AdvancedTextsin Physics This program of advancedtexts covers a broad spectrumof topic,swhich are of cczOt=demerghgHtereMhphysiO.EaGbookproddesacompzGensive=d introductioato a âeldat the forelont of modernresearch.As such, yet accessible Gesetexts are intendedfor serliozundernduate aadgraduatestnrlents atthe MS and PhDlevekhowever,researc,h scientjstsseeklngan introductionto particalar areas of physicswill also bemeât from thetitles in tlzis collection.
Springer Berïin Heidelberg N'ewYork HongKong London
Milan 17t4r: Tokyo
PhysicsnndAsaonomy
ONLINE LIBMRY
http://www-spsinger-de/pliys/
ProfessorDr- FriedhelmBechstedt Fzieclt-iczz-ssr%a'llc UniversitiitJena Instim:ffr Festkörpertxeoz'ie tmd Theoretiscàe Opdk z Max-wien-?latz Gerrnnny 07745Jena, e-rnnll: beciLtêifto.physilc.uri-lYxde
Ia-brazyof CongrmssCataloging-in-pubticatien Data: sechstedtFriedimlm.Prîndplesof s'Arfnce texts izzphysicsv ISSNz4Jp-zö74) Jndudesblbliphysics/Medolm Becbstedt. p. cm- ,- tAdmaced z. Snrfnces 1.Titlq.11. egtapbicalrieauces andindex-BBN 3-540-:0635-4 (acid-âee paper) (Php2cs) Serieszoo3oh'o6'1.t QClD.4.S94 zool stso-4lz7-dcm
BSN 143:-,.674
ISBN3-540-00635-4Springer-verlag Berl'inHeidelbergNewYork whethe.rthe whole or part of the matedal Tlziswork Lssubjed to cepydght.M right.sace reservedy recitatior,breadk concttrned,swcifcatlythezigiztsof Mnslation,reprintinpreuse ef illustrationsp casting,reproductionon znicroGlmor ia any other myb aad storagein data bnnkx Duplicatioa of tlzispublicationor partstâeaeof is perrnlttedonlyunderthe prodsionsof ttte GermanCopyrightLaw of September py*65, in its mtrrent version,andpermirqionfor use mmt alwap 5e obtainedâom tmdertheGermanCopyrightLaw. Spzinger-versap Viohdonsare liable for prosecudon
BerlinHcidelberg NcwwYork Speger-vcrlag a memberof Bemelsmnnnspringer Sdenceeusiness MediaGmbH
http//www.springecde @sprhzger-ferhg BerlinHeidelberg zoe:
h'inted'm Germany etc. hzt.hspublicadoadeesnot Theuse of genemldesczipdve nnmes, zegistered namesy trademnrksa implykeven in tùeabsence of a speec stateme'nttlzatsuchnaraes are exemptfromtàerelevaatproandthereforefreefor genetxd use. tectivelawsaad regalations bytiœauthorusinga SpdagerT> maco package Typesetrlng: Dataprepared leutersitausea Finallamut FmnkHeawegy GmblfyHeiddberg Cove.rdaslgnt#e.J;> +production Printed on add-fzeepaper SPD:k0855596 57/514:1.*a5 4 3 z I o
Friedhelm Bechstedt
Principles of SurfacePhysics WiG zo7 Figttres
L-
*-z o
Springer
To Andreas, Slzsanne,and UV
M the rivers raa iato the sea; yet the sea is not h111:unto the place from whencethe rivezs come,
thithe,rthey
M1 thina
retlvn again.
full of hbour; mn.n rvmnot utter it: the eye is not satieed WI.II seeing,nor the ear fLIIH with hfu.rlng. are
Ecclezias'te (or,The Preadzea'l
Preface
'
J.nrecent decades,surfaceatd intezfacephysicshas becomeall increasiugly important subdiscipve withic the physicsof condensed matte,r as well as a'a interdisciplico feld betweenphysics, c-z:ystallography,cltemsKtzy,biology, a'admaterials science.There are severaldriving forcu for the developmentof the âeld, among them semiconductor tenBnology,new materials, epitaxy and chemical catalysis. The (alectzlcaland optical properties of nanostructmes basedon diferen.tseMconductorsaze governedby the interfacesor, at least, by the presence of ictezfaces. A microscopic kmdemtandingof the growth procœses zequiresthe investigation of the surface processes at an atomic level. Elementary processeson surfaces,such as adsorptioa ar.d desorption, play a key role in the understandingof heterogeneo)ls catiysis. During the course of the surfacehwestigations, it has bee'npossible to observe a dzomn.tic progzessi'a tile abttit.gto stad.y surfaces of materials in generi, aa.don a microscopic scale in particlzlar. There are t'wo mai'n reasons for this provess. Fcom the expem'mentalpoiat of view it is largely due to the developmentam.dawdlability of aew types of powtkrhtl microscopes.Spectaclzlar advancesiu techniques suck ms srltnns'ngtl:nmeling microscopy now allow us to observeindividual atoms on sarfaces,and to follow tlb.e,irpatlus with a clarity qlnl'rnaginablea few years ago. bnromthe theoretical point of view(orrathez the viewpoint of simulation) progress is related to the wide àvailability of computers and tEe c'lrnmn.ticinczeazeof their power. Todayj early methodological developmentssuc,has dezusityftmctional theory allow a full Trnunttlm-mechanicaltreatment of electronsia materiab. I'a the futm'e, compate,r experiments will be able to simtzlatethe beavior of surfacesaad processE'son s'prTncesat the level of individual atomic corœ and their smzotmdingelectronswith high accmacy and Temarlçable predic-tivepower. TMSenormous progress in surfacesciencehas been documentedi.c rnn.ny excellent boolcson surfacestructmw, surfacepzocesses,theoretkal modeling of surfaces,a'adsurfacesacd interfacesof particular solids Likesemiconductors. Howevea', only very few books try to treat thc subjecttn a lpnifed and compreheasiveway, This holds true in particulaz for the ex-persrnentalaud theoreticalmethodstzsedin surface physicsand, most of a11,fo< the principles an.dcoacepts. Hence7 perceivedthe acsedfor a book dealing wxith.smface physics at the level of aa advancedtextbook. The n'l= hereis to descdbethe
V1I1
Preface
ftmdaentals of the seld and to provide a lamework for the cliscussionof sttrfacephenomenain a siugletext. Examplesof pazticulazsuzface.s of materiaks suG as sczniconductoz's or metals aze only discussedas a means of illus'tzating the fundamentalsor pzinciples. Specialtheoretical or mcperimental methods of suzfacestudies are mentioned but not describedin detail. Particulaz attention ks paid to physical approachesthat c-an be applied to the discoveryand dismlssioaof nove,lsurfacephenomena. Among them are symmetzy azguments, energetics,driving forces and elementsof gcometrick chauges,elementazyexdtationsl and other charadezistic propertics. These elementsshouldlzelpto clazsifysrface pzoblemsand to facilitate their kmderstanding.The only prior knowledgeassumedis tmdewaduate phMcsand mathematics course material. Mp.imlytextbook quantum meianics and geometrical arguments az'e used to discussand desczibesarfacesand surface processes. Graduate-leveltopics such as second qxpxntization are avoideê. Whenevezmacy-body argttments are needed,a bzief (more phenomenological)introduction is Tven.Greeu'sffmctions are intzoducedby using their' relationshipto obsezvablequautities. The use of gzoap titeor.r is restricted to geometricalazgumentsand its notations. Feynmaudiagrnml aze only shown to illustrate interactions betweenpazticlœ on s:Arfnces. An Hended mzbjxt iade,xwill help s'tudeatsand scientixs to use the book for referenceand dmu iag their every-dayscieutisc wozk. To keepformulasto a mnrageablelength, thty are written in tke lamework of cgsuaits. In Hdttion, use is vnnzieof the fad that the energiesof valenceelectronsare of the order of electron volts and atomic distancesare of the order of angstiozns. The book is basedoc lectmesgiven at the Humboldt-universttâtz'u Berlin and the nieth4ch-snhlller-univemRâtJenn.and on s'tudeatsernsnars. I would like to nrlmowledgemaay discussions with colleaguesaround the world, I aksothnunk my cozeaguesazd studentsfor their critical reading of paz'ts of the manuscript. Among others 1 am iadebted to R. Del Sole, N. Esser, J. Fnrkhrniiller, S. Glutsch, P. Kratze, J. Neugebauer,G. Onida, M. Ro1)lfmg, A. Se.himdlrnnzyz, W.G. Sn%ml'dt,azzdJ.-M. Wagner. The typing of the m=uscdpt was achievedwith competencesnd inBnlte patience by my sem'etanr Sylda Hofmnnn. Coordinationand production of the book were tmdertaken by Petra 'mdber and Angeh Lahee 1om SptingerVerlag. .
Jena, Ma.rc.h2003
'
FriedhelmSec/udeff
1.1 t%lodelS =es......................................... 1.1.1 S ace V=c B14IV..........-.......m.......-... 1.1.2 The S ace ms a Physick Objed . . . 1.2 TvmpDimRnql'on/Crys .-.-.-...............-......... 1.2-1 Lattice Planes of BT7lk-Crys-tals . 1.2.2 Om'ented Slabs---..............--..............-. S'arface. Pln.nn.rPom-t Gzoups 1.2.3 Id 1.2.4 R Smfaces:Reconstruction xnd Relaxation -
.
.
. .
.
.
.
. -
. .
.
.
. . .
.
, .
.
.
. . . .
1 1 .
.
.
. . .
. .
.
-
. .
.
. .
. .
.
.
.
.
.
-
. .
.
.
. .
. .
.
.
-
.
.
.
.
,
.
.
. . . . .
.
. .
1.2.5 Superlatticesat S'urfa' ces 1.2.6 XvoodNotatl'oa 1.2.7 S etry t'Xs:l-qx-s6cat-on 1.3 Reciprocf Space 1.3-1 D1rect n.ndRem-pro Lat-tices Zones 1.3-2 zBri'llo-l'lt-n 1.3-3 Projectionof 317Onto 2D Bm-llonain Zones *pro Space :..3.4 S etzy of Points aad Lm-es in .
-
.
. .
.
. .
.
.
20
.................................,
. . .
. . .
. . - . .
. ..
.
. .
, . . .
. -
-
.
.
.
.
. .
. .
.
.
. .
.
.
.
.
-
-
.
. .
- .
-
.
.
. . .
.
.
.
25 29 29 35 37 41
.
.
45 45
. . .
.
46
.
,
46
. .
.
. . .
.
.
.
.
.
.
.......................................
-
2.
-
-
. . . . .
.
. . .
.
.
. .
.
-
. . .
, .
. .
.
. -
-
*cS Thermod Te'cl 2.1 Km-etz-cProcesesnrd S aces m- Equilibrz-' 2.2 ThermodynzmlcRelatio'nqfor S aces 2,2.1 Thermody-aAmicPotentim.lA L 2.2.2 S c.e Moda-qcatz'on of Thermodynxmic Potentin.lq 2.2.3 S ace Tensiona'ndSTq'rface Stress 2.3 E -brixlm Sbnpeof Smnall als 2.3.1 à'nsqotropyof S ace Bnergy 2.2.2 AbsoluteValuesfor S ace Energie.s 2.7.3 'W:t1@ Clonntruction 2.4 S ace Elergy n.md lTorphology 2.4.1 Facett' and itoughe 2.4.2 3D Versu 2D Gro
.
,..............................
.
. .
. . .
. - . .
. .
. .
. .
.
. .
. .
- - - .
.
. .
. .
. .
. .
.
.
.
. . .
.
. .
.
. .
-
.
.
-
.
.
48 49
. .
-
.
.
51
. - .
.
.
51
. . . . .
. .
55
-
.
. . .
'
.
.
. . .
. .
. . . . .
.
.
-
.
. . . .
.
- . .
.
. . . .
.
. .
.
.
.
.
-
-
.
. .
.
.
.
.
. .
-
.
. .
, .
. .
.
-
. . .
.
.
.
.
.
-
.
.
.
.
...-...............-....
'
...............--......
.
........-............,
.-.--.........-...-
.
.
-
1 3 3 8 11 16 (1.9
. . . . .
.
. .
.
-
. .
.
57 59
59 60 63 66 66
-
X
Contents
2-5-2 Appro 2.5.3 Ch
-
ations for SmfaceEnergies
. . . . - -
'
Potenti
2.5.5 St :1'- of A
3.
. - -
-
.
.
.
.
.
67
-
69
.-............................
tes
or
78
............................
Bonaing nmd En etics 3.1 Orbitzs And Bondl-ng 3.1-1 OnoElKtroa Picture ding Jkpproach 3.1.2 Tl' Buaimd Thelr- Tnteraction 3.1-3 Atornl-c Oz-bitni!q 3.1.4 Bonding Hybm'ds 3.1.5 Bonds And Band: 3.2 Danfling Bonds 3.2.1 Forrnatl'oa of D=g Hybrids 3.2.2 Tn6uence o'n Electronic States 3.3 Total Energy an.dAtornl-c Forces 3.3.1 Basic Approvimatiomq 3.3.2 Potential Enargy Suface FmdForces Fkl-on. 3.3.3 S iCe 171-6713 'ty' Descn'ptionof StructlzreFmdStabzh' 3.4 Quantitative 3.4.1 Densi'tyRlnctiona.lTheory 3.4.2 Bnzld-structuren.ndTntezactionCoatm-butionK 3.4.3 lklodeli-ngof Surfaces Bond Brenn-mg:Accompanpn' g ChargeTrlmAfers n.nd.xtornic Displaceanemts 3.5.1 Charaderiqtie Chlmge m- Total Energy Due to Structtzral 3.5.2 Energy Gn.5n a'ndConGglzrational Ch es 3.5-3 Energy Gn.l-nand Electron ''lnrnznqfe.,r
81 81
.-.-.-.-............-..............
-.-.-.-.-.-....-..-..-..-.-..
.....-.....----......
.
.
.
.
. . .
.
. .
, . .
.
. -
. - -
. -
-
. .
.
.
. .
.
.
.
.
.
.
-
.
.
.
. . -
.
.
.
.
.
.................................
....................-...........
........................................
'
'
.
.
.
.
.
.
.
.
.-.
.
.
.
-
.
.
.
.
.
.
.
.
.
. . - - -
. .
.
.
.
.
-
. -
. - .
. . .
. .
. . . - . .
.
.
.
.
-
-
. . . -
-
.
.
.
. . . .
.
. .
.
-
.
.
.
.
-
.
.
. -
. .
.
.
.
.
. . .
-
-
. . - . .
-
.
-
.
-
.
.
.
.
.
.
-
. .
.
. .
. .
.
. . .
.
. .
. .
. . .
.
@. . .
.
.
.
. . .
.
.
.
.
.
.
.
. . . .
. .
. . .
. .
. .
.
-
-
-
. .
.
.
- . .
.
.
.
.
.
. .
.
.
.
.
.
.
.
.
.
.
.
.
.
. .
.
. .
. - -
.
.
. .
.
.
-
. .
. . .
. .
. .
.
.
-
.
.
.
-
.
. .
4.
.
.
. .
-
-
.
.
.
. -
-
. . .
.
.
.
-
.
.
Slements 4.1 Recoctxcty'on acd Bonding 4.1.1 et 'c BôndS..,...................-............ 4.1.2 Strong Ionsc Bonds 1-x- Co ent and Iorsc Bon 4.1.3 4.1.4 P ciples of SemiconductorSurfaceRecomstlmction 4.1.5 Electron Co trc'g Tt es econstructl*on
.............................
.........,,,...-..............-
. . .
. . -
. .
. .
.
.
.
.
.
.
.
'
. . .
--......---....-....
-
4.2.1 Z-)g-z 4.2.2 x-bond
4.2.3
'
-ons
of Cationsand
.
.
.
.
.
.
. .
.
. .
. . .
'
C
-
-
.
.
.
-
-
-
.
-
.
. .
. .
.
. .
. - . -
-
.
-
-
.
-
.
-
-
.
-
.
.
.
.
.
.
. .
.
. .
. . .
.
. . -
-
. -
. . . . -
. . .
. .
. -
.
. . .
.
-
. . -
- -
. . .
. .
. - .
.
.
.
.
*
. .
.
.
.
.
122 122
125 129
135 136
138 141
143 156
.............................,.-...
---................................,.--...*.*....
4.3 Dan-erS c 1)ztnAer s 43 1 S 4.3.2 etri c (Dil4-1 ers 4-3.3 lRetero ers
103 105 109 109 112 114
150
-.....-...-.-.--.-.------..-----*
tZ
1û2
133 133 133
...............,.................
. .
.
81 82 85 89 94 97 S'? 98 102
. . . .
-
-
-
158 158 163
1.66
7(1
nt-nts
4.3.4 Bn'd
'
Groum
168
..---............................
4.4 AdatomK xnd Adclustr ..-....-.-.-.--............-.,.. 4.4.1 bolated Adato=s.......................-........4.4.2 AdatomqAccompnnnsed by Rest Atoms
4.4.3 Adatomq Combined Elements with Other Recozustrudion 4.4.5 Cpetrn.rners . .
. . . . .
. . . . .
.
.
.
.
.
. .
.
.
. , .. -
170 . . . .
.
. -
170 172
, . .
. .
.
.
.
. .
176
-
.
.
.
.
.
. . - . . . .
. . .
. .
. . . . .
Elementary Excs-tations 1: Sl-ngle Electronl-c Quasl-particles , 5.1 Electronq &ad So1%......,.............................. o- 1.1 Excitation acd Qua-sl-p icle Charader a- 1.2 Sc Ahlnn S tr py...............-.. 5-1.3 Photo-mlssionSpectrosco AndTnverse Photo-rrll-qm-on 5.1.4 Satelpltes 5.2 hlnmy-BodyEFects 5.2.1 Quasip icle aatà'on..............,............. 5.2.2 QAlxqipicle Sbsf'tsa'cdSpectral eights 5.2.3 ScreeningENea,r S'urfaces e States 5.3 Quasip icle S 5.3.1 S ace B=I'er.................................-. 5.3.2 Chnracteristi c Energies l-zzation 5-3-3 State Lof-iml 5.3.4 Qusmip icle Bands ar.d Gaps 5.4 Strong Electron Correlation 5.4.1 ïmnageStat&...-----.....-....................... 5.4.2 hlott-TTubbardBxnœ...-....----...*............. .'-
.
. . - -
. . . .
. . . . . .
. . .
-
.
-
. -
. .
-
. , . . .
. .
. . .
.
-
- .
.
.
'
. .
. .
.
.
.
.
.
. . . .
. . . .
.
-
-
. .
. . .
. . .
.
.
.
-
.
.
.
.
.
.
.
.
.
.
.
. .
. . .
.
.
. .
.
.
.
. .
. . . -
. . .
. .
. . .
- -
. .
.
-
.
.
.
- . -
-
.
-
.
.
.
-
.
.
.
. . . . .
. . . . .
.
.
.
. .
.
.
.
.
.
- .
. .
-
-
.
.
. . .
. .
.
. . . . .
. .
.
.
- . . . .
-
Element Excitations Pnx-r nmd CollectezveExcitatl-onq 6.1 Probing S aces by Excitations 6.1-1. Optical Spectroscopl-es 6.1.2 L- t Propa tion in S aces 6.1.3 Eledron Enea'gyLosses n'n Scattelnn6.1.4 1?-,1../-f3 g s: Excitons 6.2 Electron-llole Paarction 6.2.1 Polsm*zati on l-ltoniazl 6.2.2 Two-p icle llE:4.rn
.
.
- . . - .
.
.
. . . .
. . .
- . . . . .
- . .
. . .
. . .
. . . . .. . . . . .
.
.
.
.
-
.
.
. .
. .
- . .
. .
. . , . .
.
-
.
-
. .
'
l87
194
199 2O1
................................-....
.
187 187 188
.......................*......-.-.......
.
183
. . .
.
. .
.
.
.
201 203 2O7 211 211 214 217 221 225 225 231
-.
.
-
.
. . .
.
.
.
. . -
- .
. - -
-
.
.
-
.
.
.
.
.
.
. .
.
.
.
.
.
.
. . .
.
.
-
-
- . .
. , . .
. . . . .
.
.
-
.
.
.
.
. .
. . .
. - .
-
. .
-
.
, . . . .
-
-
.
.
.
-
.
. .
-
. - - .
. . - -
. . . . .
.
. .
. .
.
. .
. .
. .
. . . . .
. -
. , . .
. . . . .
.
.
.
.
. . . . . .
. . .
.
.
. -
.
. .
- .
.
.
.
-
.
.
-
.
-
.
. .
. . . .
. . . .
.
.
.
.
.
.
.
. .
.
.
.
. .
.
. .
- .
. . . .
.
.
.
.
.
-
.
.
.
.
. - . .
. . . .. . . . .
.
.
.
.
-
.
.
-
6.2.3 ExcitonR 6.2.4 SurfaceExdton Bolnnd States 6.2.5 S Glk-iodifledB'I1.k aitoxls 6.3 P1%monK.......-.......----------..................... 6-3-1 TntrabxndExcitationq
. - - .
.
-
........................................
.
.
.
-
. .
.
.
.
. .
.
.
.
.
. .
. . .
. .
.
. . .
. - .
. .
. . .
. .
. . . .-.
.
. . . . .
, . .
237 237 237
24O 243 250 254 254 255 258
. .
.
. .
. . . . .
262 265 269
-
- .
269
.
.
.
-
-
I
Contents 6.3.2 PlnAmaOsc:'1!atl'o=....................,,......... 6.6*6 6 2Cê Od1*6cationK
27û 27S
..........,..................
6.4 PhOLOnE...........-..-.-......v..........,.....-...... 6.4.1 Hxrmont-cLattice D nzni/w-q. 6.4.2 S ace a'ndBlllk Nlodes . . . ylei gh ave s 643 . . . . . . 6.4,4 clnq-KliewerPhonons . . . . . . 6.4.5 Tnfluence of RelaxatiozzzmdReconmtmzction 6.5 Element citations for R uced Dl-znenqion. . . - -
.
.
. .
-
.
.
.
.
.
.
.
-
.
.
-
. - -
.
.
.
.
.
-
-
. . . . .
. . -
.
.
.
. .
.
.
.
-
.
.
.
.
.
.
.
.
.
.
. . .
-
. . .
.
.
. - .
.
.
.
.
. .
.
. - . .
.
.
.
. .
. . . .
.
.
.
.
.
.
278 281 285
290
.
.
, . . . .
.
-
-
.
.
-
-
. .
. . .
. . . . .
.
.
.
.
.
2*f3 273
. .
276
7. -
tl-c a'ad Ideal Surfaces 7.1 U.2 P0Ul't Defects .........................,............,... m'es ,......................,............... 7.2.1 V 7.2.2 pun't'es ...--...........................---.... */.2.3 Y-m-tes.............--..a.o......-.............. .
- . .
. .
. .
.
-
'.
-
.
.
- .
. .
.
.
.
2S3 294 294
3û0
7.3 Lu1'e Defeds'.Steps 7.3.1 Geoznet and 'NotatT-on 7.3.2 Stepsoa Si(10û) S aces S s 7.3.3 Stepso:a Si(111) D : S 7.4 X V1 Y 7.4.1 Defed, ReconstrtzctionElementor B 7.4.2 Si on Si(111)3x 3-.B
302 303
.....................................
.
. . .
. -
. .
.
.
.
.
.
. . . -
.
. . .
-
- .
- . .
.
-
-
-
-
. .
.
.
.
.
.
-
. . . -
-
.
-
. . .
. .
.
.
.
.
.
.
.
.
.
. . . . .
.
.
.
.
.
-
-
-
.
.
.
.
.
.
.
.
. . -
*
*............>....-.......
.
.
.
. .
.
.
-
.
- . . .
Propc .
.
.
. .
. . .
? .
- . -
3O3 306 310 312 312 313
1. Symmetry
1.1
Vodel Smfaces
1.1.1 surface vez'sus Bulk
'
.
Evcry re,alsoEdis boundedby surfaces.Nonetheless,the model of ac in6n7'te solid which neglect,sthe presence of sucfees works very welt in the casq of m=y physical properties. TNe remsoa is, Astly, that one usually deals with properties, suc,has traztsport, optical, magnetic, menhxnscalor thermaz properties, to wllic.h all the atoms of the soliclcontribute more or 1- to the same extent, and, seconiy, that there are many more atoms f.n the buik of a solid samplethan at its surfRe, prokided t'he solid is of macrœcopic size. Irz the case of a srdiconcube of 1 =3, for e-x=ple, one has 5 x 1022bulk atolns and 4 x 1015surfaceatoms. The surface atozcs are only visible in smfacesensitive experimentaltecy niques or by studying propertio or processeswif.ch are determinedby sttrface atoms only. Among them ate phenomenalike crystal growth, adsorption,oxjdation, etching or catalysis. They cnmnot be describedby the mode) of aa l'nflnlte solid. However, there aze also eseds wlf.c,hare detnrminedby the For interplay of bu.lk and surface (or,more stridly spexlaang)the hterface). i'astance, the clmnne-lof the cnrrler transport in âeld-efect traaetors is dcutermnenedby the slxrfnœ (interface) staœ ws well ms tb.e bulk dopbzg.H one appliedthe of tlze ftrst theoretical approaees to the feld esect,Bardeen (1.1) prtamiseof chargeneutrality at the sT:vhces/intcrfaces. TMs conditiön meaus that in thermal equilibzium the surfacebard bending adjustsin such a way that the net chargein s'nrfnne states is balancedby a space nbxrge below tahe suxfve of the semlconductorfomningthe maH paz't of 'the electzicaldevice. 1.1.2 The Sttrface
as a
Physical
Object
Under normAl conditions,i.e., atmosphericpressurean.droom temperatuke, the zeal smfaceSf a solid is far removedfrom the ideal sylems desirablein physical Gvestigations. A freshly prepared smface of a material Ls normally of vezy readive toward atoms a'ad.moleculesîn the enviro=ent. M kn'mds particle adsorptioa from strong chnmt'sorptionto weak physisorption give rise to an adlayeron the topmist atomic layers of the solid. Oneevxmple -
-
:. Symmetry
is the l'rnrnediateformation of an extremely thin otde layer o:a a frebhzy deaved silicon crystal. Usually the chemicazcomposition aud the geometrëcal strudm-e of such a contmsnation adlayerare not well defned. A.saz objectof physical izwestigations a welydeMed soace kas to 5e preparedon a particular soEd,in a special preparâtion procces, under wem deG'ned extez'nalconditions.Such a solid colzldbe a czysinlli'ne material, a singlecrystal or a mystplline filrn depositedby epitax'y in a well-controlled of suG a crystnlline sygtemmight alsobe prepared my. A rathe,rcleaaslxrfar..ta as an electrodesurfazein an elec-trrïz-ynn,l'ci cell, or a somiconductorsnHnce iu a reactor where vapor phase epîtaxy (V.PB) Ls performed at s'tandard prvttre conditions and at elevated temperature. Howeverzthe processesof the tmderlyingmethodsand the results are l'athe.r comple,xacd dsocult to characterize.The simplestways to prepare a sozd surfaceshotlld happea in altrœgh vacuum (UHV1, i.e.) at ambient pressurelower than 10-S Pa (about 10-10 to=l. There are essentiallythree ways to mauufacture cleaa suzfaces tmder UHV conditiozzs: .
-
i. Cleamgeof brittle materials in UHV. Of course, only snlrênreswhicz are cleavageplanes of the czystal nnn be made in this way. ii. Treatmeat of imperfect and contnmsnatedsurface of arbitrary orieatation by ion bombardment and thermal n.nnenling (YA), generallyin severalcycles.There are no lsrnstationsio certain materials and to certain tallographicorientations. 1. Epitaxial growth of crystal layers (oroverlayers) by mexns of emporation or molecular beam epitax'y (MRE). '
Obvioasly,a smooth and cleansurfacenn.nnot be realized i.c the ideal form, bui rathe,r oxulyto some approxsmation. My real smface wi:tl exhibit irregl'lnr deviations from perfed smoothnessand pttrity despitetEe care taken Snits preparation. A.zl illustration of such a surface is given in Fig. 1.1. J.n reality a surface conskstsof a nltmbez of irregular portions of parallel sum facelattice planes whic,hare displacedvertinn.lly by one or more lattice plaae sepratlons 'VM respec't to ev% other. Atomic steps TERRACE
xjxx
occlzr
at the botmd-
STEP
ADATOM
STEP-ADATOM
VACANGY
Jllustratîon of structural imperfeotiomsof crystal sudaces. Atoms aï.d thekrelcctronshel)sare indicated by little cubu.
Fip. 1.1.
1.2 Two-DimensionatCzystals
3
ade'sof thesc httice-plane portions whic,hin this coatext are.calledterraa. The steps may exhibit lrsnk-q.1.naddition to teztaces, steps and ksnkq, other stmmtmal irregularities may ocmzr wiziclzcaa be substtmedunder the temn isurface roughness'.Adatoms aud vacancies bdong to tMs categozy, as do complexœof thœe m'rnpleddects. ln the case of slzrfnrzxsof compotmdezystals qaite often atoms of one of the contributiag elementsare dcpleted more thn.n those of the other which ruults in an emrinbrnentof the latte,r and in a noa-stoiYometzy at the sllrfnre. The most signiâcaut fo= of chemical disttlrbauceof surfaces)whic,happliesboth to compotmdaad elementalcnrstals, is the contaminxtioa by impuritics or adatoms of anothes spœies. The impurity atoms or adatoms may be situated at regular or nomegular sites of the surfacelattice plane: at locations above and slightly below'it.
1.2 Two-Dimensional
'
Cl-ys'tnlA
A completeGaradezization of a solid surfacerequires Howledgeof not only atoms of Ewhatspecies'aze present but lwhere' they are. Just as in the bttk it Lsnot that the atoec coordinatesas suchare of pucb direct interest. Rather, besidesthe clmmicalnatme of the atoms their geometricalarrangememtgoverns tke electronic; magnetichoptical, and otNe.rpropeztiesof surfaees. 1.2.1 Lattice Plnnes of BI:11zCrystals
A geometrical constzucdonwhich Lsof particular signifcxnce in describiug czystal surfacesis that of a lattia pltme. Ipatticeplan.e,sare uslpnl'y deuoted by MCJerindicos (W) where h, k, î aie the iutege,rredpzoca)axis intezvals given by the i'atersectiomsof the lattice planeswith the three crjrstallographic axes. They have a simple men.nl'ng izzthe czaseof zectangalarcrystal systems, for example,denoteslattice plaau e.g., the cubk system. The smbol (100), perpendicularto the cubic a-axis, (111) means lattice planespezzlealc'tictllnr to the body diagonaliu the ftrst odant of the cubic unit cell, aud (110) denotes the lattice plan.e.sperpendiclzlar to the face diagonal izl the fzst (padran.t of the o-plane of the cubic lmii cell. Ustlnlly, the collection of suchplanœ t'hat are eqttivale'nt by symmetry is labelH (à,k!J. 'lnhlzs(100J ld>nds for the collection(100), aad (OOï), if these planes are (010),(001) (1.00)r (010)) equivalent. The bar aotatlon ï ldicates the correspondiugnegstive coeëciext. J.nthe case of trigonal and hexagonallatiic%, four crys-tazlographic &es are conside-red, tlzree bxsteadof two perpendicular to the c-A='q. The lattiœ instcad of three- The &st plres are then cltazacterizedby fom indices (h,kïI) three, However,are not independentof eackother. Iu fact h + k + ï = 0. The fourth n='q (correspondbzg to t;e iude,x2)is perpenclicular to the he-xagonat aze sometimestermed Bvavaisï'rlJces. basalplatae.The Lhkélj A particular geometzical plane 'can also be characterized by its normal diredion
ê. Symmetr,g
4 n
.;V1v5,1.
=
J.nthe ca-se of latticeplanesit is convenien.tto relateit to a lincar combination 1
'jg gez+ k%+ !;a)
6, =
(1.1)
of the primitive vectors b.f (/ = 1, 2, 3) of the reciprocal httice with the iuteger coescients h., k, a'n.dJ. The vectoz's bj are directly related to the prlrnl'tive lattice vectors aé (ï = 1, 2, 3)by the relation (zç . '
.
bj
=
zmAj.
(1.2)
Apart Fom the cmxe of priml'tive Bravaislattices) they are Hq'Ferentfrom the crystallograpilic axes. M:yway, a lattiee plane can be charaderized by the Miller indices(i,kJ) anGhence,a normal parallc) to the vedor Ghkt = hbï + k%+ Iba of the reciprocallattice. Eowever,as a con
Rï
=
'T-lnlsc,s H1
(1-3)
1.2 TwoeimensionazCrystals *
(a)
*
*
*
* *
wc
OOO .,,, o o o Eiccl
J
OQOOO 00000 00000
jnrl
O grfjo (ir()l O !..72:1 ()1I:I
p107
OOO O O OOO ooooo
00 OQ
OOO 00 OOO 00 OOO
000C4
O
00
(a)Cubic Bravaislattices x, fcc, IXG (b)low-indexplxnes (1K),(110), (111)in a sc cell; and (c) low-inde,xplaxtl.sresulting 1om cabic httce. Bravais lattice points are indicated as dots (+b)or sphmv (c). ng- 1-2.
6
1. Sylnnàetry Z
1 (0t'o1) ! 1
(odolq C
('î01'0) (011G)
('1XO0)
p'I2o) Xc 1 l'ooll (-1 (21.:ib) Elol'o) .J
:214
E()-I l'oj----->
y
----
.-4
X
Fig. 1-3- Charadezisticphnes in a àevxgonalBravais lattice. Certain dlrvtions h thsshtlice are alsoindicated.The vectors œz, .2 (oraa),amdc r-'m be identifv , and (0001q with the priml-tive Bravais httice vectors- The directiou (2E1Oq , (01E0! the XexagonalCartesiancooroate system. represez/ 1, 2, 3).The index I characterizesthe n'nfsnl'te6xrn51yof paratle,lplanesin a certatn ctistancefrom OC.IAotller. The plane l = 0, which contnsnRthe zero poet, may afterwardsbe identisedwith the snrfn'ceof senzil'n6nlte space. This is demonstratedin Pig. 1.2 for the low-indexsttrfaces'ofsc, fcc, ard bcc Bra>is lattices (ormonatomic metals czystalliziugwitlnt'n tllese In the ca% of a hexagonalBravakshttice suchlattice planes az'e stnztztavl. indicated in Fig. 1.3. In pradice, it is oaly izl the descriptioa of non-cubic czystals that one mu.st rememberthat the MflTer indices are the coordsnntes of the normn.l tu a system given by the viprocal lattice, ratxer tha.utàe direct yttice.'For tlzat remson, sometima for simplicit.y tzheVller lndices (hàït)are also tusedto nhnradezize the normal diredions as done in Fig. 1.3.
with integezs as
(ï
=
1.2 ToDimensional (717%*
7
à63);kt3b));qj,.'''''''''' '
''/
I I
$
I I 1
1 1
= Squlre (J$1 IJ2I Y= 90*
J1
i !
I
Y
l
I
!:
1
Rectangular IëhIXIJ: . 'r = 9D1 /2
j'
%
Y'
X$
'
-
*<
E :
XY
qy
---arau-. : 1
centeredredangular
# lJa(, cas '/ = It7a(/(2 1411 E-ct) #' #'
J'
#
'
a'1
4
?
:
z
l
z
ê
l z l
Jh l
HexagcnalIR1= 'f = 120*
1L1
J
l
.xrIa2I oxkue 1,$1 y'Rr Fsg. 1.4. The dve t'wo-dimensionnl Bravais latticœ. Besidespra-mstivellnt-t cells (dwshed linos)ts shown. lineslalsoa non-prirnitive ceE (dotted
Suc,ha l
=
0 plane represents a.two-dimensional Bzavaislattice
2
R
rmaç
=
ï=1
wîth lz azd az az the prsrnstivebasisvedors of tlzis lattice a'ad integer nn'=bers ru and mz. The three vectors 4z, 4a, zz fo= a, zlght-handcooreate rne possibleîve t'qodimensional Bravais lattices of tNe four planar crystal systems are representedi'a bnig.1.4. Apart from the redangalar case, they are prlrnitive (p).Iu the c%tered (c)rect ar cue additionally the non-prlmitive ce2 is also iadicated. h pzactice one often uses the non-priuzitive lattice for the convezzience of description. Sometimc, aksoaonprlrnstive centeredsquare me.shes are USGIin order to keep a cez'tainorientav tion of the tmit cell. .
8
1. Symmetry
1.2.2 Oriented Slabs
Accoran'ng to (1.4) all lattice pla'aesin an a'rbitrar.ghxlfspace (! = 0, -l, -2, or crystal (2= 0, +1, éc2,...)may be describedby
.&; =
...)
J''lmiö,z + ldst $=1
where J,a is
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)bulk a
lattice of the uzderlying m'ystal. The vedor Ja ca'a be deterrninedfrom the Diophantine equation 'FzAz = 1 with expressions (1.1)a'nd (1.3), as long as the vectom &1, /2 satisf.yfk & = 0 (1,2j. The choiceof la is not :tnique, of cotzrse, a'ndary vector aj which dfezs fzom aa by a vector withT'n the lattice plane can also be used. We call 4a the stackingwedor becauseit deterrnî'nes how the chosenlattice pln.nes a-re stackedin the cozlsideredBravais lattice. Foz t'wo Bravais lattices Table 1.1 showsa possible dloicc of thc vedors &:, &2, ar.d 5,a. The vectors :,: and ac span the lattice planesshown izt Figs. 1.2 azld 1.3. The selectionof the titzee vectors J,z, 42, and ö.z showsthat the primstive cell of a Bravaislattice may be ehosenas a parallelepipedvith one of its pairs of pazallelfaeesbeiug parallel to a given lattice p7stne.This implîes that sach a lattice may be c'hnracterized as consistin.gof pa-ratlellattice places which are displacedwith rœpect to enzq other as i'adicated in Fig. 1.5. ,
.
Table 1.1- Possibleprsrnitive lattkc vectors of a plane and stanlclngvectoxs for certtu'n plxne orieutationsizztke case of two Bravaislattices. Cubic(a:)and hexagonal (a,c)lattice constan'tsare used.
3D Bravaislattice
Plane
217Bravaislattice
fcc
(111) hexagonal
aa
-
tzz
oa
(110)
Dreotangular
fzl
-
c,z
tz1 + c,c
(100)
nsquare
c.a
-
ac
/,1
ag
a;
tu
0,3
tzt
tzs
c,1
d,z
ë,z
-
az a2
c.z
zzz = Ma al
=
as
=
(0?1, 1) M.z 0, (1, 1) Ma 1, 1, ( 0)
hpvngonal tz1
=
a2
=
as
=
(0001)huagonal
c(1,0, 0) ! (-1,Vi, 0) (10ï0) Dzectangulaz c(0,0, 1) (1:tJO)Drectangular
cu
-
c,z
-
az
c,z
1.2 rrwo-Dimensional Crystab
n
12 a1
Fig- 1-5. Construclion of a 3D Bravais lattice from its lattice planes.
Usuallya crystal possessesan atomic basiswith S atoras at the positions
(.s
1, ..., S) ia the uzuit cell. k'a correspondence vith the lattîce planes, atomic phnes may be constructed. The lattice plane & caa be considered to be occupied with atoms of speciœ 1, e.g., at rz = 0. The next atomic plane, ctisplacedby .2
|
=
'
10
1.. Symmetry
(a)
Zinc blende
(111)
w
*
*
*
% *
*
* 1 *
%p
œ
*
*
% N
*
FJ =
*
*
o
-1()ll E1
(110)
1:
1
$
4
$ $
4 $
$
i
IW *1
(111)
(100)
(110)
@ IA
@ IA
@ IA
O 28 * 3A
C)28 * 3A
C IB
o 4B
o
.
o
48
*
M
O
28
5A 68
F$()T Fsg. 1.6.
(a)Top view of izzeduciblemystal slabswith
certain orientations 'n, for zino-bleade cgstals.The atoms i'a filserent layers a're indicated by cln'Fe-reutsizes. Dashedlines mdicate a possible217tmit cell. The size of the Slled and open circles iadicates the layer bemeaththe stlrfnne. It ks related to the layer index -J in. the legend.The 6:1ing of the circles de-scribœthe cation or anion character of tEe conwpondjng atom. Mer (1.22.
1, 1),ao (l, 1, 0),ard c0(1)
(1,0, 0).For a t'w>atomic Eesxagonal cryl-talwîth suc,hslabs contain four for (0001), tllre for (1120), aad co
wurtzite strudum atmrni'c layezs.Projedions of suc,hirredudble s1,.:1:8s fom for (1010) of twoatomic crystals aœemœentedin Pig. 1.6. Crys'talexamplœfrom two Bravais 'and ronGlt, hexagonal systeznsare plottu: fcc with zinc blende (diamond) with 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 the numbe,rI of the primitive crystal slab, the number s of the atomic sublattice atd the hteger coordinatesrru, t'nq of a point ia the 2D Bravais lattice as '
1.2 Twfoimeasional Crystals
(b)
Rocksalt
(-100)
(111)
t'
%'
O
o
Q % * '% % o h
x
%
o
%
k &
*
o
1 1 6
Q +
D
<>
1
I I
I
1
@
l $
I
1
o
r
@
o
-1o'J g1
*
& =
4Q1 go
(110)
:
Il
II
lI
(111) (100)
(110)
@IA O 28
@ IA O IB
@ IA
=
@ 3A
*
M
*
M
&
o
4: 5A 6B
o
28
o
28
S
.-
.
o
F101 Pig. 1.6.
1
œ
1
r pRt
t)
+
œ
O IB
(b)Samems Fig. 1.6a bat for rocksaltcrystats. c
Rls,J)mt, ??zcl =
+ n. mk%.+ Jë,:$
c--cz
The complete set of atoMc sites in an irtGrtîte 3D crystal can be obteed by assigning all possibleintege.rvalztes9om -x tzl +x for !, ml? zna aad all possible values s = 1, ..., S. T:e aormal direction rz, 'apoa whic,h the construction of the lattsce planesis 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 aboverepresentation (1.6)of aa lnflnite czystalcan irnmnediately be employed i.n desczibicga C'I'yS:,aI with an ideal surfme.and normnl zz, i.e., a bltlfkpace. Such a srtem may be gencrated9om an in6rlste crystal by removing azl atozniclayers above the suzfacea'adretaining those below. The
1. Symmetzy
(c)
WuAite
(1110)
(0001) *.
+
*
*
1
%
%
%
*
x
%
% %.
.
o
&.
e
'*
I
44 %.
.. .. .. .. - -.
4
.
@
r'n & $vIw
:
,'
t 1 1
l
r-
& D
f'C 't
*
o
F1ooél
Eal-i'o!
(1OT0) (n001) (##Jo) (-î0h-0) 1
I
I
I
rD =c O c L-Z
Ff2C-b!
T'Cg-1-6.
(c)Sxrnems Fig. 1.6a but
@IA
@ IA
* IA
O 28
O IB
O CB
.
3A
@
M
@ 2A
*
48
o
28
(7
28
(4
3A
1
3A
*
38
@
3B
.
4A
o
4B
for wurtdte crystals.
atoxnic laye.rr' epresemtsthe s-urfnceor, at least, the upperrnost atomii layer of the sazrfaee reon of the resulting bx.lfqpace.Eow mrmy atomic layers are countedto belongto the sllrfnce rcgion dependson the method USCXII to ia
of the exdting and/or detKted pnrticles. Jn a 61rs-0 approe one may nv:me thsttthe atoms in the uppermcst atomic hyers, in pazticularin the topmost hyer, kcep the atomic posîtions of the im6nite cr-gsvtal. Such a czlnfgarationis usually tnmnedan ideol dtzoceTlzeatoms of a crystal having an ideat surfaceare thus located at the positions
1.2 TwccimensionalCrystals
13
However,only sites belowthe stufaceplace are A(a,J,ml, rzalgfvenby (1.6). occupied..Thue obey the relation 'h,.A(s,I,rk,
zral
=
. r. L+ 9,6, :< 0
(1.7)
with ii- given iu expression (1.1). The snrfcce oz Ast atomic layer is obtaincd if the ief-t-haudside of tMs relatiou is taken to be zero. A possible solution of (1.7)is l = 0 and 'rso = 0, so long as tke site of one atom sz of the atomic basjs is identifecl with a BraYs lattke point. Tkus, the &st atomic laye.r correspondsto the-particula,r lattice plane perpendictllar to the noz'malA, whichgoesthrotzgllzea'o au.d.whox lattice points are occupiedby basisatoms of speciessz. There msy be other vedors ra beside rso which) although not being zero them-lvem have a zero projection a rs. Tâemthe bVs atoms of this speciesa are also located in the flrst atozaiclayer.They are displRed 'Mi,II respec'tto the atoms of speciessc by a veor ra paratlel to tb.e sttrfve. Suc,kmGtiple-spectesoccupancy of a'a atomic layer occm's, for instance; in snx4hcesof zinc-blende-type the cmse of (110) I11this ca, two atoms a catioa and = aaion occtzr in eachprimitive uzlit cell of suc,ha 25 drjrstal. The (110) surfaceformq a non-polar facebecaux of nlnnrgeaeatralits wlzich is one of the reasons why the (110) pl=e represents the cleavageface of zinc-blendecrystals. The resultiag hnlêqpacewitk an ideal, bullc-terrnsnnn.ted surface (oreven a rea! stkrfaceas dîscmssed nvhlbits not only a 2D periodic'i'tyor, more below) precisey'a 2D translatioaalsymmetry with the prlrnitive basisvectors ë,1and ë,cbut alsoapoint sgmmett'y.As a cozlsequence of the trxnqlational symmetzy accordingto the 2D BTavaislattice points R (1.4), physicallyequivalentspace points ca'a be zelatedby .
.
-
-
m/
=
(sI.K:z)
=
ê;c + R
=
z
+ A,
(1.8)
â denotesthe transformationmatrh characterizi!lgthe elements. SuG whea.e points are displacedagxînKt ench other by a Bravais lattice vector .4t. No rotatioa or rededtionîs involved. This is indicated by the unit elemfmt a and the uait matrix J, respectlvely. A11elements whic,h balong to a certain tranalaional prtlu, are abbreviatedby (c1.K.However,ic addtion there can be poktgroup operations.tajo) wlzic,bu alsorelate physicAllyequivalentspace poiats œ? azd œ. The point srametry dements tz are necessarilyrotations abou.t axes whic,hare parallel to the normal Tz, and rezedions at lines withizt the surface or cell planes perpendicttlarto 'n.. Only n = 112, 3, 4, or 6-fo1d rotation axœ perpendicalarto the snzrfKemay occur. Corrcxspondingly, the smbol Jr sigoes a rotatioû around the surface norrnn.l directioa by the angle(360 mjnlowith zn. = 0, 1, ,.., zz 1. rrhe mirror planesaa also normal to the surface.Jnversioncenters) rnsrror planesacd rotation %e,s parallel to the sllrfazeare aot allowed,since they refer to pohts outMdethe surface.The pbssiblereEection lines m., z%, rzz, rza, md, ?pJ, aze specïed p%,an.dzp,â in Fig. 1.7. -
1. Symmetzy d'
mx
rn 1
I 1
1
1 I 1 1 I
,
1
I
1
t
m2
1 I 1
)
1 I 1 I
I , I j
1 I
X
mY
1 1 1 p
I
I 1 I j i ,
1 1
rn#2
4 1
ml
d
m'$
Fig. 1.7. Dmotation of refection linœ.
By combl'nlmgthe limsted nnrnhez of allowed symmetry operationsj one obtalms10 dilerent planemïrzï gronps. In the ictmvnxtional system (SczoenThe they aze denotedby either n (Ck)or nm, nmm (Ckv). =es system) zmmerals = 1, 2, 3) 4, 6 denotesrotations by Y aad the smbol m refers to reqectionsia a mirror plane. The third smbol m indicates that a combination of the meceding two operations generatœ a new mirroz plane-The 1B 3m (Gv),4 point groups are 1 (Q), 2 (Ca)) m (Qv),z'm,zn(Qv),3 (Cz), (C4),4mm (C4r),6 (C6),6mm (Qv).They are geometrically represented i.a Fig. 1.8. The plaaeBravaislattices presentedia Fig, 1.4 alsopossesspoint symmetrie-s.Howevar,tlze possiblemzzltipicitiesof a rotatîon spnmetry aadsof pMe latticœ are restrided to '?z= 2,4 aad 6. A lattiœ which only cxmtnsnqa Ffold symmetry a'ds is eithez an oblique lattice or a zectacgular one (independeut The poin.t voups of these lattkes aze 2 (tQ)a'n.d of the .p- or o-characterl. 2mm (Cav), respectively. Quadratic lattices with a çfold symmetz'y nn'q poss%s fotzr reûection line,swhic,lza're rotated through 45? with respect to eaciz other. The point group of S'U?,h a lattice is theefore 4z?zm(Qv) Eeagonal lattices with a Gfold a'ds have six re:ection lices whic,hmeet at an angle of there are thus 800.Ia this c:se the point voup is 6mm (Gk).Sh'rnrnarizîng, foar zlsFerent plane crystal svsfems the obDquewith the holohzdralmin,t ptHzztgro'up 2mm, the quadratic gro'up2, the rectangular wjth the ltoLoltedrat poïzztgro'up4mr?z,a'ndthe hexagonalwith the holohedrai with the hokohedral lattices (cf. pofnf gronp 6mzruThœecrystal systaq contain îve 217Brawaâs Fig. 1.4) The low-kfill/r-indu sndhtvtq of fKe-centeredcabic and body-cemtered cubic metal costals nvbîbit such lligh poiut-voup symmetries becauseof their stractuzal simplidty. As iadicated in Fig. 1.21these sttrfatestend to '
.
-
.
.-
1.2 TwceimensioaalCzystals
*
15
t --q(-Y -((j((( m
2mm
L
3m
4mm
6
6mm
Fig- 1.8. Sclmrnntcreprto-tation of the 10 planèpoiat groups.
havethe highestdegreeof symmetzy and tEe smalHt uait cells.Exmples aze aad fcc(100), which have tkreefold, twofold, aûd fourfold fcctlll), fcc(110), rotational syxnmetrs r-edjvely. Othe.rA'mmplœa're bcct4llljbcc(110): and bcc(100), which have threefold.,'twofold, and fourfbld rotationi s.,metry, respectively.Isolated (111) plaaes even possessa higker rotatiozM symmetry. Of comse, all thesesurface,s aksohave rnirror phaes in (sidold.) additioa to the rotation axœ.
15
1. Smmetzy
1.2.4 Real Surfaces: Recozustruction aud Relnwtion
The 2D trn.nqlationalsymmetries of ideal surfacesand halfspacesM'KII bulk by the primitive Bravais vectors az and a2. atoMc positions are aharactezszed J.naddition to potat and line ddeds (cf.Fig. 1.1), on a real smfaceof a crystal thcre yre otb.erreasons that the msshlmptionof an ideal surfaceis not valid î:agen'eral. Sucha pictm'e doesnot Allly accotmt for the bondingbehaviorof the atoms in a crystal. Sincethe forcœ acting on atoms sila'xted beneathan atomic pla'aein an r'o6nstccrystal are partially due to tke atoms located above the plane, one c=, in general, expect that the forces azting on atoms i'a a czystalwitlz a surfaceshottldri-lFe.rh'om tboseacti'agi'a a,û im6mitecmtat. The deviation fzom the ln6n'lte case, however,dsrnl'ninheswith incremsiagdiexnce of the atoms 1om the surface.Onenan thas assllme t'hat the forcesactin.g on, and hence the posîtion of, atoms deep Mside the custal bulk are, to a the same as those in au r'm6nr'teczystal.TMs is, Eowever, goodapproximathon, not true for atoms rauated near the sl:rfn.ce. The forcœ acting on tEem are appreciably deerent, resulting in displacementsSRLS, J,znt, mzlof atomic positions RLs, J,mz, mz)(1.6)vith respect to those of the in6nite crystal. Constsquently, the equilibri:nrnconditions for surfaceatoms are moeed vith respect to the sninste C.IFS-taI. Onee'xmectsaltered atomic positioas !, Tp't, Tn,cl 1)rnq, =2), + SRLS, R'Ls,1,nklrn'z) A(,9, =
(1.9)
vith
JA(s,1,rzu, mzl->
'
0 for l
-h -x,
(1.10)
aud a smfaceatomic structure that usually does not agree with that of the btfk. Thus a shprfsceîs not merely a tnlmcation of the balk of a crystal. The distortion of the ideal bulk-like a'tomic conâguzationdue to the exisprcdsely,the non-existence of formerly neighboring tence of a surface (more dependsoa thc bonclingbehndor of the materii conatoms in the vamzttml such as diamond) Si, Ge? sidered..In tetrahedrallybondedsemiconductors, Gn,Aq,1aP, GaN, etc., strong directional bonds are present, The brtmldng of bonds due to the gcneratkonof the sarfaceis expecied to llav: dramatic eseds. Syste-mswith dangMg bonds shotlld be in general tmstable,siuce rebondin g usu ally lowers the tota' l enerr of tkte bxlRpace. Sometimes , this process is acEompeedby briuging surface atoms closer together. Ozleof these me%smisms rtalltiztg in pairs of suzfaceatoms is schematically indicatedia Fig. 1.9mHowevc,sucka reazrangementexn nlltn yield roul surface layers, the stoiœometzy of whic,his changedwjth reoed to the ideal surfnr.e (see Fig. 1.9b). Lu b0th cmses, the 2D Bravais lattice of the sm'faceis cizmlged.Suc.hperttzrbations detroying the translational symmetzy of the 'reztvgsfnsctï/= Editious idcal nurface are known as snrjace atomic rear'nere are general argttments for such symmeky-bremxlring raagements. One is basedoa the impossibilityof âegenerategrotmd state,s
1.2 ToDjmezssionalCzystals Reconstrudion a)
b)
00 00 00 QQQOOO 000000
O O O O O C) O
000000
O O O C O O O O O O Q () O O
Relaation
c)
OOOOQO
000000
000000 000000 Fig. 1.9. Schematicillustrations of atoznic rearrangememtstn the s'4rfnrtez'egion. (a)Pairhg retxlns-tzazdiiop(b)m$e row recons-ction; and (c)rllxvation of the uppermost atomic layer.
of tbe system. Sucha degeneracyzaay occtm foz iustance, for (111) surfaces of diamond-stmzctmeczystats.In. tEeir case, on average there is oaly oae electron for eac'hclanglinpbond orbital pazallel to the (111) surface normal) althougk eac,horbital can accomodatetwo electronsof opposite spius. The system gwund state eAn thus be romlizM i'a numerous ways by plnning two clectronsizl two orbitis, two iu one or aa equal distzibationov< tEe dangling boads. However, the arrazgememt of these orbitals may rlsfer. Accordingto the well-knownJ'tzhn--Teller titeoremspontreoas spnmetry brenBingwill occur (1.31. The degeneracyis lifted. In the discussed(1T1) case this implie,s Jxhn-Teller clisphcementsof the smvfnre atoms which destroytheir eqttivam lence. Onemay at leaastexpect a so-called2x1 reconstruction, in a sense that is explainedbelow. 1x1stmple metals) instead one haa a gaAof quite delocalizedelectronsand cheznicalboad.swhich are faz less diredionat thr in sezaiconductors. Consequently, there are no preferred diredioms in the displaœmentsof atoms with the eazeptionof that parallel to the sadaceaormal vector itself. One thtzs expects a displacementrnnsnly of the fa
1. Spnmetz'y
18 *
* @
@
*
@ *
*
(b)
@
@
* @
@
@ *
@
*
*
@ *
*
*
(a)
*
@
@ @
* *
Fig- 1.10- Sch=atic remœentation of a metal surfaceby cores (dots) and WignerSdtz cells (hexagoms) before (a)and after (b)the surface relnvn.tion. Deformathons of hrastgonsindicate tàe redistdbution of the electfon density accompanyingits smearingout at the surface and the inward dksplacementsof the cores.
this picture woutd lead to a rapidly varying electron density at the surface: azcording to the arrangement of the b4zlkatonzic sites. As i'acticatedschematically ilf Fig. 1.1Obthe surfaceeledrozdcchargetemdsto sm00th out, which is only porible by vertical displacementsof the ion cores. Simultaceously, the electrostatic repulsion of fzrst- and second-laye,rioas is reduced, which resalts i.n the izlwairddirection of the reln.vntion(contractîon). The efect is obsezwedfor many low-inde,xmetal sarfacœ. Reconstructioas,for instaacethat of the type incticated:in Fig. 1.9b9rlm also occm at metal surfaces.However,the relnvntion is not restricted to metals but ca,n also ocmzr for surfacesof nonmetats.The cleaacleavedGaAs(l10) surfaceis one well-lcnowne-xamplein thîs respect (1.44. Due to the electrœ static neutrality of the surface unit ceRwith oae cation a'n.done aoon (cf. Fig. 1.6),t'wo danglingbonds and.t'wo e'lectronsare available. Consequemtly, there is no need for Jn.hn-Tellerdisplacemeats.Iastead: opposite vertical displacements,i.e., a surface bunkling accompaniedby a,n electron trsmqfer between the dangling bbnds, staboe the smface trnnqlational spnmetzy lmowa from the truncated bullc. Besidœthe argttments of the satuzation of dangling bonds or smoothness of eledron distributions, another argttment is related to the zeduction of the electrostatic energy of systems with partially ionic bonds. Due to their Coulombcharacter,the electrostatic i'ateraction of ions Lsof lonprange natkkre a'addoesnot show a directionaldependemce. Shce the only defned Oection in the system is still the surfacenormal, again a tendexcy to a special type of relaxation occm's. Oneexamplefor a classof correspondingcrystals concerns ArvBw semiconductors with paztiaz.y ioaic bonds and roclcsalt structure. Two diFereai types of sarfatesare possiblei'a thesecrystals,a pola,rtype with exclusively A or B atoms, and a non-polar type with b0th A and B atozns in the surface(see Fig. 1.6). J.nthe casc of the non-polar (100) surfacesone expects from considezationsof the Madelungenergy vertical disphcemeatsof the surfaceatoms but nozte parallel to the s'tzrface place. 80th sublattices, A and B, shouldrelax inward but to dt'gerent degrees,becauseof their dlFeretlt ion sizes. TMs zœults in a so-called rumplin,g of tlze surface E1.5).
*
* @
@
*
@ @
*
(b)
@
* @
@
@ *
@
*
*
@ *
*
*
(a)
*
* @
* @
* *
Fig- 1.10- Sch=atic remœentation of a metal surfaceby cores (dots) and WignerSdtz cells (hexagoms) before (a)and after (b)the surface relnvn.tion. Deformathozus of hrangons indicate tàe redistdbution of the elect/on density accompanyingits smea.iugout at the surfaceand 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 b4zlkatonzic sites. As i'acticatedschematically ilf Fig. 1.1Obthe surfaceeledrozdcchargetemdsto sm00th out, which is only porible by vertical displacementsof the ion cores. Simultaceously, the electrostatic repulsion of fzrst- and second-laye,rioas is reduced, which resalts i.n the izlwairddirection of the reln.vntion(contractîon). The efect is obsezwedfor many low-inde,xmetal sarfacœ. Reconstructioas,for instaacethat of the type incticated:in Fig. 1.9b9rlm also occm at metal surfaces.However,the relnvntion is not restricted to metals but ca,n also ocmzr for surfacesof nonmetats.The cleaacleavedGaAs(l10) surfaceis one well-lcnowne-xamplein thîs respect (1.44. Due to the electrœ static neutrality of the surface unit ceRwith oae cation a'n.done aoon (cf. Fig. 1.6),t'wo danglingbonds and.t'wo e'lectronsare available. Consequemtly, there is no need for Jn.hn-Tellerdisplacemeats.Iastead: opposite vertical displacements,i.e., a surface bunkling accompaniedby a,n electron trsmqfer between the dangling bbnds, staboe the smface trnnqlational spnmetzy lmowa from the truncated bullc. Besidœthe argttments of the satuzation of dangling bonds or smoothness of eledron distributions, another argttment is related to the zeduction of the electrostatic energy of systems with partially ionic bonds. Due to their Coulombcharacter,the electrostatic i'ateraction of ions Lsof lonprange natkkre a'addoesnot show a directional dependemce. Shce the only defned Oection in the system is still the surfacenormal, again a tendexcy to a special type of relaxation occm's. Oneexamplefor a classof correspondingcrystals concerns ArvBw semiconductorswith paztiaz.y ioaic bonds and roclcsalt structure. Two diFereai types of sarfatesare possiblei'a thesecrystals,a pola,rtype with exclusively A or B atoms, and a non-polar type with b0th A and B atozns in the surface(see Fig. 1.6). J.nthe casc of the non-polar (100) surfacesone expects from considezationsof the Madelungenergy vertical disphcemeatsof the surfaceatoms but nozte parallel to the s'tzrface place. 80th sublattices, A and B, should relax inward but to dt'gerent degrees,becauseof their dlFeretlt ion 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 accordancewith the above dsscussions, the atomic displaœments&R(st 1, ,rrzl, rrzzl i'a expression (1.9) wit?hregrd to may be divided into two clmsses theiz Gect on the tranqlational symmetr.yof tile stlrfime. V the trnoqlational symmetry Lsnot asected, the displn-ments repr-nt a relxxxtion of the smFig. 1.:1a). face (see Thea &RLS, 1,mt, rul = L'rskfor all ml, ma. Ozulythe vecton 'ral of the atomic basis in the hnlfqpacebelongin.gto tEe 2D Bravais lattice are altered. ln the case of suface reconstractton(see Fig. 1.1âb) eqtlivalct atoms in diFerent llnit celksarc not all displacedin tAe ==e mnmne'r, J,rrzzl mg)depenclson zp,s and rzz. Both the atomic basis a'ad the î.e., SRLa, BraYs lattice are nhn.nged. Jn the cnse of a reconstrttctedsarfacea new 2D Bravaishttice with prlmitîve basis'vectoz'sEz aud &a occurs. Tzlth:is situation a perioclicity is prœent in the topmost atoMc layers wlliez ii HiFereat 9om the corresponding2D trnanmlational symmetzy |
T
=
Tsfl Tb
(1.11)
.
Alternatively, ore cau say that T is the largest common subgzoupof both groups Ts and Tb. There are two possiblities. Either T only consists of the identity trazzslation,whic.h means thst the lattices dained by Q aud Tw aze non-commensnxte, or T contnsmsmore elemeatsthan just the idemtity, witie,h tlnn.t the soace (Q)axd bulk (Tb) me= aze comnzensurate-In the flrst =e, the crystal with surfacedoesnot possv any htticotramslational symmetry. Reeations of non-commezssurate snlrfnres az'e more likely izl the case of adsorption.In the secondcmse, tke lattice assodatedwith T kqe-qlleda eo-
.<7
.do
.47
a)
b)
Fig. 1.11- E'xamplesfor suzfacerelmtion (a)and sttrfve reconstruction (%)iuduendng the ftrst aud secondatoznk layers. A 2x2 reconstzuction Lsshowni.n (b).
1. Symmetz.y
incidenceJcfjice.1$in partictzlar: Ts is a subgroup of Tb, then T is equal to httice is identicazvith the httice of the sttrface.If Q, i.e., the coincidemce Ts ks not a subgwup of Tyl then T cnnnot be equz $o Q acd is necesrw-trily a of Ts, i.e., it is Krnxlle.r thn.n Ts. proper sazbgroup Iu general,the primstive vectors &z, Q of the lmrfaze Bravais lattice aad those az, Ja of the correspondingblllk-lilcelayers are relatedby 2 =
ai
m
=
i;Ja..),
(1.12)
,
j=k
i.e.: by
s .
a
2x2 matrh
=
'l4nls matrix
V
with
znzz rzzz 'mzz M22
(.1 ya)
w
'
be xxpzvlto denote the sarfRe sazpelattice structure. It romzltsin the so-rxlled matrix notation. The matrix also allows a convenient classlcation of the relation of two 2D lattices. There are tbree pezeent Twm
ca-:
L When all matrix elementsmv are integet's, the lattices of the sudace region and the bulk substrate aze simply related. The surfacelattice Ls calleda simplesaperlattice. ii. Whea all matzix elementsmq a're ratior./ ntunbers,the two lattice,s aze rationally related.The surface Lssaid to have a coincidcncc structtzre, and the s'aperstrudttre is referredto as comrnezzsuzate. iii. Witen at lewst one matrix elemezttZZUJis an irrational nlzmberj the t'wo lattices are ixrationally related, and the supezstructme is termed incollerent or incomrnenlmrate. Iu t'heXst and secondcœses, the combintdslprfncelayer andb411k mzbstœate is charactedzedalso by a Bravaislattice, azd the stkrfacelayeris in complete or partial coinddence witlz the substrate.The,three cas/ (i), (ii),an.d (iii)cap more easily be Garactezizedby the deterrnlnant of M, det V, Fheredet M is atl i'ntege-r,rationvor irrational n=ber. Geometricallydet .5.frelat% the areas of the primitive hlnnstcells spannedby the veetors Z51, Ea an.daz, lc. In to c3aracterize a reconstructed surface. aknycase the matzix V can be tzsed 'I'he correspoadlug notation is called mcfzi'r notation1.2.6 Wood Notation
As an alternative to the matrixnotadon, the more trnmparent Wbodràtltatf,on is 'IZSGI in m=y ceasesas a labnlsmgsche'me for the reconecucted stzrface E1.6) aud the occumhg superstructare. The f1.rs1 step is t:e Garacterization of the crystanographcorientation of the substrate sudace(more preclely, the (?zk!) Thezeis a simple notation pl=e)with the chemicalcompositionS by SLhkl).
1.2 Tw-DimensionalCzystals
21
for the reconstruction-inducedsuperstmctlzres in terms of the ratkosof the lengthsof the pm'rnl'tivelattice vectors of the t'wo 2D Braes lattices unde,r considezation.Sucha smface Lselnn.razterized by
(r3''l1*21)
awo. J(&AJ)s x a:I Iazl /ç ls either $ (forprimltive) Izl the notation (1-14), or
tc'
(forcentered)
accordicg to the way in whlcà the l'=a't cell of the mzrfaceBravais lattice is the prirnl'tive notation is understood deoed. Wb.e'athe letter $p'is dropyed, implicitly. The quantity tmxz;l= (14: l/1ë,l lx lQ1/(5.2 1)indicates the ratios of the magnitudes of the tusuallyl prîmstive basis vedors of the surfacelattice a'n.dtEc bnll1rbeneath One speaks about an (mx A) reconstruction. The symbolRvl intludes the possibzity of a rotation (.R)of the tïnst cell of the overlayer by p degeœwith respect to the n'nit cell of the substrate, i.e., the aagle betweenzzl and @z.If e is zero, then Rpo is omitted Fom (1.14). Consequently,typical denotationscould be '
Slhkitmx
r;,
x a), and S(hA!)(m x n4RIV. S(h,kl)c(m
Examplesare plotted in Eig. 1.12. For a couple of evnmplesof recomstractions the relationbetweenthe' Woodnotatitm (1..14) andthe matra'v notation (1.13) is givea in Table 1.2 (1.7). Severaladditionalremarksare necessaly First) sometimesthe Wood notation ks not tmequlvocal. The lattice vectors iz = 'rrzal and Rz = rzë,z are not nemqsrily primitive as o ald in y assllmedin. tNenotatâon(1.14), addiiion to prlmstive (p)zeconstmzded sarfacelattices, also centered(c)ona '
'
S. >. c) 1.12. Three dr-fTerenttgpes of surface zeconstractiozus.(a) 1x% (b) (WxW)S30*,au.d (c)general =e. The Wood notation does not apply in this the matrkx rotation do% with =zz = 5, mz2 = -1, mgz = 2, ma2 = 2. case;koweve'b
ng.
22
1. Symmetry
Table 1.2. Woodand rnntrix notatitm of recomstructed sarfaoesof cubic and hexap onal crystals g1.7j.
Reconstruction Ideal suz'faœ
Woodaotation
Mntrix notation
X 1 r(1 )( 1)z'â1 Z) (l' 1 j fœ(100), pt2x 1)A2x 1 (0î) x 2 bcc(1Q0), p(1x 2)z%1 .s0) (1 1 l diamondtlœ), c(2x 2)>(Và'x Vil.a4.5= (zJ '
blendetloo)p(2 2)*2 (2Wx W)2450 c(ax a) x :L p(zx 1)...1,1 x 2)*2x l fcctlll), p(2x 1)=%c(2 hc.ptt)Kll, p(2x 2)*2x 2 dinmondtlll), (./J x VJ'11?.:$00 zinc blendegll), c(4x 2) graputetoool) lv'#'x ./flaa,rct=t./J/sl p(1x 1)*1x 1, fcctllo), ,(2 x 1)*2x 1 dwmondtllo), ,() x 2)i1 x 2 zinc 'blendetsrto) c(2x 2) *:. x :, p(1x 7.) bcctllo) J,(ûx 4*2 x 1 x 2 p(2x alec2 zinc
x
x
2
Cj) (? l(02) lz)
(1 1 j 0) 0J (m 1 ( o)z (1))
Cc J) (( j)
D) (1o l (j t) (j ()): ()t) (à-0) 1 (-2 .) 1! (0p
possible. TH c.FJ1only take place, howeveer, for rectn.nglllar smface lattiœs- Thus, the mrvlifled aotation applies only to this cmï4ra, althouglt it is also sometimesased (formally for sqwe httices. Ic the rect ar incorredly) case the notatlon ctl x m)descibes a iype of reconstruction witic.his usually not coveredby one of the notations W x 'rn) or (n?? x c//lswo.For squaxe x m) notatioa is Justa simple,rdescription of reconstzuc'ted lattices the ctzz a reconstzuctionof type (rt'x r/)JM51. Oneexample is shownin Fig. 1.13. Second,aaother problem is related to the fact tlmt one alld the same reconstruction may be deflnedin di%rent ways. The problem is a consequemce of tEe high poiut symmetzy of the crystat with aa ide.alsurface.lf the latte,r hxq a square latticq an.done of the two poiut syznmetryg'roups 4mm or 4, the directions of the two pvlrnl'tive lattice vectoz's aze symmetrica;y eqzziw alent. A surface reconstrudion which iz=eases the smface tlnit cell in the are
1.2 Twceimensional Crystals
Ja
23
:2
(y). (y).()) --yky)s t;k;-.l O :1 C)à O (9 1 1. (3.* Oe O O1Q1O1 ..'
.
Fig. 1.13. A surfacesupeatructua'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 is equivalent to an zrz x zz reconstruction. Azl evn=ple ksgiven izl Fig. 1.14. A.u. analogous statement holds for a'a ideaz sttrface, having a hezxagonalJxttice aad one of the p'oint groups emm, 6, 3m, or 3. J.nthis case, three symmetdcally eqaivalemtdirectiotuseist (see Fig. 1.14). If there is zzo physical reason which malcesone of the geomctrically diferent but symmetrically equivalent reconstructioas more likely than anotller, they will take placesimultaaeously in dsFerent regions of the surface. As a result domahzscan be formed of othezwiseidentical, but diFerently oriented,recozustructed unit cezs.Due to the domain structme, the overalt trn.nKlationalsymmetzy of the surface is destroyed.Strudaral imperfectionsof a more local natme occur whearethe botmdn.rlesof such dornxinnmeet. In the cmse of the Si(111) surface,the 2x1 qlnl't meshesmay occar in one, t'wo or all three (211) diredions,'depending on the cleamgeconditions E1,8j.
Fig. 1.14. Symmdrically eqtzivaleni 2xl reconstructions of square and hexagonal
ideal latticu.
1. Symmetry
24
(111)
z'
a) & N. N. N N
Y
z
N
#
%. %. % N
%
/
.: x.
%.
;
ê w ;
*
N. N % % %
,' ez
h.
h.w
.,
##
%*
Nw
1
t'
1)-1o'.l
#
*
<< >
r
l
;
%h. .
c)
b)
,
i'& >' ! v- I
/
.f. q
%
& N &
+
> ., *
v
#.
r= :N1
-
v.
