisrnezasiortn: structtkres are t'ypical for rn>my reconstracted semi-jiönductor surfaces. Important Rvmples are the nhnan structmes discussecl injéc't 4 2 However, more interesting iu the many-body contex't are adsorbate-e. . *: E . .induced modp6catîons of semicoaductor spqrfaces. Self-orgn.nt'zed adsorbate-tnduced modi6cation of semicondudor surfaces is a powerful tenhnique forfàbzicati'ng suc,h low-dimensionaz nnnoscale quaatlxrn structtzres. Jmportant: : .

--

.jxxmples are one-dMettsional quantl:nn chnanq of metaz atoms, sue,h as In orlu.oc Si(111) surfacez. The nrlqozption of irtdblm induces the formation of:': -

. ''

. -

-

tmaai-one-Hnmenqional cktyunsl on a 4x 1- or 8xz-reconstructed Si,(111) surface(6k422, 6.1231. Arrays of monatomic cllnsns aze also observed for Si(2.11)5x2-ia,or stepped Si(557)-Au ((111)5x1-Au) smfaces (6.12*6.126j.

The arrays of quasi-ono-dzemezusipnal chnsnq induced by metals on theE!i(1'11) suzface exhibit a valiety of iuteresting electronic states. Theze are

,bè>oz'ts of correlation eects dastroyixtg the metallicity of suzfaces with a,!zJ' éd electron cotmt per llrp't cell (6.127-6.1292, of anomalous surface cormlga-Eiiàn eitser by cows E6.180, 6.1:12 or by large atomic displacements (6.132q,.7 .

.8?E,metxll5c nanowi'res (6.131, 6.133, 6-1341) of sudaces of rnl-ved ri'-rnfmKional-;.:''j;' . . .

#y 1(6.).24j) azd of spitscharge separation in a Luttinge,r liquid (6.1351.E.1n a Luttiuger liqtzid (6.76, 6.119-6.121) the elecvtron loses its identity and

Erçsarates into t'wo q'tlmipazticles, a spénon that carries spin without chargeVd a hoio% that carrîes the positive charge of a, hole vfthout it.s spih. T'h.eieon for the positive charge of the holon is not peculiar to a one-dinnensional'Blid. It is simply related to the fact that one cxnnot probe the e'nergy andiömentnlm of a:a electron in a solid without ejectkg it, for exztrnple as a pho-iioeledron tn a photoeznission experirnent accompn.nsed by a photohole (seejèèt 5 1 2). This leeaves the solid iu a positively charged, excited state. The-' '

t*o asgerent quasipmicles, spinon a'ad holonh have rlsFerent grou.p veloci-tteir and rkm away fxom eac,h other. T*e proble,m of spin-chazge sepvation îsjj..;: .

' <'

ianxlyzed n'lrnerically in the met>llsc phmse 41s3.r.g a one-ba'nd Eubbard Hn.rniktözïia.u (cf. Sect. 5.4.2) i'a one dimension (6.1214. The spedral Annctiozz of theiigle-particle Greea's Alnction (5.20) with spin can be decomposed icto t'woktfwsipazticle peskq that change their energies cvztk) licearly with the (wwsi-ùzùmentx'rn Flk. These two pen,k-q coincide at the Ffarrn'' level ss ozfy, whereEfh' e sphon and holon lines intersect. The spinon hms a laTger group ve-locity'8i (k)/d(Fzk) th= the holon by nlrnost a factor of two.i

The ùzcreasizlg amolmt of cozrelation bet-wto.e'n eledrons in lower dirnen-jions can be rationldized by a simple, clazsical pictttre whe-re eledrons behaveik é billiard 1m1lq (6.136j. They are forced i'ato head-on collisions i'ct one dnrnen-

à b use they cannot escape hom each other on a true one-dirnensional!si n ecati-k.. QpnnmtT:nn menhn.nically thei.r wave packets have to penetrate each otheràt some point în time a'ad thereby geneapte mnv'nnllm ovLrlap. In t'wo or threelzifnensions sucz a situation is impzobable. In one rlsrnension, the IFe'T:G suo

202 6. Elemertary Bxcitations 1I: Pair and Collechtive Excitations

face' 6v = szyatkl consists ozf.y of t'wo points at kp aud -kv (with the Fermiwave vector kp). The i'ateresting physicaz proq-es happen arotmd the sFfA=l'surface'. Consequemtly, the-re is no such thing as a single electron i:a one di-mension. When exciti/g it datrimg the meazurement process one necessarilygenerates a chain reaction that exdtes other electrons. The rewstllt is more likea cozective excitation. Consequemtly, one may visun.lsze a holon in a Imt'tinge.rliqttid as a CDW acd a spiuon as a spin-density wave (SDW) .

Kxotic phenomeaa, sucb ms sph-charge separation, have nmn'nly been dis-cussed for metallic systtarns. They allow l'n6mstesMally small excitations andare sensitive to tiny pertrbations. Recently, signatmes of spin and c,hargecollective modes j.n onmdirnensional metallic chnsnq have been believed to beobsmed withim photoemismion ex-persments (6.135). Aa array of monatomicAu dmins on a snghtly misoriented Si(111) surface (Si(557)) h.as been studied.Afte.r xnnenls'ng of a 0.2 monolayer-thick overlayer a Si(111)5x 1-Au sprrface

appears with parallel chaims cf Au atoms i'n an intrachain distance of 3.83 ihic,h are separated by about 20 i twhie,h corresponds to the tezrace widthlfW

Photommission seems to indicate a metallic character with a parlially ftlle'dband. The authors have no indications suggestive of CDW and claz'm the ab-sence of a Peierls trausition (6.118) for such a, system. Fbr one-dMensionalsystems: however, the vezy efstence of a metal is in question. Accordsngto the Peierls theore,m (6.1181 a one-rlsrnension.al c,hn5n of atoms is kmstablewith rcspec't to a pairing of atoms, which creates a'a eae'rgy gap in the bandstracttzre at the Farnnl level. In a trae ID system of idcntici atoms such areconstruction is the ozfy possibility for lowering the total energy. The en-

eror geed by lowem'mg the occupied states at the bottom of the gap exceedsthe strlu''n energy that is necessary for the displacement. Tm. fact, it is dië-cult to Snd one-dsmensional systems that a're metalc. hdeed, for rzhnsns on

Si(111)5x2-Au a Peie'rls gap is obsezved (6.1241. However, for metaltic sys-tems suc,h as Si(557)-Au a Luttinger Iiqt'dd witll a spGon-holon splitticg htzsbeen rulcd out and, moreover, details of the atomic structkzre, suc,h as brokenbonds at the steps, have to be taken into coMideration (6.125, 6.126j.

One of the signlcant featuzes of qumsi-one-dimcnsional IVSi(111) sys-tems is a reversible phase trn.nqitîon, TEe room-temperature Si(111)4x 1-1'asurface undergoes a tr>m.qition at about 100 K to a new phmse with a 4)4 $2'zeconstructiong whic.h is ddven by a ID CDW or, equivalently, a Peierls in-

'

stabitity alony the Iu chnsnq (6.122, 6.123, 6.1371. It is specttlated that thetrae low-temperatme grotmd state mig'h.t be a well-ordered 8x2 phase withthe CDWS locked in pbmse (6.131j. Electronically the room-temperature 4x 1phwse evhibits the Ferrni-iquid-iike behavior of a strongly anisotropic two-Hsrnensional metal (6.131, 6.134), whereas tke 8x2 phaxse evlnibits feature,s ofan insulati'ag system. Recent studilas (6.123, 6.137j show, however, that thesimple SD-CDW formation camnot be the dri'ving force for the phase transi-tion.

7. Defects

7-1 Realistic ard Ideal Smfaces

Otherto our picture of what a solid sttrface loolcs like at the atomic level wms

fairrly idealized. Atomic reazrangeaents and mon-stoiœometric distributiorsof cations aud atfons were only allowed to zeslzlt ic relaxed or reconstruc'tedsarfaces with a deMed 2D trauslatioaal symmetzy In realit'y, invaziably s'ar-

faces aze found to consist of a mbttt'tre of Qat regions, the so-exl7ed terraces

with a cel'tain reconstmction/relaxation bat surrotmded by Ene defects, e.g.,steps, kn'nk-s, and modled by point defects (see Figs. 1.1 ard 2.1). The ter-

races Tepresent poztions of low-inde.x sttrfaces. Only extended terraces with-i t defeds represent realizations 'of the idveal smfaces used az physicalout po n

objects =d, hence, are dlscussed cornrnonly.On the other hand, the presence of a large ntlrnber of defeds in bulk

czystals acd on slufaces is a nat'ural occurrence. This hoids in pazticalar foraative dcfeds, which on surfaces can be categorized as point defects (va-cancie. ackisjtes), steps, 'irsnks, and their complexcs. Steps are natural forviciual surfaces (see Sect. 2.3). For a given temperature native point defectssuc,h as vacancias should occtzr with a certain probabitity i'a thermodrmmicequilibritlm as in the bulk case (7.1). Despite the tTHV conditioms atoms inthe restga.s may give rise to contxmsrations. Such adatoms 9om the rœtgasrepresent impuritîcs (at lemst, msi'az the lazguage of bulk semiconductors).

One may classif.g surface defeds accoreg to their dsrnensionn.ll'ty. Three-dimensional defects due to the mosic structure of a crystat or due to'straiu

shottld not be discussed here. The same holds essentially for two-rlimensionaldefects suc,h as stacldug faults or dornna'n botmdaries. Only one evn.rnple isstuded i'a Sect. 7.4. Importaat one-dimensional or line defeds a're steps in

which the ledge (c,f, Figs. 1.1 and 2.1) separat% t'wo terraces lom each other.

The step hdghts depend on the surface ozientation, the elcctrostatic behaviorof the atomic layers, and the polmorph of th.e czystal. Jn many cases steps ofsingle atomic Eeight prevail. However, for polar surfaces double-layer steps are

more Dkely bccause of charge neutrality aad, hence, the accompauying smatlelectrostatic forces. I'a the caqe of (0001) a'n.d (0001) surfaces of hexagonal611 (4H) compolmd czystals such az SîC, steps with a height of three (+0)bilayezs occm to realize one bn.lf of the e-xtent of a bullc lmit cell parallel tothe sarface orientation.

294 7. Defects

There is go ltnivocal deBnstion of the step keight. Here, it is expressedin terms of single atomic layers with an orsentation parallel to the normnlof the low-index sllrface represemting the adjacent terraces. Accordz'ngly, thedenotation ms a monatomic (shgle-layer) or a biatomk (double-laye'r) step isased. Wke,n a step is charactfm'zed by a bilayer, usplltlly a'a electricaxy neutralcombiuation of two atomic layers is discTpsmed, e.p, a combhation of a cationand an xnlon layer in the (100q or (111) direction of a ziuc-blezdc crysti oric the (0001) direction of a he-xagonal polytype of a compolmd sach as SiC.

Zero-dinnensional or poht defects (Fig. 7.1) hvolve impurities (adatoms,ledge adatozns), i'aterstitials (adstoms or substrate atoms at a non-lattice è

site), ldnkq, vacaacies, and antisites tin surfaces of compokmds) . Other surfate rdefeds aze related to dislocations, wllic.h zepresent one-dimemsional defeds hEthe blll'k. An edge dislocation penetrating icto the smface with the Btuge,p q

tor ozieated parallel to the surface gives rise to a pomt defect at thi 5vec:surface. Step dislocations Mtting a surface also cause poi'nt defects wllic,h aréE::

uslpn.lqy sottrces of step lines.The local eations related to defec'ts may cause changtas tn all jmportani

. . (:sl:rface propelies, sac,h ms bondzng behavior, atomic coordlnation, electrordd' ''

s'tates, lattice vibrations, etc. A thorough study of suc.h defects is thereforeimportant for a hmdmental tmderstaadlng of their role iu growth nttcleu:

ation/epitaxy (evsporation, surface cliRlsion, adsorption, and desorptëonlçrE

('

'

.'

smface chemical readions (e.g. the slprface catallic eëciency), and eleistro/c device (surface carrie,r recombination, Ferms-leve,l p'nnsng).

7.2 Point Defects

7.2.1 Vacaucies

The deaved (110J surfaces of zinc-blende semiconductors (Sect. 4.2.1) are

ideal objec'ts to study vacancies. These surfaces are non-polar, i.e., equal num-,

bers of cations and aniozus occur izz one atoznic layer i'a the absence of defects; '

EThey are only relaxed, i.e.) the bulk trxnqlatioual spnmetry is coasezved 1:two Hlrnemsiozus, azld thek surface geometries, at least tlze lateral coordmatcigE:are sirnslar to the bulk ones. The defect densitjr direc'tly after deawage Ls small E?:. .

which makes f 110) sttvL.ces ideal for investigations of isolaied defects. TllçEsabstrates are AE compouuds, i.e., 50th 1-.0e,5 of vacancies, Wx acd Ws (see E

. l .ïT'ig. 7.1)) c..%'n be stuctied. Thereby a ç-fold càarge state Lq = ..., +1., 0, -1, ...)Emay occm. Such vacancies are generated during the cleavage process. IIl genzeral, the dezzsity of atomic vacandes is considered to be cleavage-depeadeui.! :.

but for A.s vacancies on (110) surfac% of p-GaA.s samples it is estimated 'tobe roughly 5 x 1011 cm-z or one per thousand surface anions (7.24. The ssg,rj) ..

aad the magnitude of the clzarge q of the vacancies is ''n6uenced by the clo/iiag level of the bulk czystal and, hemce, band bending near the smface. Ah

o . . ' :.example of an STM lmnge of defeds occuzring on the 1nP(110) surface of 'a ,r.

'

7.2 Point Defects 295

Ideal surfaces

o A atomo B atom .

* Impurity atom

Substitutional impuritiesS SA

: .

VacanciesV VA VB

(y' .

I

' ! .

: O

iikerstitial impurîty

'

) ,)r.(-. l..y

q/.AB

(Eigen-llnterstitialsIA IB

O O

?lkrltiïiitf,isBA

SPig. 7.1. Elustration of impoztant point defect,s ia two-dsrnensional elememtal andttimpound mystals, i.e., conwpoading sarface layezs.

'

2S6 7. Defects

Pig. 7.2. STM image (snmple voltage -2.2 eV) of a (110) smface of a Zn-dopedJ.'aP crystal (7.3) (copyrigùt (2003) by the Arnerkm.n Physical Society) .

Zn-doped I=P czystal is show.a in Fig. 7.2. (copyright (2003) by the Ameri-nltn Physici Sodety) I'a addition to pEosphorus wacancies (V) appexrlng ms

blnztk depressiousp white elemtions suzrounding Zn dopaat atoms (Zn) andsmaller blac.k depressiozzs related to vacsncy-impurit.y complexes (V/Zn) ca,n

be observed. The depressions and elewations aze related to nlnnmged states azdare therefore infuenced by t:e band bending,

The occuzrence of such a dafect, e.g., an acion vacaccy, strongly dependson the surrounds'ng atomic geometzy acd its eHrge state (7.4, 7.5). The c%Ar-

actezistic quantity that governs the probabslity of tba appearance in th-rrno-drmrnl'c equilibrium is the formation energ.v vf%. Jt can be fbrmulated izz am'mslar way to the sarface enerr Rs (2.45) using the forrnnm'm of Zhang andNorthm:p (7.61 as

f)f (Ws) = SIXA) Nn - 1, q/ - ELNXt NB) + #B + qlsr + fvsM).

The total energies of the syssem 'Gt,h an ideal relaxed (110)1x 1 snrfn.ceu

ELNA., A%l,-an.d of the surface on wikich a B atom Ls missing, ELNh, NB-1, q),may be calculated Mcording to (3.55). For q < 0 (ç > 0) i:l additionalelectrons (Yoles) aze considered 5.:1 the system which aze compensated by a

positive (negative) backgrokmd as in tàe jellbnrn model. However, vithin thenumerical treatment using the (repeated) slab approxn'mation the-re is a com-

plication due to the local missing of such a compensating background in thevrunm regîon. The ch-rnl'cal potential JZB of B atoms ân a ceztain z-rvoiris detmrrnined az described :in Sec't. 2.5.3. The Last term in (7.1) accotmts %rthe energ.y of l.t2k electrons Lq > 0) oz holes Lq < 0) ia the eledronic reservoir.

7.2 Point Defects 297

da db

dc

Eo01)

s p' $ oj

Fig. 7.3. Schnrnn.tic representation of ar aniora vacancy on a III-V sornlcondudor

(110) stkrface. Three dacgling bonds (Q, dN? and dcj are created dtlring the for-mation of a vacancy. Small circles correspond to atoms in the second hyer. Filled(open) smbols represen.t cations tanionsl.

The Fermi level sv (the s=n.11 temperatttre-induzed cliFereace to the chemi-cal potential of the electrons Js negleded) is measlzred 1om the mlen.ce-bandmnavlrnam evnu. Usually an alignment proceduze of the energ.g scales in theideal system aud the system with a charged vacancy is applied (7.71. Ia ex-

plicit calculations often a shift of the valence-baud maOmllm due to thecbvging of the dafec't Lq considered. lt may be calcuhted by the dlference

of the spatially averaged eledrostatic potentials Montœl + &k@) (3.48, 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'nesthe atomic relaxatioa around the dmfec't (in contrast to the bulkl and theenergetics of its formatioa. The three daugljng bonds of an anion 'vacrtzjrVGs (see Fîg. 7.3) are occupied by (3 - qt electrons. Consequently the chazgedvacancies Vs- azd V+s prefer rebonded geometric whereas the aeutral N'a-

car.cy Ws exhibits a weaker temdency for rebondtng. The positively chargedaaion vaca'acles a're dominated by an inwazd relaxation of the neighboringcations (7.8-7.101. However, the symmetrically non-rebonded and the non-

symmetrically rebonded covgwations are almost deguerate i'n eneror. lnthe non-bonded geometry the catioms a aad b (Fig. 7.3) relax inward. In therebotded geometu, the a or b cation forms a dimer (see Sed. 4.3) with the ccation. In 170th. cases a single empty dangling-bond-derived state appears ixtthe 'h:ndamental gap.

298 7. Delhects

Fcg. 7.4. Formation ener#es calculated for dl'lerently charged P vacalcies öy!.1nP4110) mlrfaces tmder P-rich conditions verstus the Ferrnl' energy. Adapted fibm(7.:3-:1.) -

Accozdîng to the formnt, iolk eneror (7.1) tEe most favorable eahazge statqdeper.ds on the doping level. TMS is klltzstrated in Fig. 7.4 for dlFere/ttkcharged P varnncies on 1nP(110) surfaces prepm'ed tmdez plmsphoruscrich

bunc '

conditions, Jzp = pzp (7.11J . On surfaces of pdoped substrates a singltpositively cttarged P vacancy JC is stablej wherems a single negatively chargèdP vacancy V- hms the lowest formation energg for zsdoped substrates. Fcktmdoped TnP the neutzal P varancy F'eo may be the most stable one. TlzevMdsmgs are in agreement with m=y ex-perlmental observations for Gavj!1nP, acd GaP(110) surfaces g7.12J.

The positions of the Fnrml' level sp at wllich the formatioa enfzror J)f 8fdefect iu two cllFe-ren.t charge states q and q + 1 become.s equal in Fig. z.iïa

or (7.1) deft'aes the éonémiion Jewel or charge-transition J6VeJ sLq + 1jt2) vitilrespect to the VBM. According to (7.1) one %n.q g7.7, 7.81

:(ç + l1ç) = ELNX,NU - 1, qt - EINX,NB - 1,ç + 1) - Sv'aM

disregarding for a momest tke alignment procedure. For the acion vacacéîr!qthere aœe t'wo gap levels, s(+I0) and s(0E-), as demonstrated in Fig. 7.4 fùt5'pv on TnP(11O)1x1. The Srst (second) one is related to a more acceptötlg

(donoz-) like state. The euerg.g c(+I0) may be interpreted as the bindiikenergy of a hole to the neutral vacancy, while X - c(0I-) gives the e'nerj/f a'a electron bonded to the neutral vacancjr. Uaforttmately, neither STV''o

a'n.d photovoltage memsurements noz ftrst-principles calculations give a uzziqttèEpictzlre for the t'wo energy levels in the gap g7.12). Even for the mode,l systelE

7.2 Point Defec'ts 299

:-01 f

(a)

-):y y j yj

Fig. 7.5. Stmzctuzal models (top view) for the C-type ddec.t on Si(10O). Dànglt'agibùnds rezated to rniAsing atoms and located at the furst-hyer aad tbhd-layer atomsE:. ' ;' . .

v . o

aàe indicated.. Bu of the z'mers Ls allowed aa kzdicated by the dlflkrent shes6f the circ'les representing Srst-layer atoms. (a) A two half-rl':rner model (7.15, 7.16J,(b) m''qsixg atom in the second layer E7.17).