IW1
'f
.,.
i
-10E1 (:1
-
D1n:l
suzfaceof diamond1.15. Two diferent 2x1 recoctructions of the (111) stmcttlre crystaks.(a) ideal mtdnne; (b) 2xl recons-tnzcted=z#v..e due to rthxl'u
Fk.
formationiand (c)2x1 reconstrudH mtrfnne due to m'rfnz,e atoms. Dots: nominal fust-hyer atozns; clple
an heqtlivalent bunlrMngof nomloxl second-layer atoms.
certai'a recorsstruction denoted by an oression of the type ca,t be realizedby rls/erent atomic coegurations. This is dezronstrated (1.14) t'a Fip 1.15 for diferent arrangements of fzrst- and second-layeratoms oî a surface.The easiestway to reconstmzc'tthe ideal diamond-likematerial (111) The daaatoms (Fig. s':rixz!e (Fig. is by bunkling the frrst-laye,r 1.15c). 1.15a) ne accompanyingdeerent becomeineqzzivalent. gling bondsparaDelto (11lJ ElJJ.Mwith electronssupports the tendencyto an inert surface.However,the bonds can alqo be rebondedif they occur at atoms which are 62stThe sarfaceatoms may be the bulk point of view). zteaœestneighbors(from a possibleexamlying nex't to eac,hothe,r (see arrangedin the form of c-lznsrtq A11atoms of a càain are coupledtogethe,r by 'r-bonds of p1eia Fig. 1.15b). paralle:danglingorbitals. The gcceration of daqglingbondsat the formerly second-layeratoms icclude zemarlcablechaazgesof the bondingtopoloa beaeath.The atoms in ballt custals of the diamoad structme aze bonded in however,fvefold azd sevenfold sidbld zings.In a rr-bondedc'hainmodel I1.9J, Hngs are formed. Fourth, the Wood noiation e>n also be used for sllrfxvte overlayerstmlc1-%% dae to adsorbatœ. A periodk arrangement of adatoms or adsorbed c'tureawhich caa be classife accordmolemtlesA.1qngivœ rise to a su a ttRmn-pA to the Wood Eowever,one usqlnlly NICIS ing to exprëssion (1.14). notation. The chemical stoichiometzyof the atomic or molecular overlayer is given by X, and T is the nllnnberof adspedesi.n tba overlaye.runit cell. susrface at a h'actional For example, CO adsorbedmolecularlyon the Ni(100) surfacecoveage of one 'Axlf formqan overlayershown by the dots in Fig. 1,13. in this case the suzfacedeaotation bcxcomes According to oression (1.14),
'PMrd,
a
'
or Ni(10O)(UfxVl)A45O-CO. Ni(1ûO)c(2x2)-CO
1.2 l'wo-DimensionalCrystals
25
1.2.7 Symmetry Classecation
In the commensurate case all the reconstractedsltrrn.ca with the tmderlying bttlk halfspacespossessa translational symmetzy nyxracterized by the four Bramis classE'swith group elements (slA). The corresponding 2D Bravais Fig. :.8), lattice tramsformsaccording to a certain planar point group (sce the so-calledholohedral pott groap, with elememts(aI0J,The 2D crystal, the atoms in the surfaceand the bttlk below, transform according to a suN group?its poict group. 'I'he combination of tNe translatiomal and point voup smmetrie,s gives the spRe group. There are 17 plnmxrspacegroups. Latliceg with a corr-onding spnmetry are shown in Fig.1.16. It is evidemtthat fur.h of the 10'point p'otlps oi equivckleatdirectior.s combiuedwith the conuponding aodated lattice gives rise dizedly t,o a so-enlled symmorphic space group with ele-meats(tx(A).The spee groups p1, p2l1, J1m1, pzrnm, p4, fmrn., p3, p3r11, #, and fmm orioate in tids mn.nner. Since tke point groups of the rectangular crystal s'ys-temare eachassociatcdwitk t'wo Bravais lattsces,prlmltive or ceatered,we 6nd two f:IVGe.r spae,evoups, elrp,l sad czmm. 1xïthe ctkse of the point group 3,m, there exist t'wo dslerent possibilitiœ of positioning t'wo reNedionlinesrelative to the hengonal lattice vectors, elther through the vertice of the equilateral hexagon of the Wigner-seitz cell as assumedin the case of p3mI, or such that they bisect its edges.J.cthe latte,r c%e oae has, as the thirteenth s'pace group, the voup p31m. The point grou.p rfAmn.l'nR uncha'agedif in its space group a glide refecvtionliue Lssubstituted foz a.c ordinary re:ection liae. One musi thereforee-xarnine the 13 spacegroupsalzeadyestablishedto determine wheGe.rthe mzbstitutioa of a reNectionline r?z by a glide reflec-tioali'ae g (i.e., with a trnrmlaf-ion'v by half of thE shortet a zefectioa irt rrù in conjtmdion lattëce vector parallel to m) 1(.)* to a new space group. Oae easily 61.1.* that th.is is not the cmse for the hexagonalcrys'tal system. Ia the quadkatic crys'tal sys'te,mit is possibleto s'ubstitaztea s'yste,mof glide reEectioa liaes for one but rot b0th. of the non-eqttivalezttrefedion lin.e systems. Tizis yields the additional space goup pngrrt. The rematniag space groups .p1.gl (lom acd pzmn, p2.:.: (lomp2mrzlocctzr in 213crystals with a prlmstive .y1m1) Bravakslatuce. They contain elements of taheform (aIA+'rJwith :rectangulaz :i- ks a Factional latdc,e translation. The ceateredrectnmotlxr aad oblique .'iotal systems do not =ve rkse to additional space voum. Cocmuently, ''four of the 17 2D space groups hwolve glide rezections,i.e., they m'e aon-
'.b#zdmorphic groups. ',
Ia Table 1.3 we sllrnrnnrize the symmetr.g classïcation of 2D crystals. ;',ïVJnternational notation is used.Despitethe fact that the atomic basksis parallel to tite negative sttrface normk direction, we use a plane Elékvnded '' è'ç8Angularcoordsnxte system with uzlit vectors ex, ev. The ozigin of the co:?. 'érdihatesys'temLspositionedon the rotation x='q if one ests. rne prlmitive ', .bzsisvectoz'sof the 2D Bravais lattice are az, Q. The secondbar inclicat'
,
.
'
:
.
'
@
@
* *
-
*'
h. 'y .
*
*
.
@
*
*
. *
.
* @ @
e
r*
*
*
* @ @ * . *. '* * * @
@.
*
*
*
p
*
*
* @
*
*
.
o
*
* @
*
e
@
@
* @
*
*
o œ
* @ @ @
* *
*
*
* *
@@
** *:*
** @*w
@*
@@ @@
*@ +*
0*
*@ @**
o. ** **
*
*@
*
*
.
*
*
@
*
*
.'o *
.
. . *
@**
*
** .'.
* *
* * ** @ % * *
@
.
* *
* *
*
* * * @ @ @* @
** ** .* *@ ** *. @. .* ** ** œ @ m o
o
*
* *
* *
* @
* *
@
* *
.
@@
* *
@ * @@ * @ *
*
* @ *
*
@
.
@
o *
@
*
* .
*
*
@ *
@
.
@ *
@* @
*
*
p
@
**
*
p3
*
*
*
e
*
*
* @ . @ @* @ * *
*
@*
*
* @
@
*
*
* M
.
*
*
@ @ *@
@
@*
**
@
*
* * * @ * @ * @ * @ @* * * * * @ e @. @ . @ * * * * * e @ @* * @ * @ * @ *
p6
. @
@
@
* o
*
*
**
o
*
a-
@
.
* @@ ** * @* * * * o @ @@ @ *
**
*
**
p3m1 * @. @ * *
*
p4gm
Mmm *
. @
@ @ @ @. * * . @@ * . @ œ@ * * * @ * A * * . * * * * * @ * * * * @ * * @ * * * @ @ . * * @* * @ * +
*
* @ *
@@ @**
Vmm
* *
* @
@
p2nann
@@ * @ @ @.
* @*.@
*
.
*
*
*.
* @ *
œ
*
.
p2gg
*
@
*
*
* *
*'o
. *
.
e
@ @ *
* * @. *
*
.
*
ol
p2mg
*
*
clrnl
@
@
*
plml
*
@
J*
* @ @*
*
@ *@
*
o *
@
* @ @
@ @ @ @
v
* œ@
*
'.*
@
'*
*
@ *
N'-
*
plgl + '*.
@ *
p211
1.
*
.
œ
@
*
@
y.
@
pt
@
@
@
* *
@ @
p31m * @@ * * . * œ *
****
* *@* o * * e *. **@*
* @@ * *0 @ @ * *@@ * * @ * * @ @ @ * @ * *. *..* *. * *@* @* @ o *. œ * œ @ *..* @.e
p6mm
Egg. 7..16. The 17 phnxr space groups representedms parts of lat-tice,ssatisfying the synzmetriesof thosespacevoups. Tîe sma:tldotsare at the corne> of the unit cells, or at their midpoints, for reference.Onelarge dot is positioned at a.c arbitrary lecation within the tmit cell, and khe other large dots are obuined non-synjmetrical 1om thts one by appl>g a:llthe relevantsyrnrnetry operatioc. ARe.r(1.7).
l.2 Twceimensional CUS'i?IS
27
Table 1.3. Symmetzyclassifcation of 'twmdîmensionalcrystals (1.2, 1.10j. Crystal Bravais system
class
Point
Space Symmetry
group
group
elements
pl pall
(cLA) Le.a), (4Ix) (s &, tznogxl
oblique DobDque 1 2 c rec.t-
p-rect=rllpr
r?z
plrzl
p1J1 2mm
pzmm
pz''n'l,p
(s1A),tznvr'r+Al,= = Ye. (cIx), (JIIAJ,(NIAJ, (m.1AJ (4IAJ,'(mxl'r+ A)','tmvrr+ A), (c1AJ, v. = 1e. (FIAJ,(JilA), frrul'r+ AJ,(mvl'r+ aJ, m .ve.+ ve. (sIR),('maIA) ..
772:: .
c-rect-
m
clzrzl
.
.
=>= zrrzzrz czmm square
p-squaze 4
.p4
(slA),(m.1A),trrh/lal,(JlIA) (s)A),(J)1A),(JlgA),(JIIA)
4nsm 4mm
p4mm
p4:%1
hexa-
gonu
p-kexagonal
3
p3
(slA)j (J11A),(J11A), (J11R), (mm1A), (Tr?'p EX),IMJIXJ,I?T4IAI (J112Q), (d1l1Q), (cIJz),(J1EJQ), + &, (m:IA+,rJ, f-l'r + A), (,41,:1-, (m4IA+ +), 'r = l(e. + ev) (sIAJ,(JI(A),(JâIAJ
6mm r37%1 6
,31m
,6 6mm
p6mm
(;JIA),frt>alal, (c(A),ftQ1A)) (mzlA))t=ilAl (JlIA),trzvlAl, (a1A),(J)1.R), f'rrzz IAJ?t'm1 IAJ (c1A)? (tQ1A), (J3IA)) fJl(J0) IJâIA).,((Q1A) (dJLA), (clA),(JàIA)? (#1A), (J1EA), (Jà1A), (m=IA),(mcIA), (m1IR),('mXIAJI t'rruIA), (ml!R)
28
1. Symmetzy
Table 1.4. Spacegroupsof ideal low-inde.xsuzfacesof diamond-,zïc-blendeo and wurtzite-type crystals.
317crystal
Surfade Sp= groum
structure
diamon.d
(111) (110)
Fit'st layer
ïn6nite First First two layers three hyez.s half space
p6mm
lfzn,l
p3m1
p2mg
,2r/ 2727p,rp,
p2mg p2mm
p3ml pzmg p2mm
p3m1
p3ml
3)3z?zl
plml
p1zrz1
plzp,l
pzmm
42mm
p2mm
,3>1 ,27/,=
p3m1
p3rz1
plrp,l
p1rrz1
yzm'm
plrnl
plml
(100) pkmm zinc blende (111) pîmm (110) plzna (100) p4mm wmtzite (0001) p6mm (10ï1) pzmm (1120) plmm
.
Spacevoups of reconstruced low-index surfacesof cliamond-kypecestals Smface
Model of reconstruciion
Fi:'S'tlayer
Tn6nlte First tw'o layem half space
,2mm
plzn.l
'm-bondedckain pzmg gubondedbucldedckain plml p2mm x-bondedmolecule
plzrtl plml
plml plzr,l plml plml
symmetrtc 25=e.r asymmetric dlmer
p2m7p,
p2mm
g2mm
plml
plml
plml
bunkqlng (111)2x1
1 (100)2x
Spacegroups
y1'm1
ing the reconstmzction ksdropped ic the followiqg. The oorrespondicgplaae lattice constaats aze tzz acd ca. The generalclassiîcationof 2D cz-ys-tals izzTable 1.3caa be usedto elhn.racterize the symmetry of mlrface of 1.ea1 Fxxamples are given in Table 1.4. Tllis table iudicates the space groups of low-index surfacesof tgpkal s/micondactozscrystnl7n'zingin diamond?zinc-blende,or wurtzite structuzes. Ideal and reconstructedmlrface are consideredand the reslzlting groups are discussedfor diferent nlzmbezsof atomic layers below the uppermost one. Table 1.4 clealy showsthat the reskfting spacegroup dependson the czystal .
1.3 ReciprocalSpace
29
orientation, the nlzmbe,rof atomib laycs takeninto accotmt, and the model of reconstntction. That means, the space voup of a 217Orstem depends on all of the above-mentionH details.
Space
1.3 Reciprocal
1.3.1 Direct and Reciprocal
Lattices
A two-dîmensionalsolid smface is chazacterizedby a 2D Bra,vaishttice (cf. Table 1.3)'ivith prirnitive basis vedors 41 and Ga. We use the vectors with only one bar independent of whether or not a reconstruction ks pre-sOt. Ia coorrlirate system with l:mit vectors termq of a rectangtzlarpla'aa.rCartesslm cx an.dev, the bmsisvectozs read as J,z = Azze. + Azzew, 5,2 = Azzew+
(1,16)
Jbzev.
The determsmantdet .j. of the 2x2 matrh
./i
-4.1:Jkz .,121.d%2
=
givesthe area W of the nlanl'tcell of the Bravais lattice. 1.nfact A = 'n' (&1x&z) = det aiwith n as the surfacenormal. A correspondsngreciprocalIctâcc in Fourier space is mssociatecl with the Bravaislattice in real space.TEe redprocatlattice, as we shallsee throughout the book, is ex-tremelyuseful an.d pertinent in all dlfrraction methods, i.u pwrticularin tlle case of low-energ,geledron doaction (EEED). h.sin threedimezssionalspace, the prt'mstive bmsisvectors 5zacd bl of the 2D reciprocal lattlceare desnedaccordîngto the orthogonalityrelatior ,
% 8J= With '
'zz as
(ï,#
2rctjk
-
=
1, 2)
(1.18)
.
the Alnit vector normal to the surface, solutions of the relatioa
are (1.18) -.
bt
27
=
J.a x .p, j
Ië1 :,21 x
%
=
2x
n x ,
a1
l41)<
. .
5,21
(1.19)
:'
= The leagtits of these vectors are 1b:1 sin (<, a2)J. The prl-mitîve 2'V(c!.î : basks vectors elm be used to constnzd the reciprocal lattice to a given 217 .;'. : network. TV is Mlmmn.tizollyryhownin Fig. 1.17. A general tnzmqlational :' vltor in rKiprocal spu is Tve,nby .
'
è.
.
gu
=
W
+
k82,
(1.20)
where h and k are iutegezs.The set of all vectors ghk #vesthe reciprocalnet.
1. Syznnzetry ''''''jk:)t!;y I
*(p2
b:
*
J
Square Iattiœ
1 l 1
,
-
-*
*
*
o
@
#
X
ve /
he=gonal Iattice
- -
-....4
-
X
#' J'
bz
% *
..
1 1 t 1
Nx
->I
X
ee' .
o
J.
-
-
.- -
4
.
*
*
-
bl
x.-
N
-
R
.......* *
@
Iattice
* *
..l
-
* x
.
e
1
* @
1
l l , 1 1 1 1 1
E7
X redangular
1 t t 1 t 1 1
1
*
œntered
X
**%
ai
redangular lattice
X
1
I
X
* 2
#
#
2
N' %
%
l
oblique latjce
-
b1
1 I l 1 l
X
Fig. 1.17. Direc't httice tleftland correspondlngredprocal lattice 217Bravais latlices are presented.
(right).T/e
1.3 Redprocal Space
kill ki
ki
n
(00):kul=kill (01):kï1II=k iI1+g()1
î J2 *
*
.
.
*
@
R e
e
@
e
.
Fig. 4..18. DiAaction of an ixtddemtplaûe wave V:.IZ wave vector 1I. The mlrFnr.e Ssreprœentedby the correspoadùzg2D Bravaislattice. Parallel momentum oomservation with any reciprocallattice vedor .gsu create well-deoed 'llpvacted beams '
(hk).
The reciprocallattice vedors havea direci physicalmea'ning.I'a a dlfh'rtion experiment, e.g., LEED, exrx dlfrrncted lvmm comwpoadsto a reciprocal lattice vector gsu =d, in fact, ea& such beam eAn be labeled by the watues h and k as the btxnm(h,k). Thks is hdicated seematically i'n Fig. 1.18. The angle of emugence of the ds*aded ben.ms is detemnimed by the corervatiou lsw 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 217Bravais lattice. A scakteringgeomeWis considerH in which tzzqmomentum conservationis 6'1G11ed with reprocal lattiœ vectors gtA parallel N) èz. '
32
1. Symmetry
'Pable 1.5. Dh'H and reciprocalJatticesof twodimemsionalczystals. Brawais class
az , ,,z
s, , àa
ura cen of dired lattice
srioui.a Z0n6
E2 ey v
lc ex Ds
(a1x, 0) lJa , 4t'v)
oblique
2* oz.
ex
*l'e
(11 oa ) a2* 2v (0,1) -
1
*j5r:
2
wb,
p-rectaxs
tralar
(c,,,0) (o,cc)
oxq .-
Fz(1,0) 2:Js(0, :) f:tF ey
ey
6z
Ja
cur ect a.u
-
galar
Mz
ex
Mz
( , ) (s.z 2 ) ea) 2 -
F, (1
,
1
-
;
XWC
bz
W 24
.
hexagonat
(a,0) !.(1, Và')
ey
q
.4m(:, au; cc c'.t
2
(c,0) (O, c)
e:
Ma)
$
3E(1.0) C L':(o,1)
e.x
bz
az
e
ex +
z,,s(:g,
1
:zj ) 2J(0, w2)
a
.
e:
q
1.3 ReeiprocalSpace
33
sufaces. Mter Fig. 1.20. LEED images of six dt'ferently prepared Gp.Aq(lO0) The stufacereconstmzction and the electron enerr are hcticated. g1.15j.
dent pazticlo is p = zki, where &. is the wave vedor. With ks ws the vectdr of the diAacted particles the momentltnn conservationreadsas
wave
ksrl = ki r + guk. (1.21) After the c7''fFractivescatterhg the parallel component of the momentllm mxy an electron) bexm, i.e., gsk = 0.' be equal to that of the incident pazticle (e.g. Were is no relation betweau the componeats of ks and ky perpendicula,r to the surfacq becausethere is no trnanqlationx7 symmetzy in this direction. However,the particles stttdiedin a difh-actionexperiment, e.g., the elcctroms in the LEED cwse, are elastically sottered. Onetherefore bn.q
Iksl Ik:1, =
(1.22)
1. Symmetry
34
K K
3x3
3x3
-.-
..,
.
.
.,
..y
(W xW)œ30o
'
tq..fj x 6'.W)a30o
Fig- 1-27..Sjquence of LEXD patterns (withnlmost the same elecron enmgy $4$130 The lx1 bulk-terrnlnatedphase e'Vlfor the Sz-termlnatedsmfaceof 61LSiC(0O01). ksstabHed by OH adsorption,wherebythe following reconstmzctedsurlacesrestklt ibllowed by n.nnealingat by E00QCAnnenlsngof the latter in Si-Sux (3x3pbnAe) about 1000*c ((WxW)2R3r and at 11D0IC ((6Wx6W).R3Ob pkase) phwse) of Erlangenlj. of J. Bqrnlurdt, U. Starke axzdK. Htain'z(University (courtesy
always efsts for given A solution of the two equatio> (1.21) and (1.22) vectors kt and gsk. This is i'a contrmst to the caqe of scattering of particles hom blllr crystals with 3D tritnslational symmetzy Coherentscattering can only occur if kk lies oa a Bragg reâection plnre. The solution of the above equations esm be readily cmied out using the Ewald construction shown in F'ig.1.19. The points, at whick the vertical lines passing throur,h the rœiprocal httice pohts ghk iatersectthe sphere!ks)= Ik:J1 deterrnimethe directions i:a which dOaction mnxn'rna occm. There is exactly one mn.vl'm'r= for each reciproci lattice vector. The reciprocazs'arfacelattice can tEus be read 1om the di*action rnn.orima oc the LEED re>tration screen. The ralation be-
1.3 Redprocal Space
35
twee,nthq Oect lattice and the reciprocallattice for the fve 2D BzaYs nets is shown expûcitly in Fig. 1.17. The dired relationqlnipsbetweenthe direct and reciprocatlattices of 213systemsare #venin Table 1.5 i'a termq of a twodirnertsional Carteziancoordinate system deMed by the vectors ez a'ad es. la more detaâl,the table relatesthe prlmltive basisvectoz's/z, J,a and 81,t)z to the Carteslxnvecto::susing the lattice constants cz, az (orcz = c2 = a)of the 2D nets. Moreover,the relatiomshipbetweenthe Wignerw-seëtz cellsof the direct lattice (i.e,,the ltnst cell)and the reciprocal lattice (i,e., the Brillotns'n is prœented. zone) '
Typical LEED irnngesare preserztedfor rectaagular (squaz'e) lattices ia Fig. 1.20 and for hexagonallatticœ 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 has
'
to mention that the construction in Fig. 1.19 Ls evnz!t only in the limit of
scatterixtg 1om a true 2D network of atoms. J.x1 a real electron dffrraztion ex-perirnent: however, the primazy' electrons penetrate several atomic layers i'ato the solid. Therefore, the mea'a 9ee path of electrons deterrnlneshow the third Eauecondition becomesmore aad more tmportant. This lto-vsto a modulation of the intensitic of the Bragg refections iu compnr'Kon with the case of pme 2D scattering. 1.3.2 Brllloxal'n Zones
'
In trauslationAu7ly iuvadaut systems the wave vedor k dténes a set of tgoodl qlTnntlzmnumbea's for eachtype of elementG excitatiom Iu the case of an orderedsurfaceof a czystal, such a wave vector, i, is zestrided to two dimeusions, i.e,j is pazallel to the surface. Wit%n'na reducedzone sche-me it îs restrided to a 25 Brillolli'n zone (BZ).The eatire 2D redprocal space can be coveredby the vectozs L-+J, where g ksa shtrfnno reciprox httice vector rne sndhce BZ is do6nedas the m'nxllest polygonin the 2D redprocal (1.20). as spaçe ïtuat'ed symmetrically with respect to a give,nlatdce point (used coordinnte zero) and boamdedby points k satisyngthe equation -
k g .
=
1 a . -LgL 2
(1.23)
The set of points deMed by (1.23) gives a straigb.t liae at a distance141/2 from tba zero point which bisects the connection to the n.e,x-tlattice poi'at .ç at righ.t angles. Sincethere are Evedlfearen.tp7n.neBravais lattices aud, hence,dve dlsezent reciprocazsuzfacelattice-s,there are also ûve HlFere'nt 2D or sarfnzwo Brillollr'n zonas. They are shown in Fig. 1.22. Their shapœ are the same as thoseof the Wignez-seitzce1 of the correspondingdirect latiices (cf.Table 1.5),since the Bzavais type,sof the dizect and reciprocal surfacelattîces alwayscoindde.
1. SylaMletry
y
7 y
û
z''
û
Rz
z
?'
z'
?'
A
Z
z
F px
r
R' A'' îR i'
R'
R
p
s.
x
x
@)
z
y
(c)
7 Y
7
Z
P
91 Z
R
2 a
h'
Z'
r
Q
E'
h
P
.:.'
D'
P
>
x
A'
Z
v
z
z
z'
9.
(e) (a)oblique) (b)peedaagalar,
(d)
Fîg. 1.22- Brillolpi'nzones of the dve pla'ae lattices: , (d)scpare,and (e)lmvltgonal.Symmetzylines a'adpohts aze also (c)c-reda shown,and tlzeir notations are htroduced. The 217Cartesiancoordinate system ts eosen so Gat the point symmetry operatîons $nTable 1.3 can be directly appzed.
Iu ng.1.22 m havelabeledjomeof the high-symmetry points of the Brillouia zones Msinglettea'sX and r. The bar indicate.ssuchpoht,s ia 2D Brillouh zones wherea points like X and r iudicate positions in the correswnding 317 We follow the convention of dmptiug Brûlouin zzme of ln6nite 317 the BZ by Greekletters, e.g., r aad high symmetz'ypoints and linçs i'?/,Wtfe 1, d, L'. Poiusts acd lines oa the bonndary of the BZ aa'e denoted by-Roman letters, e.g.) M and Z. 'I'he cente of the BZ ks alwaysdenote by r. Apart Fom the hexagonalBravais syste.m the Mgh-syznmetr.ylines parallel t,o the axes of the 2D Cartesiancoordinate system are indicated by the Greekletter to a corner 1. Iu the hexagonalcmse, E or d is ttsedto hdicate a line from 2% point of the hexagonor a midpoint on art edge.The primes on the Greek or Romaaletters are uscdto allow azt indication of Xerent symmetriesia c.%e,s wherethe point group of the 2D crystal is only a subgroup of the holohedral group of the Bravais lattice. .
1.3 ReciprocalSpace
37
Unforeately, the notation in the literatttre is not consistent.J.nthe ori#-
nal papers severalmocls6cationsaa.e wsed.WIJ.C,E of the asPere'ntpoints sho'ald be indicated by a prime or not, is not exactly sxed..For instauce, sometimes in papers about the cleavage face of zGo-blendecrystals or the (110)1x1 surfaceof group-W crystals, I'X is used to indicate the shope,r mds in the BZ, izz contrast to Fig. 1.22 where Ehn'qline is denoted by FX-/. The're aa'e xlr.ry>amples where authors use the notatâonC tns'teadof X' (1.11j In the ar.d (100) cwse of the 2x 1 reconstructed(111) sudacesof groamlv matezials, there is a tradition of following the notation of the squaœe lattice. Instead of X acd V, the notation J' and 1%is used (1.12, In the latter case even 1.13j. J and J' are lteraiuanged.. Sometimesone fmdsa paper in whic.hthe corner of the BZ is denotedby &V and the midpoint of the edgeof the rectangle by .
7 (1.14) .
Prolection of 3D Onto 2D Brilloxxx-mZones
1.3.3
The fact that the 3D wave vector k from the BZ gives a set of Cgood'quaatttm 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,xcitatsons i.n a mygtal 'Zt,'Nsurface.0n the (me hand, bulk eaxdtationsshould aksooc= in a semi-lnGnl'tehatfspacewith surface.On the other bnrd, sucil a system Lso1y Gracterized by a 2D translational symmetry. Cozusequently, by the elementaqexcitations of the ftnite s'ystem ca'n only be elnnracterizmd wave vectors k from the Brillouin zone belonging to the corresponding 2D Bravis lattice. Ia order to lzse the Bloch-likn eigenvalu.oof a btto eleaneatary e-xcit,ation, the rehtiorship betweenthe ekenvalue of the bulk crys'tal and the wave vector ha.sto be itered. To representall allowedeigenstateslumpnlly the component kjq of the 3D vector parallel to the surfacecmn be Aed, while the perpendicular compoaent k.s has to be vazied.Generallyspenving,the bulk eigeavaluemust be assignedto a 2D wave vedor L ia the suzfaceBZ instead of a 3D wave vector k in the bulk BZ. For obviousreasons, suc,ka relauonshipLs called a projection of the Blocx-like eigenvaluesofsthebalk cystal; the 3D dispezsionrelations, onto the suzfaz:e BZ. Withia = eprpûcitprocedttrecertain bulk directions and points of high symmetr.gin the 3D Brillouiu zone are projectedoato the 2D surfaceBZ. For tbree 3D BraYs lattkes and some low-index gurfacesthe relation Lsdepicted ic Figs. 1.23-1.25. Izzorde,rto illustrate the projection proceduze,ftrst the Brillotlin zonœ under considezxdon must be specised.The conwponding bulk BZ is deznedby the Bragg reoedioa planes ,
k G= .
1
c
jqG
,
(1.24)
wkere G is a cer'taizzvector of the reciprocal lattice of the bulk crystal. The The prlrnstive vecvtors àz, surfaceBZ mny. be calculatedaccordingto (1.23)1a-of tlle redprocal stlrfaeelattice neeed in derieg t%iq equation follow
38
1. Spnmetzy
fcc Iattice (100)
(110) *
-.a=
M
-
x
F
J
x-.
k W
U
k
Y
.-
W
X
J
1-
X A
W
e
L*
K
:K l
.
'
A
.
1.
:,
.
x
W
W
1 t
L
1 )( 1
.' e'
K
L.
.Ll
kk
*
W
$'
''''''!k,4;;r, ''''''jj;(;:r,
K
r
-
M K M
K
-
M K
K
K U W
KlI
I
?k
X
1
l'' l
' .e
K l t
't
etl!
#'
àX
IJ X 1 K: *
W w
L
W
Fig. 1.23. Relation betwee,n 2D Brillnuin bulk BZ in the fcc cax After I1.16q.
from
tzsing the prlrnstive vectors (1.10)
zones
of low-index RGMeS and the 3D
az, 82 of the dizect surfacelattice
considezed.Su& vectors are given lin Table 1.5 for all 2D srtems. Secozd, onto the pla'n.eof the sadace BZ as indicated in the bttl'k'BZ is projected Figs. 1.23- 1.25.We denote by kII the component of a wave vector k of the btlk- BZ paraltel to the sarface.The bouad.at'ypoints of the pzojeded blllk BZ are locatedon straight li'aes detnvnîned by the t'wo equations :1I = klI15l+ k112à2, kjj
.
G
=
1
j.IGIa
.
(1.25)
l.3 RcciprocalSpace
39
bcc laoice (100) XR
$
E
'g
û
# 'J P : F l
r
.
F
'c
y H
P
P
N
N
p
1 H
H
: ---
wy--p li
P
H
1
$
N
H
.--.
-j-
2 I
e
P
1
x*
: 1j
.
e ee
P
'
N
.
1 l
N
1
zr
.'
H
!'.
e *
N P
P H
(111) -c
R
û
R P H
1 1 Nr l 2 $
: Io
I N * -*
H
1
I
'F' ..K -
ee
xe
P
*
1
-)*-
H
P -
X B
.
K
*N
H P
Eig- 1-24. Relationbetween2D Brillouin zonas of lowede.x sm'facesand the 35 'bulk BZ izzthe bcc case. Me.r g1.l6q.
In gemez'al, the projededbulk BZ doesnot coincide'with one surfaceBZ. It kstlsllxlly larger tsee surfMe of e.g. in Fig. 1.26 the tu-ampleof the (100) azt fcc crystal)and one has to fold back the part of the projededbulk BZ 2 not conteed m the surfaceBZ onto the latter one. Sincethesepaz'tsof tlle to projectedbnllc BZ agrœ with neighboringP,DBrizouin zones bezlonging redprocal lattice vectors gtkjj)1the foldiqgts identical with a displacement by s(kgg). Consequently, all wave vectors k in tbe sizzface BZ are give.nby
k - k11+s(à1I).
(1.26)
1. Symmetry
40
hcp Iatjce (Q005)
r
-K
ë
H
H
H
1 l
A
H L
1
L
l I
H
l
!1 --**
<
e. ., F* ..
*
!'
%x
=w.
M%-
hx
l
%-.
AJ
xw
l M
K
16
.
j
h1
Ft
L
lI
K
*A H
H L
1.25. Relation beGeen the 2D FY. Brtllouin zone of the stufaces
an.dthe 3D BZ of After (1-16q
H
a
(0001)
hcp stmzcture.
.
J.ntMs mnanner certaizz re/ons of the mnrface BZ are covered two or more tHes by projectedpoints of the brllk BZ. using as au e-xamplethe We elucidate the above general comsiderations, suzfaceof au fcc C.USIPZI. The 14vectozs of the redprocal lattice de6nlng (100) are G' = 21 + eu + cz) aud G = the blll;r BZ according to (1.24) @ (Aez iMez,v,a. The prirnstive vectors of the redprocal surface lattice are 81 = G Mlev es)azzd52= 3X.(eg+ ea).The four vectors de6mlmgthe 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 3
+
(1.27)
it followsthat ea),for examplevis cahoscn,
u,
(1.28)
klll + kllz =.1,
(1.29)
kllv = aad
1.3 Reciprocal Space
(a)
41
(b)
l
F:g. 1.26. (a)Brillouin zone of a (100) suzface(shaded area)together with ihe projectedbulk BZ of au fcc crystal- Projectedcritical poinlqof the 3D BZ are indicated along a (011) directioa. (b)BG BZ ïor comparlson.
if G = Mev. For the otb.ervectors of the 3D reciproe lattice, e'ither ssmilnm relations resttlt or the associatedBragg redectioa planesare parallel to the piaae of the surfRe BZ taadthus do not inte-rx'c't it). The eeuation of the relations of type (1.28) and' (1.29) reslilts in Fig. 1.26s. One notes that the eapcoidal are.as of the projectedfcc BZ lying outside the surfaceBZ caa be folded ove,r the surfaceBZ by displacements along o=e of the httice vectors 5z,-8:, Ja,-&. The little square i'a the center is part of the Bragg reNection plane boundt'ngthe b:llk BZ and associatedwitll a reciprocal lattice vector G = +2E(2., 0, 0)perpendicular to the sudace.The mn.v'rnl:= (rn5n''nu,m) of fl k.z(= kz)is thus +3X. Outside the Ettle squikrel k.z = k.u(k:j) 'vades in a smallezhterval âxed by the 313BZ. 'I'he iaterval dependson the wave vedor kil1.3.4 Smmetry
of Points and Lines in Redprocal Space
The spaual symmetry of a crysial with slldhce itas implimtions for the'possible degreeof degeneracyof elemento e'xdtatiozus with e'aergieszflgLkj. Thc as a fnlndion of the 2D wave vecRr k #ve the so-called eigenvalcesDgLk) dispersionrelation for the correspondingelemento excitation. The set of indicesJzlabelsthe remasningqtzantumnumbers.Evnmplesa're electzxmand hole e-xcitations with J2#s(i) as the surface energy barzs and g as the band indecqsttrfacephonomqwith dispersionrelations Xlkj of the vibrational branGes g, smface plasmons,etc. The syatialsymmetry results izl relations betweenthe values f2s(i) for dxerent k values.The key for such conclusions are, in aaalog,gto the 't=6nite bulk case, the irreducibierepresotations of the space g'roup of the fven czzztal with surface. This is based on the fact tllat the fzigenAlnctionsbelon#ngto a padiclzlar energy eigeavalueform s basis set of azt irzcdttdble
1. Symmetzy
42
represcntntionof thksgroup. Suc,ha repres%tation may be characterix'd by the sta,r (V of the wyvevoctor i and the irreducible repr-ntatiozts of the small Mzzf gnmp of k (1.10J. A m'nnll poi'at group is a subgroup of the poim.tgrqupof the crystal. The poiaçgroap elementsâ of sucha subgrouptrn.nsformk neither into itselfnor i'ato a vector equivzem.tto k that difers fzoznk only by a Teéprocal httice vector g. The set of all HsfFe-rentaad nozseqlzivalentvectors (RALscazed J'tar of i. At all points of the staz (V the eneror eigenvalues XLkj have the points and lines i'n.the same value. The s=n.ll point groups of Mgyh-symmetrjr BZ are listed in Table 1.6 for the various space groups of BraYs lattices. Table 1.6. Poizti rotzps of tHe high-symmetz'ypoints and liaes of the BZ The trredudblepart of the BZ Lsindkated by the lzatchedregon.
a) Obliquelattice Symmetz'ypoint
Spacevoup P1
17211
: l
;
'
? J
2
Heducible part of BZ
b)nrectaagular lattice Symmetrypoint line
Spacegroup ylml
plgl
y2mm
nzmg
pzgg
m
m
2mrs
2mm
2mm
gvwt
srz
rgv
yyz
XXt
2mm
2mm
2mm
Z'
m
m
m
or
2%
V
m
m
2mm
2mm
2mm
ZZ' zl
m
m
m
m
m
-
-
m
zrs
m
Irredvcible êari Y BZ
(1.10j. .
1.3 Reckproc/Space
3)àble 1.6.
(contilued)
c)csrectangularLattice Symmetzypoint or
line
P
Spacegroup clml
Omm.
m
2mzp,
ytvzs
m
X
2m,m
Z
m
Cz C
m
,4
pz
m .
Irreducible pazt of BZ
d)Squarelattice Symmetrypoint
Spacegroup J>4
p4mm p4mg
4
Qnm
4=
,4
m
Az
J
2r=
2mm
Z
m
m
X'
4mm
4rzr4
m
m
or
Ltae
X
.D Irzedudble part of BZ
-
44
:. Symmetry
Tbble 1.6.
(conthued)
e)He-mgortallattice Srmmetrz
Spacegroup
.
points or lines ,3
p31m
p3m1 .3% fmm
JN
3m
3m
3
6
6mm
zs',!;œ
m
m
4z-/'
3m
3a
ZZ'
m
m
##'
m
2mm
AA'
-
-
z?s
-
m
Heduc3ble pat't of BZ
The Heducible repruentations of these point voups are gvenin (1.10) for the vaziousspacegroups of a, corresponds'ngBravais httice. TEe dimensionof the Heducible representation deterrnsnesthe degeneraeyof an eigenvalueat a given:. For the systtamKtmder considerationonly Heduciblereprœeutations with fîsmensîons equal to 1 or 2 will appear.
2. Thermodynamics
Processes and Surfaces in Equilibrium
2.1 Kinetic
The real surfaceof a solid tmder atmosphericpressureis ver.v diFerent hom alt ideal system desirable in surface physics.Therefore, surfus as objKts of physical studles are usually prepared in UlIV. Besidescluavagea'ad the combination of jon bombardment and n.nn-b'ng, acother chssical method to prepare a fresh, clean surfaceis thc euporation and condensationof an overlayer on a substrate, e.g., within a:ct MBB procedure. 'ne resulting s'urface system, more predsely tke solid-vacu7nminterface, is no static system. SeveralMnetic processesocmtr to a,cez'tainextent dependingoa the mzbstrate temperatm'e T.
A seledion of such processes is iudicated scornxticaxy in Fig. 2.1. Tltey represent elemeatary events whic.hhappen daring epiteal growth, e.g., NIBE. Hbwcver,such processesalso oc= at the substrate-vacuttm interorl a terrhaoe (a)deposition(aisorptson)
(b)depositionon an island
.
q
'
(c)
on on a
ce
.q
. .
(d)daorption
.
.
(e)cucleation
.
. .
. ..
.
,.
w
...
.
.
! .
.
.
r
.
.
'
I .
.
.
.-.
.
.
.
.
. .
'
'
'
'.
.(..
.
'..
'
-
. .
.
.
z'
vh'
.5 J ' ..'., ., . .
..
... .-.
..
' .
.. ...
..,
'
'
,
.
..
.
.. . .
'
,
..
.
. .
. .
'''F
'''
'
.
. .
..
.
.
'
7 .::4
(exchange) ($knterdlfrusioc (g)atlcmont
at an island
(h)de/chment from an is:and (i)attachmentat a step F5g. 2.1. Schfemxtic representationof R'nziame-ntal atomic procassesoccurriztgduring epitaxizl vowth.
46
2. Thnrmodynamics
face,though with
reducedprobability due to the lower tamperattlre aud the smxller number of atoms ia tike gas surrotmdiug the s:arlhce.'Ia any case, the vacaum Lsnot perfect, rather a reatgmsoccms. The atoms and molecula in the restgas interRt with the Rmrfaem.Adxrption and desorptionof atoms are observed.Bœidesthe temperatme and t'hc cknmicalnattzre of tlm atoms, the strength of 1%- proces- also depemdson the atomic sites on terrazes, happ= on or in the surfaze, e.g., at islandsor near steps. Other proca
dsfFnqionon terraces or along stem and jnterdlelnion due to exchangereo tions. Onepossibleconxqueace LsnucleatiomAaotheaone is the atttement at stem or isLandsgiviag rise to a layer and/or isl=d growth. The elementary ldnetic processesare also infuealced by the sudaze morpkologyard geometrya,s the presenceof tzenehesand stepswhich detmmnlne the local suzfaceemergy.The spatial vadation of the surfaceenergy deoes the adsorptionsites ard the difhcqion barriers. The clhn.razteristicbondhlg sites enea.a of an atom at a specialsite a'adthe enerr barriers betweea.such the probabity for adsomtion, desorptiou and difhlqson of atoms detèrrnsne at a given temperatuze. In the thmrrnodpmrnic equilibrittm al1surfaceprocesserproceedin t'wo opposîte diredions at equal rates accoreg to the Sprincipleof detailedbala'ace'. Detailed balance means that the rate constacts for the forward direction, vç, a'nd.tke back-wa'rddirection, /b, of a process satLsfythe rehtion where AE is the energy dsFerence betweenthe vtjrs = exp(-1S/kBT), iaitial an.dftnal states. Processessuch as adsorption and desorption, decay a'adformation of 11a41*,etc-, must obey the detailed balance.The equali'ty of forward azd baclcwazdrates in equilibrium is incompatible with the net epitaxx growth of a surfaceoverlayer. Crystal growth is clerly relate-d to non-equilibrium ldnetic processes.Yet the 'pzincipleof detiled balance'is still 'h11671ed. However,closiagthe shutteo of the emlsion cells, Le.t intenmptholding the system at not too high ir.g the molecularor atomic beamszua'nd temperatuzes, a ratb.ers-tatic sttrfac'e'caa be prepared.,which doesnot change the principal strudme a,s is indicatcd, e.g.: by the conselwation of a cez'tatu LEED pattern. Such a surface can be consideredto be in timmnodynnmt'c equilibri'tzm with the substrate and the surroundings, e.g., the restgas.
2.2 Thermodynnmic 2.2.1 Thermodyzmmic
Relations for Surfaces Potentials
The eqttilibxium state of a one-component system consisting of N pxriucle at a %cd temperatmv T and pressurep is the one wit,h the mn'nimum Gibbs 1ee enthalpy G(Ttp,N) (2.1), G = F + p:r
whereFLT,Jr A'),
(2.1)
2.2 ThermodynxmscRelations for Sttrfacœ
F
U
=
-
ST,
47
(2.2)
is the Eelmboltz Fee enera. It is related to the intelmal energy U = tJ(S,J,rN) and the entropy S by a Legendretraasformation. The energy conservatioalaw atd the relatioizshipbetweenheat and work n>.n be written ia the fo= KU = TdS
pdF + JzdN
-
(2.3)
for azt kTIIIiteSiIIIAIchangeof the internal energy. Variation of the numbe,rof pmicles N Lsalloweddue to pazticleKxchaagewith a reservoir characterized by the cEemicalpotentiat Jz of the particle. For an isolated system with no
heat Gchange(0 is
=
0)a'n.dparticle Gchange(dN= 0)the intcrnal energy tdy
constant at cozkstant volume
0).The corresponeg
micascopic distribution is the micronltnonicaleasembleof statistical met+n.nics. a
=
The thermodynmp'c potential G (orF) can be used to deriw the thew modynnmlcquantities of the consideredsystem at consta'at temperatm'e T, paztide aumber N and pressur'e p (orvolt'tme F). Tn6mitemnl changesof the three vadablesgive rl'- to l'n6nitesimal s of tlze poteatial, so that dG = -S&1' + V'd.p+ MI:'L?V or
(lF
=
-5'dT
-
pd7 + Jl KN.
(2.5)
1xlequillbriAlrn, F is a ml'nimllm wtth respec'tto the Her variablesat constant T, 7, and N, whcreasG is a nninirnllm at constant T', p, aud N. The correspondingmicroscopic distribution is the rltnonicazensemble.The Gemical potentia,lJzic (2.3), (2.4),or (2.5)is given by .-
(w'&)(w'') (N,x) -
S,V
-
:r'I5J
.
r,a
Under normal pzessm'eof abou.t 1 atmosphere,the dllerence betweeathe Helmholtz 1ee energy F and the Gibbs free energy G, J?= F
-
G = -pF)
(2.7)
is insîgniftcan.tfor a bnlllcsoDdor liquid. TMs holds'i.apazticular for volllrneinduced changcs-pdF. 'nus, it is mlmcient to 'ttse F for most cases in solid state physics.Thc difrerence (2.7) is Kramer'sgzandpotential J2= .O(T, F)Jz) Despiteits smxl:ness, more preciselyits vaaishingMuence on changes (2.22. irt bulk systems includizlgphmse trnnsitions, the potettial is convemiOtto use for system transformxtions that occur at a constrt temperature T9 vob:rne F, and chemicalpotential y. This may be of pazticular iuterest for tke surface
region of the estems under consideration.Togetherwith tKe Gibbs-Duhem
48
2, Thermodyaamics
eqaation, SdT
-
Vdp+Nd;&
dfz = -SdT -Jd7-
=
0, inflnitesimal nhxnge of the variablœ result
Ndp.
(2.8)
In this cmse the thermodrmrnic properties of a syswte,m are governedby the graadcanonicalstatistical operator. Comparisonof (2.5)a'ad(2.8)indicates a tqansforrnlttion law J) t= .F' /zN.
(2.9)
-
This is a consequeaceof the fact that the Gibbs free enthalpy varies lin-rly wit,h the zplmbe.rof particles (cf.(2.7)and (2.$), G = P'N vrith the proportionality factor /.: =
(2.10) for eaG homogeneous phaze(2.1j. gLT,p)
2.2.2 Sarface Modmcation of Thermodynnmlc
Potentials
A stlrfnne of az'e,aA infaencesnot oaly the spectroscopicproperties of a solid but also its thermodynnaicpzoperties. Tn order to discassthe Muence of a h'eesllzûce of a hnlfqpacc, we follow Gibbslideaof the dteqllsrnolarl dividing smfaceh(2.2,2.3). This is illustrated in Fig.2.2 in terms of tke particle demsitjr n = NjV as a fnnction of the distaacenormnl to the sltrfnce. It es gradually from its solid to its vapor vaâue.In Fig. 2.2 ihe vertical lines incticatea partition of khe total space into a bulk solid vobnrneVi, a bulk mpor vol=e 'Ui,an.da volame W of the trxnRition region,the surface.The correspondkg nz and 'pu characteriz,e dwlnm'tjes the llnlform bulk phases,the l-nal'-in6nl'tel solid and its vapor with whic,hthe solid coexists azd which occupiesthe othe,r halfspace.The sllrfnre region, whose spatial eoctentis of atomic dimeasions is thus a strongly tnhomogeneousregion sulu 10 or less atomicylayers), (about rotmdedby t'wo homogeneous phases,the solid and the vapor. Unforttmately, the partition in Fig. 2.2 is aot uitîque, siace the numbe,rof particles in eac,h
n g. 2 v c . Parucle density'of a onmcomponent system near a
surfacearotmd z
=
0.
2.2 ThermodynamicRelatiomsfor Surfaces
phase 1 or 2 dependson thc n=ber
of particles Ns in the s'arfacetransition
regitm.The same holdsfoTthe voblmes.TMSuncertatnty is of the same order of martude as the surface eFectitself. Howeve-rjiu the framework of a the partition is madeunique macroscopictheozy kere the titrqrrnodyztpml'œ, by applying the n.ataralconditions (2.1) N
ï,l = J4.+ Vi, 'nzh + nzFz.
=
(2.11)
J.ncompnriKonto the total nllmber of particles the nllrnber of partëclesNs in the surfaceregion is assumed to be negligibie. In thc macroscopic limjt one has N :2:0. The extensive thermodyzmrnic potentl'n.lq tmde.r consideratîon,the free energy F azzdthe freeenthalpy % can be written as contributioztslom phases 1 and 2 plus a surfaceterm. The ozigia of the surfacet-rrn rltn be discussed 2.2,2.41We follow tke derivation of tssingat least t'wo eqaivalemtvîews (2.1, a'adSmmjaard as we)l as that of Desjonqubes 12.2J Landau aad Twifshitz(2.14 and considerKramer's g'razd potential. Sinceiu thlrrnodpmmic equslibrillm the pressare p is the same in the tvo homogeneousbulk phazes,at least for eqaatioa (2.7)can thus be written in the fonn a plaue surface g2.l), '
.
-p(M.+ vi) +
J) =
fls.
The surfacecontTibutionJ?sto the grand potGtial should be proportionalto the surfacearea A
q
(
'
wîth y
=
(2.13)
'yA
ms the'surfacee-xce,ssdensity of
.0.
2.2;3 Sttrface Tension a'ad Smface Stress E
czcexss be identfed as the s'arjace cc energymr zlrzif area or s'udace .freee'rternyfor short (butimprecisely) Lh-' $bra oae-componentsystem the chemîcalpotential p is equi in b0th phases Witlz ihe total nttrnbe,zof particles accordiug tn thermodynxmlc eql'phbrinpnn. From our approximate description is st2l A11611ed.. G = yN (2.40) to (2.11), of the tclividiug surface' with Ns = 0, one readily obtaius Gs = O for the contribution. Tlms, M'itll (2.13) : sizrface
Tlle proporttonality factor ''f in
can (2.13)
.
:
I
Fs = f)s = JX
(2.14)
fensftm, holds. The surfacee'xcess h'eeenezr 'y is sometsmescalledthe snrj'aco althoughthis term is somewhatconhlsing despitethe corresponding common t'nst of measmement. Using the Glbbs-Duhemrehtion it cau be shownthat all thermodylmrnîcquactities of a s'arfacecan be expressedjn termK of J. The
2. Thermodynnm''cs
50
of ''f also allows the de6nition of the s'srfqce temperatme depezdence e'rcas,s entropy Ss (2.14,
'
'
Ss =
0J2s
-
OT
=
,z1,/$
-.A
X
&T
.
Aj/z
.
The surface9ee ene'rgy ,(2.14) contnalnstwo physically diFerent contributioas. Au l'n6mitesimalchangeis given by dl's = y ILA+ A d'y. The flrst term y dx represents the reversiblework doneto cllange the surface =e,a. by dA. This cllangemay happen by incremsingthe n'lmber of atoms ic 'ihesurfaceat a Sxed averagedarea per surfaceatom. lt ksthereforerelated to an ideally plastic deformation. Ic contrast, the srcondcontribution A d'y is related to a,n ideally e'lastic defo=atiom The zu:rnberof suzfaceatoras remnsnq constant but their interatomic distancesvazy. This is accompanied by a vadation of the =e,a, per atom. Sach a stretched surfacehas a modifed surfaceenergy y-j-k, whie,hleadsto azl additional contribution to the chaage of Fs The c'hacgeof the smfacearea may be interpreted in a rnnnroscopicsense as the consequenceof a (biaxial) strain ,ac'tjng on the smface atoms. With the components%j of the adual strain tensor azd assllrnl'aglinear elasticity the work done i'a the surface region can be descibed as the c,hangeof the elastic enerr .
dFs = X Voudqj
(2.17)
.iIJ'
wlth (7vJas the componentsof the s'ur.fatte siress tensor (2.54. They are deoed as the spatiatly averaged deviations o'u = J dzko(z) cz?j) of the local stress (qjtz)from the stress âeld deepin the btlllc ct. Accortiimgly,considera plane norrnnl to the sudacearld labe,l the norrnxl to the plaue as the dimction j, Ju îs the force pe,r Ilnnit length whicll the atoms exez't across the line of intersection of the plnnaewith the smface iu the Ftk. dtrection.The dements of the strain ttmsor qJ are dqflned in direct analogywith the corruponding bllllc quantities. With d.â = A E: dqç, comparison of expre-ssions(2.16) a'ad -
leads to (2.17) *3i = '/V +
(2.18)
.
'
witk %g =
ru
,w
.
tlevLt
(2.19)
TMS Lsthe venerable Shuttlewoz'th equation (2.6). Sinze in a liquid there is no resistaace to plutic deformn.tion,the second termq in (2.16) and (2.18) vanish aud the surfacee'nergyand sudacestress becomeequ.at(2.5).
2.3 Eqailibdum Shapeof SmallCzystals
51
It1 the case of surfacœof czystalsthis is usually not the case. The deviatiozusvij play an important role for the reconstruction or relaxation of metaz smfaces (2.74. The stabitity of the 7x7 reconstruction of snrniconductorsurfacessuchas Si(111) sçemsto be a consequenceof stressesin cllfbrent atomic tensor '3'dependson the surface symmetry. If a layers (2.$. The secozd-rn.nk sllr'hce has tikreefoldor lzigherrotational s-immetz'y, then the tensor becomes is So the smfacestress (2.18) diagonz with equal components, no' = 'ré%. isotropic in a surfaceplane. Anlsotropic smfacestressesmay haveau Muence on the step geneaatioa on surfaces.For instance, the 2x1 spnmetrsc tasy'mhas only ml'=or symmetry with the metric)dimer reconstmtdion on Si(10O) point group 2mm (m)(cf.Tables 1.3'azd 1.4)=dl Nence)the surface stress with a height of one atomic teasor is anisotropic. Single-hyers'teps(steps layer)may occur witlz dlserent orientations with respect to the Hsmers.The o:a dsferent tezraces formatioa of b0th 2x 1 and 1x2 domaias (cf.Fîg. 1.14) is likely, since tEe sarface becomesisotropic 9om a macroscopic point of view. For semiconductorssmface stresses have beea direcvtlymeasured (2.9) or calculated(2.10-2.12). 'A liquid Lschaœaderized by a vauishingresistanceto a îow of atoms 9om the bulk to the surface,a'ad'dce versa. For solid surfacesone may therefore coasiderthc tenmr '? as a driving therrnodpmmscforce to move atoms h'om the balk iato tlze stu'facelayez. The signs of the diagonal elements deterrnsne the strain stste of a surface.When 'lnrê> 0, then the smfaceteads to accw mulate more atoms. When Trê < 0, the opposite tendency sho'aldoccur. As a consequcncethe surfaceatoms prefe,rsmaller or lrger lattice spaciagsthan in the bulk. This arzangementis accompaied by compressiveor tensile stzrtinq. In the cmse Tr'? < 0, a'aothe,rcozzsequencecokfd be a (srnzlletr.r-brenb'ng) smface reconstruction of the type schematicmllyiudicated in Fig. 1.9b. Stuw facerearrangements like atomic dislocations and elastic buckling of a sulface colzld also be possible.ln a metal surfaceatoms might be placed in unfavorablebonding posltions with respect to tàe next laye,rof the substrate. shouldbe large enoughto compe=ate for this enerr expense. Thereby,1Trê1 The important role of tke Hi/ereace be>een sllrfnzte'energyand surfaces'tress for the actual smfacestracture is generallyaccepted.Howeve'r,there is a contmveray about this dl'ference az a drivi'ag force for the recoas-truction of at surface (2.11, 2.13,2.14J. leazt metal surfaces,e.g., of the Aa(110)
2.3 Equilibrium 2.3.1 Anisotropy
Shape of Small Crystals of Stu-face Enerr
The smface free e'nerg.rper llnit area 'y of a cez'taincrystal surfacevaries with or the its czystallograpkicorientation characterizedby the surfacepla'aeLhkLt izl general.A plot of tMs energy surfacenormal zz, i.e., 'y = 'y(h,kJ) or y = 7('n,) plot (2.15!, plays azl importa'at role in versus orientation, a so-called W'IzlF
2. Tkermodynmics
52
Un! ro1(1
(ln)
Ez-g.2-3- A
ce,
whic.h -:s slightly
mus-om-ented Fom
the
(01)s
ace.
the theory of the eqllsFlbri:'mshape of crystals and morphological stabillt.y a two-dimensional soltd To illustrate this orientation dependence, (2.16,2.174. 1om the gO1) direction is shown in Fig.2.3 for a squaze which is mirw-a'gcted surface(1>> 1)represents a lattice with lattice comstant a. Its nomsnal (1n,) 'dcsnal (01)stzrface.It consists of a Mgh n'lmber of (01)ten'aces separated by atomic steps of height c. With f 'w A as 'kke azlgleof orientation of (1p1 orientation, the step density is given ms tan ela.If Js is the agnsnqtthe (01) emergyper step aud g(0)is the energy of a (01)face,the mndhce energyof a (1n)surfaceis
'y(8) =
,
cos
9
g(0)+
taa 9
ps
(2.20)
.
c
The prefador cos 0 g'uarantee that the relative amount of the (01)terraces to the total surface=e,3, reduceswith hcrewsing azzgleJ. The htcaction of steps IhRAbeeaneglectedin (2.20). Asmlrnsngthat tbe s'tep mode: also holds for large.rangles8, tlze reslzltîng siu 81is plotted 'm Fig. 2.4 for angles # in h'ndion .,7/) = g(0) Icos :1+ J.r-I the fokzrquadzantsof the plane. Suc,ha polar plot of àhesurfaceenerr cac the above be dz'awnfor 0e,1:place 'wit%x'na 6n5te crystal. Even msmlma'ng trivial dependence of 'y on 3, it gives a WG plot of the surfaceenerg.g(2.15J. Y z
Z
z'
> > - e
.- -
.--w
N
/ î
N
N
N
-
!
''k
$. N.
( (X
/ l
z
z
j
0
hx .
t
'> >.
N N.
!
' e'
N
N ..ee
N
N
y;
...
-A
A
Z
/
1 l
X
Fig. 2.4. Polar plot of the sarface ezkergyfor a vinlns'l s'urfRe augle 8. The with a 'mzsalignment simple energy exwession (2.Q0) has been USCXCI for several ratios
J%/ h(0)4q= 1 lsolidlinel, Aa (dashed linel,and ) (dotted linel.
2.3 Equilibrsum Shapeof Small Crystals
Polar (W:)1'FF) plots of the smface auerg.gare possible fol. arbitrazy plres in the crystal. lt is obvious that the variation of the surfaceenergy with orientation is of great Lmport=ce for a a:lmber of pzobleznsrelated to sttrface inhomogeneitiesor surfaee morpholoor. Notice thst the surface energy in Lsa conthmous gmction but that it hmsdiscontinuousderimtes' Fig.2.4, 'y(J), at angleslimiting the quadrants,e.g., b = 0. J.nfact
-.2
'dù
dê
:=+0
z
=
-
*=-0
.
c
That means, there is a cusp at 8 = 0. The increase of the acgle 9om : = 0 iè large vazuesis azcompazkiedby an increaze of the s'tep density. A proper ixpression%r 'y(û)must hencei'ncludethe iateraction betweensteps. Ia this èase it has been sho'wnthat
'y(p)lzasa cusp at
eveo- angle wilic,h corresponds
to a set of Msller indices whoseratios are rational Iplmbers(2.181. The shat'p.
npessof the cusp Ls a rapidly decreasingfnrnction of the index
mw
Yig.2.3).
1/a4 tsee
rone exnmple, whiclz can be interpreted as a renlization of the model above: could be a surfaceof a diltmond-stzudmem'ystal, e.g., Si, EètsEctuqsed àfiented between (001j an.d (111). As the sxmple orientation is tilted 1om j1'j 01)to g111j, i.e., û = 0 to 54.7o , the smfacemorpholor varies h'om (001) .7g r
.
''.
' .
.
.''':''' :':''' .' .' '' .'7* .'.' '' .' '' .'
('1*:':':'5* :' i':' :' :''' E'
''
''
:0012 , ...
.
r
)..t... .,.j.
'
.. .
j..(:è.t'Ii 1 ' .: . .
1.
()a,!
. .
11a
el-ls
.
-112
,
r3 :
:
i:('
11S
:1112
337
r5
55 12
.
..
el.:$-1
'
'
:E'
-!19
114
335
.
117
.
..' .. ' '
.. . g
plsnes. The and (7.7.1) #lkkE i''!7 'a'à sideview of a disrnond lattice betveen the (001) of the lattice li'aesrepresentprojections of atomk positions onto the jkflsèçtions ;E iï'd''' ;,E(. .)..pl%e The connections of the dots with tbe circle indicate the surfaceplpne àtï' oiedby (hk2). After (2.l9J. t@'' p
.....
.
-
-
'
-
Thermodneœ
54
(011) (111)
+
(137)
(11i) (113) (114)
(001)
Pig. 2.6. Stereovaphic trirmrlè for cubic-czys'talsurfnrm ahîbiting the low-index corner points (111), (011), and (001).In addition, one hig,hindex smface.is indicated in the i'atedor.
to (113) to (5512)to (111) as has been mexqlpred for stli.con (2.19j. (114) Possibleimk-mmediate surfu orienvtions are shown in Rg.2.5 irt a (lï0) to
.
plane. The Mlller indicesof these snxrfacesare indicated. Given a set of Mnmer then tke polar aagle8 can be obteed 1om tan 9 = Wvk. iatlices Lltkl), The sllrêxce orieatation r-qn atso1yeVaHH Fom (oO1q in anothe,rdirection thnan(111). Arbikary orientations of high-index surfacesof cubic crystals are visaalizediu a sterovaphic tzi=gle. The construction of the stereovaphic tregle ks based on three-dsrnensional consîderatiozus. M crystallovaplkic cnn be given by poàt.s on a sphere. If one connects all points of direc-tiozus the upper (north) half vith Ge south pole, eve.ry direction can be marked by the pokt of inteoection in the meridiaa plre. Tile aazest are'a of noneqdvalent points is givea by the s'tereopaplzictriangle plotted in Fig. 2.6. TEe three corners are givea by the three low-inde.xsurfacw. The surfMes whlch Iie on the co=ection lines be-een the low-index eoz'ae,r points are formally lin a bulk-tmlncatedview)composedout of the corner planes.Any plane in the inteior of the trinmrle kscrmposedin the same sense out of the three come,r place's.TMS shows that the compleYty of the diferent planes increases 9om the corners to the connection lines to the interior. The variation of the sllrfnne energy betweenthe low-hdex surhces (1.11), acd (311) , (110), may also be representedversus a polar angle 0 iu a (100) plane as izl Fig. 2.7. Witk a surfaze normal E!l0q for $ = 0 one observ' e's (110) the variation 1om (ï12(,(I13q, (11:35, E1ï0), (1ï15, (001)) (1îl),(1IZ!,(00i1: Axvllmsng thn: the (110) and (001) surfae are b0th mirror gïllj to LI1J). ezm be represemted plauœthe measuredvalues'y(#)(2.29) as a relative slrrênrza Wlnlfrplots have aksobeen energyverstus the polar angle as in Fig.2.7. Sn'ml'lnr ex-tractH by other groups (2.21).
2.3 Equllibrium Shapeof S=xll Czystals :.20
glch
4. W
.
ve w
G
1.-1.0a
r
w
'-N
':. ns & %-'' >'
*
pp s o
Z
o
s
5 u
;
*
t (lot)l k
z
#
120
*
wg .
p
N
) )
= o
>-
(3.11) (a:1)
=
:.1.j
r:l
55
au an
p
:
:
m
=
':
.
u
$f
:
e
J (4(* I
r
Z
% /
()
90
'I8O
040)
u u a
p
%
t t ! (A11h (117.)(I:.'lJ
:-95
r u
t
(1111 167û
,36:
Fig.
Slnrfnzteenera plot 7L9j.The surface energy ratios havebeenextracted by a reverse Wtzl'm constnction fxm voids 2.7.
(2.20J .
2.3.2 Absolute Values for Surface Energies
The absolute'valueof the sarface9ee euergy of solid materials is a fn'ndamentally importan.t energetic quantity which is neededfor the tmderstandsngof a large numbe,rof basican.dappliedpheaomemwsuc,has crystal growth, smfwze fafettklg, growth and stabllity of thln fzlms,the shapeof sma:tlmystctes in a suppoded catalys't, and ma'ay generazmaterials sdence applications. Dœpîte its wemrecogOed sigadcance) there are relatively little reliable primazy data of ecperimentalsurfacefree energiesbecausethey are vezy diëcult to memsme. k. 1u.contrast to ftzid interfaces,where the surfaceenerr or tension r-an usually be obtaizzedquite emsilyby capillar.y and similar eyperimental tech-
zliques)the determsnation of g(a) for solid-vapor inte'rfaces(i.e.1 surfaces) is eAremely diec-ult. Thereforc, at present not much reliable expem'rnental information cau be fotmd about J in the literatme. TMS is in parrticulartrue for the Anssotropy of the surfaceenergy'.Only a few ttvthnl'ques,such as zeroaud cleavagetechnsques(2.244, havebeen USHrepeatedlyin camep(2.22, 2.23) the past to obtain (pln.ntitative valuesfor a, lirnited nlzmberof solids,mostly metals. Irt other exmerimentsthe equilibrium shapesof voids izt crystals, suc,h as Si (2.201, have been measmed.The void shape, to a good approximntioa a tetmknldecahedronlsee also Fig. 2.10a), is related to the eqtlilibrblm czystal shape, and the surfaceFeeenergy is extracted via the W111fF construction (see next sedion) of suchau equilibrittm shape.Resultsare lis'tedfor Si sarfaces i'a Table 2.1. Absolute surface 9ee eaea'giese-q,n also be calrnllltted..However, ârstprinciples calculations are dlëc'alt for numerical and methodological reain detail ia Chap. 3. The slab coogarations usually used sons as dkscussed
56
2. Thermodynamic.s
Table a.1- calculated slxrfnce energie.sa' (1 J/m2)of low-isdex surfacœof fcc semiconductors cmtnlllzing in the diamond structure (C,S$ Ge)or zinc-blende and of bcc (Mo,W)and fcc (,M,Au)metaks.Reconstraetedlrestaxldttre (.fazAsl Iu the (311) column (metals). laxcdlsnrfxces lzaveyyens'tutliM for sezniconductmrs of TnAn Mo, W) Ls listed. ln t'he the value for tke (ï11)surface 1(211) Se'rGne) compotmdcaa the aaion cllemic.alpotemtialis fLXI.dat Jzas= Jolk 0.2 eV. In G't.'XIvaltlt.s (2.204 are also given in. parectkeses. the cmse of Si, expe-ram' -
(311)
Rderemce
(100)
C Si Ge
W
4.06 5.51 5..r1 5-93 (2.24 1.41 (1-36)1.70 (1.43)1.36 (1.23) 1.40(1.38)(2.25) 1.17 1.01 0-99 1.00 (2.251 0.66 0.67 0.78 0.75 (2-121 3.24 3.11 3.34 2.92 (2-261 4,18 4.01 4.45 4.64 (2.271
A1
1.35
1.27
1.20
-
Au
:.63
1.70
1.28
-
I'nAK
Mo
(110)
(111)
Crys'tal
(2-27J
(2-274
possesstwo sarfaces.Izl the cmse of crystals consistingof only (2.12,2.25-2.27) one element the use of symmetric slabsinducesconvergenceproblennq(2.25).
of slab 111the case of compotmdswith partially ionic bonds,i'a the malorit.y the elecorientatiozssone buasto deal with t'wo inequivalent surfacesto A11611 Moreover, in th.is case the surtrostatic neutratity coztditlontc.f. Sec't.3.4.3). facefree energiesdependon the preparation..conditioas or, more precisely,on the chernical potentials of the constîtuents (cf.Table 2.1).Neverthelv, suck calculationsare =ow possible.Resultsfor covaleatand ionlc semiconductozs with a'a fcc Bravais lattice as well ms for bcc and fcc metals are listed in Table 2.1. A rath.er compléte colledion of data for the llnrelaxed surfacesof 60 metals Lslisted in (2.27). The absolute valuœ aze of the order of 1 J/m2. Eowever,the values vazy with the surfaceorientation az.d the bondingbehavior. For smicondudors a surfacereconstrudioa may considerablylogez the smfacefree enera. .An intuitively reasonable,but rough estternxtionrelatc the surface energy 'y of Camaterial to the cohesiveeaergy per bond? Le., apprnvfrnately half of the value given in Table 2.2. Together with an sttrfacœof area of about c2/2per atom (avalue whic,his correct for (100) ziuc-blende/diamond ) one Mds for econ 2.5 J/m2withou.t reconstmzctioninto accm'nt. The absolutevaluœ of the surface9ee ener#es dependon the dnn-qity,in particular on the elmtron density n, of the m= terîxh. This is cl-ly demonstmted in Fig. 2.8 fqr snbonded metals. With the sttrfRe eneir decreasingdis-tanceof the eledrons rs = (3/4m)w (2.28) iucreasesan.dvice versa- The behaviori'a the tow-densit'ylimit rs k 4 follo= tlle predictton of smface energieswitlnsn the jel'tiummodel of a metal surface(2.291. '
2.3 Eq:lsllbrblmShapeof Sma'tlCastals 1.4
57
AI
1.2
D
E 1.o
R
: >-
> 0.8
Ga <:)
P o
6,t)o.6
In
ol
cd
:
*w
.
eb g ug s zn
o.4
't
@; =
ca Li
o
(E;:I 'E
E':' '' ''.'
2
.'.''' '' .'.' .' .' '' .'
q':'E' r' :':':' :'
:E '
t
.
Na
L Hg
g.g -'
Ba
o
nz
:
sr
u
o
3
4 .rs
KO
Rbo cs
5
6
@s)
.
q.:,dài:. a.B. Surfacepeeenergy of snbonded metals (2.27) versus
tke averagedistance the electrons (2.282. 'ijii'ïnL' The amuotropy of the ezzergiois negleoted.ortly valuesfor
are plotted. fcc: rj'(zztlr,, (cubicor tetragona: cystats)or (0001) (hexagonal crystals) bcc: cdrcles,hcp: tHaugles,a'ndbct (body-centered diamonds. 'àkttirès, tetragonal): '.,:;Lj . .. ' .. ..
.
é''..'
:'''.; '''r. ... , 2:'.
(;(''(:.y'..; .'
.
iE/'Lï:jrjj,,à) tjt wujs construction .
' ;f'L.
.
.
. .
. .
: k (...k y.,:. ;:'ry.! .j( j Lq(E .k .: 5 L :'
.r.
. . .. .
..
' '
..
.
..'
k 'àè''lhzajuotropy of the suxfacef1.%eaea'gydetermiles the equlibri:rrn shape 2111 ziti' ,costals at a particuur temperatme z, rrhe czystal is wsm:rnedto : ( ''! 1%'.l':J' size so that edgear.d ap> egeds ; rf rnnnroscopic (orat least mesoscopic) ;yL c%t the eqttilibriïlm 9k1.4 (2.8)ar.d (2.13), t keglected.Accordingto expressioms shape (scs) at constaat temperature 7. witu fuxedvoblrne >- and '$'f'! ( kt'û; Vkak'' 'èxl potential p, is determc'nedby the zairsvnxl excess suzface9ee eneroI.l i'' Liîzukebhèct to ths surfacea4.
'
'
)ik: f ':
-.
.
.
:'.
'
..j.,,...
..
.. .::..' .
'
.
,.
.
'
'' '
;
,.:
'
..
,
.
E i.1.ttè.t);E,E... .lr' E .:..)( .
@' q.( '
--.
.
'
.
:.,
:.
u jiiyi #,
,
.
'
:
....
..
,
'E .
jjz 'y('n,)
.
.A(F)
(; gg; .