ö? a single .ks vacancy at a cltoltn GaAs(110)1x1 sl:rfbnrte the leve,l positiozlslait' 4 the intezwetation of the chaapder of the levels va'cy (7.8-7.10, 7.13).k wever, recently remaruble progress ia the calculations bnx been aœeved. ,, 9t Jncerns'ng convergence and treatmemt of the c-harged states (7.112..'LL? '

.

k Vacaudes are also discussed for sttrfaces of covalent smmiconductors, for

'Li the Si(111)2x1 surface (7.14) and Si(100) suzfaces (7.15-7.171 . Oneacce

pvn.nnple is the so-enlled C-defect on Si(109) (7.15J wllich is always obsetwedi/ room-temperatttre STM lmltges. At positive snmple bias it appears as a

Vgh protrusion adjacent to a depression in the surface layer. Two interpre-Vtions tsee Fig. 7.5) have been given for these STM images. Izz one case theyake interpreted to inclicate a dimer ddect in. which t'wo adjacent Si atomskong the dsrner row are rnr'qsiztg, i.e., like t'wo 'hn.lf-fssrners (7.15,7.162. Consc-ktzently? this defect should represent a mcaucy pair in thc fkrst atomic layer(Fig. 7.5a). Ciculations (7.171 have showa tha,t the optirnized structtzre start-

. .L E,k#j Fom the two lnnlf-dimez cougttration is llnnble to rèproduce the observedj' 'FCM ima es. Tnqtramzl a C-type defect structme izl whic.h only one Si atom i'ag

(b)

> Q (

3OO 7. Defeds

Fig.. 'J.6. Lltustration (top view) of a mlqsjng-dimer stractuzej a sty-nn.lled type-.4 defed, on a Si(l0O) smface; (a) idea,l dimer row; (b) cll'mer zow wiih a dimervacaztc'y. Large (small) open circles represent Srst-layer (second-layer) atoms.

the second layer is mlqskzg, a,s shown in yaig. 7.5b, IhAA been proposed. rIn1np'scorresponds to an isolated vacancy in the atomic layer bene-ath the dimerlayer.

Many other defeds disturbing the dirner rows along (OI1j have been dis-cussed for Si(1O0) surfac%. lmportant ones are related to missing dimers,i.e., to patt's of vacancies izz the 6mt atomic layer. A dsrner-mcmmcy modeli'n whic,h t'wo adjacent Si atoms along a (O11j direction are absent is shownin Fig. 1.6. suc,b. a model has beea proposed by Pandey (7.18) to elurpln.in thetype-A defect obsemrecl on Si('1O0) (7.15, 7.16j. Other defect complexes con-sisting of t'tvo oz more missing dsrners are observed.) e.g., the so-called type-.sdefed (7.15,7.16j, an.d give rise to confgurations with not too lmgc formationenergies F.19).

7.2.2 Impurities

The abil!ty to kcorporate reprodudbly dopant atoms, in particulaz shallowimpurîties as acceptors and donors, with precisely controllcd concentrationsand. spatial distriimtions is essential in vadous de<ce applicati8ns of semi-

7.2 Poht Defeds 301

conducti'ng materials. The geaea'ally acceptecl view ks that the charge of a

shallow impurity is smvencyd by the charge cn.rrsez's ixl tlze doped semicoxduc-tor, restllting in a lonpracge repulsive screened Coalomb interaction betweertdop=ts. Suc.h a repulsion ia turn leads to a rather homogeneo!zs distribu-tion of the dopant atoms in a bulk somsconductor. Surfaces disturb such arepdsive interadion and, as a consequence, the bomogeadt.g of the spatialdistribution of nlnnrged dopant atoms is destroyed (7.20J .

Scnnnsng tntnneling microscopy is particutarly well smted for investiga-tions of near-surface dopant atoms and their defect complexc, because STMrnxlces it possible to identify mdMdual atomic defects and to explore thekproperties. TMs holds in particular for (llol-oriented III-V semiconductorsurfacc witk their simple atomic relavxtion and their reproducible prepara-tion by cleavage. A prototypical impttrity ks Si in Gal.s 27.214. I'a bulk crys'talsslcoa on a Ga site, Sica, is a perfed example foz a shallow donor with hy-drogenic s'tates witb. a small electron bindiug energy of 6 meV sn'rnilar to thatof the Wxnnier-Mott exciton i'a this material (6.46). On an. A.s site, Sipks, itis still a shallow imptzrity with a hole bindiag enera of 35 meV (7.22). Theformation energy of such an impurit.y atom X on a certain lattice site nxlledB also remarltably deperds oa the Fermi lcve.l sv. Then

Of (XBV) = E(Nht NB - 1, X, Q - E I'ZVA, A%) - lzx+JzB +ç@F' +svBM) ('DZ)

<th the total enerr ELNX, NB - 1, X, q) of a system with aa X atom izt thecharge state q on a B substitutioni site. yx denotes the chemical potentialof the Mpkzl'it,y atoms in their reservoir.

A detved high-resolutîon STM study of the (110) smface of as-grownGaAs crystals with di/eren.t Si coacentrations 9om 2.7x 1018 to 6 x 1029 cm-3

in the bulk eebits a variety of surface defects, b0th nstive and Si-tuduced. Aconection ks represented i'a Fig. 7.7 by their STM images. Five major defectswere observed izz a,tl samples witiz =,1-.7,111g concentratjon F.21q. These are

(a) suzface gallblm vacaudes Voa, (b) Sioa donors, (c) Sias acceptors, (d)Sioo-vca complexeas, as well as (e) rathez phnnm Si clusters cut by thecleavage plane. The Sioa defect iu the-(110) cleavage surface shows ï s'tronglylocnlized etectronic featuze due to its daxgling bond. ln contrast to this, thesubsmface Sioa donor has a less locnll'zed eledronic strudure with a tgpicalspatial ex-temsion of 2.5 = i'a STM images (7.23). This value should approae,hthe Bohr radius of the shallow bllllr impurity (7.221 with iucreasing depth ofthe lattice site below the sttrface.

The STM measurements allow tus to cotmt the Si atozns in all defects acdto detemm'me the surface concentration. Jn the absence of any d-lm:qion at

room temperatmc the surface concentrations reoect the bulk concentrations.The n7lmber of Si atoms with doaor (Sios) or acceptor (SiAs) Garader ca'a

also be deterrnsned. &71k- calculations (7.24) let 1zs eocpect a strong increaseof the concentrstions of donors a'ad acceptors with Si dopmg. This is not

obselwed using the STM data of the GaAs(l1.0) suzface,s of substrates with

3O2 7. Defects

diFerent Si incorporations. Foz smnller Si conceatrationsy Si atoms are ini-tially incorporated only on Ga sites as donon. With incremsing rstype dopiugaccompanying the incremsing Si concentrations, however, the formation en-

er> (7.3) of Sixs acceptors decremses (7.25J a'nd Si is incremsHgly Lncorporateêon A,s sites as acceptors. Cozlsequently; compenqation eFects happen. I'a ad-dition, tlb.e attractive i'ateraztion of Sioa and Sihs shallow impurlties suppoz'tsthe formation of Si clustea's. Suc.h a view e-xplnsnm the obsenation of nearlyconstau.t donor and acceptor concentratior (7.21J itl contrut to theoreticipredieionsj whie,h usllxlly study uornînally Ssolated defects and do not takéthe defect-defect iateractioas hto account. Nevertheless, the electrical 5nAuLence of the other defects in the system is reasonably modeled by tqb.e chenlicalpotential of the eledrous.

7.2.3' Antisites

Mtîsite ddbcts are not identïed ia Fîg. 7.7. However, the iccotrpozatipyjEéfexcess arseaic at low vowth temperatures leads to the forrnyktion of a hij

tration of defects. Cross-sectional STM shows diretztly that the Yfé/' il''concen

fo=ed by the incorporation of Acess arsenic are mostly isolated Ge2i:, ytjantisite (Asoa) defects, aud that tllese antisite defec'ts give zise to an iatènb ,bazd of midgap states (7.26,7.271. One of the most stm'kl'ng featttres revéalèàb the STM images are the spatially extended local DOS of the defeds (i.jd)r.: u ëBased on the symmetr.g aud the intensit,y of the local DOS as imaged by.STV)Asca defeds in dllerent sabsurface layet's were distiagaished.

The syznmetzy and intezlsft.y of the Iocal DOS also allow the ideutifcatld:' (

of ardon antlite defects ën the topraost surface layer of (110) surfaces éf

Fig. T.ï. STM ''mxges of occupied (upper panels) aud empty (lower panels) elec-tronic states of the major defects on Si-doped Gn.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 (per-pendictzlar to the surface). Tke tnmneling voltages are -2.4 (al), -2.0 (b1, cl, d1);-2.2 (e1), 1.8 (V), 1.4 (b2j c2, d2), aad 1.5 V (e2). Adapted from (7.21) (copyright(2003) 'by the lmerican Physical Sodety).

7.3 Line Defects: Steps 303

a-doped GaP(110) surface.Four Poa antisite defects(copyrigît (2Oô3) by t:e

GaAs: GaP, azd J.IIP using STM, in pazticular, by compariqon with resultsSf DFT calculations (7.28J. The aaion antisite defect in the surface laye,r of

(110) smfaces b.as a vezy localized DOS, i'a contrast to the extended DOSE .

$f the sabsurface a'atksites (7.26). As an mvnmple the STM image of a'a zz-

Ediped GaP(110) cleavczge soace with four Poa antisites is shown i'a Fig. 7.8.(T he constaut-cunent STM image eebits rows of occupied danglhg boads

E:locnlszed above the P atoms of the lx1 surface stracture (compare with

k: Fig.4.9). In addition to the regularly spaced dangling bouds, one obsmesi additional mnorima between the rows (indicated by arrosvs in Pig. 7.8), which

may also be identifted with completely Slled P danr;ing bonds =d, Eenze,with Pca aatisite ddects.

'

The lack of any apparen.t Eeigkt chnange or long-range voltage-depezden.tcontras't around the defect featmes in large-scale STM images indëcate thatthe kmdealying defect type Ls electhcally uachazged on a11 investigaied zt-

doped substrates. Consequently, the surface aaion antisîte defects aze elec-t21 ically iaactive a'ad do not induce a Fermi-level pinning 'lnlike bullc aatssiteddects. This holds not ozkty for GaAs but essentially also for Gap and Tnp.

7.3 Line Defects: Steps

7.3.1 Geometry and Notation

The most impoztant ckmss of liue defects oa surfacœ are steps i'a whic,h theledge separates t'wo terraces from eac,h other. Steps 'are impoztaut i'a theformation of vicinal surfaces (high-inde,x mmhces), i.e., surfaces whic,h are

oriented at a sma;ll angle I (i 100) with respect to a low-index sarface. Suc.h

3O4 1. Defeds

vicinal surfaces are formed by small Eow-inde,x terraces and a 1nltg. h deusi'ty ofregnzla,r steps (see Fig. 7.9). Thetr surface energy should be srnnller thaa thatof the high-inde,x surface with a certsu'n atomic structure (see Sect. 2.4.1).As an evn.rnple, a 2D kut of a simple cubic stzuctttre with a suzface norma,lslightly misoriented with respec't to the (901) direction is sbown in Fig. 7.9.

The peziodic successjon of terraces and s'teps on a vici'nal surface c.ark

be specfed by their corresponding Mille.r irdice.s, for i'astance (5,57) for thesituation of an fcc crystal (without atomic basîsl in Fig. 7.10 with a tilt angle0 = 9.40 of the surface aormal with respect to the (111) direction. However,this notation is not ve,zy convenient since it does not indicate, at Sz'st sight, thetmze geometrical stractme. Such rstaircEtse' stzazcttlres ms shoqm i'a Fig. 7.10are more conveHently referred to tzsi'ag the fmicrofacet' notation (7.2%

p(AkI) x Lhtkllî), (7.4)

in wizic,h h'kk and Wk'l' are, respectively, the M511e.r indices of tbe terracesand of the ledges, an.d p gives the wzmber of atomic rows in the terraceparalle,l to tlve edge. Hence: an fcctlll) surface may alternatively be la-belled 64100) x (111), i.e., a series of six-atom wide (100) terraces separatedby (100) x (111) steps. Sinzla,rly, a'a fcc(331) place may be referred to as

3(111) x (111), i.e., three-atom wide (111) terraces sepazated by (111)x (111)steps. The suzface in Fig. 7.10 is1 thusj a 6(11:) x (001) stepped surface of azl

fcc czystal with a lattice cozsstaut co. The geometzy of the (111) ar.d (001)planœ of aa fcc crystal are described in Fig. 1.2. The steps a'e propagatingalong the (EE21 clirection. The coordinstes of the lattice site,s are given blTable 1.1. Periodicitr in this dizection happens for L = 17cc/(2W) with adistance c = Wcc/W betvee,n t'qro adjacent atomfc rows fa a tevace, thestep height is d = (U/W, and the horizontal shifk bet-ween t'wo consecutiveterraccxs nmouts to g = co/A/d.

We mention that, contrary to low-index surfaces, the sign a'ad the or-

der of Miller indices in the notatio:a pLhklj x (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, showing terrace atd ledge atozns inproftle. The nominal (014) suzface decays into (001) terrace and (010) steps. Theincline angle 8 to the low-inde.x (001) plane is gfven by tan8 = 1.

. 4

7.3 Line Defeds: Steps

(EI,.fE1 5 OqE-l ï21

00:)

periodiclty L;

(5541 E1 -1 1.1

(b))'h'jl ,;

*--- a d

9L

Fig. 7.10*. Vicinal (557) smface of an fcc czystal. (a) Positions of lattice point.s in a

6(111) x (001) s'tepped surface. The characteristic lengths and heights are indicatedin (b).

(001j diredion., in which the suzcessive atoms are second-nearest neighbors,whtle for the second step it is pazalle,l to (ï11J , in whic,h the successiveatoms are flmt-nearest neighbors. Motker problem occuzs for orientationsin the centrat part of the stereographc triaugle (Fig, 2.6). For example, at

(311)*2(100) x (111) = 2(111) x (100) it sllotf.d be noted that steps and ter-

races become indistinguishable. Traveling from the (111) pole abng the jo:ïjdirection to tke (100) pole, the size of the terraces decreases and (eveniuatly)the size of the (100) terraces becomes large'r. Therefore, (311) is referred to a,s

the t'lrning pokt of the zonç. The stereographic triaagle may also #ve otherîzlformation. Points witbsn the triangle (away &om the edges) wsually cor-

respond to k-inked high-inde,x plaues (sïtrênces containing regular non-linear

steps). However, recent studies have shown that high-inde,x surfaces on aa

edge of the s'te-reographic trsn.ngle, e-g., Si(311) (7.31j ) or Mrltbs'n the stereo-

graphic triangle, e.g., GaAs(2 5 11) (7.3241 can be s'table agn.inqt decay into

terraces of low-index surfaces separated by steps.For realistic cabic czystals, for example for dinmond-stnzcture crystals,

the stntcture of (211), (311), and (331) vichlal surfaces haa been discussed byChadi (7.33), partimzlarly for steps propagatiag along (îI21 and g11l). One hms

306 7. Defects

to mention that for viainxl surfaces of such cprstals sometimes akso a deviatingnomenclatme is ased (7.34). Sig3(111) - (112)q =d. Si(2(111) - (ïï2)) suzfacescorzis't of regtzlarly spaced terraces tkat are three/t'wo Si atoms wide iong theE11I) or g1E2) directiop separated by siugle boyer steps. The nomenclatareis also simplifed aûd s'ac,h stnzdttres aze zeferred to as (11R acd (1ï2j (ormore genea'ally (11i) aud (ïï2)) steps.

7.3.2 Steps on Si(100) Surfaces

We coasider steps on Si sa'rface.s as prototypical objects. Silicon is available izithe fo= of the most perfect single crjrstals. lt is used as one of the most com-

mon substrates for homo- a'ad heteroepitax'y. There are severi remsons fors'tudying the geometric and electronic structure of steps on atomically cleaziEvicinal Si(1O0) stzrfaces (perhaps after n.nnen.ling at high temperature for sktfascient time) ms well ms of steps on cleaved Si(l11) surfaces. Fit-st, dedactioyof the azomic s'tructare of the steps is izztrsmm'rm.lly interestiug. Second, an ux/derstanding of the step s'tmzct'aze leads naturally to possible insights iato thestructure of high-index smfaces, whic,h are themselves periodic arrays of Low (

d teps (7.33). Third the heteroepitaxial growth of other snmiconductozp '

i'a e.x s ,

suc.h az GGs or Gap ozl silicon appears to depend on the step stnzd'ure 6fthe sllrfnne.

ln tlle case of vicinal Si(1O0) surfaces the atomic layers on the terraces arqat 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 should be possiblqt '

'lnwo distinc't tgpes (.4, B4 may be distinguished according to the orientatio/of the dirners on the adjacen.t (100) terraces. The steps are labeled as Sz,Ss, Dz, and Ds accordi'ag to the notation of Kroemer (7.351 . The three moàt

De- - - -,z,,,, ss

HH-He - - ezzzmzz TTTTTTT -

- - - - zzzz zzz -- - - -ZZTTZTZ

e-- e - ezzzzzT 2222222 - -

H--- e-- - - -

zyz, zzzzzzz -JZ - -

- - - ezzzzzz 221r122 - -** - - -

ZZTCTTT -

SA

Fig. 7.11- The three most impoz'tant single-laye,r (Sz, Ss) and double-layer (Dsksteps on Wcinal S6(1O0) surfaces (sche-matfc) . The step height and the orientationof the dimers oa tlte terrres a'e iadkated.. The terraces show 2x 1 ar.d 1x2 rlsrnéfreconstructions.

7.3 Line Defects; Steps 307

1*

Fîg. 7.12- Top view of tlsmer-vacancy strudares used to model Sx, Ss, azd Dsiarface steps of Si(100), The size of the circle.s w.1-1e.s with the depth of the threecomsidered atomk layers beneath tke surface. Anom Mdlcate lime,r Sup' atoms:dotted lines the resulting surface uzkit cell, and dashed lines approm'rnxtely thesteps. After (7.362.

. *

importaat ones are sczematically shown itt Fig. 7.)1. The âguze makes.evidentthat the subscripts .4 and B denote ,whethe,r the dimerization d/ection on

azf upper tezrace near a step is normal (A) aad pazallel LBt to the step edge.Possible atomic stmzctures kzsed to model the steps Sx, Sxv and D.s oa

'a Si(100) sarfacej at least for a high dtmqity of steps, aze shown in Fig. 7.12(7.36!. The vicinal Si(100) surfaces are assllmed to be tûted about g011q and,consequently, the steps appear along the g0î1j diredion. Monatoxnic stepsseparate 1x2 an.d 2x1 domains of dimerhation, therefore Sx aa.d Ss steps'bltmz-nate. The Sx s'teps do not require the formation of new or the break-ing of existiug bonds. The Ss steps, however, aze more complicated audnviqt izl three mriatiozts. M titree types az'e observed in snAnning txlnnel-Lng zaicroscopy, even tllough the rebonded version has the lowest formntionanergy (7.37j. The comrnonly observed biatomic steps are of type Ds andrebonded, Tile neighboring terraces show a 2x 1 reconstrudion (for rot toolarge terraces and not too low temperatures) .

'

308 7. Defect,s

Experimentatly the occmrence of a ceztnl'n step and hence the stepheight depend on the temperatttre ar.d tkc misorientation. Surface steps havemonatomic height below a miscut angle J of 1-20. For larger miscuts.moreand more biatomic ste/s are observed. The steps aze zlmost exclusively bi-atomic for 9 e-xceedl'ng 6-80 (7.37) . J.n the gn.mework of the tight-bindl'ngapproach the formation energy 'yf = qslwith step) - Stideal sttrfacelj/fzs ofa cerwin step pez llnit step Iength Ls haB been ciculated (7.384 . The valuesare yf (Sx) = 0.01 ev/fgs, %(Ss) G 0.1: eM(Ls, 'g/f (Dx) = 0.54 ev/fs, artdyf (Ds) = 0.05 ev/fs, where Ls = cc/uf = 3.85 A Ls the lx 1 sarface latticeconstart. 111 reaâty these vczlues depezd on the step-step intezaction, i.e., on

the periodicity L of the large surface lnnit cell used irt the cazculations (7.394.Thc tendcrc'y is boweve,r obvious from the above-mentioned values. Steps Sx,Ss, a'ad. Ds sholzld occur on the tilted Si(100) s:lrfaces to a certna'm extect intlmvranody.cLnml'c eqllilr'brilIrn. However, their occunence is also l'n6uenced bythe smface preparation.. For (100) sllrfaces misoriented toward (O1lJ oz (011)the eaergy situation is completely chamged in compadson with the :at sur-'

faces prepared by a repeated c'ycle of sputtering a'n:d nannen.lsng h'om a (100)Si wafez. For a tilted surface it is fmpossible to have only Sz steps. If such astep does occur then a Ss step ks tmavoidable across some botmdar.v betweenterraces. For vazkishing spadng of Sx and Ss steps, howeve'r, a biatomic stepof type Ds is created (see Fig. 7.11).

Steps s'nflttence many surface properties and, hence, are measurable iumany mzrface measlzrement tenhnsques suc,h as LEED, STM, RAS, etc. Thedslezenï combinations of 2x 1 and 1x2 dimer reconstrudions and the elec-

)l I1 g1 gTr !i 1

1 : : @)1

L' l :h l :/ '

:i o.co2 tter , . s)I 12 : (;.n q : r :

uq k T z lGh-o t

4j ; (y; jc '

: ;S :- : d)t : 1 (, l

' w l %1 l

: 2 3 4 5 6

Photon energy (eV)Fig. 7.13. Refectaztce n:n''qotropy spedra measured for clean Si(100) sarfaces bya'-merent poups: (a) (7.402: (b) (7.412, (c) (7.421 , and (d) (7-43). Dmshed lines markthe posîtlon of the bulk matical pokn.t energes Eï and F'z. After (7.361.

7.3 Line Defects: Steps 309

l : sA

'B l lc :.P : Ss (a)m : lK1. 2 oï s : : E

1. l :(1(1.()()2 ) :

l @0 d >* lQ = -1 .'

f- -.An?.c?. .. ... >... ,1 .. * 1 ---.v..

-P 1 1 ê

o - I -- #

.2 *.w. A'w >. .A- . . w.QG *6 RWW''

q... .p :8 î N...w...m ?(:: t:p , ' I(:;:p .-ztI. .. zp

,; 'j ()j:, ;j* % we 4 1%< ex e* 1o xe w- 1 '

&; -6 s 1 I1 IT It pt j

0.001 ' E . E1 1 $ .21

1 2 3 4 5

Photon ener (e

Fk. 7.14. Step'-hduced opiical Jmsqotropy Re((FIoïz: - Fyzsl/rol: (a) RAS calcu-lated for Sx, Ss, and Ds steps in Fig. 7.12 (K36J; (b) RAS memsured for cllokrentmiscut angle,s 9 (7.41q. Terrace contributions have been subtzacted.

troaic strttdme of the steps thcselves sholfd particulazly iHueace the sar-

face anisotropy wlzich can be deteded using optichl reectance aaisotropyspectzoscopy described in Sect. 6.1.2. In. Fig. 7.13 we present the reûectanceaaisotropy (see Sect. 6.1.1) of fo'tm dferently prepved Si(10O) surfaces

(7.40-7.431. A1l experimental spedra show comrnon featmes: sach as a max-

im= at or close to the & critîcal-point eneror ard negative auisotropiœaround the Ez enezgy. The spectra (c) an.d (d) obtained for 'vinsnn.l umpleshave a rninn'mttm around 3 ev denoted by S in. Fig. 7.13. The spedra (a) and

(b) of tEe higitly oriented samples, instead, are charadezized by msnima atc

1.6 (:71), 3.1 (S*2), and 8.6 eAC (7*3).The featmu T1, T% and T3 can be related to smfRœslate-related optical

trausitio:ns of :at, sizlgle-domain surfres with essentiuy c(4 x 2) reconstruc-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 kscleady related to steps. This is demonstrated in Fig. 7.14. J.n ordez to extraswt

the pttre step contribution also for Sx and Ds steps) for wïc,h the contribu-tions of tlte upper ard lower terraces do not omnce,l eac,h other, the norvnxln'zedspedra of singlodomain surfaces have bem subtracted. Given the lsznitationsof the caèettlations (a)s in particxllnr with respect to the modeled step density,whic,h is far higber than at the sample.s studied experimentally, the compari-son witit the experlmemtal data (b) is gratifging. The surface steps, especiuySs an.d Ds, #vc rise to a broad negative Jtnlqotropy (S feature) belov the

310 7. Defects

.% energy w1t21 a minirnam at about 3 eV. This agrees witk the expprt'men-tal ftndiazgs, whic,h show increasing AnsRotropy with increasing miscut angle1:1. Experimentatly a positive acisotropy between 3.5 aud 4.0 e'V is observedfor zniscut angles j9I hrger than 4O. A s5m57= featme appears iu the IRAspectrum calculated for Ds steps.

7.3.3 Steps on Si(111) Surfaces

If siticon Rxmples are ealeaved dn situ along the (2114 or (711j dizections, 2xl-reconstruded terraces occur. These terraces are botmded by steps. LEED aûdSTM studies obxrve predomlnantly gzlîj-oriente s'tel)s with a step Eeight

s(c) 4 6 z4 -j '

c@ a

y 2

o() lc 20 zo 40 6:

scan distance alocg (g1t'z (&Fig. T-15. STM Lmage of a step on Si(111); aquized at sample voltage +1.2 V anda comst=t tttanneling curren.t of l nA.. (a) perspective view; (b) top view of tEe samestep! an.d (e) crosmsedional cut alorg the line indicated 1. (b). The step edge isidentifed by tick maz'lcs at the border of srnsge in (b). After 17.45).

7.3 Liae Defeds: Steps 311

(2 -1 -((a) , .. - . (b) . .

. Nh ' ) 7*. , 1

f .:; , '.

, m'.

, '.

... .! .ji? . ); ,.Li:

Fig.. 7.16. Stde <ew of two po%ible con4gurations for a (2-1-)1 step lnmning iong(0 l 1) .

of az/s/z = 3.14 .t correspoading to a'a atomic bilayer i'a thd (111q direction(7.44,7.45). A typicat STM hnage of such a step on a Si(111) surface (7.45) isshown in Fig. 7.15. Two cegions of the step edge are apparent in the uppe,rand lower parts of tke fLgure. 80th regions have tlnst pedodicity along thestep 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 inthe ttpper pazt of the image. The cornzgatsoa along these two rows is wealcA cross-section througlz this uppe,r region of the image is shown in Fig. 7.1:c.The 'kwo rows observed along the step edge give rise to Fmxn'rnn. in the STMcontotzr, with the rows separated lateratly by 4.5 + 0.5 A.

The STM images indicate that there are two possible classes of stepsalong (01î) and a bilayer height (7.33, 7.461. Conceivable models are shownin Fig. 7.16. The outermost stom at the step edge f->tn eithe,r be tvofold-coordinated as sitowa in panel (a)) or threefold-coordlnated as slmwu in panel

ng. 7.17- Two possible modelsr-bonded cknn'n (ândicated byges-tion in (7.45) .

of step reconstruction (side view) , iztcludz'ng a

arrows) reconstrtzctioh of the tercaces. Afte,r a sup

312 7. Defects

(b). The ideal structtlrej the flrst class, with only six-member zings also occms

on high-inde,x surfaces sar,h ws (211) and (311) (7.33q , which may be inter-preted as s'tepped soaces with shozt terraces. The daagling bonds of theouterrnos't edge atom form a drabbît-earttype azaangement, ideatical to thatof the Si(100) surface (Fig.3.11, Sect. 4.3.1). Consequently, a rlsrnerizationalong the s'tep edge is expeded. Howeverj it leads to a double or quadrupleperiodicit.g along the edge, in coatradiction to the STM obsezwations.

The seconâ class of steps in Fig. 7.16b is Garacterized by a geometrywhere the closest zizzg bn.q lost onc atom resulting in a sve-member ring a,tthe step edge. Suc.h structures are also discussed for (211) suzfaces (7.33,7.464. They cltn l'rnrnediately be combined with a 2x 1 reconstruction withs'nthe r-bonded nhnsn model (Sect. 4.2.2) of the (111) terraces. Two clanglingbonds remai'a at the two step-edge atoms ar.d may be completely Slled or

empt'y after recoastnzction. A11 other dangling bonds are satttrated during theformation of tEe new tezrace topology with Nve-member and seven-membe'rT>. Two possible stractmes are shomt i:rt F'ig. 7.17 (7.45J . The dl'lerences

izl the fmrö models are due to the topology of the rin.gs with whic,h the upperand lower terraces approach the s'tep. Tite step extents in these models of5.1 at.d 3.6 i come close to the value of the lateral sepazation of rows on

'R t tezraces of (4' .5 ::E; 0.5) i. However, it sllould be mentioned that alsorl: eren

more abnpt steps have been observed by STM (7.454.

7.4 Planar Defects: Stacldng Faults

7.4.1 Defect: Reconstruction Element or Bulk Propez-ty?

Stanlrs'ng faults are one of the most cornrnon types of planar defects in crys-talline dinmond-type azd zinc-blende-type semiconductors, suc,h ms Si, Ge,a'ad GaAs, as well as ia many fcc metals. They have vezy low formation en-

ergies (of the order of 20-70 mJ/m2 (7.471) and a're created when chacgesof the atoznic plane stacking sequence in the perfect czystat take place arongthe E111) direction, without breltking bonds. For evnmple, dislocations can

dissociate f'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 crystalwith a stanlrs'ng sequence . . . ABC. . ,

. The corresponding layer sequences are. . .ABCA ICABC. . . (1SF) or . . . ABCAICEBCAB. . . (ESF). For zinc-blead.e-

(dixmond-lstractttre semiconductors in att attmmaxti've way one cazs view theintrinsic (exetrinsic) stanking fault a,s consisting of one (>0) layem of tetramhedra twisted by 180D. The periodic arrangement of such stacldng fatlltsgives new htexagonal polytgpes, 211 (wuztzite) for ISF, 411 for ESF, and 611for triple stacking faots (not shown in Fig. 7.18) as Mdicated in Fig. 7.19.For compouad semiconductors with stroager ionic bonds sttch polytypes re

7.4 Planar Defects: Staddng Fattlts 3l2

ISF ESFideal (3C)++++

+++

+

A B C A B C A B C

Fsg. 7.18. Staclcing sequence for an ideal fcc structtzre tleft panel), fcc withu an

htrhsic-stacldng fault (JSF) (middle panel), and fcc 44th an ex-trlndc stacldngfazklt IBSFI (right q=ell. A, B, C zepresen.t the three inequivalent positions withc'na (110) plar.e with'p an irreducible crystal slab (d. Sect. 1.:.2). In the case ofsemîconductors a dot repraents a pair of atoms with the connecting bond parallelto (111).

++

++

+++

A B C A R C A : C

+

++

+++

A B C A B C A B C

e'aergetically more stable tha'a the 3C (ziac-blende) one, for evnmple 2H

(wartzite) for ZnS, GaN, and .A.IN as well as 41-1 acd 6H for SiC.Stazlrs'ng faults also occur on mIrfaces. They are not ac-tually a tfault'

but a,'a importaat reconstrudion element of the d'lrnemadatom-stpcklng faultmodel of the Si(111)7x7 surface (see Sect. 4.4.3), in contrast to stanMng faultsin blll.k Si czystals. In the case of halogen-terrnsnated Si(111)1x1 smfaces theintroduction of a staûlrimg faalt aloag a (111) step edge atlows those bilayersteps rin the two czystallovaphic directions (ïï2) and (111) to have the sn.rne

atomic structure g7.48). '

7.4-2 sî on si(111)xdx,vY-B

Interestingly, Si homoepitaxial vowth on a Si(111)WxVFB m:rface whic,hincludes j monolaye,r of boron shows a tendenc'y for rotation by 1801 of the Sitetrahedra i.'a the interface, i.e., for the generation of the flrst paz't of a stackiugfatllt (see Fig. 7.18), a so-called twin boundar,g (7.49, 7.501. At 1ow tempera-ture, the surface reconstmzction is partly prcxserved, buried unde,r an epilayer,ar.d the homopitanial Si layer gwws rotated by 1801 with rcsped to the suNstrate. This situation is indicated in Fig. 7.20b. The rotation is in coatzmstto all other cazes of bomoepitaxy. Usually the epilaye,r is crystatlographicallyaligned with the substrate, irrespective of the surface reconstrucion, impu-t'it.g segregation, oz other Gects at the substzate surface. Tlle origsnnl smfacereconstntctiox is always reordered iuto an thmreeonstructed iaterface betweeathe substrate and. 61m, since epitak'y requires a snTmciently lligh temperaturefor surface difhlsion to occttr. That means, in prindplc, there is the possibil-it.g of growi'n.g hexagonal Si polytypes (7.51) on a Si(111)WxW-B surface,

314 7. Defects

2H

+

+

+

+

ABCABCABC

4H

++

++

A B C A B C A B C

6H

++

+++

A B C A B C A B C

Fig. 7.19. StnrNsog sowemces in hexagonal pclytypes 2H, 411, and 6H.

because they are strttdmw incladiag twin botmdaries perioically along the

(111) direction tsee Fig. 7.19).A quatitative analysis of the system Si(I11) (WxW)A3O12B shows that

the boron atom pe,r surface lnnit cell occapies a, subsurface subsitutioaal S5adsorption site tsee Sec't. 4.4.1) because of the smmll covalemt rnrh'uz of boron.Adsorption in thé T4 site (Fig. 4.35) would lead to B-Si bonds much shorterthan the substrate bonds, giving rise to considerable strnan'n (7.52-7.55). Tkèright panel of Fig. 7.20 makes evide'nt tkat the mcontn.msnated surface servejas a template upon whic.h the new orientatioa of the twi'a crystal (layer) is

(a) (b)

p ,1 1; )Eif1.lq

Fkg. 7.29. Interface stracture of a homoepitaxial Si layer and a Si(111) substratei

(a) witkout B coverage; (b) after ) monolayer boron coverage. The Wx W reconu

tion of the interface (dashed lm' e) is introduced by occupying every tlnird sitestrucizz a single atomic layer. The bond-stnrllng directions are indicated by thsn solidline. Open tfzlledl mrcles: Si (B) atoms. Mer (7.491.

7.4 Plnnn.t Delectm Skackiug Faults 315

enugetically prefeaa'edj at least during the nudeon stage of the 61m gwwthnear 400 QC (7.49, 7.501. Eighe,r growth temperatau'œ of 800 OC or nnnealillgst 1000 OC make the B atoms mobile and #ve rkqe to a normnl st

'

of theSi bonds and tetrahea- The third-nearest neighbor atmms of the tetœahedraabove the contxmînxted layer te'ad to ocmpy a dte directly above the B atonxsj.û Ss sites. TMS is only possible by a twist of the Si tetrahedra by 1800. Two:facts may stabilize such a situation. The smaller B atom reduces the repulsiveforces between the third neighboz's. The eledron trnnqfer bemeen boroa andSi atoms may give zise to an attradive Coulomb hteraction between B andji atoms above an S5 site. -

References

Chapter 1

1.1 J. Bardeeml Phys. Rev. 711 î15 (1947)1.2 E. Bechstedt, R. Enderleitu Semicondnator zAr/'oce,.s and fnterlace.v

(AknHemie-verlag, Berlin 1988)1.3 R. Kubo, T- Nagnml'yac Solz z6'tcte Plzysics (McGraw-Hill, New York 1964)l.4 A.U. MacRae, G.W. Gobeli: J. Appl. Phys. a5, 1629 (1964):.5 C.B. Duke: J. Vac. Sd. Tec%nol. 14, 870 (1977)1.6 E.A. Wood: J. Appl. Phys. 35, 1306 (1964)1.7 M.A. V= Hove, W.HL Weinberg) C--M- Ckau: Lomœnergy Sleclmrz D6pc-

tden: Ezjmrimont: neozwp and Arftzce Structnrn Dcfgzwzïrzcdtm (SprWer-Verlag, Berlin 1986) .

1.8 J.J. Landm', J. Morrison, F. Unterwald: J. Appl. Phys. 34, 2998 (1963)l.9 K.C. Pandey: Phys. Rev. Iet%. 4T, 1913 (1980)

1.10 T, Terzibaschi=, R. Enderleh: phys. stat. sol. (X 133, 448 (1986)1.11 H. Qu, J. Knunqld, P.O. Nzlsson, U.O. Karlsson: Phys. Rev. 843, 9843 (1991)1.12 J.E. Nortbmlpj M.L. Ccèen; Phys. Rev. Lett. 49, 1349 (1S82)1.13 D.J. Ckadi: Phys. Rev. Lett. 43, 43 (1979)1.14 L. Miglio, P. Santiai, P, Ruggerone, G. Benedek: Pkys. Rev. Lett. 62, 3070

(1989)1.15 P. Drathen, W. Ranke, K. Jacobi: Sttrf. Sci. 7/ L162 (1978)l1.16 H. Lûth: Solid Syrlaces, Jzztczk'coe,.v and Thin Fdrzus 4th ed. (Spzinger-verlag,

Berlin 2001)

Further Reading

Bechstea F., Enderlein R.: Semkonductor S'ur/'cccs and Jnzerfcce,s (Akademie-Verlag, BezH 1988)

LGth H.: Solid s'tface-:, Interfaces cné T/z1/ Flrus, 4th ed. tspringer-verlag,Berlsn 2001)

Wydcof KW.G.: Crystaj Sfrucfuras 2nd ed. llntersdence, New York 1963)

318 References

Chapter 2

2.1 L. Landau, E.M. Lifsbitz: Staistical Jlhtl/sfcs, Vol. 5 (Pergnmon, Oxford 195$,Chapters 2 and. 15

2.2 M.-C. Desjonqutres, D. Spanjaaad: Concepta ïn S'ttrfaa Ahv.dc.s (Spzinger,Berlin 1996), Chaptezs 2 and 6

2.3 J.W. Gibbs: ne Sdenttsc Pgers, Vol. 1 (Dover, New York 1961)2.4 A. Zangwilt: Physic.a cf Suvfaces (Cmmbridge UzkiveDiï;y Press, Cvnbhdge

1988), Chapte.r 12.5 H. Ibvht Surf. Sd. Repoz'ts 29) 193 (1997)2.6 R. Shuttlewozth: Proc. Phys. Soc. A.63, 445 (1950)2.7 R.J. Needs, M.J. Godfrey M. Mn.nqâeld: Stn'f. Sci. 242, 215 (1991)2.8 R.D. Meade, D. Vauderbtlt: Phys. Rev. B40, 3905 (1989)2.9 R,E. Martintzz, W.A, Augustyaiak, J.A. Golovchenko: Phys, Rev. Lett. 64,

1035 (1990)2.10 X.-P. Li, D. Vanderbiltt Phys. Rev. Lett. 69, 2543 (1992)2.11 W. Wolf: Phys. Rev. Lett. 70, 627 (1993)2.12 L.G. W=g, R Kratzm', N. Molk M. ScheEer: Pitn. Rav. 862, 1897 (2000)2.13 R.J. Needs: Pbys. Rev. Lett. 71 460 (1A3)2.14 M.Y. Chou, S. Wei, D. Vanderbzlt: Pàys. Rev. Izett. T7., 461 (1993)2.15 G. WtllfF: Z. R'riqtallogr. Mineral 34, 449 (1901)2.16 C. Herring: nc Us6 o.f aasir.al Mcrmsczefc Cortcepu ïn the Svrfcze-fnerr

Fmblcrzh in: Structure and Fropeïe.a of Soàd lnterface-h ed. R. Gomes, C.S.Smith (Uaivezsity of Chicago Preas, Chicago 1953), p. 5

2.17 T. BMineau, D. Iofe; Y. Velenilc J. Mathemxt. Phys. 41, 1033 (2000)2.18 L.D. Lardau: Collede.d Pcyer.s (Pergnrnon Press, Oeord 1965), p. 5402.19 A.A. Basld, S.C. Erwizt, L.J. Widtrnn.n'. Surf. Sci. 392, 69 (1997)2.20 B.J. Eagleshn.m, A.E. White, L.C. Feldman, N. Moriya, D.C. Jacobson: Pbys.