58
2. Thermodynaaics
subjectto thE constraint of âxed volttme kr
=
Yd'Z.The WIO
theorem
v'(z)
the ECS is not nec-arily that of tEe min:'mll= smface area. It may be a complex polyhedron witll the lowest total surfaceenerrr for a given volume. A minimal suzfaceonly occurs for a perfectly sphericalWhtlfr plot, i.c., an. isotropic excess sarfacefree qnergp The corzepponding ECS is a sphere.Tikis hasbeenexperimentallyshowafor ia the absenceof gravity. ln the case of czystals the variatjon water dzoftets of y with the normal .rt will produze, on eac,hsurface elemènt dA, a foree proportional to àyjonwhie,hwi.ll tend to alter its direction a,t the same time as ''/ tends to shtr'tuk5ts area. Consequently,the ECS can no loager be a sphea'e. Figure 2.9 sehematicallyshows a res'alt i'n.two dlmensiorus.It indicates shapeard the morphologithat the WttlF plot c('n,) governstke eqlTilsbrblrn In the case of real three-dimensionalC'Iy'SC;aLS cal stability of a c'zystal(2.20j. the sîtuation ks more compncated.Besidesthe variation of 'y with the surface normalhalso the strength of the vaziation plays a role. This is demozustraîed in Fig. 2.10. The ECSis construded for silicon tytking mto account the surface and (111). In Fig.2.10a energiesof fom' orientations, namely (0011, (1131, E011), but < 'y(100) < 'y(110) < 'y(113) and b the same eneageticordorlngof 'y(111) dl'ferent values (see Table 2.1)have been ased. The ftrst paramete,r set is derived 1om meaaurementswhile the secondone %nAbeen calculated by azl the shape,skz thc t'wo âgtu'es(a)and (b) ab initio method. Qaalitatively aze the same. However)the relative areaz of the cnrstal facets vaor with the absolutenllmbe'rsof the various energles. Tn.kinginto accout only the two orientations, the cazbicsyma'n.d(100) lowest surface energiesfor the (111) facets ocwtalledron with (111) shapeis a regnzla,r metzy suggeststhat the W111fF states that 2.16,2.30) (2.22)(2.15,
.
Fig. 2.9. A polar plot of the stau face free energy for a 2D mystal (solidlinel and the ECS based construcion (dotted on the W111fF in (2.30J. Seealsof1g7.u.% Linel.
59
2.4 SurfaceBneav and Morphology
(b)
(a)
.41% o'w. ''q t.
.
.
.
jh
.
s .h. ;. ykayyjo
... . .
, ,
.
.
.
.
.
-
% # ..
l's .
y
,. ,.j
.
.k. .
'
.v . 4 ;
.
j;.. N.-K
'
.
.
'ê
x. . . .
.
.
.
x :fk
.i; j. .
! ,,,.J ï. >
), b
.
1 'Il; ?* Wr' (' .
. dbài. .
i.
r
..
.
,
...
-
t &.g'
.
v q. +i ! .
)k4',''',
'è, <4k
.
.i :x'. .
..
-g ':L-
J'ç ' .èIt
t)a Et.. k.jtqt.. .
,
qvk.lt., f'hl,g
..
'
qh; j.. 1.1. ,p'x$ $
'.k ,
%.! . --*t5 2;.1'!.$M G$1q% :!jr%%!ujjy'J)j'jj
'
.
.
.-
'
consiruction Si cryst.albased on the W411F lzs'tng (a)experlmental values or (b) calculated values. Fottr surface orientations are consioered.The surfaceenergiesare taken from Table 2,1. The azems with the and (111) ozieatation sequence(100), va,ry fmm blxztk-to white. (311),(110),
Fig. 2.10. Faquilibrîumshapesof
a
planes perpendicttlar to the cube ames at the tznlncatedat eachapœxby (100) center. The inclusion of the (113) sn.rne distance from the oc'tahedrron eae'rgy the octGedral shypediscussedgequently for homopolarsemiconêeastroys facets in the tetrpkmsdeem.heclron ductors (2.31). Oa tlze othe,r hand, the (110) disappear completelyin Fig. 2.10b iudicating the l'mflaeaceof the (Fig.2.10a) ''
values. constzuctitm are' tlmrrnodynlkrnn'CzystazpLanesthat are part of a W1:1'fF cally stable (2.301. Sinceall foar oriectations comsidered appear on thc ECSS to a cezdnin ex-tent în Fig. 2.10, the fo'tzrreconstmzctedor relaxed sïlrfn.ces azd (311) az'e stablesurfacesi.c thts resped. Of co:tme, (100), (111), (110), the (110) areas in Fig. 2.10bare negligibly s=n.ll becaaseof the high smface give,nin Table 2.1. The inclusjon of the 16x2 leconstruction eaergy'J(110) surfaceshould,however,lowe,rthis vallp. of t*e Si(110) a bsolute'y
2.4 StuYaceEnetgy and Morphology 2.4.1 Facctting and Roughening ne vadation of the surface enerpr with the normxl caa already Muence the shapeof the surfaceof a semi-i'n:nite crystal as indicated by a possible buelclimgof the surfacein Fig.2.11. This buclding happens on a mesoscopic Let tts compare the smfacee'xcess thaa atoraic distances). leagth scale(Iarge,r energy of a âat sudace linnlted by a plane A with its normal e = ()Oa'ad.a sucfacewith a s=n.ll polar bucklingpreserdng the ave-ragesurfaceorientatiolu Then .
Fa/=
,
'y(8)d.A A/
=
(lA a?(p) cos 0
-
al
(2.23)
2. Thqrrnodynlmlcs
so
Fig. 2.11. Sml1.llbunlêsngof sudace.
a
The azmtrnption of a, weak valatio:a of 'y 'with 0 yields aa exlmnqion of the integrard up to secondorder, '
F,
=
d -2
J(0)A+
dl
x
dyl + p=o
1
92
'j -4
d2'y C1.A. + ,7(0) d, z :=o
(2.24) V
The &st term givesthe eleror of the Qat sarface.The secondone vaaishesfor symmetry Teasons. The third tnrm #vesthe enerr due tô surface bunkling. Oneconcludesthat for e/(0) > () the fat surfaceis stable (or + (d2'y/d82):so whereasfor 'y(0) at least metaztablc), + (dW'-t/d92)#=a < 0 the bucktedstlrface
is more si'able. Izza lzighly nansqotropic crystal with a strong variation of 'y with the smface orientation the buclclimgesed dlscussedaboveindicates that such a czystal will ml'nlml'ze the surfaceenergy for a given Itornsmal normal by the formation phenomenonwaz Emt discussedby Herriug (2.16,2.301 of facds. This jacetti'ng constntctiozu with the hèlp of a geomctzsc.al However: at :5Up to noo tempeaxture efec'tshave not beenconsidea'ed. aite temperatures the discllmion must be supplemented'by the Melusion of emtropy esects. At low tempezature a stretched surface Ls:at on the micl'oscopic scale. Whea the temperatme incremses,thermal factuations appear: tlle surfaceis no longer Qat and may buclde. These thermal âuctuations are related to the mean square deviatioas of the atomic positioas with respect to the averageposîtions. Their strengths dependo:a the actual temperature. The fuduations may remsn'n Gnsteor diverge.J.nthe fzrstcase one speaksabout a sm00th surface.J.nthe secondll'rnstk one says that the smfaceis rough. Above transition. a criticat temperatare the sarfaceundergoesa roughœnén.g 2.4.2 aD Versus 213 Growth
I'n Sec't.2.1 we stated that, i.a thermodrmrnl'c equitibril:m, there is no net growth. All elementar.yprocesse.sproceed in two opposite directioas according to tEe priaciple of detailedbal=ce. The czystal growth must be a nonequilibriclm kinetic process. The xesulting macroscopk state of the system dependson the reaction paths in the covguration space ms tnclicatedin tlle obtained state is not Fig. 2.1. Siacethe result is ldnetiem,llydetmmnsned, pal'ts of the overall processmay necessarilythe most s'tableone. Neverthelcxss,
2.4 SurfaceEzlergyand Morphology
61
'be ldnetically forbsdden,whereasothers may be in loci thermodynazrdc equilibrillc). Consequently,equilibrblm arrlments may be applied locally, even though the total growtiz processis a non-eqz:c'l'lbzlphrn process. Exactly these ideas are used in modern Emt-priaciplesstudies of epitaxia1growth (2.32-2.341. The activation energiesof the processesocclzmqngon the growing surface,such as diffusion, nucleation,and attaeAment ot detachfrom ab i'aitio : ment of adatomsto existing islands and steps, a-rc ezxtracted total-energy calculations, The correspondiug atornic processeshappe!l in the length and time domain of 0.1-1 mm and femto- to picoseconds.Hencethe èazclllntions for modeting atomistic aspects of growth have been hampered : by thc needto bridge length an.d time scalesby many orders of mn.gnitude. First-principlesmoleculazstudies, while being powerful tn the irwestigation ipri)findividual events on a time scale shorter tlza'a 100 ps, a're not suitable Yraccessingthe time scalesi'nvolvedin epiteal growth, nor cnn they tackle i,lq.r',iih: statisticalinterplay of the nttrnerous processesthat are respoMiblefor tke ë.: Tùtitcome of a growth expersrnent.Howcver, ldnetic Monté Carlo simulations g,tfger an emcien.taad accurate way to cope with tMs dilctzlts J;).y rz Viteadof following such atomisttc approaches one may consider the ) of an overlayer more pheuomenologically. ln the cmse of heterocpikrsàrii$th lsfiky,the overlayer grows on a substrate in one of thTeedilerent growth i:'illbdes as indicated in Fig.2.12. In tlle bwnk-va.n de.r Mezve mode (2.35q, :i>ibzizs or moleculesgrow layer-by-laye,r in asequentialfashion.Jn contrastj the è:''V' yliher-weber mecûanism(2.36) leads to the fomnntion of indMdual threih.EE..)L, irtéljliéùsional Lslands.The third mode is a combinationof tke two an initial sltkisdiiensional process creates afew monouyezs, the so-canedwetticg layez, ;'':' tttu it t. thèn udivi4ual islazïdsform'wsmore matarial is deposited. This process i r,.tkskàkuws ws t:e stranslu-xrastauov modeg2.aq. It migbz 'beinduce: by sew ij'''''? 7tjktE'sfacsors. , (i) A cezta- lattice msmatcz betveen the substrate aud the Efi'' :v cnmnqtcompletely be accomodatedin an elastic mauer as vithin j;Jjrsktzlayer q.2.j J'ljhëkck-va,n der Merve mode. (ii) Alternatively, the dieezencein the czysEs'p)j;t1k. ji liltjj lrE)i or the orientation of the overlayerwith resped to the subs-trate :((12 (( ' ( jinzfzttzy )!).i. ?(!!FEVd2'' tt'sjucll a growth mode. As we will see below, the Strxnski-Krastanov ''' @t'y: gj.gts! (jjuckatsr tue selpassembledgrowth of an azray of quantnnrndots on Pr)P '
'
'
. ..
..
.
.
-
'
'
'
. .
.
''
'
.
..
'
:. '
'
.
.
.. .
..
'.
.
,
'
'
.
..
.'
.
.
.
li(#.jtt' àitiutb.
h',)ip?'k of the loc.alsurface1ee energy per t'nit )j.. jjgtilryEfjuckssioc jj.E....L-.,.ijjE. .j
area eyalsoazlowsa kk !(:;di'' 'àizin iti6n of the growth modes. One has to introducethree dsFerent energiœ, : .t,.!11.' .tf.t1(; ''a:ljj.!j jgu s:o energ.gof the sttbstrate-vamr:rn intezface,x, that of the ' ''
. '
: . . : i. . .. ' .::..
.
.
,
kEtt )kJ lav(tijjk jjg9uyym yatesaae,yo , aud.tkat of the substrate-overlayer interfacc, :(yï't)i#)) jiyl'''k.. (;;..'-i!. tgkdjkg to the discussionin Sect. 2.7 the quantîty'.'y ca,n also be ,l(t' .-rrr! 'kël.-2kq,.j.:;ïj(q',L; '.,;rj' -)tëi.. .k :.2.: a force per Trnlt lengt.h of bolmdary. Disregakrztmga possible bt,è884tks .t j: .: 'J one may considera. situation as showaia Fig. 2.13, where the tiltkitt/kllayvr, :,.. Jfllfijtp) #,. k't jj .m uuuu u assn,medto be depositedintla.eform of i'adividuilislands. s:'r.I ; rE.h.jp'r'yttu ë toucz? the equilibrium of rj rj ujore substratej isln.ndj and vacuum )! ..:..jzjgEEg(l iï. j''') j. E
-7j.j;';E .
... .
Jsiti
lii! .
.
.
. .
.
'
'' '
.
'
.
. . ' '.. .. '
.
,.
' ' .
.
....
..
zs J:kiy:'àè1ls rltlb' 1: .
k.' ::''...... ;.:.(..k. ,. . . .. ft' ';.j:'.E:
..
..
62
2. Thmodynamics
Frank-vander Merwe
Volmes-Weber
Senski-Krastanov
mFk; 6?2
'b#'tqîh' kè'f '... .
'CCj Sjt'isjy t l(y''i ;';i),?' ypiiil L...@ #hï$)j:) t''k. 4)4 'îh. ,, , ...i;j.,-!.A.,A . ,:...1. ?.rh .. . sjî.',. i :.:;.'L ; ..f. -.r . ;.;k . ''.-;1qk,-;;.
'
jllEh
.
. .
.
. . . ...
%.>. 9s'j)' .3'*. k3x 4 u.. j.t;.' ...qo s
Mjy6% Iù'J'i:!l N ': h f t$%;à 5it' .'(p(hîF a(K %69 û'% klu, ' ' : . 1. b .';ï . qrh'fl 9p qjj). C;I'b :.r ..Lg ïplj. E.k' I. :!.jk (x.' 1Z #xF ulép
'
;qt
.'
'x
.,j;' ,, r:,: y.. uts' 0..44 'Vi 1V't4tl WqQ watj.%
-.. xj xt ;j
-
s,)r<j)'='.&5k s:
jj,e..kà? ..
. i; k. 'i2h qkktgr . 14 q,
.
..
..
;'
, 34.'. sk,tqkjjk/y. j;.yr.g jtqkzdkti.iz n(+% ( ..y.)t . ?,. .()jkL.,,kék .. . ,1.g ... ..t.? . .. , -as.. ..jt. ,,41.-;....,.i..b,i ?..... kr-:, ., i.,,.-:1. , .s
, q;;(1j. 'k;i.q.è. jj()s:. j/ j t.<).,(k:.. -st,ijtyf.. t.r-. :, ).jllqtk..yvyjyk fk.!. ..ty q u.2y .?f, ! ....1). ,3%
-.
..
... ..
'' ' k !LfI'' lîL $?:û)1 .>..vn.r. jqpzjk. gk.: çç,./jy j;v '. cïY11? ' ajj E::$ jk <j'z . a jiyk jjz.. , . ...N> . . y,tkjjs ajyg .à;)R, r,.;kj 42j,/ tt W)k%t
i-?
...
..
.
.
.
.
.
.
rllmerent modes of epitxxial growth tseailezztatie--a.llyl Fig. 2.12. T for dilerent coverages:below one monolayer,betweenone and two mono ezs, above t'wo moztolayers.In tzhe217Fknmlc-vander Meme mode, layers of mxtfameAlgrow on top of eachothet'.In the Volmeer-Weber mode, separate3D Is'hnds form on the substrate. 1.nthe Strsknmld-nutanov mode one or two monolayers(wetting form fzzst, laym') followedby m' dividual islands.
Js = Js/o + 7o cos
(2.25)
$
the angle betweenthe overlayer-vacuumfaceand the substvatevacuam hce. The two lsmiting growth modes,the Frank-vaa d. Merve mechxniRm with / = 0 azd the Volmer-Webermode with 4 > 0, can be charac-
with 4
as
Yo
, :
.. .
.. .
. .. . .
2.
...
overlayer
:7.,' ./' . ! :. z :' : '.L'. . .?J''j7 ::J f.''' t . . q : :..t. ' .' ' ' . . ( .. . . L.: . : : . .. .:. .'.. L.? j. .. 7 : .7l .: .'. ,,' '. .. . ,., ,.:jr .. ,. : .: . !.. : ., , p,jjujpr .. , .Ey . . j' .. , g. : . / ; j j . . .g.. ., . . .. ;. . .( t . .y. .. g a. y ; g,: i.. .g :. , .g .jjj j.;.,a... ty . jtggjy . . . .y. .. .. y :
'
l'diu/'r
',
jjrjjji
.
s
,!
,
.
jj,.. t.sy.
,.y
g...j y
;
y...
st te
4 Eig. 2.13. Faquil*lbrbtm of forces (schematically) at substrate, depositedislazïd and vactmm. (overlayer)
2.4 Surface Baergy aud Morp/ology
63
terizedby ï'y f 0 or Ay > 0 with 17 = yo + wo %. For the intermediate Siranski-Kraàtanovgrowth; A.f S Ofor the flrst atomic layers (wetthg layex -
and Arf > 0 for the islands. 2.4.3 Formntion of
Dots Qv3Arttaarn
long time, it was believed that the of 3D kslaudsin b0th the mode and Strnmld-Krnstanov modeLsrcompanied by plasVolmer-Webe.r tic rehvxtion, for instancq by the formation of dislocalons n.ear the island base.In recent years, it 'hxq bee,afotmd for severil heteroepitaMalsystems islaudsform' vith a lattice miqf.t 2: 2%that dislocation-hv(i.e.,coheremt) in Stransld-Krastacovgrowth on a wettiug layer. Suc.hsysttarnqare Ge on Si Tn As on GAAKwith 7%(2.39,2.40): and1nP oa JAGa,P with 4% with 4%(2.382, akisst.Eventhe system Ge on SiC with a norninally mueh high.e,rlattice mlpreûevemucph showssuc.ha growth mode. Thœe 1s1a11(1s ft of aboat 23%(2.41) of the czisst-inducedelasticenerr by ckaqgingthe implane lattice const=t in the island for layers away from the intezfacè.Tllese naztoscalecoherent islands,which are often fouud to have a narrow size distribution a'adto be are accompaGedby a spatial quanarrangedin a regular array (2.39,2.40j, tization of electrons aud holes i'n three dirnensions. Hence,the islandsbaaed aze promisicg for on direct-semicondudor combinationssuc,has Tnls/Gn.A.K diodes (LEDs) azd lmsers.lmages of such use in quantum dot light-vitting whicà havebeenobteed by sc-annîng ttmqu-tum dots or nanoczystaltites, neling Mcroscopy (STM), are shown in Figs.2.14 and 2.15 together with a schvxtical represOtation of their facets.The examples are an TnAs pyrxrnld surface(2.43J on GAAq(OO1.) and a Genanoczystalon a 4H-SiC(0001) (2.421 For
'
'
a
.
(b)
(a) T' '
'
r
(%cl
. .
W W,''k. $A% > :%'&'% é.: :u ' 'k qY l e %.' L..xs;: z--.w kiB'?: -. m. V ''+ ' ' . . val w . . 'e'zt : x#&. '
'
.
'
.
u
;''.
'
.
. r.h g T#j<
6.
..
l r
'
'* .
FAt ' '
.
.
i
'
' . '' .
'
.
4. @.. ,
'< ' .
'
'
t.y
'
'x@. , w ?.* h : : *s . < NxW'à
t
.
.r
' '.
I
.
:' @
.
.
l .z
.1<'
..
.
1',
' '
:;
Z! ,
.
%
'''
. ' ' . ..
..
s
.
'
'
''ie
'
'
.
:'
'
1)'.'
1
. c t.d ;/ .
'
.
. .
$1 .
. '.
.
. : '' '' '
.
'
j
'k
on (a)Three-dimensionalSTM image of an T=Aq quantum dot grou a Gn.An(001) smface.The oblique objectsin tke foregrouadare due to As rlimers. (b)Seematic reprœemtationof surface facetsof the nauoczystallite shown in (a). eachother. R'om (2.421. However,the two representatioztsare ivgted agnsanqt
Fig. 2.14.
64
2. Thermodpmmlcs
(a)
(451) (131 ) (541) (r54) (145) (111) (:.;,: .:) '
(113)
F'îg. 2.15. Ge nazlocustal on a scbematicrepresentytionof tke dot
(514) (415)
surface. (a) STM 4&SiC(OO01) (topview).After (2.43).
image,
(b)
The mecxnfnimm of forznation of coherent island arrays iu ltigiuly lattlcemiRmatchedEetrroepitaxyis not fally tmderstooduin partictzlar not the selfFrom a thqrmal-eqplillbrillmpicture, it is understoodthat the mssembly (2.441. is energetically more favorable compared formation of 3D coherent 1s1a.n.c11 to a aniformly stzn.ined6lrn ms discussedabove.However?even when the formation pf azl isla'ad array is not consideredand interest is focusedon azl isolated 'nn.noçrystallite,a completenmwer is still rnlssing. Suc,ha, strnsned, dislocationeee jsland vith a c'haradetris'ticextent of about 10 nnn represents a quant:'rn dot. Eledrons in suc,hsystems containing severalthousandatoms tmdergoa spatial quantization, which 4'nfluences the electronicproperties. The size, shape, and stabilit'y of such a quatzttnm dot grown on a substrate should be governedby the energetics, although klnneticinfuences cnnnot be excluded as, e.g., in the case of the nanocrystallite shown in Pig. 2.14. The totaz-energy changedue to the formation of a Iarge, isolatedquantttm dot of the type showain Fig. 2.16can be characterizedby three contrëbutions The enerr gna''nper tlnit voltlme due to islrding with resped to the (2.12). situstion of a homogeneousoverlayer is
Zêlsland= Zfsurf
Zfrelax Zâlyyer + + , (2.26) v' V' P' pr where lfusla.d is the total erergy gnln of an island with' vobnmeJ/: Zssurf is the energy of the additional surfacegeaeratedby island formation. Accordimg it is give,nby and (2.17) to exmresions (2.$6) Zfsu'z.f =
Sï
'/ + 17YklfkA$ 'ylayertoôlxo. -
(2.27)
J,à
with surfaceenergies''l The stnrn over f rtms over all island facets jrnRtrained) azldareas Ai. The efect of the straiu on the facetsis describedby the second
2.4 Surface Energy and Morphology term in (2.27). The tensors t'/ and Li characterizethe islaxd smfacestressa'n.d strain aversgedove'r a facet. The esed of the coveredwettiug laye,rsuzface with area .A0 has to be substzacted.'ytayerto0)is the smface phs iaterface glvea i.n enera of the wetthg layer with a nom'nnalthckmess 0-n (usually ltnits of monolayers). The diference in elaztic enera betveen a situatiox with an island aad a straiued 61rn witit the sn.me Arnotmt of material is given by homogeneously the secondcontribution ic exmression(2.26), ïsrel,ux
7
=
Rilrv. .
&'
.
(z.za)
gqjm j
o
.
wEerefeilmst is the elastc energy of the isiand a'c.d. ssm ks the elmsticenerg.y dmnqityin a homogeneously,Al'nsforrnly strssned 61=, az obtained from elar,descdbesthe changei'a the tidty theory. The third. contribution to (2.26) formation enerr of the wetting layer as a restllt of its t g. lt is desncd '
Iswer
=
(-.)yP) -
lwy.tel
-
'
wyert6blq
with the combined surfaceand interfaceeaergy of the wetting layer, 'ylayerto), cocespondsto tNe aud the are,a dezssitjrof iélands,zz. By de6mition.,'ylayerto) surfaceecergy of the substrate. 1zzorder to understacdthe energeticsdriviag the formation of TrtAs islands total enengygn.insof the type on a Gn.AKsubstrate as a prototypici e-xnmple, are calculated for situatio'ns az shown schemxticallyin Fig. 2.16. The (2.26) pymmids with (110)facets (2.45, studie,sbegr with sqaare-based 2.46)but were latez refned by consideration of (distttrbed) hexagon-bmsed praxnids idth (111)and (110)facets a'adtheiz trlpncation parallel to the (100)smface DespitetheseactMties with increazing efort fuzther work is (2.12,2.46,2.47j. neededin order to explna'rtthe detailed facetting and the e-xad odentation of
'
!
.
(001) Lh-c'fz :1q1 lrtAs
(014 .
:
(ccj)
Wettin;layer GaAssubstrate
'
:., .:
L.
1o$
x
joo)
.
(2 :' E' .
.:.,, :
>nsg.2.16- Square-basedTnA.s pyramid with lbyerdepositedon a GaAs(001) surface.
(101)facets over
As an 177
wetting
66
2. Tàermodpmmscs
the TnAs islandsbowain Fig.2.14. Realdots are slkhtlyfatter and also possesshigh-index facets such ms (2.37) or even
2.5 Stoichiometry
calculated
(2.511)(2.42j.
Dependencé
2.5.1 Thermodynnmlc
When only
as
Approac.h
single specie.sis presemt in the syste,m cha-ractezized by the soEd-vaporeqailibrblm, we have seem in Sec't.2.2 that in the fmrnrwork of the Gibbs çclividl'ngsurface' it is always po%ibleto choosetizeposition of the s:qriimce suchtha.tXs = û. Whea thereaze severalatomicor moleculazspedes, this conditîon cnannot be reAlizedsimultaueotzslyfor all species.lt is gezzerally chosento satisàthis conditionfor the majorcomponent (e.g. the atoms of an oneucompontent substrate) but not for the rnsnnorone,s (e.g.atoms of a.c For that reason we genernllzethe result (2.14) of the Tdivids'ng adsorbate). surface' to Ns # 0 also for a one-components'ystem. The surface excess of the rand potemtial uOs, 'y, should not dependon spedal Goices and is given by 'y =
a
1
1
xfzs= 1(Fs G)
1
'
(;ï(Fs p,Ns),
=
-
.
(2.30)
-
wherep. is the chemicalpotential. Fs and Gs are the surfacee'xcess Helrnhoitz free enerr and Gibbs free enth/py, respectively. For a mttlticomponentsyste,mwith speciesf = A, B, C, ..., the valueof '? .
becomu v
=
(2.48, 2.491
1
-x ss
-
)-)g : Nu
(2.31)
,
i
where /zç ks the ckernicalpotential of the componentf and Nu denotesthe n'lrnbe.r of particles of that species i'a the sktrfaceregion. The uzderlyiag surfaceexce'ss thermodynnmscpotentials are J?s= 'yz, J's = JX +
V/z:No,
(2.32)
ï
Ga =
S'J ANU
for Iframer's gra'ad poteatial, the Helmholtz free energy, and the Gibbs fxeeenthalpy, respectively.latrodlzc,ing the correspondingsurfaceentropy Sa ss=5lazlythe surfacee-xce-ss internal energy is (2.15), Us= Fs + TSs = TSs +
V IsiIhS+ qA. 'J
(2.32)
2.5 Stoiœometry Dependemce
67
This quaatity dependsoa the variablessatchas exc- eatropy &, surface a'rea A, and the partide numbers Nu. Theqefore,oùededucesimmediately that
dgs = zdss + c dz + Deereatiating
Ef la dms.
(2.34)
yields a'n.didentiyngthe result with (2.34) (2.33)
ssdz + xdg +
E Nïsd;ss
-
(2.35)
o,
01*
d'y
=
1
-
1
-xSscl?F -x -
Si Nu dpi
(2.36)
.
No contriThks equation L6calledthe Gibbs ndKorption equatioa (2.2, 2.501. ln the case of a bution correspondingto defomnatioasks iacluded izz (2.34). solid, one may thereforeexplicitly Gtrodace the esect of smfacedeformatioas accordingto tlle discussionin Sçct.2.3.3 (2.5052.5.2 Approximatior
for Surface Energies
The eqlTipibrillrnstate of a surface ms a htnction of compobitionks deterThe mined by minîmlzing the sttrfaaeexcess free enera y = .%lA (2.32). conwponds'ngcontribution to the grazzdpotential depeadson T, A, and p'x, one derives /zs,.... Fzom expression (2.33)
X, p'x, JIB, .--)= Fs(T, X: NKn,Nss, -..) f2sIT, -
Jllz:,?1s, j
'v
(2.37)
where the surface exce-ss 9ee energy Fs = Uv T& is a function of the numbersNxs, Nzs, in the sudaceregion. In the caze of a solid system the fxee energ.yF in expressionsof the type (2.37)n.= be replazedby the 96.eenthalpy G = F+pV. For novmnl pressme, càangesin the surfacestoicometzy propordoual to AN. are accompaaied with 1)2Om as a cyxraderistîc volume of by enezr variations pt.4zvkrlt=l b ia a certain bondingcoaNg=ation.Witâ a characteristic a'a atom of specie,s value of r2Om= 16 A3 azd normxl pressme of about p = 105 Pa one ftnd.s enerr changesof about 0.01 meV pe.r aom. Heace,the direct Maeace of .Xsm will pressure variations on the surface energy cau be neglected(2.511. see later, tlle pressure may howeve.rplay an importact role in the determlnation of the clzeakical potentials, at leastdescribhglocally the situation aftez .-
..,
an
epitaxy step.
In prindple, all contdbutions to the surfu excv cauonicalpotentik (orsurfaceene-rgy,ifit Lsrelatedto
a
part of the graad certain stzrfacearea
68
2. Thermodynp.mics
dependon X) Os in eopression(2.37)
temperatme, the internal ertergy,the eatropy and the chemicalpotentials. The term -T;% contairs the contribution of the surf'aceforaation entropy to the surfaceeaergy'.lt is governedby the lattice vibrations and, hence,its detailed itwestigation requires lmowledge of tlb.ecompletephonon spectra. However,a simplifed discussionis posssble, since (i) s5mllarcontributionzappear in the chemicalpotentials and (ii)agsu'n only chrmge,shave to be consideredin peciple. A rough estimate of a threefold coordsnatedsurfaceatom in compazlson to a fotlrfold coordinated bnl'katom yields Ss $:k$1.75 ks per atom in the case of GaAs (2.521. Ia contrmst to the pressure iuuence, this contribution cnannotbe nveglected,not at room temperature and not at a11for highe.r growth temperatures. Another estirnnatefor the metal oOde surface,Ruoc (110), gave a vibrational contribution to 'y less than 0.16 J/m2i'a the temperature ruge up to T = 1000K g2.51). However,compv'ng the stability of two surfacesvith. deerent compositions, oaly di/erences of suc.hentropy terms should appear whic,hare disregazded iu the following. There is another', .in m=y cases more impoztant argttment for the neglect the of the lattice contribution i'a the discussion of static smfaces.Wheereas absolute value of the iaternal energy Vs is essentiallydeterrnr'nedby the electrortic contribu'tion,tthe efect of the electronson its temperature dependence can widely be neglected.An apper liznit of the lattice contribution follows h'om the equipartition theorem. One fmds
Nss, ...)+ 3/cars-llrïs t&(T,X,N,xs,Abs,...) ELNA.., =
svith B beiag the total enera of the electrons and the non-vibrating ions at zero temperature. ln the classicalb'mlt fhe linear contribution 3:sT EçNu to (2.38) also governs tbe ettropy term TSs in the smface frec energy Fs. Therefore,a wide compensation of the lattice-dpmmicat contribtttioas to Fs can. nearly be replaced is exmected.As a result, the fzreeenergy Fs in (2.37) to the internxl enerr Z. Accordl'ngto by the leads'ngcontribution E (2.38) the disctussionof the emtropy tmmn, the contribution of tke lattice vibrations to tlle internal eneror cxnnot be neglectedinvall cmses, in particulr not at growth temperatmes where the atoms staz't to move out fz'omthe eqpxilibzixlm positions. Neveztheless, in order to e-xtract the domn'natingphysici processes stabilizing a certna''nsurface phmseor to s'tudy a static suifacestrudme, the entropy may be disregardedsimultaneously in both the Us and TSs terzp and Fs i:a (2.37) Lsreplacedby E. alsocontains the efect of the zero-point The mnt'n contribution E izl (2.38) 'eibrations.Within the Debye approvirnation their eFect is given by zàs/o s of Despite per elementau cell with eo a.s the Debye tempeaxture the solid. the fac't tlwt the Debyetemperatmes vazy for dlferemt materials (2.281 2.53), oni expects a compensation of the efects of the zero-potnt
2.5 StoichiometryDependence +. .
P+ .+ '
.F:
?
5 '
i.
...
;
1
? !*
..,
c(4x4)
'
'
(p;j:?$;)k::!: )
,.,
.-.
'
69
r,11())
...
02(2.:4) (2x6)
mlxed-dimor
'v7(2x4) .
t
.
(1$n1
surfpcestrtzctures of Gaks (a)and TnP (b),ordered Fig. 2.1Y.. Top view of (001) accordingto the amoant of catio/ coverage.Empty (f2ed)drcles represeat cations Posiiiozls in the uppermost t'wo atomic layers are indicaàedby lrge,r taaionsl. symbok. The layer closestto the bulk contains catiorus.
Consequently)kt an exmlicit calculation usually E is replaced by the total enerr of the electrons a'adthe classicalrepulsion enerr of the nucld (or and the apprordmate exprerssion(2.54) coresly .N'z,.-.) )CP'GN. /zB, .--) ELNX, fzst/zA, =
-
(2-39)
is used to discussthe relative stability of surfacephaseswith dl'flbrent stoieometrg and/oratoznic structme. The inde,x ;s' to the nllrnbezsN6s is are automatically restricted to the sttrdropped..Cakulations or discusssons face region with a 6nite nt:mber of atomic layers. The total-ene'rr calculatiozzshave to be performedfor severalmoctelsof the smfacestructme with varying nllrnbersNh and NB. Suckmodelsfor relevant suzfacestrudm'es are smfacesin Fig.2.17 (2.55). ao.d1aP(001) sho'wmfor GaAs(001)
k.s.3 chemical Poteatials The chorn'lcalpotential A = ihlptT) (2.6)is defmedto be the derivativeof the Gibbs free enthMpy G for a g'iven phmsewith respect to the nztmber of and fzxedm:mbersjNjj. of other pazticlesof type ï, p.l = Lt3GjlelNàtpbvfqxauj 'particle,sapart from Nn. Since in eqlpllibrpzmthe cornîcal. potentiaz /.4 of a given spedesis the same in. all phaseswhich are in contact, eac: /wJcac be consâderedas the free emtbn.lp.yper paz
70
2. Thermodynamics
Si substrate
. .7 .% . .. .$.. T .j.?:;>>.$.k cQijux. 1lKjè.An' %o .. &' yyktiopjtobjwx.. /nk)gy .;).k.) k-jjuç khjajfxl ..u
.
.
ï
klt; s:7
ï
k .L J& W.
4':t .;T'.%2; k' .%!' (.(. ' u?B. , ' ' ' Q'. iy o . . . 9 .2' w &' ' , , r ' ' h . 'Tsh k' 'f é J ' ''rf' rk ï'. EF E ..L;t k $)' f '' . ' t! fd:'k';ï' Tt: k .1 $2 hilk k4f. .LL''k.Eëçl L(l? i;r . '. '5;' b'''6ç''7' lh w) u;i. owV/kyuz W qg ' ' 7 ;: hkl.hl i-l '-1i-l i'': 7'' 'b.bLkê'Lt-A' ikkkikh îikii'ik.ui/' lkuz./dskik)' sLzil'qs W '
H
H
k?kJ. .;4 kz c, .
.zZ
.
;
Er r'L@'
'e
.
.
hy.. , , H jj y s ; j,
$
H H -$-...?,. j.y,çLq
F
,
'
yj
H
o. c
f . .. w? w' . m !&'
H
u ell
a
'
'
H
.
/ AX
chemicalvapor deposition(CVD)
.;
yy yy v. .
xo
qs
yy yy yy 1+' '' 4
Ge $ u-s-y,y
hh-rs-hh-r--sh-slg
molecular beam epitaxy (MBE)
Pig. 2.18. Schematicrepresentationof two diserent epstmdaltecAnl'ques,CACD aud MBE, tzsedto prepaz'ea mrr'ace or to deposite au overhyer, here Ge on Si.
The actual chemic,alpoteatials dependon the surfaceprepazatype ï (2.56) tion conditionsor the kn'ndof epitaxy usedto deposttea surfaceoverhyer.The latter is schematscallyindicated in Flg. 2.18 for the depositioa of aa nzlqor'bate (here: Ge)on a subarate (here: Si)using clsleren.t epiteal tenhnreques. ln the fozowing we excludeadsorption,rather we study thc preparation inâueaceon the eqlzilibdnlmsmface of an AB compound ms a'a exnmple. The relevant chemicalpotentials are p.x and #s. Eowever,i'a addition the clzemical potential Vxkô of the corresponcllngcrystal bulk also has to be take.ainto accotmt: sirze the b'll1r solid Ls a reervoir whic,he-q.n e,x e atoms with the surface (2.541. For condsmqed states, e.g., the GM.AS sudaces,bllllc Gadts,or blll1
.
,
2.5 StoiYometr.y Dependence pntentials pthlk of elementsï =A, B data 1. Càeznical Table 2.2. Tkermodmrnl'cal in ev/atom.Tkey are deined a-s negative cohesiveenergles, The cohesiveenergy 5sthe energyrequired to separate the element into neutrf aioms at T = O K The Tn'mentalvaluu (exp.) are from (2.31, at atmosphericpressure.The expm' 2.5/4. are cakulated b#means of ab Jnitio methods. tkeoreticazvazues(calc.) -A yulk Blement
Czystal structtzre
exp.
talc.
B A1 Ga In
rhombohedral
5.77 3.39 2.81 2-52
6.11 (2-58)
C Si Ge
diamond d-txmond diamond
N P As
4.52 Na molecule white mod't6cation(cubk) 3.43 2.96 4,12 (2.52) rkombohedral
fcc
ortûorhombic tetragonal
4.19 (2.581 3-58 (2.58j 2.89 (2.58)
7.37 10.12 (2.25J 4.63 5.94 (2.25) 3.85 5.20 (2.251
'':k for the corstn.lline phasesconsisting of only one the chemicalpotentials p'%% spedes ï are given as negative cohesiveenergie. Calcalatedvalues are listed iu Table 2.2 aud comparedwith experimental value,s.j2.31, 2.57)The calcuTMs has mnsnly lated values are substaatîally larger than the mexastzredhones. with spi'a two reasons. Firsth the use of total energiesof free atoms calcmlated polazizationwould reduce the discrepancy.Howover,such a renormalization The second of the energy zero doesnot in6uence the actual value J)s (2.39). remson is due to the local deasity approvsmatsonusually uzed for exchauge Improvaments of the and correlation izz the total-emergyexpression (2.592. exchange-correlatîoaqnearr exmressioadramaticalty reduce the discepauHowever,aLsotheseeseds due to des betweentheory and experiment 22.521. the local evhn.nge and corzelatioausedcancel eac,hother 'widely considering JlFerences of expressions (2.39) for #)S.ln the case of the bulk group-lr e)ements a ftzrther improvemen.tcan be obtained by aoticing that sometimes the bulk phasewith the lowest energy is not consideredas a reservoir. For instauce, the potmd state of the Ga metat is orthorhombic aud not fcc as asThis lowers the euergy by an additional 0.2 sumedin some calculations (2.5$. of Slmilar remarlcsare valid for the chmrnicalpoteatiaksitbgkqïk ev/atom(2.541 givenin Table 2.3.)Thebttlk chemic.alpotenbulk compoundsmrnscozductot.s tials of the crystnlllne phasesof tke compotmdand the i'adividual elements of this compomld.: a're Telatedby the heat of formatîon I.WAB .
.
bulk
#AB
=
bux
llx
bulk + Jzs
-
zsAe J
,
(z.,o;
2. Thermodpmrn:'œ
72
Table 2.3. Thermochemâcal data I1. Chemicalpotentials
yVVand heats of for-
mation ZSrAB5nev/pair crystnllszingin zinc-blendeor wttrtzite of AB compotmds stracture. The chemical potential is defned as negative cohesiveenergy. The heat of formation is given in (2.40). Values taken fwm experlrnental data (exp.) and taken 9om ab 'imtio calculations(ca1c.) are listed. t
-;LVk Coumpound BN Am GaN JZaN AlAs
GaP Gn.Aq InP SiC
IJZ'/'B calc. ex'p.
ex'p.
calc.
13.36(2.60)18-95(2.581 2-65 (2.58) 11.52 (2.60q15.98 (2-5$ 3-30 (2.62) 3.28 (2.58) 8.56 (2-60)13.61 (2.5%1.11 (2.62) 1-28(2-58) 7.72 22.60)1:-01. (2.5:1 0.38 (2.58) 7.56 (2.6û1 9-14 (2-52q1-20 (2.63) 1.01 (2-52) 7.12 (2.60; 0.91 (2.63) 6.52 (2.60) 8.26 (2.52)0.94 (2.6!21, 0-74 (2.63)0.66 (2.52) 9.96 (2.50) 0,92 (2.63) 12.68 (2-60)16-65(2.61)0.72 :2.621 0.58 (2.61) -
.
-
-
Tke comparison of the mcasmedand.calctzlatedZA/Bvaluesi'n Table 2.3 shows very good agreememt. This fact corroboratesthe wide caucellation o?inaccuracies in the calclzlation if one considers quantitics suc,has I,WA'B or f?swllich aze defned by 8sFerences(2.39) If the surface is i.'a and (2.40). equilibritlm with the bulk substrate,pairs of A aztdB atoms ean be exchanged witiz the bulk, for whiclz the chemical potentixl is Jzbxtjk. The equilibrblm phaserulelreads as u conftion (Gibbs p,.'v+ JZB=
b lk
y.x% .
It1 principle: (2.41) may also be written in the fo= of a mass cctïtm Ja'tr,if the chemical potentials yx aud p.B aze rewritten as Apmctionsof the partial pr%sures m aa.dps an.dtemperature T. The advantageof the eqln'libri:lm condition (2.41) is that JIA and yz are linearly dependent.Hence,the fozowing considerationsonly have to be madefor one element,e.g., A. This is of particular admntage i'a cases of mnrface prcparation., in whic,hmolectzlessutzh as Nz am involvedue.g., ia plasma-assisted MBE of groumlll nitrides. Restridîng ourselvesto trends in the stability of s'tzrfacesfor rllFereut stoixometries, the dezimtion of prope,r chemical potentials p,x (or/zB)can be avoided. Without detailed knowledge one cazl still set rigorous bounds on the rarzgeof ih values ms long as the surfacesystem is izl eqTtilibrirpm. ln u1k the particulaz, each/wJmtsst satisfy /.q :K ;ttulk l since when pg = ybt gt)s phase condensesto fo= the elemental bpll'kphmse.Sometimes,the strlzdtzre of the bt:llr pbn.qeis not really clear.For instance, elementalphosphortuqshows a wid.estructural variel, the most cornmon allotropes betng wMte (cubic),
Dependcnce 2.5 Stoickiometry
73
and red (polmeric) phosphorus violet (monoclirdc), black (orthorllombic), as well as some amorphousforms.Forttmately,their cohesiveeaergiesare rot very di/erent. For that reason, in deriving the AHIBvalues in Table 2.3 the formatîon of TnP and GaP is consideredfzom white phosphonzs,the solid high-temperatttrephase. However, the ase of a moiecular phmseinsiead of a solid may sometimescotusiderablychangethe results for the stability One and Ascmolecules(2.64, exnmpleis the alternatîveuse of solid As (2.54) 2.65) by means of MBB. discussingthe stability of GKA.Ssarfaces yrepared Tly variation of the surfacepreparatLon conditions may be represented by deviations of the actua: cheznicalpotentiats from thesebulk values,
1/,/: = iq
u1k phx
-
(ï
=
A, B) .
and (2.41) gives (2.4û)
The combinationof this defmition 'with Ayx + zlpz
=
.-ZSfAB.
(2.43) 0 tZï/zs
= preparatiou conditions are flxed by Ayx = SinceA-rich (B-ric,h) with its surrounctingsonly 0),one ûztdsthat thc surfaceca'a be i'a eqp:slibziqlm if the chemicalpotentials are within b0th upper and lower botmds
-s5.%*B < ztjs,j.,gO
'
.:
(2.44)
A, B).
,
lkNs abulktwk Ns) f4(1/zA) ELNg, Ns) Jzx'ùb AjthLNx #s). -
=
'
=
Thus, the variation of each Jz4is restric'ted to be in a rrge given by AHLA.B belowits respedive bulk value. Surnmarhhg this discussionone Mds that the preparation conditions of a in a ratber certai:a soace can be i'adudedin thermodrnmic corusiderations eslegant mn.nner. Even withoat a detailed knowledgeof the chemicalpotentials allow one gé as a A'nction of pressare and temperattlre, the reiations (2.44) to esstablîsh rrges for pç which are relevant for the deternûnation of the suzfaeestzuctures Imder equilibrbTrnconditions. The values .-AHIBaud 0 defne ls=1ts on the allowablerange in equilibri'amwith all possiblephascss. I'tl pazticular, the chemicalpotential for eac.helementcn.nnot bc above that = Omeans that there is in general of the bulk elemeatalphxse.The cmse ztî/.t.à bulk material present and the surface is in equilibritlm with the elemental condertsed bulk phase. The case Ayé < 0 means that the bulk is not st>ble aindthe surfaceis in eqtzilibrplm with the gaseousphmse.Both cases are : schematicallyindicated in Fig. 2.18. can be writAccording to the abovediscussiomthe surfaceenergy (2.39) of the tea for a'a AB compotmd ms a Annction of the variation A;h (2.42) one hn.q cahemica,l potential of one specias,e.g.1# = A. Hstead of (2.39) J
.
Li
-
-
-
-
4.45)
J.nthe liznit of a monoelementalsystem, e.g., a groumW saiconductor or a ssmplemeta:, one observesA = B with the equilibrb'nn condition px =JIBzX changesinto the surfaceenergy (2,39) irtstead of (2.41)Consequently? .
2. Thermodynazniœ
74
q(/zA)= ELNA.)JAAUAZVA -
(2.46) JA/Xizl
with Nx atoms in the sarfaceresom The surfacefree energies'T = Table 2.1 kave beencalculatedusiug 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 surfazewith tlke low%t ettea'gyfor a #venc'hemicalpotential g,x (ormore strictly, given prepration conditions), one has to compare energiesf)s (2.45) detezrmlnedfor surface models with vazying smfacc stoiclziometryand geometry but for a ftxed potential gx. Suc,henergiesf)s derived for the surfacestructttres of GaAs(O01) and 1nP(001) in Fig. 2.16 are plotted in Fig. 2.19. The msninulm surfaceenergy X correspondsto the most stable sarfacephasefor a giveu ehmmicalpotential or, more precisely,its deviation from the bulk value. Sincethese'em.erarplots compare the energiesof dsFerent smfacephasesfor a, g'ivenc-h/rnicalpotential, one sometimœdenotesfgtzreé of the type presentedin Fig. 2.19 already a,s phase diagmrns of surfaces. The stabtlit.gof the surfacephmseof a certain reconstruction ar.d stoiclliometzy is, however,not absolute.At fmite temperatures a ceztain surfacephasewith energy f)s per (1x1)tmit ce2 occurs with.a 6n5te probability G
N
ex
trp,x rzlf?s
(-
yy
(j .
(2.47)
For that reason the ene-rr Lmx zîlfls(measmecl with respect to that of anoth.e,rsmfacepbnxe) is sometimes called the formation energyof the tznxzzl reconstructedsuzface.Of course, in generalthis probablty is also Muenced by emtropy eects (2.36) whic,h may modify the stability of a cez'tainphase at a given temperattzre.
I'n.contrast to condensedstatœ, for gmseousphpsesthe efec't of temperature T aad pressure p upon the chemical poteatiakscaacot be ignored. Accordiug to gas theory the c'hemicalpotentials Lh depend logarithrnl'cally and the large mriatiozls in p.i crtn be used to control upon .p azd T (2.5611 the state of condeasedp'h>mesiu ecplilibdclmwith the gas. Titey ace related to tNe change of tlle Gibbs 1ee enthalpy whema pazticle is transfezred1om the ga.sphaseinto the condensedphase of the deposited 41m or surface layer (cf.,e.g., Fig. 2.18).U this trn,nqfe'roccurs exactly at the eqllilibritkm vapor pressure po:(T),then no emergy is needed. If, however, the particle of the f-th component ch=ges over h'om the vapor to the solid at a certain paztial pressme Jv, the h'eeenthalpy changesby = 6.1.3* /cBTI'n.IA/PO6I.
(2.48)
The spqm of the partial pressures Eçpi yiezdsthe total pressure p. For is reideal gases the eqlnilibrblm 'vapor pressme poï = kBlnjkqie-#i/kBT lated to the vobnrne ls3ï dfaAnedby the thermal de Brogtie wavelemgth
2.5 StoiclliometzyDependence
(a)
o.2
As-dch
75
Ga-fch
.
â.2:4. S% T. .,
(2xtf I
0
B E
(utogl
y
(2x4)
.e
(
mzed-dime'
'
2) -0.2 qq
a
'
p' o
'#
cmx4
)
-0.5
0
l
-0.4
(eV)
lgoo
(b)
cy
ln-rich
P-rth
0.2
q4.p S* .t,
G2(2X4)
0
07
rhzssj
a
,&
&
!; Y @-0.2
c(4x4)
CSV
x.
,
:m4)
i
muedsxmer
I
tp o
1
ct
-0.4 -1
-0.5
0
Ahzln(eM
Eig. 2.19- Relative smhce energy fzs ?e.z. (1x1)llnlt ce,llfor various sttrfacere(b) constracions in Fig. 2.17 versms the catzon Gezaical p8teatial. (a)GaAs(001), lirnp'ts of Dotted linesmark the approvlrnate auion- and cation-dc,h JnP(001) (2.55j. the thermodpmmscallyallowedrange of the deviations Ala,A.(A= Ga, 1n).
76
2. Thermodlmmlcs
with M6 as the partsde mmss au.da thermal activaR(2zr/JQksT)è tion factor (determined by the chemicalpotential lh of speciesï in the solid phase) (2.662. Expression(2.48) represents the chemicalpotentials of asomic or molec-
,Ls: =
ulaz spedes in a vapor izl the 1owpress'urelimit. J.nthe cxse of matezf.aks consisting ollly of one atomic speciesthe index ï can be dropped.The suy limation of a pure sold at eqltslsbritrmis given by the condition that the dmmlcal potentials of the atoms in the solid and the mpor are equal. tu other words, b'y = 0 and p = po hold.. The zatio s = p/pcelm thereforebe cced s'upersatnratéon.Expression (2.48), b'y = kBTIILs? characterizesthe driving thermodrmrn:'c force for the formatiozzof a thin 61m deposited9om a'a ambient vapor pressme (cf.the sehemein Fig. b'Jzis clearly zero in 2.18). equilibzglm,is positive dttrimgcondensation,acd negstiveduring subMation .
or
evaporation.
Taking the vapor phaAe into accotmt, the condition for layer or island g'rowth (2.25) has to be moeed. The characteristic quautity is Arf. = A'y c*kaT l.ns with zb''/'= Jo + 'ys/o cs and c, as a certain constaat. .ây' K () (,Ag*> 0)staudsfor layer gzowth in the Frank-va,uder Merve mode tisla'ad growth in the Volmer-Webermode). The Garacteristic quantity .4J* for growth of a ceztaizbmatcrial on a substrateis no longer a, constant material parameter, but caa be c-hanged with temperattu.e and pzasm'e. The use of relations of thc type (2.48) allows one to relate the prepara-
-
tion conditions to pat-tial press'tzresand substrate temperatmes. In principle, the mse of theseqppnrtities also allows the determination of the surfaceemrgy and,hence)the stable surfaceph%e for certain preparatioa conditions. (2:39) Examplesare given ia Fig.2.20 for the MBE preparation of GaA,s(O01) (2.67j and 1z1P(00:) (2.6$surfaces.Htead of a partial pressurea beamequivalent js ttsedto account- for the presemceof molecttlar beams.lt is pressure (BEP) a pressttre whic,his equivalent to the ftux of molecules or atoms impinging on the suzface.Togetherwith the measurement of the s'arface reconstruction by refledion high-energyelectron rllFraction IR.H%EDI at a given substrate temperatme, the beam eqaivalen.tpresstzresallow the construction of BEP-T p'hnxediagra'ms.Jn tke cmse of GaAs(0O1) the BEP ratio for As molectzles aud Ga atoms is varied, wheremsfor ToP(001) tEe ûux of Pa moleculesis varied in the presenceof an atmostzero ln ûttx. Iu the languageof the theoretical p'hnse cliagrn.rnqin Fig. 2.19 an increasiug BEPASa/BEPCa ratio (increasing BEPPZ ûuxlas well as decreasingsubstratetemperatme T correspondsto a tendency toward more As-rich (P-rich) preparation conditions. The opposite tendency, decreasingBEPAw/BEPOa (BEPP ) and incremsingtemperature, describes ? Ga-rich tln-z'ichl preparation condstioms.The meastkredphmsediarytrn!x in Fig. 2.20 and the calctfated surfaceenergies versus the cation chemicalpoteatial i'a Fig. 2.19showthe same trends for the most stablereconstmzctions. For example,in the GaAscase the stable4x2 reconstmzctions in Ga-rieh conreco' (2t i ons changeover into a c(4x4) nstruction undezAS-HC,h preparation
2.5 Stoichiometl'y Dependence
rr
(0())
ryr'z
800 700 600
(a)
1()()
..? ' .-
...z''z' !J
'ZZ'Z''W ''VE
tu
LD n 1:I 1 10 rn ..-
----
-'.,,'e
*
<
- --
l11(( I 1
-
.
'-
: :- :- :.: t (IX 3) : : . % .' K'. ''. 'J : ''.'- '. -
: : J: : . . - - -
s -.
. .
G>(4 = x4') * I
--
I
: ) :- : * :- :* - -
.-.
, .
-
. . . .
'. z'.-e :: : . -.
. .. -
(axm
(zxr) maz(zxs)
, (4x1).(.1x1)
. -
- -
. . . . . . . . . . ..-
/ Az -û; V/;G 1.E
-
(4u)!(3x1) ; > W/ k 6r -h
xr
n
l'.
I
*-'' M.:
:5h z'zJ., me''zz' .j'''z'z 'é E zz'.u. = e'ez % z;:,4
400
500
...'..........
*'
-.-. .-
:
: .. :. :: ..
z z; zxs'.%v.1 '''VAJ*'%*'T T'=rl-'71r4h1'û;m, Ga dro pIel:s a.q 6 pe'lgW.v :kN%.). rpliu' X1) a'4xl 4x 4x2 ( .1 %%%s%+
.
.--.
.
1I I
<
fn
' ' '-
...
-
1
'
.'
-
' ' ' -
;:..
facetting
1.2
1.0
1.4
'.103 I T
(b)
(K-1)
V*
10
1.6
4
c(2x8)
<
X
n:'
c(4x4)
2
(2x1)
(M) 300
4O0
500
T (oC7) witk perng. 2-20. Phase diagrn.mnfor GaAs(001) (a) (2.67) (copyright (2003), surfacesprekared by means of M3RB. rninsionh'omBlsevser) and TnP(0O:) (b)(2.685 The stable surface reconstmzctionsare shown in beam eqmvalent presstae (BEP)the ratlo of the BEP for azsenic tempezature (T) ctiagrams.In the case of Ga,A.s moleculœand BEP for gallium atoms is varieo, while for HP tke BBP of Pc atoms. mol-tles Lsvaried at constant BEP for .1.zk
2. Thmrmodynnxnics
coaditions.In between,i.e., uaderintermediateprepazationconditions,there is a large re#onirt which 2x4 recocstructions are most stable. 2.5.5 Stability of Adsorbates
with varying The theou developedto discussthe stability of smfacephmses stoichiometry and reconstruction can also be usedto discuss the stability of adsorbatestractures dependingon the preparation conditions.Accordiugto and the specfcations for a'a JkB the deftraîon of tlze surface e'nea'g.g(2.39) ole Mds for the depositîon of a third element C, compotmd(2.40)-(2.45), the adsorbate,
.f1S (zJu,zvc) - E -
Nz, Nc) gbxktxzJlusktxaNs) (2çk, pbculkyc Alsxllqh A's) Aitclic (2.40) -
-
-
-
-
-
v'ith Nc ws the nnlmber of adatoms in the surface region. In m=y experimentazsituations the c'hmrnt'cal potemtial of the adsorbate only underlies the constraint gc S p/culk S 0. Apc = 0 de) i.e., Agc sczibesadatom-ric.hpreparation conditions. ln the caae of ecporation in CIW, e-g., of M-BE, for instance, it may mea'n that the shutter of the efusion cell for the C atoms is open azld that so rnxny C atoms' are deposited that they stazt to fo= clusters with a blllc-lilte crystal stractme. The om menmq that practically no C atoms nrrsve o2 the posite 75=it Ayc -> -x sttrfnce. Howeve'r,tlze interwat of the variation of Jzc has to be remn.rkably
modoed if s'table compoundsAC, BC, or ARC (vith porssibleadclitional variations of the stoickiometry) exist. One evmple could be tke adsorption of C = Lî on a zinc-blende crystal Znse (A = Zn, B = Se).An' upper botmd on the cxemicalpotential of the adsorbateis fouad by eloring tlle vaziouscompoundsthat the adatom nltn form in its interactioa with the sys-tem. For Li, a possibletppe.r botmd on ;tu is of couzse imposedby Li metal. Howevu, the most stringent constraint arises from the com(bt17k) pound Lizse;whie,h leads to the constraint jm the chernicalpotential of Li, b lk = 2 b!znc 2 6gj 2/:L2+ #se = Jilse Jzl,z + p's E Bjk zxnaase f For a certain smfacestructure or, more precisely, for a cez'tain adsorbaàe Thus, structure, the ntlmbe.rsNx, NB, and Nc are fzxedin expression (2.49). f)s only dependson l/za aad zlpsc. Two adsorbatestmzctlzres are in ,eqlzilibrbtm for equal %. TMs condition allows the coastmzctionof the phase boudarie. ln a region of Zï/IA and Apn wbic,h Lsbouuded by such botmdaries, the adsorbatephasewith the lowestsurfaceenergy f)s Lstlle most stable -
'
'
One.
In order to iliustrate the stability of adsorbatestructm'es on compolmd thc cleamge face of Iztlsis suzfaces,the deposition of As atoms on TnP(l10), studied. Foz simplidty the translational symmetry is pstricted to the smallschematicallyi.a Fig. 2.21. They
are
are (2.701
presented related to four dilerent [email protected]
est lx 1 surface tmit cell. The fou.rfavorable structmes
S.5 StoichiometnrDependMce
(a)
:'
'.
JL.
'' :''
: '(::.' ly . :
(b)
(c)
Side views of st:rmces with A.s/Tnp(11O)1x1 dl-lerertt As coverages e. (a) 2.21-
Fig.
Buckled
cl-n
.
l InP(11O)1x
0), (b) eacchauge-reacted geometr.g (8 1z),(c) epitax(&
=
=
contûmed layer stntcture Le = 1),-(d)ewxck=ge-reacted surface 'wath a;a extra -&s oveau layer Le= 2z) P (In,Asj atoms are denoted by full tempty, smbols (2.70j. shaded)
ial
.
coverage0- = 1 conwponds to one monolayer of arsemic.1c the case of a (110) cleavageface e = 1 3srelated to t'wo adatomsper surfacetmit cell. The clean stufacewith O = 0 is describedby a reiaxedzig-zag cha,instrudttre. The &st s'tep of art adsorbatewith a coverage @ = àzrepreseat,sa'a exchange-reacted geometzy The upplrmost phosphorusatoms are replacedby arsenic adatoms resalting in an ï= AA monolayer. In the nex-t step of coverage,0- = 1, an As smface.Severalstructuzal modelshave monolaye,roccurs on top the ïnP(110)
been suggested.The most stable one is a so-calledeyïfczriclcontinzedzcper The largest coverage@ = .j is representedby an stractur6 (ECLS) g2.702. geometrs ECSL on top of a,n exchacge-reacted of the fottr stradmes tmder coasiderThe results for the surfaceener#es in a m/z:n,-Ayxs phasediagram with two triple points ation are slzrnrnarized i'n the allowedregion in Fig. 2.22 g2.N.Despite the existemzeof the stable
80
2. Thermodpmrnics
> S
1 '
ex oange-readed
=.
exch.reacted
SUOCO
'<
-K
= GZ * c c.
+
.
1
-0.5
I I
clean1nP(110)
sudace
%
=
1 I I
2
tp
r
ML As
.j.g
(2
ECLS -1.0
M)-5
0.O
Chemical potential Agws (eV) Fig. 2.22. Pkaaediagmrnfor the As/TnP(110)1x 1 sarface.The dashedlinesenclose the thermodynltrnlcallyanowedregioa (2.70) .
Tn As compotmd?only an upper bound Ayxs
O is considered..The occtu'rence of the clean 1nP(110) surface,the exchange-reactedgeomet:y a well orderedAs.monolayerwith ECLS or the ECLS coveredexehange-reacted sarface depends4ensitively on the preparation conctitions.For a low amotmt of azsenicin the rccipient bat P-ric,hconditions the clean TmP(110)1x 1 surface is stable. Uzzder1- P-rich but more In-ric,h conditions there is a tendency for the formation of the exchange-reacted geometr,yas lortg a,s As is presOt. This corresponclsio tize fact titat dlnrs'ngthe Annnltlsngprocedarea pbospllorasd.epletion from TnP occms. Tn a wide raage of chemicalpotentials the exch=ge reaction seems to be the preferredprocess. On2yfor very As-rich and P-rkch conditionsdoes the prepratiou of a,n orderedAs monolayerseem to be possible. However,becauseof the volatitity of phosphorussuc,ha structme may be Hl'ëct'sltto prepare. Under extremely AS-HC,h conditions the formation of atl BCLS monolaye,ron top of atl exchange-readed geometryis ecergetihally
preferred.
=
3. Bonding
and Energetics
3.1 Orbitals and Bonding 3.1.1 One-Electron Pictare
Eventhough the sarfare of a crystal may appeazver.y sm00that fzrst glance, exmezimentaleddenceshowsthat it is heterogeneouson a microscopic scale. On tllis length scalethe thermodpmrnic treatment of smfaces in the previous clmpter Lsno longe,rsecient and,hence,must be reHed by microscopic considerations.However,suchstudi% on a'a atomic scalehavc to considerthe bondângbetweenatoms in the surfacelayer and of sudaceatoms wit: atops beneath in a butk-like eaviro=ent. The bonding behavior is governedby the valenceelectrons, kusuallys and p electronsof the outermost elecvtro/c shells. 1.nthe case of metals, e.g., transition metals,but also compotmdsemicondqctgrs,suc,kas GaN, semicore d electrons have aksoto bc studied. ticle picture, the electroGcstates #@)with energies Witht''nthe single-yar' s obey a one-electronSceödinger equation
H#Lm) c4(œ) =
with
Hnrnsltonian :2 H = ,4. + 7(œ). a
-
2m
(3.2)
repnesentsthe total poteat#al Et'athe shgle-electronHn.rnsltonian(3.2)J?r(z) with mass '??zin the strictly, a valenceelectron) 'energy of arz eledron (more : electrons. This pot'en, ield due to the atonlic core,s and the othe,r (valence) brieoy poteatial) dependso:a the treatm>t of tile electron: ùal energy (or èlectrozz interaeion. In generi, besidesthe efect of the core and the Hartree due to tite clmssicalsnteraction with the otlzer electrons, it coatxsnq Sotential $ certa in contdbutions of exchangeand correlatiou. The spin varisble of the that it is inLelectronis not e'xgocitly consideredhere. Rather, we msmnrnç cluded in the space coordinate œ. The spinor character of tke wave hlnctions e' : i s thereforealso not xplicitly tikea into accotmt. ln cases where the spin importaat, if e.g., the spG-orbit hteraction strongly modifes the becomes electrozlicstmzeure, the spin variable and the spin quartttlm nllrnber will be explicitly memtioned. .
82
3. Bonding and Bnergetics
of an orderedaad comrnensttrate sttdacesystem the total poteutial in (3.2) obeysthe trn.nqlatîonal symmetry of the halfspacewith the surface.Then J.1lthe
case
+ .&)= F@) V'(œ
with R beiug a vector of the 2D Bravais lattce of the suzface.As > conse+ A) = ebuR#Lœ), quence the eigenslnctions obey the Bloch theorem, '$Lm and an eigemstate ca.a be classiâedi:a terzns of a 217wave vector k &om acd a b=d index v. The the corresponasng2D Brillouin zone (cf.Fig. 1.22) scbrHnger-like equation (3.2)takes the fo= '
(3'4) JfW:@) fy(i)#y:@) with 2D Bloc.hA'anctions#vx(z)a'adcorrespondingBloe,hbands zv(:) de=
vector i. Onehmsto mention that the theory presented in (3.3)and (3.4)ks also valid for a system with 3D translationat symmetzy and.wave vedors k from a 3D Brillolnl'n zone.
pendikgon the 217wave
..
3-1.2 Tight-lBinding
.
Approac,h
I.zlorder to tmderstaudthe bondmgbehaviorin the smfaceon a length scaleof the order of neazest-neighbordistance-s,it is convenient to expand the' singlepazticle wave Alndions #@)in (3.1)izt termq of orbitals 4.(z) centeredat aa atomic core at the origin of coordinates. The inde.xa labels the kiud Tizis approacll to the electronic of atom azd the atomic quantntm nthmbez's. structure requires knowledge of the atomic positions A... The solution of i'a Sect.3.3. Eere we adopt a locnlivedthis strucvtmalproblem Lsdkscussed orbital basis set (4o(z .&..))and assn'rne that their locnlization centers Rs, are lmowzuWithn'n the so-called linear combination of atomk orbitals method.,the following equali'ty hole for an arbitrazy polyatomic (1uCAO) system u'ade,rconsideratioa '
-
#(=)=
V %i$=LT.&)' -
(3.5)
l,$
where in explicit calculazionsatomic mMe fn'nctions, hybrids, or bonding and as /.(œ).ln fzrst-priuciple or semi-empiric,al antibonding orbitals are IZSC.CI electronic stracttu'e methods the hmdions should be lmown a'ad may even be adaptediu self-consistentc'ycles.In more or lesse'mpirscalapproaGesthe =alytic fo= of the basis Gtmctionse-qn remain n''nk-nown. Mther, one introducesthe matrix elemeats of the Hxrnsltonian H and the overlap integrals and tries to S'adexplicit expressionsfor thesequ=tities. With the a'asatz (3.5)the Schrödinger-likeequation (3.1)tmnqforrrminto an imfnite system of homogeneous algebraicequations. The expznqion coef-
fcients restttt from tlze eigenvalueproblem
3.1 Orbitals and Bond'mg
?pj .RJ)! R,.) sksabLltz' (J2u(a., I'''.2 -
-
b,j
(3-6)
0,
=
83
'
where
HzsLmt .7) are
J
d3tr/o'(œ m'IHSb# AJ)
=
-
-
the matrh elements of the one-electronHn.rniltonian S
tem, and
(3.2)of the
s'ys-
'
%sL.% S.J) =
-
j
d.3z4.* (œ m.I6LT A#) -
-
The intra-atomic iâtegralsare Krodenotethe intenatomicoverlapi'ategraks. of the oae-cente,r necker smbols becauseof the asgprnedoz-tbonormnlbguation orbitals. I'n.ab izlitio calculations, the multicenter iutevals in (3.7)and (3.8)are Apart evaluatedexplidtly for a givea S and a given set of %Lm .Rz+). requests suck a procedarerequires heav,gnuh'om possible self-conskstence -
merical calculations because,in general,the matrix elements do not converge rapiclly iu spsce. Therefore,m=y empideal and semi-empiricaltecbnlques have been developedto reduce the tplrnerkal efort (3.1, 3.2).For eAample, assllrnes that HC.bLm', the extended Hfic'keltheory (EHT)(3.31 .Rj) is proportiona) to S=SLK.&) if i # j. Besidesthe enormous' reduction izl the computatioaal esoz't,empirical methods give muc,hinsight iu'to the chnmical bonding processe involvedan.da deeperunde>tandi'ngof the trends .in properties 9om oue system to azmther.A majordevelopmenti.n thq history waswthework by Slate.raud of the empirical tight-binding method (ETBM) who suggesteda new and waiuablerole foz the LCAO method, Koste,r (3.4j, that of azl intezpohtion scheme.First, they suggestedto treat the Hn.m51tozzia'amatrix elements ILVL.%, .aj) (3.7)ms pazameters aztdto ft tkem to one-electroneaeates.Second:they invoked lœown tmeasttred or caklzlated) accordingto whic,ha,n orthonormal a theorem,ftrst proven by Löwdia (3.5), set of LCAO orbitals could be de6medrigorously suc,hthat -
XbLm XJ)= Qbéa.p -
.
(3.9)
The Löwdl'n theorem statez that a set of non-orthogonal orbitals located at dferent atoms ca'a be trxnKformedinto â.new set of orbitals whicit are othe,raad preserve the atomic geometzy orthogonal to e'a.C'N The advantagesof the empirical methods becomeobvious when considericg an isolated bond formedby two orbitals c = 1, 2 localizedat neigbboring .atoms. Examplescodd be simple diatozaicmolemzles,e.g., the hydxogen moleculcH: with a pare colminntbonda'nda I;iI'I molecalewith a, heteronoIJr (orsimply, polcr)bond, for which, howevez,the consideredorbitals are restricted to the most importau.t s-like ones of the valence electroils.Without the dgenvalueproblem.readsas the neglect of ovezlap (3.8),
3, Bondingand Bnergetics
84
antibonding ca
+(AE)2 IH12l2
Ha
AE AE
1-11$
IH1zl2 +(A:)2 Eb
bonding Eig- 3.1- Formationof bonding and antibondingmoleculeleveb fromatomic level.
Szz e Nlz czsa S-lc sS* 12 Sa2 e -
-
-
-
-
:1
Q
=
(3.10)
0.
Neglectiugthe overlapof the t'wo orbitals forrnlng the bond, i.e,, S1c= û,
12+ the eigenwlues sap = I ::i: are the eneates of the antibondingan.dbonding orbital combinations, respectively. They are schematm ically reprœented i'a Fig. 3.1. The abbreviations ë' = (fftz+ Szz)/2and zâe= (JA2-Szl)/2 defmethe averageof the two atomic ener#es ard one-half of their dileaence, respectively.Tn the case of diferent atoms in the molecule with Jêza> Joz, the atom 1 (2)represents an aalion (cation). Co=equently, ls n-xn alqo be called the yolar cncrr g3.6) , in contzwt to 'the colmlenie'rJern 1Sz2 1,whic,h îs also nonzero in the lt'rnl't of equal atoms and orbitals. The polr energy determinesto what extent the electron densit.yalong the bond is deformedtoward the anion. Iu. the point-nhnrgep6dure, zîc indicates how many eledrons are traasferredbetweO the catioa and anion alongthe bond. A quaatitative me=zre is given by the bond,polczwzp (3.6)
glsza(zk)j
Gp
=
Iffzzlz+
(3.11)
.
(2!42
àz = It is related to the eigenveeors by IQ12 j2= lz(),::F:aplfor (1+ap)ar.d 1?2 the bonding/anbibondzng statc. Ia the votmd (bondsng) state of the molecule one Ods the rœtllt that the probability of an electron appering ou atom 1 The is (1+ (zp)/2 and the probability of Mdtengit on atom 2 is (1 &p)/2. = ap. Sinceihe coYent dipole of the bond is proportîomnl to (1ê1I2rJaJ2) enea'r is a, Alrctiozz of the distanced = lRz Az I of the t'wo atoms, the polarit.y aud the ezmrg.ysplitting of the molectklelevels dependon d. The bond length deqfollows fz'omthe condition of mypvlrnl:m enera ecltsqsbrihtm -
-
-
3.1 Orbitals acd Bonding
85
gain of the t'wo electronsdueto bonding. The enero- (.FJ1z12 + (1s)2shotzld be a mprrim:lm. However, to perform sach an optlmlmation e-xplicitly, the overlap interadion ($n general,a short-raagerepulsiveiateraction) has to be
takeu into aecout (3.1,3.64. I'n the case of trn.nslationallyinvariant systems of the type defned by (3.3) and (3.4)with atomic positions m' = .R + rz (Bravais lattice vector A aud atomic basisvector .rv)the LCAO exnmnsion(3.5) can be speaifed as = '?Jb:(z) 'h7cvc,ililxt@l
(3.12)
c,ï
with Bloch
sllrnq
1
s:.)z$
:kf@) Gn J((2 /.(z A;) =
e
-
constmzctedfz'om the locatized basis ozbitals. The R-sllm exteads over N elementa'cycells of tbe hlndamemtalregkn of the 2D (or3D, theaythwave vector k) crystal. The Bloc.hbands ss,(k) at a.a azbitrazy poiut k az'e then obtainedby solvingthe eigenvalueproblem
c,aj.(:) gl:l'oçe''t:l J-'-J ,u(i)JoyJvq =
-
0,
by.#
wherethe lbrnsltozsianmatrix reads ms Jfoïby(i) EE:
=
1
y
c -i:(a-.a,.).s ab (a. .%) V AyR/ .
,
of the orbit/ a localized at m. = A + 'but t:e overlap (3.9) ;. itocnlized at Rj = R' + o is neglected.
Ju
aad orbital b
é' '
j
E.321.3Atomic ;
'
;
.::.':. :
.
Orbltals aud Thelr Interactcon
.
'!Eè
'Let us considerelamcnts A an.dB belongingto colllmns Nx=N aad Nz = (8'':,X) (N = 1)2, 3, 4) of the Periodic Table and formi'ng compolmdsA.NBs-x.
Vsuallythey
crystnlllze within the cubic crystal syste,m with two atoms A i and B s:a the primitive cell. The occurring crys'tals posses the dsamond, qzinc-blende, roclcsalt,or cesblrn-chloridestructure. Sometimesthe resulting LïKNBZ-Ncompound crystals belong to the hGagonal crystal system, e.g., i ïhey possessthe wurtzite structure with %ur atoms pe,r'lmst celt. Th.ebondiug Tbfsuch compozmdsis governedby the N and (8-N) valenceelectrons of the Ao atoms of a cation-azzion pair. However, L'V-m compolmds,e.g., the lead I'ubiltsPbS) Pbse, and Pbrl'e: cau be descaibedi'n a &'=1'1n.rmxnne,r allowiz.g Lïheformation of a lone pair of electronsat the grottp-Rq atom. 7 Tlle s andp atozaic orbitals (or,more strictly, the Löwdin orbitals) of the !:: oute,r electronicshell with orbital euergiescs, cp acd radial ftlnctions Ra(r), .
.
.'
86
3. Bondinz and Bnergetîcs
coordinxtes m = polq Usiug bondiag. the clmmical most to contxibute .Rp/) the Arll wave cos 0)and sphedcal hnmnonic,s w) 8 sinw, cos siu ybmtp, $ p, rtsin rephced by tlle,ir combinations are umzally fllndions 4c(m)
1
1J) X04:,W)&(r) .zw.&(r)' =
=
-1
= J'zz (p, pj 4&-z(p,p)) a,(r) ( 1yu) W
f
yp) w -
-
yi z (9,w)+s-,(:,.,)q as(r)E
W)&(r) lpa) Fzûlû, =
=
Z-J.w(z.;, .
=
4x
v
-tX.w(r), 4.
p
(3.16)
t3 z
s-c&('r)'
where t:e corresponeg atom is asstanedto be sitaated as the origin of tb.e coozdinatesystem. Suc,horbitals are represeatedin Fig. J.2.The new p n='R. orbitals are directedalong a Cartesia'a fktledd la more complicatcdcompouudsor even in mtztals,inzompletelf and I shellsmay contribute to the cheraicalbondi'ag.Jmportaatexamples three''series in the Peziodic itwolve t'ransition metal atozas. These belong to to of 3d (Ti to Ni),4: (Za. Table whic.hcorrcspoudto the progresive 611n'ug Pd) and 5d (Hfto Pt) sbells. Real combinations of the d orbitals with a'a enexgyed and a radial part &(r) aœe z
Pz
S
Py
9
Px x
Fig. 3.2. Scàemltticrepresentation of s and p valemceorbitals.
3.t Orbîtals tmd Boadinz
87
15 Az
ltfwv) y,,.g 2Q(r), Ta =
IGz)=
15.:, s r a&(r), 15 zz
Idam) spadtr), =
z2 v2 15 = Idz2-#a) 16zr r z .!Q(r), 3z2 r2 15 = Edaxu-ral .TQ(r). 16c r a The atoznic orbitals givea ict (3.16) nnn be usedto calculate the and (3.17) matzix elements.J.ILyIJk., at lemst,to calcalate their smmetly and Aj) (3.7), geometzicdependemce. We illuswtratesuc,ha calculation of the Hxrniltozzian matrix in a more generalFamework.The following assumptions are made: -
-
i. The basis set is restricted to one s orbital and the tkee p., pu, and pz ozbitals of the mlence shell of eachatom tsee Fig. 3.$. 5i. Orbitals on Werent atoms are assumed to be orthogonalso that (3.9) is flgfllled. iii. Only the nearest-neighboror second-nearest-neilbor iuteractions S0:(.&,Ro'4 (3.7)are retainetwith the additional asslfmptiontha,t the Hn=sltonia'a lnAAlocal cylindricazsymmetry around the n.='K couecting a pa,ir of interacti'ngatozns, so that J.Qs(A,., Rj) H IILSLR..Rj). -
The adequacyof the tuse of s and p orbitals of the highest (partially) occupied shell (i) seems to be quite clear from cizemical azgamertts,Le., that only valenceelectronsessKtially participate in chemicalbonding. This is obvious for tetrahedrally coordinatedhNBs-x compolmdsbut also for simple metals suchaz: e.g.j A1. J.naddition) for trnmqition petals or transition-metal compotmdsthe d orbitals (3.17) have to be taken iato accolmt. Of coul-se, all thesebmsissets are incomplete. NevertheHs, electrordc stnzctmes axd total eaergiescazz be deduced,at least, usi!zgthe Harniltonian matrix elementsas parameters to be ftted.. Foz iustance, the restdction to s a'ad p orbitals is qlzite accurate for the desciption of both valencea'ad.conduction bands of s/miconductors. Howeverjbaides the nearest-neighborinteraction (3.7) also the second-neazet-nejghborinte-raction has to be taken into account, Only by the inclttsioa of suc.hmatrix elements beyoad the fzst-nearut-neëghbor interaction, cau the Dlike condudion b=d msmima of the i'adired materials Si, Ge aud GaP be obtained near the X and L poiats of tNe Brilloqlln zone (3.8) The improvement of tike condtzctionbandsca,'n alsobe acllievedby an enlvgement of the b%is set wilbsn the fzstmearest-neighborapprcfm'rnatioa. For instance, the periplleral atomic states o-xn be approximatedby e.n excited s state (denoted The additional s* statc couples with e*)(3.9, 3.10). .
88
3. Bondingand Bnergetks
the aatibonding p-like conductionstates near X or L and pushesthesestates towazdlower energy. The neglect of overlap (ii) has often been viewed aa a seziousdrawbac,k of the ETBM, since simple estimates fnd SabLm.Aj) betweeaatomic orbitals on nearest neighbors to be of the orde.rof 0.5. However, according in au.y case a'a ortkonormat set of basis Atnctioms may be to Löwdt'n (3.5)) constructed without loss of local symmetry. The restrictson to frst- ar.d perhaps second-neazest-neighborinteracvtiozls(iii) in the Hamiltoaiaa (3.7)is also aa mssllmption. A justiâcationcltn be given by viewing tlle JQ:(.R., .&j) efecvtiveaveragehteradions Jueto the Mterpolation scheme.The îttîng procedure of known electronicbands or othe.rquazttitîes lduces the partial Snclusionof longer-rangedinteractions izl the Xective quaatitites. The smmetr.g restridiol in (iii)means that the intra-atomic matzix elementscan be replacedby one-center integrals wheremsthe interatomic rnntrix elementscan be treated as t'wo-cente.rintegrals (2A. or Witk localized wave faactions /atœ Az.)of the type given in (3.16) can the intra-atomic (i m j) matrix elememts of the J'hmiltonian (3.7) (3.17) be written with the symmetr.v restdction in the fo= -
'
-
= Hab(A4 , K' )
Xb
eu +
j
d3z<*(z Az.)(F'(œ)Fk.tmAz.)j/ts@ .R;) -
-
-
-
,
wlke'reFk. (2)is the one-electronpotential of the correspondingfree atom ocmzpyiugnomimallythe site Rs.. The second term on the righvha'nd side of (3,18)represents the eFectof the crystal âeld on a certnsn atomic level. The of an ETBM inclusion of the crystal-âeld efect destroys the transjerability Hnmlltonian to Tmknown atomic azrangcments.1z1the sense of the introducthe crystat-feld shifis are usually negleded, tion of a psetztb-Ehnnsltozf.an
S'obll%:l NJ) Jïc'ékôgo? =
(3-19)
and ceatnsnt'nsversalor cazmnicalparxmeters ea are used as Kective orbital eael'gies(3.1,3.$. The interatomic matzix elementsof tke Hlkrn'lltozlian,the hopping integraks.!.QSIAZ., Rîj , aze responsiblefor the spDtting of the discrete atomic or for their brooening into energy levels iuto molecule states (see (3.10)) They ca'a be a;)bands ic trxnslationalty iavariant s'ystems (see(3.14)). pzovlrnn.tedby t'wo-center integrals. Tize distance dependenceof the hop. ping mlttrix elements between orbitals at ril'fîerent sites can be obtained by vaziotts methods (3.1, 3.2j.In the simple cubic case a l/dz-dependenee Ld= IAJ' K' E:atomic distance,bond length in the nearest-neighborcasel The geometric dependenceof the Hamiltonian matrix cazt be derived (3.6). elements is give,nby the Slater-Koster relations (3.41 -
3.1 Orbkta'lsand Bonding
sg
Ilis (R.,Rtt = Kag., Ssp.(,!Q, .&j)= 'ruljstsptr,
EptxaLm', AJ) '-rsarrlsptr' Hvop,(,&,RJ)= (TG1l + ('D=2. npIrlvppcr 'O-LIIV',' =
'
'
for the ss, sp, an.dpp iateractions. The hat on Zp. indicatesthat tiuis spintaraction is aigerent 9om k:sp.in the case of atoms of dslcrent chemical nat'tuw at m and Rj. The direction of a pœorbital, rzcz, is decomposedinto two components, n.. = p..s + zurl, into a vector perpezzdiclzlar(.1.) au.d a vector parallel (11) to the vector d = Jzj R,. betveen the two interactiug -
atoms.
The Slatez-Ko<mparameters
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)
=
-
-
-
given in terms of p orbitals (3.16) parallel or pezpendicalarto the connecting vccto'r d. The physical menmsngof the prazneters Ls itlustrated in ng. 3.3. The zelatiozzslzips and (3.21) are bvenin the original paper (3.20) of Slater and Koster (3.4) and in the book by Hazrison (3.6j not only for s andp states bat also for d orbitals. Geaeralformulasa'adexplicdtexwessjons involving .Jan.dg orbitals caa be fotmd in (3.112. The pazametezeWscr, Wm., 'Uaw,5fw, and 7/,,. depend on the atomic distance d. For typical nearestneigûbor distanzes they have the signs Xse < 0, Kpv,Zp. > 0: $$w> 0, F's,rr< 0. J.ngerzeral,the absolutevalues of the c iuteractions lKwl, Iï%.I, and I7pp, is smalle'r. 1V;,.1, (are of the same order of magnitude while IV's,pxl Basedon a comparison with the gee-eledronbandwidth ard with empidcal tight-biudjrtg parmeters, Harrison (3.12 deducedthe following set of tmiversal az'e
i'wo-ceuter interactions,
:2
T'kv rltzbtû M =
s
(3.22)
with Tsu = -1.32, Taw = hspc= 1.42,Tppc= 2,22, and r/mg.= -0.63, where d is the bond length. 3.1-4 Bonding Hybrids
Ic solids with partially covaleat bonds the valenceelectrous are maimly 1ocnllzed between atoms along their bond direction d. Therefore,it is more
90
3. Bonding and Energetics
atom A
c interadion
P#
,71interadion
Eig. a.3. Iltustration of interatomic matrix elemenu for atoms & B coupled by s =dp orbitals. The line connecting A aud B Lsassamedto be t/e z-es. The matrix dement betweentwo arrows denotesthe interaction of the respectsveorbitakskThe sign (+ or -) inclicatesthe odentation of = orbital.
3.l Orbitals aud Bondiag
basisset ill which..the atomic orbitals are direded azong The new orbitals az'e linpa:r the bond, bzsteadof simple a and p orbithls (3.16): combinationsof s and p fltnctions and, hence,called spà hybrids. They #ve rise to probability distributions Scdsngan 'eledron that are pointed i.a the direction of the p orbital entering the respedive hybzidr The resulting fottr
advisableto
use a
ftlnctions in the directions d)
(J=
4.(11+. l r
1, 2, 3,4)a're
'
The hybrids fulEll
,k.)(1+
31/ jopj
+
j.j the orthonorrnn.lstyrelation $,j j
y
(1+
za =
.y;(tAstr)
/a,.(x)
=
&)
1+
ASAJ
dja.j
.%(r) =
.
(3.23)
1, 2, 3, 4)
.
(dylldjl
=
&g.
(3.24)
If three bonding directions dj/ll, I (# = 1, 2, 3) are given., one is able to deterrnsnethe four constaats kj describingthe hybridization stage aad the from the six orthonormality relations (3.24) fourth bondingdirectioa d
Cspn
1 1+ y
=
X
+
&f>l
'
Iu tetrahedrally coorainated solids the fom bonding direcvtionsare smmetrjcal)y equivalent. They point Kxadly iu the ideal tetrahedron directions dz = .J.J 1, 1, ), Jz = .-g.c -1, -1)1 (1, (1) 4 4 -1, 1) dz = .l.a(-1, 1, -1)1 84 = .--EJ (-1) 4 4 with c,o a,sthe cabiclattice constaat a.s showni.n.Fig. 3.4a.The a'nglesbetween two of them are eqllxl to 109*28: ard the orthonozmality relation (3.24) are glves 1, = 3 (J = 1, 2, $ 4).The s and p atomic wave Alnctions (3.16) trxmqformedinto fom spo hy brids 1
+ lpzll, + lpz)+ 1pv) l8p31) j. (L.s) =
l
1'S1/2)é'(1-9) + 1pz.)kpv) kpall, =
z
-
-
(3.27)
+ 1.p,) l.pz)) L.spS3) , 'j (Is) 1.pz) 1 I.s#4) .j (ls) Lpm)lpv)+ Ipa)J =
=
-
-
-
-
.
sketGed izzFig. 3.4a. For tetrahedrally coordinatedcompotmdsthe sp3 hybridseftnlflllthe princéple ()/''rrltzlrlrlzcmo'verlap.Sincetlle hybrids st They
are
92
3. Bondingand Energetics
(b)
Sp1
sp2 -*.'. *d.
z '.:+ rrl. @ .# *:@: J..)4 ia :. = 4 . - . + .
Fig. a-4.
es
of
A
**q ,ô;+:; v ' . 4 : d +*.'*J.'JQr2e.'#;)
hybn'
'
titm: 2
(a)fottr spDhybn'
pom'tm' g
tke corn
of ide tetr e on', (b)t 'ee hybrids with anglesof 120* to each.other 1-n a plaae- (c) o sp hybrids polmtn'ngl-n. opposite ectiox)s ong the s e -
'
.
the neighboringatorcs point in the negative tetrvedron dîrediou -% the stronge-st overlap of two hybrids happensalong the connecvtionline of two atoms. The bondlngis mn.xn'mized when the exten.t to which the t'wo hybrjd sites overlap spazially is mAvsmszed. orbitals ozt adjacent The corresponding interatomic interadson yz (see Fig. 3.5)becomc largez thxn the other interactions. The intra-atomic .andinteratomic interadiohs in AB compolmds
with tetrahedralstrttctuze
(3.12),
3.1 Orbitals and Bonding
7zh= 'nb = 'h*
=
.nB= '-tz
-
74
=
%
=
'ys
=
76
=
1
A
A + 3&,
) 1 s s 3sp + ) E (cs 1 x .. ï (G o ) 1 B (ca epB) ï
(G -
,
,
,
-
,
l
k Ysvv'5>%.-Rvspc 3/7,.) -
1
4 1
ï 1
ï 1
k
(3.28)
-
1
+ .--srep.+ lspv Mse, v'àvrsw W -
-
93
1
ltss. + .--ï4p.
-
W 1
Ws. + -Kx.
+
w
Wac+
1
-Vlptr ygj
+
Vijhp.+
k-m. 1
1
'--vspv jlip. -
yé 1
-V. xgé
-
,
,
+
4
jlipcr
,
8
1
ïïrppc ï'f';zwr1 -
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 of according to (3.2$.The other nearest-neighbo:r about Ja = -3.22/à2/(r12) and ,./6 = Zteractioc are 74 = ,%= -0.1852/('md2)j ys = -0.325,2/(md2), Their absolute values are indeed much smaller th= 1% 0.31h2/(md2). 1. For other bondl'ng geometria tlze principle of madmllrn overlap is f7T1617ed by hybz'idswith anotller character. J.na thzeefold coord-lnatedplm.narbond-ing :coniguration sp%hybrids point to the atomic neighborsin the same plane. 'Each atom is at the verte.x of three bonds at 1201to eac,hother formiag a hexagonalnetwork as, e.g., in the bazal plane of graphite. ldentifying this plane with the zy plane the directio:csare dz1 0, 0),Jaoi1(-1,+W, 0)/2. 1(1, are trrmqformed l'hen ./1 = àc = zv = 2. The atomic wave Gnnctions(3.16) 1to (see Fig. 3.4b) are
lsyp:) -
c
1
W 1
+ (Is) v'ïlpzll ,
).
1.sp 2) VJ Is) W =
ksp23) =
1
-
1
lp=)+
vq 1s) Uf lpz) -
-
3
glpv) 3
glpv)
.
dtrection 84 perpenEdictzlar to the bondingplaae, in. the consideredcase the pz orbital. k..'. Ia a ltuear cb.ainthe bonding to the neighbors is governed by two oppositely dâreded(d: = -da) hybdds with sp cahakracter (lz = la = 1).The
)T.hefotzcthflcnction is a pure p orbital (l4 -> x) in
:: '
.
(3.29)
,
a
3. Bonding and Energetics
3.5. Directed sp3 ùybrid orbitals on two adjacenttetrahedra ar.d the.ir intra-atoznic aad Mteratoxnic (nex-t-nearestinteractions (3.28). ndghbor)
Eig.
two other Alnovtiorusin the directioas perpendicutazto the chain direcvuon are buik from p orbitals (,V= ,/4 -+ x). If the CI:a,Yis parazel ((%,dpa.Ldll to the z-aads,the,n 1 = + I.pg))' , 1's.p1) .ya(1,s)
1
lpxl'.l E.s'>2) (ls) G =
-
(3.30)
for the two sp hybrids. They are plotted in Fig. 3.4c. 3.1.5 Bonds and B=ds
The consequencesof the hybridization stage (Fig.3.4)and tke interactioa cff hybrids (Fig.3.5)for tEe azlowedelectronic states can be eazitydemonstrated for tetrahedrally coordsnatedsystems, e.g., covateator partially ionic spmicozductors c'oestn.lll'zingin the dinmond structurre or the zinc-blendestrac= follows with the fcc Bravais ture. The lbmiltonian matrix Iïlàbllk)(3.15) lattice po>ts (A) and neighborsof an atom along the positive or negative Siucet'wo atoms A and B are i:a one J'n:'t cell, tetrAedron vectol's dj (3.26). one ftnds an 8x8 matrix (3.7)
3-1 Orbitn.7qaad.Bon
el
0 O
o
SA
Egagz(%) Eapn
0 () c
-
(k) Bvpgz(A)Espgh(1)
Aaagz (J4.E'irs4l/o) Erugzqkj -.Esp:z(%) o .-àspgzlkt.Ebvg4(%) ,EbzJz(J4 EnvgnLkj o o o .E)z.vgatpl scvgzllèz'==.vz(k) s) .-k.pg.iLkt eBO (1 0 0 .*l.,#2(b)-Xap>;(%)-Xop#1(A)-Xx,p9:(1) &Bp 0 O 0 BspgqlktSwzp''z(h) BzwgL (1) Ecug-o(D) cpB o 0 c Bnpgllhsfwvp't(h) Em=g*z(â') fa,yv;(=) cD 0 Exp.q.z(k) E=ugq 0 0 (*) .%og*L (%) .FQp,1(k) P c c
S'A >
.
(3.31) hteractions with the composite matrix elemeatsof the ftrstrmearest-netghbor
(3.2:) zas = 4vuv, Eap= 4K,./7i
2ap=
-4!V./V%
.pu
=
En
=
4(k;p.+ 27,.,.)/3, 4(vr,w *9p=)/3. of the phasefactors gjLkj(J=
(3.32)
-
The stlmq Bloc,hwave vedor k, 1
ipa,
are
+ e kud, +
.çt(k) .,.jge 1 spaz .ç2(k) '; Ee + =
=
::
ipaz
J3(k) 'k Ee 1 l/od.z :4(2) 4 (e =
=
-
dto-fnedby
-
-
eipaa
-
eiaas
es/oax
eioaa
aaa
eipla
e
:aaa
+
e
-
1,2, 3, 4),which dependon the 3D
.j- (/Tea.j ,
ppwj
-
,
eipaij
-
(:.a3)
,
+
emaxj .
ca'a TMs LCAO I'Iackiltoaian matrix for the zinc-blende structure (3.31) be easily diagonnlizedfor certnsmhigh-symmetry points and directions i'a the bz,lk fcc Brilloplin zone. For instaace, for the direction k = k(1,0, 0) Fig. 1.25) along a cubic nan's betweenthe Mgh-symmetry poizztsF and X (see one obtnsnsin the lirnit of a dbmond-stract'are crystal (A= B) 'with lattice
(3.7)
cozustant tu
c1/2/3/4(k)ep =
:i:
c
2
-% *1
(f.2.cos2 1-Jj + Qv sin (k< )j 2
(3.34)
for the two pg/z-lilte bandswhich are twofold degenerate,azzd
) (s-
ss,e?,,,s(p) -
1
jp -
:
+sp +
'+' (,-,-û )q 4 l sin2 + .ss2p (k-% 4 )j tk-t)
(s-
+
s-)
cos
,
sa
+
(s..- sss)cos
a
notation is tha,t fo'ar for the Btr spz-like bands. Here the meaning of the *:':F They are + + +: + +, + -, combiuationsof the two signs n.='.qe in (3.35) for the bandsu = 5, 6, 7, 8. an.d .
-
-
-
-
-
3. Bondingand Bnergetics
> > U1 u X c
UJ
Fig. 3.6. Ba'adstmzctme of a sinconczystal witk.in the d:st-nearœt-nerghborapprovt'mationand mstngan ap3 basis set. The valenceband mnr'm= (r1s)is taken 1uszero energp Tke re#onof the Aamdamental gap ls shnzled.
There
are
m=y
sets of frst-nearesfmneighbor tight-binding parameters.
one fnds describedin (3.22) Withsmthe tplmsvezsal' moêel of Harrison g3.1! Sss = -7.28 eV, Bsp = 4.52eV, Szz = 1.76eV, az.dEn = 5.24ev for silicon bond length dqq= 2.35A. Usingsa = -14.79 ev and sp = -7.59 with a b177k an.d(3.35) are plotted in Fig. 3.6. The uppermost vatence ev the bands (3.34) b=d varying Som J'cs,to ,X4 symmetry charader is twofold degenerate.Togethe.rwith the lowest conductionband vary'ingfrom Jlz to X1 it surrounds the fl:ndn.rnezttaâ gap region. Wheremsthe valence bands are reasonably desczibed by tlle Ast-nearest-neighborapprovsrnation and the spn bmsisset: the cpnductionbands show features it disagxeemeutwith experimental observations. Thls holds in particular for the position of the conduction b=d minirnttrn in the BZ, which is expeded to be situated at about 0.85TW.J.ustead, the lowest conduction band îs înevitably predicted to zise in going frozzz .;' to X. However, with the inclusion of a peripheral .s* state in the bae set interactions tke dispersionof the lowest or of tlle second-nearyst-neighbor couductionband can be describedcorredly. '
2.2 DanglingBonds
3.2 Dangling 3.2.1 Formation
There is
97
Bonds of Dangling Hybrids
natlzral quasi-chemical view of the creation of a surface, as a, by brexkeirgof interatomic bonds. Such a simpMed process accompstnied quasi-cimmlcalpictme can conveniently be discussedin the fmmework of the tight-binding appronn'h described izl Sec't.3.1.2 =d, hemce,for homopo1a.rand heteropolar semiconducztom with n.7=os%directional bonds.' I'a sue.h crystals the eledronic structuze is deterrnînedby hybrid ozbstakswhidz are directed toward eacz other and fo= bonding orbitals as sEownin Fig. 3.5 for the t'wo hybrids coupled by %. When creating a surface with a certain orjcntation suchbonds and bondir.gorbitaksare truncated. J.nthe ideal cmse there (withbl:lk-terrninated atomic positions a'cdno electron redistributîon), appear hybrid orbitals which are directedout of the stufacear.d remaiu ttzlbonded. These orbitals are to.lled danglLnghpérrld.v(Fequeutly, but less apFor the discusslonof these dangling hybrids propziately,also danglingbonds). or bonds, prototypical systemq are the (111) a'ad.(100) surfacesof din.mondtype crystals. TMs is illustrated in Fig. 3.7 for silicon. The âgure showsthe a
(a) .'
.,.-.-.'
,'
.'..', '--.,-.--..'....'-'' ...-,'
'---------''''''''''''''
1.( E1
t
VV X# .%py,; .-----... -----,,,,,,,. .y. -----
-
C
------.
,
.'
-,-/
..
E.l
,
)
y ygjyy, j
Fig.3.7. Contour plot of the total electron density itz a (110)plane intersecting l (111) plazzeat zight a'ngles:(a)b4'lk-siticon; (b)ideal Si(111) sttrface.A contour ppacing of 0.10 1/A?is used.
3. Bondlng and Energetics
98
(b) e' t' Tb
#' #
e
2 e
e
1 '
1 .
-&'
-
A
*
2e .------mw-
I
------
:'
l 1 1
k
3
1
. '
I '.
I
1
,
.7..i.
j j
e
?' #' l1 e' # I 2e I
1
e
1
1
1.
I t
.p
1
1 z
1 1
)
.F
I I
1
I
I
I 1
j
1 l
'
2
1
I
'
1
.*
t
v..
4
e
e
'
'
ç A#'
,
J 2 sp3 hybrids in the Pig. 3-8- mustration of bozd cutting durin.zsmfaceformation. four bonds surrouztdingan atom in a diamond-or zhc-blende-sirudme cryst/ are sarface.The surface planœ are ùatchedwithin surface; (b) (100) shown: (a)(111) the little cubes.
total electron denzity in
a
(1ï())plane for a
btzlk crystal and an ideal
(111)
shlr6nce.
The cutting of bonds darizzgsurface formation is d=onstrated in more detai) in Fig. 3.8. It showsa little cube with edge lengkh co/2that contains orientation iu a climond or the four bonds around one atom. For the (111) strudme, two types of surfacesaz'e possible.Oneof thesehwsone zinc-blende dangling orbital per surfaceatom while tile othe,r haz three daagtinghybrids. here only the fzst type, the singledacgliag-bondsmface,whic.k We conside,r is thQnatural cleavageplaae of diamoud-structmeczystals.ln ma'ay cases it surface,sinceonly one bondhmsto betruncatedper seems to be the favorable 1x1 surfaceuzzit cell. Bac,hsmfaceatom hmsone dangliug hybrid perpemdicu1arto the surfaceand three bac,kbondswith atoms i.a the furstuuderlayer.In. tEesecondcase, the formation of a triple daugling-bondsurfacewoltld reqttire tHe cuttmg of thzee bonds. Three danglinghybrids would occar per smface atom which Ls bonded to att atom in the naxt atomic layer by one back bond pmlle,l to the sztrfnne normal. Por diamondtllllsuc,htriple êangling-bond m,rfnzveshave indeedb*n dlrœulr.d?d i3.13!. cax two bondsbaveto be tmmcated dlxring the formntion of ln the (100) the stxrfn.ce.Thts is reprxted in Fig. 3.8b. Each sarface atom has twolsp3 y bulk-lilcedangling hybrids.It is bondedto atoms in the aext atozniclayerY. two back boads. 3.2.2 soduence
on
Electronic States
The allowedeledxonic states of a, hzlfspaceare ve,rysensitive to the presence of dangling azd bac,kbonds.This can be demonstrateêby tbe investigation
3.2 Dangling Eonds
99
of the band structuze due'to the presenceof an ideal sttrof the c'hanges face. For two orientations the projectedbulk bacd strtzctures are plotted izz Fig. 3.9 for a, silicon crystal. Accordlng to the procedre describedin Sect. 1.3.3the alloled ban.dsof a Si crystal are show.afor wave vectoz'svacyingizl the correponding surfaceBrillo':sn zone. Besidesthe Alndamental gap regiou between the occupiedvalenceba'adsand.empty conductionbands,one n.lrxn bands ms a consequenceof the forbiddea observespockets in the projecvted regions îzzthe bulk b=d structme. M1 theseregiozzsbeing forbidden for bulk states. states allow a, clear identlcation of surfaceuderived 1 surfacewith half-fttledda'n.In spite of the faet that the ideal Si(111)1'x gling boads has never been observed,we discussa Si imlfqpace vith sucb a surfacefor the purpose of illustratiom Tke llnit cell and the BZ are sbown in Fig. 1.6 and Fig. 1.22ej zespectively.The irreduciblepart of the BZ for the space group p3m1 (Table 1.4)Lsindicatedin Table 1.6e. The most prominent efect of the smfaceis the occuzrence of a band td' in the A'ndamental band structme sllown in Fig. 3.9a. The corresponding gap of the projected electronicstates are rnninly due to the drgling sp3 hybrid orbitals as shown i.n the backin Fig. 3.10. Only a small part of the wave hlncwtionksloenllzeè (a)
Si(111)1x1
(b)
Si(1O0)1x1
10
217wave vedor
2Dwave vedor
Fig. 3-9- Band stmzctures of the ideal, bulk-termsnated Si(111)1x1 (a) and are calculatedby means of the empMcal tightSi(100)1x1 (b) surfaces.They Tile probinrling method inclualng second-nearest-neighboriuteraeions (3.141. aud the bar.dsof botmd surfacestatœ jeded bulk band Gtructure (shaded rebozbs) lînes)of types $d'1 tbr' ; and %' are shown. Surfacc resonxnces and antizescs(soDd nances in the shadedretonsare not plotted.
100
3. Bonding and Bnergetëcs
((l11J
l
Y
3.10. Contour plot of the wave-Gpnction squarefor the td' state at the # point in t>eBZ of the Si(111)lxlsurface(cZFig. 1.22). A (110) plane tntersecting the (111) surfaceis shown. A contou.r spacing of 0.031/â3is uged. Fig.
The dangling hybrid energetically resides between the bonding bands.Its energy ccT= npx (3.25) is a'adantiborzdt-mg (valence) (coaduction) vet'y closeto the top of the bl:lk rca, valencebands at T' vith energy sp -S= svith the three hybrids in the back bonds The interaction 'yc (3.28) (3.34). Fig. 3.5)shifi;sthe dangling-bondband slightly to higher energies.Siuce (see the point of view the (111) surfaceatoms are secozd-neal'estneighbors(from tlle interadion of the dacgling hybrids at an ideal surface of the blllli-czystal), is weak, =d, tims, the dispersion of the sttrfaceband td' is fairly smaz. TMs bazldpins the Fermsenerr. It is partially Oed, and therefore the ideal (111) Tn .adclition surfacesof group-W crys-talsshould have a metxlllc charactea'. to the dangling-bond-derivedband i.n the fhindamentalgap, the surfacealso rise to b=ds %' in pockets of the projected baad stracture arcund the gjve,s K point in Fjg. 3.9a. They are mn.l'nly related to bonding a'aê antibonding combimlntionsof the t'wo hybrids fomninga b>ck bond. The bandsdb'related to the bondr'ng (antibonding) combinatioM occuz i:a the projectedvalence iateractions bands.Sincethe Emt- azd secoad-nearest-nekghbor (cozduction) of the bnnt- bonds with the environrnent are ed in compa'dsonto the situatson of bonds in the bu)k crystal, such an energetic splitting between blk7k'statœ ar.d ba'ek-bondstat% Lsunderstazdable. 1.n.the cmse of the (100) s'arface(cf.Fip3.9b)tEe suzfacemodiâcatioâ of the electronic structme is more drmsticcompared with that of the (111) smface.First, i:a the (100) cmsbtwo bondshave to be brokec per surface atom (cf.Fig. 3.8).This resdts ia two danglingsp' hybrids.Sineenow thesehybrids are localized at the sn.rne surface site, for symmetzy reasons they dehybridize tuto bridge-bond (tbr')orbitals which are pazalle:to the surfaceand danglinp As'a bond (td')orbitals whic.hare pezwndiculazto the surfaceplane (3.15j. resalt, the orbital chazacterof the correpondsng smfacestates strongly difers from tha,tof the origiaal sp3hybdds.A possibledebybzidizationof the tvo sp3 bond
a'rea,.
3
i otq
3
S7
Sp .u -
De hybn-dQatiOn
..z.
Eo1 14 Bulk
ldeal surface
of the surfacesp3 dattgling hybrids on Eig. 3.11- De-hybridizatioxp (schematic)
a
(â00)suzface.
hybrids into a p orbitaèacd an sp orbital is Mdicated in Fig. 3.11. Formxlly, and (3.27) the zzew orbîtals can be written as with the de6nîtions (3.16) autisymmetric and symmetric linea,rcombinations 1
:$
:$
1
+ pa)) 2)) .a (1pv) 1 1 a s + I.pz)1 + 1.) i'S.p 2)1 vg EIs) 1d) v,j. ((.4:.'P
lbr) xgj (ldp1) =
-
sp
=
=
=
,
(3.36)
of the danglinghybzids, whea'etke surfaceatom is asstlmedto be located'at tite origin. The two states with eaegies =
Sapz -'- ,72,
d;d =
hox + 'ya
cbr
(3.37)
by the interadsotl malonger degenerate.The splitting is determn'ned also Fig. 3.5). of two sp3 hybrids at the sn.rne atom (see trix elements'ya (3.28) The strongest iuteradions be>een bridge-bondaad dangling-bond orbitals cemteredon dz'fbrentsites of the square lattice of s'arfaceatoms mxsnly happen via the substrate and lead to a broadfmingof the two levelssbr a'cdgd (3.37) i'nto surfacebands in Fig. 3.9b. Sincethe smface atoms are second-nerestthe broaderiing neighbor atoms (fromthe poi'n.tof view of the blzlk crystal), comes essentiallyfrom indirect iateractions betweensurfaceatoms locatedat They are due to the coupling of tke surface the same atomic row aloztg(0ï1). of the srst underlayer atoms pointaad.the sp3hybrid.s (3.27) orbitals (3.36) ing along a bond. Nevertheless,the Id' band i'n Fig. 3.9b showsoruly a weak dispersionbecauseof the small efective rr-like interaction of the (d)orbita,ls The esective interaction of the lbr)orbitals is much stronger, iu par(3.36). dtredioa. TV correspondingbaad therefore developsa tictz:ar in tke g()llJ quasi-one-dimensional character.Accordsngly,the dispersionof the (br' band directions. The interaction of the bridge and .1Re is strong along thc 13..11 aze no
102
3. Bondtng asd energetic,s
is the is of r type. The consequence bondsalongthe chaia direction (0ï12 weak dispersion of t:e (br; band ioag tke J'J an.d#J' directions. Tke redacedintexaction of tize sp3 hybrids i'a the back bon.dsZvesrks: Their strongest to states locnqizedat fzrst- aud secoad-layeratozns (3.161. cha,i'adirection, i.e., parallel to tite ,highiuteractioa happens in the (0î1q symmetr.r lines /'J aad J'R. Consequently,the correspondingsurfaceban.ds appeaœia the # and J' pockets, ic particular izl the stomac,hgap arolmd # bnxllcband stmzcture in Fig.3.9b. of the projec'ted
Forces
3.3 Total Energy and Atomic 3.3.1 Bmsic Approximatîozls
Jn Chap. 2 we fokmdthat the smface free e'aergy and tàe thermodpmmic potentiaksof a imlfqpaceare domînated by the total energ.yE of the syste'm, witicll is nearly the interni energy at zero temperatare. A sttrfacesystem coasists of atoznic nuclei and electrons.The electronsare fisFerently bound to the auclei. Therefore,they c,an be divided into tightly bolmd core electrons aud valence electrons.Sincethe chemical bonding and, hence,the smfaceprocesses aze governedby the valenzeelectroits, it is usefutto regard a solid with shprfn.cein terrns of a regular cozectionof ion cores and valence eledrons. For ex-nmple,the free silicon atom with valenceZ = 4 hmsa'n. ion core of càarge 't 6 with the electzonâccoMguration 1s22,,22p6 ) acd a valenceshell of chazge Only in the case of atoms, with -4e with electronic covguration 3.623p2. shaltowtt J, o:r g electronsdoes the picttzre need to be made morè compûcatedby the inclusion of seMcore states that should mostly also be treated as Gence states. One example concerns the Ga3deledrons, in particular in compoundGaN.They are energetically closeto the N2.s the seznicouducting states and, hence, conkibute partially to the chernicalbonds. Apart from of ion cores in thks compication, a hn.lRpaeecaa be regardedas atl ensemble distributed and a valence electron density 'rz@) equilibrpxrnpositions (Rz.). among dsFerent atomic cores. The spatial distribtttkon of the valenceelectrons is usually clmssled i'n.terrnq of metallic, ionic, covalent, molecular, or hydrogen bonds. For the salceof brcvityt here we will refer to ion core.s aa ions and valenceelectrons as eleerons. Anotker basic approximation coazerns the vibrations of the fons with In Chap. 2 we have already learnt resped to their eqpnilibrbxrnsites (.Jk).. that the eiect of the vibratjng lattice on the s'arfacefree energy is s=''n.l!(at for not too high tempeaatures. least, 1tsvadation 9om one phaseto acother) the eqailibrblm geometry of a syste,m Oa thc other hand, i'a detemmlnlmg one lnnmto study many confgtzrawith Nx, NB, atoms of ldnd A, B, tions (Ao.J vith respect to the rnsnl'rnlzationof tEe toti energy B, i.e., the total eneror ELNX, Ns) ...; (.&,.)) ms a hnnction of the atoznic confgaration, aud the accompanying forces(-Vaf(NA, N.B,2..9(A..)))acticg on the ions. ...
...
3.3 Tota) Bnergy and Atomic Forces
103
Tikis couideratiow however,stazts from the assumptionthat the dyaamks of the electrons and the ions can be decoupledso thatj vhatever the tlyzla'a!ics of the ions are, thè eledrons are ia the eluronic ground state of the icstantaaeous geometrs This is the Born-oppenimlrner adiabatic approfmation (3.17,3.1$ It is usnlltlly a good apprnvlrnation becausethe eledroa mass is much smaller t'hn.n that of the ioas. 'I'he electrons respozd idmost instsataneotzsly to chaugesin the positions of the ions. As a consequezme tb.e eledronic an.dionic deree of leedom can be separated.More suctly spensdng,the positions of the ions are parametea's;the total enerr depeencks .
parametrkally
the positions. The Bora-oppenheimerapprov'tmatîonis well jtzstifedfor all stat'ic surface problems. However,it zaay sometim.es breA do= for excitation phe'cal reac tions. For e-xam'ple, irt rnnny clkfprnl'c.al nomena aad clmml reactions aa electronjumpsfrom one enerr surfaceto anothe,r in a lon-adiabatîc way. The approzmation also f>.'11s for non-radiativetrn.ngitionsin solidswherea;n electronfalls from a high-energjrsatrfv.e to a low-enerr surfre not by emitting a photon but by emitting phonons.Electron transpol't (electron curreat, electroa ttlnneling) in surfaceregionsmlty aksogive rise to a Wolation of the Born-oppfanbeimerapproxn'rnation. on
3.3.2 'Potential
Enerr
Surface and Forces l
ta the iirnl't descibed abovetEe total enerr E = ELNX, NB, ..:; (.R./),when is ofïen called.the studied'as a hlndion of the atomic coordmates(Rz.), p0jentïc! energy dur/tzce in the 3 E:=As,.. Jvk-dirnemsioaal. con(PES) (atozaic) fglzrazionspaze, becauseit defmœthe podezztkal enera laad=pe on whic,h that the motion of the the atoms A, B, travel. Since oae hmsassnlrned nuclei is not vezy fast, that the temperatme is very low, aad tlzat the electronic states are at the grotmd state, the termq adiabaticpoterlfùl s'ucface or Bocrt-oppenheimer are atsoused. Jttr/cce Usually a complete PES cxnnot be repr-nted graplzic-allybecatzseof the many coordinates, However, important infovmxtion about the sttdace stmzcture azd energetics can be obtained whena test atom iq.displacedover a surfaze'M:,K a norninn.l geometzy Sucha test atom may be a real adatom or aa atom of the same spedes as i'n the bulk. Then, the coordlmntes of the test atom are âxed in the sudace plane. However, the normal distanceof this atom and tEe coordinatesof the surfaceatozns are allowedto relax. One obtains a spedal PES, that of a suzfacewith a test atom. Axl nvnmple of a resulting total enerr surfre is #venin Fig. 3.12for a Gn,AK(110)1x1 surfu aud a'a Sb test atom. The plm of the energ.yof the tmcoveredstkdaceand the energy of the 1ee (Le., Sb atom is usedas eneror zezo. For that isolated) remson the negative total energy in Fig. 3.12, ...
Eaa =
-
-
lSttestatom + suzface)ftcleansGace) Elkeetest atomll, -
(3.38)
:04
3. Bondingand Bnergetics Ga
--12!
.
.$ .,
''
'
>
ap
x
-
3
(j-1(jq
. ,
3œc
,
F- œ -4
.
Iloïl
4)
As
t)-
Tng. 3-1.2. Total energy smfaceof an Sb test atom on a GaAs(11O)1x1 smface plotted over an m.e.a of two lx 1 surfacetmii cellsas a three-asrnensional perspective view (lef1) and as a contour plot (right). The smface atoms are indicated by fIILIC.II and empty cizdes.From (3.195. represeats the adsocpiionepz6zrgp of the test atom. Tlb.eslTrihceatoVc structure determl'nestite shown PES. The most stvik-s'ngfeatttre in the plotted PES Lsthe deep'channelwlf.ch Lsquite rectllinear and parallel to tlle (1ï0j direction, i.e., the direction of the Ga-A.szig-zag chnln. 1n.eac,hlx 1 surface llnc't cell two eqtlivalent flat =ln5mn. occ'ar. They correspozdto a lonpbridgebond position of the Sb tes't atom betweenthe Ga and .ks atoms of ch'gcremt surface chains. There are no isolated minima in gont of Ga or As dangling bonds. Seaz'chingfor optimal atomic structuzes (As.)i'n tEe cortsidered lowtempezature lr'rnit wotzld necessitate a calculation of the electronic grotmd state for a given consgaration (.R:).The driving forces,sometimescalled
Hellrnxnn-yneynman forces(3.20, are 3.2.1.j, Fi
=
-Va.&
.Ns)...; (a./) LNA.,
.
(3.39)
Besidesthe electroniccozltzibutions twlzic.h can be calculated by the HellmxnnFe theorem (3.20: the forces (3.39) also contain a repulsivecon3.211), tribution due to the ion-ion iztteraction.For a given eomposition Nx, NB, and a tven confguration f ,&J,the mltgllitude and the direcwtionof suc,han atomic force (3.29) #ve izzformationabout how far a ceztain atom îs from a position i'n a metutable or stable corïguratiom Tlze corresponding 'eqnlilibrittm' atomic geometu Lsidentïed by ellrnsnatinga11forces(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:lrndoes,in general, not necessazily need to be a global rnn'nirmlm. In order to ftnd it, usually several optimal covgurations haveto be studiedand comparedwith respect to the resulting total-energy vatue. tEe THePESazd the resultingforcesare not oaly importaut to detemmine equilibrhlm atomic geometrybut also for the dpmrnlcs of surface systems. Onedynnrnicalesect is relatedto the vibratlonal properties. Their treatment witln'n the hn.rrnorticapproximation requires a liaeadzation of the forcesia the positions. Other dynnmscal displacementsof the atoms from the eq:nsl-lbrilpm efects may be related to the time evolution of the atomic positions mltj. Suc,han esect may be caused by heatîng up the system a'aê can lead to of the correspondingmolecalao a meltiug of the crys-tal.The perfommnmce dynxmscs simulabion rzeedsthe k'ategration of Newton's equatîon of motion (12 Mi a m.@)= Fy dj
(3.41)
for eac,hcore with atomic mass Mi. A Srst approae to the driving forces is The classicaldescfption of the motion of the given by tlze expression (3.39). atomic cores is usually jlzstiûed.A simple criterion is basedoa the thermal It shouid be smaller than the de Broglie wavelengthlst = h( zx&-sTz5Q. At a tzpicat MBE growth temperatme atomic dist=ces ill the slprface (3.221 of 60OK 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-oppfmlneimertotal 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 reEected irz the disucssion of Fig. 3.12. The adsorption energy Ezve(3.38) governs the desorptioaprocess.The probabilit.yfor the desorptioaof 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 of atoms. Surfaced'fhtqioncan be coasideredas an isolatedproccs for systems with a high energy barrier for the penetration of au atom on the smface into the bG. However,in many cases a sharp rlilereatiation botweensttrface =d. balk layers is decult. In the cmse of a metal with an I:nteconstructect sttrface the built-in of an adatom is energetfcally thnfavorableizï both the upporrnostsuzfacelayer and i.zl a 'bulk layer. The sëtuatsonis ds/erent in semiconduztor surfaces. The clzeznicalbonds in the tlze case of recomstructed surfacelayer rcay clsf'er 1om those in thc'bullc. Therefoze,a built-ia of an adatom i'a the surfacelayer cnnnnotbe excluded.NeglectMgsuc,hpenetration
1O6
3. Bondir.g and Energetics
efects, the PES determinedfor the movement of a test atom
or
adatom
the smface plxne may be 'LUSCXI:I to i'aterpret the elementary processes contributiug to the surfacediRtniom lt is obvious that the local rninlma on a PES should play a'n impolant role, since the dsfhnsi'ngatoms tzy to reach suc,ha position. ChennisorptionEappcnsat a minirmnm and the compbsite sy stem gains energp TEe results of a moleeular-dmnmical s'tudy of the atomic motioa based on (3.41) ca'a directly be tzsedto descibe the process of surfacedifhlsîoa. The trajcetoryAtesttflof the te-st atom or adatom on the surface is related to the difhtqion. constant D by menz,q of the Eiastein relatioa. ln the cmse of isotropic systems one has ove,r
'
lsryh ilztesttf) J?.<.v(n)l2 . -
jEI
uuz
41
z-yoc
(a 4c) .
Sucha trajectozyis sGematically representedt'a Fig. 3.13. Most of the time the te-st atom 'dbrates azotmdthe boadimgsites, 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 surfacedimlqion. A.n elementary step in the Hifh:sion of a surfaceatom may be tmdemtood izl terms of transition-state f/zeor.p(TST)(3.23) . The path in real space bet'ween 'two sites on the surface correspondsto the reaction coordsnate. The its f-zzalposition atom with its mlrrotmdingsis the reacvtaat and the atom 54).
after
an
elementary process, a hop, is the reaction product. Proceedingbe-
X X
X
X
X
)M: X
X
ft ttlst(t)
X
X
Fig. 3.1a. Schernn.tic zepreserzatioz of tke tralectoryof a test atom on a suzface. A satfacewith two eqtkivalentbondcngshtes(czosses) pe.r smfacellnst cell is shown.
3.3 Total Energy azd Atomic Forces
107
treea the two sites the atozû usuanycrosses a certaùl energybarrier Méth a tmnsition state as in the case o'f a bawier height EB, whic,h tzlzarac-tfm'sro a càemical reaction. The bn.rrier energ,gLs defmedms the rlilerence of the iotal emergiesof tXe t'ramsitionstate aud the lower minl'mtlm. Thereby,the transition staie gives a saddlepoiat at the PES. The trnrKitson-statetheory asstzmes that at eacà bonding site the test atom is iu local thermodpmmic Thea, the trxnsition rate eqnllibrblm with the heat baih of s'ys'te.mphonozss. betwee.ntwo bonding sites c.an be calculated az a local tkermodynamiceasembleaverage (3.24, The condiiion of the applicability of the TST is a 3.252. high energy barrier EB :>.>kaT. Only in this'limd'tdoesthe particle stay long enough i.!l oae dfe, so that equilibration eltn occur. The pzobability of such an elemeuta'cyhoppi'agprocess ks given by the '
behavior Arrhenius (i.e.,activation) r
=
(3.43)
rcexpt-ls/ksTl,
whereihe prefactoris related to the hoppi:agrate'vithout thmrmalactivation. It ksdeterrnlnedby the frequendesof ihe loc.allattice vibrations, thdr interzzal ene'rgyand entropy. Estîmates relate ro to dmaderistic phonon frequencies used are the Einste.inhwuency, t:e Debye of the system. Tjmical valu.e.s vibration (3.26,3.27j. Sucà flwueaqy or ihe fxequencyof the highestsubsirate vith those a proceduregives a n,5nK motion along the (001) direction reqees that adatoms clîmb over tke zig-zag chxinq.In any cwse, the considerable variation between&1: aad EBS.shoid lead to a strong Ysotropy of the sYace dimnsion. Smfacerlllrtlqion also remarlrablylmfluence,sthe homoepiteal p'owtk of a material. This kolds in paztietzlarfor the molecular beam epitaxy of compotmd semiconductorssuc,has Gn.Aqaloag a cubic a'ds. Dependingon the .
lû8
3. Bonding and Eneagetics
growth conditions,whethe,rthey aze Ga-richor As-rich, diferent reconstructions appeaz'.UndermoTe Ga-rjc-hprcpration conditionsone of the most important reconstructions is the GaAs(100)((4x2) Fig. 2.17) smface(sce which has been discoveredrecently (3.29). The potential enerar surfacescomputed for the adsorption of As a'ad Ga atoms (3.304 are plotted irt Fig. 3.14. One fmds the adsorptgonbehnuviorfor ad-cationsto be very dxereut from that of adsorbedanions. The preferredbonding position for a Ga adatom (ACin Fig. 3.14) is located in the tzenches,fourfold coordinated betweenthe doubiy occupieddaagli'agbondsof the Ast-layer As atoms. The calculatedpositiou is supportedby a recent X-ray analysis (3.31), wizich givcseWdencefor a 19% occupation of AC sites by Ga adatozns.However,eve.n for extreme Ga-ric,h surface preparation couclztionsthe adsorption of a Ga atom in the AC posi'
(01k
plïj Fig. 3.14. Potential energy surface for the adsorption of As (>)and Ga (b)on tke Ga-rich GaAs(100))(4x2) surface. The contom spaaing is 0.15 eV. Bright =d, hence, favorable adsorption po(dark)regions indicate m''nlrnn. tmn='mal sitiozzs(trxndtion-state Open tfzlledlcircles represent Ca (As)atonzs. regions). Large'rsmbols indicate topmost atoms. The rn''nn'rnn. AA and AC are describedin the text. From (3.304.
Descriptionof Structareand Stability 3.4 Quantitative
1û0
tion increasesthe total energy by 0.1 eV. This type of adsorption does not thereforecoutitate aa eqpll'll'briltmsurfacestrucime. Thc most favorableposition of the As adatom (denoted .&A)correspondsto threefold coordinatioa with three empt.y Ga dangtingbonds.The occttpation of the AA site inueases the total eaerg)rby 0.3 ev tmder Ga-richpreparation conditions.This eaezgy
indicatesa reducedmetaztability of the As adsorption. The PESSin Fig. 3.14 have consequencesfor the dlfhlsion charaderistics of the adatoms.The dlmlsion is rather diFerent for Ga and As atoms. Ga direction,where ehergy atoms preferablymigrate in trenGes along the (011) barriers of only 0.2 ev need to be overcome. The =lnsmpnrn energy barrier for dl'mxsionalong the (01ïjdirection is 0.6 eV. The motion of As adatoms direcdon. is somewhat less anisotropic. It preferably occurs along the L01î) The miniml:rn eneror barzkitrin this direction is O.5eV. lt is smallerthan the
directiom The adatom difh3Ksondepends barrier of about O.7ev in the (011J remarlcablyon the surfacereconstrudion considered.The previouslyaccepted J2(4x2)surface strtzcture (cf.Fig. 2.17)gives rise to a completely dxerent landscapeof the PES for 50th Ga and As adatoms (3.32, 3.334.
3.4
Quantitative Description of Structure aud Stability
3.4.1 Demsity Rlnctional
Theory
Most of modern suzface-strttctttre,stability, ard electronic-structme calculations deali'ag'with the complicated coupled atozaic and electronic problem 'mnllreuse of deztsit.y Gtmctionaltheo'z'y (DFT)(3.34) and a certna'napprovimation for the fhxcitazzgeucozvelation contribution to the total electron-electron approfmatnteraction. Its standarddescziptioais bazedon tize local dezssit.g by adding gradient co=ections However,geuern.lseations tion (LDA)(3.35) or by calculatingthe e-xad exchangeusing the DFT-LDA wave hlnetions aa'e of the electron gas. more hwuently usedto accouat for the im%omogeneity Due to its foz'maland computational simplicitw as well a,s êue to its vezy impressivesuccesses in describi'nggrotmd-statepropertie.s of manpparticle systezns, the DFT-LDA has becomethe domsnn.ntapproackfor calcuhting stmzctttraland electroaicproperties of b'nllcsolids and theîz surfaces.Withia Sect. 3.3.1) for the descdptionof the enthe two basic approvsmations(see of interactiag coz'es and dectrons) the total energy of a sttrfacesystem sembles is divided into .
NB, ---; (.&,.)) -E%n-:on((.Rz.)) + .Ee:ta, (A$)), ELNA, =
a s'Tnrn of
the classication-ion iateraction enerr
1 ./ozz-iontta'll 2 =
,
-
zLzjel Ia. n.i -
ï.?
I
11O
3. Bonding and Bnergetics
descibing the repulsion of the bare ions at the positions m cisely, of the
.
or, more
pre-
vith the valence Z.t and the pure electronic contribution Seltrz, (Rz.))dependkg oa the electrondensity. The electronic contribution can be calculated witbln the DFT (3.34,3.35). Tkis theozyis basicatlybuilt on. the Hohenberg-Kobntheorem,whicah proves that the em.ea'gy of a many-electronsystem izzthe Jrotmdstate c.n,m be obtained For a give'n ato'mic confrom lmowledgeof the corred electron density zz@). âgttratiozz(A,.J,the quantity Seltn, is a lznsquefllndional of r.t#. lt (.R,.)) value when obeysa variationi pzinciple in the dtmqity.It acqturesa n75=14731:173 the eledroa densit'yis the corect ttmzel density.The extremum conditjon allows one to map the Mteracting Ne-electron proble,monto the deterrnination of Ne single-pazticleorbitaks.This me--mq eac,helcctron is movi'ag icdependently of the other electroms,but it experience,sa,'n elective potemtial 5r(z)' whic,h emulates att the Mteractions with othe,r electrons. One obtaius the station.ary Kohn-shnm equation (3.354 cores
SkS#J@)= JJMC' (m), :,2
Z= + 1'r@), (3.46) Jm foz' enr'ln independe'atsiugle-peicle state in analog.yto (3-1)but with the Nxs. Here %'(z) and s.f are wave Glnnctionsa'aê Kohn-slmm Hn.rnsltonîn.n eigenvalues,respectively, of norsinteracting âctitious single pazticle,ssuc,h that the correct electron density is obtnlned axs
J:os
=
-
'
zztzl -
J7'zjl#J@)l2,
(3.47)
j
wllere ni denotesthe occupatioa nptmberof the eigenstate that is representedby the one-pazticle wave fhpnction#j(z). I'a the cmse of a 2D trn.nsformxlly replaces the oneiationat system, the e'igenvalueproble,m (3.46) electronScbröasngerequation (3.4)wit: a sjt of one-particlequantdtm nzlmbers J = ?A. The egectiveKohn-shnm potential is deMed as -
(3.48)
+ Vh(œ) + kkctœ). F'(z)= Mo=@)
= The extm'rnalpoteatial Mon@) E( 'tijonlœ R.)is gemeratedat spacepoint due to tEe cores at m with chargeZq.ln z by all Coulombpotentials '?J,Q@) at As' the case that oaly valenceelectronsare studied,thesepotentials 1(oa@l However,sacha stmplefo= of have to be replacedby pseudopotentials(3.36). Fiontz)is only valid for local pseadopotentials.For the non-localpotentials towa'rd non'adedin the majorityof modern applications, a geneaaDzation locat operators is reqaired.The Eartree potential -
F's@)=
c
2 .
j
d3a/
VtQU) j:r a:, I -
azd StabiDty Deschptionof Stmzcture 3.4 Quaniitative
111
coais directly givenby the electrondemsity.The emxc-llaztgetxl-cola'eiationtc) tribution to the total potential follows as the Anmctîonalderivative of the XC contribaiion to the totaz electroaenergy SeT.In.general,thc XC contribution one b.as is ''nsrnown. lu the Famework of the comrnonly used LDA (3.354
and, hence, J dSam@)Exc(n(=))
Wc @)
=
with exc
d. dn
(zz)as
t
hexclzzlg
a=(.)
the axchange-correlationenergy per eledron of
a
tmiform
electzon gas of densit.yrz. The XC eue'rgyoxcln)has been calctzlatedby severalapproaGessuc,has and quactttm Montc Carlo metkods perturbation theoty (3.371 rnnany-body is expressedas an acalytical'htmction For practical calcGtions, excln) (3.381. of the electron density.Tiutsca'a be illustrated by the LDA of the exc,hauge term. Witbsx the X. approirnation (3.392,
9
c
exctzz) '-û'je =
3
(3.51)
-?z K
fozowsvwitiz the parameter tz adjustabiein the intervat 23 S (z S 1, in orde,rto accotmt partially also for tEe electron correlaltiop..Freqientlythe is tzsedfor Exctzzl paramdrization of Perdew and Ztmge,r (3.40) Many mtems such as magnetic transition-metal suzfaces,reconstmzcted semiconductorsurface with remnsnl'ngdaugling boads or the dissociated moleculeson a smface involve unpaired electronsoz molecllln.'rradicals and, th'tzs:require a spin-polarizedmethod.1.nthis context it is the spin demsit.hr hTnctionaltheot'ywitlnp'rlthe local spin densit,gapprovsmation (LSDA) (3.41). of the correlation eae'rgy on the spin polarization is Usually the dependence replacedby the same htearpolationas fotmd for the e-xclzauge energy. For the majorityof surface calculatiorusthe LDA or LSDA gives a sufûdent description of exchangean.dcorrelation in 'ltke gro'tmds'tate. Surface geometriesaad the nattzre of surfacebonds a're reliably described.Eoweverl there are also cases where the original nozslocality of the XC energy lzas to be talcen i'ato account, at least paztially, alread.yfor grolmd-statecalculatioas. E'aa frst step, corrections related to the pactieat of tllc local elHron densityaze addedto the XC energs and the geaeralizedgradientapproximaLs employed. Gradieat-correcteddensity Glmctionalshave been' tion (GGA) Beclce(3.z.t4), and others. For eqltz'librblm suggeted by Perdew (3.42, 3.43), surfacegeometriesthe GGA usually gives results sinnllar to the LDA. Oaly hencel the overestMation the tiny tmderestimationof tlle bond lenzths taad, is lifted an.dthe bondsare weakened.For the descziption of the bond energitws) readions on a stzrface,in particttlr of tmnsition states, where of cl.tarnt'caz bren.ld'ngof old bonds and mn.lrs'ngof new bonds occur, the GGA appears to be supedor to the LDA, i.e., the LDA often gives even (ptnlltatively incorrect .
3.. Bondingan.BEnergetiœ
112
exampleiu this respec't concerns the physisorption of pazticles.Unfortunaiely, vau der Waalsinteractionsnear smfacesare not corzedly describedic b0th LDA a'nd GGA. Moreovea,tEe DFT-LDA and DPT-GGA elmmot cozrectly describeucited electronic states (d. Sect.5.2). In Sec.5.4 anotb.erdeîcit concernicg the imageDke behavior of the singlepazticlepoteutial will be discmssed. resalts
Mothe,r (3.45).
3.4.2 Baud-structttre
ard Interaction
Contributions
The Kohn-en.m theory (3.46) atlows a splitting of the total ene-rg.rof the elHronic sylem ia the groundstate (fora given atomic covguration (.&)) into two physical contribuiions
.E'olt'z, (.R.)) .ELt'n, t.R:)) .àu(0l=
(3-52)
-
T'he band-stmzctureenergy
f'bst'z, (a.)) =
J7zzjv
(3.53)
?
#vesthe total eneror of Ge ron-hteracting system of electronsoccapying the Kohn-shm states j. The sccondcontzibution '
.j:v's(œ) wd:xc (s) z= (pz + EoeLn) ,.:(.) k
j
j
=rs@l
(3.54)
describesthe e:ectron-electronicteraction whic,h acco=ts for double cotmtill (3.48) icg (atletkstin the Hartree contribution (3.49) but also paztially f.n the XC contribution) în the one-electronenergies sj (3.46). The total energy of the suzfacesystem, B, depenas'ngorz the particular atomic conâguradonIJQJ,was de'ned originally as the lrinetic energy of valence eledrons plus the potential emergiesof electronvectron, eledronion, aztdion-iot icteradions. From (3.z14) and (3.52) oae obtxsms + fost'n,: F(.Nk,NB, ..-; fAs'J)= Sbst'n,) (As'.).) (Rz.))
(3.55)
with the elearostatic enerr of the sudacesystem
.E'estn, (.&'))= lion-iontta.l) .feetzzl. -
(3.56)
This splitting of thc total energy is rather convenient for nlpmerîcaltreatment using both a plane-wavebmqisset or a basis set of locldl'medhTnctiozus.The electrostatic energy is partimllarly impohant for the behsvior of surfaces s:ldxces. of compoaads.Examples are the vdzious reconstrtzcvted GGs(100) Their reconstructionsaire remarkablydriven by a temdencyto mimlrnise the loztg-racgeeledrostatic interactions (3.461.
Descriptionof structuzeacd Stabitt 3.4 Quantitative 2D trn.nqlationalsymmet:cy the moment'qmcaa. be apklied.The bazd-stracture ene'rgy space formulation (2.47-3.40) with J = vkt the baad iadex v an.clthe wave keeps its representation (3.53) vector L-i.zzthe smfaee BZ. A vczy powerq:7 approac.his the repeated-slab which allowsone to use a'Et artiîcia,l 3D transapproxqmation (cf.Sec't.3.4.3): is expandedinto pla'n.e Eationalsymmetzy The electrostatic energ.y (3.56) wsves defned by the vectoo of the correspondsngreciprocal lattice, G. One fmds 1 = fzat + VxctGl ëxcllT Jcolyzlztt7l (1 EosLn, (AJJ)
For ordered slTrfnces with
a
V'5,(G)
-
-
-
o
+
akx + 'Yswald
wsth the vobnrneperx atöm, flzt, aud the average valemceIrtrnber,
(3.57) X. The
overcotmting of the electron-electron interaction is govnrned by the Fouzier of the electron densit'y, the Hartree potential, the XC potential trn.nsfoz'ms aud the XC enea'gy'per iectron. The constant a:
=
1
)7 .%% :
d3z
z$e2
+ kqostm) 1œ1
the degreeof repulsivenessof the ionic potentials or pseudopotentials averagedover the atomic basis in the unit cell. The Ewald eaergy measmcbs
v=ld
=
1
2.+:J7 y
.Ekn-ozz((R;)) -
Z?e2 d.3œ*
(3.59)
(œl
is obtained by zemoving the divergenceh'om the ion-ion Coulombrepulsion. tn the emseof a dinmond-stzuctureczystalwith t'wo identical atoms i.nthe 'tmit cell Jswala= -2.6936 Zzeljaoholds with c,: as the lattkc constant g3.50j. The momentlnrn-spaceformn7sqmof the total energy allows a simpMed representation of the Hellmg.nm-Feynman forces(3.39) adiug on 3.21) (3.20, the atoms in the case of non-erlslsbrbtrn atomic coMgtlrations (a.). There are 'two rlilere'at somces contribttting to DmE. One is due to the explicit dependemce of the total energy on I.R.Jand the other to the implicit depenThe dencetbrough the solution %(z) of thc Kohn-slna.mequation (3.46). latter contribation vanishesfor a self-consistentsolution of the Kolm-shn'rn equation, if the bac;isset msedto reprHent (#j@))is complete (3.47, 3.51). Ihis can easily be show'atusingthe LDA, more exactly an XC potential (3.50) The restridioa to tbe 'flrqt contribution corremsponds in an Xo-like form (3.61). to ïhe spplication of the Hellrnn.nn-Feyrtmarttheorem (3.20, 3.21)However, Usthe rcult correspondsakqoto the older force theore,mof Ehreafest(3.521. for the representation of tke eigenvalues; ing the Kohn-sbnamequation (3.46) a rather simple expressionfor the forces one fnds with (3.45) .
3. Bonch'ngand Rnergetics
zszjcz (x; Rt ) Rs% 12IA,' Ri I 3,/ iA,' ,
.% =
-
-
-
-
Re isa,
5-25-2JgJ(G)goo(G)ciGAJ J
with
(1.60)
c
bebzgtke Fom'ie,rtransform of the tonic potential situated at 'J:oa(G)
term ca'a be evatuatedusiug the Ewald plmmatioa method . describesthe force on the ion core at m. d'aeto the bare ions at sîtesRj. The secondte= representsthe forceof the esedive electricEeld
.%. The 'smt 3.53!lt (3.50,
of the density on the (pseudoldemsity indaced by the ionic (pseudolc>n.rge Kence electronsor vice versa. Therefore,the atomic force Tk (3.60) #ves the total Coulombic force excertedoa the ioa at a.. For a given covgaration (.R,.)all quantities apppltrlng in (3.60)are knowa. Only the valenceelectron llas to be calcalated self-consistentlysolviug (3.46). (pseudoldensity 3.4.3 ModelMg of Surfaces
.
Ls lœowp one has to solve the Eve,n if the actual atomic geometzy (.Rs.) for a semi-ln6nstesyste,mself-consisteatly.Since Kohn-sham equation (3.46) the respective uzkitcell is i'n6nitely long in the direction perpendicular to the surface,it contzsnK sn6nitely many atozns. Thus auy standazd bulk bandstractm'e method leads smmediately to (x x c;c matrices that need to be diagorxlszed.Sincethat cauaot be achievedone resorts to either substitate geometriesto simulate a sudace or to alteraative formal approachcswhic.h or do not necessitate the diagonxh'zationof a Harniltonian of the type (3.7) The model of a sernNin6nite soûdis mostly studied together with the (3.46). simple jelliclmn model for the material. In the jelliummodel the positive chaz'ge of the atoznic nuclei is simply represemtedby a llnifoz.m constxnt positive backgmundimsidethe solid a'n.dzero outside an âp'propriately Gosea surface Plane. Maay, but by no men.nm all phenomena iu surface scieace are rclatively shorwrange h ttature normal to the surface. Usually, the surfaceregion cau be restricted to a few atomic layers the nlnrnberof wilicll, howevez,has to be tested carefully depending on the surface pheaomenonstudled. The restriction e-qn be used to model the surface directly in space or by certain pemturbations. lt is possible to choose geometric models wikich aze small enough to be tradable by todayls electTordcstracture methodsbut aa yet still lazge enoughto be physically meaningful.Systemscontnimimgof the order of 10@ l:'nl't cell can be treated a,t a frst-principles level usually atoms per (repeated) basedon DFT 'with today's progrnrnq and compater 'hltrdware.To deal directly |
3.4 QuantitativeDescription of Strueure aad Stability
of
a
surfaceregion in
a
115
normaûyin6nite soDd.Fotzr of these metàodsare
brie:y described below.
Slab methods. This method simulates solid suzfacesby studying relatively thin 61rnRof about 5 20 atomic layers embeddedin a vacuqlm reton. Depending on the electronic-structtzremetbod used either isolatedslabsor periodic repetitions of slabs ia the dâredion perpeadiculazto the surface are studied. In the centrosymmetric case t'h.esingale slabs are ckoseato be thic,k enough to approach b''lk-lilte behaviornear tlze center of eac,hîlm. Iu noasyxnmetric cmses one slab side is pazsivatedto 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 vacullrn region betweeatwo s1abs are m''nimszed.M' a reasonablen'lrnezic.al test of the 4.&m:41773thinkness one may demand that the total one-electron potential (3.48), averagedover the plane perpendiculazto the surfacenormal, showsa plateau in the vacuntm region (cf.Fig.5.17 i'a Chap. 5).About 10 20 ,l. are usually plmcient to f''l611 tisis reqeement. The slab approvsmxtion waz alreaadyused together with the ETBM for The trblmplnn.nt advancecn.me electrooc-structt'trecalculationsi'a 1969(3.121 with the generalizationto the repeated-slabapproximatioa and its combination with a pseudopotential tvbnique to describe the eledtroaic stmtcture (3. à4).The periodic arr=gement of the slabs (sVFig. 3.16)recovers ata artfcial 3D periodicity. The new crystal represents a superlàttice with a lazge -
.
-
.
(100)
(001)
t
E01'l >
Fig. 3.15. A slab consistingof staclcsof atomic laye!'sin the surfacenormal direcsmfaceof a ziuc-blendemys'talksshown.The small tion. A slab modeling the (100) shadedspheresin the lower part indicate a possible passivatiol of the lower slab side.
116
3. Bondsngand Energetics formallattice period c
slab
vacuum
slab
vacuum
Fig. 3.16- Periodk arrangement of slabs.A sitaation suitable for is sîown.
slab
Si(l11)sudaces
T:nlt cell consistin.gof a slab and the vacul'rn paa't.The formal lattice constant in norrnnl direction is givezzby the s'lm of the thicknessesof the slab a'adthe
batd-stmzcture region. For such a geometrjr, any three-Hsmensional for the 317band-structuze methodca'a be used.The most common appToaches method, the fh111calciations are the pseudopoteatialplxne wave (PPPW) potential linesrszedaugmented pllme wsve (FLAPW) method.,and the linmethod with its genera7lzationto F1'11 eazizedmnzGn-ti'aorbital (LMTO) potentiaks. The 6mt t'wo methods use au expansion of the eigenhlnctions of in terms of plane waves, at least in certaiu the Kohn-shnmnproblem (3.46) spacexegions,in partimzlarinbetweenthe core regions. Pradical applications of the combination of tZe repeated-slabmethod and 3D eledrovcstructure catcalationsare lsmstedby thc n'trnber of atoms in the 3D supercellboatded 1.5, 1.6). by the lattice constant c (Fig.3.).6) ard the surface 'nnl't ceE (Figs. Thus, a compromsqeneedsto be foun.dbetweeathe sla,bthickness:the spacing betwee'athe slabs,and computationn.l egoz't. Izl the slab approac'nnationthere az'e t'wo surfacespe.r um't cell on opposite sides. These t'wo sides introduce severe problmmqin all cases, even for centrosymmetric slabs (see e.g. Fig. 3.16)with eqtln.l non-pola.rsurfacesof compotmds or physically eqlzivalentsurfacesof simple metals and group-rksemiconductors.In the lattèr case surfacestatœ, if they endst,appear in pairs. Becauseof the 6mite thieknessand sepvation of the slabs'there is generally an iuteraction between the tails of these smface states azldtheir ideally degenerateener#es are split. The correspondinglevelsbelongto symmetricand antisymmetzic combinatîonsof surface states localized at dslerent slab sides. TMs situation is usectto decide on necessary slab thinlcnessesand also to identiàsurfacestates i'a calculations. J.nthe limn't of delocalizedweak surface resonance.s the approvimxtion of nominterading surfacesbrexkq dow.aand. states becomeb':lk-lilce with a qu=tization shift due to the dparticle i.n.a box' efect. The level spacGg due to this q'rnntization Gect is approvlmxtely kwersely propordonnq to the square pf the slab thinkncrss,au.dthus can be vacuTpnn
3.4 Q'lxntitzttve Descriptionof Sttazd'tum and Stability
117
reduced by choosîng slabs of appropdate' thinLmess.Special care haa to be takea itt calollxting absolutesurfaceenergiesby menm of centrosymmetric slabs 23.554. Mn'ny layers itave to be taken into accotmt and, becauseof the two physically eqaivazentsoaces, the reosultof euemession(2.39) h.asto be divided by 2. Unlike slab systems with electrostatkally neutral atomic layers, slabs Inimie4zmgpolaz sbprfacesdâsplaya net chazgeon evezylayez, and thus also on the surfaceitself. A slabwith an A and a B surfaceappears. 'lnypicalexn.mples tx.e the slabsto model polaz (100) and (111) surfacesof ziuc-blendesemkcozs dudors. In the ideal cmse one simultaneouslyobserves a catiotsterroinated surface, (100) or (111), and a,n n.nn'on-terminxted sudace, (XO0) or (111). The diserent polarities of the two surfaceswill introduce a spurious eledxic ûeld in the vacullm region: whic,h will Wect the surface reconstrtzction. J.n addition, inequivalent dangling-bond states associated with the two surfaces would, artifcially, give rise to chargetransfer from one surfaceto another, whic,h*11 prohibit converg:nce of self-consistentcalcalations. The esed of the spurious electric Neld can be sappr%sedby applying a dipole correction to the calculatedelectrostatic potential (3.564. However,one still has to deal with the geometrg,the electrordc s'tructtzre, an.dthe absolute smfaceenergy of two completelydeereat surfaces. To oveercome such a situation, Kaxira,set a1. (3.57) employedthe dslcillful stab-ttonbnîque'. By putti'ag two identix slal)s together'rith the cation11J E'j
I
Vz'7. .'f.///.Z'ZSZZSZZ'.X jj
Fig. 3.17. Schematkrepresentatiozzof a sympetric SiC sla'busedfor the simuhtion of a Si-temncnstted
surface. SiC(111)
ll8
3. Bonding aad Energetics
surfRe fachg eachother,the electricseld izt the mcaam (anlon-lterrnsnxted
rezion coutd be elsmt'nated.It does, however, imply that the t'wo central atomic hyers consist of identical atoms, a'ad thus that the coeent boads are bejng brokenin betweenthe twp slab halves.Heace,electronic stzatesdue to titks cemtralb%yer could Muence the iuteresting ene'rgyzacge of surface states. 1:athe case of the Si-terrnsnatedSiC(111) surface (see Fig.3.l7)the method wozksmuc,kbette.r g3.58j. The artifdal C-C interface gives rise to elcdrozzicstates far 1om the fandamentai gap of the zmc-blendeSiC and, hence,the interesting suzfu staœ in this e'aergyregion.Eowever,larges1a,1:z aze neededto avoid the (particle in a box' efec't izt OC.'E slab balf separated by the artiâcial central bilayer. Consideringthe lont end of a slab to contaia the polar smfaceof intere, the back s'ttrhce rxn be suitably pnzqssvatM.Fozowing jh'is idea Shiraishi (3.59) introducedan altemaxtivemethod basedozl saturation with fzactionally cbnrged pseudohydrogen as showniu Fig. 3.18 for (100) stzrfacesof X-V semiconductors.By czoosblg a hydrogezsbkepseudopotential with a valence nhmrgeZ = 0.75 or 1.25, the dangling bonds ozt any (111) or (100) IRI-V surfve ztnn be fZIJMand convez'tthe back of the slab %to a perfect neutral >-miczmâucti'ngsuzface.Similar apprnnryes hold for IE-W compoundsor group-rv crystals but vith pseudohydrogenof au appropriate Meace chazge. For group-N atoms tEe dnrgHg bonds 'c-q.n be satmatedwith tnte hydrogen, Z = 1. T'he bottom laye'rsof the pseudohydzogen-covered slab sidesare kept Fozenduriug tite surfaceoptl'mlzzations.They simuhte the bulk regionsof the under cozusidezation. snmsconductors The boading and antibondingstates re.