Rev. Lett. 70, 1643 (1993) '

2.21 J.M. Bermond, J.J. Mltois, X. Bgtf-a, F. Floret: Surf. Sci. 3301 48 (1995)2.22 E.D. Hondros: Proc. R. Soc. M86, 479 (1965)2.23 V.K. Ktlrnlk-ov, K.B. Kkokonov: J. Appl. Phys. 54, 1346 (1983)2.24 J.C. Bûetto, D. Dew-Hughes, A.T. Pucizto: J. Appl. Phys. 54, 1821 (1983)2.25 A. Stekolnlkov, J. Pttrtltmiiller) F. Brkchqtedt: Phys. Rev, 865, 115318 (2002)2.26 J.G. Che, C.T. Chan, W.-E. Jt'v1, T.C. Leuug Phys. Rey. B57, 1875 (1998)2.27 L. Vitos, A.V. Ruban, H.L. Slciverj J. Kollm': Smf. Sd. 4.ï1., 186 (1998)2.28 N.W. Ashcroft, N.D. Mmvmio: Sotid Stafe Physics (Saundem College, Philadel-

pMa 1976)2.29 N.D. Langl W. Kohm Phys. Rev. Bl) 4555 (1970)2.30 C. Hezz-mgn Phys. Rev. 82, 87 (1951)2.31 W.A. Harrixon: Exntentary Slcctrozzfc Strv,ctk= (Wor1d. Sci=tisc, Singapore

1999) .

2.32 T. Sbitara, D.D. Vvcdensky, M.R. Wilby, J. Zhattg, J.H. Neave, B.A. Joyce:Phys. Rsv. 846, 6815 (1992)

2.33 A. Ishii; T. Kawamura: Appl. Surf. Sci. 130-132, 403 (1998): Surf. Sci. 436,38 (19.:5)

2.34 P. Kratzer, E. Penev, M. ScheEer: Appl. Phys. .*.75) 79 (2002)2.35 F.C. bnrxnk', J.H. V= de-r Merve: Proc. R. Soc. London, Ser. A198, 2G; (1949)2.36 M. Volme, A. Weben Z. Phys. Chem. (Leipzig) 119) 277 (19Q6)2.37 I.N. StrxnKk, L. Krastanow: Sitrmlngsber. Akad. Wkss. Wien., Math.-

Nattttwiss. Klasse 146, C97 (1937)2.38 Y.W. Mo, D.E. Savagea B.S. S'wartzentruber, M.G. Lagally; Phys. Rev. Leu.

65, 1020 (1990)

Cbapter 2 3l9

2.39 J.M. Moison, F. Houzay, F. Barthe, L. Leprince, 5). Aitdri, 0. Vatel: Appl.Phys. Lett. 64, 196 (1994)

2-40 R. Heitz, T.R. lumachandran, A- N'xlbarge, Q. Xie, L. Munametzhaaov, RChen, A. Mldzi>ptlrn'r: Phys. Rev. Lett. 7'S, 4071 (1997)

j.4). B. Scleter, K. Komlev, U. Katser, G. Hess, G. KJ-Mdzeu W. Richte-r: Mate-rials Saiencq Fortrn 353-356, 247 (2001)

j.42 M. Jacobi, L. Geelltaar, J.M. Mârque'z: Appl. Phys. A1'5, 113 (2002):0.43 K. Komlev: PK.D. rrhesis, FriedrieâknsrYlller-uzLivemrs'itàtj Jena (2X2)j.44 V.A. Shch'llrin, D. Bimberg: Rev. Mod. Phys. T1, 1125 (1999)2.45 B. Pebllre, N. Moll, A. Kley? M. ScheEer: Appl. Phys. A65. 525 (19g7)2.46 N. Moll, M. Scheeer) E. Pehlk'e: Phys. Rev. 858, 4566 (1998)2.47 L.G. Waug, P. Kratzer, M. Sckeëea., N. Mo11: Phys. Rev. Lett. 82, 4042 (1999):7!48 J.M. Blakely, J.C. Shelton: ir. Scr/tzce Plwsics ()./ Matyals, VoL 1, ed. J,M.

Blakely (Academic Press, New York 1975)'j.:9 M.J. Spwmaay; Surf. SCL Rep- 4, 101 (1984)i''8o J.-P. Bozel, A. Châtrzxin: Smf. Sd. 1.56) 572 (1985)'

)l.t1 K. Reuter, M. Scheser: Phys. Rev. 865, 035406 (2001).2..52 A- KJey: PE.D. Th'ym's Technscal Un'w' v'ty, Berlin (1997)2.53 K.-H. Hellwege (Ed.): Landolt-Bsrnstein (Springer, Berlin 1982), Vol.

HI/1'Ja+b'2.54 G.-X. Qi= R.M. Martin, D.J. Chadi: Pllys. Rev. Lett. 60, 1962 (l98E); Phys.DRev. B38, 1649 (1988)'.$55 W.G. Snbmidt: Appl- Phys. A75, 89 (2002)lj 56 F. Reif: Fundamentala o.fzshfkqtfc'al and T/zcrmcl Physics (McGraw-'Rh'll: NewT lj .

York 1965)'j.57 C. Kittel: Introduction bo Snlïd S'tae Phgsïc.a (J. Wiley & Sons, Hc., New

York 1966)$t.à U. Grxner, J. Ftuthrniiller, P. Becbstedt: Appl. Pll>. Lett. îA, 3851 (1999)2.$9 B. Parid, R.J. Needs: Phys. Rev. 845, 1067 (1092)1.@60 W.A. HrLson.; Blcctronic Sfruct'tz're and thc Properjïe.s oi Solïd.s (Dover? New

York 1989)/:61 A. Zywietz, J. Parthmiiller, Ft Beestedt: Phys. Rev. B59, 15166 (1999)2262 0. Kubaachewskk C.B. Akock Metailnrgtcal' Tlscrwzeemxs'trp' (Pergxmnn

Press, Oxford 1979)2:63 D.R. Lide (Ed.): CRC Zandbook JJ Chemistry and npa-scz, 77th ed- (CRC

. Press, Boca matozt 1996)2.5t E. Krsxiras, Y. BK-Ya. J.D. Jonnnopotklos, K.C. Pandey; Phys. Rev. 835,

%Z.S + 9633 (1987)2.65 P. Kratzer, C.G. Morg=, M. Schexer: Phys, Rev. 859, 1$246 (1999)j 66 J.A. Vtmxbles; lntroductçon to Stfcca and 77drl Fftm Pvocenes tcambridgeL.*' '

UzkivmsEty Press, Cambddge 2000),j';jf, L. Dâweritz, R, %ey: Smf. Sci. 236, 15 (1990)'..j'cd8 V.P. LaBellay z. Dix, D.w. Bunoclc, c. Emers P.M. Tikibado: J. Vac. SCL

'

. Tenhnol. A18 1492 (2000)

)'

:2140 C.G. V= de Walle, D.B. Lx.k-.q, G.F. Newmark, S.T. Pautelides: Phys. ltev.: g4z g.g tyggg;1

:2t70 U. Grossner) W.G. Szthrnsdt) F. Behqtedt: Phys. Rev. 856, 6719 (1997)

320 Refezences

ntrther Reading

Desjonqueres M.-C., Sprjaard D.: Concepts ïrz Snrface Jqrsïcs, 2nd ed.

(Sprhzgewverlag, B'erlin 1996)

Zangwzll' A.: Physics at zskr/tzcas (Cambridge Uztiversity Press, Cambridge

1988)

Chapter 3

3.1 W.A. Hnv'lzon: Slezpzrtcrp flectrtrrsïc Strnctnre (World Sdentec, Singapore1999)

3.2 K. Ohno, K. Esfarjari, Y. Kawaaoe: Compntaiional Mctcricls Sciczzce

(Springer-verlag, Berlin 1999)3.3 R. HoFmnnn: J. Chem. Phys. 39, 1.397 (1963)3.4 J.C. Slater G.f'. Koster: Phys. Rev. 94, 1498 (1954)13.5 P.O. Löwdm: J. Chem. Phys. 18, 365 (1950)3.6 W.A. Harrkson: Electvonic StrucHre o.f i'sblids (Dover, New York 1980)3.'F D.J. Chadi, M.L. Cohen: phys. stat. sol. (b) 68, 4O5 (1975)3.8 S.T. Pamtelides, J. Pol7mnmn: J. Vac. Sd. Technol. 16, 1349 (1979)3.9 S.G. Louie: Phys. Rev. 822, 1933 (1980)

3.10 P. Vogl, H.P. Hjalmarson, J.D. Dow: J. Phys. Chem. Solids z$4, 365 (1983)3.11 R.R. Shnmna: Pbys. Rev. B1.9, 2813 (1979)3.12 K. Hirabayazèti: J. Phys. Soc. Japau 27, 1475 (1969)3,13 A. Schohe, W,G. Szthmidt, F. Bechsiedt: Phys. Rev. 853, 13725 (1995)3.14 Y. Li, P.J. Lin-chlmg: Phys. Rev. B2T, 3465 (lS83)3.15 J.A. Appelba4lm, C..A. Baraf, D.R. Hamann: PTNr. Rev. B11, 3822 (1975)3.16 P. Kriigez, J. Poplrnxnn: Prog. Surf. Sd. 35, 3 (1991)3.17 M. Born, R. Oppenheimer: A'nn. Phys. 84j 457 (1S27)3.18 M. Born, K. Huang: Dynaméœz T'Jleo'rg o.f Crps'tttl Lattices (Clarendon Press,

Orord 1988)3.19 W.G. Srlnmsdt, B. Wenzien, F. Becbstedt: Phys. Rev. 849, 4731 (1994)9 W.G.

Sclnrnl'dt, F. Bertheedt, G.P. Sdvxxstava: Surf. Sci. Rep. 25, 141 (1996)3.20 H. Hellmxnn: Einflihcun.q irz d6e Quczztczzclzeznée (Deutickec Leipzig 1937)3.21. R.P. Feynma'n: Phys. Rev. 56, 340 (1939)3.22 G.R. Darling, S. Hozoway: Rep. Prog. Phys. 58, 1995 (1995)3.23 S. Glasstone, K.J. Laldler, H. Eyrilg: ne Theorp o.f Jlate Pvocessing

(McGww-Hi')l, New York 1941)3.24 C.H. Bennett: in Alio6thrrts .J/r Chemïco,! Comlmtatçon, ed. Ik.B. Christofer-

son (zrnezicaa Chemical Societs Wmshington 1977)) Chapte,r 43.25 ().. Wnxnerömt jn Ratn Jkvtzdonxs Scte Conmttmts an'd S'ur/cce D'tkbsio'rh ed.

by V. Bortolnn:', N.A. March, M.P. Tosi lpletutm Press, New York 1990):Chapter 16

3.26 G. Bhrlicl:, R. Stolt: Am'n. Rev. Phys. Chem. 31: 6O3 (19S0)3.27 E. Zeitler: in D@usion ct Surfack: Xitrm,cceïc Conczpth ed. G. Brtl

(Springer-verlag, Berlin 198$, Chapter 113.28 A. Kley: Ph.D. Thesss, Technical University, Berliu (1997)3.29 S.-H. Lee, W. Modtz, M. Sckeëer; Phys. Rev. Lett. S5: 3890 (2000).3.30 K. Seinoz W.G. S'nhnnidt, F. Beckstedt, J. Bernholc: Appl. Surf. Sci. 190s 264

(2002)

Chapter 3 321

3.31 C. Kumpt L.D. Marlcs, D. Ellts, D. Srnllgies, E. Ludemarkt M. Nielsen, R.Fe,ideztllaazbsh), J. Zegenhagen, 0. B'ank-, J.H. Zeysing, Y. Su, R.L. RobHon:Phys. Rev. Lett. 86, 3586 (20û1)

3.32 K. SeMo, A, Lshii, T. Alsnlen: Sttrf. Sci. ,438, 43 (1999)3.33 K. Seino, A. Ishii, T. Kawamttra; Jpn. J. Appl. Phys. 39, 4285 (2000)

- 3.34 P. Hohenberg, W. 'Koimt Phys. Rev. 136, 8864 (1964)3.35 W. Kolm, L.J. SlB.=: Phys. Rev. 140, A1133 (1965)3.36 W.B. Picket't: Comput. Phys. Rep. 9, 115 (198S)3.37 L. Hedin, S.J. Llmdqvist; J. Pllysique 33, C3 (1972)3.38 D.M. Ceperley, B.J. Alder: Phys. Rev. Lett. 45, 566 (1980)3.39 J.C. Shter: Phys. Rev. 81, 385 (1951)3.40 J.P. Perdew, A. Ztmgert Phys. Rev. 823, 5048 (1981)3.41 U. von Barth, 1w. Hedin: J. Pkys. C3, 1629 (1972)3.42 J.P. Perdew: Phys. Rev. B33, 8822 (1986)3.43 J.P. Perdew: in Electvonéc Jtmcftzre o) S/li& *1, ed. P. ZiesGe, H. E

'

(Akademie-verlag, Berlin 1991), p. 113.4.4 A.D. Becke: Phys. Rev. A38, 3098 (1988)3.45 B. Hnxnmer, M. Schele'r, K.W. Jacobsen, J.K. Nyzskov: Pîys. Rev. Lett. C3,

1400 (1994)3.46 J.E.. Nortizrup, S. Froyen: Phys. Rev. B50, 2015 (1994)3.47 J. Fhm, A. Ztmger, M.L. Cohen: J. Phys. C12, 4409 (1979)3.48 M.T. Yin, M.L. Cohen: Phys. Rev. 826, 5668 (1982)3.49 G.P. Sdvaztava, D. Weaire: Adv. Phys. 36, 463 (19S7)3.50 W.A. Harrison: Pscmdopotential.s ï'?z tâ,e Tlteo'r'y o.f Mqtal's (Benjnrnl'n, New

York 1966)3.51 J.C. Slater: J. Chem? Phys. 57, 2389 (1972)3.52 P. Ehremfest: Z. Phys. 45, 455 (1927)3.53 A.A. Maradudin, E.W. Montroll, G.H. WeL% I.P. Ipatova; Solid State Physics,

Supplement 3 (Academic Press, New York 1971)3.54 M. Scltlûter, J.R. Chelikowslcy, S.G. Louie, M.L. Cohen: Phys. Rev. Bl2, 4200

(1975)3.55 A. Stekolnlkov, J. fïu-thrnit:ic, F. Bechstedt: Phys. Rev. 865, 115318 (2002)3.56 M. Bockstedte, A. Kley, J. Neugebauer, M. ScheEer: Comput. Phys. Cornmun.

10C, 187 (1997)3.57 B. Kaxtrasj Y. Baz-Yhm, j.b. Jon.nnopoulos) Phys. Rev. 835, 9625 (1987)3.58 B. Weztzie'a, P. Këckell, F. BehKtedt: Surf. Sd. 331-333, 1105 (1995)3.59 K. Shiraisllit J. Phys. Soc. Jpn. 59, 3455 (1990)3.60 N. Chetty, R.M. Martixu Phys. Rev. 844, 5568 (1991)) 845, 6074 (1992)3.61 A.T. S'Flkrebtii, N. Esser, W. Ricbter, W.G. Schrrddt, F. Bechstedt,'B.o. Fim-

lazd, A. Mey, R. De1 Sole: Phys. Rev. Ldt. 81, 72l (1998)3.62 I.P. Batrw S. Ctraci: Phys. Rev. Lett. 36, 17O (1976) .

3.63 W.A. Goddard 1r1, T.C. McGil1: J. Vac. Sc'i. Teclmol. 16, 1308 (197S)3.64 M. Seel, P.S. Bagms: Phys. Rev. 823, 5464 (1981)3.65 J.D. Jonnnopoulos: Pbys. Rev. B16, 2764 (1978)3.66 F. Beckstedt, R. Enderlein: Szmzçcondnctor S'tzrlcces and Jhïer/bce,s

(Akademie-verlag, Berlin :988)3.67 J. Pollmpnn, R Kriiger: in Handbxk ().J Surjace Science, Vol. 2, ed. by 1(.

Horn, M. ScheEer (Ezsevier, Amsterdam 2000), Chapte.r 23.68 F. Yndtzrair, L. Falicov: J. Phys. C8, 157 (1975); Solid State Commtm. 17,

855 (1975)3.69 E.J. Mele, J.D. Joannopoulos: Ph'ys. Rev. B17, 1816 (1978)3.70 G. Moster, J. Slater! Phys. Rcv. 95, 1167 (1954)3.71 J. Koutedol: Adv. Chem. Phys. 9, 85 (1965)

322 Fsference.s

3.72 J, Polprnnnn, S.T. Pantelides: Pàys. Rev. B18, 5524 (2.978)3.73 D.J. Chadi: Pbys. Rev. BI9, 2074 (1979)3.74 D. Vanderbiltj J.D. Jonnnopottlœ; Phys. .Rev. 822, 2927 (1980)3.7,5 F. BecMedt: D. Reichardt, R. Enderlein: phy's. stat. sol- (b) 1a11 643 (1985)3.76 J.F- Janalc Pkys. Rev. B18z 7165 (1978)3-77 H.A. Jnhn, 1. Teller: Proc. Roy. Soc. A161, 22 (1937)3.78 R. Kubo, T- Najayima: Soud State Phv.We.a (McGraw HiQ New York 1969)3.79 D. Hanemam Phys, Rev. 170, 705 (1968)

ntrther Reading

Hnrrison W.A.: Electrœic s'àzvftsrc of Solftfs (Dove'r, New York 1980)

Srivastava G.P.: Tlteoretoal Mcdelën,p ofsemiconductor Stzr/cce.s (World Sd-entiâc, Shgapore 1999)

Chapter 4

4.1 S. Andersson: in Snrlace ScMce, Vol. 1 llnternnt. AtoMc Enerv AgencyVienna 1975), p. 11

4.2 E. W''mvnea', A-J. Fkeemx.n: in Handbook o/srtzrfac,e Scïenceo Vol. 2, ElectwaicSsrtscïurq ed- K, Hozm, M. ScheEe,r (Elsevier, Arns-terdam 2000), p. 1

4.3 H.L. Davls, J.B. Hn.nnon., K.B. Rav E.W. Pbnrnrner: Phys. Rev. Igett. 68,2632 (1992)

4.4 A.R- Sandy, S.G.J. Mochde, D.M. Zehnerà G. Grûbel, K.G. Haang, D. Gibbs:Pkys. Rev. Vtt. 68, 2192 (1992)

4.5 K.G. Huattgj D. Gibbs, D-M. T-bne'r, S.G.J. Mon%rie: Phys. Rav. Lett. 65,3313 (1990)

4.6 M.A. V= Hove? R.J. Koestam., RC. Stair, J.P. Bibiria'a, L.L. Kesmodel, 1.Bartfxs, G.A. Somorjaj: Suzf. Sci. 103, 189 (1981)

4.7 D. Gibbs, G. Gziibel, D.M. Zehner, D-L. A'Fwrnnthy, S.G.J. Mociaie; P/ys.Rev. 6T, 3117 (1.991)

4.8 S.Q.J. Monhrsej D.M. Zehner, B.M. Ockoz D. Gibbs: Phys. Rev. Lett. 64,2925 (1990)

4.9 D. Wolf: PEN. Rev. Lett. 70, 627 (1993)4.10 M. Mnmfeld. R.J. Needs: J. lZ%N. Condens. Matter ;, 2361 (1990); R.J.

Needs, M.J- Godley, M. Mxnsdeld: Surf. Sd. 242, 215 (1991)4.11 R.C. Cxrnrnxrata Surf. Sd. 279, 341 (1992)4.12 R.J. Needsl Phys. Rev. Lett. 71? 460 (1993)4.13 M.Y. Chou, S. Wd, D. Vrderbzlt; Phys. Rev. Iaett. 7l, 4s1 (1993)4.14 C.B. Duke: J. Vac. Sd. TMhnol. 1% 870 (1977)4.15 J.E. Nortbn'p, R. DiFeDce, J. Neugebauerl Phys. Rev. B55. 13878 (1997)4.16 G. Renaud: Surf. Ati. Rep. 32, 1 (1998)4.17 X.-G. Wang, A. ChaG M. Scàemnr: Phys. Rev. T-t. 84, 3650 (2000)4.18 W. Weiss: Surf. SCL 377-379, 943 (1997)<19 X.-G. W=g W. Weiss, S.K. Shxikbutdlnmr, M. ltitter, M. Petersen, F. Wag-

ne-r, R. Sclzl8gll M. FscheRea'; Pbys. Rev. Lett. 81, 1038 (1998)4.20 P.R. Watson, J. Miscllenlto 111: Phys. Rav. 842, 3415 (1990)4.21 Y. Ynnmrnoto: Phys. Rev. B50, 8534 (1994)

Ckaptzr 4 32

4.22 N.W. Jepps, T.F. Page: Progz. Crysi. Growth Cbprac't. T, 259 (1S83)4.23 L. Panling: Tlte Nszzure o.f t/za Clscmieni Stlnt$ 3zd. e(L (Cornell Universit

Preu, Iihaca 1960), pp. 2:1-2644.24 J.C. Phd'llips: Bands 4z)3 Bon, fn Semicotductors (Academic Prœs, Ne<

York 1973)4.25 Sargent-Wele: rasfe olperiodic Pzpperlïes ofLhe Slemenfs (Sàrgent-Welcb

Skolde 1980)4.26 A. Garcia? M.L. Cohen: Pllys. Rev. B4C, 4215 (l9â3)4-27 K. Kxrtth, F. Bechstedt: Pbys. Rev. 856, 7404 (1997)

'' 4.28 C-B. Duke: Chem. Rev. 96, 1237 (1996)4.29 G.P. Srivhstavxx Rep. Prog. Phys. 60, 561 (199:1

-

. 4.30 D. Hanernn.n: Rap. Prog. Phys. 50, 1045 (1987) '

4.31 W. Möncz: Somicondnctov Surïacv c'n,d Interfacea, 3rd'ed. (Spriager-vezlagBerM 2001)

' 4-32 R.J. Pbxneuf, M.B. Webb: Surf- S()i. 164, 167 (1985)4.33 R.S. Becker, J.A. Golovchmnlo, B.S. Swmzentraber: Phys. Rev. Lett. 54

2678 (1985)4.M D. Reirthnrdt: Progr. Surf. Sci. 35$ 63 (1.991)4.35 M.Y. Chou, M.L. Cohep S.G. Ouie: Phys. Rev. 832, 7979 (1985).'4.36 P. K>''n1ce11, J. nlrk'hmiiller, F. Becbstedt: Phys. Rev. 858, 1326 (1998)4.37 K.C. P=dey; Phys. Rev. Lett. 49) 223 (1S82)4.38 R.S. Becker, B.S. Swartzemtruberj J.S, Vcoers, T, Klitsner: Pkys. Rev. B39:

1633 (1989)4.39 N. Talceuclzi) A. Selloï, E. Tcsat-ti: Phys. Rev. Lett. 69, 648 (1992)

.. 4.40 C. Kressj M. F'iedler, W-G- Scbrnidt, F. Bechstedt: Phys. Rev. Bsû, 17697.: .