'
.
(b)
(a) As
As
Ga
Ga
G
0.75 chargedpseudohydrogcn
1.25chargedpseudohydrogen E'ig. 3.18. Lower part of a GaAs(100) slab with satvration by lacdonally charged hydrogen.ls=xlldots). 80th the lowe,rM-temnsnated(a)and Gatemnlnated (b) surfacesare shown.
3,4
Description of Structuzean.dStability Quantitative
119
latedto the cationtanionl-pseudohydrogen bondsshodd beremoved9om the
gap region. This haz to be checlcedin detail. For instance, for cltun diamond slxrfn.cesthis requirement is not valld for the CC-Hantibonding states (3.55). rfhey appear in the Alndxnnentatgap zegion neaz the projected bulk conduction bands. After passivationof one shb side the geometry artd tike electronicstate of the surfaceof i'aterestcan be stuciied.Eowever,two problemsremain. The t'wo slab surfacesare still inequivalent and electric âelcls(although much smn7ler) '
rematn in both the slab and the vacuzlm regions. For thick layers, however, their infuence is small. 1n.the majorityof applicationsthe electronic singlepaztide states àre we,llseparated euergetkallk. Thrtsallows separate studies for the t'wo slab sides.Unfortunately,this is not possiblefor the total sudace energies.According to (2.39) oae calculatesthe s7lm of the absoluteenergies of two diFerent surfaces.In order to divide the sll= i'ato the contributions from the isolateduppe,raad lower surfacesof a non-symmetric shb the eaerr density formalism of Chetty a'ad Maz'tin (3.60) ha,sto be used. Only in cases of lucky coincidencecan absolutesurfaceenergiesbe derivedby combinatioa of results obtaiaed foz diferent shbs (3.551. The iaeqzzivalentsurfaccssalso izdudeprobleznsin tlle calculation of other surfaceproperties: e.g., the opticat propelies of surfaces.Iu this cmse the contribution of the pseudohydrogencoveredslab side to the optical propeztiœ is separated by the Lutroductionof a linear cutoF hïnc'tioa in the optical trnmqition opvator (3.614.
Cluster methods. A sllrfnzte s'yste,mcaa be modeledapproxirn'ately using a seciently large cluster of atoms. In prindplej ezstiug methods of computational qttanttlm chemiqtryor straightfozwardgenerxlizations of sac,hmethods can be employed to deal with .suc,hclusters (3.62-3.64). A cluster-type approac,hh,asthe basic advaatagethat ab initio calculationscan be pedbrmed and total energies zwzm be rnsnsmszedwith regard to atomic covgarations for s'ys-temswith not too rnymyatoms. DMculties, however,arise in studying between genuine localclustersof suocient siz,eto enable the ctiscrlmn'natîon ized sadacestates and states whoseloe.xllzatibnis merely a zesaltof the 'Fnite size of the c'luster. In particalar, clusters whic,h enn be handled in practice d.onot give good bulk referoce energy value.saud the resulttng ene'rg.ygaps are too large. Identifying the enerr position of a given surfacefeatare on a,n absolutescie may be problematic. The enlargedenergy gap in small clusters results Fom the (particle in a tktrcxe-dl'rnemsiolli box' behavior of the eleerons whic,h are reQectedfrom the cluster boundariesgiving rise to standiugwaves insteG of true localizedsmface states. A secondclassof cltuster-typeapproaGes hmsbeen developedto avoid the shortcomingsof 6nite-cluster methods:the smcalledefedive ield methods. The interaction of a Mnsteduster with its real environment is representedby an esectiveîeld to replacethe botmdades(3.651. The clttster-Bethe-lattice method is the most comrnon method used i'a dexlr'ng with this embedding problem (3.65). lt hastmmedout to be a powergtll tecbrllqueespeciallyin the
12O
3. Bondingand.Bnergetics
of amorphous matezisls,for instance Sioa, but has never been directly usedto model slTrflmesor, at least, substTatesbelowsurfaces. periodicity of surfacesor interNote that the remslnî'ng t'wo-dsrnezzsionat thus facesis not exploited $a any of the possiblecluster-type appxoaches, oaly iaformation of an integra: nature (inegectintegrated over the sudace ca'a be obtnlned by thesemethods. Local dcnsitiesof states Bzillollsnzone) in ternns of the Greea's'hmctionof the cluster can be calculated,but not the dispersionof the bands. case
rrrarqfer-matrix method. More iadirect simulations of surfacesare f1-1. quently basedon a representation by layer orbitals azd Green'sRtnctions. The layer orbita'b take advaatageof the 217trn,nnlationn.lsymmetry but varjr with the Ipprnberof atomic layersbeneaththe surface (3.66, 3.6'4.The layerorbital representation therafore combincxsthe advartagesof localizedorbitals solvesthe The Green's'hzncvdonG@) (3.5)with those of Bloch snt'rnq (3.13). = 1 belonging to a homogeneous inhomogeueous equation (S cS)G@) Sfthrödinger-likeequation of type (S cS)#= 0 (3.4,3.6, 3.46)vith the siugle-particle HstrnilioaianS and the overlapmatrh S. Trn.nsfermatrices relate diferent rnxtzix elements of Green's fundions to MC,IIother! itt the layer-orbital representationof surface calcttlations, they couplesuc,hmatrix elementsfor digerent atomiè layers.Using trnrsfer matrices of this type, the Gzeen's Alncvtionof a semi-inGnite crystal can be calculated layer by layer. The trn.nnfer-matzbcmethod avoidssome of the problemsof the slab method and the cllzstirmethod by dealingdirectly with a semi-lr6nlte crystal. Details whereit is of the method hivebeenêescribedby Yndttrnsnand Falicov(3.682: who employed for a modelHpmiltonian, and by Mele an.dJoannopoulos(3.691 surface.Bound states must be found applied it in a study of the GaAs(110) as poles of the Green'sAlnction in the gap amdpocket regions, in contrast to the more direc'tamat-once mspectof the slab method.However,the transfersurfacestates. I'a principle, matrix metlloct is not restricted to wemlocn'.lp'zz,ed. the method also gives an accarate descriptionof resonances, antiresonances, an.dother featuresin the bar.d continua whateve,rtheir localization properties may be. However,identifyin.gsuchfeattzresinvolves subtracting the local densities of states of the semi-snflnite atd sxtfmitesolids wllic,h are large and nlmost equal qlllmtities. The trazmfer-matrix metbod has beea developedto attàck the elecvtronic-stractureproblem of layered systems. lt caa aksobe used to solve the surface geometry problem exadly. Its implementation îs, a, slab method. however, considerablemore cTlrnbersomethaa, for e'xaample, -
-
'
Scattering-theoretionl approach. A.n exad solation of the surfacegeometry problem cac also be obtained by anothe,rmethod which is bmsedon the for localized pertmbations of mystalliae origi'aal Koster-slater idea (3.701 The sohds.This idea was 6r!4t extendedto ideal surfacesby Kouteclc.f (3.711. method starts with an ''nBnite blllk' crystal, whose eigenstatescan be reltraaslational ative'ly emstlydetfarmlnedby exploiting the three-fqsrnension&l symmetzy azd then e-mployingthese solutions to constract the bplk Gzeea's .
3.4
Descdpticn of Strudureand Stability Quantitative
ltmction G0(s).Finally free surfaccsare createdby introducirg an sppropriate short-rangepertlzrbation.The Greem's Atmction G0(s)of the pefect crysta1 entirely contaiu the electronic structure of the unrelaxed surface.Tlms, the method hcorporate,s and retalns all the bllllc propeaties suc,ha,s band continua, ban.dgaps, etc.) and the alterations resulthg from a cleavagecan
be obtainedOectly 9om scattedngtheozybasedon G0@)a'cdthe cleavage potential without subtracting large quantities. Both conceptttally a'aclcomputationalty tMs is a rns.joradvaatagecompared to the othe,rmethods,in wié.c,hone attempts to ft'td the blll'k as well ms the sudace-inducedsolutiors dizectlyj from either a slab, a cluster, or asemi-im6mitesolid.Sizzce the method requires the Green'sRtnction for the perfectin6nite solid.,wllie,hcan be folmd with moderateefoz't by stnrnrningover (partsof)the bTtlk'BzilloTnsm zone, it is computationally Iess clrrnbersome thaa the transfer-matr.ixmethod..The implementatîoa îs not as simple as in the case of the slab metbod.,but once implementedand if thq surface geometr.g is lmown, it is more eëcient and
reliable.
C)
@
@
O
O
*
@
C)
O
O
#.* Nl
O
C)
l
(a)
O
@
@
C)
O
@
@
C)
*
@
C)
èC) '
.
(
u -w cK)
(b) ::
j
Fig. 3-19. SGematicgrapiusshowingthe creation oftwin surfaces:(a)bond-cutting hethod; (b)removal of atomic layers. The resulting uppnvost surface atoms are shaded.The arrows indicate (a)which interatoznic matrjx elementsel'e set to be zero or (b)which intra-atomic matrbc elememtsm'e shifted in eneror by the value u. Mter (3.661.
l22
3. Bonding and Energetjcs
The Mvantagœ of the methodhaverettlted in cation in the theory of relued
a
relativelywide appli-
reconstructed suzface. Ia pazticulaz,Pollmxnn and coworkers(3.6% have employedtlle scattering-theoretiY method in dealing with ideal aud reconstzuctedsmfacesof smmiconductors of (liamond, Gc-blende or wurtzite stzuctttre 'by mtmns of the ETBM ms well as tEe deusity-functioni fozrmlsqm.The basic idea of the method (3.66:3.67) is to divide tàe Hxml'1t,0n1=H = SD + U jnto a bulk ezmtribution S? arzda pertmbatfon U simulatiug the presence of a sarface. Then, one has to solve a Dysonequation G = GQA-GDUG. The b&t reprcentation of the pertmbation U Lsthe layer-orbiial representation. lt allowseazyphysical approaees for U, The e.g., the bond-omingzp,efzWaud the layer-romovdrzzeszoé (3.66,3.72.. tèo approachesare illustrated in Fig.3.19. J.utke 611% method (Fig.3.19a) the ftrstv and second-nearest-neighbor izzteraztioztsia the bdk T-TmiltoGan atomic planesare switGed (3.7)decribhg interadiomsbetweent'wo adjacent o5. Jksa result two equivaleat surfacesare formatly cweatH. In the xcond case (F$g.3.19b) aiomk layers aze forrnnlly (i.e,, with resped to the eaea'gyof the related electronic states) removed fxom the spRe by Klnl'fiingthe atomic eigeaxalues(bya lazgevalue %)far from the eneror region of interest. or
3.5 Bond Breaking: Accompan>g and Atomic Displacements 3.5.1 Characteristic
Charge Transfers
Chnnges ln Total Energy
In order to understand the efect of boneg of orbitals aad brealdngof bonds oa the total enerr (3.55) of slzrfax systmnnn, sevezalmodelshave beendeveloped.l'a the force consta'at model of Chadi (3.734 the cùaage of the electrœtatic emergy (3.56) is daqnribedwithin th.e hxrrnoaic apprnvimation for is exthe geometry changes.The contribution of the geometrical n'hnanges pandedinto powers of Factio:aal bond-lengtàchaqges%j. T:e vadation of the attmber of bonds dkNsondis desm-ibedby an additional erergy contribution with respect to this nl:mber. The fradiom-klbond-leagthchangesof two atoms at m and Hj are deMed as nearest-nefghbor qj
=
1.R; Rj I/Qq -
-
1.
(3.61)
In the presenceof a perturbation, e.g.: a surface,taheralative nearest-neighbor displacementsand the brealdng of bonds nhn.ngethe eledrostatic eaea'gy formally accordlng to (3.56) zâfes =
+ WzNsond, + tfaetl S(rAe< b<J
(3.62)
if orly one coatributioa to the second-orderterm, the raclial one, is hzcluded. Three-bodyterms suchms tkose proportional to 6çJEfa(J# k) a.re ignoredfor
3.5 Bond Brealdqg: Accompanytng ChargeTransfea's
122
to azcoantfor the simplicity.Vaaderbiltaud Jcœmnopottlos suggested (3.744 wiation of the number of bonds in the total eyergyby the tezm W/Nbond. The constant W has to be identiâed with the cohesiveenergy pe,r bond. is calculatedfor a given covgttration The band-structttreenerr Eyn (3.53)
and the dezaitioa of the isotkermazbulk modulus B. From small perturbations of the bulk bond lengïhs one derives The pazameters Uz and Ug foilow 1om the equiiibrium conition (.RJJ.
Uz =
2% with F'
=
-
Dms 0ê
957
ms the
-
,
2=o
:2as 0a?
(3.63)
d.o
volume of the consideredtmit cell and ê as tbe tensor
of (3.61)
the Fadional bond-len#hchauges. Lsbasedon the ehxrge-self-consistent Anothe,rinteresting mod,e,l(3.66,3.754 treatment 'tight-bindingmethod.The basic idea of the charge-self-conskstent can be easily illustrated formal)y asizlg a Hartree-llke (ora Hat-tree-Focklikel treatment of the occupation dependenceof the total e-IIG'gieSand of the slgle-partide eigenvalues.Consideeg a free atom or 1ee ion with a valence coMguration s'Hfp, accmdingto the splitsng in (3.52) one has a total energf Sltom Lslnyn.p )=
swy
(tnj);
t--.a,p
..-
1
ngjnyri.j
-
2
*,i=s,p
with the single-pazticleenergies
sv(.(zz:J) c2+ -
('z.f 'zbôlr-&,f, -
(3.65)
J=ayp
the orbital ener#es i'n the couguratîon s'Zprz; trzoa = ation of the neutral atom. Each c.0n551T> Z), i.e., the grotmd-stateco tion js characterizedby certain occupation nl:mbers na and np. The energies intra-atomic interactions of vaUé; (ï).i= s,p)can be idemtifed with egecwtive lenceelectrons in the same or dsFerelt angulawmomentplm state. They may be calculae as Coulombintevals using the correpondsng wave Amctions. f1llfll1 the Jaaak theorem (3.761, a'ad (3.65) The expressions (3.64) Where &Dsazd ooare
c:
=
+z4
0 stozn nw .B' (.s p,zml &n%. .
(a.66)
The eigenvalueeé equalsthe e'hn.ngeof the total energy q'ith the clkangeof the kccupationnumber zzç of the level ï = s,p. Exw-ions l4smilnr to (3.64) can also be derived for tlke total eaerbes of solids or solids with surface.A system for whiG this can exsily be (3.55) demonstratedis a tetralzedrally coordinatedsemiconductor, e.g., a group-N
124
3. Bondingazd Energeticg
stnzcttzre. The most important contrirnxterial crystnllszing in the din.rnond. butioa to the band-stracture eneror is due to the cherniealbonding gî.1). Thks foEowsapprovimately as the sl'rn of all bonding ener#esof the bonds of two sp3 hybrids pomtim.gto eac,hother along a bond direction. Withim the molecttlemode,l(3.:0)the bonding energymny be written as sbond
=
1 + 3%) V'h+ -4Lâs -
s'Ti
with fs and.G as the orbital enargiesof the hybridizedgroup.N atoras in the covguration skpn. The Xect of the overlapp'mgof the t'wo hybrids) S H Slz, has been taken 'mto accotmt in the lowest llinearlorâer. A representatbn liuear ic the overlap integral has beenGosenintroducing an eective matrix The energy W = -(Ws.- WIV. elemeat5rh= (SzzNlzJ.G)/(1-S%). of t'wo ,sp3hybrids. is the negativeinteractiozt energy(3.28) sfjxpc-svppcl/4 In a itting procedme the te= Skk is usedto model also the repision of the cores. Near the eqllilsbriplrn positioas: Ji rxp 1/J2holds for its vaziation the same Followingthe Hfickeltheory f3.1p with the bond lezrh d (3.6â. 3.32 is valid for the overlap iategra). With the Cottlomb dependenceS rxp 1/(J2 of of sp3 hybrids the total enerr (3.55) iategral U = -Y + Guap+ 9Uvp) 16 (Jss a blhlk'covatentczystalconsistingof N atoms is given by -
-
ba:k(x,(a))-
N zs +
so
-
(deq)2(deql' (a.6s)
43-,:-
d
i4J'J>h
-
su
.
fi and é' The eqttilibrintm bond length doq has been iptroduced. t'a (3.68) denote the corresponkding quNtitiesat the equilibrbnm distances. The parametez'sW and S fcoow 9om the conditions of mechecal equilibrbxm in the bnllk, vauislnn'ngbydrostatic pressure, and the relation of the with respec't to the volzzmeto the btûk modulus secondderivative of (3.68) As a result one fmds S = la The dilerence Scoh = SMOm(s2p2) (3.63). deter.&balktN, (4eq))/Ndeqnesthe cohessveenerr per atom. lt is mainlycsO s0s zninedby the covalent bonding 'w Q and the sp promotion energy -
.
-
Scoh
=
A
2Q @p0s0s)(j.tr7s, 2%p+ Uppt -
-
-
-
.
describes the chaage in the intra-atomic electronThe last term i.zl(3.69) electron interacuon due to electron promotion lozrl an .:2.p2confgaration to the skp3 elec'tron confguration. The remaining fou.r parameters follow h'om the ftrst four atomic ionization poteatials deuecl as ciiFerencesof expxessions (3.64). The resulting tight-bincling paràmcters are lksted in Table 3.1 for diamond,Si, and Ge.
3.5 Bond Breaking: AccompanyingChargeTrausfers
l26
Table 3.1. Tilht-bindtng pammeters es-timatedfrom bulk eqllêibrittzncônditions ard atomic ionzzationpotezttials. M values are in eV. R'om E3.75). Pammeter
71: -Vks. Vip.
Si
Ge
8.73
6.86
6.06 1.94 2.55 4.49 1.12 3.86 11.05
2.79 2.19 3.67 2.89 6.47 5.08 1.61 1.27 4.71 4.05 13.15 11.34 16.61 11.67 11.49 13.20 8.14 8.71 13.11 8.19 8.05
V'pp.
-kip,r -o0 -cD
C
.5
Uas
Usp Upp
3.5.2 Enerr
Gain Due to Structural
and Confgurational The total energ.gof
Chuges
system of electronsand cores is esseatiallydetevnined by the caheznieal bondsng.Apart from plomotionazd changesin interactior esécts,the accompanyingbond energy 'U'his usuallytaken equal to tite cob.esive energ.y (3.69) , i.e., the heat of atomization, per nearest-neighborbond. Conscquentlythe sarface energy.of a halfspace (2.39) is therefore given by the excess envergydue to the broken bonds (see atso (3.62))Such a picture hdicates cozrectly tEat the msml'nulrn-enea'gjr surfaceof a covazentdixrnondstructme crystal is the (111) surface tseeFigs.1.6 a'ad 1.15)whch is tite cleavagesurface and.an jmportant growth plane. The simplicity of suc,ha boading picture makes it fmscinatmg.However, lt cxnnot dkectly give details of the smface behavior, such as sudace reconstnlction. Ncvertheless,one may also use it to de-scribe surfacechanges. a
.
The
surface of a (111)
homopolarseraiconductorwith one dangling hybzid pe,r surface atom in the normal diredion allows aq easy Gltzstrationof the eimplifed pictme. Using (2.39), and S = âzone f-nds a surface (3.67), (3.68), enezo-per atom (per1xl tlmit cell)of
A
=
lw -5i 2
Eqfor the 'aaperturbedideal sllrfnne. However,such a
smfacewould be tmstable , Ebecaatuse a single electron is placedin eac,hof the danglinghybrids despite its Ereadivity,Suc,ha surfacerepresents a degenerategrotmdstate. Neglectiagfor : a moment the eledron-electron interaction in neighboeg bybrids, one could alsoequallyput t'wo electronsi'a some hybrids andleaveothersempty without :.
3. Bonding aud Energetics
changingthe total energy.Accordingto the Jnhn-Telle,ztheorem(3.77,3.78) a system having a degenezate Jrotmd state will spont=eously deformto lower its symmetzy unless the degenez'acy is simply a spitt degeaeracy. Iu this simple exmple the surfacerelntion doa not moclifyihe smface Gymmetry.The plane of surface atoms tinds to be displacedslightly inward or outward. TIIJSis accompanied by two efects: a rehybridizationof sllrFnce atoms aad a variatioa of the bonds betweenthe fzst aud secondatomic layecs. The reybridization mnc'nly Mueaces the dangMg-bcndbaad in Fig. 3.9. The sAlrfxce atom displacedoutward (iaward) ends with an a-like @x-likel dangling hybrid, whereasthe three equivatenthybrids in the bnnkbondstend orbitals (cf.Fig.3.20). AII orbitaksare characterip becomeplike tspz-likel with lz for the danglinghybrid and la = hed by the hybrid expression (3.23) la = A4 = A foz the hybrids in the back bonds. Without rehvxtion (v = 0), displacementof tke sudace 1, = .k= 3. In the case of an otztvard linwardl atoms by 'udeq |
edb
=
as
=
(1+ 3'$2 ês , (zp ) % 4 + âp (1 1441 3a)(% ?a), 4 -
-
-
-
(a)
-
(3.71)
(b)
4v +
Flg. 3-20. SchematicMdication of the dehybridizatîonof atoms on a (111) surface bybrid is shown wheremsthe displace outvard (a)or inward (b).The UIX'XZJIW ûybrids in the back bonds are only indicated by straight lines.
3.5 Bond Brealdng: AccompanyingChargeTransfezs
Fig- 3-21- Hybrids forming dangling bonds and back bonds in tke lx 1 cell of a (111)surface.
and izz a lenxgthof the bac,k bonds of d = deq (1 u)2+ 2.(1 + tz)2/W. Izdeed the exwessioc (3.71) correctly desczibethe llrm'ting cmses. For a v=Ls'hlngatomic displacementthe bulk struct'are aad the sf hybrid enectes displacement of 'tz = l (u= -j ) the are recovered.For au outward tinwardl J$ dangling hybrsd Jlmsa pme s (p)character, while the hybnds izt the back orbitals. bondsbecomep Lsp%j The raisaligne'atof the hybzids in the back bonds gîves rkseto (bondbending' interactions whereasthe variation of the length of the back bondsis relatedto 'bond-stretchmg'forces(3.61. Tile changein length of the threeback bonds is seen to be udzgjh to lowest order in zI. The conopondiug change in enerr per bond is tims ACo('u/3)2 vith the radia) force cozzstant Q. The z atomic isplaceme/tmisaligns the t'wo hybrids mxkn'ng up a back bond. It atsoreducesthe hybrid covalentemergyaud therebyraises the e'nerg,y of enftb bonding electron. Iu lowest order the a'agleof rotation of the back bonds is -
j'u.lf oaly a singlesurfaceatôm is moved,MC,IIof the backbondsLszotated
while the oth.e.rthree bonds at the atom in the secondlaye,r by th:is angale betweea remain Sxed. The changein energy due to the changeiu the angales to be added for each of the bonds meeting at eac,hbnr!k atom is 2C:12/3, tla.ee back atoms. One must also add the angular ecergy i'n the three bond angleshaving apexesat the smfaceatoms. lt totals 4C:'u2.Consequentlythe H jCu?. total elwstic energy tpersmface atom)amolmts to (C0/6+ 6Q)'u2 or Tlze force constants r-q.n be relatedto the tight-binding prameters (3.22)
3. Bonding and Energetscs to the elaztic constants c11 and c1a of the cubic crystal, For silicon Q
=
55.0
hold appro-vsmxtely. eV, Cz = 3.2 eV, acd C = ,56.7ev (3.6) The described model for the deformed bac,kbonds aud their energy is cot suscient to describe the details of the i'award relaxation of a g'roupIc this ease the details of changesin the 1 sltrface (3.54, 3.55). IV(111)1x orbital hteraction have also to be takcn into account. Moreover: the elastic deformationof the atomic layer beneaththe surfacegivesrise to all increasei'a the surfaceenezgy:Nevertheless,simplifed consiêerationsallow one to discuss the 2x 1 recoxxstractioaobsez-ved ota cleavcdSi(111) surfacesjn the fmrnework of the b'ucklin,g Z/JIIIf:Jlseealso Fig. 1.75). This wms sttggestedoriginally by
Hanernn.n(3.791 and was the most cornmonly acceptedreconstruction model tmtil 1981. Todav it allows a didactic iscussion of the phenomenaaccompanying the pairwise buckling of atoms. Witbl'n the bunklimgmodel rows of surfaceatoms are, respectively, moved up a'addown ms indjcated by Fig. 3.22. the bunk-ll'ngreconstruction is accompeed by a rehp Accozdîngto (3.71) bridization shown by the transformation of the hybrid characte.rsbetween Fig. 3.22aand 3.22b.Consequentlythe d.aûgling orbital en.ergywill decrease atoms, respectively,proor iucreasefor the raised ('tzzdoq) and lovered ('tzzdeq) ducing a splitting of the dangl'mgbond surfacestate band in Fig. 3.9a. When is sue,hthat this removalof d.ethe amplitude of the bur-kling,(%: tzzldeq, generac'yis large compazedto the initial dispersionof the surfacestate band, an absolutegap will occur inside this band, the lower bard being ûlled azd the upper band.behg empty. The consequencesare schematicatlyindicated i'a yqg.3.22c. Negleding the changesin the electron-electroninteradion the sudaccenergy pe,r lx 1 htanitcell is given a-s -
ns
=
1
1
yï't + j. ).7q(w) i=1,2
with the contributions due j o veztica.latomic displacements 1
f4(u) k =
g(-1)611 911aj :
-
LG -
êa)+
1
z g tz j.
.
The bunkling correction J'àis given by the emerc changeof the dangling added to the elmstic enerr 3.Cw,Z.Tlle t'wo corrections bond cdw (3.71) ! The mimmizatîon predicts that the are plotted ia Fîg. 3.23. energy 5ztzql surface atom with the doubly occupied daugling hybrid is displacedrtlmost 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 BondBrealdng:AccompanyingChargeTraafeas
j
(a)
(,)
129
+
(u, Uz
1
(u1 1
U2
3.5.3 Enerr
GG
Fig. 3.22- Efects of buaklîng of a stlrface)e-g-, of Si(111)2x1, on danglingkybl'ids and their occupation. Norrnxl atomic clisplacementstsl aud ua are assttmed.(a) Unbuckled surface with half-fzlled0p3 hybrids; (b)bucldedsurface with dehybridization; =8. (c) buclded surfacewith Qlleds-like ar.d empty pa-like danglirg bonds.
azld Electron Transfer
Ia expression(3.72) a'actFig. 3.22ccomplete electrontrn.nmfer h'om the more plîke danghnghybrid iato the more s-lilte dargling hybrid dllm'ngthe buclding 2x l reconstruction ha-sbeen assnlmed.Tlzis wa.s oaly possible becausethe eEedof the electron-electron iateractioa izl tlle two daugling hybrids, 1 aud 2, has been neglected.In. order to discussthe efect of the întrabybrid electroneledron interaction on the electron transfer, we restrict the considerationsto the two danglin.gorbitals in the 2x 1 tlmst cell of a (111) surface. lnstead of sp3 hybrid energiesthere are the danglingbond energy cz - sds: of the raised atom ar.d the enezgyE'c H aabz of the lowered atom. Asslnrni'ngan electron trlmsfe.rb'n(0K Jn K 1)and, hence;ocmlpation zrlrnbersnz/zàl= 1+Jn and = 1 Sn of the t'wo hybhds, it becomesobvious that tlze danglingn.a'(Jn) bond energies ss a're diFereni fzom the energiœ & bebnging to the slp3 -
confgaration.
130
3. Bonding and Enezgetics
>
X
m
c>x c) = (D c
1 to tite surfaceenea.r of a &4111)2x Fig. 3.23. Correcwtions fzztut)and f2zL%<) suzfacedue to imckltng. The atom vitll the completely Slled dangling hybrid Ls g bond displacedoutward by 'mdoqwhereasthe othe,ratom with the empt,g d i:a dksplacedinward By .-ltluldeq. '
the sur'faceenergy may be describedssmslazlyto the behavior of eledrons in 16.e for the staface energy one fmds a au.d (3.65), atornR. Accordsmgto (3.64) clzangeduc to the electron trxnmfe,rJn the GeC't of the electron-election interadion Quantitatively
I''-.I
éfzs =
1 ? m(J'n,)&:(éh) (5n)& -
:=.1,2
y'rk
1
rzf(0)sé(0) j-zk?(O)rJ -
-
on
(3.74)
with the abbreviations 1 (-1)fJn: '?u(é'?z) s:(Jzz)êi (-1)frJWa. =
-
=
-
(3.75)
For the sakeof sMpllcit.y the orbital dependenceof tlzo interactioa interals Un = Uap= Upp= U has beenaegledcd.The minlrnszation of the resulting. e'aerr change(3.74)
-(fa ê:)(h+ U(J?z)2
Jfzs =
-
(3.76)
gives au electrontrnmqfer 1 éa Jz -
ôn =
ï
rz
.
(3.77)
3 s Bord Br-pka'ng:Accompanying chacze Transfezs &
'
131
It is proportion/ to the dxezencein the hybzidizatioa stagesof the two dangli'n.gbonds. In the llm5t of a norrnltl U beha 0j the repulsive eledrou-electron iateraction hinders complete electron trnansfer.For the Si parameters in Table 3.1 one fnds b''rtQ$ 0.5, i.e., incompleteelectron transfer. to the lowering of the sudace energy Coasequeatlythe coatributious (3.73) are slightly overestimated. (3.72)
(
4. Reconstruction
4.1 Reconstruction
Elements
and Bonding
4.1.1 Metallic Bonds
The most widely studied surfacesystems are thoseof metals. Oneimportant aim of the eaharacterizationof clean metal sttrfacesis to produce suitable substratesfor adsorption studies. Sequencesof low-inde,xsurfaceshave bee,n ùwestigatedsuchaas(111), of fcc metals (e.g. Cu, Ni, Pt)and and (110) (100)j a'nd (112) of bcc metals (e.g. W).I'a the early studies, for (11O)j (:.00), (111): most cleaametal surfacesthe lateral pedodicity hmsbeenfoundto be identical to that of a parallel plane ic the blllk-.TMS conclusion was in gemeralbasedon the observation that tlze LEED dOaction pattern did not contnrl'n any other dsAactioc spots than those e-xpectedfrom the Sdealperiodicit (4.1). lndeed., smfaces of Cu show dl'sradion patterns the clean low-index (111) and (100) with hexagonaland square symmetries as expectedaccordsogto Fig. 1.2. The clean low-inde,x (110), (100)and (211)surfacesof W yield patterns which are also cbytracteristic of ideal lateral perioclicity. The conservation of the lx l tmmqlational symmetry for the abovementioned metal szzrfacesis accompn.nledby surfacerelration. A correspondl'ngmeeanism which is cbractezized by a smeazingout of the electron distribution and a'a inward displacementof the eores in the topmost atomic layer has been discT:.qqed in Sect. 1.2.4 (Fig.1.10). The relative spacïg of tlle top layers is typically reduced by c.p to 12%.For e'xample,in the case of the W(10O)1x1 surfacea contraction of about 5% is fotmd for the top. The eledronic origin of the reluation o:a a trn.nKition-metal most layer (4.2). smface sue,hms the W(100)1x 1 seace may be unders-toodby considea'ing simaltaueously the efects of bondsngof localized d electronsand delocalized sp electrons.Ia the btllk of a transition metal, the bond formation driven by d the interatomic distanceswhile the free-elec'tronelectronstends to deearease like sp electrozzs minimize the,ir contribution to the total energy by expaadimg the lattice. The balance betweenthe two mec-haaisass leads to the eqltilibrilrrn geometry. At a smface,the d-d bonding betweensurfaceazd subsurface atoms is emhnanced., i.e., the bond clistanceis shortened,since the sp electrons can be pushedinto the vacullm region above the surface.However,there are exceptions. Akc;ooutward relaxation may occtm The densestmetal surfaccs
134
4. ReconstructionBlements
goc 1(I 1QJ Eo Fig. 4.1. Possiblelateral atomic displacementsin the (01ï) direqtionof atoms on a reconstructed w(100) smface.The open circles inclicate buût-like positions.Dashed lines desczibea (WxW)JM5Q cell. '
sometjmœ xlm show azz expausion of the tomlayer spazing. One mxxmpleLs stu'fruv (4.31. the Be(0001) Onereaszm seems to be relate to the strong directional bonds formed in the Be bulk due to the hybzidization of : and.p bands. meti smfacesoften show beautiful Adsorbatocoveredor contn.mimated zeconstracwtions.However,since a few years slTrGce reconstractions are also stuu lmown for cle,atkmetal sttrface. For iustance, upon cooliug the W.(100) Lqc,e belowroom tamperature a p'hxqe transition 1om a lx 1 iuto a c(2x 2) The reconstmzctionis describedby alternatiug structure is observed(4.2). diectioa to Vm zig-zag lateral isplacements of W atozmsabng a (110) The structural nlnm.nges can witk a (WxW)M5*geometry (Fig.4.1). c,hn.''nm Smface be explalned by the lowering of the band-structtlre energy (3.53). for the surfazelayer at relamatioabarely reducesthe dezzsityof states (DOS) the Ferml'euergy sv. The sur%cerecozzstructionis, however,accompaaiedby opening of a sarfaceband gap arou'adcp azld,hence,a sigvcan.t reduction of thq DOSin kbiq energy rauge. oriented Pt and Au sttrfazes(4.'4, Recently,reconstnzctions of clun (111) 4.51have beea fotmd. However,the (1û0)surfacesof Ir, Pt and Au also The apparentlywidespreadmvlntence display sarfacerecozlstradioa (4.6-4.8). of surfacereconstmzdions h.asevea raised the qucstion tslzolzldall sarfaces It wms suggestedthat the dzivir.g force for a surface be recoazstructed'(4.9!. of surfaceatoms is the dsFerexce'bet-w'een reconstruckioh changingthe dertsit.g aadthe sttrface ihe surfaœs'tzessnj (intmits of forœ per unit length) (2.18) Le., the strain derivative enerr J (in tmits of eaea'r pe,r tmit area)(2.13), Consequentlya suzfacewouldtend to of the smfaceen'ergy qj = (2.19). reconstruct towxrd a state in whicktthe surfaze stres is equal to the s'Irfnne enera. Whea the strain derivative of the surfaceenerr is positive, there is '
yks
4.1 Reconstractîon axd Bonding
135
:
tendencyfor the density of surfaceatoms to inœe-ue,andvke versa for a The cxmclusionconcerning the Aegative value (cf.disctlssionin Sect.2.2.3). (:11-1v1)1g forcefor surfscereconstruction is ssrnilar to results of other studies of However,in those stttdies it was shownthat surfacc densiscation g4.10,4.11q. this driving force does not resalt in acy reconstruction. The reason for the obviously cligerezlt fmdings could be that the abovecriterion follows witbin linea.relasticit'ytheory and?hence,appliesonly for s=n.ll tlnlibz'mstrnainqwhen the surfaceLayerand bxlllr are strna'nedtogether. A reconstruction iuvolving strain:ing the surfacelayer alone necessarilychangœthe surface-substrate bonding:whic,hgenerallycosts energy. The application of the abovecriterion surface should therefore fail (4.12, to the case of the Au(110) 4.13).Tikis in which alternate surfaceundergoesa missingrow reconstruction (Fig.1.9b) rows of atoms are removedfrom the sudace.The removal of one half E1E0q of the smface layer of atoms cnunnotplausibly be discussedwithz'n linear es elasticjt.ytheozy. The missingrow reconstraction involvesvery large ic the local environments of some atozzus,wlzich cn.nnot be descaibedwitbln homework. the li'ne,a,r a
4.1.2 Strong Iozlic Bonds
Despktetke completely diferent bottdGg behavîor ia ionic crystats with on mnny surfacesare rather strong ioaic bonds,the atomic reaazrangements weak aud keep the 1x1 trrslational symmetry. Surfacerelaxation or even are favored a.s for rnnry metal surfacu surfacerttmpling (seeSect. 1.2.4) Among suchionic systemsare the leadsalts PbS)Pbse, and PbTe, the (4.14). metal ozde.sMgo?Niol Zn0, A12oa,and Fealb,and s.lkxl:'halides,e.g., LiF. halides,except for CsC1,CsBr, and The lead salts, MgO, NiO, aud the n.1kn.17' Csl, wbic,hare most stable in the cesitfm chloridestructure, crystnllize uzder Fig, 4.2). stmzctme tseq normal conditions ic the sodillm chloride (rocksalt) The other metal oxidesfavor tlze hexagonalcrystal system. The a-F'ezo.sand osAlzoa coimpozmdsczystn.llize iu the hexagonalcom:ndl:rn structme whi)e ZnO is a wuztzite materjal. In contrmst to covale'atbonding the relatively strong ionic bondîng forces act not only between nearest neighborsbut also between atoms witich are situated further away, becauseof the long-rrge natttre of the Coulombi'ateraction ilwolved. Furtlmrmore, one hmsto take into accotmt the Alrnost directional indepexdenceof ionic bondiag as a consequenceof the hiler coordination tbxn in the rocksalt case (Fig.4.$. As a result, mltlnpy small structmgl c'hxn.ges parallel to the sllrfytce normal appear. This holds in paw ticular for nonpolar surfaceswith an equal nplrnberof cations aud aaions in the sttrfaceatomic layer. This is valid for tlle (100)facesof the sodip:mcbloand (l0ï0) facc of the wurtzite structure tsee ride strudure and the (1110) Fig. 1.6)Oneimportant argtzment is basedon the electrostatic enera (3.57) roclcsaltfacesboth sublattices wjth a strong Ewald contzibution.For (100). relax Snwardbut to dlFerent degrees,becauseof the dlFereztt ioak rnr155of '
.
â3$
4. Recozetnzction Elements
r---ca.--/
Eoo-lll
Eo't oq E1 onll Fig. 4-2. Schematicrepresentation of the zocksalt czystal strtzcture. A prlmltsve fcc Ilnlt cell, a cube with edgelength ac, is shown. x
non-
'a
cations and n.nrozls. Tlzis results in the so-calledrumpling of the sttrface(4.142. I'a the e-qmeof polar faces,e.g., (0001) of the wuztzite structme, one and (0001) expects larger rearrangements. This is iudeed the case for AW(0001) (4.15). However,this is not the case for the polsr (0001) facesof hexagonalcommcbnrnwith -2LXO:$-MXO3stacking (X = A1, Fe)in the (0001) direction. The oxygen atoms fqrm hcp hyers, aztdthe metal atoms ftli two-tbirds of the octahedral sites betweenthese layers. 'T'woXaoa formttla tlrlts (10atoms) are izl tlze primitive rEombohedratllnl't cell. J.na description by a conventioni he-xagonalI:mit cell six Xatx formtzlallnits (30atoms) occur. Usually lx 1 trauslationf symmetr,g is observed(4.16-4.19). The phenomenonof surface mtmpllingdoesnot seem to be restricted to ionic crystals. R7lrnpliughas abo been observedfor nozppniqueTi(10ï0) surfaces(4.20) and is even suggested to 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 nerestaeighbor bonds are the tetrakedrally coordinatedsemicondudors cnrstal:izing ill diFtrnond,zinc-blendeor wurtzite structures (oreven in othe,rhexagonal polytres, e.g., 4.11or 6H (4.221). Among them are the e'lemeatalsemiconductoz.s C, Si, aad Ge aud the compouzd semiconductorsSiC, III-V, II-W, and I-WI. The amotmt of ionic aharac'terof a single bond can be successfully describedusiug diferent quaatities ard scales.One qurtity is the bond polxm'ky ap of Harrison (3.11); others are the eledronegativit (Ak/B)scate of Pallls'ng(4.23) or Phillips (4.241. The ionicity is related to the dference of the atomic electronegativitie-s Xx and ,X%.Meaawhile, sLightly revised Ns1-
4.1 Reconstrudionand Bonrlsng
137
Fig. 4.3. Valenceiectron densityof cubic AlN along tEe (lllj bond direction (solid (fazhed) line).lt is decomposedinto a s'ymmeiz'ic(dotted)aad an antoetdc pact. Drawn ushg results of (4.274.
A of Pal:lsng's electronegativitiesca'a be found ix the literature (4.252. has beem scalebasedon ab initio calculationsof the electron density (3.47) The ionicitie,sof this scaleaze recently proposedby Garcia ard Coheu(4.26j. coedents g tn.king the squa'reroot of thc computed az charge-msymmetr.v ratio of the squares of the antisyrnmetric and symmetric parts of the electron density htegrated over the vobnrne.These contributiorusare plotted i'a Fig. 4.3 for zinc-blendeA1N resaltin.gin g = 0.794. ia Fîg. 4.4 for tetrvedrally coThe bonds are illustrated seahe'mxtically czystals. Each of the,mcontnsrs two spin-paired ordinated (i.e.,.5p3bonded) electrons.When a soace Lsformed: some of thesebonds 5dll be brokea,leading to lmpasredelectrons. As discussedin Sect.3.2 tdangling bonds' a'ppear. dependl'ngon the group They are foed with a fractiona,l zp:mberof elec'tro:as of the elementA or B in the Periodic Table form'lngthe AR compotmd.The of electrozts.amounts to Nx/4 (orNz/4)for atoms A (B)from the nTznnber = group NA LNz=.8 -XA) becauseof thc conserqationlaw, XA/4+.N's/42. Suc,ha btllk-truncated smface hmsa high surfaceeaergy,and is therefore vea.ylpnKtable.Hence, the atorns in the mIrface region relax 1om their blllk positions 5n order to reduce the surface free energy by forming rew bonds occupiedby t'wo electro>. Reacldnga structure with a defced s'toichiometzy wllich cxhibktsa local minirnl:rn of the stlrfaccfree energy, implies that the of the smfacespecies(orat leazi of most of thcse species) chemical valemcies are satisfed i'a the recoustructed geometrp Consequently,the observedroconstructioustemdto be instfating. Surfacebonds or d=gling bondswhose energiesfall hto the bulk enerr gap, crossing the Pemnilevel, woald i'acrease Sect.3.5). the sttrfaceenergy (see Besidestlle geometzy alsothe sarfaccstoichiometu may vaty Sometimes it is useful to regard the surfaceregion as a new clmrnlcal compound whose ues
l38
4. Reconstruction Elements
crystal. Open and 5l1ed. Fig. 4.4. Batl-and-stic,kmodel of a zinc-blende(cliamond) are i'adicirclesin the mzberepresen.tcations a'adnmions, r%pedivezy.The 'bozzds cated by doubleljnes (i.e.,the sticlts).
the stractm'e and composition may be evaluatedtheoreticallyby =5r,5=5,z111g free energy (cf.dkscussionin Sec't.2.5).Multiple msnirna occyzr for most satrfaces,difering i'a both compositionand structare. Whicb.minimttrn is accesed e'xpersmentally depends upon how the surface is prepared. Thus, a typicat compotmdsarfaceexhibits multiple structtzres depeadhgupon prepazatfon conditions as well as tlle ambient temperatme a'ad pressure. 4.1.4 Principles of Sezniconductor Sttrface Reconstruction
Surfacereconstrudion aad the specialcase of surfacûrelaxationfottowcez'tain A surfaceproduced in detail by Duke (4.28,4.291. guidMg pzinciplesdiscussed cleaxrageprocess, a growth process sputtnrsng a'n.dn.nnenlsng'tmderliesa either by
a
or even
repeated cycles of
stmtctnre os,sezved'?All bn the l/.t(7c.# Basic Prirzcïple; The dtûr/ccc structure kfrlcàfctzîî?/ acccssibln czàer the prqmration conJree-enerlp tfftïonxs.
a'ad Apart from hozennon-equilibrblrn situations whic.haœe also obseawable by the growth kinetics, this prindple re:ects surfalcegeometries detemnsnned the fact shat a certain surface may be generated by a variety of processe's =l'ns=plrn and, therefore,correspondsto a local (butnot necessarilyglobal) to the thermodyAmmt-c of the fl'ee energy. J.na sense somewut genernh'zed considerationsin Sed. 2.3, we witl also call such a surfacetstable' ia tlze following. The càeareste'xamplesof the valiclity of the BmsicPrGciple and tlle its îuence of the preparatioa process are the (111)cleavageface of Si and
4.1 Reconstraction and Bonding
139
Ge.Low-temperatmeor room-temperature deamgeproducesa 2x 1 structlzre, wheremsn.nnealingto Mgh temperatmes prodacesthe ture
(4.30, 4,31jand the
strucSi(111)7x7 surface (4.31-4.33), both of whic.h Ge(111)c(2x8)
stable upon subsequentcooling. The 2x1 structuze obtahed by lowtemperature cleavageis a locally stable structme, but not the lowest fzeeenergy stracture whic,his obtained for b0th smfaccxsby o.nnenlingand subsequent coollng. As we will see in the following sections,hir,h t-rnperattlres are necessazyto overcome the eaergy barriers between deerent structuzw aad are
to generate the nAntoms iu the lowet free-euergystrudures.
There is an iuteresthg hypothv'A about the generation of the 2x1 reDuzing eleavagein the /211) direction a construction duiing cleavage(4.34!. Its formation eaergyof about 0.02 stanking fault is generated Fig.4.5). (see surfaceemergyin Tablez.l. ts small comparedto the Si(111) J/m2(4.35,4.361
A spedal bondingtopole is suggeste. A doublelayer of six-memberrings is rephced by an alternate arraagement of âve- and sevea-mambezrings. Exactly such a ring stradure occurs in the topmost atomic hyers of a 2x 1 reconstmcted salrfve of a dimnnd-structure mystal wiàht'n the widely accepted r-bonded CA:J.II model of Pandcsy24.37). For electricatlytmclzargedsystems, autocompensate,dvste'ms or systems with no space Garge, the Basic Principle may be reducedto three maia 'principlœ whic,hex-plain the ddving forcG of recoxsstaatction(including the of smmiconductor surfacesia more detail 1 x 1 reconstruction, the reln.vxtion) and are consistent with a mlnsmtlm of the suzfacefree energyp. Pcinciple 1: X sur/bcetends @0rzinïzrlizc fâ,c nnmber of dtmglfp,p dangkén.q 50.n,#,.: bondsby the jomationJ.Jn'e'tr ùo'ztrs.The rerrztzfnï'n,p tend to ù6 satnraied.
(a)
7
I)111
(2ï11 plxne: Si czystalprojededonto a (01E) Ft. 4.5. Bond structuze of a (lllj-orlented
(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 stzrfacedaztgling bon.dsvia rehybridizationor convert them into non-bonzlngeledronic states whic,hmay be fzlledby a lone pair of electromsor be completely emp'y Prominent evnmplesfor the minimiv,atio'n of the nurnber of dangling bonds are the and 1l1-V(110) 1x1 s'urface.Jm.the Ge(111) case, Ge(111)c(2x 8),C(1.00)2x1, six of the eight danglingbondsper c(2x 8)cell are saturatedby t'wo adatoms in Tzl positiozts(4.38, This covezageresults in a total nttrnber of four 4.391. dangling bonds. The four êangling bonds of the two C atoms per (100)2x 1 ce2 are saturatedby forrnlng a symmetric fqsrne,rL4.4O). The fottr electromsoclx 1 surfaccxs cupy the c atld x bonclinglevelsof the dsrner. The cleavage(110) of IH-V zinc-blendecompounds indicates another menbn.nneKm of the passivation of the two dangling bonds (4.412. There is a tendency for > transfer of three quaz'ters of au electron leaving the cation dangling bond empty and 611ingthat of the n'nioa with an electron pair.
Pziriciple 2: ./1s'ttdace àontlsto compnnsatecàarges.
An important constrxint whic.hls=l'ts the possiblestoiclkiometziesof compound semiconductorsmfacesgsthe reqttirement that no chargeaccumulates at the surface(4.424. J.nthe limit that sarfacedefectswhich accotmt for the
compensation of charge $nthe spacechargelayer are negligible, tMs holds also on the length scaleof the bond lengths. The relative stability of two surface reconstructions 1511611:1g Principle 1 follows to a large extent from the minimization of the electrostatic interactions i.n such a smface structure (4.43). The tendency to small electrostaticinteractions may somettmesalso drive an interchange of stoms across an interface (4.u). The 'vacancyreconstraction 24.45) snrfaces of IïI-V semicozducof (111) tors (see Fig. 4.6)may be considezedas a consequenceof Principle 2. The removal of one cation generatesarz equal nlpmbezof cations and anions i.c a 2x2 surface pnst cell, a,t least cotmtin.g their dangling bonds. hstead of folzrdanglin.gbondswith a total of three electrons,one obsezves three catiort dangling bonds a'adthree Jmson dangli'ngbonds a,1e.1. formation of a. cation The sidbld rings vacancy. The eledrostatic neutrality is locally g'uaratateed.. of atoms surrotmdl'ngthe corners of the lnnit cell in Fig.4.6 conskstof alternatmg threefold coordt'natedcation and n.nson atoms. The chainsof attezing danglingbonds allow a slmslar electronic s'tzazcttlreas in tie (110)1x 1 =e. ln agreement with Principle 1 the cation tanion.l dangling bonds are empty tflledwith an electron pais.
Pclrzcfple3: A semicondnctormzr/acc tendsto 5eizssulaténg (or.semb-
nonducting).
In principle, all surfacereconstructions f7,1.15111:),g Pricdple 1 and Principle 2 evhibit a tendencyto obey automatically Principle 3. Howevez,there are m=y cas%, in pazticular for quasi-pno-dimeasional surface structtlres, for
and Bondir.g 4.1 Reconstzuction
surfaceafier formatsonof a Fig. 4.6. Top view of a Ga-termlnated GnAR(111)2x2 ae indicated Ga vacancg. A possible2x2 surfacetlnl't cell ksdrawn. Cations taazionsl by open tftlledlecles.
whicb ad.ditional atomic relnvn,tionor recozustructionleadsto a semiconducting instead of a metn.llic eigenvaluespedmpm. For srteras with exterded electronic wave fllndions, metallic grotmê states do not occm in one dt'=en.sion (4.464. One clear examplefor the operation of Principle 3 on tetrahedrally and uctor s'trfaceuris the tilting of the dîmers on Si(100) coordinatedsemicondk Tlb.einteradion of the dirners aloztgthe rows in the (01ï1 Ge(100) (4.47-4.494. direction gives rise to a remarkabledisperstonof the %' and 'ir- bands. If the dimez's are untilted, tke 'zr and g'* bands associatedwith the clsmem overhp i.zïenergy for certxsn 2D Bloch wave vectors. The chnsns of dt'mea'salong the surfaceform a snrplmetal.According to the Peierlsesecvt,semirnetals are unstablein ttrael1D. The l'esulting grouzd state is charadezizedby Peierlsdistoled (i.e.,tilted)dimez's.
4.1.5 Electron Cotmting Rules.
Prindple 3 arzdthe part conceratngthe pusivation of the daagling bonds in Principle 1 aze automatically f7al6lled,if the studied surfacestntd'tzre obeys Suc.ha mfe states that an electron cotmting rule (ECR) 4.50-4,522. (4.42, bonding and non-bonding sttrface states that lie below the Fermi leve,l at the surfacemust be ftlled, whereas the non-bondingaud antibonding states whtc,hlie above the Fermi energy must be empty. This criterion is irt 'clear behaviorof agreement with the condition of the insttlati'ng (semiconductlg) the surface,i.e., Prlnciple 3. Am.ECR cazl be applied directly to detmmnîme
4. RecozzstructionElements
142
m4z
Fig. 4-7. An acioreterminated (100)2xr?z surface of a zhc-blende crys'tal with one are indirnllsing dimer. Aniozzs (cations) cated as full (empty) circles. Hatcheddangling bonds are ftlled.
21)
allowablestkdaeecompositions in agreemèntwith Priaciple 2. Only in order to accotmt to dealwith dopedsemicondudorsmust theserules be genernlszed for defec'tsat the surfacewhich compemsatethe space charge (4.531. surThe electron colmting is well tllustrated for the prototaical (100) The surfaceenergy of thE'se facesof ziac-blemdesemiconductozs(4.51,4.52j. suzfacc is loweredafter d''rnerlzation amdpassivation o.fthe daagling bonds in agreement wîth Prhcipie 1. More precisely,the danglingbondsare ftlled on surfaceaxkionsand are empt'y at surfacecations. Only dsme'rs fo=ed solely aad 2xm (ormx2)reconstrudions by catîorts or aaions (i.e.,homorlirners) are considereêtu the original fo= of such an ECR. The 12'LAa co>equence of the Hsrnerization, while the zrz-foldperiodici'ty is aamnrnedto arise from Lm D) rnsqsi'agdlmeas, leavhzgD fqirnep per qlnit cell where D :; m. Sue,h a reconstruciion is sbown 1. Fig. 4.7. J.norde.rto determsnethe relatioaship tu the aaionbetveen D and m, me cotmt the nlzmbe.rof electrons (4.51). termsnatedcaae of Fig.4.7 ennlntop cllrner requires six electrons, t'wo i'n the c. clinnerbond and two 5.ueach of the remnlnlng dangliugboads forming .n' a'ad z'* statœ. The total number of eledroc on the Hsmezized surfaceshould be equal to that of the blllk-tlnlncated surfacewith a missi'ag aztion pair. This yields -
6.D= 4Ja.D +
4.fcLmD) -
.
He'refradions of electrons
Jc= NA/4' A = NB/4
(4.2)
4.2 ChJ4.lrlA
i'a cation or aaion daaglingbondsllave beeniutroduced (.&+ Ja= 2).XA and NB = 8 Nx are the nxlrnbersof walenceeledrons of the atoms i'a the surfacesone has .A.Bcompound. In the case of cation-term4'nated(100)z?zx2 a'a opposite ftllsngof tke daaglingbonds resulting ia -
2D + Both
8(r?zD) 4#cD+ 4A(rr&D). equations (4.1)and (4.3)give the =
-
(4.3)
-
same
res'alt for the n:nrnberof
aimel.s
D
=
1 Nx
ï Nh
(4.4)
zn..
1
-'
For cliamond-stracturecrystakswith Nx = 4 no iateger solutioncaa be fotmd. This is in agreeaent with the fact that missing dimer reconstructioms are and Ge(l00) sudaces.ln the ease of 111for C(100)j zmt obsemred Si4100) V semiconductors (NA= 3) the rehtion D = 8a/4 is fltlflled for 2x4 and 4x2 rcconstructions with three topmost dsmers. Hdeed, tNe resulting # has beensuggested to explain expnrsrnentalMclings reconstmzction(4.43,4.541 surface.Mfvnwhile, we Hlow that other for the As-ternal'nnnted GaAs(100)2x4 Fig. 2.19) aad that the idea of reconstmction modelsare more favorabletsee Fig. three topmost As Hsrnea.shas surdved in the rough.e,r72 struct'are (see 2.17),where these dlrners are distributed over the flrst a'ad third atomic The 72structure is more stable th= the .precomstraction layers (4.43,4.554. becauseof îts lowe,relectrostatic e'aerr lpdaciple 2). holcls vith NA = 2 and D = m. No top For II-W semicondactom(4.4) dirner should be removedan.dthe smallest possibletlnl't cells shottld be 2x 1 and CdTe(100) surfaces2x 1 a'n.dc(2x 2)reqor 1x2. Hdeed, on ZnSe(100) Larger reconstrudions suchas the constructions were obseznred (4.56-4.591. one are describedby combhed vacaccpdsmer Zn-t-rrnsnated ZnSe(100)4x2 The condition (4.4) of a missing-dsmerteconstruction can structares (4.60j. also not be htlAlled for I-WI compotmds LNx= 1).For instancc CuBr(100) The surface structme is desurfacesshow a c(2x 2) LEED pattera (4.61). scribed by t'wo atozns and vacancies izz tbe top layer of a non-priraitive of electromsiu the dangling bonds is c(2x2)lnnl't cell. The total nApnnbe,r 2 2 Je+ 2 2 A = 8. This nthmberseces to inslzre that fottr anion azld cation dangling bonds will be S'LIHand remain empts respedively. Therefore, a genvalized electron cotmting rule is fctlfllledin the s'pirit of Principle '
.
.
.
.
1.
4.2 Chains 4.2.1 Zig-zag Chains of Cations aud Anlons
EEI-Vazd H-VI compotmd semiconductors,whic,hcry..;Binary W-LV (SiC), planes.1.zlthe bulk taltize ia the cubiczinc-blendelattice, cleavealong (110J
144
4. ReconstrudionElemenis t;l()
k:k$1
E:i'oq 11 gflo
JQ
22 : : : l' :
:
1
:.
1 ; : :w : : 2 :
)1-
: : :. : : :
:' : : 1 ; ; 1
Jc
2
-....
ds
o
1()! E1
.
(joslj
Fig. 4.8- Arrangement of atoms at relxved (110)1xl sutfaces of ziuc-blende semiof bonds in the Srst atomic layer are indicated by conductors. The zig-zag c'knsn!c thiclc lhes. The dotted lines in the top view 5.ve a surface :lnlt cell. The most impozantpairamete,rsof the bord-rotatzon/bond-contration model the rotation angle:zJ.!!tke c'haizz-bunklsng amplittzdezds.kand the bond length ds witht'n the zigzag nhnzns are show'nin the sideview (lower partl.l'ntons: fzlledcircles;cations: opemcircles. Hatching indicatesf1!5ngof dangling bondswith electrons. -
.
-
of these semiconductozs(110)layers consist of plana,r zig-zag clln.7'nq of alanions along a (ï10)diredion (Fig. teznating cations a'n.d. 1.6).Ic a blllk--ltlce covguration eacbv soace atom b.asthree nearest neighborsan.done broken or dangli'ngbond. Two neilboringatoms possessa bond in suc,ha (110) plxne parallel to a E1îî) or (E1R1 direction. The cation tanionl dangliug bonds arc partially Ved vith A = .&4 (A = 'V) eledrons. For (110) surfacesof the zinc-blemdecompolmds the genezalizedelectron cotmtiug rule (Principle 1)can be satisâedwithout chn.ngingthe ) x 1 trauslational symmetry. The Gazacteristic chxngesof tlze atomic geometzyand the bonding behador with respect to the bttl'k termination e-qn be s'Immadzedby a combined sontf-rotcfionvetaantionmodel (4.62,4.63) an.dbond-coniraction
4.2 t7%s.lenq :45
In order to reduce the smfaceenergy tBmsic Prinreïcaation model (4.64J. at tba surfacelayer ciple)accorduto the bond-rotation relaxation model, the n.nson moves away Fom the b'nll- in favor of a p3 bondlng vith the three neighbozingcatiomsrestûtingizl a local pyrnmldalgeometry with bond angles tending to 90o (see Fig.3cz,0a). The cation iu a surface''nit cell, on the othezhand.,moves into the btllk in favor of au 4J2 bonding with the three neigbboringanlons restllti'ag i'a for a loci pln.nltr geometzy with bozd angles dose io 1200 (see a tezzdenc.y Fig. 3.20b). The correspondingbucldiag of the surface zipzag clmlns'is aecompanied by d-like erson dangling bonds and p-like cation dangnngbonds. i.n Sect.3.5.3 the n.nion dangling Therefore,in agreementwith the dkscllrwqkon bond îs completely Xled 'witil eleetrons whtle the cation dangling bond remn.l'ns empty and, hencevPrinciplœ 2 and 3 are also f111611ed. Thjs sftuation Lsindicated in the lower part of ng.4.8. 'nis picture is con6rrnedby t'hemeasurements of fzlledaud empty sttrfn'ce states by mp-emKof STM. The comhination of both state lmngesinclicatesthe formation of zipzag chairts of cations and n.nlons in the FI0)dtredion as sudace (4.654. A czaracteristic demonstratedin Fig.4.9 for the 1nP(110)1xl direction featttreof the zîg-zagchp.5nsis their lateri extent lzrl in tbe Eû01) A the ideal brllkwterrninatect case) Zllj = co/4. also Fig.4.8, top view). Jm. (see further low-ri'ngof the smfaeeenerr (Principle 1)may be a consequenceof a shozleningof the bondsin the zig-zagchains due to the attraswtivehteraction of catiotusand anions.This restfts in bond-contractfonrelaxation. A spatial' represantation of the zipzag cllains is given i.u Fig. 4.10.
a
.
. . .'
j, %.. ....
'
l
& .1 k.';kp 1$. k2$..k ;kii'lë jrlt '
'à';/
o
ljk, . .2 .
o
.
,'.
k
,..
'
:rfi'?#. ' zC I' . .1:::::/. ' t't. lt m's .1 lk1)27. -:; ' !a!h t-? 1;i.s iE ..
.
,.. t..4qj xq)pj y, k.:t y 'o ,vu k-httv. ..
.
.
.
.
' '
.
:,.*'! kivt)'jr . t14 ' ' ' ' 1':h
,:
. '
ïl'
.
fi Iled dZ rl ; I1rl g bOn t
emp
danglingbond
Fjg. 4.9. Zig-oag chains of empty' and FIII?.IIdapglingbonds at the rehxed surfaceas measuredby STM for bias wlth opposite signs (4.65q (copp 1'nP(110)1x1 from right (2003), wit: permlnsion Elseviez').
146
4. ReconstructionElements
surface with the resultFig. .4.7.0. Schematic indication of a relaxed GaP(11O)lx1
ing zig-zageahsGoq.
Att key 3tructural paœametersof a relaxed (110)surface are showa in Fig. 4.8. Tïe relative displacememtof the cation and acion dtmnesa vertic.al sllear ïza., the b'anklingamplitade, and a correspondingrotation angleLo ill the (1ï0)plaue. The deviation of the length ds of a cation-anioa bond fzom the bu)k bond length deq= VVo/4characterizesthe Coulombinteraction along a zig-zagchain. The z'ig-zagcharacteris governedby the deviation of In addition, the cations zlllj 9om its blllk 'value co/4(Fig.4.8, top view). azd aaio:as in the subsurfacelayer show a cotmter-relaxation,i.e., cations move away h'om (toward) the bux albeit on a much smaûerscale. taaionsl For coaveationalIII-V compoundsthe vertical separationbetwee,nthe smface interplaaar layer and the subsurfacelayer becomesshorter than the (110) distance ao/(2W)i'n the bulk. relaxValuesfor key parameters of the bond-rotation/bond-contration ation are listed in Table4.1 togethe.rwith the bulk httice constants co and the charge-azymmetrycoeëdeats .ç of the b7llk-bonds. The values are take,a 4.66, 4.67) a'n.dmeasarement of strttdural h'om ab initio caktzlations(4.41, The key parameters indicate that parameters LEED measurements (4.681. the surface rele.xntion is strongly Mueaced by the interplay of covalentand ionic bonding aud, to some degree,the ratio of the atomic radil. The rcu laxation prameters can be arrauged ia t'wo groups. The conventional III-V compotmdsare ard II-W (ZnS) arseaides,aad phosphide-s) (antimozzides, chazacterizedby bond-rotation relaxation with a.n plmost bond-lengthconservation. The layer-tGtanralesul vazy in a relatively small mterval of 25-320 1 surfacesof the while the elnm'otilt 11.u. amouats to 0.6-0.8 A. The (11O)1x compokmdswith anions belongiagto the fzst row of the Periodic Table (zzirela=tion. The trides, SiC)exhibit a mixed bosd-rotatîon/bond-xntrRtion
4.2 Chstl'nq
l47
Table 4.1. Charaderisticpazametersof bulk bonds @o, J) aud geometzyof The stracl surfacestlz-u,az, G/doq 1)for zinc-blendese 'mlconductoz's. (110)1x or LBED 4.66,4.67) t'aral parameters are taken from ab jnitio calculatioms(4.41, The càarge-asymmetry coeëcient .q is taken from (4.26). meaamements (4.68). -
(A)
Compotud
c,o
SiC GVb Insb
4.286 6.095 6.475 5.66 5.559
Alàs
GeA,s TnAK
5.861 5.45 5.359
AJ.P GaP ï:c.p BN AIN
4.34
4-46
HN
4.97
ZnS
5.409
0.475 0-169 D.230 0.375
0-23 0.77
15-4 30
0-78
28.8
0.65 0.316 0.67
27.3 30.2
+0.D
0.450 0.75
32.0 25.2 29.2
-1-2 +2.9 -1.9 -1.8 -7.8 -3.6 -5-3
u;
0.371 0.61 0.506 0.67 0.484 0.18 0.794 0.17 0-780 0.23 0.853 0.24 0.673 0-62
3.60
(O) gs/deql (%)
lza.
0.425 0.63
5-662
GKN
(A)
g
-
30.1
15.7 11.6
14.3 13.1 25
-5.7 +0.4 +0.0 -1.3
-4.9
+2.9
,
Re-f.
(4.66) /.68) 14-64 (4.68) (4.411 (4.41) (4.68j (4.414 (4.41) (4.67J (4.671 (4-67) (4.67) (4-6$
4.
-
surfaceneazwt-neighborbond lengths are shortenedby 4-8 % with ropec't to the bulk bond lertgth, and muc,hsrnxllez tilt angles tzg = 12-160occlzr. The buckling ll.z of the surfaceclmz'nsis reducedto values of about O.2i. The key parameters are not independentof eachother. Tn fac't
i suw, %/ (ds 8) 2 , ao/ (ds 8)i (1 sias'l.
Zluzzll
-
z
=
v
-
-
(4.s)
-
For that zeason we only discusst'wo pazameters in terms of covalent and ionic bondicg in Fig. 4.11. A.scan be seec tzl Fig.4.11a, there is a fairly good licear dependence betweenthe bunldsngamplitudeïz-u and the lengthof the bonds ds in the sudaceahnsns,except maybe for BN. The gene'raliêea that the bunkls'ngamplitude is Gversely proportional to the bonding strenkthis con6rmed.This relation, howevea.,is somewhatmoeed by the ionicity of the bonds, the atomic sizes and electronic correlation efecwts.One esect of the ionic part of the chemicalbondimgis demonstratedin Fig. 4.11b.The depen.dson the strength of the Cotllombinteraction alongthe zipzag cahnsns of cations alld aztiorts(4.69j. egectiveatomic chargesZc/a = ZVA/B 4(1:F.t.?) Thetr diference Zc Q = 8.: (Nz NAI governs the strength of the bond-lenlh contràdion in the cimsnq.There is an Mlrnost linear dependence. The exceptions BN, AIP, and ZnS are probably a consequenceof the fact -
-
-
-
Elements 4. Reconstruction
148
1.0
(a)
Q.8
jnxs
O
I!1!:.
R w
GaV
: 0-: <
=
E
o
sap
o 10
AMs XP
zns
0' 4
.; c
SIC 02 '
C) BN
0.0
1-3
1-5
oo o
GaN INK
xN
2-0
2.5
3.0
surpacebondlengthds (A)
Dference Enionic chargesg-(Na-NA#8
Fig- 4.11- GNrnical trends i.xvstractural parameters of relaxe (110)1x 1 surfaces of ziac-blendeMmiconductors; (a)nbnlu buekling versus mvrflw bond lemgtk; (X bond-length contrn-ion versus Herence of esective ion c%nrges. Data are taken from flrst-prindples calculations14.41,4.66,4.6% and LEED measurements (4.68).
4.2 Ckains
that deformationsin
muc/ elasticenerr
strongly bondedsyste,m (inparticdarly BN)cost too that the ab icit'iô ionidty scaleskhtly overesb'mn.tœ
a
or
the electron trxnsfer
149
(e.g.in AlPI.
Besidesthe tendency to end with. completely ftlled or empty dangliug bonds (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 opeusurfaces. jnglceaa be exploredby studying the atomic rehxation of (110)1>t1 Due to the natmal builçin civge trnrRfer beGeen neighbozin: cations azd anions even the ideally tmrml'nated,i.e., llmrelaxed, (11O)1x1 smface of a ec-blende material remnsns semiconducting(4.701. Two bauds appear ia the fntrtdamentat bllllc bard strucvtme. The lower band As is gap of the projected Jmlon-derivedaz.dthereforeOed. The upper bazd Cz correspon.dslargely to '
X
.
.
-H
5
>
t::i3
'
.
BN
A5
0
.'M
.
pi
v .
6 ,--w
>
...
-. .
zy
ck.j/'
0
.
Q) c
'
>
.
'''
.w '' '
'<.w
.
wx
.
A5 ..
-
.
.
m rp
M.
AlN -.-
<*
>
'%'
c
..-
F'
.V
.
.
N
-5
. .'
..av.
NSX
=-
.
5
.-
.
' Hn'
.
.
'
>ev.
.-
..
.
xx
n'=.''' '
'''
...
GaN .
''
'-= !
.
.
0
.
.
AS
.
. '
..
<.
'ilkr
''7k .
. .
v..
-5 .
1!5;
.
..-.
u' .
..
N=.
r
l..rxcuuaz=' ..
.
.
'
Ië nM M n
.
w
....
-!...,e
..-z
-va'ze
...
p : '%
0
.5
xaqw
...
.'.=. -.
mv#
'
.
..
M5
-. ' wtu=
v'Jv. .
X
.. ' '== 'xu
w
.
M
X
5.
..
r
Fig. 4.12. Occupied (As) and empty (Ca)surfacestate-relatedbands tthick solid 1inœl of relued l11surfaces wftilin N(110)1x1 the projededbtz)k band s'tacture. R'om (4.67).