. (1994) .

4.41 J.L.A. Mves, J. Heberstreit, M. Scheler: Phys. Rev. 844, 6188 (1991)4.42 W.A. Wxrrison: J. Vac. Sci. Tenlmol. I6, 1492 (197$: 14, 883 (1977)4.43 J.EL Nozornlp, S. Froyen: Phys. Rev. BK, 207.5 (1994)

,'

4.44 W.A. Harrkson, E.A. Kraut, J.R. Waldrop, R.W. Grant: Phys. Rev. B18,:.. 4402 (1978)' 4.45 S.Y. Ttmg, G. Xu, W.Y, Hu, M.W. Ptzga: J. Vac. Sci. TecHol. B3j 1076. (jgss)24.46 R-EL Peierls: Qnantum neozr.!/ of SC,Jï/ (Clarendon Pressl OAord 1955)147 D J Chadi: Phys. Rev. Lett. +3, 43 (1979)9 Phys. Rav. B19, 2074 (1979)4.48 J. Dzm, M.L. Cohe,nv D-J. ChHi: Phys- Rev. B21, 4592 (1980)4 49 M.T. Yin M.L. Cohen: Phys. Rev. 824, 2303 (1981)

( k.50 J.A. Appelbqalm, G.A- BaraR D.R. Hamxn'n; Phys. Rev. B12, 1623 (1976)4.51 M.D. Pashley; Phys. Rev. B40, 19481 (1û8S)L;

' 4.52 D.J. Chadi; m nc zs%'ruclv.re o! S'uzfoce.s HL ed, S.Y. Tong, M.A. V= Hove,K. T@kny. anagi, X.D. Xie (Springemverhg, Beerlin 1991), p. 532

4.53 M.D, PasMey, K.W. Habevrern: Psys. Rev. Lett. 6% 2697 (1991)2:.4.54 H.H. Farrek C.J. PnlmAtrlml J, VR. Sd. Tehnol, B8, 903 (19S0)

. 4.55 W.G. Scbrnidt: Appl- Pbys. A7'5, 89 (20û2)r 4.56 K. M=da, 1. TpBeayasu, T. Mh-nxto, M. Kawaslnp-mx: Jpn. J. Appl. Phys. 26,

.

zjlaz6 (1qa,r)4.67 H.J. Cornelisser, D.A. Cltrnrnaclc? R.J. Dalby: J, Vac. Sci. Tenbnol. B6, 769

,. (:gs8)' 4.58 Y.S. Wu, C.R. Beclcer, A. Waag, M.M. Krauw R.N. Blnlcn/alr-Tassjtks, G.

Landwev Pkys. Rev. BZIA 8904 (1991) -

.': 4.59 S. Gundel, A. Fles-zar, W. Pmscllinger, W. Hanke; Phys. Rev. 859, 15261

(1999):4.60 C.H. Pazk) D.J. Chai: Phjrs. Rev. 849, :.6467 (1994)

324 References

4.61 U.M. Dassnrxyake, W. Chen, A. Knbn: J. Vac. Sci. Tecbn, o1. B11: 1467 (1903)4.62 A.U. Mxztlue, G.W. Gobelit i:a Snmicondnctors and Sezrzïr/zetqls, Vol. 2, ed.

R.K, Willardson, A.C. Beer (Academic Preas, New York 1966), p. 1154.63 J.D. Levine, S. Fieemam Phys. Rev'. B2? 325 (1970)4.G J.E. Jafe, R. Pandey, R Zapol: Phys. Rev- Bs3, R.4209 (1996)4.65 Pk. Ebezt, G. Cox, U. Poppe, K. Urbaa: Stt',T. Sd. 271, 587 (1992)4.66 B. Wertden, P. Kâclcell, F. Bechstedt: Surf. Sci. 307-309, S8S (1994)4.67 U. Grossner, J. Fkzthmiiller, F. Bec%tedt: Phys. Rev. 858, R1722 (1998)4.68 K. KA.hn: Surf. ScL Rep. 3, l93 (1983)4.69 W.A. Harrison: Eleziwnic Strucfure and f,??,c Pmpcrlics @.f Soiids (Dover, New

York 1989)4.70 M. Sablvb, P, Kriiger, J. Pollmxnn: Phys. Rev. B51 13367 (1995)4.71 J.C.H. Spence: Phys. Rev. 838, 12672 (1988)4.72 J2. Nozorn'p, M.L. Cohen: J. Vac. Sd. TeBnol. 21, 333 (1982)4.73 K.C. Paudey: Phys. Rev. Listt. 4C, 1913 (1981) .

4.74 K. Takeunht, A. Selloni, A.I. Shlcvebtii, E. Tcsatti: Phys. Rev- B+q 13611(1991)

4.75 M. Rohl6ng, M. Pablrnmo, G. Orwidw R. De1 Sole: Phys. Rev. Lett. 85, 5440

4.76 F. Beckstedt, A.A. Stekolnskov, J, Fttrthmiiller, P. Kekell:lphys. Rev. Let't.897 016103 (2001)

4.77 A. Schohe, W.G. Snbmldt, F. Becbstedt: Phys. Rev. 853, 13725 (1996)4.78 mA. Stekolnikov, J. Pnrkhmûller, F. Bechstedt: Phys. Rev, 865, 115318

(2002)4.79 J.A. Strosdo, R.M. Feenstl'a, A.P. Fein: J. Va2. Sci. Tevhnol. A6, 499 (1987)4.80 R.M. Feenstra, G. Meyer, F. Moresco, K.H, Rieder: Phys, Rev. 864, 081306

(20D:)4.81 R.I.G. Uhrberg, G.V. Hansson, J.M. Nicholks, S.A. Flodström: Phys. Rev.

Lett. 48, 1032 (:982)4.82 A. Cziteûti, S. Selci, K.0. Magnuyson, B. Reihl: Pbys. Rev. B41, 12908

(19K)4.83 J.M. Nicholts, G.V. Hxnxqon, R.I.G. Uhrberg, S.A.' Flodström: Pkys. Rev.

B2C, :594 (1983)4.84 J.M. Nicholls, B. Reihk Surf. Sci. 218, 237 (198S)4.85 J.E, Nortbmtp, M.S. Hybertsen, S.G. Louie: Phys. Rev. Lett. 66, 50O (1991)4.86 X. Zim, S.G. Louie: Phys. Rev. B44 12146 (1991)4.87 S. Jarlori, G. Galli, F. Gygi, M. Pnrmello, E. Tcatti; Phys. Rev- Lett. 69,

2547 (1992) '

4.88 W.J. H '

M. Lobrneier, H.A. van der Vegt, J.F. Peters, S.A. de Vries,B. Weg, V.H. Blgensl T.B. Dea'zy, J.F. 'vr. der Veen: Surf. Sci. 396, 241

(1998)4.89 1t.P.'C%1'ny J.Y. Huang, Y.R. Shen: Phys. Rev. 852, 5985 (1.995)4.90 D. Fauderbilt, S.G. Louie: Pbys. Rev. B30, 6118 (1984)4.91 W.G. PveAmldt, A. gabol'zz, F- dt: Surf. Sd- 351, 183 (1996)4.92 G. Kem, J. Hnfmer, G. Kresse; Surf. Sd. 366, 464 (1996)4.93 F.J. ABrnpsel: D.B. Eastrnn.n, P. Hezmazm, J.F. van der Veen: Phys. Rev.

B24, 7270 (1S8l)4.94 G.D. Kubiak, K.W. Kolashxsld: Phys. Rev. 839, 13Sl (1989)4.95 J. Pollmnnn, P. Krùger: i.n Handbook o.fsur/'t?,cz Scé.enc.en Vol. 2, ed. K- Horn,

M. Pvnlmme.r llsevier, Amqterdam 2000), Chapter 24.96 R. Seiwatz: Surf. Sd. 2, 437 (19M)4.97 R. Feder, P.P. Auer, W. Mönch: J. Phys. C12, 1412 (:979) .

4.98 X.M. ZEeng, P.V. Smith: Sarf. Sci. 253, 395 (1991)

Chapte,r 4 325

4.99 H-G. Busamnn, S. Lauer, LV. HertG W. Zsmmermxnn-Bdling, H.-J-G'-nntherodt, T. Frauzm%eim, P. Blaudeck, D. Porezagl Surf. SCi. 295, 340(1993)

4.100 W.A. HA.v'soa: Eiementary flectrtmic Stmtcture Aorld Scientïc, Singapore1999)

4.101 W.G. Snhrnldt, S. Mlrbt, F. Bechstedt: Phys. Rev. 862: 8087 (2000)4.102 W.G. Schmt'dt? F. Bxhqtedt: Phys. Rev. 854, 16742 (1995)4.103 R.E. gcb7le.r, H.E. orth: J. Chem. Pkys, 30) 917 (1959)4.104 A. ScEolze: Diplomx Thesis) Friedriclt-scitiller-unieut, Jena (l9S6)4.105 V.M. Bermudc: pkys. stat. sol. (b) 202, 447 (1997)4.106 P. Kâckelk J. Fkzthmiiller, F. Bechstedt: Appl. Suzf. Sd. 104/105, 45 (1996);

Smf. Sci. 3322354, 55 (1996)4.107 N.N. Graenwoodo A. Emrnqhaw: Chemistry o.f Elcmnnts (Pergamon Press,

New York 1984)9 p. 6434.108 W.S. Yang, F. Jona, P.M. Marcus: Solid State Comrrntn. 43, 847 (1982)1

Phys- Rew 828, 2049 (1983)4.109 S.J. Jenlrins, G.R Srivagtava: J. Phys. Condens. Matter 81 6541 (1996)4-110 A. 'Rxme-''u!. G. Brocks, P.J. Kelly; Phys. Rev. B51, 14504 (1995)4.111 K5. Hnmers R.M. Aomp, J.E. Demuth: Pko. Rev. 8341 $343 (1f86)24.112 J. Dabrowskb M. Sc%efer: Appk Sud. Sci. 56, 15 (1992)4.113 A.I- Shlcrebtl, K DGelice, C.M. Bertrzzi, R. Del Sole: Pîys- Rev- B51j 11201

(1995)4.114 R.A. Wolkow: Pîys. Rev. Lett. 68, 2636 (1992)4.115 1-1. Tfvthsltxrao T. Amnkntsw M..ïwatsuld: Phys. Rev. B50, 12262 (1994)4.116 T. Yokoyama, K. Tpkmya. nagi: Phys. Rev. B6l, m5078 (2000)4.117 Y. Kondoj T. Amxkalsa, M. Iwatsal:i and H. Toknmoto: Stkd. Sci. 453, L318

(2000)4.118 K. Haia, Y. Sainoo, H. Sbigekawac Phys. Rev. Leit. 86, 3084 (2001)4.119 S.B. Healy, C. Flpps P. Krafzer, E. Penev, M. Scàeëer: Phys. Rev. Lett.

87, 016105 (2001)4.120 W.G. Sn%mldt. F. Beckstedt: Suzf. Sd. 409, 474 (1998)4.121 W.G. Sckmidtt F. Becistedt, N. Emzar, M. Pristovse.k. CM Schultz, W.

Richter: Pkys. Rev. B5T, 14596 (1998)4.122 W. Barvoswfartmy A.S. Bracker, J.C. Culbertrzm, B-Z. Nosho, B.V.

S%lmabrook, L.J. Wbitmam H. K5m, N.A. Modine, E. Knvlrms: Phys. Rev.Eett. 84, 4649 (2000)

4.123 L- Pattkey, E.L. Bullock, T. Abnuw.x S. Kono) L.S.O. Johanœon: Phys.Rev. Lett. T5, 2538 (1995)

4.124 K. Liidge: P. Vog't, 0. Ptzld, N. Esser, F. Bechstedtl W. Richterf Phys. Rev.B62, 11046 (2000)

4.125 C.D. Macphezson, R.A. Wolkow, C.B.J. Mitchell, A.B. McLtx*.n: Phys. Rev.Lett. 7T, 6Sl (1996)

4.126 P. KM- kell, J. Ruthrniizer, F. Bechstedt, G. Ifresse? J. Hxfne.rc Phys. Rev.Bs4, 10304 (1996)

4.127 J.M. Powel's, A. Wamder, P.J. Rous, M.A. V= Hove, G,A. Somorjai: Phys.Rev. B44, 11159 (1991)

4.128 J.P. Long, V.M. Bermudez, D.EL Rnmn.ktAr: Phys. Rev. Let't, 76, 991 (1996)4.129 H.W. Yeom, M. Sht'momtzra, J. Kitamura, S. Hara, K. Tono, 1. Matsuda, B.S.

Mun, W.A.R. Huf, S. Kono, T. Ohta, S. Yoshida, H. Okushi, K. Kajimttra,C.S. nzlley: Phys. Rev. Lett. 83, 1640 (1999)

4.130 F. BreeMt, P. KM'' lretl, A. Zywietz, K- Karch, B. Adolpk K. Tenelxn, J.ntrt%mûller: phys. stat. sok (b) 202, 35 (1997)

4.131 U. Grossner, J. Fkrthmiillea', F. Beczstedt: Phys. Rev. 864. 165308 (2001)

326 Refemn-

4.132 J.E. Nortbmlp, M.L. Coàen: Pàp. Rev. B50, 2015 (1094)4.133 J.E. Northmlp: Phys. Rev. Lett. 5;, 154 ()986)4.134 R.D. Meade, D. Vanderbilt: Phys. Rev. Bd0, 3905 (1989)4.135 P. Becitstedt, W.G. SnBmsdt, A. Schohe: Ezzrophys. Lett. 35, 5S5 (1996)4.136 J- nlrkllmiiller P. Kn-nkfo, F. Beheet, A. 'naSG K. Pfnnn-tglza'as, B-! . y z azja (:ggs;Schröter, W. Rachtezc J. Electrozsic Mate-rn. q 2 ,

4.137 J.EL Nortlmnp, J. Neugebauer Phys. Rev. 852, Rl7001 (1905)4.138 M. Sabisck P. Krftger, J. Pollmnon, Phys. Rev. B55, 10561 (1997)4.139 J. Farthmiiller, F. Bechstedt, H. Hœken, B. Seöter, W. Ftichter: Phys. Rev.

858, 13712 (1998) .

4.140 L.I. JohmnKson, F. 0wm=, P. MMensson: Sttd. Sci. Let't. 360, L483 (19S6)4.141 J.M. Themlln, 1. Forbeauxa V- Langlais, H. Bnllrhir, J.-M. Debever: Europhyq.

Ott. a9, 61 (1997) '

4.142 V. lurnsmhltndraz:, R.M. Feenstra; Phys. Rev. Lett. 82j 1Oû0 (1990)4.143 M. Robl4ng, J. Pollmxnn' PYs. Rav- Lett. 84. 135 (2000)4.144 K. Tnonynnztgi: Y. Tazkishiro) M. TAkmhashij S. Tn.lcsyNKSM; J. Vac. Sd. TeGr

zzol. A3, 1502 (1985); Surf. Sci. 164, 367 (1985)4.145 G.-X Qiaxb D.J. Chacti: Phys. Rev. 835, 1288 (1987)4.146 M. Koike, Y. Einaga, H. Hirayama, K. 'TnDnyxos%: Phys. Rev. B55, 15444

(1997) è

4.147 K,D. Brommer, M. Gavu, A. Dal Pino, J.D. Joxnnopottlos: SuH. Sci. 314,57 (1994)

4.148 S.H. Ke, T. Uda, K. Teiakllrm Phys. Rev. 862, 15319 (2000)4.149 R.J. Hnmen, R.M. Tromp, J.E. Demuth: Smf. Sci. 181) 346 (1987)4.150 G. Chiarottî (Ed.): in N '

Data cn# Fuzlctixd.l lïezationxnl6iits ïzl Scbiiezzce and recànniopg, Landolt-Böznzsteic, New Seriu Group 111, Voi. 24d(Sprhger-verhg, Berlin, 1994), p. 3?9

4.151 T. Stimpel, J. Scxtzze, H.E. Hoster, 1. Eisele) H. Battmgârtner: Appl. Sart.Sci. 162-163, 384 (2000) û

4.152 J. 0l%6+, F. Florœ, A.L. Yeyati: Phys. Rev. B58, 4584 (1A8)4.153 D. Vanderbilt: Phys. Rev. B36, 6209 (1987)4.154 C. , J. Fttrthrniiller, F. Bechstedt: Phys. >v. 866, 075201 (2002) .

4.155 N. Takeuchi: Pkys. Rev. 857, 6255 (1998)4.156 H.J. Gossmztnrh, J.C. Burl, L.C. Felzlrnaw E.G. Mcltaaj I.K. Robinson; Phys.

Rev. >tt. 55, 1106 (1985) .

4.157 F. Becbstedt, R. lnderlein: Semiconductor s'ur//c.e,.s and Jszerlcce.atAknrliAml'e-'Ve.r1ag, BerH 1988)

4.158 L.C. Snyder: Sud. Sci. 140 101 (1984)4.159 M. Aono, R. Soudao C. Osh:=n, Y. Lshizawa: Phys. Rev. Lett. 5lj 80l (1983)4.160 F.J. Himpsel: Phr. Rev- 827) 120 (:t983)1 F'.J. Himpselv 1.R Baea: J. Vac.

SCai. Tecuol. M, 952 (1984)4.161 E.G- Mfmno: Pbys. Rev. 828, 2305 (1983) -

4.162 D.J. Chadi: Phys. Rev. Lett. 52, 1911 (1984)4.163 D.K. Biegelsem; R.D. Bringans, J.D. Northrup, L.E. Swaztz: Phys. Rev. Lett.

65, 452 (1990)4.164 J.M.C. Thorton, P. Weightrnlm, D.A. Wooll, C.J. DttnKcombe: Pkys. Rev.

Bsl) 14459 (1995)4.165 T. Nxknzla, T. OsAkx; Phys. Rev. Lett. 67, 2834 (1991)4.166 EL Kxvl'qms, Y. Bar-y=, J.D. Jon.nnopoloms, K. P=dcy: Phys. Rev. 835,

9625 + 9636 (1987)4.167 U. Stadqe, J. Scharctt, J. Bernhardt, M. Frnnle, K. Reuterl H. Wcxdle-rt K.