150
4. ReconstzuctionElements '
empty cation d bonds.Dlxm'mg relaxatinnand hencethe rehybridization of the dangling bonds, thae brds are displacedtoward the bulk valence bands (a%) or bulk conduction bands (Q) aad e their orbital character. TMs situation is displayedin Fig.4.12 for therelred (110)1x1 smfaca of grouml'll aitxides.The ba'ad s'tates As correspondto mostly p-zlceazzioa danglingorbitals,whereasthe G baud getsa strongerspz-lilqecharacter.The relaxation-hduceddownwardsïfts of the occupiedAs bands amouat to 0.09 0.24 (GaN), 0.21 (1*) W at tke .j%point of tùe slnrLzte (BN))0.11 (A1N), BZ (4.67). This gives rise to a relaxatiomizducedgair of band-structure enThe corrcpoading shiftsof the empty G bandsare considerably ergy (3.53). larget. They are 1.40 (BN),1.03 (AlN),1.10 (GaN), and 0.85 (TmN) ev but have no dired inhuence on the tota) enerr. Howevez,as a consequencethe Cz band of JnN disappearsfrom tEe Alndamentalgap ia Fig.4.12. '
4.2.2 Jr-bonded Chnlns
The (111)stzrfacesare the aleamgefacesof diamond,Sij and Ge. An ideal trnncation of a buzc crystal perpenclickfazto a (111) direction would result in a surfacewith either one daaglingbond (DB)or tltree d=gling bondsper surface-hyer atom dependingon whether the trlmcation is made above or betweenone of tàe voup.l'v double'layerssteickedalong the special (111j dizection în the bulk (see Fig. 4.5).Correspondingly,one distinguishesbet'ween SDB and TDB (single and tziple daagling-bond) surfaces.The TDB =face is energeti-lly 1- [email protected] SDB surface,siace the for'mer hmsthree broken bonds while the latter ha,sonly one. We thereforedisctuss txe SDB surface1st. The ideal bonding topoloa of a gzoup-Nlllllsurface is displayedizl Fig.4.13a. The dangling bonds belong to atoms wllick are second-nearest neighbors iu the bulk. Qnherefore,they elmnot form bonds to gain emergy accordingto Principle 1. Reamugemeat of the atoms seems to be likely, at least to eVplainthe stable (C)or metmstable (Si,Ge)2x1 rronstruction. One possibility is the formation of Evomemberaud seven-memberrings'in the stu'facebilayer (see Fig. 4.13b)right panel). This may be induced directly duz'ing lln.q been mzggœted(4.71) cleavage(4.A. Even a special Mnk mcschanlsm Suc,ha ring strttctttre cau be nnhievedby displacementsof atoms ia a (1ï0) plaae. Norlmlp and Cohen(4.72) have demonstrstedthat the new conîjw ration caa be obteed in a simple my fz'omthe ideal structm'e by applpng a'low-energyshear distortion to the top doublelayer of atoms. Break-irtgtke bond betweeaatoms 1 and 5 (Fig.4.13+ right pl.aellaad 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 arzangedin the fzrst atomic layer, wbile atoms 3 aad 4 are in the secondlayer. ne interatomic dis%ances are close to the bulk bond length, and ennh of the suzfaceatoms possesesa dangling' bond. The dcct'ibed displaoamentss pr&erve the zmderlyingbulk mirror reNectionsymmetzy The threefold coordinatedatoms 1 and 2 becomenearest .
4.2 Chzu-nK
Ej1f E1 .1:1 2
2 $
1
3 W
(a)
4
5
3 6
8
7
2 r 2 1
(b)
4
3
3
4
5
6
B
z
7
Fig. 4.13. Top and side view of a groumlvtllll sudace: (a)ideal lx 1 coafzgmation; (b) rr-bondedchaiu model of the 2x l reconstruction. Large open circles indicate ârst-layer atoms. ,
neighbozs.Their daagling bonds can satmate eac.hother via a rr-bondinglike mechanismaccordiug to Pzinciple 1. The surfaceatoms 1 and 2 are eMzt'h bondedto two other surfaceatoms a'ndform zipzag nhxins dtrected along the gI10) direction. The resulthg surfacestructure is itz completeagzeement wsth the reconstrtzcvtionmodel of Pandey (4.73). A perspedive view of suc,ha sudacewith zr-bondedzipzag chain.son top and the new bondhg (111)2x1 topoloa is shownin Fig. 4.14. The aew bonding geometr.yh.msconsequencesfor the surface electronic stzucture. To gain a cel'tain priacipal tmderstandicg we Vitzmethat the surface-statebands ocolrrlmg in the projected Alndamental gap can be descdbed by the interaction of the dangling bonds formhg the cizain and by neglecing their coupling to the bulk. Let sz aad ca be the orbital energies of the dangling hybrids 4:(m R (k) (ï = 1, 2) on atoms 1 and 2. The twmdl'mensionalBravaislattice (1X of the (111)2x1 surfaceand the vectoz's di (i = 1, 2)of the atomic basksare iadicated ia Fig.4.15. Wèthin a nearest-neighborapproximation of the tipt-bindîngdescription (3.7)one derivelan interatomic hlteraction matrix elemen.tSta of orbitals 1 and 2. Wit: kr = Sz2((d1 1)+ Jzblida())one has a Fave-vector-dependent matrix elementof the Hn.rnsltonia.a(3.15), S:a (i) = 7S(i), with the genernls'zedgeometric.al structure fador -
-
4. ReconstrttdionRlements
Fig. 4.14. Pezspectiveview of the A-bonded chain mode,lol
a
surface.
S(/1 )=
kéeïLd'i
-
group-Wtllllzx1
(4.6)
f=l,2
of the r-bonded chni'nswith X = Slatldz' 1)/2atld 11+ Aa= 1. Dimerization = of the chlu'n bonds, dz # dz (dx' Jtfz' I), and the bunk-ln'ngnrnplitude ztA= Fig. 4.15). The infuence of (dl dclzof the clminqare generallyallowed(see the dimerizatiou on the absolute valaes of the hteraction betweenda'agting of the structure bonds has been taken into accolmt by the jeneraWation factor (4.6). The diagonitlsmmtion of the resultmg 2x2 problem (3.14) yields the t'wo s'arfacebands -
(i)
s::lc
=
1
1
2
a
z
jg j. (sz+ sa)+ .k.(sz s2) + 1RI1S() I -
1-
1 surfacesan appropriate Caztesiancnordinate system is (111)2x a'n.dzg , v1 zll E11R I Eï101, I (11lqThe two vectozs chnracterizicgthe bonds of an ideal cha,iuwithout Hbmerizationand bucldiug a-re dïp = co(1, ui:vx0)/2V&
For tke
.
1oq E5
q%az W
E1 121
Fig. 4.15. Top view of the 'Jr-
bondedclml'nqon a (111)2x1 surface.Bond dimerization or chain tîlthg îs atlowed.
4 2 Chaizus =1 J -
r
.x
-
1r
J
ar
(b)
-1 J
r
K
Fig. 4.16. Schematicrepresentatiox of the sudRe bands(4.7) withln tke gwbonded t+xin model: (a)tilted chainswith ideal structme factor but tql'lYea'ent splittings of the dangling-bondenergies.lal s2 (/IVI= 0.0 (solid 0.1 (dashed line), line),and 0.2 (dotted linel;(b)dimerizedchltlns= with dxerent interatomic matr!x el-ents Az and .:2 along the czain, ilz Aa1 O-0(solidlinel,0.1 (dashed linel,aud 0.2 -
-
(dotted line).
= A s'trucime fRtor IS@)I parallel cost/cn/zWlrets. A chaia bu to the z-aadsdoesnot changethe structure fador. Accordiagly,tlze bands are dispersiortless for k direct#onsparallelto (11Iqj i.e., along X,Fand JX- tsee BZ irz Fig. 1.22). Stzongb=d dispersion occurs along the chain Aection (ï10j, i.e., along P%ïazd ..FZ i'a the BZ. ne t'wo bards (4.7)are mb=atically showniu Fig.4.16a for severald'l/erences of the daaglizzg-bond For ener#es. ideal rtbna'zusand identical dangling hybdds ihe surface system represents a zezogap semiconduckm A d''Ference i:a the dnnrlinr-bond energies,however, opens a'a ene'rgygap along J# aud,àence,makesthe mtrfnre seznicondading. 1n.any cmse, the formation of delocalizedzr bonds and correspondi'ngzr and ')r* bands (4.7)gives rise to azL enera ga,in accordi'agto Principle 1. A gap opsmsngadditionally lowezsthe band-strudure eae-rg)r(3.53). A gap opening along J# due to rllFeren.t danglinpbond enengiesis related to a loweeg of the cb.ainsymmetry iïz the sense of a Jahn-Tellerdistortion. One possibility is a bunlclingof the eyn.inn as slmwrt in Fig.4.17. The raised '
(a)
iA t
(b) -A
i t
Fig. 4.17. Sideviews of zr-bondedcakainstruciuzes with positive (b)burklsng of the Pandey chains.
(a)azldnegative
4, ReoonatructionElements
154
A
atom of the chaiu again tends to be p3 coordhated,wheremsthe lowered atom tends to axt spz coordl'nation. TMs bu i'adeedlowers the total They dz'il'eroaly energy.However,two rlieerent isomel'sare possible (4.741. with respect to the sign of tke buclch'ng(Fig. 4.17),i.p., whether atom J. or 2 (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 surfaceatoms in the severemember rings are at the same heilt. Starting from thâs,one generates the t'wo isomers by titting these two atozns izl clockwise or anticlockwise dkectiom The two choices in Fig. 4.17 are not equivalent with respect to the third atomic layer. The istandard' choiceis the structure with pcksitive bunk-ling.However,the two isomers are nulrnost degOerate in enera. For 1 the chain-leRtorchaizehigh) isomer $.74) semmq to be the most Ge(111)2x stable geometzy (4.75, while for Si(11l)2x1 tEe eae'rr dlFearencets 4.761, '
'
'
smaller (4.76). For C(111)2x1 no bucklingLsfoa'ad(4.76,4.77) of the because enezgy loss due to strain in the substtrfacelayezs. For Si =d.-Ge the bu amplitudes az'e large, ztï = 0.2-0.3deq,whereas the chain bond length dz = d2 is shozïer thnmltke bulk lengtk deqby a factor of about 2W/3 (4.78j. The =ompanying gap openlng is 'isplayedilz Fîg. 4.18 for Si and Ge(111)2x 1 surfaces.The empty and Rled states forming the zr* aad rr bauds have deazly bpe.nidentled in scazmiag ttmneliag spectra (4.79, whic3 also mclicate a negative bunlcllng for Gc. The ctis4.80:) persion of the ban.dscompa'resfavorably witk data 9om angtzlarresolved photoemn'xsion and k-resolvedinverse photoem:'xqion spedroscopy IARPBSI spectroscopy(KRIPES) The cakulated absoluteenerr position :4.81-4.842. and the separation betweenthe two bacds show the tzsual DFT-LDA shortcommings whick however, can be resolvedby qllxqipartide band-structme calculations(4.75,4.85,4.86). Chnn'ntzttug is nmfavorablefor diamondtllllzx 1 24.F, However, 4,784. thea'eis anotherpossibility foz a suzface-state gap at the JR line by moclify'
.
,'
''.'
$ ' -16, 7'' ''lî%4 ?tk.:4. ) jf; rt.'t))h 'itrqot. .: L.y ,, i4L;1 tr 4-:$) . t ., . I k!.t). 'q' f' ' :. . )/-'h(($ z f17 <. ttkt' lt , xo i . ..y .
.
.
.
.
yt.,yy.
x'.%.
4
1--1,
(12/
$::1
ë'rrr!
...'
.
'
'
. . ..
. .
.
.
'
..
.
.
(21..... %'.'
.(;.,. ' h tt4xzbtv' sër);. )%yq kk z
-.
.
' '
y j.
,.-rrjrjr'ljl;:j:rr
.. '-.
..
-
,.
'
N
k
y.'.,.)y.y ryjj.
t jr sj x j:y) ''vj kN.i.!..>#;i.; .',k;ru q$. Fxqttt bbàvqqbn--bb..b '4k ,'L kt$, -$:? , (1t. ,y,s :--,..,k.f.., hyk b.btt. %a(. '$(,fl,tt :/..) stklr.&. xtstg.? b' ''s ,tk $r.. u.s,y? '--Nw avjj J.-j)a.t rt k'p r;. );)y jf. -)j4, j
,. .
..
61-;i3b..
'
.
.
a.(...
'
.
.
'
q7
x0 t::::j
' Eh 1!1j). 7k//*..18 ' 4;. krJiph - $ $rjà' sj :1,st& k .,4ry-,à xpX. tssyq.-. l,,.f'' - ), .A. -Nç;l:'.ky SIattjJ $ -t.:f jk tj q%% s /:/ j'xlés @@, xy .s-.y .,N Ctjz. s
piëlt.''l!cèf ';?E@qil)'-. tCiii'''''.f ':244!1/:--'* 'lljiqi'.;.f .
:
'.
.
-
.
.
'
..
,
.
.
,
.
t4. h'
!F
lqqtk
0s,;1ïk 1:E2h yJ.-...s,sxs, : krr.)g l.t( IJ k)11 ?! z%à ,p)y.r. )). .1,; s. .(.. blbbb'''. -L vjk,z-jtj, 1,. v k!!. I)yh,(t, : ... yy! 4).yI. ' :$s hj;,, .. :.ilrsll,'.?t î.%'.%..%.. 'kt, .î; . k o, p,lrfy t4k-tj. ' '.'j :qj,' 'kl .ii) '.:'t b p ' q- 't,.-.qF:,) .;j; hj ?h$-',,-$/sîk$.p,A qîtl''xï,%kï '-:hk#;-?'ï.' 'tk s rlêîd .:-..-'t-!(!:r-k t lb'h $. .. *mn lrt#hl -1F# #;. p'y x. 6. '' (It'/ lq'--îLtï - gylhlki --1 ïi dp,' j:zs-tr. !i,, !t ..3.2.:3 y.,y zo. :s x-st-titgy,y lk '-?ir' st .'t ,t-. j))( 'w+).. k,, - .t .l i'sNrl .. , $. , qt v ,v slz.xp,,iygs-ktdqyt.-, ; 4'$?; y s '>7(1îzf5-? .3.. '.')k;'.h7rt . -. j) jtt.' a?hy...r; . '.' . ., 7 t.' ,'J. yt y.lyj j /.?? y.z .yyt, 'i.t, ttq'' 'L :41 k,lo.,,. 'fy ht7kt' j J ,,..,jt.--,'-da.ù,œty--y..y.,7%. ..,,e.-.. .1h.y%:;7' TA'+')k:= , . ç, sg ,# t:. jtu-, , ,y, a: . ;t.ift'vxvvk.,.k. s4-k -,,, v .L..'X.. 4x.v.bu. 5'f ..,%.n,.s., $-, *s. uj # As jzliuzv qslllàAsxlssttttllkNllh yL./j ' hkci .,t-è '.k. f<.b zA. ' .y ';k )t. .
.j(k), ,: :p: ,.)j..;t; r >)j ,k /lpik.h 78. .
.
.
.
.
.
.
'
'
.
'.
.
..
?,),-h... ''zthk'Sttkas'k j,..;.,t')',.
-8 r
.
J K
j). )
.
.
.
.
'
.
-
.
.
-2 J T' f'
.
'
'
'
.
.
.
.
J K
.
.
.
.
J r
.
'
'
.
.
.
.
..
..2 r
J
.
J' r'
Fig. 4.18. Band structures in DFT-LDA of tEe (111)2x1 sarfacesdescribt.dv'itb.izz tbe rr-bondedchaiztmodel tchain-left isomez) (4.76j.
4.2 ClmlnK
155
ing the strudklre factor. This caa happenby a conjugation of the zr-bonded c'hainsms for hydrocarbonsof the type =CH-CH=CH-CH=CH-.... This is accompanied by a dlmerization aud, hencetfliferent bond lengths dz and d, Fig. 4.15). The eFectc,a.rt be describedby assuming Alp = j (1+ ,4.'t)irt (see The geometrical strucwture factor becomesIS(i)I2= cpszt%co/zWl + .(4.6). along ?.1 and LS(i)j= gzâà) along <W.The gap open(1A)2sinzfkvaojzsfk iug along J'# is given by 2I1AIfr.TMs egect is clearly demonstratedia ...
Fig.4.16b. For Si ard Ge no indication %r dimezization hmsbeen obsen-ed.In the 1 surfacedirnezization is lmder discussion(4.871. cmse of tlle C(111)2x Performs'nggrazing-in'cidenceX-ray diAactîon studiesHuism= et al. (4.88) favor Shim-sequenc'y tilted chnsmq. experiments (4.8Sj genezationspecwtroscopy e,xpla,in the observedselection rule,sby (I':G buckling. Apaz't from only one
reference(4.87j: convergedAst-principlescalcalations(4.77, fotmd 4.00-4.92) almost undsrnerheda'ad unbuckled -C-C- chains on the diamond surface as a minimlnrn stzrface-free-enerastnzctme. Bucklsng and dimerization of the clmsnsresttlt i'a a deformntion of the tmderyngseven-member azzdfvemembe,rrings and, hencej increase the elastic energy whic.hreducesthe enr erg.g gain due to reconstmzctton.Ths eFect Ls strongest for diamond. The wave înnctions resutting for the ideal x-bonded chai'n structtzre are shownin Fig. 4.19.In qualitative agreementwith the tight-binding estimates (Fjg. 4.16) the accompanyingband structttre in Fig.4.18 resaltiug within DFT-LDA indicatœ a sernsrnetatticbehavior. There is a need for fuzther studie,sin-
1U111
./*
I'ttz:l *
r'i--la:l
1
O
(:r-;c2 (,1::1
1
Fig. 4.19. Plots of the square of the wave llnctîon of the 'C-artibonding and gw bonding surfacestates of C(111)2x1 for a wave vector 0.4 Jt%.The up#erpanels show ihe projectiononto the chain direction and the lower panels a top new of tàe surfacec'hain.Lines of equal densit.gare separatedby 0.075boizr-'. Fmm (4.774.
4. ReconstructionElements
cluding electronic exnhnangeand correlation in a bette,r way (4.40) On the other haad, the data poiats for occupiedsttrfacestates fxomARPES (4.93q a'nd t'wo-photon PES (4.94) have been shown (4.95J to be i.n good agreement vith the b=d dispeoion derivedfrom DFT-LDA resutts for ideal x-bonded chains (4.771. .
4.2.3 Seiwatz Chnlms the (111) smface of diamondstruckt'are semiconductorsmîght be expected if cleavagereslzlts in a triple dangling-bond (TDB)surface.I'n this case the cleavageoccurs between two narrowly spacedlayez's(see Fig. 4.5) , izï coatrast to a single dangling-bond (SDB)smface, which sqpazatesa widely spved bilayer at wlzic,hthe bozlds direction. Ir. the TDB case three boae az'e orîented exactly i:ct the g111)
Qkliteauothe,r2x1
arrrgement
of atoms
on
r rEl E1
:.:':t.l
(b)
12,11 n'.I
4
E11Q
Fig. 4.20. Formationof Seiwatzcllairts of surface atoms resulting in a 2x1 reconstruction of a TDB grolzp-lYtllllsarface:(a)ideal TDB surGne (side view);(b) and (c)Seiwatz chn.lnq(top'eiew). Seiwatzciminq (ssde'=ewl; .
4.: Ckalns
Eig. 4.21. Perspectivcview of the formation of rr-bondedzipzag chains on dangling-bond (114) sudace of a diamond-stmzcurecrystal.
a
triple
brokemper smfaceatom (Fig.4.2Oa). Two of them might saturste when neighbozingrows of the atoms are movedtowardseac,hother tmtil the distance of atoms of one aad the other row approachesthe bullc bond length deq and 4.20c). A perspectiveview of the resalthg stntctme is shown (Figs.4.20b in Fig.4.21. Jt is consistent with the Seiwatzmodel of x-bondedc'haizus (4.964. A 2x1 reconstruction occms. The surfaceatoms form zig-zagczhnt'nq alongthe direction. Two atoms cha,inare paz't of Ae-memberedrmgs. Eac,hc'haiït (ï10q atom is threefold coorrlinxted.The remaining dn.nglingbonds form zc-bonded e'hnsnq similar to that discussedfor the Pandey chains on the SDB sttrface. They aze accompaniedby 'zr-bondikg and x#-rtibondin.g suzface-statebands in the ftmdmental gap of the projectedbulk band strttcture (4.771 Until the discovet'yof the Pandeymodel the eazly Seiwatzcb.ainmodel was usedto explain the Si(111)2x1 surface,thougll it did aot completelydescribe the LEED fndings (4.97). Since 1981a11the properties of tke metastable Si and Ge(111)2x 1 sufaces are explained itt the gamework of the x-bonded ckaiu model of Pandey.Only i'a the case of diamond are the TDB surface azldSeiwatzchnsnsstill tmder discuasion.Hydrogenttarmimatîonmay play an important role, in pazticular fo< C'O-grow.a diamond samples.A tllzoe-step model hmsbeen suggestedfor the chenzisorptjonand desorption of hydroTDB smface,which makesthe creation of a correspondsng gen at the C(111) clean C($11)2x 1 stHaceenergetically possible(4.984. The outstaradingrole oî hydrogenis also obvious from the preparation of diamond61mAsvitlnin a CVD saperstructures have beenobservedafter heats.ag process. (WxW).R3OO ttp surface of polyczystnqll'neBl=s. They are related to TDB s-udaces for (111) spnmetr,y reasons. However,STM ha.salso shownthe coepdstenceof 2x 1 domaizsswith IWXWIJBOO structure.s (4.9% From the thezmodynamic point .of view the SDB C(111)2x ! surface is euergetically much more favorable thaa the TDB C(111)2x1 surface.Their surface energiesdfer by 1.35 ev The TDB smfaceis not likely to occtlr upon per 1x1 sudacelmit cell (4.774. are
.
'
I57
.
158
4. RecomstructionBlemexts
preparedby CVD growth and cleaxingdiamoadbttlk material. For smface,s subsequentAnnen,lingthe situation may be difFeyent,however.
4.3 Dsmers 4.3.1 Sylnlaetric
'
Ilirners
The ideal; bulk-termlnated (100Jsurfaces of diamond- azd zinc-blendeFig. 1.6). stzmcture crystalshave a square ttnit cell, with one sarfaceatom (see Thc seace atozns posseD two dangling.:.p3hybrids. Eowever,suc,han ideal experimentally.Accordângto the fmdingsin structure hasnot beenobsezwed surface a dehybridizationtakes place yieldi:ag a bridgSect. 3.2.2, on a (100) AccordMgto Principlq 1 the a danglingspx orbital (3.36) ing ppjz orbital ain.d. arlmber of dangliag bonds elm be mimimszedby a pairiug mechanism, more s'tdctly by the formation of dimers witk a bond length Q, as iaclicatedia Fig. 4.22. Two atoms that aze secozd-neazest neilbors fxom a bulk poin.t of They form a cr bond emanatiug from view approaG eachothe,r along (011). the two bridog ozbita,lsIbr)in betweenthe dz'merizedsurfaceatoms an.d poHted a weakerzr-like bond emanatiug 9om the t'wo danglin.gozbitals 1d) of the sicgle-electron away from the dimer. In the orbital representation (3.18) Hmn'ltoxf.an the t'wo interatoxnic interactions are approvlmately givea by where the Fjl = (brlslbr)= Fm. and Vk = (dlsid) = lz(Wa.+' V'ppxl, have to be ta,kenat the dsrner bond length Q. matrix elements F'oys(3.21) Neglecti.agdeformations of t:e d=gling-bond orbitakqdnlrsngthe dimemrization and interactionswith tke surrotmdingorbitals) the moleculeHnmiltonian (3.7)e-n.n be written in the form .
0 0 a'd -
s
5Q
W ca
-
e
TV resttltiug aatibonding aald qrlth the intra-atomic matrix elements (3.37). bonding Ievelsare ec-jc = ssr + 1$61 1and s.:r.fr = ca ::: lFki. =
!àooli
we-X'>.
-
x=>
E()r'Il on a Fig. 4.22. Formatîon of 35me,.rboadsalong (011)
surface (schemxtically). (100)
4.3 Dimers
l5S
Ga tr
9
V
G
6
.x.
C.i
%t.3
g: .a
>x
61
P ;
:
o.e
cbr
cur W
c%
-3
:1 r
cd
Fip 4.23. Molecule levels of a climer of eqcvalent surface Ga,,As or C atoms. The sf' hybrid energy' of cazbonis taken as zero energy. The correspondiugmolecule levels of isolated Ga, A.s aud C dimers are = representedin Fig. 4.23. The orbital enertesn (Ga) -11.55 eV) o(Ga)= = -5.67 eV, aa(As) -8.98 eV, ss(C)m -19.38 eV, s= -18.92 eV: sp(As) = -11.07 ev have beentakea from the Solid State Table a'n.dcp(C) (4.1004. The interacticm mnnutrix elements(3.22) have been (nlculated with'dlrner bond lensh Q = 2.49 i for Ga on Gn.AK(1O0)2x4 Q = 2.50 A for As on E4.101),
1.37.1.for C on C(100)2x1 The (4.40,4.95j. energetic ordrktg of the dsme'r levels i.n F1g.4.23 and the nlnrnber NA/s of avazableelectrons determsnethe level occapation. 1n the crase of a carbon 1 surface with Nzvs = 4, eachof the c and 'c' dimer on a din.rnondtloolzx bonding states is occupied with two electroaswith spitsap ='d spia-down. Prindple 1 is also G'lfllled vith respect to the passivation of the dazlgling bonds. I'n the case of the As dsrners, the three lowest levels o't rr, azd x* should be Slled to obey Peciples 1, 2: azd 3. This requires one additional electrop NB + 1 = 61in aveement with the electron cotmting rule (see Sect. The sitaatîon js less cleaœfor Ga dsme'rs becauseof the high-lying 4.1.5). bridge-bondorbital energy sbr, The absolutelevel mlues howeverindicate a tendeacy to donate electroas,at least oae, itt sprfaceAs dimers resulting in Nx 1 = 2 and a'a occupation of the lowest bonding cl5=e.1.level. Thereby, the ECR can be hllfllled. Mtuitively, a 2x 1 reconstruction is obtnsnedfor goup.Nmatezialsby an in Fig. 4.22 (mght arracgeme'ntof dsmers paralle,lto (011j partli'a linearc'hains iong a g0ï1q d/ection. This dimer model was Srst proposedby Srllnllerand Fhrnqwoz'th(4.103) to explnsn2x l reconstructions of (lool-oriented surfaces of silicon and germanblrn. It also chazactfm'mes the clean (100)surfaces of 1, neighboring surface atoms fo= double-bonded dhmozd. For C(100)2x dsrners with a bond length of about 1.37 A (4.40, 4.95j,whiek Lsvery close to the doublmbondlength of 1.34i in, e.g., the Cam molecale.The C=C double.boad is formed by the c- axd 's-like orbita,ls of the C dizner atozns
and Q = GaAs(1O0)2x4 (4.102j,
-
(see Fig.4.22).
Because of the strong bonding of the surface C atoms, the c' and c* states eAnnot modify the eledzonic structme arolmd the projectedAln-
:60
4. ReconstruciionElements
Fig. 4.24. Filled acd empty surface-state bandsof C(l00)2x1 in the fandamentalgap of the projected bulk baudstructure (shaded
area)ku DFT-LDA (4.7$.
dnmental gap. Only surface bands belonging to the %-- and x'*-like states may occuz iu the Atndltrnezztal gap as shown in Fig. 4.24. The dispersion i.e., of these bands is a consequenceof the dimex interaction along (0ï1q, along the dimer rows. Such an interaction V.Sof two neioboringdangli'ag qrbitalsis smallerth= Vkbecauseof the largerdistance.Then approvimately $4$0.37.1.is valid. According to (3.15) the corresponding Vk = 7.1.(QVVc0)2 tight-bindîr.g Hamiltonian is .
.#,r-,r(:)=
cd
+
ccost:aal'/..u
0 -ixtza!z&
aa +
eiEzfavr.k
(4.9)
acost:aalp.u
with @,z= co/W(O, lc 0)and dd = Qtl, 0, 0)and a new Cartesian coordinate and zI1 1ts diagonnlszation gives parallel onesystem zlIp11),:1IE011), (100q. ;:!z Vkl. They show dimeasion.albaads c..y.(i) = cd .k.2 cost/cvco/v/llvk Fig. 1.22) strong ctispersionalozgthe J'X-and J'15lines i.n the sttrfaceBZ (e,f. bbr Q > 0 (the 27.1.one observesquantitative agrecmentwith the enviromment) results of the DFT-LDA calculations iu F'1g.4,24,The indirect gap betxeen the #+ bar.d along #T a'adthe x band along 13.1 is dpBne' d by 2(lFk I 2JQ). The orbital character of the two band states is shown for the X point in Fig.4.25. Their bonflimgor aatibondingnata're is clearly visible. Symmetric dimez'shavenot only been obsezvedon dixmondtlool surfaces, smfaces of zinc-blendematerials. One exampie is thc Sibut also on (100) termiaated surfaceof cubic silicon carbide (SiC), whic,hbesidesthe c(4x2), The 3x2, and 5x2 reconstructions also shows a 2x1 reconstruction (4.105). bonds are partially ionic. TMs is related to a nhnrge tranqfer from dlrner Si atoms to C atorms in the secondatomic layer, which malcu dsmer buckling ltnlskely.J.uaddition, when %surfacedlmers form, axlgular forceson the -
4.3 Dimer:
161
*
ElKI
1
Ct-----â's p
1)0û1
)
:()11
.
q> :
$:1 Eol
Fig. 4.25. Sqlrnad wave fn'nctions for the occupied'r state and the empt,y zrv state at Ge poht K ir. lhp BZ. of the C(1Q0)2x 1 sudace (4.1041 . A pleme contnsm'ngt:e dime.r and the surfacenormal is used. The contour s'pacingis 0.05 bohr-Y.
secondrlayer C atoms
izwolved.They prevent
too strong atomic rearrangement. Moreover, khe lsttsce const=t of SiC is about 20%smaxer than that of silicon. Consmuently, the Si sutface-laye.r atoms move only slightly tovctrds each other (witha dlmer leng'th of yboutda = 2.73 i (4.95, 4.102) = ânsteadof the atozaic distaucq co/'/f 3.06 A on the ideal sudace). The di-staace of the two paired Si atoms is lazger thaa the typical Si-si bond length of abou.t2.35 i. Therefore,one rnnnot really speak about the forrnntion of Si sl:rêlme dMers in this case (4.95,4.1064. The polar (100) surfacesof III-V compoundsalso tmhibit a dlmer reconstzuction to hll6ll Peciple 1. For Hstance, As surfacedimers are impolta'n.t building bloclcsto e'xphln the various reconstractions of the GaAs(10O) smu face (cf.Fig. 2.17a). The surfacedimqrs shouldbe almost tmbuckledto avoid caharging esectsof the surfacein contrmst to Prixciple 2. One important example Lsthe 2 x4 reconstrudiop whic,h aksocontains the basic unit of the are
a
2
nj >
& >
P
Q) c
m0
Fig. 4.26. Surface-statebazlds(solid sarfaceverlinœlfor the GaAs(001)72(2x4) sus the projecwted bulk band stmcture (grayregfons). From DFT-LDA calculations (4.1O2j ttsing pseudopotentio wltîcètsomewhatopen tZe gap.
4. Reconstraction Elements
16t?
that the 72stractme c(2x8)reconstruction. Now, there is ageement (4.55)
2x4 reconstmction appen.nlngkmde,rintermediate As-ric,lzpreparatiou conditions (see Figs. 2.1.9a,and 2.20a). Under more AsZ'iC,h conditionstke c(4x4)structme is stabilized.For lessXs-ric,hconditiotus 2.17a a'ad 2.19a) :Lsstabilized. The top As dlmers the ('(4x2)structme (Figs. mplitude is not of the 72 structttre are @lrnostsymmetric, i.e., the bu The larger thn:n 0.02 .l. acd no twisting in the sudace plane occms (4.1û2j. Hirner bond length of Q = 2.50 i is ver.y closeto the bond dkstanceof 2.51 -t found for bllk- .&swhere tEe atozns are threefold coordinated E4.107q. The /2 structare ia Fig. 2.17a contnsnq two As dinners in each2x4 mkit cell of the top layer. One pair of Ga atoms is removed from the second atomic layer, aad aa additional hs dsrner is formed i'a the third layer. Tlle the electron cotmthg rule (4.4)derived for a missing As p2 structm.e 6111611n dlmer ia the toplayerin a somewEatgenerstlsaed sense. Origiaally,there azc 12 As danglingbonds with A = 24electronsand 4 Ga ctaaglingbon.dswitlz Jc= 24electrozls.The total 18 electrons allow the completeoccupation of the c, '7r, and x* leveltsof the As dimea's,whereasthe Ga dangling bondsremaln empty. ms shovn by the b=d stntctm'e in A semicozducting suzfaceresults (4.102)
exeplnsns the (Fig.2.17a)
-
C1
o
o
@
Q
Ct
e
.
t7
C2
.
o
o V2
V1
too (g;
y,-g.y'g. .
L-
=
'
. '
.Q' V4
73
% So Vè
>.
Q* Uca
(ï*
Fig. 4.277 Cmntoklrplots of the squred wave Glnclions at X for surface states of stmzcttae as indicated in 57.4.26.The ccmtom spacing is the GaAs(100)72(2x4) 10-3 bohr-3. A)l plots are drawn parallel to the surfacenormal. C1is plotted along C2, V2, and V4 are at a tomhyer d''me2. V1 and 5J3 the x4 dizection, i.e., (011). smbols iudicate As (Ga)atoms. are shown at the third-layer d''rnea..Filled (open)
4.3 Dime'rs
163
Fig.4.26. TEe smface-state gap is la'rgerthan the btllk gap hdicating a, gain of batd-structme enera due to recomqtruction. The two lowestempty surface states C1 aud C2 as well p:ksthe four highestS'II?'CI surfacestates V1, V2, V3, and V4 at the # poin.t are plotted irl Fig. 4.27. The highes'toccupiedstates V1 a'ad V2 are related to aatibonding rr' combinationsat the topmost aad third-layer As dl'mers. The states V3 aud V4 represOt bondiugcombinations. The lowestempty state C1 is related to Ga daagliagbonds.Eowever)the nex't one, C29is mnsnly built by antsbondisgc* combinations at top As dsrners. For the GaXs(100)((4x2) s'tzrface the b=d stradttre and the djscvion of the orbital characterof the surfacebacds cau be fotmd in Ref. (4.*54. 4.3.2 Jksylnlnetric Ilitaers
In. contrat to C(1.00), for Si(100) a large nttrnberof reconstrudionshnAbeea fouzd. The most impoztant ones are the c(4x2)a'ndp(2x2) reconstnzctions obsezvedat 1owtemperatmes, and the 2x 1 stzazdlzreat room tempezatme. The revezsiblec(4x2)-F 2x 1 pitasetrn.nKition takes place at a'rolmd 200 K. The Ge(100) surface shows siznilar behavior 24.28, The large,rrecon4.311. structions shoutd be possiblefor crystals vith covalent bonds weaker thau timse iu diamoad.J.nsuchsysorns the surfaceenergy may be further lowered by allowing the d'lmers to buckle o'at of the sT:rf'nce plaae (4.47j or to Mst the dime.razs in the surfaceplane (4.7.081. The c-haractezistlc parameters are zîb a'ad dk (see Fig. 4.28). The accompaayiugenergygain is larger than the correspoaclingloss due to the elastic energy of deformatloa. The driving force for suc,hadditional atomic rearrangements is rehted to Principle 3. For symmetric Si dinners the surfacomormatdangling-bond orbitals form a nearly degeneratepair of rr acd x* bands.A Jn.lnn-rfeller-like distortîon is expected to open a surfacegap between the 'r. and =. bam.ds Totat-energycalciations (4.95, indicate a bunldsngof the (4.47, 4.9,$. 4.1091 du
4àb
T :511
ElDf! -
E01 f
,
:1 f
Fig. 4.28. Top and.ssdeWew of an msymmetric tî''=e.r geometzy Cbnractedstic lengths suc,has the dimer bond lengtà Q: the bu#ingamplitude ïs parallel to and the twisting nmplitude ztk parallel to (011) are indicattd. (100),
164
4. ReconstructionPlements
Fig. 4.29. S'urfaceelectronic states itl the bttlk b=d gàp reglon of the Si and 1 stracture within the asymmetric dime.rmodel. The DFT-LDA bands Ge(100)2x are
taken from
(4.7$.
dirners but no twisting, i.e., lt = 0.'-1naccord=ce with the discuession in Scc't.3.5, one of the asmer atozns is pushed away from the surface,becorning moTe pyrnmz'dal(p3-1ilcel bondedwith a fdled s-lilce dangling bond Dup. The other dimer' atom is pushed inward into a nearly planaz (spz-like bonded) cooguration. The accompanying pa-lilce daagling bon.d Ddonmdonates its electron. The restzltîng dimer ha,sa weake,rbond with a:a iucreaseof the boad to Q = 2.25 k (asymmetric Si length from Q = 2.20i (symmetric Si di'rnea'l dimer).1ts veztical buckqlng is ztlla= 0.6 A, malcing the tilt angle nearly 169. The xsymmetric dimer geometryresults in an energy gsin of approxn'mately 0.12 ev per dlrner (4.10% F1:11re4.29clearly showsthe exmectedband-gap opening betweenthe surfacebandsbui)t mainly by Ddownor Dup orbitals. The eHstence of azymmetric dimezseasily e-xplainslazger reconstrttctioas snd p(2x2)as dlferent covgurations of left- aud right-tilted suc,has c(4x2) dime> (see . They give rise to small additional total-energy gainsof Fig.4.30) 0.05 ev per dimer hom knte-racwtions betweennaighboring dimers in a dsmer rOW an d 0.003ev per dimerâ'om iuteraetions of nearest dimersin Vo aeighIt turns out that the smface eneror is minimal boring dimer rows (4.1101. if tbe Jimer-tilt direction altnrnates along and is perpenclictzlarto the dime,r Tom. The resulting c(4x 2)reconstradion is indeedobservedat 1owtempem .
stures.
Despite ex-tensiveexpmrimentalaad theoretical bwestigatioa, the nature of the Si and Ge(10O) surfacesis still subjectto debate. of the reconstvction This holds in pazticular for the asymmetrjr of the Si dimers studied by me.ans of STM. In early STM expe-rimentsthe dirnezs appearedto be symhetric
4.3 Dimers
165
yX1
l'ïj j $' .y' y' p'
ê'
P(2x2) l 1 1 .1
1
d 1 1 1
.p
I I
I
11
I I I
1
1 1
I
1
I I I
1 1 1
Fig. 4-30- Surfaceunit cells of variotusreco>trudions of (100)sudaces.The formxtîon of dimers is indicated by horlzontal lines. TEe lazge empty and shaded c'irclesindicate the inequivalent tomlayer atoms, e.g., the Cup'and sdown' atoms i'a atorns. an aymmetric dimer. Dots representsecond-layer
Tlzisexperimentalfmdsng,however,is not necessmrllyilz coatradictiou (4.111j.
to msyrnmetric dsrners. Studies of the dynsxnl'csof the sttrface have shown that the dlrnez's may Sip betweeatheir opposite tûtiug diredions (4.112, At room temperabme, thia dimer Nipping happens on a t.l'me scaleof 4.1132. 10-10-10-Ss. Most meaemements do not havethe respective time resolution. At room temperature, most dimersappear symmetric due to tb.ei:rdplnrnlc flipping motion, ar.d only dlme,rs close to defects are plrned in a buckled The nlrrnbe,rof symmetric dz'mez'sdecreazesbelow 129K coMgaration E4.114).
aclddirrlersbuckle alternately witkin eac,hrow, witit the formation of p(2x2) rows inuidezztkalor opposite domains correspon8ing to adjacent âztdc(4x2) ozientations(4.114, 4.115J.
166
4. Reconstruction Elements
Thîs picture has recently been chatlenged. A seriesof low-temperature STM studies b.a'sbeenreported that, while the c(4x2)structme is obsezwed below 120 K, farthe,r cooling below 20 K causc the dimers to again appear tha't.this is a drmmical phenomenon symmetric. It llas been argtted (4.116) causedby a loweeg of the potemtial energy barzie,rbetxreenthe t'wo backled cougurations, which allows the dl'rnez'sto resllme the Eipdop motion czharacteristic of room temperatmz. Other authozz(4.117) claim that the observed symmetz'icdirners are static sincetheir imagesd.oaot exhibit the noise associated svitb.the Sipping motion obsezwed ita the same sample at 11OK. Howevcrl there is also evidencefor a sigocant imAuenceof tlxnnelingprocesses(4.118j. To cla'rpy the low-temperatme bu behaWor1om a theoretical poin.t of view, the subtle apects of electronic conelation near surfaceshave to be taken izzto consideratiom The syznmetric dimer with two g' dangling hybrids, eac,hfllled with one electroa, representsa biradical state. Therefore, static cozbrelation beyozd LDA and includicg gradient corrections, (perhaps, see Sect. 3.4.1) must be properly talcen into accolmt. The clime,r bu is accompeed by electron tzansfer.Thus, the net stabilit.y of the buckledconfguration depen.dson the devee to wilic,htite repulsiveCoulombinteraction (cf.Sect. 3.5)of the two electrons in the apper Si daazglingbonds is dynamically screened.. To accoaat for this dpmmical correlation esect, qttnantllm Monte Carlo studies have been perfo'rmed E4.119J. However, they qualitattveiy gavethe same favorizationof the bttcltled couguration as DFT-LDA or DFT-GGA calculatiou, '
'
4.3.3 Hetetodkmers
On the GaAs(001) surface b0th. Ga and As dsrners f.t geometrically to the underlyiag bulk. The covalen.tradii of Ga, roa = 1.26 i, and As, rxs = 1.20 i, are nearly the same (4.25). The situation is HiFerent for TnP(0û1) with atoms of dl'gerent size, nn = 1.44 1.ard re = 1.06 X An Jn-P mived Jlrner or heterodirne.rshould ft muc,hbette.r to the bulk. Consequently,an In-P e-xchangeshoald be energeticluly favorable. TMs ide.ahas been used by Scbmsdtet al. (4.120, to explain the 2x4 reconstruction occlzrrlng 4.1211 i'a a large range of In-zic,'itprepration conditions tsee Fio. 2.19 and 2.20). Heterodirnezsaksoappear to play a'n. important role i:a Imdeationa'adgrowth sttrfnce.s (4.122) on 1DSb(001) as well as for submonolaye,rgzowtll of Ge on
Si(OO1) :4.1231.
The suggestedmodel for the 1nP(001)2x4 recortraction is shown in Fig. 4.31. The sttdaceis ternninnted by a complete In layer. ln tlle ideal case, eac,h1'n.atom possessest'wo dangling.6p3hybl'ids along (111) and gïï1) with a 6115ng fraswtiom.al Je= QuThe corresponding2D square Bravaislattice is given by a1 = .g-o.a (1,-1, 0)and s,c = e.z (1,1, 0).A mived In-P dirner on a 2x4 cell adds eigb.tclectronsto the available12 electrons in the 16 Iu dazzgli'ag bonds aud forms four back bonds whic.h are occupied by.etht electrons,The top. layer In atoms fo= further four bonds with additional eight electrons. The .
4.3 Dimez's
I
167
(110q -'
'
Eig. 4.31.. Top and side view of t;e rnlxed-dsrne,rmodel of the TnP(001)2x4 suzface. Emqtytslledlcirole'srepresemtln (P)atoxns. Large (srnM1) smbols indicate positionsm tàe &st aad second(thizdand fourth) atomic layers.
dangliug bonds at the I'n atoms being fa'r away from the heterodimer remain empty. Two electrons occupy the c' bonding orbital of J.n and P states of the buckledmsveddimer. The remnsnn'mg two electronsSl thee-lilte dattgling orbital of the P atom in the topmost heterodlrner.One may' concludethat all reconstructionprinciples,Prvciple1, 2, and 3, aqreA116l1ed. Consequently, this is also tzme for the electron counting rtzle in a geueralizectsense. Total-emerg.yoptsmizations witlnl'n a Mt-principles appronnb (4.120,4.121) com6rm the aboveconsiderations.The bond length Q = 2.57 i of the mived fimer is close to, the sum of covalent radii 'ru + 'rp. TH dimer is buclded witiz an angleof j.7b.The ln atom is 0.43 i close,rto the substrate than the P atom. Ss=l'ln.rresults are obtained for the mlxed-clime.rreconstractioa of tïe GaP(OO1)2x4 suzface(4.124) The Ga-P dlmer is characterizedby a bond .
V1
V2
Fîg. 4.32. Contour plots of the squm'edwave functions at R for the two highœt occupied surface states of the mixed-dimer reconstrudion of ïnP(001)2x4. The plots are clrawn in (001)planes 0.8 â below and 0.8 .â.abovethe uppemnost P atom
respectively. (fzlled circle),
168
4. Recoustrudzio'a Elements
1nP
GaP
(2x4)MixedDimer
(2x4)MixedDimer ij
.'.
)!!
i7 J :î .jjj l jk tJ (l1 (' ï),i jJ
.
. p:r ;('E 'j:L... .... k E: .t,. yy. y,;;r ::5.: ):.; è'q:.. I.k
'
111121/ c:
.
ro
f8
v-
. .'.
L??(.j).: .;n::.!'ti. q'':i' ;,; u
; .L; 'i'!!.t .f :?..:.
&
.
1.66 nm
'L
..
lll,-')
',t
? Ct !.. ,j: ...
....
; ï .).éq ).,' ,':LjL.fj4j.LL.
.
:
,.,.,;j,,
;.. <'%<'% =
R ': ?.jjî .:.( .k j qq $!..L,,;ïi (). ;;k...@.,.,... . .
.
. .
1.53 nm
:
E1 10J
Fig. 4.33. Filled state STM t'mngescalculatedfor Ir.P(001)2x4 and GaP(001)2x4 in the mived-dsmer reconstruction using DFT-LDA (4.124J.
length Q = 2.36i ('rca + rp = 2.32 i) and a bucldsngangleof 9.50. The P atom is 0.39 i above the Ga atom. The resulting electroMcstractme is that of a, smmicondudingsurface The occupiedsurfacestates are below the mlencoband (Principle 3)(4.1202. maximlpnn in tlze b'Ilk. This fact infiicates-'-again of band-stmzcture energ.y due to reconstrudion. A variety of surface-state bands occurs iu the proG7ndn.mental jechted gap closeto the condaction-bandedges.Most interesting aze the two uppermost occupîedszzrface states at # in Fig. 4.32. They are O.1and 0.4 ev below the blllk- valence-bandmaxkrptm. One is relatedto the c-like bondsbetweenthe top In atom and the two cations below. The other one represents the dangling bond of tlte top P atom. The laterazpositions of the resalting three msvlrna in the electron density due to thœe states form an isosceles triangle, ia excelien.tagreementwith the corrugation measlzred by STM (4.121,4.124,4.125j. Theze is no needfor an asslTmptionof P trimers on top of the s'trfnne as ori to explain the STM images.The y suggested STM imagesobsezwed for negative bias ca,n be simulatedassllrnlng a mixed Hirner as demomqtratcd in Fig. 4.33.The single triangles observedconsist of a center struct'are (lhead') and two side stractmes (1e=7), the origin of whic.h is shovm in Fig.4.32. '
4.3.4 Bridging Groups
A nlnrnbe,rof stractm'al modelshas been suggestedfor tîe Gterrnsnntedpo: 1= sarface of cubic SiC, SiC(0O1) or actuazy SiC(0Oï) 4.105,4.1261. (4.95, Becauseof the strong C=C double bonds (see Sect, 4.3.1) oae mlty expect a dimer reconstruction, in particular to explaia the obsezwed c(2x2)trax)slational symmetzy Jt may be desczibedby a staggeredarraagemect of the
4.2 DMers
:69
Fig. 4.34. Top 'dew of the C-term''natedSiC(001)c(2x2) surface:(a)staggered circles 1ndimer geometzy;(b)staggemdbridog group arrangement. bqull(o?en) dicate C (Si)atoms. The squares of dotted lines zive tke surfacelnn:t cells.
C dn'rnez's(se!Fig. 4.34a). Despitetlle ver,y dîffe-renthttice constants of SiC = = 1.54 i) C dimers with a C dou1.88 A) aud dinmond tdeq (deq (4.1001, ble bond a'nda bond lezlgth of about Q = 1.35 .t$(4.95,4.126j can be eazily formed. However, becauseof the large SiC lattice constant a complete reacrangement of the top carbon layer is also possibleaz indicated i'a Fig. 4.34b. Instead of au azrangementof >C=C< dt'rners on squares of four s'ubsurfaceSi atoms, bridging Ca groups, -CHC-, in perpendictf.ardirection betweent'wo Si atoms are possible.The bond lengtil of thcsebridging C pairs is about 1.23 i. This idca is consistent with the obseawationof triple-bolded C atoms in organic chernsqtry.It seems to be in agreement with a LEED =alysis (4.127J The formation of the anomalous bridgeand spectroscopic EndA'ngs(4.1281. bonded dimers is 5.11 agreement with the recortstnuction Principle 1 and the electron counting rale. Becauseof the triple bonds, all surfacecazbonatoms are fourfold coordinated. The second-layerSi atoms are atsopaired in a direcFig. 4.34). Tkeir remnimingdangling tion parallel to the bridging voups (see bozzdsform a'a occupiedbondlng orbstal. From the point of view of the reconstradion-inducedenergy gnn'rlone cannot favorthe acetylenc-likebridge,s-CHC- versus the ethylene-like>C=C< dimers. The corzespondingvaluesare aearly the same kdepçndveatof the deHowever,there are tails of the ftrst-prindples calclllntiozus(4.95,4,1061 4.1264. drxmatic dsferences i'n the resiting surfaceelectronicstrucvtme. Sttrprisingly, the staggered dimers give rise to a nazrow-semiconductorbazd structtue with a.a e-xtremelysmall gap of about 0.033ev (4.126). The gap betweenthe surface-statebands in the projectedf'nndnmentalgap is remarkably opened for the triple-bonded Cc bridges abeady witl:dn the DFT-LDA. The hlnd.amental gap is n.lmosth'eefrom sudacestates. This fad is in good sgreement Mdth photoemissionmeasmements ac.d STM studies (4.128, as weX as 4.129) 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 and.(0û01) sar(111)surfaccsof zinc-blendematea'ialsas well a,s the (0û01) facesof wultzite stntdures (see Fig. 1.6)possessone dangliug bond parallel to the surfacenormal at eack surfaceatom. The same situation holds not only
'
for the (llll-sarfaces of cubic 3C-SiCbat also for the (0001) and (000ï) surfacesof the hexago:aat polytypes 411or 611of SiC. ln the polytype case only the bond stacldngiu deepersubsurfacebilayersis moeed (4.130q. The surface atoms are second-neazest neighbors9om the bulk point of view. Therefore, they wranldyinteract =.d. thus cau only be satmated aaer remarkable reazzangementsof the surfaceatomic geometry (see Sect.4.2.2). A minirnization of the number of dangling bonds (Peciple1) e>m be ïmbfeved easily, however, jtzst by adding a threefold coord-lnxted adatom, which co'ald be a,n atom chemlcatlyidentical with a blllk spedes or a subs'titutional atom. Sucll adatoms on (111)aud (0001) surfacesmay occupy two types of sites wlzie.hare illustrated i'a Fig.4.35. These geometries a're distinguished ms hollow (Ha)aud atop sites (T4)dependingoa whether the subs-trateatom bebw the adatom is fotmd i'a the fout'th or secoadatomic layer, respectively.i:a eacZof the geometriezthe adatom is in a threefoldt3latom bdow symmetric site. J.nu the T4 case the adatom a'adthe second-laye,r wemklyinteract what gives neady a foulfoldtzl) coordiuatioa.I'a the geometry Ss the adatom occmpiesa substitutional site in the secondatomic layer beneath a T4 adatom of a bulk constitaent. lt possessesfour neares't neighbors
Fig. 4.35. Adatoznson (111)surfaces of zinc-blendeor diamond czystals i'a Ta, Ha, aud Ss sites. Adatoms are shaded. Top (a)aud side (b)'dews are showa. The coorasnate systems are given in Fig. 4.l3a.
4.4 Adatoms attd Adclttstes's
but is also Huenced by a flfth (forthat reason, index 5)atom :in the T4 position. The illustration of the adsorption sites in F1g.4.35is not only valid 4.11and 611polytres one for (111)surfaœs.ln tke case of (tooly-oriented of the bondsizt the appermost two bilayezsas may expect tie same st shown in the fgure. The Ss geometry Ls only pbservablefor adatoms chnmle-q.llydiserent 9om the substrate atoms. Typical esxamplesfor the renalization of Ss are '
aad ozl recomstruc-tiozlson Si(111) surfaces (4.315 B-induced (WxW).R30O surfaces(4.131). J.nthe pme silicon case the incorSi-termlrnted SiC(111) reaction. The adsorption of poration may be consideredas a'a exclnn.nge other group-lll elements Ak Ga) a'ad Ia lcads also to the formatioa of stractmu. Shce they are tzivalent, eachgroup-/l atom sat(WxW)-R30Q tzrates three dangling boncls of the group.N atoms and, hence,leads to surface passimtiom The T4 site is favored.This pictme of adsozption also remains valid for constituent atoms. Total e'nergy studies of the gcoumW tmnKlational smsurfaceswitbl'n the (Wx W).R3OO adsorption oa 1V(111) show that the T4 adsorbatesite is mue-hfavored metzy (4.130, 4.132-4.136) over the Hz site. Despite usicg diserent calculational approaches,there is agxeementfor Si and Ge that adatom Vqorption in a T4 site is exothermk 1 smface.This is in contrmst to diamond with respect to the relaxed (111)1x smfaceson whic,hadatomsare clearly energetically llnfavorable. (111) T4 adatom geomeOn the other handj the Si and Ge(111) (W x W).R30O tries give zise to ê=gling-bolld-related surface bands itt the Gmdamental gap. They are ha'lf Ved. and pin the Ferms level near midgap. This fact violates Priaciple 3 and the electroncouting rule. Moreover, r.%adatoms on (111)2x2 Sect. 4.4.2). cells shottld be energeticallymore favorablefor Si a'ad.Ge (see However:there is a mystery about the widmband-gapsemiconductorSiC.The same type of,violations occm.s for the Si-termsnated SiC(111)(WxW)&0O
111 E1
(1-:21
E1 l'oq
0* surface (W3xW3)RV F'ig. 4.36. Perspective view o f the si-terrnsnatedskc(1:1) circles. with si T4 adatoms. si (c) atoms are uicated by open (f:1led)
172
> *
>
P *
m
4. Reconstruction Blements
2.5 2.0 j MwX 1.0 0-5 0-0 -0.5 -1.0 -1.6 F -.
**>..
'**
œ v
'
...
wwus
.%
> +
. *
*
.**+
..
l -
M
..-
K
-
F
Fig. 4.37. Ezectronicstructure of the 3G.SiC(11))(WxW)&02 sttrfacewitht'n DFT-LDA (4.139). The projected bulk band structtlre is shownaz the shadedregion. The dotted line representasthe Si danryling-bond-dedved band. surface taad%r the corresponding(0001) ones of the 411and 6H polytypes). Nevelheless,tEe L adatomgeometry shown iu Fig. 4.36 gives a stable reconstructiou for not too Si-ric.h preparation conditions (4.130, TMS 4.1364. result is iu agreement with othe.rtotal-enerq cakulations and e'xpezimental data meastlred for the (WxW).R3OQ recdnstruction of 3&SiC(1l1), 6Han.d4H-SiC(0001) s'urfaces(see also (4.95)). SiC(0001): The stkrfazeb=d structures of the Si(T4) adatom (Wx W)S30O geometries of the Si-tcrminated3C-SiC(111), and 4H-KC(00O1) 61FSiC(00û1) suzfacespossessa dangling-bond-relatedhalf-oed bar.d in the AJndamental One exampleis shown izl gsp, at least withim the DFT-EDA /4.137-4.139). F1g.4.37for the 3C,-SiC(l11) sudace.55='2a,rban.dstmzdares (W x WIA300 are obteed in the cmse of the hexagonalpolytype,s 611and 4H. Only the prolectedfttmdamentalgap is videned by about 1 eV. The theoretical results obtained witlnl'n DFT-LDA are in disaghree-memt with a comblation of ARPES (4.140) and KRIPES $.1414 inve-stigatiomqas well ms scnnml'ngt=nelîng spectroscopy (4.:421. For the hexagonal polytrpesthe expem'ments 'indicate b0th a'n empty aad a ftlled danglinpbond b=d separatedby a surface-state g-apof about 2 eV. The discreparcy is solveé by the mss:nrnm tion of a, Mott-Hubbazdinsulator golmd s'tate of the SiC(Wx surWIS3OQ faces (see Sed. 5.4.2) with a.c on-szte CoulombGteraction paramete,rU rz 2 ev (4.137, 4.139,4.1431. This is in agreementwith Peciple 3.
4.4:2 Adatoms AccompYed
by Rest Atoms
Although e,azhadatomreducesthe dangling-bonddensityVcording to Pri'n.ciple 1, tbe (WxW)J?,30D Tecoastruction of group-lvtllllsufaces csmmot G11611 Priaciple 3 i'a tlze original sense. However,this principle fxltn be obeyed
4.4 Adatomsand Adclcsters
(b)
Fig. 4.38. Adztoms in T4 (a)and Ha (b)sites formsng2x2 lattices on W(1:1) cir dots (second layer, adatom) surfaces.The atoms are kdicated by circles (fzrst cel'ls Adatoms are shaded,rest atoms aze hatched.Rectaurllar (Nmcagonal) layer). lizïes. are iadkated by thizzsolid (dotted)
considering a, 2x2 reconstlmcioruH tMs cmse one adatom occurs for eveayfot:r sarface atoms accomrnodating75%of the broken bonds =d, conseqaently, leaving aa adatom density correspondlng to .24of tha original sarface.The pm'rnl'tivecells have four times the azea of tke ttnrecozlstructedlx1 smface. There are two suc,hcoverings,one with hexagonalsyznmetl'y, the 2x2 stlrface, aud one with rectangular symmetry, the c(2x4)reconstractiom 80th cell types are hdicated iu Fig. 4.38. The hexagoaalcells an.dthe T4 adatom pocitions usuallyTveziseto a lower surfaceenero'. The essentialmechanisp is howeverindepeudeatof the cell shape. 1'atke 2x2 case of gaining enwetrgjr the adatom bonds to three of the surface atoms, leaving one atom with a and Ge(111) the adatom danglingbouduknovn as the rest.atom. For Sî(111) is displaced towaœdtlle btttlc, whereasthe rest atom tends to be pobonded and.,hence, is displacedaway from the bulk. Tlle dangling bond becomes TV rearraugement is acfor the adatom (restatom). more p-like (&1ikel companiedby an electrontr=sfer from the adatom to the rest atom forming Table d.2. Recoastmtction-inducedenera gaizzin eV/lx 1 cell with respect to the clearzrelaxed smface for the 2x1 x-bonded chain modek2x2 Hatom model and.the 717 dimer-adatom(hexagonal cell),the c(2x8)adatommodel (4.38,4.39): from a: izkitiodezusztyGlnctioual calculatioms.' stacking fault model (4.144,4.145) 7x7 c(2x8) -0.18 (4.76)-0.23 (4-76) C(l11) 0.80 (4.76) 0.83 (4.136)-0-10 (4.135) 0.27 (4-76) 0.30 (4-7$ Si(111) 0.24 (4.764 0.28 (4.13410.27 (4.1341 Ge(111)0.22 (4-761 0.20 (4.134J 0.26 (4-761 0.26 (4.76) 0.28 (4.74j 0-27 (4-74j 0.32 (4.742
Surface
2x 1
2x2
;'
4. Reconstraction Elements
lone pair in the daagiing-bondorbîtal of the rest atom. The resultiug sarface geometzy fnllfllls Pzinciples1 aud 3. A surface-state gap appears (4.134). The accompanying lowezingof the sl4rfnne smera comes in the range of values calclzlatedfor the x-bozded c'hn.in reconstrudioa (cf.Tab1e4.2). Only for dlamoadtllll are adatomslessfavorablebecauseof tlle stronger carboa bonds in the bulk. Consequently,the 'r-bonded e'hain reconstruction repreents the grouxd state of the C(111)2x 1 smface. Otaecaa gemerateother coveringsof the adatom density iu? suchas c(2x 8), by decorating larger pm'mz'tivecells. Suc,haa exmmplewitlz T4 adatoms is shownin Fig.4.39. The two pairs of adatomand rest atom pe,r'nn:'t cell oger new degreesof freedom, which lower the total energy. On Ge(111)c(2x8), ab iaitio calculationson the simple adatom model have indicated that the msymmetzy of protrusions fotmd in the STM srnn.gesis mxsnly e'atzsed by the a
.
:1'tlll
ptol
Fig. 4-39. Atoznic structtu'e of a group-IV(111)c(2x8) surface (topview).The simpleadatommodelLsrepresented.The atoms are indicated by cirdes t5.rst layer) or dots (second Adatoms are shaded while rest atoms Erst layer). (ontop) tin Layer) zlnlt are hatched. Possible2x8 and c(2x8) cells m'e demotedby dotted lines.
4.4 Adatoms and Adclustel's
175
Fig. 4.40. Sm'faceb=d structures versus high-symmetrydirec-tionsin the 2D BZ surfaces(simple adatom model) of C, Si, acd Ge. for reconstruded(111)c(2x8) The shadedareas represemt the projectedbullc band structures. From (4.76;. '
The bualdlng bu between the t'wo rest atoms in the llznit cell (4.39j. makes a 2x2 subnlnstmore electron-ric,hthn.n tEe corresponds'ngc(2x4)subltmt't ia the c(2x8)recozlstruction. Thks stabiGcs the c(2x8)trauslational symmetzy ove'r the 2x2 a'ad c(2x 4)suzfacesaccordingto Principle 2. Tlle bunkllng of the rest atoms alsohappensi:a the c(2x8) orderiugon the Si(111) surface (4.76, The chn.racteristic energy gn.s'nsin Table 4.2 clearly in4.146). stabilization of relaxed Si(111) and Ge(111) surfaces dicate an (even bette'r) chalns. Tllis is izl by adatoms in comparison wxiththe formation of zr-bonded dear contrut to dixrnondtl t1) A'n idea about the energy gytlns due to a, c(2x8)adatom reconstnzction surface violates Prhciple 3. In is indicated in Fig. 4.40. The C(111)c(2x8) However, contrast, surface-stategaps al-e openedfor Si a'nd Ge(111)c(2x8) in the Ge càsethe occupied surfacestates are much lower i'a enerr tha,n Therefore,for Gemore baad-stmzctureenergy is the valence-bandmnv'=:lm. gainedthan i.n the Si cmse. More precisely,the adatom-inducedelastic energy '
.
.
(a) J'
(111)
1- (11:
.
f'/e
e
F.
.a o
..
z
(b)
s:'s .
G
.
.
.
qsl .
.
4 CED .
.
:
0.
co .
.
.
Fig. 4-41. Contpmplots of surface s'tates of tke adatom-coveredGe(111)c(2x8) plane contaMng a.u adatom aud a rest atPz-ty sttrfacein tke (110) (a)state of tize lowest empty sarfaceband (Fig. 4.40)at a wave vector on thq CYline; (b)state of in the higllest ocompiedsurface band. at a wave vector on the rF/ lice (Fiz.4.40) the smfaceBZ. The distanceof the Ge atoms to the piane of the adatom and rest atom is mclicated by the varying size of the dots. Fcom (4.78J.
176
4. ReconstrudionElements
is obviouslyovercompensatedmore strongly by lowerirtg the band-stnzcture of the densityof daaglingbords'in the Ge case. eaergydue to tEe redvuctioxz The orbitals beloagingto the lowe-stempt.g smface bacd and Mghest occupied surfaceban.dare plotted iu Fig. 4.41 for the Ge surface. Indeed, as discussedabovefor the 2x2 adatom situation, the wave ftlnction of the high.est occupiedband is mnr'nly localizedat a rest atom (Fig.4.4lb).'The snme holdsfor the localizationat the adatom of the wave Atmctionbelongîngto the lowœt empt.y surfacebaud (Fîg. However, there are also contdbutîons 4.41a). from the neighboring atoms. 4.4.3 Adatomm Combined with Other Reconstmzction Elements
The energy gains in Table4.2 showthat the grotmd state of the Sî(111) surface corresponds to a huge 7x7 reconstmzction. After long and controversial disputesit is nowadaysexplainedby a dsrneoadatom-stacldng fault (DAS) model with corner holes (4.144, as representedizt Fig.4.42. Each 7x 7 4.145) unit mesh contn.ins (i) a stanlrimgfault in one of its triangular subunits, (ii) a corner hole correspondingto one mlrwqiagatom i'a the secondatomic layer and, hence, leaving a dvgling bond at its center atom ill-the third atomic laye'r,(iii)aine dimersforrnsngdomainwalls abng the botmdar.gof one of its t'wo triaugular sabunits: (iv)12 adatoms in T4 sites in a 2x2-1ike environ.ment, and (v)six rest atoms in the ftrst atomic laycr the danglirlg bonds of wlkic,hare not saturated by bondingto the adatoms. Consequently, 42 atozns remah in the flrst atomic layea',among them the rest atoms, 48 atoms are ia the layer beneath,a'ad 12 adatomsi.n the top layer decorate a surfacetmit cell. Tlle DAS model also follows Pzinciple !. The 7x 7 surfacemixtimizesits dangling-bond densit.gby the formation of dimer-row dornnsn.watls whiczh are euergedcallyfavorablebecauseof the relatively 1owenergiesof stackinp fault au.dcorner-holeformation. The decorationwith 12 adatoms passimtes 36 d=gling bonds in the ftrst atomic laye'r. 12 dangling bonds remain at the adatoms,sbc more at the rest atoms, and one at the center atom of the corner hole. However; furthe,r dangling-bond saturation happens via the mechaaismdisccsedin Sect.3.5 in accordancewith bunkling/charge-trxnsfcr Priuciple &. Danglittg bonds locsll-zedat rest atoms acd cqrner-holeatoms corssidera' b1ycontzibute to occupiedsudacebandsjus'tbelow the Ferml level. The contributioa of daugling bon.dssîtuated at adatoms near corner holes (CoF,CoU)or the center of the llnz't mesh (CeF,CeU)are mu' c,lzsmaller, independent of the occurrence in the fautted (F) or nlnfaulted (U)triaugle The empty surfacebands mns'nly adse fzom daugling bonds of (4.78, 4.147). adatoms. Thîs situatioû is indicated in Fig.4.43. In. gece-'ral the loe>lszation of the surfacestates is lesscleaz.Partîal minring of dangling bondsmay occur. Due to the reducedCsv point-groap symmetry the adatoms ar.d zest atoms are no longer equivalent. They HiFer with respec'tto the position, Co or Ce, in a triaugular subllnst ard tlze occurrenvce of the stnrxing failt (4.148) (see '
4.4 Adatomsand Adclusters
*
.
* *
* . '
(a)
@
* *
@ '
*
*
@.
*
-
110
*
.
.
. *
@
.
*
*.
*
.
*
*
..
*
R*
*
*
G
* *'
R
*
e
@'
.
'
.
*
*
é
*
Q
.
R
..
@
.
*
R
. ..
*' @ @
..
*
*
*
* *
q-2y (lq
CeF CeU
CoF
(b)
CoU R
R
1f q1
1'
faulted
>
unfaulted
fault Fig. 4.42. Top view (a)and sideplane vievr (b)of the dimer-nrlntom-stacldng smface. J.nthe top view (a)the shaded circles desip (DAS)model of a Si(ll1)7x7 nate the adatoms.The circ'les'with a letter R designztetite rest atoms. Large open cîrcles designatetriply Bonded atoms izzthe ftrst atomic layer below the adatoms, whereas small open 'mrcles designatefourfold coordfnatedatoms in tke lower part of the same bûayer.The dots designateatoms in the third and fourth atomic layet's beneaththe adatoms. The lower panel (b)correspondsto a plaze view o: the nearest-neighborbonding in a plxne normal to t:e s'arfacecoat 'aningtke long dîagonalof the sttrfaceunit cen. A possiblesmfaceunit cell can be describidby 'lznes connecting the atoms with a dangling bond in the centers of the fottr (notshown) corner
holes. Afte'r (4.14z1J.
Fig. 4.42b) However,there are excellentexpem'mentalstudiH using H''gerent witiclk give information about the methods, e.g., STM (4.1491 or PES (4.:501, dangling-bond-relstedstates. For tlznnelklg out of rnn.ny occupied smface states the STM image (see Fig. 4.44)showstwelve potrustons ia the smface lTmstcell which caxl bc traced bnr,'t-to the adatoms.The corner holes aad the regions betweendimers ixï the domin walls aze also clearly visible in Fig.4.44. but alsoARPES (see, Scanningtl:nneling spectröscopy (4.14% e.g., (4.312) found a baxld whic,h extends up to the Ferrnl' level and pins it at 0.7 ev at abovethe top of the valencebacds. The correspondsr,gstate's are loem7szed adatoms. Naively, the Si(111)7x 7 strttdm'e may not seem to satisfy the re.
.
1'J8
4. Reconstruction Blements '
.
-
y
(a)
k
.
.
.
..
.
QO :'
'
.
.
:@ - . ..
<
.:
.
.
u
.
.
..
.
.
.
u
-
' .
.
'
'
J5,.. *() .. i e @ @@*@@*@@**@*@@
:
-
.
.
-
.
ô
*
:
l
'
.
-
..
.
.
.
Q
œ@e*e*e*@*@*@@ *@***@****@**œ
@ p
@)
ï
>
.
*
e
.
.
W -*
.
*
.
.
ê
*,
@ . e :@ * * *)
..
*;
*
*
-
.
.
*
.
l
.
@
.
A Vœ * @ * '
,
4
:
.
@
.
@
->%
*
*
e
. . ***@******@e*@
*
.
,
:
.
.@.*.*.*@@.*@* @*@****@@*@*@*
@ p
(c)
*
.
@
*
i .
o
.
l .@
.
.
.
l .,
.
@ .
@ .*
>
.
.
@
.
:*
# m
* .*. .. pe
e@@@@@w @*@*@*@
B1.)1
e
.
'.
.
.
*:
.
.
,
*
:
' > % F *.
e
*
.
.
@
.
.* , @. .
@ *
o
*
.. .z% .* .% .. o. *@*@e@ @@@@*
E11X)
surface. T:e Fig- 4-43- Contour plots of the sttrface states Pf the Si(111)7x7 planefor tlzreesurface bands squaresof wave hlnctiomsm'e representedin the (110) near the Fermi level (for detKs see Fig. l in (4.76j). (a)Peially FLIIGIsudace band in the gap, (b)au.d (c)ocyupiedsmfacebandsjust below or above the 'bullc plxne contxl'nq the long diagonal of F1g.4.42but valence-bandmaximl:rn. The (110) is ceuteredon a corne.r bole. Fkom (4.781.
construction pzinciples. The seemsnglymetxllic c'hn.racter indicates residual tmsatisâedvalenciesat the surface,ia contradiction to the spirit of Principles 1 and 3. One possibility to solve the puzzle could be related to the a-s obsenred in the cmAe of occuzzence of strong electron correlation (4.152J A scp.lr'ngof and 3x3 smfaces(4.137,4.139) SiC(111)/(0001)WxW 4.:431. vith the amo'ant of Si iu the surthe Hubbard pamrneter & (cf.Sect.3.5.1) facere#onmay indicate a value of about a tenth of an eV. The accompanyin.g surface-stategap shotzldbe llardly nmmzqtzrable (seealso Sect. 5.4.2). The DAS model described for the 7x7 trnanslationalsymmetzy can be The generalizedto (2n+ 1)x (2n+ 1)reconstractionswith n = 1, 2: 3, 4, corresponaing pnnc'tcezscontns'n (27z+ 1)21x1 surfacellmit cellsand are decorated by zzlzz + 1) adatoms leaving 2a rest atoms. T*e resulting adatom '
....