Hekkz, J. Furtbrniiller, P. Kxsrlrelia F. Becbs*dt: Phys. Rsev. Lett. 80, 758

(1998)

Chapte.r 5 327

4.168 T. A.n, M. Yosblmure.k 1. Ono, 1f. Ued.a Phys. Rev. B6i, 3006 (2000)4.169 W. Kx.nk'e: Phys. Rev. B41, 5243 (1990)4k170 J, Dabrowsk, H.-J. Mûssig, G. Wof: Phys. Rev, Lett. 'r3$ 1660 (1994)4.171 W.B. Pankxrdo J.E. Dcw: Phys. Rev. 855, 15643 (1997)'4.172 M.A. Kulakov, G. Hezm, B. Bullemer: Sud. Sd. 346, 49 (1996)

Pkcther IR emdlng

à' eryeodt F'., Endezlein R.: Seméconductor s'vrlcee,s and fnfezïccea tàk-MdAmse-k erlag, Berlia 1088) -

#

Mönch W.: Semicnnductor Sur/cce: tmd Intnvjqces, 3rd ed.. (Springer-verlag,B erlin 20 0 1)

Chapter 5

5.1 G. Bl'n'nlg, H. Rohrer: Helv- Phys. Acta 55, 726 (1982)àk2 J. Bardeen: Pkys- Rev. Lett. 6, 57 (1961)à 3 bx l'Nnq'rna J. Tersos: J. Appl. Phys. 61, R1 (1987)5.4 N.D. . Pbys. Rev. Lett. 5s, 230 (1985)9 56, 1164 (1986)5.5 J. Cerda, M.A. V= Hove, P. Sautet, M. Splmeron: Phys. Rav. 856, 15885

,, (zgg7)5.6 J. Tersof, D. R. Hnmxnn: Pbys. Rev. B31, 805 (1985)5.7 L. Hefit'n, S, Ltmdqvist: Solid State Phys. 23, 1 (1969)5.8 K. Lûdge: P. Vogt, 0. Pulci, N. 'Fkqer, F. Bechstedt, W. mchtmn Phys. Itev.

B62, 11.046 (2Kû)5.9 R. mesendangerh H.-J. Gfmtherodt ('9Mq.): Scannqng Tunnelïng Micvoscoby1, II, 11T Springer Ser. Surf. Sci., Vols. 20, 28, 29 (Springemverlag, Berlm

'' 1994, 1995, 1996)à.10 J,A. Stroscio, R.M. Feenstra, A.P. Fein: Phys. Rev. Lett. 5,6 2579 (1986)5.11 S. Hiiher: Phoialzctron Silcctrtucom, Sprsnger Series in Sohd-state Sdences,

Vol. 82 (Springer-verlag, Berïin 1H5)112 F.J. Hh-mpsel: Surf. Sd. Rep. 12p 1 (1990)5.13 J.B. Pemdry; J. Phys. C14, 1381 (1981)5.14 J.P. L. Hopolnson, J.B. Pendzy, D.J. Titteegen: Comput. Phys. Gornmun..'

19, 6: (w8c): 5.15 C. Solterbeck, W. Se-hxttke, J.-W. ZnAlrnnnn-Nowitzlta-, K.-U. Gawlzq L.

Kipp, M. Skbowsks C-S. Fwlleyy M.A- Van Hove: Phys- Rev. tett. %9, 4681:

'

(1::7)116 C.N. Berjltmd, W.L Spicer: Phys. Rev. 136, A1030 (1964)è5.17 A. Einstem: Ann. Phys. 1'6 132 (1905)'

5.18 D.C. Laaveth: Phyj. Rev. B1, 4D. (1970); Phys. Rssv. Izet't. 26, I-ZZ9 (1971)5.19 L.1. Johxm-n, 1. Lmdaa; Solid State Cornmlm- 29, 379 (1979)5.20 D, Chastenet, P. Longe: Phys. Rev. Lett. 44, 91 + 903 (1980).5.21 L. HTA3n'n, J. Michiels, J. Ogltssfeld: Phys. Rev. B58, 15565 (1998)5.22 R. Graupner, M. HoDering, A. Ziegler, J. Risteim L. Ley, A- Skxmpî: PYrs.

Rcv. B55, 10841 (1997)$.23 H. Cazstenxn, R. Claesseu, R. Mxnsnke, M. Sldbovold: Phys. Rev. B41t 9880

(1990)

328 References

5.24 U. H8fer: PL.D. Thesis, Teclmicaz UniveDity, Miinchen (1989)5.Q5 P. Steiner, H. Höckst, S. D'lfner: Z. Physik B30, lQ0 (1978)5.26 F. Bzvbmtedt, 1<. Teneksen, B. Adolph, R. De1 Sole; Phys, Rev. Lett. 78, 1528

(19S7) -

5.27 P. Minnhagen: Phys. Lett. 56A, 327 (1976)5.28 F. Becksedt, M. Fiedler, C. Krem, R. De1 Sole: Phys. Rev. 849, 7357 (1994)5.29 A- Fleszaq W. lhnlre: Pkys. Rev. B56, 10228 (1997)5.30 W.-D. Schöne, A.G. Eguiluz: Phys. Rev. Lett- 81, 1662 (1998)5.31 L. HM. in: Phys. Rev- :1.39, A796 (1965)5.32 G. Strinati) La Rivksta del Nuovo Cimento 11, 1 (1988)5.33 P. iawan, 0. Gxlnnarsson: Rep. Prog. Phys. 6l, 237 (1998)5.34 M.S. Hybertsen, S.G. Louie: Phys, Rev. Izett. 55, 1418 (1985); Phys. Rev.

B34, 5390 (1986)5.35 0. Pulci, F. Gchstedt, G. Onidw R. De1 Sole, L. Roinsng: Phys. Rzv. B60,

16758 (1999)5.36 C. Blomberg, B. Berg=en: Can. J. P/ys. 50, 2288 (1972)5.37 B. Bergersen, F.W. Ktts, C. Blombep Cam J. Phys. 51? 102 (1973)5.38 F. Behstedt: Adv. Solid State Pbys- 32, 161 (1992)5.39 W.G. Aulbtm L. Jönsson, J.W. W''1Vr-ns: Solid Svte Phys. 5d, 1 (2000)5.40 N.D- T,xrig, W, Kohm Pkys. Rev. B'6 3541 (1973)5.41. D.M. Newns: Phys. Rev. B1, 3304 (1970)5.42 F. Becbstedt, R. Bnderlein, D. Reichardt; phys. stat. sol. (b) 117, 261 (1983)5.43 L.D. L=dau, B.M. Lehitz: Electrodynamia QJ Colstin'uav Vol. 8 (Pergamon,

OAord 1959)5.44 T. Andoh A.B. Fowler, F. Stezm: Rev. Mod. Phys. 54, 437 (1982)5.45 F. Beckstedt, A. Bnderleip D. Reinhxrdt: phys. stat. sol. (b) 118, 327 41983)5.46 A.G. Eguiltzz1 M. Heinn'chsmeier, A. Fleszar, W. Hxnke: Phys. Rev. Izett.

68, 1359 (1992)'

5.47 N.D. Langj W. Kohn: Phys. Rev. B1, 4555 (1âT0)5.48 N.W. Ashtrofi N.D. Merrnl'n: Sol6d xtxaàr Physica tsaunders Collegej

L6)Philadelpha 19 f5.49 F. Bfvthstedt: M. Scheëer: Surf. Sd. Rep. 18, l45 (1993)5.50 U. Grosmer: J. Fuztbmihller, F. Bechstedt: phys. stat. so1. (b) 216, 675 (19â9)5.51 U. Grcssner, J. RAbmiiller, F. Bechstedt: Phys. Rev- B64, 165308 (2001)5.52 A. Scimlze: Diplomx Tllesis, n'icdricll-scla-lller-uGveauitât, Jena (1995)5.53 C.-O. Alrnbhdhl U. von Barthi Phys- Rsxv. B31, 3231 (1985)5.54 M. RohlGng: PKD. Titœks, Wœti-alische WiRLektus-l/nivezvitât, Miinster

(1996)5.55 F. BMhqtedt. R. De1 Sole: Solid State Cornrnun- C4, 41 (199Q)5.56 W. Mönck Somiconductr Sîtrfc,cez and Jnteryaceyl 3rd ed. (Springer-verlag,

BerDn 2001)5.57 U. Groiner, J. Fkrtbrniiller: F. Becb-stedt: Phys. Rev. BS8, R1'C22 (1998)'

5.,58 F. Beqbeedt, R. De) So1($: Pkys. Rev. B38, 7710 (1988)5.59 1. VvrgxRrrnan, J.R- Meyer, L.R. Rnm-Mohan: J. Appl. Phys. 89, 5815 (2001)5-60 A.W. Maue: Z. Physik 94, 1k1 (1935)5.61 E.T. Goodwim Proc. Cxmbr. Phi1. Soc. 35) 205 (1939)5.62 S.G. Daeon, M. S'MDCH; Ba-* neprp oj s'ldlce Staiea (Clarendon Press,

Ozbrd 1992)5.63 V. Heine: Sad. Sci. 2, 1 (1964)5.64 V. Heine: Pbys. Rev. 138, A1689 (1965)5.65 F. Bechstedt, R. Bnderlein: Szmzconductor Sur/c,ce..s cnd Intrdacp

(Altademie-vezlag, Berlin 1988)6.66 M.S. Hybertsen., S.G. Louie: Phys. Rev. 838, 4033 (1988)

Chapter 5 329

5.67 K Zlm, S.B. nang, S.G. Louie, M.L. Cohem: Pkys. Rev. Izett. 63, 2112(1989)

5.68 J.E. Noztlmlp, M-S. Hybertseë S.G. rxuie: Phys. Rev. Lctt. 66, 500 (1991)5.69 J.E. Northmlp: Phym Rev. BK, 10032 (1993)5.70 C. Kress, M. Fiedler, F. RecbKtedt: in Pvoceedings o/fhc Fourtlt I 'onal

Con,j'erence tm fze Fovrnaton of Senzïernducttrr Jnterfcms, ed- B. Lengeler,H. Lûth: W. Mönch, J. Pollrnann (World Sdentéc, Singapore 1994), p. 19

5.71 C. 'Kwtquq. M. nedler, F. Beckstedtc Rnrophys. Lett. 28, 433 (1994)5.72 C. Kzus, M. ncxcller, W.G. Snhmldt, F. Beckstedt: Phys. Rev. B30, 17697

(1994); Surf. Sci. 331-333, 1152 (1995)5.73 M. RolsBng, P. Kriiger, J. Pol7rnn.nn: Phys. Rev. 852, 1905 (1995)5.74 M. Ro , R K'tiiger, J- Pol:mxnn-. Phys. Rev. B54, 13759 (1996)5.75 0. Pulci, G. Onida. R. Del Sole, L. Relnsng: Phys. Rev. lxxtt. 81., ,5374 (1998);

pzivezte comrnllnscation5.76 M. Rohl6ng, S,G. Louie: Phys. Rev. Lett. 83, S56 (1999)5.77 M. Ro , M- Pabtmmo, Q. Onidap R. Del Sole: Phys. Rev. Lett. 87 944.0

(2000)5.78 R.LG. Uhrberg, G.V. Nhnqson, J.M. NiGolls, S.A. Flodström: Phys. Rev.

Lett. 48, 1032 (1982)5.79 P. Perfetti) J.M. Nicholls, B. R>i'h1: Ph>. Rnv. B36, 6150 (1987)5.80 J. Wu, R- Cx, X. Yang, P. Piandtw 1. Lindau: J. Vac. Sd. T'xehnoL A11,

1048 (1993)5.81 G. Franczt P. Kn.n'tl!.l G.. Ganther, H. Stupp, P. Oelhafen: phys. stat. sol. (b)

'

1s4, 91 (1â96) ,

5.82 F. Beehstedt, R. De1 Sole, F. Manghi: J. Phys.: Condrenn. Matter 1, 75 (1989)5.83 0. Pulci, M. Pal:tmmo, R. De,l Sole: in prepazation5.84 L. Sorba, V. Hinkel, U,U. M-iddmlrnn.n'n, K. Horm Pkys. Rev. 832, 8075 (1987)5.85 Z.M. Nicholls, K.O, Magnusson, B. Re11:t12 Sm'f. Sd. Lett. 243, L31 (1991)5.86 W- Drube, D. Stzaub, F.J. Hlrnpsel: Phys. Rev. 835, 5563 (lS87)5.87 A. Criceati, S. Selci: K.O. Magatzssor, B. ReM: Phys. Rav. B41, 12908

(1990)'

5,85 J.M. Nickolls, G,V. Hanssow RJ.G. Uizrberg, S.A. Floclströmc Phys. Rev.827, 2594 (1983); J.M. Niclmlls, GX. N'xnRqon, U-O- Karltnanl RJ.G-Uhrberg: Phys. Rev. Lett. 52, 1555 (1984)

5.89 S. Solal, G. Jezequel, A. Barsld, P. Steiner, P. Pkchatm Y. Petrof: Phys.Rev. Lett. 52, 360 (1984)

5.90 J.M. Nicholls, B. RtM'h1: Surf. Sci. 218, 237 (1989)5.91 R.M. Feenstra, G. Meyer, F. Moresco, K.H. Rieder: Phys. Rev. B64, 081306

(2001) '

5.92 R.J. Hamers, U.K. Köhler: J. Vac. Sd. Tebnol. AT, 2854 (1989)5-93 M. Sldbowsld, L. Kipp: J. Electron Spectrv. Relat. Phenom. 68, 5576

(19S4)5.:4 L. Kipp, R. Mnnzke, M. Skibowsld: Soiid State Commlm. 93, 603 (1995)5.95 J.A. Kubby, J.E. Groth, R.S. Becker, J.S. Vickers: Phys. Rev. 836, 6079

(19S7)5.96 P.M. Echeaique, R.M- Pendzy: J. Phys. C11, 2(%5 (1978)5.97 P.J. Jemnlngs, R.O. Jonc: Phys. Rev. B34, 6695 (1986)6.98 G. Binnig, N. Garcia,, H. Rohrem J.M. Soler, F. Flores: P/ys. WM. B30, 4816

(1984)5.99 G. Blnm'g K.H. Fïonk', H. Rzclts, N. Garciaj B. QA%e, H. Robnm F. Salvano

'.:A.R. Wilhxmq: Phys. Rixv. Leu. 55, 991 (1985)5.100 U. Höfe-r, I.L. Shnmay, C. Reus) U. 'Chomrb.nn: W. Watlauer, T. Fauster:

Sdence 277, 1480 (1997)

330 Referenca

5.101 F. P=ek, M. Donath Phys. Rev. Lett. 69, 1101 (1992)5.102 M.V. Coleu M.H. Cohem Pbys. Rev. Lett. 23) 1238 (1969)5.103 R. Eknderleip N.J.M- Horinj: Fundamentau O.J Srmoondnctor P/zvsïc,s and

De't)ice.s Aorld Scientfq Smgapore 1997)5.104 L.D. Landau, E.M. T,!'Eqb1tz: Quantum MezLmic-s VoL 3 tpergamon Pre>, .

OAord 1958) ,

'

5.105 D. Straub, F.J. Nsmpsek Phr. Rc'v. 833, 2256 (1986)5.106 RM. Bcxelzique, J.B. Pendry: Prog. Suzf. Sci. 32, 111 (1990)5.107 KO. Jones, P.J. JfAnnino, 0. Jep-n: Phys- Rev. B29, 6474 (1984)5.108 P.J. Jennings, R.O. Jores, M. Weinezt: Phys. Rev. B3C, 6113 (1988)5.109 T. Fauster, W- Stelnmxnn! iu Plmtonic T'robea o! Juzpcas ed. P. Halevi ' '

(Elsevier Science, ArnKterdam 1995), Chapter $5-110 A. Golamlmn, V. Dose, G. Borste.h Phys. Rev. 832: 1971 (1985)5.111 J.B. Northrap; J. Neugebauer: P/ys. Rev. B5T, 11,:1230 (1998)5.112 J. Farihmiiller, F. Bechstedtv H. Husken, B. ScNröter, W. Richter: Phys. Rev.

B5s, 13712 (1998)5.113 M. Rold6ng, J. Pollmnzm: Phys. Rev. Let't. 84, 135 (2000)5.114 1a.L JohxnAqon, F. Owmanj P. Meerusson: Surf. Sci. 360, L478 (1996)5.115 L.S.O. Johausson, L. Duda, M. Lauznnzis, M. Mrieftewirth, B. ReM: Sud.

Sci. 445, 109 (2000)5.116 J.-M. Themlln, 1. Forbeattx: V. Izanglaiw H. Be7lrh-lr, J.-M. Debever; E'uro-

phys. Lett. 39, 61 (1997)5.117 C. Beuesciu PL.D. Thesis, WMfn-llqclze Wizlrgnhs-Tlzûvcsitâtp Mt'inqter :

(1998)5.118 V. RlbmMltl'.ndr=, R.M. Feezustra Phys. Rev. Lett. 82? 1000 (1999)5.119 V.A. Gmsparov, M. Riehi-cimdoba, B. Sckröter, W. Riczte-r: Eutophys. Lett.

51, 527 (2000)5.120 N. Sieber, T. Seyller, L. Ley, M- Polcm D. James, J.D. Ridley, R.C.G. Leckey:

Materiab Science Forltrn 389-393, 713 (2002) '

5.1.21 J. Hubbardt Prx. K Soc. London M76, 238 (1963); 283., 401 (1964)5.122 W. Jones, N.H. March: TAerrctic,tzl Sotid State Pvaïcd (Dover, New York

1S&5)5.123 W.A. Hxrrlson: Phys. Rev. B31, 2121 (1985)5.1M W.A- Harrison: flemerer!/ flectmrlïc Structure (World Sdentœc, Singapore

1999)5.125 N.F. Mott: Proc. R. Soc. London A62, 416 (1949)5.126 R. Sn%sller, W. Nolting; Phys. Rev. Lett. 86, 3847 (2001) '

5.127 KLG. Uhrberg, T. lfaum'lx, Y.-C. Clun: Phys. Rev. B58, R1730 (1998)5.128 RJ.G, Uhrberg: private cornrnunicatîon5.129 F. Bechstedt, A.A. Stekoln-tkov, J. Fkrtltrniiller, R Ka-nkez: Phys. Rev. Lett.

87, 016103 (2001)5.130 F. Flores, J. Ortega, R. Pércz: StTrf Bzv. Lett. 6, 411 (1999)

Chapter 6

6.l W. mchtea': PMos. T. Roy. Soc. A344, 453 (1993)6.2 D.E. Avspnœ: Sttrf. Sd. 309, 1017 (1S94)6.3 J+. McGiip, D. Wp--ll're? C.H. Patterson (Eds.): ktioptica. J/ïnear and Xcrlà'n/-

ear œfrzr' nl Szectmscoyv o.f s'wrlc.c'e,s an'd Jxferïccas (Spzingez-verlag, Berliz1995)

6-4 W. Richter, J.T. Zettler: Appl. Surf. Sci. 101, 465 (1996)

Chapte.r 6 &3:

6.5 D.E. Aspnes: Solid State Commtm. 191, 85 (1997) '

6.6 R. Del Sole: in Photonic T'r/he.s o! Suzfacas ed. P. Halevi tELsev/.r ScienceAmsterdxm 1995), Chapter 4

6.7 P. Cbiaradia, R. De1 Sole: Surf. Rev. Lett. 6, 5l7 (1999)6.8 P.V. SaatcG N. Erxse-ra J. Groenem, M- Cardona, W.G. Schmt'dt, F. Becitstedt

Phys. Rev. BsA 17379 (1995)6.9 P. Chiradia, A. Cricenti, S. Selci, G. Chiazotti: Phr. Rev. Izett. 52, 1144

(1984)6.10 J.D.B. Mclntyre, D.B. Aspnes: Smf. Sci. 24, 417 (1971)6.11 K. Hingm-l, D.B. Aspnes, 1. Kxrnlya, L.T. Flores: Appl. Phys. Lett. 63, 88t

. (1993) -

6.12 D. Pnlhllce, M. Aremsj N- Esser, W. . Xchtèr: Surf. Sd. 352, 66 (1996)) NBsser: private communicatioa -

6.13 H. EhremreiG: Electromagnetic Trnm.qport in Solids: Optic.al Properties aniPhsma Esects, i'a The Ogtfccz Properties OJ SOD&, Proc. Mternat. Seool o:

Physics C>). Femni', ed. J. Tauc (Azozlemic Press, New York 1966): 6.14 S. Acller: Phys. Rmr. 126, k13 (1962)6.15 N. Wj=: Phys. Rev. 129, 62 (1963)6.16 V.I. Gavrilezzko, F. Becitstedt: Phys. Rev. 855, 4343 (1997)6.17 L.D. Landau, E.M. Lifschitz: Electrodynamia J./ Conbinut (Pergmmon Press:

OAord 1959)6.18 A. Bagclti, R.G. Barrera, A.K. Rajagopal; Phys. Rev. B20) 4824 (1979)6.19 R. +1 Sole: Solid State Commum 37, 537 (1981)C 6.20 K Del Sole, E. Fiorino: PNys. Rev. 829, 4631 (1.984)6.21 Y. Manghi, R. De,l Sole, A. Selloni, B. Molsnxm': Phys. Rev. B4lj 9935 (1990)6.22 H. Stolz: Einshmtng fzz d6e Ffeîezet-trtlp.enfzzsode d6v A'riztclle (Akademie-

Verlag, BerM 1974)6.23 H. Ibach) D.L. Mills: Elsctron Ensrgy .!,tu.6 Speciroscny cnd Stdace Wàro

tiez?.s lAe-qriemic Press, New York 1S82). 6.24 B.N.J. Perssoa Phys, Rev. Let't. ö0, 1989 (1983)6.25 H. Lûth: Solid S'tsrltzccs, Interjace.s and T/ùn Films (Sprhger-verhg, Berlin

2001)6.26 A. Selloni, R. De1 Sole: Stzrf. Sd. 168, 35 (1986)6.27 U. de,l Pennt'no, M.G. Betti, C. Marlmn'', C.M- Bertoni, S- Nxnnxrone, 1.