4.4 Adatoms and Adclusters
179
stzm Fig. 4.o4d;.STM image of a Si(ll1)7x7 nm2) sxrnplebims; face (image ex-tentt 12x12 with -3V). R-om (4.151) (copyrig'ht (2003), permsssionfrom Elsevier).
densit'y is reduced by 14 ,) with respect to the derusit'y.14in a c(2x8) structuxe. The domaia wxllg consisting of dimer rom with 3%Jsmers f:.ll 2/, the sarfaceLsfaalted. A rel1x1 tmit cells,the corner hole 6111one, and %M,16 atively simple attempt to tmdezstandthe variotzsstructures is that suggested by Vanderbilt (4.1531. For Si this author esttmatedthe euergy costs per 4 x 1 nlnst cell with rupec't to thearelaxedsurfaceof the s'tnztlringfault, A.f = 0.06 eV, of the domain wall, zl'tzl= -0.655 eV, and of the corne,r hole lc = 1.40 eV. The enerr of the formation of a'a adatom is a = 0.28 eV (cf.also Table 4.2).Thtus,the total energy gaiu per 1x1 smface region is
zo)
AE :2n+1)x(2a+1)=
1
'j/.f
+
2n,11/2+ mc + ntx+ (2a+ 1)c
1)c .
witiin the simple adatom model fotmd for Ge(111)c(2x8) the energy gain nmounts to
ïfctcxs)
=
1
zc.
allowsthe eonstrttcand (4.11) The comparison of the two energie,s(4.10) tioa of phasediagrams,e.g., 'usingthe relative star'zng-fatzlt formation enerr lj.fj6lj'trLaud the relative corner-holefommyttionenergy Ic/lzl'trlms indepenadatom dent variables.Suc,ha phasediivam is shou in F1g.4.45for a fLXe'd eaezgy.lt exbi'bits a series of DAS stractmes tf I.J is s=n.ll, which have iIs creaskzg(2zz + 1) periodicity as lc increases.At lazge.rvalues of ztu, the stackizlgfault is n'rfn.vorable,and tiere ksa trxnsition to azl orderedadatom stracture, notioaally the c(2x8)onc. .Xn inczeaseof the adatom formation recoastmzctions, e.g., 5x5, more farenergy malcesthe lower (2a+1)x (2zz+1) phmsewith the parameters vorable. The star indicates the stableSi(111)7x7 givea in the text. If the cost of fommingthe stanlrlng fatfts and corner holes ts too high, simple dangting-bond-removing adatom structttres are formed, like the c(2x8)stracture of Ge(1l1) shown in F#g.4.39.lndeed; the formation of stacking fautts cos'tsznore energy in Ge thxn ia Si (4.154). ,A.sa result
18û
4. RecoustrudionElements
0.6
O.5 0.4 ,..SR>
c(a8)
.t1 0.3
Y
.S 0.2 (/)
.%, 3x3 0
0
5x5
1
7x7
9x9
2 3 Cornerhole Ac /IAwl
4
'Xg- 4-45. Phmse(liagram of a (111) surfacewith c(2x8)recorbstzudion(sknple an.d (2zJ adatom model) + 1) x (2r;+ 1) DAS reconstructiozls.The reduced formation energiesof staddng faltlts =d corner holes are taken as variables. The = reducedadatom formation energy is fxed at c/1zl01 0.427. The star indicates n. the Si(11l)7x7 surfacewith the prameters listed in the text.
c(2x8)yeconstruction is more likely tlnn.n for Si, thougk the enera is close to that foy 7x7 (cf.Table4.2) However, the c(2x8)reconstruction can also be obsezared for quemchedSi(111) surfaces(4.146, On the othe,r 4.155). the
.
hand, strnsmedGe(111) overlayersgrowa on Si(111) substrate also show a 7x7 reconstruction (4.156). 80th fac'tsindicatethe outstandimg role of surface stress for the forrnntion of the lonprange reconstructions 7x; and c(2x8)ms disclnrvqed in Sect. 2.2.3. 4.4.4 Trimers
In. order to accotmt for the 12 protrusions
seen
in STM images of the
suface, a few of more complicatedadatom modeks'have been Si(111)7x7 proposed (see the pyraamong the,m tEe milk-stool model (4.1584, (4.157q), midat-cluste,rmodel g4.150), a'c.dthe trimer model (4.160', However, 4.1611. suzfacedismlssedthese models.'T%'me,rs subsequentstudiesof the Sî(111)7x7 on
group-W surfacesagain extered the discussionafter the obselwationof
(WxW)R%Oreconstructions on diamondtllllsurfaces of crystallites in CvD-growa 61mq (4.991. Oneremson was that STM imageshaveshownthat 2x l domnsnqcoexist witE (WxW)A30O strudmes. For (WxW)S30ô reconstructions of the single dangling-boud (SDB) sur'
face, there eldstsno model that allows the half-ftlled dangllngboads to become nearest neighbou. One conceivableway is to reduze the density of the danglingbonds by adsorption of one addition.al carbon atom.pe,rthzeesurface
4.4 Adatomsand Adclustezs atoms ia a Ha or Tz site, still leaving one dangling bond per
(WxW)A3OO
tlnst ce2 tsee Sed. 4.4.1). Suchadatomsare howeverenergeticazytlnfavorable (4.135). Iit contrast to the SDB surfacej the triple dangling-bond (TDB) smface to reconstruct by the suface provides a nattzral way for the (111) l'nst cells without any additional adatom. The formation of (Wx W)S300 reconstmction obset'vedia STM foT dixmoacl curstxl7stesis (WxW3)A30b suggeted to consist of trimer structures that are centered at a hollowll-llsite position on the TDB mtrface. Such a stnzcture is showain 1Rg.4.46. The hollow-site position is characterizedby the absenceof atoms uitderneath the trirner ic the secondsubstratelayer. The atoms fomning the trsmer are bonded with one bond to one earbonatom ia the top substratelayer. The other three dangling bonds per atom pardcipate in a 'very s'trong bonding withp'nthe trjmcr. lt forms an isoscelestriangle witk eqlllllbrip:nnbond lengtk 1.39 .â.of the tvo equally spaced bonds. The elongated bond length equals TEe distortions 1.52i. The angleat the vertex of the triangle is 67O(4.77j. The odd apnmberof dangling bonds on the substrate are ex-treme'lyMmM,11. suggestsa metallic charader of the stuface i.c contrndiction to Priadple 3. Neveztheless) tEe surfaceenera is lowe,rthan that of the Seiwatzclzainstsee Three smfacebands occttr i:a the projectedGlndsrneatalgap. Sed. 4.2.3). The wave llnction of the b=d pinning tbe Ferrn'' level i'a a midgap positiol consists of dangling bondsat the three t='=e'r atoms. Til.ts b=d is very sat, kence,perhapsGdicating the possibitit'y of a Mott-Hubbard metal-insutator transition (4.136,4.138, also Sed. 5.4.2). (see 4.142) Q
'f01 E'I
l .
-
4.46. Top view of a hollow-site trimer reconstruction of the C(l1l) triple dauglirzg-bondsurface. Carbon atoms in the uppermost sur(WxW)A30O facehyer are iuclicatedas large open drclœ, whereasdots and small circles descdbe atoinsin the two layers beneath. Fig.
4. ReconstrudâonEle-ments
182
Other exnmplesfor trimer formation cotkldbe polar suzfacesof compound safaces of 111on (111) semkoaductors. The formation of vacanc'iœ (4,45) V semiconductorstseeFig. 4.6)seenls not to occur at the As-teminated surface,of'tenreferredto as Gau&stlïïl. Whtle the formation of GaAs(111) is exothermic, the formation of As vacaccies Ga vacandes on GaAs(1I1) Unll'ke the situation for Gn.Aq(111), is endothermic(4.1621 on Gn.AR(ïîï) xs WxW3, 2x2, for (>.A.s(1I1) a vaziety of dîferent reconstractions, s'ac-h 3x3 and Ui-fxx/i-f,occuz's depemdingupon the procerssingconditions (see The 2 x2 strucwtm-eis generally believedto be the As trimer strucvture (4.31)). illtustrated in Fig. 4.47. Most of the evidencefor this reconstntction model Iu particlnlltr, the triangular potrusions is obtnsmedby STM (4.163, 4.164). surfacecaa be explainedby seen tu the STM imagesfrom the GauYs(ïîï)2x2 the chernhorptiozz of A.s tz'imezs on the As-terml'nated stvface. Mnn.nwhile, the trimer model has receivedfurthe.t support h'om a trausmisstoneledron micoscopy kwestjgation of Iusblïlïlzxz (4.164. Witbi'n the tzimer mode.lan n.nsoa-terrnl'nated(ïïï) suçfaceis decorated by a'a n.n5oaoverlayercorrespondingto a e = ) coverage.THee anions withâa .
/ J
r
/
?
X,
z'
'>s s 's.,
/
,
gjj.y
N
/
% <.
d'. ..'
'y.
J'
.C
%k
N'%.
/ z
K..
z
y
z'
vn
..*
%, zz'' A
Ayz
reconstruction within for tke H1-V(!II)2x2 Fig. 4.47. Structural model (top'view) the Ant'on-trirner model and mss:tmlngthe T4 site for the trimer position. Dotst n.nn'ons in second-laye,r catsoxxs, opemcircles: Gmt-layera'aions, ar.d hatchedcircles: the top t '.nrn ers.
*
4.4 Adatoms and Adclttstezs
183
2 x2 uait cell (see Fig. 4.47) form a trimer that saturates tkee dangling bonds but leaving a rest atom with one dangling bond.The trsmer can occupy a 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 pziaciples of stamscozductor reconstrucvtionand thereby the electron cotmthg rule. Six out of the- 15 valenceeledrons 9om the anion tm'rne,r aze tzsedto bond the '' trsmer atoms together and anoth.er sh aze IZSCSCI to ftll the thzee dangli'ng orbitals of the anions forming tke trimer. The remnsningthree electronsFom the trime,r are transferred to ftll the dangli'ng bonds of the fou.r topmost substrate arkions.ê 4 of a,a electron is transferredto form a loae pair in the dangling bon.dof the res't atom. The othe,r3x .?4eledrons complet,ethe three azd substrate.Sincethe daagling bonds of the rest bonds betweent'Z'IEn.e,T atom aad the tlzreetrimer atoms a're occupiedwith lone pnsrq, the smface a
is passimted.It becomessemiconducting(PrGciple 3).The BmsicPedple in Sed. 4.1.4 inclicatesthat the consideredrecoastruction must be accessible in an As-rich enviroaaynt,since the trimer is preided to be a minsmum free-eaezgystrudure izz such a'a environment (4.163, Jndeed,thks is 4.1661. conqvrnedexxpsrirnenully (4.163, 4.1641. 4.4.5 Qzetrnrners
Besidesdimersor trimers JtTqnlarger clusters of adatomsmay occur on surfaces.SuG cluste,rcoHgurations could be tetramers to e'xplainSiC(111)3 x 3 or SiC(0001)3x3 surfacestrudures (4.136, or reconstntdion elemeats 4.167) on a Si(110)16x2 sttrface(4.168). Larger reconstrudion elementssachas peù' tagons 'or evem hesxeonsmxy nbnmaztarizethe 3x1/3x2recomenlctions of lkigh-index snrfxtxas of Si axd Ge, such as (113)(4.16% or also the 4.1704, sutface (4.168, Under very Si-rich preparation condltiorts Si(110)l6x2 4.1711. the (111) surface of 3C-SiCas well aci the (0001) smface of 4H-SiC or 6HSiC twhibit a 3x3 recoztstruction(4.167). Despite their slmllarities with the Si(11!)suzfacea 3x3 DAS model contaiuing two adatomspe,r unit ce2with dxe-remtstack'ingorientation fn51R, lince only a siugle protrmsionis fotud e.xpfm'rnemGlly (4.172) Even a variant of the DAS model contain.ingone nzsntom cluster on a silicon adlayer is not in agre=ent with total-enerU mxMlms'zatioms (4.l3s) and LEED fndiugs (4.167J. A completelynovel structure expln.s'nstite 3x3 reconstructedSi-terrnsoated sarfaces.This reconstruction includtxsa twisted Si adlayez SiC(111)/(0001) abovea bulk Si-terrnlnatedSiCsubstrate with a Si tetramer adclttsteron top. With e = 13/9the Si coverageis large,r thaa ona monolayer. No stlmsrlng fattlts, Ht'mea'sand corner holesappear. The centralelementof the reconstruction ks a Si tetramer on top consisthg of a Si tm'mer aad a'a additional Si adatomj as shou în Fig.4.48. Titis stmzcttzrewidely satisûesthe requirement of a fourfold coordlnxtion or of dangling-bondsaturation (Principle 1).The adatom cluster (tetramer) saturates 'nîne of 27 dangliug bonds of the adlaym'. The Temnsnsngdangling -
.
'
184
< Reconstmzdson Elements 1:(I E1
(a)
(b)
E1 -111 13-ljl = .
.
j'
''
a .-
.'
.
-
'
t'
r> ) Xm .fl#t ' .tc&'J '.i .. '.P .zz Q....%6'!
#.'
'
' -
w
'*'
'#
'
>' Q-' .
'
''
-.w
:
L
' -
.
iA' ' ''MW -'
TJ
F/g. 4.48. Perspective view (a) and top <ew (b) of a Si-rich Si-terminated surfacewithin the tetramer-adlayermodel. Dots desigzza'te C atoms. SiC(111)3x3 circ'les designate Si atoms in the subMrate (tom Large open tfzlled, katched) most substrate layer, adlayer). The larger hatcked circies indicate Si atoms izï the tetrnrnez.
hybrids fo=
a
threefold coordination within the Vlayer. There is a tendency
for a reybridization to spl. and p orbitals for one'part of the nrllayer atoms'. Theiz .%2 kybrids form bonds witlnl'n the ftrst nrllny.er? aud the rernnsns'ng p orbital of, a,n adlaye,ratom formq a bond to the substrate. The adlaye.r atoms possessiug oae bond to the trimer basis of the adcluster evhfbit a tendency toward t'wo sp and two p orbitals. The adatom itself appears to be lmhybridizedand bondedby p orbita!s as judgedfrom its bond a'aglesof 90ö. Jn efect, this allows azl energeticallyfavorablef-like drgling bond or orbital. The fm'eence of this half-flled daugliug bond seeaus to contradid b0th Principle 1 and Principle 3. However,it haeeen shownthat the more Si-ric,h 3x7 reconstnlction forms a Mott-Hubbard insulator grou'nd state as the less Si-rich (xCxVi).R30B rcsconstructionof the SiC(111)/(0O01) surfaces(4.13% Scct. .5.4.$. Only the Hubbard Mteradioa parameter U a 1 ev is mue,h (see srnnller becattseof the larger mount of silicon on the surface.The fad that the Si-rich 3x3 reconstrucvtionsatisûesthe Basic Priuciple is iudicated ia accordlng to the Fig, 4.49. This pbn.qediagram b.asbeen constmzcted (4.13$ rules describeê in Sect. 2.5.4. The tetrnmer-adlayer surface is predicted to be the rnsnlmllm gee-energy structme ttnder Si-ric,hpreparatioa conditions. This is con6rmedexperimeatally since this stntct'are is accœsiblein a Si-rich enviromment (4.136,4.167,4.1721.
4.4 Adatoms and Adclusters Nx
.
x
0.0
N
5X1 clean
Nh z'.y x S
x.
X 1./N.
9
& o
-
hhx xwx
o
m
x
v
.-'
1*
-'B' -().:$ *
cl
'r41x:j
(Cz a)
x
z) * r œ o t: c
.
N *%î% x N. W.x wx N'4 p*%
-
öi
x%
,.c mz.: Jz>
x
,9-: .4 :.>. C'vo > .o
0. 6
h.x
x.
.
K
x hx a+qp%'h. +lzh-4w ''N. ..>
PTQlr>x
%o
-0.8
,dho
A*A. '%#' x. -w
tp x h. x 'h z o x 4w. N %. >g%% N h% Nhv -Nx
=
m
+u> x
-
(i- 2
185
Q'o z
1
I
C-ricb
pbsf P.c '
Sl-rich
Fig. 4.49. Phasediavam of a Si-termlnated mzrfacems a function of the SiC(111) càemicalpotential of the Si or C atolns. Relevantrecozzstrudionmodelshave been seiected.The enerr of the rathe,rdisordered 1x1 struct'are hmsbeenlowered by azz emtropy term assuznirg Sa= 0-15 ks aud T = 1000 K, in orde.r to explain tllc sequenceof pllasetransîtions Wx W -> 1 x 1 -F 3 x 3 with increasir.gSi coverage. From (4.136) .
5. Elementary Excitations 1: Single Electronic Quasiparticles
5.1 Electrons
and Holes
5.1.1 Excitation
and
QllnqiparticleChnracter
Surfacesare many-body syztems consisting of interacting
and electrons 3.3.1).In order to descrîbe rnxny propertiesj in pardctzlar, vound(Sec't. state propez-tie, it is stnlcient to l'eplazethe system of interac'ting electrons by a system of independentparticles (3.2). One exn.mpleis the density Gpnc.tional theor.y (Sect. 3.4.1)witbs'n the local approvsmatioh for the exchange aud correlation contributioa to the total eaergy (3.50). Using the KoMSham equation (3.46), the grotmd state of the electronic subsystem can be particles, more describedby Mdipendent(i.e.,eiectively non-int.eractiag) An stridly eledrons moving in an eobcwtive single-particle poteltiat (3.48). Cindependent'electron i'a a Kohn-shnm state possessesa âxedsingle-particle this eledzonin space. enero- and a defmedprobabilit'y distributioa of 0c111)g Eowever,exdtations of ac electronic system cnmnot correcvtlybe described by the independentKohn-sbam particles in (3.46). A 1ot of experbrnentalstudies are associated with spectroscopies a'ad, A.n elqdron may be added therefore:evitations of the eledronic (sublsystem. to the system or an electronmn.ybe taken away from the s'ystemand.,hence, a h,oleis created.The e'xcitedelectron or the kole strongly Mteracts with the many other electrons of the system. The electronic subestem is polarized and reacts with a redistribution of the e'lecvtroa deasity. Consequently, the enerr of such an electronic exdtation will diler from those for nozsiuteracting particles. lt is renormmlszedwith respect to energy azd to behavior i'a time: distribution. If azt excitation hnuqa, stnlciently long lifei.e., to the specwtral time, it howeverbehaveslike a particle. There-fore, it is calleda cutkdpc,ràïcle, moze strictly a quasielecvtron or quasiholedependingon the occupation of the corresponalng single-particle state before excitation. The properties of the cores
quasiparticleare better describedby a spativy non-localspectralt-weight) hlndion thau by a'a eigenenezgyand a wave hlnction., Suc,h quaaipazticlesare adually observablein severalsurface-sensitive spectroscopies.
188
5. BlementaryBxdtations 1: SingleElectronicQttasiparticles
5.1.2 Scamling Tanneling Spectroscopy
The Xst scn.nningt'Tnnelizzgmicroscope was built in 1982 by Binnn'g aad Rohrer (5.1!. The physical phenomenonat the ol'ignof this new inldrralment is tEe brnneting of electrousthrough the vacuplm. J.nsuch a microscope a shar.pmetallie tip is positioned at a distauce d (ofthe order of a few i) h'om the suzfaceof a conduding smple t'Fig.5.1).l'a this way there is au overlap bet'weeathe electronic wave Rmctions of the tip and substrate. A voltage F' is applied to the t'wo electrodesresulting in a tllmne.lcurren.t IT. This can occur 1om the metal tîp to the smface or vice versa, dependiug on the dkedion of the bias.Structaral icformation can be obtainedby sc-nmning, i.e., by moving the tip over the lmrface, e.g., witlu'n conqunvcurreat mode. by vazying the The condition of a constant tllnmel ctzrrent I.c ca'a be A116E.ed dkstanced betwee,nthe tip and the sample. The rœulthlg corrugation hlncoon contnlns information about the sarfacetopography.A topograpidc5Vage of the tke suzfaceis also obtained wit'hT'nconstant-heightmode, by memmnvlng zaartudeof the ttxpneling m3rrentas tke tip is movedacross the st-nce at a microscopy (STM) Lsusaally performed fzxeddistanced. Scxnnlz.gt:tmmfRll'ng ic the constant-current operation mode. More information aboat the eledronic s'tructare of the surface can be obtaiued by studying the dependen- of the STM signat on the siga and magnitudeof tlb.etip-sample voltage. By varying the bims,stonmlng t':nmeo'ag can be done. The sign of the voltagedeterrnineswhether spectroscopy (STS) occupiedor empt,ystates are studied, ms showai:a Fig. 5.2 for a sernsconducvtor snrfnme. For posjtive bias (a),ttmne,ling of electrons rA.n only occur from occupiedtip-metal states inio empty suzfacestates or conduction-bandstates i:a the substrate. lu the opposite case (b) with F < 0, elmsticttnnneling of electrons1om the metal iato the semiconductoris not possible.Only a current with oppYte siga is measurable.The measured'ialnneling curzent I.r originates from occapiedsucfaceor valeace-bandstate,sin the semkonductor. .
1 Tip
V d
Y
Surlœ d
Scan 7
Diredion
Fig- 5-1. PrMcipleof a scltnningtunneling Mcroscope.
5.1 Eleckronsa'adHoles
1%
(a) Energy
Eneay
CBM x
CBM
x
.s-1
sF
eV
-1.y-. .eV
VBM
y/E,,,
a---
----------------------------------;, /,
/
.-,ç
-,''>--u
Ffg- 5.2. Electronic band schemealong a sudacenormal of a semiconductorsttrface an.da meta,ltfp trv,h' valuesof the 'biasvoltagein (a)and (b). (1eft) t) fbr oppos-ste The energles of the conduction-band rnlnlmum (CBM),valezlce-bandmn.vlmum (VBM)a'adFerml' level (ga)ak well as distributions of possiblesurfacestates are indicated for the unbhaaed seMconductor.
By meastlri'ngthe dependenceof the curren.t I'j: on the applied voltage F, one can obtain an image of the energy distribution of the elec-trordc states ia
the surface region. Sizlceone is more interestedi.a the generalspectralbehaviorof the tl:nnel current and not i'a its exact absolutevalue,a simpMed appzoac.h caa be used. At the surface-tip separations of interest in STM or STS, of the order of 4 i or more, the surface-tip interaction is extremely weak. It is natllral theh' to calculate the tlmneling current using time-dependentpezttzrbatioatheozy The result for an elementazytrlnnel processis given by FerrxstsGoldenRttle. Axsplrningnon-interacting electronsJ.nthe tip materii and the sxmple surface o'r of the type chazacterized e.g. ia (3.1) the probabûil per tmit time (3.46), for an Lsoenergetic tptnmelprocess across the barrier betweena surfacestate with energy ss = eu (i) and a tîp state #.r@)' with enerr s,.r #s@)(SL?A) is givezzby
lWs
=
2% ,-
.: 8 @'r+ ekI'J.'Ts I -g-
-
&s))
wherethe trn.nnition matrix element
j
r'as -
d3z4,:t(z)>#s@)
i s intToduccdfor the taaaeling operator t. It may be issentiallyidentised M'ith the current densit'yoperator (5.21. F' denotesthe appliedbias. First we considera 'Fnste negative biks 1//,although tilis is relatively small on the scaleof the tip work fllnction or iomszationenerr of the sample.The tqtal current for tpznnelingfrom the sutface into the tip (7 < 0) is give:a
by
(5-34 JT
-
gi ?.(cr)) wys ceyyltss) -
T,S
190
5. Elementary Excitations 1: Sinéle Electronic Queparticles
J@)mq the Fermi dltzibutziomThe factor 2 accolmts for the spin decan be rewzitten a.s generacy.With (5.1)expression (5.3) with
I'2 =
4rre s
+*
L.
a dcltc)%-ulçe eF')J J(c-cv'-e,r)J(s-ss), (5.4) 57IT'z'sl -
T,S
using the properties of Dirac's élhlncwtion. Resukssfrn-llarto (5.3) or (5.4) Eave beenexploited by mrtny groups. However,genernlizationsto obtaia an explicit formllln.for the cl:rrent density have alsobœn usedvand real atoms of the tip
and surfaceinstead of modelpotentials kave beentékea into accout (5.41 A more or lesse-xact way to determl'nethe tltnneling current vithin the singleparticle picture is possible asing a Green'sAlnction formalism aud solvir.g zplmericallythe Lippmaa-seahwiuger equatiou (5.5). However,the zplmerical efort is large and,moreover, a detailedMowledàe of the tip shapeksneeded. The theory of the tnlnnel currem.t (5.4)makesno distinction betweensmfaceand tip states. Howevez,ixl STM or STSthis distiaction is cruciat. Ideally, one is interœted to relate a'a STM image directly to the stlrrlme propezties, wheremqin the abovedescription the current mvolves a convolution of the electronic spectra of surface and tip. nerefore, Tersof aud Hhmn.nn (5.6) proposed= approvsrnn.teway to elirninate the tiphpzoperties. They consideredau ideal tAlnneliagOcroscope with a model for the tip that woald have the highestpossibleresolutiomThis goal is best aœeved assnprnlng that the tip is a rnnthematic,alpoint sotzrce (s-wave 1ts potectial a'n.d approvsrnation). wave Atndions are arbitrnHly locnll''zed.STM experimentsobviously n.1mat atomic resolution. Therdbre, intuition suggeas that the tip must have atomic dimenm'ons as shown in Fig. 5-1.Iu oth.e.rwords,the tip is not point-like on tshe length scateof the axperirnents. Nevelheless,the point-probe appzoxn'rnxtion leadsto a reliabledescriptionof expersmental data, thougk it is not c prscrï cleaz why thks approvirnation holds. Witlnl'rt tlze polt-probe approximatioa for the tip at the positiou œt:p, the trnêndtion strength is given by .
:x: (#s(œt$p)121T$s!2
(5-5)
This approvsmxtion lfo-qrlm directly to the local electrozkicdtansityof state of the surfaceregion of the snmple.Without cotmting the sp% degen(DOS) eracs $t reazitq
p(œ:i)=
1#v/:(z)I2é@ ::.(/2)). X--2
(5.6)
-
vvk
This quautity represents the diagonal elementsp@; c)= A@,tr: sj of a more geuezalquantity, the single-pmicle spectratt-wtAirht) h'ndion A@,zT;e) of the electroic s'yste,m(5.7J. This hlnction also accouuts for the qua-siparticle character of the' exdtations. =* Comsideringthe lirnl't of low temperatures witk J@)(1 .f(s cF)J O(as>c)O(c ey sp)and introduchzgthe giobat electronic DOS of the -
-
tip
-
-
-
5.1 Blectrons and Holes
sy)
=
191
EJ(s cv), -
T
one
falds for the voltage dependemceof the t'lnneling curzem.t
dsots ekrlptœtyp; s). &F+W
J,r =
-
Low temperattzres me-qn that the thermal energy kBT is small comparedto $he position of cp with respect to VBM and CBM. 1.ntlze llmn't of small applied bias voltages or nearly constant DOS of the tip in the region of the Ferznienergy, this expression becomes .
Je
JT =
DLsvj
/ JFVW
dsptœt:p; s).
A ssrnilitr result holds f6r a small positive voltage (7 >
0)
el?+eV'
DLep)dF
I.v (x
'
dcptmup; c).
Eoression (5.9)showsthat for Jzr< O (7 > 0) the tunneling cltrreat îs accompaaiedby the generation of hole (eledroas) ia the rebozlof the spmple smface. Correspondingly,the tl:nneling clTrrint locnlly probes the pazt of of states of the surface that is occupied (unoccapied) the electronic dezusit.g for zero bitzs.Expression (5.9)is derived withsm the single-pacticlepicture, in the tip with the The attractive Coulomb interacbtion of electrons (holes) holes(eledrons) generatedj.n the surfacere#onha.sbeenneglectedfrom the One argument justifyiugthis neglect is the distance of tlb.e vep- be particles, rother the dmmatic redudion of the eledzon-holeiateraction by the metn.llsc screenlng in the tip reson. is proportional to the local eledronic density The tlmneltug current (5.9) of states (DOS) taken at the position of the tip. The local densityof stata is integrated over an energy intezval for F' > 0 (7 < 0)above (below) the Fermi energy, the length of wlzic,his given by the applied voltage. Consequently the restzlting 'STM images depend on the siga artd the magnitude of the applied voltage. Exadly thks dependenceis damonstratedin Fig. 5.3 for Garic,hGaP(001)2x4 surfacereconstructions (5.81. A constrt-height mod.ewith a tip-surface distance of about d = 4 â. is assïlmedfor the simulatiom To accolmt paztially for the nqnideality of the tip, the specvtraare averagedover a small interval Zz = 1.5 A of the normal distances. Two llmstations in modeûng STM images witbin the Tersof-Hn.rnltn'n approach (5.9)have to be mentioned.The tmderlying perturbation theory requires timsn.mpledistances larger thac the decay length of the wave hlnctions into tlle 'vacullm. Larger currents are modifed by multiple scattering of tite eledrons. Mother problem for small distaucesis related to the wavehlnction overlap of tip and sttrface.The resttlti'ag lbondsng'atld t=tibondsng' '
'
.
192
5. Elcmertary ExcitatioM 1: SingleElectronicQ'aasipartides
GaP (2x4)
M'IxedDimer (:.:j
Top Ga Dimer j( 1.t. y: àLt y .tj u:? . . ,r , . (.?s ;.(..'p't
.
)
..
.
!
.
.., k:'';.Jz )..i.)f . :
: '.
. . ..
(tL''.
%%
z ...t. / :(.: .
.
lk'
., ;..(y: ë. .'r' ..!,
.
.'
'
'
1
..,..t. ... j:l ,,. ..' . . .
'
: .. ? s......) .. . .:-.
(
J
'
.
i
.
12
:
i
Ii
'
,
i
I
1 l .
j..v ,.
....'..
t,..,'.
'
G' +-
:
.
EI :1 i!(
.
)
.
!
.
l
f
.
i ,.
. .
1.i'
é.xq
L
1+-d ' '
1.û1 17:.
7.1.53nm
'.
'
i
;
$ 53 nm .
Fig. 5.3. Calctzlated STM imagœ for two dimer geometries of the GaP(OO1)2x4 sklrface.In the cmse of the tomGamdlrner str'ucttlre the P atom in the hetezodsrne.r of the mhed-dimer geometry (shown ita ng. 2.17)Ls replaced by a Ga atom. The negative (yosijive) voltagœ are measured=th respec't to the CBM (RM). The ''brightness indicates the magetude of the 10c,alelectronic DOS. From (5.8).
linear combinatîonsmay #ve rise to a completechangei'n the STM contrast (d. the remarlcsin (5.9q). The memsmement of the spatial variation of the local DOS for a given voltage is a powerfool tool for obtaining local stntctltrat 'Oormation about metal or semiconductor mnrfaces. Moreover, the spectroscopicmode of STM, irt which the tlmneliag current f,r is recordedas a fhlmction of the applied bias 7, gives direct information about the loci electronic strtzctttre of the suzface.Asmtrnsngthat the prefactor of (5.9)is nearly independentof the voltage,diferentiating thks expression with rezpect to F yields the diferential conductance,dfan/dvr (x: Dtsslptœtsp; se + e5rl.As a function of F' this qt,n.ntit'y roughly reproduce.sthe local densitjrof states of the surfacein the neighborhood of the Fermi level. Density-of-states featares in I-V curves appear as various lcin'k-sand b:tmps.They aze obscmedby the fact that the '''
$.1 Electronsand Holes
193
'*'
*
6
4% % 1.
*
r.r . .
zs.
a
..*.
..e .
m
.
.
.
Jea
jj
* @
2
2lb..
%*-z
*>
.
*w .
*
'.
Nà ,. z,
2
.-
hwx lw-k
a
;.ee sv< =
x
*>
A*
**
&
*
0
..4
a
. .
.
*
T:
4
-2.
-1
Tr
0
1
2
3
4
Energy(eV) Fig. 5.4. Relative conductancevezsus electron (hole) eaergyrelative to the Fermi energy s'v measuredby STM on a cleavedSi(lll)2x 1 sttdace.After (5.10).
tunneling cuzrent dependsexponentially on tlb.etip-sample separation and. in a norl'lnea,r mnrl'ne,r on the applied voltage. Most of this dependencecan be removed by computiug the ratio of diferential to total tTnnnelingconductance,
c'1.f 'r dy
rrf = = 'f/'
plztip,Ev + dV')
1
fF
cl'U'qssicv
dz#lttip,&).
Eereonly the restûtfor v' < Ois givem J.nthe opposite case 1/r> 0, tEe energy integratioa hmsto be Ganged accordsng to (5.9b). hdeed, (5.10) gives the local DOS of the surface at a hole (electron) energ.ywhic,h is given by the applied voltage. It :is normalized to the local DOS averagedoveer a,n eneror interval of length elFlTvenby thks voltage. Am examplefor the z'esttltof such a procedttreis shown ia Fig. 5.4 for a Within the region of the G'nndxrnental cleavedSi(111)2x 1 s'arface(5.10J gap .
of bpllk Si two surface-state-relateddoublmpeak stnzctttres appear with peak energies -1.1, -0.3, 0.2, and 1.2 ev with respect to the Fmrrnsenergy. The spedTalfeatttresat higher or lower energiesare due to bulk states. The fotzr central pen.lrKcan be ex-plxinedusing the model of tûted em-bonded citains (see resulting witlnln a simpletight-binding Sect.4.2.$.The band structure (4.7) approxsrnation gives a (normazzed) electronic density of states of zsbonded càains as (é'= 1s1 sz 1, sl + sg = 0)
j
-
'
(5.11) The meastu'edpeakpositions in Fig. 5.4 cac be identïed with the square-root singulHties in thjs DOS.This gives a surface-stategap of about rs)-s2 I = 0.5
194
5. Blementazy Excitatioms 1: SingleEledzonic
Quasiparticlc
ev auda.tè averagebandwidth of a rr* or rr bandof about 0.9eV:i.e., Ikrl= 1.0 W with F' bemg the averagedinteratomk Mteractioa matrix rlement. 5.1.3 Photoemission Speeroscopy and Iuverse Photoeml'ssion' If one bombardsa samplesurfacewith electronsor phètons,electromsand/or photons will be emitted, whick have aa enerr spedmzm. Heuce,Hormation is obtained dkectly or indirec'tly about the jurface electrozzicstates. The most important and widely used experlrnental technlque to gain iaformatioa about occupiedeledronic surface states is photoemlnqionspectroscopy TNe solid surface (PES)(5.112. (or sometimes photoeledzon spectroscopy) is irradiated by monochromatic photons with energy ruo, holE'sare generated izt the sample, and the emitted electrons are analyzedwitit respect to When photons in the tûtraviolet (UV) their eetic enerr ck:u = :2:2/2m. spectral rauge are uset the teelnniqueis cmlledUPS ('UVphoterniAqioaspecBesid%the kiaetic eaezgyone may also use the enzissiondirection troscopy). by the angle9 with the suzface sin/sin8, cosl)de-sczibed kjk = (cos/sin8, normal a'adthe angle / in the surface plaue to charactezizethe geometry of the exp-rime'nt (Fig.5.5a). Vatyiag $ and/or / the method is then kmown as angle-resolved(AR)PES or UPS, ARPES or ARUPS. hwerse photonmsssionspectroscopy' ImESIcan be regarded as a timeIt therefore probes the tmoccupied reversedphotoemissionprocess (5.12J surfacestates. Ia this tenbnsquea be>.= of electronswith ene-rgysktn and wave vector k = àtcos ysinù, ss.a/siap, cos ;) is kcident on a smface (F)g. 5.5b). The electronstrn.nqmittedinside the solid decayto states with lower enerr through tlle Auge,resect or by emitting photons,'which are detected.There are tvo operatbzgmodes:either the energy &zJof the detectedphotons is held .
(b)
(a) /
Ekinl k
ho
/
Z Ekin, k
T'sg. 5.5. schematicrepresenzation of a photoemission sion (b)process.
(a)and inverse photoemis-
5.1 Electrons and Holes
195
collstaat ard the spectrtlrn is obteed by varying cku (isochromat mode), or sk.ln is kept coneant and the spectranrnis taken as a funcwtioa of hu'. 1f, izï additîon,oae takesadvantageof the k-vector raolution, one ca,llsthe method k-resolved(KR.) DES, KR.TPES. A rigordtls theoretical approachof azl elementar,gphotoem7'Fmion (invezse 6111 q'aantllm-mechxnscaltreatment of the process requires a photomml'ssioa) complex coherentHteraction processesstarting with a photon (eledron) and Enishin.gwith an electzon (photon) in the detector ms well as a hole (eledron) ia the smple. Titeoretical approaGes of this k'sndtreat the pkotoemission for instaace a.s a one-step process (5.13-5.151. A more instracvtiveapproachis The optical excitatioa of an electron ts.r. the so-catledthree-step model (5.16). a 6149,1 state fk) and a hole (iLthe izzitiatstate ïi) can simply be desmibed agnin by Fe-rmils Golden Rule. Withln the dipole approvirnxtion aud the iadependent-pazticleapproximatjon the trxmqition probability for the ftrst step is given by 2r
a
i t î&h(&)= -..àk) aytjg)spl (k)( j @.f( 1.,,/.2 '
-
-
.
Withi'n the single-pazticlepicture the pezturbation operator Xint) the lightmatter interactionj is given by the eledron velodty operator 'v = ihlV,a(- tiI1 the cornmutator represeatation) oz' the m'omentamkoperatorp li:athe case of local potentialsi.u the single-particleHxrniltonian Hj and ilzevector potentiaz A of the iuddent electromagneticwave as tin.cgs llnits)Xfnt'= --G.A .p with mc matrix elementé t
J42(:)
e
-
--A mC
.p.s(:),
py:@) d3œ#),@)p#ns@).
(5.13)
=
The gauge of the electromagneticâeld is here chose.nsuch that the scalar potential vanishes. The vector potential .A is nearly spatially constant iu the long-wavelengthl'l=5t (inITPSAhe wavelengthof rmiiation is still à > 10û describesenerr. conservation dlpring a direct A). T:e J-fttndioa in (5.12) optical tmnsition from the Oed initial state sç(i)into the e'mpty ftnal state sJ(i) in the surface bands. For photon energicslazger than the ionization energy of the system, f.nal states with energiesabovetke vacullm enerzysvuc occm. In the caze of blllk states) as a secondstepj the e-xcitedeledroms propagate to a certain extemt to the smface.'This trn.nqpozt probabilsty dependsalso on the eaergy and wave vector. J.naay case, for egLL-) > svac electrons may escape through the stkrfaceinto the vacmtm with a probabiiit,y T/kin,i) as a thhd step. They appear in the vacuum with kinetic enerc slqia a'admomentlnrn hk. The escapeprocessesare elmstic:i.e., dkia = sy(k) emic holds. Together with tlze ettergy corzservation in (5.12), this equation results in the well-lcaown Einstei'a law of the photoeledric eF.ect (5.171 Becauseof the 2D tr:mqlational -
.
196
5. ElementaryExcitations 1: Shgle Electronic QlxMipadâcles
symmetry of the system, the electron transmlAqion through the suz'faceinto form kl g= the vacqrlm requiresmomentl'rn conselwationin the generxliaed k + g for the wave-vector component parallel to tbvesurface,where g 'xs a vedor of the 217reciprocal lattice of the surface.On the other hand, the wave-vector component parazlelto the surfaceoutside the crystal is relatecl sin ù ard )ç:g= to the experimentalpazmmetersclcis, 0,tj by kjj = zmskin/D,z kgk(cos 4,sin/, O).Consequently,not only the energ.yof the initial state (Mth i.a the surface b=d structme but also its wave respect to the vacutkrn level) vector can be determineâfrom the expem'rnentalparameters FlaJ,skin, 3, and
/.
In s , the three s'tepsylelcl an approvsmate expression for the photoelectron current in the vacmlm or for the measurednlTmberN of photopointing in a specfc electronsemitted into a cone (witha small solid a'aglel
di'rection
.
lJ2'y''lt::)l2 NLeysn,u), klI)cc EEltszn + svac Fpx;l -
t'kl kiys
x x
+ sm-sc r'zs) w4o.ét/l,sulu cusa + cvacl ék!!,:+sT@kia, Y. Wyyti, -
Here the diagomxlnantrix elements Avv' (k, s) =
d3z
tr'; s)#v ,x(œ?)' q?W#. 1 .: (z)./.t(œ
(515) .
of the single-electronspectral-weightfunction of the sl:rfn.ce system are takeu with single-particleeigezzstates '(u@)(withquantzlm nllmbersIzil that have been calculatedin a certnsn approvlrnation of non-interactiug particle-s,e.g., to interactiug withsn DFT-LDA (3.46). Thrusallows a genernlscationof (5.14) electrons ms indicatedby tke replacementé'(c cv(i))--> Xvvti,s)of Dirac's matrix elememts of the spedral-weight by d-iagonaz J-fïtnctionsin (5.14) (5.15) hlnction A(m, a/; s).Jn this my the idemtécation p@; œ;sj of the s) > .A(œ, becomesobvious. local eiectzonicDOS (5.6) with v = #: izl which ortly 1.cthe second spectral factor Jks izz (5.14) occupiediztitial s'tates are considered,it is allowedthat the exc-itedholesinteract with the remairdng electrons. On the way to the Apacutlm the oatgoing photoelectron, v = j', also interacts to a certnsn erent with the electroas in the surface acd barrie'r region and the remnsnlng holr (adiabatic approxHoweverj this ezect shouldbe weakenedwith increasing1dimation (5.18)). so that the replacement of the electron netic energy (suddezz approvlrnation), spectral hlmetion AJJ by a élAlnction is oRen a good approxn'mation.The i'aterpretation of expersmeutsis mostly baseclon the sudden approxn'mation, expressing a photoelectro:a specwtrlprni.zzterms of the one-pvticlespectral hlnctiozls of the iaitial s'tates,Aç(E-, s).However, in genezal,with the excitation interacting hole and electron quasiparticla occur (5.19). -
5.l Eledrousand Hole,s In the simpliîed expression
197
(5.14)'with
tite factoeed representation A:(i, c F?zslWyyt/1, s),the electron-holeinteraction hmsbeenneglectedin agreement with its derivation witqhsmthe origizlal picture of non-interactiug partscsles. Sincethe e-xcitation energiesr?ware muc,hlarger th= the energy of the fl:ndnmemtalgap: the cledron a'nd hole excited i'a a photoemissioc procv are energeticallywell separated.Their coapling in the surfaceshould thereforebe smltll. This holds even more for the interaction of the oatgohg photoalectron with the remainlng hole. Nevertheless,for not too large klnetic intemsit'y vafatiozus of the enertesof the photoelectrons (adiabatie lc'rn-lt) plazmon satellite of the main photoelectron peaakversus energy have been and have beemt'racectbnr!lcto the eseds of the electron-hole observed(5.19) interaeion, at least to vez'te.xcorrectious (5.20, 5.211. stdctly a photoelectron spectznlm in the adiA spectral Glnction (more abatic limjt) for a surface-state band is shovm in Fig. 5.6 for the occuyied The enezgy Hssereacesv Ev(k) is surface(5.221. r-band of the C(111)2x1 intezpreted a,s the binding eaergy of an electron with resped to the Femmi = 50 ev has been performedunde.rsttrfacelevel. The measurement with FlnJ seasitive conditions. The measuredazimutlnxl direction is fxed parallel to (ï10).For 0 = 33.30 66.80the wave vec-tor L thereforevaries along the ?R li'ne from its middle to a,a eqttivalentpoint in the neighboring surface BZ Lg# 0).Despite the broadens'ngof the spectra, the peak positions should be identiîed with surface 'bazd energies, here with those of the cr-bandof the Pandey chnsnmodel.The observeddispersionis in qualitative agreemezzt w'ith that of the calcltlated c-bands in Figs.4.16 and 4.18. Near # the occuied x-band comes closestto the Ferrnl'level whereazin the directionstoward Bz .a strong dispezsionof t:e xr in the sxrne Bz aud.J' ic the adlacent baud towa.d lowerezzergilas is observed.Thc measaredvalue of cx(A)at lewst -
-
-
0= 33.30 K = c =
37.40 41.60 45.60
r=
Q)
49.70 54.00 58.20
c
62.50
w
>h
M
(D
c
T(
5
4
3
2
66.8o
1
0=EF
Binding energy (eV)
-;2
Fig. 5.6- Angle-rœolved photoelectron spectra taken frop a:a Aonealed surface. The photon enC(111)2x1 ergy Ls 50 ev at normal incidence of the incorning light: and the measured =imuthal direction is (ï10q.
F'rom (5.22).
5. Elementmy Excftatiorss1: SingleElectroaic Q'aimiparticlu
198
z'
PES i;tk
,k()!!.
.-.
ut qk
*û.i!.:,)z
.
m Jixsiyqçs.;
%
,
IPES
3.5eV
N.
.
/
. . . '
J
.
z
GaP
I
hx'
x
k!' . .k
<*
:
:
k m-Z
.r ev a
-
& x
;
&
J..
N..... Nx z
X
ql
I
-
A
's
InP
T ) t.%
/
:.
/
:
*
,
v
.
.,( .0eV
.= c
.
Q >-. (1 we
,J
.
i
2 4ev
? 't
j1
c-...
y
r
4:
. ,
: . .
-:
IDAS n
/
. . *
>'
.
j.
'
G%s .
t ,
-
;: .
N
> G)
-.'
.6 v ..&
=
(D c
.
2.QeV
N
y
.
x; :
$'
GYSb
1. V :
I 4
l
*
l n Sb
-
/'
'
.1
Energy (eV)
Fig. 5-C. Combined photoemission and inverse photoernlqqion spedra for the determsnation of surface band gaRs at the highsymmetry point X? ir the surface BZ of s'zx IH-V(110)1x,1 surfaces. The energy zero is given by the VBM. The photon eneries have been chosento be >= = 21.2 eV (PES)and &zz = 9.9 W IIPESI. From (5.23).
0.5 eV bclow cs hdicates that the C(1l1)2x 1 sarfaceis stamiconductitlg,iu contrast to the DFT-LDA result i'a Fig. 4.18 tle:ît p=el). A.n expression s7'=57arto (5.14) IPES may be derivedfor the tve-reversed spedrnTrn is dorninated by the main peak in tlze process. The corresfonding spectral 'hlndion of the electrons in the empty ân.al state. Neglecting the iniuence of the trnnqition matzix elements and the veatex corrections, the spectral variation of KRIPES is governed by the empty i-vector-resolved density of states Ey AULL, c). 1n contrast: in the cmse of ARPES/ARUPS' the spectra aze governed by the occupiedpart of the Lvector-resolved dettsity
s.1 Blectrons amdHole.s
199
of states of the smfaces'tates,Eç.3ksti, the combinationof $. Conseqaentiy,
ARPES and KRDES allows one to detemnlnethe complete i-vector-resolved smgle-paaâcledemsityof states of a sllrfnme slrstem. An exampleis shown in Fig. 5,7, in wbic,h combiaûdphotoemsAsion aud invezse photoèmlssionspectra (5.23) aœe presOted for the XT point iu the s'arfaceBZ in the enerr region of the ftlled anion-derived dxnglblg-bond band Jls and'the empty catiozl-derivedCa bard (see Sect.4.2.1,Fig.4.12) for slprfnrtes.For tkase surfacesthe expercleavedEFI-Vsemiconductor(J.10)1x1 iment has the aduntage that a cornmon energ)rreferencefor the two applied spedroscopiescoald be es-tablishfd.J.nall cases i'n.Fig.5.7, it is e'ddentthat the anion-derivedsurface s'tate gives rise to a cler peak, whilè the cationderivedfeaturein some casœ is a rathe,rbroad line with a large slope at lower 1 surfaceof these materials energiescThe surface band gap for the (110)1x at the X' poin.t of the surfaceBrilloTlin zone is consideaxblylarger thn.n the fltndamental bttlk gap, bttt the hcrease 1om I'asb to Gap hmsl'oughly the same slope. 5.1.4 Satellites
The genern.lizationof Dirac's élhTnctionsfor electron acd hole excitations in to shgle-quasiparticlespectral flmctions (5.15) allows a moce complex (5.14) view of photoeelectron excitations beyondthe trivial lifetime broadenimgof a J-l:nnction (as,e.g., a porssibledeascriptionof the spectra iu Fig. 5.6).ln the spitit of Sed. 5.1.1 such a geaeralization allows oae to account for egects of the complete electron-elecvtroninteracvtion at the leve,l of single-partide excitations. The accompanyingm=pbody eseds al'e best demonstratedfor the spectroscopy of strongly locltll'zedcore states and a wider enera range. As an exmple, the Sizp and Si2.sphotoeledroa spedra of a Si mystal with (111)7x7 surface are presentedin Fig. 5.8. They are measuredwith a photon energy of Fttzp= 1486.7 ev IAIAQradiation) Tlle large ki(5.24). netic energy mdicates the sudden lx'ml't. The escape depth of the outgomg photoelectronsis varied by cblmrimg the escapedirection from normat emission (#= 00: mav'='lrn depth eqtln.lto the mean 9ee path Arafpof elecvtrons) to grarjing emsxqioa(ù= 80Q,smxll depth 'w àmfpcos 800). Iu. the flmt case, the spectra are dornsnatedby blllk losses.1'athe secondcase, it is clear that the sltrface plays atl esserttialrole for the PES. Beside the miu'n quasipam ticle peak desmibingthe core-holeexcitation witkout losses,the positiou of P wlliclt is usury used to d'R6 ne the core electron bindsng enera, or p QP W ith respect to a referencelevel, .-ggs j e.g., the vacul:rn level avac or the Fprmi level ek- (here: ss),the spedznlrn h.asan incoherentsatellite structme. In dnbonded solids the satellite stractures are mainly due to shake-up of sllrface aud blllk plmsmons.At the side of higher binr?îngenergies, multiple (n= 1, 2, 3, ...)lossesby plasmonsare Hdeed visible. The strongest satellites are due to blllk pMmons at about ssizl .-nop (J= p, .s) with Fopprts 16 eV. In
-T
Si2p
Si2s
Aop O
t= =
pmtop.
71
rM
=
:3 pdhop-è
<-
-i;5
I
.
I
j
c q)
ecqs
-
bhœp- I pzhropj,l
8c0
j
l
111 fûs
I 1
I
200 15D Binding energy (eV)
250
l
100
Fig. 5.8. Si2pand Si2sphotoelectronspettra of a Si(11l)7x7 sarfacemeaured for a latge escape depth aad under surface-semdtive condîtionsretative tr the Fetm! level. From (5.242.
the lower ctu'ves of Fig. 5.8 (memsmed tmdersarfMe-sensitiveconditions) the broG satnlllte featttres s110w shotûdersat the position of the sarfce plxqmon Figure 5.9 r'howsa similar photoelectronspedrum for energy M?sF4$Mzp/W. a qirnplemetal, Mg (4.29).
20
Gh e= c =
Mg2s
15
.d
:
%'-e
A
5'G
VV2?
10
*Q c
2
N% 5
0
80
60
40
20
0
Binding energy relative to cî; (eV) = Fig. 5.9. Mglp and Mg28photoelectronspedra excited with AIAQracliation (F?zd 1436.7 e% relative to t:e binding energy cf Mg2s -sQaP = 49.8 eV. BG and
surfKe pbnmon losxs
are
ixdicated. Dom
(5.25)-
() .J
Mèllly-otlkly
rllecb:
Suc.hsatellite stzmctmescm.nnotbe observed in spectroscopiesprobingthe energy reg-ionarotmdthe hpndnmentalgap az in the cmse of STS.The satellfte structures i'a photoeelectron spectra of core electronsare dueto two dferent mechazzisms.The main lossasoriginate &om Ax(c).TEey are a cozzsequemce of the polairization of 1he electroic system in the presence of the core hole and are probed by the photoeledzonbefore it leeaves the solid, This gives rise to intzinsic losses(5.18, On its my to the sttrfaceand csscape9om the 5.211 system the outgoing photoelectronitself pohrlze'sthe clcctronicsystem. This mfvtbxnhm givœ rise to extrlnKîclosses.Between these two 1'.0% of losses there is quant:trn-mecllanical interference (5.20, This hterference (the 5.211. vez'textcorrections beyond the single-pazticleapproac.hin (5.14)) results in a at lemstin the lsrn5t strong suppressionof the satexite stntctures (5.20, 5.26), of 1owklnetic energies,as compared to the predictions witbx'n the three-s'tep .
mode,l(5.162. Neglectingextzinsic and surfacelossesthe experimentalspectra iztFigs. 5.8 Alnction in the fo= azld5.9 may be representedonly by the core-kolespecvtral
(J s,pj (5.27q =
*
r
p
sozy c-7 + zzhfggplj WaTzltc) J@ )((; V =
-
(5.16)
a=0
apart 9om a broadenin.gof the main spectral li'ae and the sateltite dae to fmite lifteirne and plmsmondispersion.The pcsition of the ma,in line, QP by tEe same interactions that generatethe gc; = fa2 + z:ï2I, is zenorrnn.li'zed satellites, henceJ9= lcl/&azp. The center of gravit.y of the spectmnm (5.16) $sidentical with the 'lnrenormnlsqedenerg,gszl. Meanwlkile,rather complete shgle-particle Gcitation spectra havebeencalcttlate for semiconductoz.s aud metab (5.28-5.30q, at least for electrons ar.d holes excited in bulk states.
5.2 Many-Body s.2.1
Efects
QuasiparticleEquation
ln Sect. 5.1 we have seen that in important surface-secitive spectroscopies sucà as STS, PES, ard TPESthe single-particle spedral-weight flmctdon af; c) of the system or, at least, parts or matrix elementsof it, are .A@, probed. As a specetralhlnnctio:ait is a Herrnstiaztqurti'ty:
X*(a/) z; c) = .A(z,tr/;&). It 'F!11671i' importani stlnt z'alcwsThe completenessrelatsionca'a be written in the form +x
z'; c) = Jtœ z'), dsaitœ, -
-X
(5.18)
202
5, Elementaz'yExtitations 1: SingleElectronio Quasipmiclo
whic,hdescribesthe couervation of the nllmber of skgle-particlestates independent of the treatment of the elecwtzon-electrozl interaction. The consmation of the nltrnbe,'rof electrons gives the dectron densit,y(3.47) irêthe form .
+œ
'
J
n,@) 2 e-X =
œ;s) d&.f@)X@,
with the Fermi functon
(5.19)
fLejand the factor 2 cotmting foz'the sph
degener-
a'LF.
The spectral-weightfunction is directly relatedto a more genez'al function, œ?;s) of tile system of ixé-rnftfaingeledrous. 'l-%im the Greelfs 'hmcdon GLT, e>n be seen from the spectral repzœentationor Lehmn.nnrepresentation of the Gzeen'sftmction,
G>hm/1f)
+co =
z;
' a;), dE?.g.(m a .et ..yjy
with n = cié (J-y +0)for energies above (+, electrons) or below (-, holes) the Fhrmt' enerr, at leas't for zero taperature. 'I'he Green'sR'nction descibes the dylmrnp'cs of the No -> Ne + 1 e-xdtatiozzs, whereNe is the nlxrnbe,r of eledrons in the syste,mgrotmd state (5,72. From its equation of motion with the m=y-body Hstrniltoml'ym i'acludi:agthe f4117 longitudsnaleledron-eledroa inttaraction,the Fourie.r trnnnnformationwith respect to the time wiable leads to the Dysonequatioa 25.31-5.33) e
:2 + ip + zazrk -
-
Mon@)5k@) &@,Z; &) -
=??;s)J@M = d3a//.)J@, 1 a/; &) 6Lm=p) -
with tke external potential of the ions
and the Montœ)
(,5.21) Hartree potential
z/; s) accounts for the exchange (X)and The self-energyELœ, V'H(z)(3.49). correzatioa(C)efec'tsoa the single-particle level It represeats a non-local, complœxtnon-Hermitsxml, and Gergpdependent operator. Equation (5.21)
the correspoadhghomogeneoms oae is e-qlledthe quasipmicle equation. The occarrence of tEe self-enersryJ7 mctieatesthe better treatment of XC efects for excitations th= for iudependentparticles in the electronicg'round or
state (3.46, 3.48). Due to the complicated self-consistentdependenceof E' on G, solviag is a dlec''ult task even for the simplest electronic model systems, e.g., (5.21) jellilxrnwith s'zr-lhœ. Moreover, E Yntnsnn the bare Coulornhinteraction n(m :F) = eajjm a?(to atl orders. The expn.nsionof 2 in the 677d:nonvazsisbl'ngorde,rwith respect to 'o results in the Foc,koperator, the non-local exclnnngeicteraction. The ecpxmqionof E in ttarrnKof r Lsslowly convergent. Anotb.ezapprolmh is more promising. The electronic system kmdezconsideration reacts to the pzesenceof an excitedelectronor hole wsth a redistribution '
-
-
'
5.2 Many-Body Efects
203
of the electrondensity i.e., a polarization of the electroaicsystem. For that reason
it is
more
conveGent to conside,ran epqmnqiozlof X i:a tnrmq of the
dynmically screenedCoulombpotential z'; rlaJ) T4r(œ, =
d3a//s-h(z1 œ&;w).sLm'' m')
(5.22)
-
hwersedielectric h:ncwitll the spatially non-local and fzequency-dependent œ/;aJ) of the system. The reducvtionof the iuteractioa potential tion c-l (œ, by screening makesit obdous that an expnnRionof E i'a g/mis more rapidly convergent.
The self-consistentdependenceof 11'on G is provided by a closed set of coupledequations, the so-calledsystem of fhlndamentalequatioas (5.% 5.31J which, ia the simplest approximation that iacludesdyzmmicalpolarization processes,are decoupled'by neglectiug the vezte,x corrections. !(nthis case E can be wri'ttem as a convolutionintegral œJ;c) =
E(œ;
i
s
+x -K
.
dec-oo
+
z'; &s), (5.23) G@,m'be &,J)W'(z, -
where the vez'tex corrections have also been negiectedin the calcqzlationof c-1. This correspondsto a descriptionof the screensngwitbs'n the raadom 5.30, 5,31). Tlb.eresulting approvirnation phmseapprovlrnntion IRPAI(5.7, is called the GW approdmation. Unfortunately,> strasghtfoNardtm(5.23) n pf the self-energy provement of the method is Himcult. A furthe,r ex-pn.nqio' i'a powers of the screemedinteraction may yield tmpbysicalresuks such as I1l fact, the expxnsion itse,lfis only condinegative spedral h:nctioxxs (5.33). tionally convergentdue to the long-rauge natme of the Coulomb potential. So f a,r there is no systema'tic way of cboosing wizic.lkdiag'ramsto sum to go beyond the GW approxlmntion.The choiceis usuatly dictated by physical iutuition. Fortunately,already the GW approimation gives extremely good reutts for quasipartide energiesand spedral distributions. 'WhenW' in (5.23) is replacedby the bare Coulombpotential r, the rœult,The energy irg modiâedexpressioaleadsto the Fockoperator for exelns.nge. essentially gives the integral of the single-particle GreeapsAlnctioa in (5.23) exchangedemsi'ty, whicll is related to the of-cliagonalelrents of the spectral A'mction in contrrt to the eledron density (5.19). Consequently,the dfe'rallows one to exctrad emsilythe correlation part emce of the t'wo self-energies of the shgle-particle self-enerr =/; s)= .S'C@,
5.2.2
$ 2x
-swo+gtw dFwJe , g? a.xo,y jw.ta,ttggss;
-
z,tz a/)! (5.24) -
.
QuasiparticleShifts and Spectral Weights
is similar Apaz't from the inhomogene'itythe quasipacticleequation (5.21) to the covesponding homogeneousKohn-sh.am equation of density 'htnc-
2û4
5. BlementaryExcitations1: SingleBlectrozkicQaasiparticzas
Only the XC self-energyoperator X@j:r'; s) has to be (3.46). replacedby the Kolm.-shil.rnXC potential Fkc@ll@m/)(3.50). This sugtionnl theory
-
gests the reprexiation of the swtial dependenceof all q',smtitie in tez'ms of the oabonormallzed aradcomplete set of eigenAtnctioas (W:@)Jof the including aksothose of the empty statas, for a Kohn-shnm equation (3.46), surfacesystem with 2D traaslationn.lsymmetzy.Another reason is the wide availability of the Kolln-slmm (KSIeigenfnndions. The majorityof the opt'rnsqations of the st:Hace structuzes and the cazculationsof total en4rgiesis bmsedon DFT i'a LDA (orGGA), The Kohn-shnm equation is always solved using these Rnnctions. TMS lp.1rl!;to the represeatation ..
Glm,tF;s)
=
t?vzzzli, s)#,:(=)#s!zx(œ'), X'.I7-.7: ukvl
(5-25)
jp
m'; for all other quaatities, e.g., A@,œ';s) and J7(z, s). The og-diagonx!esementsof Gwv'(/7, ej azzdrvrzzlijs)Mth. respect to the b=d index are respousiblefor the fact that, in general, the quasiparticle (QP)wave Ganctionsare dieea'e'atfrom the Kohn-shnanones. However, for of excitations near the h'ndxmentalgap of b:pl'lrsystemsthe near equivalence QP aad KS wave G:nctions has been sbown (:.34j.There is experieace that the equivalemeremains 'valid if the energetic ozdezingof the KS eigenvaluœ sv(i) as well as of the QP energies CQ.,P (:), i.e., the posjtions oî the main peaksin the spectralA'nctions .âs,vti, c),is the snme. However,also in cases o' 1 smface, of violation f tahisordering, as in the case of the GaAs(110)1x the eect remai)ks small due to the small hybriOation of surf-related aud b'ttk-relatedKS states (5.354. In generaljhowever,the QPwave ftlmctionscorrespondingto nearly degeneratestates with diferent degrec of surface/bulk loemlszationmay be h'nfluenced by a stronger hybridization, mnldrtg the QX wave h'nctions signlcazttly dfaent 9om the KS ones. Dxerencesbetweea KS' and QP wave îtmctiozzs can be eoected in any noxsb::l'ksystem, i.e., when there are resonsof space where iEe electrondensit.ygoes to zero. 1:a thœe zetonstlte LDA generaœ an XC potatial with iacorrect asmptotic a rnsmingk/r tail izl the caze X atoms). behavior (e.g., the resultizfgDyson equaDespitethe asmnrnptionof band diàgonnulîaation tion can only be solved approvîmately via an iteration procedure, shce the pertatrbatlon itself Ls a, A:mctional of the Greelpsfl'ndion. The dhgrstmrnxtic representation of the self-energ.g contzibution iu the GWapprov'mation (5.23) Ls showu schematically iu Fig. 5.10. The solid electronline. represen.ts the Gleen'sTnmctionG, while the dnAhedline indicates the scrïvmed Coulombinteraction W. ln orderto begia witù a Green'sG'nction G0 VV (i,c), the spectral fltnction of which is close to the true oneh a shl'ft 1v(k)of the as
.
KS eigenvalueto sQv P(:)=
sv(i) + z;z(t-)
loding to is htroduced (5.36, 5.374,
(5.26)
5.2 ManpBodyEfec'ts
J
#
'
+
<
+
205
%.
h
xu
h
h
%
% *
2
#' #'
%
N
;
v'k'
vk
vk
f'ig- s-10- Scllematicrepraentation of the self-energycontributions v'itlzin the is representedbyla solid line, whereas the GW approvsmation.The eledron (hole) dashedline indicates the dynnrnscallyscreenedCoulombmteraction. l
G7v(:,s)s sr(&) + iw(:) -
-
w(i)
with
+J
=
(-J),J ->
(5.25)for the
representation
With +0, for electronstholesl,
and the (3.46)
Green's 'hlnnction and the seklenergy the Dyson
reads (5.21) Gu(:;s) G2s(t-, s)(1+ pz'z,.(:, s) zv(:)jctv(&,s)) with the diagonn) elementst'EvvLk, c) of the rtarnnlnsmgXC self-energy(irL the GW approvsmxtion) equatiol
=
-
zJ; =/; c) ELm, JJ7@, c) J?kctœll@œ') =
-
-
zzot included i'a the KS muation.
Restridicg to the 'Ilrst non-vinishing order 'with rezpec't to the pertmu in the bation /JCv(i, , in whîc: the Green'sfnnction G (5.25) s) z1.(ï,-)q has to be replacedby Gz (5.27), ozle obtnsmq a ptmmaad self-enerr E (5.23) being quadratic i:a GD. ozt the right-bnnd side of the Dyson equation (5.28) 1n.order to avoidata ucphysicaldoublepole of the zesulting Gzeen'sAtnction -
at c
=
aQw P(i) in this terrn, one hmsto set
zv(&)=
Rc è'Euuv @;cQ.P(&)) .
(5.30)
ht this way the nlnnlrnown quasiparticle sbiêk in (5.26)is now defmed by The restriction to the Green's which.llas to be solvedself-consistently. (5.30) fntnction GQ (5.27) 'with the shift (5.30) cau be identfed with a Srst-order pertmbatjozstheozy treatment of the pelurbstion 5.8 (5.29) beyond the exwession (5.30) azd (5.28), givez an exDFT-LDA. Together vrith (5.23) KS eigenplicit de6rition of the QPshkft, whic.hcorrec'ts tlle corrcxspondsng
.
1r.addition.,îzl many practcal Gculations the energy dependenceof the Thjs seY-energy matrkx elementsis asstïrnedto be linea,rarolmd c = cQvP(J&). '= gives a renormalizatioa ,t1v(:) a.(:)ReJ&0.(i,s.(:)) where zv-l(i) = 3,e.,Avitha denominatorof nearlythe spectral t'iRefvovti,
(1 -
g)/&Ie=a.(:)1,
5. ElemeutaryExcitations 1: Single Blectronic Q''nmiparticles
206
0.40
>
..Q 0.35
,
MainQPpeak
c'
X' 0.30 w
Nzvtk)
xs peak ,.1
gsgg
2=
0-20
9 :e$ c ls c .Q -6 0.10
'w (ufetirnel-l
> <
'
a;z
Oo &.
@)
Plasmonsatetlite
0 05 -
0-0g
sîotkl
cp
EnergyE;(arb-units)
of a quasiparticle spectral fntnction in the Fig. 5.11. Schematic representatioa GW approvîmation for a smgle-pardcleKxdtation belowthe Fermi levez(hole)For compstriqon, tke spectral fanction of the non-interadin4Kohn-sha,m particle àsalso shown. The spectral weights of the main p'alm are Mdzcated. .
Fig. 5.11). Details weight of the mn.5'n QPpeak in tEe spedral 'hloctioa (see of the computations of the qumsiparticleshsfts for bl:lk aad surfaces'ysteuls c--m be fouad ixl sever? redew articles (5.33, 5.38,5.39). the spectral Annction .k4.zz(i, Much in contras't to the QPshifk (5.30), s) is strongly s'nBuenceê by the dmnmics of the of the Greea'sfllnction (5.20) leads to screensng.The flrst iteration of the Dyson equation (5.28)
s'
g, aessts,slj
zs-(s,a)-
+
:i:
,,?
v
s (s-sr(:))
1
p p
-xJ=zJ2-(:, c)
(5.31)
-ss (x) for electroas (+) or holes (-). The smbol P iudicates the Cauzhy principal value. Suc.ha spectrat Alnction tsee represeats a sharp peak at the Fig.5.11) QP enerar sQvP @)with a reduced spectral weight, 1 + waReycvti, s), and s
additioaat broad stzudures x, Imfvovti, (cf.the satelte s) at shlled ener 'gzes structme' in Fig. 5.11), whic.hmay be lterpreted as replicas of the ma.in peak due to additional losse's. The additîonal pealcsdescribe satellite stntdmes related to the shakeu.p excitatious (5.7, 5.28)disctussedîn Sect. 5.1.3. A set of spectral Alnctions conduction bards or calculatedfor electron and hole excitations i:rlthe lowEsst valence bands of bulk Si is shown in F1.5.12 (5.29). For the hole exdtations the satellite strudmes are visible at large.rbinding energies.This was already show.afor the core-holespectral 'hxnction (5.16). Neglectingthe band and a'ad (5.31) is plasmon dispersions the agreement of the entpressions(5.16) obvious,at least in furs'torde.rwith respect to dynxrnical QPeeds. '
'
5.2 Many-Body Efects o.2 k = (0-5,0-5,0.5) 0 0.2 k = (0.4,0-4.0-4) () c.2 k = (0.3,0-3,0-3) 0 c:2 k = (0.2,0.2,û.2) ' 0 0.2 : = (0-1 ,0.1,0-r) 0 0.2 k = (0.0,0) 0 0.2 k = (0.2,0.0) Q G.2 k = (0.4,0,0) 0 0.2 k = (0.610,0) o o.2 k = (0.8,04) () O-2 k = (1.Q,0,0) 0 -10 -35 40 -25 Q0 -15 d0
i
i E
E '
l j
EF i
.
! i'
.
! ë !
l @ i
.
Energy s
207
t @ E
I
E i
i
j: i j: h
-5
0
5
(eV)
)
in bulk Eig- 5-12- The spectral-weightflmctions for electron aud hole quasigartides Si. The Bloch'wavevector varies iong ihe LT and T'x directioas.lt ls given in.uzzits of the topmost occupiedDFT-LDA of z'Vco.The enerqzero ksfzxedat the ezterg state at T'. A'rrows mdicate tmdamped quampaztklepulcq. They are J-fanctiozus with a spectral weigkt smaûe,rthan 1. h the lower J'X part the positions of the peaks essenticy represent the band emergiesdescdbedin Fig. 3.6. From (5.29j.
5.2.3 Screensng Near Smfaces is the Most important for tlle calclzlation of the GW self-energyJ;' (5.23) TMs is deterrnined knowledgeof the drmmscally screenedpotential W' (5.22). by khe inverse dielectric R'nctiop which js deMed by
a?';aJ)s-1(=& d3œ''s(z, 7 a/; (sl J(= œ'). =
(5-32)
-
dielectric 'h'nction The correspondi:ag(longitudinal)
eLm, œ';fzJ)J@ m') =
-
-
a/; tt)) d3acwtœa//).P(z&,
(5.33)
-
îs rehted to the so-cnlledpolazization flhncvtionor Heducible pohrization propagator, P, of the polarizable electronic sys'te,m(5.7, 5.33)lt contnlns the îzreduc ible cliagrxrnqof electron-hole paiz'sexcited virtually or physically as the response of the system to an externf pertttrbation. .
208
6. Eltmentaor Excitations1: SixgleBlectronieQuastparticles
mydiuy q
',,'X',,'X c * b
vacuum
,
yyyy?)
s.13. Iuterface betwee.tt the vacuum azd a polvhable medittm Fig.
l
(schematically).
The combiuation of the three equadons (5.22), attows azd (5.33) (5.32), the derivation of an inhomogeneotzsintegral equation for the detnrmînn.tion of the dyaamîcally stzeenedCoulomb potem.tialW' (5.7),
a/iMzl W'@, =
(5.34)
't7(mz') + -
d3œ''
œ'' d3œ''%(œ '; x''')PLW'' 1U;jWLm'' , a?;F=). -
not interested in the details of the screent'ngre-sponse. Local-held efects dueto the atomic natlzre of rnxtter arc thereforeneglected.Mahfy the spatial iohomogeneity due to the presence of the surfaceshouldbe takextinto account. We considera semi-imqnite solid suchas indicated in Fig. 5.13. The polnHzable'meclblrnoccupies the negative lnnlfxpace, arzdthe interface to the vacunlm is takd'n as the zv-plan.e with z = 0, in order to simplify the coasiderations. The positive cores aa'e sîtuated witlnt'mthe polarizable electroaic s'ystem. Wit%snthe jelliummodel the positive jellillmedge$sdisplacedby z)m below z = 0 (5.40) T.nthe classicalelectrodyzmrnlcsz = 0 desne tite image plane,whereasthe plltne z = .-a:m nearly deMesthe positionsof the topmost atomic corem Sometimes,anothe,r notation with z = 0 for the plane of posi jtive cores and z = zizn for the t'rnngeplnre is used- Exnmplesa're Figs.5.14 a'ad 5.15.
Herej we
are
.