Abbatk L. Braicodcx, A- R!-'zd: SOIJZ State Commuxu 60, 337 (1986)6.28 A. Balzm'ottij M. Fstrfoni, 'F. Patella, F. Arcipretej E. Placidi, G. Oaidal R.

De1 Sole; S=f. Scî. lrett. 5241 L71 (2003)6.29 F. Arciprde, F. Patexa, A. Bahrotti, M. Fsmfoni, N. Mottw A. Sgarlata,

A. Boselti, G. Ozlida, A.I. S%k-rebtii, R. Dez Sole; Phys. Rev. 8581 R1013S(1998)

6-30 N. Fgsser, J. Gettrt's: N-eimxn Spcvtroscopy, ia Oyhcal C'//zzmcferlcïilm, o:' Bpitaziul Sczpâconducl/r Layors, ed. G. Bauer, W. Ricizter (Spriager-verlaz,

Berlin 19S6),,6.3: R. Loudozu Proc. Roy. Soc. (London) 4.2C5, 218 (1963)'

6.32 P.Y. Yu, M. Cardona: F'andavrœnials ()J S6micondnctors (Springer-verlag,;J Berlin 2996). 6.33 R. Loudom Adw Phys. 1a, 423 (1964); n-.tnm ibid. lzl, 621 (1965)6.34 K. T'Tlnricio, A. Scbierhorn, P. Haier, N. Fzsser, W. Riclzter, J. SM.h=: Phys.

Rev. Lett. T9, 1094 (1997)6.35 W.G. Scbrnfdt, F. Becbstedt, G.P. Sdvœstan: Phys. Rev, 852, 2001 (1995)6.36 J. b'Htsch, R Pavone, U. Snhröder: Phys. Rw.-852, 11326 (1995)6.37 H.M. Tûtiincik G.R Srivmstava Phys. Rev. 853, 15675 (1996)6.38 G. 0011a, L. Reiuing, A. Rubio: Rev. Mod. Pkys. 74, 601 (2002)

332 Reference

6.39 L. Hedin, S. Ltmdqvist: Solid Stzte Phys. 23, l (1969)6.40 G. Strinlti: Ea Rivista de,i Nuovo Cimen't.o 11, 1 (1958)6.41 B. Adolph, V.I. Gavrilertko; K. Tenelsen, Lbn. Bechstedt? R. De1 Sole: Phys.

Rev. 853, 979.7 (1996)6.4.2 L.J. S1t%.m, T.M. Wce: Phys. Rev. 1.44, 708 (1966)6.43 M- Robl6ng, S.G. Lottie: Phys. Rev. B62, 4927 (2000)6.44 R. Del Sole, B. Fiorino; Phys, Rev. 829, 4631 (1984)

' 'ller-universitât? Jena (2001)6.45 P. Habn: Dlploma Thesis: Friedrich-snhl6.46 W. Hn.rtke, L.J. Sham: Phys. Rev. Le'tt. 43, 387 (197$; Pkys. Rev. 821, 4656

(1980)6.47 F. Becltstedt, K. Tenelsen, B. Adolph, R. De1 Sole: Phys. Rev. Lett. 78, 1528

(1997)6.48 S. Mbrecbt: Ph.D. Tke-ris Vole Polytebnique, Pn='u (1999):6.49 F. Beclzstedtj W.G. Stthmtdt, P. Hlthn; phys. stat. sol. (a) 188, 1383 (2001)6.50 V. Olevano, L. Re'tning: Phys. Rev. Lett. 86, 59œ (2001)6.51 F. Bassxnl', G. Pastoz'i Parraddni! Electronic Sfcle.e avtd O>$çccl Transitions

f'?z Jolida, ed. R.A- Bxllinger (Pe-rgn.mon Press. OAord 1975)6.52 B. Adolph, V. Gavrileako, K. Tenelsen, F. Bechstedt, R. Del Sole: Phys. Rev.

B53, 9757 (lS96) '

6.53 W.G. SnBmt'dt, P. Hsxh'n, F. Becbstedt: Bxcitonic and Local-Field Efects inOptical Spectra fxom Real-space Time-Domnc'n Calmzlations, itz Ifiglz Per/br-rzcnc: Computing ïzè Sdencq cnd Engin6erin.g 'tM,: (Springer-verlag, Berlin,

2003); W.G. ScAm''dt, S. Glutsch, P. Htthn, F. Becitstedt: Phys. Rev. 867,085307 (:003)

'

6.54 D.B. Aspnes, A.A. Studnm Phys. Rev. B2C, 985 (1983)6.55 S. Albreht, E. Relnlng, R. De1 Sole, G. OMda: Phys. Rev. Lett. S0, 4510

(1998)6.56 L.X. Benedict, E.L. Shirles KB. Bohn: Phys. Rev. Lett. 80, 4514 (1998)6.57 M. Rolnlflnj, S.G. Eottiel Phys. Rev. Lett. 81I :37.2 (199E)6.58 R. Erderlem, N.J.M. Horing: Fnndameniais ol Jer/zfemdllztor Pàysir.s cnd

Devices (World Sciemtïc, Singapore 1997)6.59 G.H. Wxtnoier: Phys. Rev. 529 191 (1937)6.60 N.F. Moti; Trnm. Faraday Soc. 34, 500 (1938) -

6.61 R.S. Knox: Tlzemy of E=*n% Solid Svte Pbysics, Supplement 5, ed. F-Seaz, D. Ttzrnbull (Academic Press, Ncw York 1963)

6.62 D.L. Dexter, R.S. Knox: Escjion.s (Wiley, New York 1966) .

6.63 R.J. Elot: Phys. Rev. 108, 1384 (195*46.64 M. Shinada, S. Sugano: J. Pkys. Soc. Japa.n 21, 1936 (1966)6.65 S. Fliigge, H. M=snhp.ll: Rchenmethode'n 46'r Q'ulmtnnihaorés tspringe't-

Veriag, Berltn 1952), Problem 246.66 C. Goletti, G. Bussettl, P. Arciprete, P. Chinmziz'a, G. Cbiarotti: Phys. Rev.

866, 153307 (2002) .

'

6.67 M.A. Olms-toxd, N.M. Amerc Phys. Rev. Iaett. 52, 1148 (1984)6.68 M. RbhlGng, S.G. Lotlie: Phys. Rev. Lett. 83, 856 (1999)6.69 M. Ro , M. Pabtmrno, G. 0**% R. Del Soie; Phys- Rev. Lett. 85, 5440

(2000)6.70 S. Glutscb., D.S. Chemln, F. Beclwtedt: Phys. Rzv. B54, 11592 (1996)6.71 T. Yasuda: D.E. Aspnes, D.R. Lee, C.H. Bjorkmnan, G. Lucovsky: J. Vac. Sci.

Tecuol. A12j 1192 (1994)6.72 P.H. Hs'%n, W.G. Selnrnidt, P. BebKtedt: Pkys. Rev. Lett. 88, 016402 (2002)6.73 M. Mxm'ni, Q. Onida, R. Del Sole: Phys. Rev. 864, 195125 (2001)6.74 J. Ltndhard: Kgl. Danske Vdensk. Selsk. Mat.-fy's. Medd. 28, 8 (1954)6.75 W.A. Hxrrlsonl Solid Staie Thearg (Dover, Naw York 1980)

Chapter 6 333

6.76 G.D. Mnuhnr: Many-pnrticle Php.dc.s lple.nntrn Press, New York 1990)6.77 E.A. Stern, R.A. Ferrell: Phys. Rev. 120, l30 (1960)6-78 P-J. F'eibelmxn; Phys. Rev. Lett. 3ô, 975 (1973); Phys. Rev. B9, 'O'N (1974)6.79 K.-D. Tsuei, E.W- Pln-mer. P.J. Feibelma'a: Phym Rev. Iaett. 63, 2256

(1989)6.80 A. Liebsch: Physka Sadpta 33, 354 (1987)6.81 H. Liith: Sklrf. Sc2. 126, 126 (1983)6.82 A. Ritz, H. Lûth: Phys. Rev. Let't. 52, 1242 (1984)6.83 W. Krcs: F.W. deWette (Eds.): Surface Phymsra (Springer-verlag, Berlin

1991)6.84 A.A. Maractudsn, B.W. Montroll, G.H. Weiss, LP. lpatova: SoDd State

Physiœ, Suppl=ent 3 (AcM>m''c Press, New York 1971)6.85 K. Ktmc, R.M- Markin: Ph>- Rzv'. B24, 2311 (1981)6.86 M.T. Y1, M.L. Cohem Phys. Rev. B26, 3259 (1.982)6.87 K. Kunc, R.M. Martin: J. Phys. (ParLs) 42, 06-649 (1981)4 Phys. Rev. Lett.

4s, 406 (1982)6.88 'P. Gixnnozzi, S. de Gironcoli, P. Pavone, S. Baroni: Phys. Rev. 843, 7231

.

twgz;6.89 S. Baroni, S. de Qironcoli, A. Dhzl Corso, P. Glpnrhozzi: Rev. Mod. Pllys. T33

515 (2001)6.90 M. Born, K. Huang: Dynanxical ne'orv ojt Crg:tal Jaz>ce.s (Oxford Uzzive=ity

Prœs, Oxsord 19-04)8.91 G. Km=, G. Kresse, J. Hafam': Pbys. Rev- 859. 8551 (1999)6.52 F. Bqchstedt, U. Grossner: J. Rzrtbrniiller: Phys. Rmu 862, 8003 (2000)6.93 C. 'lfattel: Jatmdtzcàïtm to Solqd State Physia llob.n Wiley, New York 1966)6.94 L.D. Landau, B.M. LifsMtz: T/zcor,y of Slcsdcïjp, Vol. 7 (Pergamon Press,

London 1959)6.55 Lord Rayleigh: Pzoc. London Math. Soc. 17, 4 (1887)6.56 E.A. Ash, E.G.S. Palge (Eds.): Scpleilh-Nat)e Theor.ç and Xppzicaen,

Springer Sez. Wave Phoom., Vol. 2 tspringer-verlag, BerH 1985)6.97 M.G. Cottn.m D.R- Tilley: Inbroduaion fzl S'tdcce arltf S'tqwwrlattia fzcfza-

?ïfmn.s (Cambndge Univeat.y Prœs, Cnrnbridge 198$, SeG- ?.4

5.98 S. Lehwald, J.M, Szeftel: H. l'bnmb, T.S. *.$t%=an, D.L. Mills: Phys. Rev.Lett. 50, 5lS (1983)

6.99 B. Hulpke, J. Liidecke: Phys. Rev. Lett. 68j 2846 (1992)6.100 R.H. Lyddane: R.G. Sachs, E. Teller: Phys. Rev. 59, 673 (1941)6.101 R. Facizs, K.L. Kliewer: Phys. Rev. 140, A2076 (1965)6.102 W. Mönclu Sqmizondnctor Scrlcce.s and Jntr/ac> (Springer-verlag, Be'rlin

6.103 H. Ibnnht Phys- Rev. Lett Mp 1416 (1970)6.104 H. Nitmhaus, W. Mihciu Phys. Rev. B50, 11750 (1994)6.105 H. Nienkaus, W. Mönch: Stuf. Sci. Lett. 328, L561 (19S5)6.106 H,M. Tùtiimcii, G.P. Srivmstava: J. Phys. Chem. Solids 58, 685 (1997)6.107 H.M. Tiititncîi, R- Miotto) G.P. Srivastam, J.S. Tse: Pbys. Rev. 866, 115304

' (200$) G.P. Szivastavœ private commtmication6.108 H.M. Tutiincû, G.P. Sdvastawu Phys. Rev. 859, 4925 (lS99)6.109 M. Zitrlsperger, R, Honke, P. Pavone, U. Sceöder: Surf. Sd. 377, 108 (1997)6.110 U. Haaten, J.P. Toernfes, Ch. Wö'tl: Pbys. Rev. Lett. %T, 2947 (1986)6.111 L. Miglio, P. Santlni) G. Benedekc Phys. Rev. Lett. 62, 3070 (1989)6.11.2 F. Ancilot-to, W. .Aneexm.', A. Selloni, R- Cap M. Pxrrsmel1o: Phys. Rrv-

Lett. 6s, 3148 (1990)6.113 H. Ibaciu Phys. Rev. Lett. 2*6 û53 (1971) '

&34 References

6.114 N.J. DWardo, W.A. Thompson, A.J. Sttell-soroldn, J.E. Demutiu Pkys.Rev. B3,4, 3007 (1986)

6.115 H.L, Stöz'mer Rev. Mod. Phys. T1, 875 (1998)6.116 R.B. Lauglnlln: Rev. Mod. Phys. Tk, 863 (1998)6.1:7 G. Griiaer; Dnnsity W'cwe.s in Solïffs (Adison-Wasley, Readhg (MA) :994)6.118 R.E. Pdea'ls: Q'uant'um Thenr.y of sblïtf.s (COendon Press, Oxford 1964)6.419 J.M. Luttinger: J. Math. Pby;. (N.Y.) 4, 1154 (1963) ,

6.1.20 J. Voit: Rep. Prog. Phys. 18, 997 (1995) :

6.121 M.G. Zachea'j B. Avrigozd, W. El'.mk-e, J.R. Sr.hm'efer: Phys. Rev. B57, 6370(1998)

6.122 0. Blxnk'j G. Fmenberg, J.H. Zeysing, L. Lottermoser, R.L. JohnKon, M.Nielsem F. Berg-lbzqmurssea, J. Baker, R. Feidtmhxnq'l: Phys. Rev. 859, .

12228 (1999)6.123 C. Kllmpf, 0. Bunls J.H. Zeysing, Y. Su, M. Nielsezb KL. Jobnqonj R.

Feidenhxnq'l, 1f. Bechgaard: Phys. Rev. Lett. 85, 4916 (2000)6.1,24 1t. Losio) K.N. Altmo.nn, F.J. Rmpsel: Phys. Bzv. Let't. &5, 8()8 (2ûO0)6.125 R. Losio, X.N. Altmnrn, A. Kirakosiac, J.-L, Lin: B.Y. Petrovykh, P.J.

Hirnpsek Phys. Rev. Lett. 86, 4632 (2001)6.126 I.K. Roblnson, P.A. Bexmett, F.J. Htmpsel: Phys. Rev. Lett. 88, 096104

(2002)6.127 F. Flores? J. Ortega, R. Plrez: Surf. Rzv. Lett. 6, 411 (1999)6.128 H.H. WettmznYg. X. SM, RD. JohnRnaj J. Chen, N.J. DiNaœdo, K. KemN.

Phys. Rev. Izett. ;8, 1331 (1997)6.1.29 VlGmnzGazkclran, mM. Feenstra: Phr. Rev. Lett. 82, 1û00 (1999)6.130 J.M. Carpinezi, H.H. Weltering, E.W. Pbtmrnm', R, Stllmpf; Natme (Loa-

don) 381.) 398 (1996) ( '

6.131 I-LW. Yeom, S. Takedal B. Rotenberg, 1. Matsada, K. Horikoshi, J. Schaefer,C.M. Lee, S-D. Kevaa, T. Ohtw T.Nagx, S. Hasegawat Phys. Rsv. 1,e1..82,4898 (1999)

6.132 A. Mxem.raque, J. Avilw J. Alvemez, M-C. Arhtarmio, S. Ferrer, E.G. MicheliPhys. Rev. Lett. 82, 2524 (1999)

6.133 S.C. Erwim H.I-L Weittm'no Phys. Rev. Lett. 81, 2296 (1998)6.134 I.G. R'11 A.B. Mnluaan: Phys. Rev. Lett. 82, 2155 (1999)t6.135 R Scgomm D. Purdiez M. Heqgsberger, Y. B=: Natme (London) 402, 504

(l9â9)6.136 F.J. 1'T4'm>e,1, K.N. Altamnn, R. Bennewitz, J.N. Chraip A. Wirakosian, J.-L

Lin, J.L. Mcchesney; J. Phys.: Condens. Matter l3, 11097 (2001)6.137 J.-H. Cko, D.-H. 0h, K.S. Wim, L. Kleinmam Pbys. Rev. 864, 235302 (2001,/

Chapter %

7.l C. Kittel: lntvoôuction f,d Solid Slcje Phusics tlobm Wiley, New York 1966)7.2 G. Lengel, R. Wilk'snx, G. Browa M. Wdflrnery J. GUICO, R.E. Men: Pbysk

Rev. Lett. 72, 836 (1994) '

7.3 Ph. Ebert, M. l'Ieinrsdb M. Simoa, C- Domkp, K. Urlwm, C.K. Sh1'%, M.B'.Webb, M.G. Lagally: Phys. Rev. 853, 4580 (1996)

7.4 Pk. Ebert, K- Urb=, M.G. Lagally: Phys. Rev. Lett. ;2, 840 (1994)7.5 Ph. Ebez-t, K. Urbam, L. Aballe, C.H. Chen, K. Horzt, G. Schwarz, J. Neuger

bauer, M. ScheEer: Pkys. Rew Lett. &4. 5816 (2000)7.6 S.B. Zhang, J.E. Nozthrnlpl Phys. Rev. Lett. 6T, 2339 (1994)T'i' A. Zywietz, J. Fkrthmiiller, F. BehKtedt: Phys. Rev. 859, 15166 (1999)

Chapier 7 335

7.8 S.B. Zkang, A. Zaager: Phys. Rev. Lett. TT, 119 (1996)7.9 H. Wim, J.R. Chelikowsliy: Phys. Rev. Lett. 7'6 1063 (1996)

7.10 H. Tf'tm, J,R. Chelikowsky: Sutf. Sci. 409 435 (1998). . :

7.11 G. Schwarz: Ph.D. Thesis, Telantr-clae Umversitât, Ber;in (2002)7.12 Pk. Ebert: Appl. Phys. AT5, 101 (2002)7.13 S. Aloni, 1. Nevol G. Hnxqe: J. Chem. Phys. 115, 1875 (2001)7.14 F. Ancilotto, A. Sellozti, E. Tosatti: Phys. Rev. 843, 5180 (lS9l)7.15 R.J. Ha.mers, U.K. Köhler: J. Vac. Sci. Tenhnol. AT, 2854 (1989)7.16 H. Tortln''hm'a, T. Amakusw M. Iwatsuld: Phn. Rev. B50, 1.2262 (1994)7.17 T. Uda, K. Ternbnra: Phys- Rev. B53) 6999 (1996)7.18 K.C. Paûdey: Froc. 11th Ini. Con.f fm th6 Phvdic.: ofs6micondnctorh ed. D.J.

Chadi, W.A. Hmison (Springer-verlagj New York 1985), p. 55 '

7.19 J. Wang: T.A, Arias, J.D. JMnnopotzlous: Phys. Ezv. B4T, 10497 (1g93)7QO Ph. Eberk T. Zhaag, F. Klugej M. Simon, Z- Zhaag, K. Urbam P/ys- Rev.

Lett. 83, 757 (1999)7.21 C. Domke, Ph. Ebert, M. Heindch) K. Urbam Pitys. Rev. B541 10288 (1996)7.22 0. Madeltmg, M. Sckulz, H. Wekss (Bds.)) Landolt-Bövrtstein. Akmersctzl Dafc

and Jtmctïtmt?t Retationsltips ïn Scdcrtca cnd Tcchnology, New Series, Vol. l7,Smrdconductors tspringer-verlag, Berltn 1982)

7.23 K. Sauthos, M. Wenderot)h J.K. Garlef, T.C.G. Reusck, R.G. Ulbricht: Proc.26th Jnf. Cons ()1 ihe Fhs/sïc..s o.f Semïc/ndvct/rd, F'dlnbttrgh 2002

7.24 J.E. Northrup, S.B. Zhang; Fhys. Rev. BJC, 6751 (1H3)7.25 R.W. Jnn.qen, O.F. Sankey: Phys. Rev- B39j 3192 (1989)7.26 R.M. Feemstra, J.M. Woodatl, G.D. Pettit; Phys. Rev. Iaet't. 71, 1176 (1993)7.27 B. Grattliulier, H. Chen: R.M. Feezlstra, D.T. MnTnttuf, P.W. Juodawllds?

S.E. NM1ph: App). Phys. Lett. 7z1, 1439 (1999)7.28 Ph. Ebez't, P. Quadbedcj K. Urbu, B. Hennlngerj K. Horn, G. Schwarz? J.

Neugebauer, M. Sclm/er; Appl. Phys. Lett. 79 2877 (2001)l7.29 B. Lang, KW. Joyaer, G-A. Somorjai: Smf. Sm. 30, 440 (1972)7.30 M.-C. Desjonquêr%, D. Spaujaard: Conzepts ïn Sur/acc Physics (Spdnger-

' Verlag, Berlin 1998)7.31 J. Dabrowski) H.-7. Mikssig, G. Wof: Phys. Rev. Lett. C3, 1660 (1994)7-32 1f. Jacobi, L- Geelhnnry J.M. Mequez: Appt Fhys. AT5, 113 (2002)

. 7.33 D.J. Chadi: Phys, Rev', 829, 785 (1984)'

7.34 W.G. Snlam''dt, J. Bernholc: Phys. Rev. B6l? 7604 (2000). 7.35 II. Kroemer: J. Cryst. Growth 81, 193 (.1987)

7.36 W.G. Schmidt, F. Becllstedt) J. Bernlnolc: Phys. Rev. 863, 045322 (2001):. t.37 J. Dabrowskk H.-J- M'iissign Sûkcon zgllriace.a and Fomwtïtm o! Jrztcr/ocas:.' (World Scientdc, Singapore 2000)

7.33 D.J. Chadi: Phr. Rev. Lett. 59, 1691 (1987): . :.' 7.39 T.W. Poon, S. Ykp, P.S- Hoj F.F. Abrxbltm: Phys. Rev. Lett. 65j 2161 (199û),'., '

.: 7.40 R. Shioda, J. 'van der Weide: Phys. Rev. B57, R6823 (1998)'

il 41 S G Jaloviar J.-L. Lizt, F. Liu, V. Zielmsek, L- Mcca , M.G- Lagally:', Pàys. Rev. Lett, 82, 791 (1999)

,

E .7.42 L. Kipp, D.K. Biegelsen, J.>l Northrup, L.-B. Swaztz, R.D. Bringaus: Phys.' Rev. Leti. 76, 2810 (1996)6 7.43 T. Yrudav L. Mantese, U- Rossow, D.E. Aspnes: Phys. Rev. Lett. 74, 3431)

'

(tggs)' '

'7.44 M. Henzler: Surî. Sci. 36, 1.09 (1973)'. 7.45 R.M. Feenstra, J.A. Strosdo: Phys. Rev. Iatt. 59, 2:.73 (1987)' : 7.46 K.C. Pandey; Pbysica 117'B & 118B, 761 (1983):'

.7.47 P. Kâcke )11 J. Furtbrnftller, F. Bechstedt: Phys. Rev, 858, 1326 (1998)7.45 B.S. ltchkxwitz, M.T. BNtrem, J.J. Boland: Phys. Rmr. Let't. 78, 98 (1997)

336 References

7.49 R.L. Headzick., B.E. Weir? J. Berk, B.S. Freer, D.J. Eagleshnm, L.C. Feldman:Phys. Rev. Le'tt. 65, 1128 (1990)

7.50 H. Hibixo, K. Snrn''tomo, T. 0g1.110: J. Vac. Sd. Technol. A16, 1934 (1998)7.51 C. , J. Fkrtln. miiller, F. Bechstedt; Phys. Rev. 866, 075201 (2002)7.52 R-L. Headricko LK. Robinson, E. Vlieg, L.C. Feldmaa; Phys. Rev. Lett. 63,

1253 (1989)7.53 P. Bedrossian, R-D. Meade, K. Mortensen, D.M, Chen, J.A. Golovckenko, D.

Varderbit: Phys. Rev. Lett. 63, 1257 (1989)7.54 S. Wang, M.W. Nmzlny, P.W. Sznith: Phys. Rev. 859, 1594 (19SS)7.65 P. Baumgârtel, J.J. Paggel) M. Hasselblatt, K. Horp V. Fernandez) 0. Schaf,

J.H. Weaver, A.M. Bradsbzw, D.P. Woodmal, EL Rotezzberg, J. Denlt'nger:Phys. Rev. B59, 13014 (1999)

N

Index

c-bonded chaân model 24? 139, 153k-resolved inverse photoomlssion

spectroscopy IKRIPBSI 154, 195X. approvsmatton 111

Adatom 3) 78- Hs site 170- Ss geometzy 17â- JL site 170Adatom model 174Adiabatic approvlmation l96Adsorption 46- enev 104- groumltl etemeats 171

Ar(100)- zmage states 229Angle resolved photoernlAsion speo-

troscopy IARPBSI 154, 194Antisite 302Arrhenius behaWor I07Atomic position 16Au(110)- driving force for rcconstructîon 51- mlqsing row recoztstruction 135

Bnr.k botd S8, 126B=d structttre 96- prolected 100Band-strudttre enea'pr 112, 123, 134Bethe-salpeter equation (BSE) 255Blocà ftmciion 82Bloe xnnT'n 85, 120Block theorem 82Btmd- coeent 83- heteropolar 83,- polaz'it 8tlj 136Bond-contraeion reluation model

144Bonâ-rotation rtalnvxtion modet 144Born-rppenhehne.r approxqmation

:03 '

C(10O)- hand stauctttre 160- dlnae,r 15:- qumsiparticle bazd structure 223- wave flmctions l61C(111)- anglœrœolved photoelectron spura

197- hxnd structare 154- electrostatic potential 213- surface siates 155- TDB surface 157- trimer recoMraction $81Chxrge-asymmetnr coe/cient 137,

146, 147(Xlargo.fo.nm'tion level 298Chemical potential 47, 66, 69

Bound surface state 218Bravais indke,s 3Bravais lattice 4, 7- cubk 4, 5- he=gonal 6, 14- oblique 14- plahe 14- quadratfc 14- rectang'ular 14- scuare 166- surface 20- three-dl'mensional (3D) 9- twcxx'rnezusional (2D) 7, 19, 30Brielp'nr group 169Bzillouin zone (BZ)- high-syzmnetry line 36- bigh-szmmefay point 36- irreducible 42- projectiozt 37- threcxasmenssonal (3D) 37- twmclirnensional (2D) 34BuaMing 51, 59, 128- amplitude 147, 152- gap opensng 154

338 Iudc'x

- compoud s-lconductor 71- ezem=t 71chornd'c,al trend- strudural pazameters l48Chenlical 'vapor depodtion (CVD) 7%

157Cleavage 2, 138Clustc method 119Cohcsive energy C1, IMCoMcidezme lattice 19Cmmplax Kqnd stracture 220Condactance- aieerential 1:2- relatîve 193Constant-carrent mode l88Comtant-hetght mode 18*, 191Comer imle 176Correlation 203, 209- strong 225, 232Crys'tal slab- Heducible 9, 10- primstive 9Crystal sjrstem- plane 14Cu(100)- lrnxge statœ 220- two-photon pkotpamsp-qson 231

- rnl-ved 167- molemée levels 159- symmctl'ic 140- t -z1 t m g 14 1- twisfmg 163Dime-ruzlxtom- ' faatt (DAS)

mode) 176, 178Dimer-row domru'm wall l76Dimerization 142- nhJu'n boncls l52Domain 23Dynamical mair:ix 274Dyunrnlc.ally screened Coulomb

poteatial 203, 208Dyson equation 122, 202, 2OS

EEective-mass approvlnnrttioa (BMA)227, 260

Binste.in 1- 195Bizzstein relation 106Electron non'-ty 214ElHron cotmt'mg rule (ECR) 14$,

l83Blectron density 110, 202, 211Electron emergy loss 243Electron energy loss spectroscopy

(EELS) 246Electroc trn.nscfer 129Blectron-hole interactiozt 197, 25:, :

263Blectron-hole pair- Hsxnqtonian 257- stai,e 25lEzectron-phonon Mteraction 251Electron-photon inteavtion 195, 251Electronegativity 136Eledzostatic energy 112, 122, l35Electrostatic potctial 211Empirical tigxht-binding method

(ETBM) 83, 88) 115B'knerpr d.ensitjr %m'nA.l'lqm 119BpitaMlzl g'rowth 2, 45 .'

('

Ekujlibrî'tzm crystal shape (ECS) 52,:.57, 59 '

Ewald construction 31 :)Ewald energy 113'Rvabnnge 111) 255, 257Bxchange-correlation eme'rgy 111Exdtation 187, 237Fwxdton 258- binding energy 261, 264- F'renkel 265- surface 262- 'kwo-dlrnensional case 262

Dangling bond 9'tt 100, 109, 126, 140,158, 163, 176, 183, 232

de Broglie 'wavelength ?4, 105Delta-self-consistemt ûeld (1SCF')

method 234Density fnncdonal pertttrbation theory

(DFPT) 275Densîtjr hnnetional theory (DFT)Denst'ty of siyates (DOS) 134- electronic 193- local 190Desorption zlG 105Diezedzic fxpnditm- balk Si 259- intraband contribation 269- inveDe 207- longitudinal 207, 240, 254- rnxrrroscopk 254D 1*' mion 46 , 105- blrm*er 46 107Dimer ,51- asymmetric 163- burlalng t63- fippùy 165- formatzon 158

- Wxnnier-Mott 261, 265

Facet 58, 65Futtinz 59Fermi's Golden Rule 18S, 195, 25lFock operator 2O2Fbrce constaztt model 122Formatioa enera 296S'renlcel udton 262Frozezephonon approack y75bnucbs-lcewer phonon 284Fkll-potential line--m'zed augmented

plane wave (FLAPW) l16Ftmdamentat gap 06, 9â, 150, 21$

cm xqtjlfï)- M trimers 182GaAs(10O)

- ((4x2) 108, 162- hnna s'tructure 162- EELS spedra 247- LEBD 34- phmse ctiagram 76- potertii eneag sudace 1û8- refectaace Anmotropy 241- slab 118- surface structare 69- topmost .&s asmers l43- wave Alnctions l63GxAq(11o)- total enera surface 103- HRMELS spectrttm 285GaN(111)- electrostatic potential 212

I

Ga-P(100)- STM image 168, 192Ge(100)- bands 163Ge(111)- adatom model 173- b=d structme 154, 175- cleavage 139- isomer 154- swface suates 175Generxliqed gradient approvirnn.tion

(GGA) ll1Gib% adsorption Gpation 67Gibbs 9c.e enthipy 46, 66) 69Gibbs pbMe rate 72Gibbs-Duhe,m equatson 47Gram's flmction 120, 202) 205- fzrs't iteration 206- perfect crystaâ 121

Index 339

Ground state 187'Growth 76Growth mode- 'hnrxnv-v.,m der Merve 61, 76- Strausld-Krastaaov 61- Volmer-Weber 61GW approvlmxtion 203, 255

Hûcke,l theory 124- extended 83Half-siab polarizability 243Hnmoaic apprnvlrnxtioa 105, 273Sartree potentM 91, 110, 202Heat of formxtion 71Hellmnnn-F'eynma.u force 104, 113Hellmxnn-Feynman theorem 104Hd/mholtz 1ee enerr 47, 66Heterod'trner 166Hohenbezg-Mohn tEeorem 110Eubbard paramete,r 184, 233Hybrid 91- .+ 94- syz 9:3- sp% 9l: 59- daagling 97, 126: 158Hydrogea problem 228, 260

I1R-N(11O)- band stractme 149- ionization energy', electrou xmnity

216- stnlctaral parameters 146L11-V(110)- combhed photoemt'skqlon aad inverre

photoemission spectra 198- phonon enera 289- strudu'm.l parameters 146Image charge 20gTmnge plane 208, 226Tmnge potemtial 210, 246Trnxge state 226lmpm'ity 3, 300Icdependent pmicles 110, 187Odependemt-pxrtscle Appzovsrnation

19$, 251, zeoy 259Independent-qumsiparticle appzova'rnn-

Eion 254, 258, 2591nP(100)- msved-dimer model 167- pàase cliagram 76- STM imaje 16E- sarface stnzcttzre 69Tn'p(ll())- deposition of As atoms 78

340 hdex

- qloxswarde-le baad skrudure 224- Rxmxn spectra 253- STM 145- surfnz'o phonon dkspeon 286Mb(l1O)- EELS syectra 273Interatom)c force constant 274Interdifusion 46Intemml energy 47, 66, 68Hterstitial 294Iuverse photoemlmqion speciroscopy

(œF-3) 194lon bombardment and n.nmep7lng (1BA)

2, 45lonization enerv 214- semiconductozs 216Islrd 46: 64Isomer 154

Madelunj enera 18Mass xctlon law 72Matrix notation 20Metal surface 18- excbxnge-correlation potentx 210- image states 23l- reconstraction 134- relaxation 133Metal-induced gap states ImGSI 221Metal-orgec chemîcal mpor deposi-

tion IMOCVDI 237Mg met,al- pkotoelectron spectntm 2O0Mllk-stool model 18OMiller indices 3, 4, 53Molemzlaœ beam epitav (MBE) 2, 45,

70Mott-Hubbard msulator 172, 184: 234

Neaarewneigkbor interaction 87> 95,151

Nearly-freGelectron (N5V) approvlm-tion 219

Ni(100)- image s'tates 229- Myleigh phoaon 2.80

Optical absorption- bulk Si 259- vith electron-hole attruion 26lOverhp integral 83, 124

Jxhn-Teller- dbplacement 17, 18t l63- theorem 17, 126JAMAV theozem 123Jellb:m model 208, 2ll

K/M(111)- eleeron enerr loss spectra 271Kiak 3, 294Koha-askam (KS) eigenXues 110Kohn-sham (KS) stata 1l2Koha-shxm equation 110Kohn-sham poteutial 110, 204Kramer's grand poiential 4% 66

Löwdin theorem 83Lattice plane 2, 3) 6, 8Lattice vibration 68, 274 .

Layer-orbital zepreentation 12ô, l22Lebmam.a representation 202LifetMe 187Liuea.r combination of atomic orbitG

(LCAO) 82, 85 '

Linearized muen-tin orbîtal (LMTO)method 116

trocal dtansrr't.g approvlmafion (LDA)71, 109

Local spin density approzmationII.,SDAI 111, 235

Local-ield efects 208, M1, 255, 268Lone paiz 140laow-ene'ra eledron doaction ILERRI

29, 33- mewsmement 146

Pardal pressure '74Peierls eAec't 141, 291Peripheraz atozaic statePkase diagram 74, 80Phonon- eigenvector 287- macroscopic mode 2?7- mëcoscopic mode 27T- surface mode 276Photonmlxqion spectroscopy (PES)

194Piasmon- losses 199- skake-tzp 199- scface 271Pocket 100, 228Polnt defect 295Point group 42- kolohedri 14, 25- plaae l4, 15Polarity 9

Pohrization flmction 207, 208, 2.54,258

- Bloch representation 256- macroscopic 254Polytype 136: 312Potentia,l energy surface IPF,,SI 103- vr'nlmxlm 1O4Potential-enm'> sudace (PES)- saddle poht l07Prlm''tive bmsis vec-tor 7, 29, 33Prlml'tive lattice vector 4, 8Prindple of detailed ba7xnce 46Pseudohydrogen 118Pseudopotential 110Pseudopotential pla'ae wave (PPPW)

method 1l6Pyrxrnldal-cluster mode) l80

Quazielectron 196Quasihole 196Quasipaztide- bazd 22l- energy 204- gap 225- holon 291- peak 199, 2O6- quasielectron 1E7, 257- qumsiàole 187, 257- sMlsl û05- spinon 291- wave 'h'nctio:a 204Quaasiparticle equation 201

Rxman scattering 250- by optical phonons 251- selection rttk 252Random phase approvimn,tion IR,PAI

203, 255ltaqleigh wave 280Remprocf lattice 4, 29, 32Reconstructîtm 16, 19, 24- driving force 135- missing row 17- pairing 17- pzinciples 138- vacancy 140ReEelztacce anisotropy (RA) 243Refectance azdsotropy spectroscopy

(RA.S) 238Refec-tance diference spectroscopy

(1tDS) 240 '

''

Resectiott Mgh-eue'rgy electrondOaction I'R.N%EDI 76

Sagittal plane 278Satellite stnzdme 199, 206Scaxming tllnneling microscopy (STM)

:S8- Ge nanocrystal 63- T n A s pyr a 'mz d 6 3Scattering-tkeoretical approach 120Sclzrödirtger equation- eoccitoas 260- l'mxge states 227- onmcllrnensional 219- siazgle-pardcle 81- two-particle 257Screening 207Seiwatz model 157Self-enera- exchango-correlation 202, 2O4- GW approvlznn.tion 203- near surfaces 210Self-hteraction 274Shuttleworth equation 50Si(1O0)- bud structure 99- reiectance anisotropy spedra 308Si(110)- 1tA spedzum 266Si(111)- baud structure 99, l54- buclding model 128- cleavage l39 '

- a''eerential resectacce spectrum238, 264

- electron e'nergy loss hmction 248- photoelectron spectra l99- quasipazticle b=d stmzcture 223- 1kA. spectzum 262- relative conductance l93- step 310- sTM l'mxge 1zs- smface phonon modes 290- surface states 178SiC(000l)- LEED 35SiC(111)- adatom geometry 172

lnde,x 241

Rehybridization 126, 140ltelaxation 16, 17, 19, 128, 135, 146- outward 133Repeated-slab method 116, 274Rest atom 173Ring structttre 139Roughen''mg 59R''rnpliag 18, 135

342 Hdex

- b=d strudare l72- electrostatic potential 212- pllue diavam 185- slab 117- tetzamer 183- t'wisted Si J'rlrxyer l83Single dangling-bond (SDB) surface

150, 156, 180Slab method 115Shter-Koster pazamete.r 89srnxll poinz voup 42Space roup 10j 25- plane 25Spectral ellipomekry (SE) 238Spectral welght 2O5jpectralt-weight) ftmction 187, 190)

196, 201, 2O6- core-hole 201- Si 206Stnnklng fault 139, 176, 312st,ariang vector 8, 9Star of m.'ve vector 42Step 2, 46, 52, 303- biaemic 306- Eeigsht 293- Dicrofacet aotation 304- monatomic 3O6Steawgraphc trin.nrle 54Strain 51Sudda approvimation 196Superlattzœ 19Supersatuzation 76Surface 4S. 51- entropy 50, 66) 68- excass free energy per tmit area 49- free enezo 51? 55, 68- nonpolar 135- polar 136-. stre 65- stress 50? 65, 134- tension 49Surface dieeautial reiectance (SDR)

237Slxrlhce energy 67, 74 128J- snbonded metals a7- jellium model 56- metals (table) 56- sexniconductors (table) 56Surface loss ftmction 246SlxrTsrm pkn.- 74Surface photoabsorptioa (SPA) 238

Smfaze plaae 13Sudace reconstraction 16Snlrfnce resonxnce state 218Surface rougkue% 3Sudace-state gap 225Symmetry

- point 13- roiational 15- txrazslakiona: 16, 82

Terrace 3, 46, 52Tersol-Hxrnxnn approach 191Tetrakedron dfrect:on 91Tetramer 183Tetramer-xdlnyer model 184Three-layer model 239, 247Tilzee-step model 195, 201Tight-binding method 83, 123, 151,

160, 193Total energy 68, 103, 109, 1.12Traasfer-matrkx method 120Trnndtion-state theozy (TST) 106'Trxrmlational g'roap 13, 19rrrirner 181Triple daugting-bond (TDB) surface

156, 181z'nlnneling curreat 190'nlnneling miœoscope- ideal 190Twin boundazy 313

Ultrahigh vacuztm IUHVI 2, 45Ultraviolct photoemission spectrcvpy

(WS) 194

vv-xncy 3, 294Vacuum level 2l4Vapor pltn.te epitaxy (WE)Vi 'cma1 surface 52, 304Virtual gap states (ViGS) 221

W(100)- reconstruction 134- relnx-xtion 133Wlmnser-Mott exciton 261Wi>er-seitz cell 25) 33Wood notation 20Work ftmcdon 214- typioal metaks 215WUIS comstzuction 57WUIS plot 51