The bnllr polvization propertiœ of cubic czys'talsare describedby an isotropic wavmvectmr- and frmuency-dependentdidecizichtmction s:(c,&7) with the 3D wave vector q or by a correspondingpartlally Fourier-trn.nsformed poln.rizationhlnction .&(Q)z; fzJ) with a 217wave vector Q. The polarizar tion probtem Nmscylindrical symmetry. Therefore: one writeesœ = Lp,z) and q = (Q,qz),wherep and Q are vectozz parallel to the soace. For metals the semi-t'n6msteJezblrn, an electron gas conGaned by a'a l'n6rllte potential barrier? Lsa lequently used model q5.41). TMS approaGcan be generalizedfor other polarizable electronic systems. A good s'tarting point is the mssllmption of specular qlectron re:edion at the surface (5.42). U the contHbutio:as of the quant'lm-mecalmnical interferenceterms to the polarization are neglected,the polarization Alndion of the semi-in6aitesystem z'; a7) z, z?;fgJ) &(-z)Oz + .!%tofz + z':u))J PLQ, (-z') (2%(Q, (5.35) =
-
5.2 Many-BodyEfects
209
emsilybe exwessedby that of the inBnite bxxllz.J-n. tMc way the dielectric respoMe of the halfspaceLsrelatedto the respon- of the homogeneouabtTptsystem. TMs approvsmatîon (5.35) satioes the J-pïrnrale (5.42). 'I'he representatioa (5.35) allows a closed nanxlylicalsolutîon of the eqaation (5.34) for the dyn.nnht-cally screenedCoiomb potential. Oue obtnsnq caa
(5.42) 14r(# #', z, V; M) -
=
+
d2Q bQ(p-g)2xd.2 (2,zc)2e Q
J s
(t-76z)86'''.j. (y.x;e-o(a+z)j z'! wqe-Qz + eL-z4eLz')a.LQ, (-z')c(:, (5.36) g/tzlOa;td)e-QJ') ,
-vjx-arj ) qe
z z + z'pa7lsaLQ, z';=)+ aLQ, gctQ, zqu?lato, z';w)j ),
+ 0-(-a)&(-z') where x
=
-
2/(1+ a(Q,0;t$).The quantity 2Q txl dçz ccstozl a(Q,ziuzl r qzsutr, w; =
j
=
(5.37)
is dkectly related to the Fourier trnrqform of the screenedpotemtik i'c the bulk. In the limit of a local dielectric Aynction,c:(t2, H sbtts1, one Mds 'Gt,h :J) the same screeaedpotential W'(p p', z, F; LJ) mq c(Q,z; :,J)= e-QI>p/sb@) calcalated using the method of clnnqicalimage charges (5.43,5.444. -
Inter&tingly, the screeaedCoulombhteraction of t'wo pazticlesoutsidq the poladzable medbxmz, z' > 0, is !e,511 l'mfluencedby the pohrszation in the halfspace,IEnthe vacu.um xegëoa(z,zê > 0) the chaugeL'E of thû XC selfeaergy wîth respect to the XC esectsalready i'acludedict the XC potential (5.29)is dornsnatedby the conelatioa part EC of the self-enerr (5.24), 62 rks EC. Negsedingthe movcation of .DC due to the dyzmrnicsin the screening, asslnmlmg,e.g., cstqawl - &s(ç, 0),the corrdation xlf-ene'rgy XC is detemninedby the diFerence of tNe scree'nedand baze Coulomb potentîal
(z,z/
>
0), ,
N@, 0) nlœ œ;
-
-
,
m
)
=
-c
=
1 -<(z+z,) p l)e dQJo(QIp1+ o
2
-
,
a(Q,0: 0) J(Q,0; 0) (5.3b)
with the zeroth-orderBesselfhlmctionJo. In the case of a dielectric constant e sb, (5.38) leadsto -(sb 1)/(ss+ 1)x e2/ Lp p')2+ (z+ z/)2. sb(t2) . The accompanying changeof the ener'gg of a point charge located at a distanceRz from the image pln.ne'z = 0 irt the vacqlllm ks#venby (5.45) -
e2
Xcpoëattxg) :F-g =
=
1 do l + o -
-
a(Q,();0)e npg. %0) t$(Q'
(5.w)
for dectrons (-) and holes (+). For large dist=ces from the image plnane, resultsin (5.39)
2l0
5. Elementary Excitations 1: Single Rlectronic Quuiparticles
1 62 = :!: ..E'pco:attRzl cs + 1 4az cs
-
(5.4û)
.
.
This Lsthe classical image potential ei/-rar of a point charge at distance Sa Fom a poladzablehn.lfqpaceelnxractezized by the static electronâcclieledric constant sb. In the case of metalswiti as -+ tx) the prefactor (cb + 1) -,'1)/@b in (5.40)4 the image chazgemeasmedin ltnlts of the elementarychazge,appzoaches1. The image-potential-like behavior of the self-emerg.g dl'seren.ce (5.29)îndicatesthat outside the electronicsystem mry-body esects, such as exchaageazd correlation, cn.nnot correctly be describedby the XC potem tial 7xc (;r) (3.50) of the Kolm-slnltrn tlleozy Becauseof its dependenceon the local electrondensit'y,tEe XC potential slmws a rather abrupt variation near 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 enezgyfundioai that can be used in explicit computations. .An bzspection of the elecwtron self-energyE (5.23) ia the GW approvsrnation shows,howevez,that the electron correlation contained in E automaticazy produces an imagelike potentîal energp The Coulomb correlation outside the surface is more long-rangeth= suggestedby Fxctz). The exchangepart still vanishesmore rapidly for large distancesfrom the image 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
>s
.---.
.P'
-0 1 '
m
ê
X = . c c)
-0 .2
O rs
e.3
/
z
#
J
t d l
:
(.)
>< .0.4
1 ' I
.
J
I I
-o 5 -1.00
'
-0.50
0-00
Dislnce
Piy. 5-14.
0.50 z
1.00
(%y)
Exckange-correlationpotential at
k'
sHple metal surfacecalculatedfor jezrl!'n with am electron gaz paramete.r rs = 3.9318. Tke solid curve represents a local XC potential extracted from the GW self-enera (5.46J1 and the dotted cmwe is the correspohaingLDA potential. The dashedcun'e is the image potential -c7/4(z mm),where zim = 0.07As.is the position of the Gective image p7xne. The distance z fxom the surface is measurediu uztits of the Fermi wavelength Atp= 2a(4zr2/9)T7Z (here:ls = 12.9 aB).After (5.464. a
:
-
.
:ê I ' .
5.3
SurfaceStates Qumsiparticle
correspon.diu.g to the GW self-enerr becomesimagelikeoutdde the surface, wherea.sthe LDA form gives rise to too large potentiat values Lu this reTon.
5.3
Quuipvticle Surface States
5.3.1 Surface Barrîer
'
The escape of a'a electron from > crystal without excitation or ex-ternalperturbation is usually preventedby a surfacebarrier. The formatîon of a stlrface barrie,r ca'a ewsily be ex-plnsnedwithsn the jellizlmmodel of a simple metal (5.471. The positive ion cores are smeare out to a positive background. h the surfacere#onit c-xn be modeledas a stemlike density (Fig.5.15). The electronic wave Glmctionspossess tn.!1Kinto the vamulm. The accompacyiug éxponential decay of the electron density is rœponsiblefor a redttcvtioaof the electron concentration in the region of the positive b ouzd near to its step. lusid.ethe crys'tal the electron density approache,sasymptotianlly the bltlk value tn the form of Fhedel oscitlations(5.484. .Xs a consdquencea net smface dipole layer appears (see Fig. 5.15). The fomnxtionof a smfacedipole layermtu'ns that the electrostaticpoteniial far in the vaclrnrn is higher than the mean electrostatic potential in the bulk. The mn.croscopicdipole and the accompanyin.gelectrostatic potential cu directly be obtxsnedh'om the microsèopic iectrostatitpotential 5Q@): ùr evnmple calculated withsn DFT-LDA (3.48). Jt is defmedby e%
O
ê
XM j.g (U >
V
--
.
...
xxxz
j
X
l
lI t j
:'*-R
g& =
0.5
;H7 Q)
ï d % %
'EP
;'7.(U
.'4n '
&
.-
c.
0
. ..
+
jy
r
a
% X
j
zx
:
zx
X
x N-
-1.0
-0.5
0
gjsjance Z
0.5
1.0
/ Fj
Fig. 5.1.ô. Blectron densit.gproile (dashed liaeland positve backgzound(solid at a jelliumsueace. A metnllsc system with a 1owelectron density, rs = 5ûB) nlikel is considered.The FezmiwavelengthAp = 2rs(4*2/9)1/3 Lsttsed as the rthxracteristsc 'length. After (5.47J .
212
5. ElementaryExcitations1: SîngleEledronic Quadparkicles + kk@), Vklœl= 5(on@l
(5.41)
the local part of the ex-ternal potential Moatœ) and the Hartree potential As one is mnlnly interested in the macroscopic spatial depen5'k@l(3.49). denceof the electrostatic potential in the surfacenormal direction z, it is convenfen.tto perfozm a pln'nar average 1
Ks(z) ; =
A
dzdp7estœl,
where z1.correspondsto the area of the surface tmit cezl. T jte plan.ar averaged fnnction can therz be passed throagh a âlter of lemglh L in orde,r to ex-trad tb.e macroscopic changesof the electrostaticpoteatial (5.492, HTIZ dz/ t = Q Cesl.Zl ). UStt ally' L is chosento be tNe rnsn5rnum thic'ks Jx-zw 5%IZ ness of atomic layers 'glvingclzazgeaeutrality or, as a mncdrmlm, the extent of an irreducible cvystal slab tsee Sect.1.2.2). Examples for the averagedelectrostatic potential are given in Fig. 5.16 for zinc-blendeGaN(111)1x 1 with diFerentGaor N overlayersand SiC(111)Wx W surfaceswith group-lff adsorbates.In the GaN case the surfacestoichiometry is changed.The secondeexample represents the Muence of dsFeront a& sorbates and adsorbategeometries tsee The graphs in Fig.5.16 Fig.4.35). clearly show the ivuezzce of the compound,the surfaceoriemtation, the stoiclliometry oz coverage,a'adthe geometzy on the actual surfacebn.rrser.This holdsixt particula'r for the trNsition rebonitself. 0n the other hand)tlle efect of all these'detailsis weakenedfor the potential step .'A?= Zstxl -'Q(-x) (a)
(b) Q
----
œA
A
ê*
-5 l +4(1
>Q) we
K w'
.
a?
-$5 ,
.
fâ
>
k.
/ #
* -2tl
l @
d
1
tI j
-:ts
J ' 7 't
.
l
t1
*30
$
'-'e.-
.-.
,
? ,
--'----
In (T4) c.o(p n (.q) ; (s$) B tsub.œ
-35 41
:l.,: z':
Distancez
c,i
(atomcc Iayor)
Pig. 5.16. Averagedelectrostatic potentials of semiconâuctorsudases'with difeerent stoickiometry (coverage) sudacesof and/ordiserezztgeometry. (llll-oriented the zinc-blendepolytype of GKN(a)and SiC (b)are chosen.The bulk valeace-band mmvt'vnxrn kstaken as the energy zero in (a),whereasthe vacuum levql desnesthe zero in (b).From DPT-LDA calculations(5.50, 5.514.
5.3
SmfaceStates Quasiparticle
Atomic bilayer Fig. 5.17. Averagedtotal one-electronpotential U(z)(dnmked tineland averaged elecrostatic potential 5Q(z)(solidlinelfor a diamondtlllllx1 surface. The sla'b a'ad'vamrnrr, regbns
are
shown. From
a
DFT-LDA calmzlatîon (5.52).
which ks a dired coasequenceof the macoscopic smface dipole. The detn5lq of the groupqll adsorption(element, atomic geometzy) have pradically no imfluence on the surface barrier. Additionnl overlayezsof Ga (N)on the the surfacedipole by sevâ sttrface)however,reduce(hcrease) GaN(111)1x eral tenths of an ev (several eV).t'a addition, many-body efects, e.g., the modif.gthi absolutevalue of the surface barrier felt image potentii (5.40), Fig. 5.14)but contribute practically nothing to by an escaphg electron (see c'hangesof the surface dipole with geometry and stoickiometzy Exch=ge a'adcorreïa' tîon efects increasethe surfacebarzier, ic pazticu!ar within the DFT-LDA ms indicated in Fig. 5.17. There,besidesthe electrostatic also the total one-electron potential (3.48) averaged over potentiat 5$s(z), the surface plane (5.42) is plotted. 1n.the bltllr region it is much lower in enerr tha'a kk(z) as a consequence of the laœgeattractive contribtttion of e-xchange an.dcorzelatiom I'n the b7llk region b0th potentialsshow oscilhtions normal to 1he sttrface. The wjdths of the oscillations are equal; only their nmpDtudesvary sligEtly. lu this region the dxerence of the (macroscopicilly) taken at the average averagedpotentials is governed by e-xpremion (3.50) eledron derusity.Tllis pammeter is a blllk quantity. It is therefore smportant for tke absoluteenergy position of the energy levels,but it doe not smflueace tlve vadation oî the surfacebarrier wîth orientation and atomic covgttration
(5.481.
2
.
5. Elementary Bxcitations 1: SingteBlectronic
214
(b)
(a)
Energy
Quasipartic!e,s
Evac
Evac
X
*
(p l
CCBM &F 'E---' -'-EE);;-''',,,,,------''''''''-''''',-,,,,,,,,,,,,,,,-----,'--''''''''' '
'-
CVBM
0
0
Z
Z
(b)near a suzfaace, Fig. 5-7.8.Baud diagrams of a metal (a)and a rveml'coztductor showlng the de6rition of the work hmction *, electron aëzkity xj aud ionization en'era 1. A possibleband bend-tngîzt tite semiconductorcase is not visible on the characteristsclength scale of a surfaceof a few monolayers. 5.3-2 Chntaceteristic Ene-rgîes
.
suc,hEtsthe The surfacebarrier can be characterizedby characteristic ener#es * of a metal or the Lonization ezzergrI au.dthe eïectronclrzfàp 'workfzmction by Narious x in the caze of a norrnetal. Thesequaltiti% can be measured the ln the case of a metal (Fig.5.l8a)z spectroscopiesincluding PES/HVS. energy of the highestoccupiedelectronic level withp'n the crystal is stûl ev, where ev is the Fermi energy calcttlated for tlke ideal l'rfln'lte czystal with the tota,l periodic poteatial V'(z).The lowest enea'o-of an electron outside the curstal mizht be assumedto be zero, siace Ftzl and 1ts(#approe a eledron cazl be constant outsideof the mystal, and the kz'neticenergy of a $1.% madearbitrarily small. We call this level U(x) = Ws(x),the nacmum Jcpreô cvac, au.dusuaxyz'eferthe absolateenerpr positkonof sGgle particle states to 9om emc. Therefore, the rnt'nirnlxrn enerr 1 required to zemove an electron the intedor of the czystalto a point outsid.ethe crystal wéuld be +
=
cvac
sp.
-
(5.z13)
Figure 5.16 icdicates that the position of fvac is ivuenced by the surfaco barzier. Consequently,atso * may be influenced by sttrface relaxand nziqorptiozt of other spedes. In the case of nonation/reconstraction metals, i'n particular for sezniconductozs,it may also depemdon the backon tEe gxolmd doping. Howeverlit is better to avoid a strong dependemce doping level and to defne otb.e.rcharactezisticener#es .
X
=
1=
fvac
cvac-
-
FCBM) svsv.
(5.44)
5.3
SurfaceStates 215 Qumsipartîcle
The clectronnmn7't.yx 'vacuptm
ionization energy J)is the energy of the (photoelectron level referredto the bottom of the conductionbandsrcshc (topof the bands
vateace Iu tMs way, the energiesof the condudion-bazd rn1'n5svBM). mn.r'mthm are generallydefmeêaz quasiparticlevalues mplrn a'ndvalence-baud Startin.gwith the Kohn-sha.mvalueszQPcorrediozss (5.30) have to (5.26). be added. The dferettce of the t'wo band-edgevaluo, Sg = scsM evnu, defmesthe fundamental QP rnerr gap. lt exn be directly comparedv-ith vasuesderived 1om measureddata for x a'ad J. Rmdamental gaps dezived from optical mvuzements are in general somewhats'mxlqer êue to e'xcitonic binding eFects.The b=d line-u.pin Fig. 5.18brepresents tlle situation for a rtxlds.qticseznicönductorfor which defects and fmite temperattziea,re allowed. For ideal systems the introduction of a quantity * (lileren.t from I would not be useful. Witbin the axact DFT of an ideal imsulator at zero temperatllre, the highest occupiedKohn-sham eigenvaluereproents the position of the chemjcalpotential of the electrons (5.53). J.nFig. 5.19 the work hmdions for some typical metals az determlmedby vario'asexperimentalmethods (including PES)are plotted vers!zs the average distance rs of the electrons (5.48). The small variation in the work hmction 6vertlsferent cnrstallographâc faôesLsnot shown.There is a cler itlcreaseof + with i'acrewsiugelectron dtanqî'ty r,). This tread js well pronounced (decremsing for nlkn.li and p.11m.15 eartk metals but 1% signifcant for group-x metals. No clear trend ca,n be observedfor the other elements,e.gk, noble metals. Their -
4-s
F
s 4.o o
g t= ?
.uR
Yo
R . .
:
x
(p o ysrx
zjzxf x
. x
3.5
4.x
i ''v
Nx x
so
N.
.
*
2' s
o .
N
.
x
x
o
2.0
x.
w
N
N
xox. o Nx
N
2
3
4
5
NQ 6
Eledron distance r.j(cs)
.;i' ;E :..
)qFig- 5.19. Memsm'ed work fklndioms * of selectedmetals verstzs averagedistance '(,E:/of the electrozlsin the system. Data are taken from A11m.1't metals: opeu (5.482. .2 tilrcles,nll4n1leartk metals;fllled drcles, noble metalm squares,groupqll elements: siars, group-nr =d. -V dements:open tmnngles,trxnsition metals: 511Mtrianlles ..
.
ind group-lo metals: diamonds.The dashedline iuclicates a quadratic variatïon ihf ô witk ra on average.
ElementaryExcitatiozts 1: SingleBlectronic Qumsipardcle.s
2l6
ZnS O N
N
''---
Znse
Sh e, -
6.5
#; @$ g 6 : 55
'ïs
'
'
-
b'x
.v
5
5 4.5 4
N
'--..-
x.x
1nP si
KR lx
x
GaM * -------1/1
+
x
x.
œ-re zn-ry-s x. x.w o
x.
o
N
*
oe xK
V T
.
6.0 S.5 Lattice constant co(A)
6.5
and theoretical values f'ig. 5.20. Ex-perimentalionization ener#es (opea syzjbols) or an'dpprovlmaàe calculatedtusinga 61111 GW zpprovsmatlon (5.64) (flledsymbols) verst'ls cubic httice constant,s cm. The data collectzonis taken self-enerr (5.50,5.55) 1 sttrfaces only data for cleaved(11O)1x from (5.564. For zinc-blendeseznicouductoz's valuesfor (100)2x1 are presented.In the ca-se of elemeatalgroup-l'v semiconductot's and (111)2x1 are shown. The dashed lines indicate a lineaz (trianrles) (squarc) variatzon of the experimentalvalues witbln a compoundclass.
+ valuesvctry only izt the range 3,83 4.52 W with elecHzongmspazameters ra gs 2.12-3.0241. The tendency of a:a increaseof * with the electroa density Ls in agreement with the simple pictttre of the fozmationof a surface ctipole izl Fig. 5.15.1ts strength should be proportional to the dezudt'yof particles. Measmxda'n.dcalcalatedionization energiesare pbtted in Fig. 5.20verstts the bulk httice constant. The ionizatkon energies I belonog to a certa.in III-V, or I1-W)decreasewith increasing lattice group of compouadsIIVLIV, of the ppc and constaut. TMs is rnna'nly a consequenceof the l/oo-variation whic,h donninatethe valince-ban.d ppr hteratomic matrix elements(3.22) ms.vp'm:xo Table 4.1) However, the ionicity of the bonds (see sc4(0)(3.34). a.lsoplays a role. For group.W s-sconductors and III-V compoundsthe experiment/ ard quasiparticle valuesfor I agree excezettly. In the caqe of the II-W semiconductorsthe theoretic/ values tmderestimate the measured are not correctly vczlucs.The reason is not clear. Perhaps the QPshifts (5.30) ctescribedby the applied oversimpMqddescription (5.552. Ionization energiesa'adelectron Alnities are plotted in Fig. 5.21for g'roum cation size.rfhe signs of these energiesare chosen HI nitrides versus hlezremsing an' d in sucha my that the plotted levelsrepresenttlze valence-baudmnvirna m5'n5rna. Their eaea'gydistancef 'J' Tvesthe quasiparticle couductîon-band dependweakly on the group-lll atom. This gap Sg. The ioaizatioa eneyrgies is in agreement with the fact that the valence-bandma-'dmum of strongly -
-
,.
5.3 Qumsiparticle SurfaceStates
217
Pig. 5.21. Negative val'aes of the ionizatîon emeroe 1 (valeztce-baud mnvlrnum, and of the electron n.mn''tyx (conduction-band rnlnimtlm, squares) for cubic circles) groumTH nitrides. The level positions wah respect to the vammm leveâhave been determ':ned for (11O)lx1 surfacc in the framework of DFT-LDA (5.57j. They are sllifted by the relevant QPcorrections (5.50) computed within a simpliûed mofel
(5.58) .
ionic compounds is essentiallydetenml'nedby the anion. The electron xmnlty 'varies dramatically with the cation size. Tlb.evariation mn.lnly reûectsthe hugevadation of the fandamenti ettera gap along the seriesBN, A.IN, GaN, IZlN 25.591. The vatuesfor tlze elec'tron aEnity of AIN aad, i'a particular, BN are smalt but not negative. Onemay, however,eomecttbxt a negative eledron n.mnlty can be realizedby the adsoation of suitable atozns. 5.3-3 State LocnliRation
The surfaceof a solid,brexlrsmg the thrce-dimensionalpeziodicit'yof the cvystal, leads to strong modiûcations of the electronic stnzcvture in its viciaity. Beasides the fact that oaly the 2D waye vector k fro:ai the surface Brillotaim sone instead of the 3D vector k, represents good quautlnm nllmbez's(Sect. the actual surfacestnzctme azd chemsqtzymod.if.g the electronic stntc1.3.3), ttzrc as indicated in Fig's.5.16 ar.d 5.17 for the one-electron potential. lt is obviousthat in the neighborhood of the surfacethe eledronic wave Alnctiomq are dferent 9om the Bloch waves of aa i'n6nite czystal. In the presenccof a staface the wave fttmctions of valence electrons are of two maia types: i. The electronic ekgenvalues of the system with surface do aot correspond to Bloch ene,rgiœof the l'n6nnstebulk crystsl. Thei.r enezgieslie iu the forbidden regfon of the. projectedbulk b=d structure. The assodated electron states are accordinglylocn.lszed at the pertmbatioa, the surface,
218
5. Elementary Exdtations 1: SingleElectronic Quazipartides
) (ét
(th)
N uû
W œ
X
1M w
..-**
7*
> (:n u'o c
.
surfaceresonance State
.
e
=
--*
--.-
c q..;:5
-
.........
œ >
R bou
surface cta
wave vector E ----.
coordlnatez
---->
Fig. 5,22. The tlzree type of eledron state of a trystal witk surface below the vacuum leve) (sckematically). Electrordc bands(solid are skown(a)together lkinesl with the projec-ted butk b=d structure (hatched Re,alparts of the cmrerebons). spondingwave ftmcdozusare also sketchedversus the H'leance z &om the surface
(b)-
and deOy exponemtially1nt,0the bulk tsee These sùates are Fig.5.22). catled ùo'urdstöac.esfxees and result i.n botmd surfacebandswhen var./ing the 2D wave vedor throughout the sudaceBZ. Evnmple for real bozmdstates are #vemia Figs.3.l0, 4.18, 4.24, 4.29, aad 4.37. ii. The eigenvaluescoindde with energiesiu the allowedreson of the projectedbulk band structure. The amplitudesof the associatedwave ftmotion are usually onhaucedat the badacelsee Fig. 5.22) due to rœznaace betveen surface azd bulk state. Mlvlng or Hybridizatioa of thesestatH is axowedfor appropriate symmetries. TKe resulting states describes'urJueeresoncr?,ce (sometimes statœ. They decay into the antirosonanaj bulk. The degreeof the decaydepeadson the actual state and n@n vazy betweentEe two extreme cases also showr in Pig. 5.22. .
The criterion of the euea'gyoverlap of states atlowedin the sezni-sn6nite crystal and balk Blochstates is Tathe,rqttabtative.ne loexlizzktionof dectron stata near a surfaceis essentially êetnrmlnedby the surface bamn'e,r axd it.s mate.idngto the potential enezorof the electrons in thc bulk tsee Fig.'5.17). This e-'m be dronstrated in simple model Yculatjons. For a fzxedk m use a onedimensional repr-ntation to model the total potential F(z)felt by an electron. According t,o Flg. 5.17 tke potentîal deep inside the crystal is describedby a cosine variation 244costzrrz/cl with lattice ,cozustanta azd
5.3 QuMiparticleSurface States
2l9
(b)
V(z) Vco
2Vo 0
-2tz
o
-a
z
-2Vc
-a
0
z
Ylg- 5.23. Model potential enera of an electron with Và> 0 in the presenceof a smface. wko diferent match>g positions z.o = O (a)a'ad zo = -c/2 (b) of the periodicpotentialand the stepike bmier are cahosem.
amplitude 2W.1ts averageval'aemay be usedas the zero of enerr. Onernny
imagine that the attradive ion cores are located at the minirnn. positio:as z = (zz + .1 zlc,oLn= 1, -2. , ) if W > -1 or a = na (n = (),-1, -2, ...) if Fh < 0. The surface around z = 0 îs modelledby matcht'ag aa abrupt potential step V'x = kr(x) (wlziciapproachesthe vacut'rn level at cvacl z = zo to the blllk potential. The resulting total potential is '
-
7(z)
...
V'x
for z > 216costzrrz/c) foz z <
=
zo zo
.
(5.45)
lt Lsevident that the sig.aof 'Uiand the matnbfng position zo are extremely im>ortantfor the locplization behaWor.T'wo emxn.rnplcs aze sketchedin Fig. 5.23. W > 0 is asmtmedand the matchsng coaditgoztls chosento be p = 0 (a)or zo
=
-c/2 (b).
The motion of the cledron is describedby a one-dinnensionaz Schrödinger
equation of the type
n,z d2 + 2,,.,dz2
(-
(3.1)or (3.4)
V'/))
#6Z) fV7(I)' =
(5.46)
The eigenvalueproblemis solvedwithin the nearly-lee-electron INFEIap-
proxsmation (5.48j. In the weak bondl'ng l:'ml't the eignnflmctions and eigenMues are oaly Gghtly flifkreat som. those of a 1ee electron îf the wave kector k (along the z-direction with -x(a < k %A'/aif restrided to the BZ) ls not near a Br a gg rvection pln.ne, i.e., near a point k = ::Ex/a,, +3x/c,... If howeverk is neaz suc,ha point, e.g.?'Vc (aswe vdll asmlme i:a the fozowing), the f:r% electron states e iAz and ei(A-2x/c)aare xlmost degenerate.The true state is nearly give'nby their linear combination .
#v.(z) Yherculting -
Aekkz+ Beïlk-anlntz
approximate eigenvalueproblem readsaa
(5.47)
5. Elemextat'yExcitations 1: SingleElectronic Qumsiparticles
220
21Vcl
Wave vedor k F'ig. 5-24. The 'two lowest eaea'r bamds(solidlines)of a ID m'ys'talin the NFE The dashedlines indicate the free electronbands. approvlmation (schematically). :' .
2 -4-.7:2
am
-
'
svtk)
Kz k m (
Vi
,4 B
11
z
-
REo2
)
sv
-
(k)
=
(5.4s)
()
with Fo is the matzix elememtof the bulk potemtial with the cosinevariation (5.45)azd the t'wo h'ee-electronstates (5.47)This problam can be easily Near the zone b.otmdar,gk = j+ t%the two bandstake the solvect(5.60-5.621. form .
a
acu
a
('j q ). (j ) +
+
-
nz
2
urp,
( vol
+
+ 4a2
v
x
2
(-c )
(:.49)
.
The degenerayis lifted and the free-electroabards spht. There is an enerpr gap of 2(W ( at k = E (s = 0) betweenthe t'wo bands. The two bands are shownin Fig. 5.24, The correspondiageigenstates are
#cu.I+x(z) =
Aeinz eioz + 1 s:lc -vo
=
(-u,s) +
-
5,2 zm
x
2
(-czc) +
e-i',
(5.50)
.
and The resulting banct-edgestates foTn = Oaud 11 > 0, 4:k.:(z).x, cos (m.z) sbl (Iz),can be describedby trigonometric sanctions. ln 'contrwst to the ba,l'srcmse, in tke presenceof the surface altowed eigenl z states can occ.ar witbi.n the .hlndamentalgap with energiess, -D=a.(I) 1%l < 2 2 s < -#-zw,(I) + L5i L.They can be determiued by applying the method Allowing complex wave vectors 3, 5.64), of the tomnlea àarlé stmtctnre :5.6, ,
-
'
'.
5.3 Quasîparticle SmfaceStates
(0
rr)
22l
Pwq.s) i.e., k = -i(2 < q < L'Zoj inside the ij iq, the solutions (5.50) crystal z < za clescribewaves decayiag L'atothe czystat (z< O).The continul:m of states (5.49) is sometimesalso called the conthmlprnof drtucz gap dfafcd (ViGS) They play an importaut role in the explanation of the (5.561. electronic properties of metal-semiconductor contacts by meial-çndnced .:c7) state.s(YGS)(5.561 WMCJIof thesestates with energies(5.49) actuatty 5.641 exist, dependson the botmdary conditions at the interface. Ia the surface caze represented in Fig. 5.23, the wave Amctions outside the crystal (z> zc)must be exponentially decayingfxlmctioxtstc:1< V'x) tz
=
-
.
.$e.(z)
=
2m,
D exp
-
(1rx sslz -
a
The t'wo general solutions for
z < za
(5.51)
.
(5.50)azd for
z > zo
(5.51)have to
be
matchedat z = zc. OT1chmsto assltrne a particular potential defmedby zc and W. Exn.rnplesare plottedirz Fig. 5.23. The matching conditions require for the wave Rlnctions and for their dezivatives
o;!v/a,-sçlzol'/u (.zn)) =
d'lizzbmla-bq = tf#a.+ -
dz
Z =zo
dz
(5.52)
.
a=z,
The two equations azowtlle detomniuation of two freeparn.meters,the enerr eigenvalue(5.49) or the wave vedor q and the ratio of the coeëcientsA(D. Extemdedstudies of the possible solutioms of the mat problem caa be found in (5.62, 5.65j.For the kwo situations representedin Fig. 5.23 one fmds: '
i. If the surface is positioned smmetrically with respect to tke atoms (Fig.5.231, zo = 0) a botmd surfacestate exists for Fi > 0 but not for Và< 0. T*e bolmd surface state (ex-tsting for 'U8> 0)is deziveâ from the analytical contizmation of the lower baad. ii. The situation is revereedif the surface is not symmetrically located (Fig.5.23b, zo = -c/2).'A boan.dsurfacestate only exists if Fà< 0 and not 7i > 0. This state is also derivedfz'omtlze analytical continuation o'f .
the lower band. Anyway the trivial ex-numple describedby the one-dimensionalSchrödinger csquation(5.46) with the potentiat enerr izzFig. 5.23 makes it obviotls that smface bolmd states, which decay into the blllk- and possess eigenMues withs'n the Apndamentalgap of the bulk, may exîs't, depending on the actual sttrfacebarrie-r. 5.3.4 Quasiparticle Bauds au'd Gaps
The grotmd-state calculationswithin the DFT-LDA (orDFT-GGA) allow for an accuzate deterrnl'nation of many soace properties, in particular stuu facegeometdes. For these geometdes the Kohn-shnameigenvatues(3.46) are
222
5. E:ementat'yExcitations 1: SinryleElectronic Quasipaaticles .
automatically detvmlned. However,there is no rigorous justlcation for the interpretation of the Kohn-sham eigezwaluesas single-particle excitation enas STS, PES and EPES,are related ergies. A)l spectroscopiesdiscussed,'sue,h to the removal or adclitkonof an electron. The cozaupondiugexcitations are the rnn5n quasiparticle peaksof whic-h desczibedby spectzalfnlnc'tions (5.15) Ttather)the peakpositiotasdeaz.c Imt locatedat the KS energies&;z(i) (3.46). whic,hare shifted by .X(i) agnsnqt fm.equasipatrticle ener#es s,zQP(i) (5.26), the KS values.Thesesllifts cnn be computedin a pertmbatîon-theor.gmaze Mcanwhile,there eldst many calculations of such QPshifts for ner (à.30). semiconductor surfaces:including the detemmlnationof complete QP band strudmvs
(5.66-5.77).
with exparlmentalobseo The Kohn-shltrn surfacestate energiesclksagree vations since (i) bacd gaps bet'wee,nempty a'ad occupied surface-state band energiesare too smals(ii) the dispersionof the DFT-LDA smfacebandsis too small i'n some cases, too Iarge in others, acd (iii) the placemen.tof occupied surface-stateenergies is in some cases too high by 0.5 1.0 eV relative Thre'e missing pkysical rnnixn'lrnlrn to the bu.lk vcklence-band I'VBMI(5.66j. efects are crucial for the corzect'energ.gposition of the QPstates and, hence, must be consideredto remove the DFT-LDA failmes. First, the spatial nonis more sensitive to the localization locality of the self-energyoperator (5.23) p roperties of surface states tha'a tie only density-dependentXC potentiat of the DFT-LDA. This requires a proper accoant of the nonlocality of (3.50) This nonlocality leads to a modifed disperthe Green's htnction G (5.20). sion of the quazipeicle enerr bands teoughout the surface BZ. Second, the inclusion of local ûe.l(tsdue to the preseneeof the surface i'a the bwerse and, hence,in the screenedlntezaction W' (5.22) dielectric Annctîons-1 (5.32) is crucial for the QPapproach, siace these local Vids describethe strongly at versus vacullm, see expressioa (5.36) imbomogeneotzsscreeniag (bulk-like Third) a'a adequatetreatment of the dyaamical efects în the tlle surface). to d.owith contriscreenl'ng is more împortant than i'a the bulk cue. This %M.q butionsfzom botk bttlk and surfaceplasmonsand the smaltezenergydistance of botmd s'arface states to the Ferm7'level. A11these efects are importaat. of surface states, ennnot Fbcusingo:a only oae esed, e.g., the loewqlszation #ve a gegerallycorrect nanmwer for the QPshiAs (5.55). b=d s'tractures of intrinsic surfacestates i.nthe prototypicat Qumsiparticle elementalsernlconductorssilicon and diamondare comparedwith DFT-LDA electroaic structttres in Figs. 5.25 and 5.26. The 2xl reconstruded Si(I11) A bucldedzr-bonded sktrfacu are selectedas evnmples(5.72,5.761. aad C(10O) or a symmetric-dsmermode: modelwith a positive bacldizk c'b.a.iu 4.2.2) (Sect. Fig. 5.25als: showsbo'and is applied.In the cmse of Si(111)2x1, 4.3.2) (Sect. surface-statebandsmeasmedby direc't and Yverse photoemission(5.78,5.794. are not izzcluded expevlmentaldata (5.22,5.80,5.81) 1.cthe case of C(100)2xl izk the îg'ttre becauseof the presence of hydrogen axd contradidory Sndiags. Figare 5.25 makes the priucipal situation obvio'as. Just as it happeas -
5.3 Quasipartkle Surface State.s
2 >
-6!2--1 >
p)
>-o r
Lu
.
0 -1
a)
2
b) yy.jg 1IIpgIj
-
. ..
:)down
rJ D up
uq,
Enlil
K
1 0
223
. r:,down
J zn utyu
,
Dup
-1 :
Fig. 5.25. Kokn-sitam (a)and quasiyarticle (b)band strtzcttue of the Si(111)2x1 sudace. Thc hatched areas denote S1bz'llc states. Fkom (5.764. Th: dots denote experimeptal data (5.78, Dup and.Dcjowuare explnsnedjn Sect.4.3.2. 5.795.
h: bltllr semiconductors,DFT-LDA is unable to provide ac accuzate descdption of the ba'nd structure of these sarface êtates.Ozf.y the inclusion of a 6111masy-body treatment of the singvpazticle problam by usicg the GW approvlmation (5.30) allows the reproduction of the expersnneutalelectronic stmzctme. ïn fact, usually QPcorrqdions may be even more important tbn.n in bulk semicondudors (5.821. The igurcxs,Fig. 5.25 and Fig.5.26, iudicate general tendenciesbut also specialities for botmd sudacestates of elementalsemiconductoz's.Ln generak the quasiparticlesblh:sof empt.y (f1led) smfacestates a,rçpositive (negative). However,with resped to the bnl'lr VBM the result depeudson the magnitude of its negative QPshifi:. The empty surface states Ddo'wn and r* are shifted towa'rclshigher enerdes,while the relative shifts rwith respec't to the blll'lr VBMIof the occupiedDap an.â'zr bandsare small. The sign of the net sb7'f'h for Dup (x)is positive (negative). E(nany cwse the indirect surfaceband gaps at J ,-+ 0.57.8azkd0.25,/./1--> X-are openedfrom O.4ev to 0.7 eV (Si(111)2x1)
:
#ig. 5.26. Kolm-sha.m (a)and ql4asiparticle (6)band structtue of the C(100)2x 1 suface. The shadedre#onsindicate the projectedbulk band stracture. From (5.72).
5. Elementat'yExcitations 1: SingleElectronic Quasiparticles
224
.
' '.
'.
. '
..
'.
.i. :;: (j; * 1 k
. ..
< 2.w
'
. ..' .'
: ,' 2, .
r
,.
.
:: a . ..
'
.
p
.
.
' ' . '
t:l
s . . .'.s71.. k ... .
.k '
<
mx
: . . .. : .....:..:... ' . .
.
. .
..
.
.,
. .
.
, E. .
.
5... .'
,' .. . .
.
. .
!
.
>
. ' '
'
ç
:. k
.. j :.ù. .7. . I osj .t b .J.(M G# .'.'
:. '
... ;... ..r. : ' ;... , . >J /?I .: . '<e k.'. ,.V . ' .: . .g
qk
.'
., .
,
:F
. . . .
.
.
E
.
.
.
.
2
.
x' V:.kè a ; . .
':
'...
.
.
...
.
.
:
..
.
.T:
b
4:9 %. ..z > ..
.: .
.
j
.
v
.
.
'
<'e
' .
(jj '
: .,
$.5
O
.
.
l
. . '. . k : . '.u.i:.J'p .' ::i7. '. 2....'. .1: . 1.'. ,' ;' ..... Li. .:.. . .. ;... .x. ;:=... .. ::. . v' '. . .. : œ . :( :': ;'. ,.
.. ... .'.. .
. .
c3
C3
>
U).. 0. 5
(D c
a
.
l.u -0.5
a
.
''k5
s 'a wà.
'
a
i
e
h
X
M
:
la
.:
A
'
o
:
:
x
.
-5 l .:
,
;
M :'
: : a ::
Ga :
' .
,
a
.
FSL. .
-1.5
F'
X'
F
r'
X
M
X
r'
1 Fig. 5.27. Kohn-sham(a)and qumsîprticle (b)bard strucure of the IxP(110)1x band structure. From (5.83). surface.The slmdedregbns indicate the projeded61111: azzdcirclœ (5.86J denotemeasuredband energies. The tziaugles(5.842, squares (5.851,
from 1.6 ev to 3.7 ev (C(100)2x 1).The openlngsof the indirect blllk gaps are somewhatlarger (0.6 eV)or smaller (1.7eV).The dispersionof the sudace baaclsis aksoiHuenced by the quasipeicle character. The effed is rather 1. However,there is and the g'* band of C(100)2x small for Si(111)2x 1 (5.68) of about O.2 ev for the 'n- band in the diamond a re duc tion of the dispèzsion or
fase.
'
The situation is somewhatdlferemt for rnn.ny sTlrfacœof compoundsemiconductors. As examples the KS and QPba'ad structures are preseatedfor s'arfacein Fig.5.27. Thks surface is nhnracterized the relaxed TnP(110)1x1 by almost resonant C& and Xs smface-statebaxlds(see e.g. Fig.4.l2).TMS energy overlap between the empty Cz b=d and.the bnllk-contdudionbands is somewhatenforcedwititin the quasipazticlepictme. Efectively the empty than the blllr Cs surface-state band i.sslightly more shifted to Mgher ener#es conduction-bandedge.There is also a small downwardshift of the energies of the occupied surface states. Deospitethe energy overlap the Cz and Jls suzfacestates essentîallykeep thejr localization at the smface.The orbital chazacterof the occupieddangling-boydstates ié p-like (atleast at X), a'adit is localizedon the smface anions. Thi tmoccupieddanglinobord state is also primarily p-like and is localized on the suzfacecations. However,izt general a hybridha'tion of surface and blllk states due to the of-diagonal elements of cxnanotbe excluded L5.351. the QPseklenera (5.23) Characteristic QP gaps between surface states are Jzted Jn Tablc 5.1. They are comparedwjth gaps obtained within the Kolm-s>nm theory and gaps measuredby combhmtion of PES and DES or STS. The small direct gaps of botmd stlrface state within the flhndameutalgap öf the b:tlk elemental semiconductoz's a're considerably openedby the QP corrections, The gap changecould be 100%or more. In most caaes the zesulting QPgaps are ia excellent agreement with measttredvalues. On average the discrepancyis about 0.l eV or less. The situation is dsFerent for the more or less resonaat
'
5.4 StrongBledron Correlation
225
Table 5.1. Almost direct surface-state gaksfor selectedsemiconductor surfaces.A trlmKition energy is characterizedby the k point in the smfaceBZ. Three values (a1lin eV)are 'gwen:the allerence of the KS eigenvalues,the quasipaakiclegap and the experimentalvalue obtained 'by combination of ARPF.,Sand R'THPBSresults or from STS.
Surface
'
,
l Si(111)2x Ge(lll)2xl
L point KS
Expern'rnent
QP
./
0.27 0.62 (5.6% 5.79,5.87) 5.76J0.75 (5.78,
J.C'
0-38 0.66
(5+77J
1 Si(100)2x Si(l00)c(4x2) Ge(100)2x1
0.2: 0.70 (5-731 0.39 0.87 (5.69) 0.80 (5.74)
X GaAs(110)1xl
1.8
2.7 (5.75)
X
1.9
2-9 (5-75) 3.2 (5.75)
TJ k' P 1 TnP(110)1x .%
2.2 2.0
2.9
1.8 2.0
V
2.2
X-'
2.4
.
(5.75) 2.5 (5.754 2-8 (5-75) 3.l (5.75) 3.2 (5.75)
(5-88-5.90) 0.54 (5-911 O-9(5.92) 0-9 (5.92) O-9(5.93, 5-94) 0.9 (5.95j 2.4 (5.23) 3.l (5-231 3.3 (5.23) 3.0 (5.23) 2.5 (5.231 2.9 (5-231 3.2 g5.231 3.1 (5.23) 0.61
Cz aad As surface states on relaxed H1-V(110)1x 1 surfaces.The absolute values of the QPgap openingsof abput l ev are much lazger than for the s'arfacesof elemental snmîconductors. Howevea',their relative contributions to the totat QPgaps are much smalle.rthan the KS gaps.
5.d Strong Electron
Correlation
5.4.1 Image States
I'a classical electrostatic's,a,n elecwtronat a location z outside a polnrszable system inducesa surfacc chazgeaxd in tllr.n experiencesall attradive image potential whose aslrmptotic %nn for large z is given ms ,x? -e2/(4z)(5.40), where z = 0 is the image plane. This imagelikebehavior ca,n be explained clmssicallyby the polarization izducrd on the suzfaceregioa by axl ex-ternal eleçtron, and it hasits analogouscountezpart in the quantllrn-mechn.nscal XC potential. Exactly tb.is has been disçussedin Sect. 5.2.3. 0n a microscopic level, the rearrangement of charge.sat the surfaceis due to long-rangeeorrelatioa efects, wlkicll are absemtin the DFT-EDA becauseof tke e'xponential,
'
226
'
'
5. Elementary Excitations 17Single Blectronic
QRxqiparticles
rather than hwersepower-lawdecayof the DFT-LDA ex mcorzelation potential outside the s'lrGce (5.401. For pazticles outside tlb.eslnrface in the vacua'm the real exnbnnge-correlation potential (asthe self-energy (5.40)) shouldhave atl imagelike asmptotic behavior.It is likely tkat this behavsor hasve,zylittle l'nfuence i:a total-energy calculations, at lemstfor bonded particles,but it is cruciat in tuderstandi'ngthe states of electronsat a certn''n dkstance from 'thesurface.The severe DFT-LDA failtzre iu this spaceregion 1e* states to a poor description of suchsmfaceststes azd to aa absenceof Smage The image behavior is also imfortaat bound by the ''rnnge potentii (5.961. for a correct interpretation of surface-sensitivetèchnlque suchms low-energ.y srAnnlng tplnneli'ngmicroscopy (5.98,5.991, electronclsAactîon (LEED) (5.97J, and inverse cr t'wo-photonphotoemissionexperiments E5.100) 5.1O1q A.tanvxrnple for the importance of such efects is the nvistence of additiozlal surface states due to the image potential, the image-potentiz states or, in short form, the image statc. Thqse statœ are not dedved in any way from btrllr states or from the symmetzy-brealdng eFect of the surface. For their discussionwe study the surface of a simple metal with ap.l'n6nite electronic dietectric cozsstantss -> x or even an elementalsemiconductorwith a Amlte diezectrk constant sb. We know 'that the surfaee bltrrser of the system in the grotmd state depeadson the geometric, clmrnîcal and bondlng details TnKid.ethe crys'tal it should of the atoms in the sudace regba (Sect. 5.3.1). smoothly matah with the czystal potentii. Outsidethe czystaâthe potential becomesconstant with a value spon. Studying the localizatioa of crystal states near the sudace (Sec't. 5.3.3)we have approximated the s'arfacebarrie,r by a stemlike potentii. Thks does not rnrnnsn valid for a'a eledron present or exdted outside the crjrstal. The surfacebarrier must havethe image-poteatial the total singleWithim a '?e,rycrude model (5.1021 asmptotic fo= (5.40). particle potential outside the crystaz (z > 0) clm therefore be describedby .
V/)
=
fvac
-
1 ea ss + 1 k-(' cs
.
-
(5.53)
An electron placed at a distaace z tu front of a surface genezatesan electdc âelcl. This ûeld leads to a rearrangement of the chaz'gein the crystal in suc.h a way that the parallel component of the ûe'ldvaaishesa,t the surface, In the vacul:rn the electron an.dthe image charge -@b 1)/@b + 1)placed at -z produce a'a eledric felct whic,his perpendiclzlarto the surface in the wbole between the smfaceplane. This results i:athe attradive interadion in (5.53) electrozzaad Rs image charge.To avoid conhlsion the image plane ksidentiEed with the plnanez = 0, i.e., approvimately Mriththe electroric surface. in (5.53) lt may be consideredas the pl=e at a distancezsm from the last atomic hyer of the cores) in the hallpace. The.n zim is approvlmately given by (position half the atomic spadng in tEe normal directiom The attractive potenti/ (5.53) can give rise to a self-trapping of the elec-' tron with s < e=c by its own image, as long ms zzo allowed states favlqt inside crystal the crystal for the energy of the electron below svac. The projected -
5 4 Strong Electron
Correlation 227
Energy
// Eg
Gap
.%.2 4z
Eu EF
XXX
'xx. Mz
0
Fig. 5.28. The potential energg of an electron in front of a metal surface.The metal the z < 0 imlfkpace::s assumedto possessa gap arozmdthe vacuum level occuyying withm the empty'bands. The gap Sscharacterizedby tYe upper (U)and lower (L) band edgeswith eneriesa-u aztd ss, respedively.
band stmzcture should have a gap or, at least, a pocket for an appropdate electron momentttm paraMe: to the sudace.S'acha situatipnis representedin Fig. 5.28 for a metal. For energies e < svuc (butlazger th= the energy of the highest allowedconduction states as below the vacultm level), tke electron clmmot overcome the vaclrtrn barrier (withappropriate parallel momentum) and becomestrapped in gont of the surface provided that there e-qn be sn a'llowed(image) state. It can only exponentiaEydecayinto the bulk. Neglectiug alsothis penetration, the crystal is taken to be in6nstely zepulsive.Henee, the potential may be approtmated for z < 0 as
7(z)
=
(5.54)
x.
L'
Outsiclethe cryb'tal the motion of the electron parilel to the surface is nearly :Nee.The remaining Gect of the czystal potential on the motion witbl'n the zp-plane e-an be taken into account in the ânmeworkof the Xective-mnss approxsrnation (EMA)(5-48,5.103). The f:-%e'lectronrnnzqs m is replacedby an eective rnxqs m., whic.h is asplmedto be positive am.disotropic for simplicity. The motioa of the elechtronoutside the crystal obeys a Schrödingerequation Then the wave Glnnctionbelongingto a s'tate with enerr j'.' of the fo= (3.1). f',: e is of the form exp(ikq(p)#(z), #(a)being a solution oî the ID Schrödsnger : equation .
.
:2 d2
t-am V/)1 dza
+
#/)
=
hlkl E'
-
amtl/(Z)
5. BlementaryExcitations1: SingleBlectzonicQuazipartiales
228
Insnite barrier
'Y>z
0 izl the vidnand (5.54), V'(z),(5.53) hyclrogen-likeenera levels (dotted ity of a surface. The accompan#ng lines)aad mve functions (dazh'ed lines)aa'e also shown. Fig. 5.29- SGematic single-particlepotential
witlz the poteatial enegg.g givea in (5,53) and (5.54). .'I'he energy c =.d. the wave vectof :, kjl = k + g, should lie in a Ipocket or gap of the projected The Annctions#(z) obey the boundary conditîon b1:7kband struâtttre.
,11/) =
(5.56)
O
k' For realistic sarfacedescriptionsthe condition (5.56) can be weakened.J.fthe bpnl'lr band energy and wave vector fall icto tke allowedzegionof the projected structure, the wave 'hlmction of an imagestate slzotzld'fapidly decayhto the 1c. l))a1 is formally eqttivczlemtto that of the determa'nation The problem (5.55) o f the radial part Ii (r) of the a orbitals of the hydrogen atom (5.1041. This becomes obviotts if one sets #(z) = rl(r) ir=z and replaces e by with ,g5 Thus: the bound states of (5.55) 5.106). @b 1)/(sb+ 1)c/2(5.10 energiessL < s < evzc + L2k?jr/(2m*)are rendily obtltimedfrom those of the hydrogen atom. One gets a Rydberg series, the image-potential states. The resulting levels and wave hmctions are iudicated in Fig. 5.29. More precisely, not only levels but two-dimensiomalbands s H sntkjl)('n= 1, 2, ...)result: 52k2 L = cvac + s,z(kj1) . + au, 2r4 2 1 RH sb = fa 16n2 sb + l -
-
'-
with the Rydberg constant Jlu
=
of the hydrogenproblem. meLjLzhlt
5.4 StrongEzectronCorrelation
229
must fall in a gap For a given k;i = L + g the baud energies (5.57) or pocket of the prolected btllk band stnlcture. For vzztishingmomen.tltm k11= O acd cs -F x the Rydberg series of botmd states varies in the rtmge -0.85 ev K c cvac S 0. For Bnste parallel momenta also energies above the 'vacullm level are allowed. The forbidden region in the projectedband stmcture should be arotmd the 'vactstrn levei. Tlze localization of the image from states varies rapidly with the quantllm Iglrnber 1z. The averagedista,nces + the surface, (aw)= possess the Gues (za)= 6n2cs(sb dzzlk,ztzllz, 1)/@b 1).The minzurnm distrce amounts to ab ou t 3 k and inc'reases dramatically with n. lt is obvious that for stzr,hdistance,sthe treatment of or DFT-GGA fails. exchangeand corzelation witlqt'm the DFT-LDA (3.59) Only the msymptoticly cozrec't trestment of XC ia the form of the selfyields a correct description. energ.g(5.40) -
.(*
-
In order to describe the Lmagestates, XC elecwts,i:a particular the electron correlation, have to be treated beyondthe approvsmationschemesof DFTnot included in the LDA or DFT-GGA. OrulyXC self-enera esects (5.40) local XC potential (3.50) #ve rise to a correct description of the eledrons ia plre. Therafore, one may interpret the vacunlrn dose to the smface/image the occurremce of image states a,s a consequence of esects of strong electron
cozavlation. We memtion that the inouenzeof the trae surface barrier on the energies can be talcen into account by replacing of the imag-potentia,l states (5.57) the quantllm number 'n by (zz + :) witlz a quaatttm defec't0 S ; K 0.5 (5.964.
(1h) T
Ni(001)
X P
8
F
6
8
S2
sj
jjj
&
;
ps
4 .e'
,
X
r
Sq xx h
S2 h
S2
R
*<
B
0
E
CA 7-
4
B
P
*
2
2
B
R
6
6 4
.
Ag(O01)
8
S1
--
2
Cu(0O1)
O
F
E
X
r
k
X
bands Sc on the Fig. 5.30. Dispersionof smnge-statebands Sï and sudace-state (001)sudaces of Ni, Cu, and Ag altmg the syznpetqline rX. B labels denote obsezvedbulk interband trnnxitions. Hatched regnonsmdicate the projectedbulk b=d strudurœ. The ene'rgyscale is refet'redto the Fermi level ep. The vacuam leve) is markedby a hozizontal arrow. From (5.1101.
230
5. BlementaryBxdtations 1: Single'Electronic Quasipartîdes
The image potential doesnot exhibit the asjrmptotic 1/(4z)dependence i'a the vinsnit.yof the cr-gstalterrninxtion but, rather, exhl'bitsa saturation iu its z dependence.For mvltrnple,the behavior izz this vicizkity is better described by (5.107: 5.108!
7(z)
=
2 z-fzl.,-
-2
t1
e-N.''P (-l(z
-
-
zimll)for z >
zun
for. z <
zim
-
l+zexp
r-aim
(5.58)
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, aacl l are three parametezs describing the dkstanceof the image plre, the inner potentialj a'adthe inverse distanceove,r which the imagepotential saturates,
respectively. Tke image states desczibedaboveaz'e empty in the system grouzd state. Therefore, the/ detectionneedsa spectroscopy in which, iu a f11.s+ step, electrons are excited iato the image state,sor, in gealeral,i'ato unoccupiedsmface states. Such a spectroscopy could be the IPES or the two-plloton photoeznission (2PPE) spectroscopy (5.109q. Figare 5.30 shou measmedsurfacestate bacds Satogethe,r with image-state bands Sz for Ni(001), and Ca(001)) surfacesalong the symmett'y liae PX (5.110J. The 2D bands Sz are Ag(001) vezy sensitive to cozztamination. J.ncoxtrast, the image-state band Sz does not disappearaaer adsorptionbut only shifts izzenergy. The brds Sï show
(b) >
O
w
gu (j ggj
YG
Ekin
x
..-w-==.
;
.--
n
œb
.------x.
1.6
10 nwx
----
A:k...x
w *
----...jyp N
n=3 n=:
% N
1%
Nx
h Qöa
12
Nx
'
x
N
x
1.$ .
.
.4(t.i
n=1
(;.ty
2PPE signal
(a) Schematîcenerr diaram for the excitation steps in two-photon photoernz'mion (2PPE). (b)Enmo-ze-solvedZPPE spectrtzm that recordstke emitPig. 5.31.
ted electrons as a Rxnction of their kinetic eztera. It is obteed after evitation of a Cu(100) surface by photons of emergy ntda = 4.7 ev and F/zzo= 1.57 eV. lnrom (5.1001.
''
5.4 Strong llectron Correlation
Table 5.2. Bindsng,energies,-ers (ineV)an'd efective masses Mage states for various metal surfaces (5.1097 Surface
-s1
-sz
Ag(100) 0.53 0.16 Ag(1l1) 0.77 0.23 Au(1ll) 0.bO Cu(100) 0.57 0.18 Cu(111) 0.83 0.25 Pt(111) 0.55 0.15 Ni(100) 0-61 0-18
-as
I
tin.ml of the
m-
0.08 1.15 l.3
-
Ni(111) 0.E0 0.73 Co(OO01) Fe(ll0) 0.73
m*
231
-
0-9 1.0
-
-
-
1.0
-
0-95
9.25
0.10 1.12
0.1'8
-
' -
0.18 0.05
-
.
öhe expeded parabolic depeademce on k;I (5.57). At kjj = 0: it albws the detmrminationof the bindi'ng energy, .-ca, with respect to the vacuum level. Witltsn 2PPE spectroscopyone ttltraviolet photon with e'atargy1.,). exdtes au electron out of an occupiedstate below the Ferm! level cp into the imagepotemtial state with the quant'zm zmmberrz (below cvacfor kII = 0).A second photon with energyhulsixl the ieared (1R)or visible spectral region exdtes the electron to a'n enera above sxac (Fig.5.31a). The eledron leaves the For normazescape,ckis = Flso-l-ca surface,and its l-snetic energyis meastzred.. holds.The escapingelectrons are recorded.As an e-xamplea 2PPB spectrct'm is plotted versus the k-lnetscenerg.gin Fig. 5.31b.It îs measttredfor a Cu(100) surfaceusing tlle photon energiesXaa = 4.7 ev and Tun = 1.57 eV. TV ptu'kpositions allow the determinationof the bindiag energiœ -sp, of the image states ms well as of tite e'Tecdvemasses 'rn' for meazmements with Anite kI1.Correspondingvalu.esare sllnnnnarizedfor several metal surfacesin Table 5.2 E5.109). They indicate that the model calculations (5.57) give the correct trends and correc't order of magzzitude. Howevera quantlam defed 0 %; %0.5 is nececysazy to accotmt for the shapeof the tru.e stMacebarzier. Tùe paraboEc dispersionis con6rrnedfor the electrons in the image-statebands.However, thea'eare only small deviations of their esective masses from the leo-electron kslae.
à.4.cMott-subbara
Bazas
Xb Mtio tefinsques suc,has DH-LDA or DFT-GGA have been employed V-titg'rea,tsuccess to descdbe the electronic stzazctureof weakly correlated .:
materials like semiconductorsor simple metals and their surfaces.There was
232 only
5. RlementaryExcitations1: singleElectro/c Quasiparticles
necessit.gto take into accotmt additional XC egects in the case of excitations.For more stronglycorrelatedsystems suc,has d- and J-baudsystems, cuprates, etc.: on the other hand, the concepts behind available ab iuitio tecHques are sometimes too lsrnited to correctly descibe the comple.xmanpbody eFects.The question nrsqes whethersurfaceswith electrons i.n rather isolated dangling bonds cazl also show Xects of strong electron correlationbeyondthe esec'tsi'acludedin the standard technlques. D=gling-bond-derived surfacebandswith partial occupation havebeen predfctedfor seve'ralc-lean,reconstructedgroup-l'v surMes | |
a
eV.
The metmc nature of the suzfaceband is in clear contrct to experiment. Photo/mirssionspectroscopy of the SiC(O0Ol)Wx 5.114, W sarfaces(5.112, showsone f:411yoccupiedsudace-stateband which ksabout 1 ev lowe,r 5.115; tha,nthe Fermi ievel at about 2 ev abovethe VBM. In the 3x3 cmse this baad is slightly s'hifhedtoward Mgher energy (5.112). Fkrthermore,inkersephotoemission specvtroscopy(5.11,5-5.117) showsthe e'xistence of an empty surfacestate b=d at about 3 ev abovetlle VBM for the hexagonalpolytypes of SiC. I'n agreemertiwith DFT-LDA c/ctzlations of the d=gling-bond-zelatedband, the dispersionof b0th the occupied and the em' pty stkrfàce-state ban.din the projeciedhlndamental gap is fotmd to be small. The bandwidths amotmt to lessth= 0.I ev (3x3)or 0.2-0.25 ev (WxW); The band.ml'olma and mn.vima occur in the WXW cmse at the same positions aa calculated for the hamfdled dangling-bond band. The tmcertaintiG i'a the measurements do not allow suc,haxï identifcation in the 3x3 case. Sclmns'ng tnnneling spectrœcopy on the 6H-SiC(000l)Wk W aad 3x3 surfacesco'nAm'nthe evleeuce of a smfKe-state gap of about 2.0 ev or 1.2 ev (5.118, Interœtingly 5.11$. a gap free of Rmrfacestates around t'tte Fmn't level and a'û occapiedmarfxcestate b=d have also beeaobservedfor specïcally preparedùmSiC(00û1)1x1 smfaces (5.120q. TEe quœtion arise,swhether the opposmg electronic-strucvture results of DFT-LDA and the eztpem'mental methods with respect to the band occupation and, hence,the metallic or uonmetallic surface characte.rcontradict the surfacereconstruction models, a Si tetramer on a twisted Si adlayer (3x3) or a, Tz-site Si adatom tWx W), which have been verled by various experimental aud toti-energy stttdies (Sed.4.4).In tlle caze of the 3x3 and Wx W surfacetranshtional symmetries of grolzp.l'v matezialsone slwaye '
5.4 StrongEledron Corzelation 222 cuts an oddattmberof bonds FitAin one unit cell. Becauseof the fou.rvalence elecirozlsper aiom all bonding states shoaldbe cömpletelyElled vith electrons, whereasmore oz less noa-interacting d.anglîngbondz should be occu;.'L pied with only one eledron. As a consequence,the DFT-LDA b=d s-trlc'tllre
predict a metatlicbehaviorof the surface.The discrepancywith (cf.Fig.4.37)
the experlrnentalSndsngssuggestseects of strong eledron correlation beyondthe scopeof the one-electrontheozy at least that within the DFT-LDA. The eMremely small bandwidtbs of the measureds'urfacebands suggestthe importance of stroag correlation efects on tke electronic struct'are in the sense of the Hubbazdmodel (5.121). I'n order to include s'trong correhtion egeds on (zlectronsin Hmnglingbond states locaced at the top Si atoms of the addusters (3x3)or Tzl-site Si nzntoms(WxW) we considera ono-band Hub.bard Hamiltonian 15.1221. lt consists of two parts. One descqibe the constderedband in a Mt-nearestneîglbborTB approximation (3.15), while the other term represents the Coalombrepulsion of electroashl suchband state,sbut at (me httice slte. ne Hprni7tonia'ais therefore governedby two parameters. A hopping parameter t of the type (3.7) i.n the tight-binctingpictu're descdbesthe interaction of the S6dangling bonds iu dferent sudace llnst cells. lt is 'asually ll'rnited to the interadion of nearest neighbors.rne electron-electron iuteraction ks'limited to the Coulomb integral, wllick Ls the largest one and not corzedly taken ipto account withi:rt the local dœcription of the DFT-LDA. 'laheparameter U (cf.also (3.64) and (3.65)) describ/ the esectiveCoalombinteraztion between two electronswith opposite spin on thc same Si dxngling bond, which is however e'mbeddedkn a polarszable medb'm. Since the dangling hybrids are strongly locnlsmzed at the adatoo, they do not overlap sigaiftc=tly. The hopping parameter Ià1 ks smatl compared to characteristic energies,e.g., the Alndamentalgap. The surfacebaud formedby the dangling hybridsis cottsequently very îat, and its energy can nearly be takea to be a constant, equal to the correspoadingorbital eaerr (3.71). This flatnerssor nnmowness leads to > strong electron correlation Garacterizedby the on-site repulsiontmrrn ew U in the efective Hmlltonhn. For axl iateraction parameter U large,rthan the bandwidthof the dangling-bondbaud, this co'rrelationefect beyondthe DFT-LDA becomes jmportant. Lf a dangliug bond on a'a adatom Ls fdled with two electronsof opposite spias, the Cotzlombinteraction betveenthem contributes to the electron energy aa indicated in (3.65). A rough estimateof the Hubbard paramete.r t) follows from the orbitk properties and the ezectronicpoladzation hduced ia the vicinity of the dar.gling bond by aa Mditional electron. The relation ?-/= U/aee nearly hoids with the atomic Coulomb integral U and an electronic dielectrscconstan.t &ce of the Gective medblm. n'om the tight-bindi'agutimate of the cohesive energy m âerived a aralueU = 8.39 W for isolated Sî.s.p3 bybrids (see Tab1e 3.1).In the Rlid State Table of Hn.rrlmn one fmds the value U = 7.64 '
'
7
ev
The Xective dielectric constant (5.1231 5.1241.
of a, sudacemay be detem
5. Elementary Excitations 1; SipgvBlectronic Quasipardcle
234
msnedby the mean valueiefs = lz@b + 1)of the bulk atd vacullm constrts. Fbr SîC with sb = 6.7 15.124) an esective hteraction parameter of roughly V = 2 W is estimated. For more Si-zsc.henvironments, e.g., for S#sutfaces with ckk= 12j values of thc eFectiveCoulombinteraction slightly larger tlnxn L-t= 1 W m'e predidH. Tt is also'pceble to estimnte the esedive interRtion pxrxmeter C by mo-qznqof total-energy rligerences betweendsFerent ocmmations (zthn.rge of the dangling-bond bands witbln the Famework s'tates) of DH-LDA calculations. Using sue,ha 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.,negatively charged,and netttral su'percell, respectively.The chazgeshave to be loo>15m.ed at the dangling bond giving rise to the surfaceband of hteerestsFrom th: -
total-enea'gy diFerences E5.112j ozze œtimates a vatue U Q$ 2.1 W D#T-LDA (Vss 1.ûeV)for the Wx W (3x3)suzface.However,the calmklationof total energiu of chargedsupercellssuf'ez'sstrongly from spurious electrêstatic in'
teactions between the supercelks. Hence,the values reprmenta rathe,r cnlde estimate. For the more Si-rielk3x3 surface the value of U appzoaœesmore closely that of pure Si as a consequenceof the rather lazgeSi coverageof tbis strttcture and, hence, incremsed screeaing. The daagliug bon.dsare azranged in hexagonal latticu. For C > 0 the cliagomnlszationof the corres/onding tight-binding l'Ixrnlltozkian Tves a dangling-bozd band with a clispersion a(i)= 2t(1+ 2 where c-ostrc,slqt the parameter s describesthe variation of the V vector iong .f'V, or along the Z'V line (0; s K 1)in the + cos((4r/3)sj) c(k)=2J(2cos((2r/3):) corrupozkdsnghexagona,lsuzfaceBZ. A ît to the dispersionof the dauglingbond b=d calculatedwitbin DFT-LDA yieldsa hoppMg parameter t = 0.014 ev (ï = 0.05 eV)for the 3x3 (WxW) stznzcture. With electron correlation rw C the single-particle problem belonogto the Hubbard Hxmiltotkiaa cn.nnot be solved exactly (5.121) However) in the atomic llmit t << t-/ (5.1221 a'ad.a nearly equal distributiop of tbe electrons in the dangling bonds ove,r the spizzstattxs (i.e.) in the parmagnetic groa'n.d the band with dispersioneLk)known from the uncouelatedlimst splits state), hto two bands '
,
.
s+(#) -
'
) ts(:) /@)+ t-;2) ) +
?7+
.
(5.59)
Oneobtains the approvimnte dispersionrelauons 1z&(l2) and lzs(:)+ t)' |
emerpr b=G a gap rw U is opened. The smfves undergo a Mott trn.nm'tion 6om a metallic to an izlsulating state (5.125j. 80th Si-richsmface reconstructions repr%ent a Mott-Hubbard insulator. Thîs picture azïdthe C and 1 values dezivedAboveexplaâathe oerimental observationsand other calculatiozls. Thy gap between the two bands is deMed by the eledron correlation eneror U. For the 6H-SiC(O0O1)W3x W narrow
5.4 Strong Blectron Correlation
235
suzfaceThemlt'net a1.(5.1161 have alignedthei.r k-resolvedbwerse photoernismiondata to the occupied baaclin an enerpr distaaceof about 2.3 eV. A
possibleuncertainty 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 surfaceis in good agreement with the DFT-LDA estimate. Northrup and Neugebauercomputeda somewhatsmalle'ztheoretici value of about 1.6 ev (5.111). Correpoadiug combhed PES/DES data are not available for the 3x3 stmlctme. Another argumen.t for the validity of the strong c-orrelationpictme is the conservation of the ctispersionin the empty aud 6.lled.correlatiort bands and the reduction of the dispersion of the measuredbands by about a factor 2 in compn.rsqon to 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 uaoccupiedband of about 0.35 ev g5.116) seems to be slightly laœger 5.xlthe W'kW cwse. On the other haztd, for the 3x3 s'arface,thfeozyfmds a measmabledispersionof 0.13 eV. The ARUPS value is srnnlle,r than the exptm'mental tmcertainties, i.e., smalle.r than 0.1 eV. There is another way to obtain the bazd structure givem in (5.59) from '.. an ab initio meareftelêapproaœ (5.1132. lt Ls necessanr to accmately incorporate tlle long-rangecormlation a'aclscreeningefeds in the electronic self-energy operator (5.21). This can be done in a higltly rehable my by the However, as a basis for the GW calculation one GW appyovl'mation (5.23). hmsSrst to trea,t the sadacesystem withi'n the local spin density approfmation (LSDA) to obtain the fltlly spin-pohrszed covgazra(seeSec't.3.4.1) tion. This already leads to a splitting of the former metatlic DFT-LDA bazd (Fig.5.32, dashedlinelizlto t'wo b=ds separatedby a direc't DFT-LSDA gap of 0.6 ev for Uj'xué tsee Fig.5.32, dot-dashed1i'aesl The reslzlt(5.1131. -
-
-
:'.
:((..:. : Fig. ii'q .
5.32. Quasipazticle band stractme (solîd linœ and hatchedregions) of t:e 6H.: jîc(0001)WxW stkrfve calculated ix a fully spin-polarizedGW approximation. ; For comparison the dangling-bond-relatectbands in DFT-LDA (dmshed 'E' line)or izz :;.bFT-LSDA (dot-dmshed lineslare alsoshovn. From (5.113q. '
236
5. ElemeataurExcitations1: SsngleElectronk Qumsîparticles
QPba'ads(Fig.5.32, solid iines)a're fmther separated. Comparedto the lowerDFFIJSDA band,the ocmpied majority-spin baud.is s'hiffeddownto ing
lower e'aergiesby 0.2 ev while the empty zninority-spin band is s'hlfted up to higher ettergies by 1.15 eV. 'I'his is accompeed by a sEght increase of ihe band widths. 'I'he mpzm direct gap betweu the two bands is iZLCZ'O'XSGI by 1.35eV due to the QPcorrections and amolmts to 1.95 eV. This value is itt good areement with the on-site interaction parameter r) of the Hubbard
model. 'No remxrks are necvary. Fizst, at a real &C(0001) surface the s'pin confguration may not be fltlly polazized.I'n fact, wîthin the DFT-LSDA calculation the totaz energy of the spin-poladzedsmface is neazly the same ms that of the urzpohm'zedone. lt eltn thus be expected tut the spin 1701=hation, if Evorable at all, is easily brokcn by noazero temperatme oz other pertmbations, so it senms likely that the real suzfaceis not spin polrized. Second:suc.ha metal-insulator trlmsqtion.due to strong correlation effecvts and ms dâscussed for SiC(0001)/(1l1) stufazo may alx occur on mxrfnex of ferromagnetic se-mlconductors.One example seems to be the EuO(100) surface(5.126!. Fblally: in ligb.t of the results fotmd for SiC surface.sthere are also some doubts tlla,t tùe Si(111)7x 7 swface tseeSet. 4.4.3) should be the only true metallic surface of a semiconductor. The sîtuation is sirnn'lar to the SiC(0001)Wx W a'ad 3x3 surfaceswith their adatoms or adatom clusters. Only the averagedistanceof the zemnsnsnghalf-fzlleddanglingboncksat adatoms is somewhstlarger than the distaaceof the dangling hybfds in tEe SiC case rd.: probably more important, the surface syreening is mttch lvger. One msy exmect that the drxstic reduction of the U parameter h'om 2 ev to 1 ev (SiC(0091)3x3) is enforcedfor the Si(111)7x7 (SiC(0001)WxW) sttrface,rasultiygizï an extremesy small Mott-Hubbard gap. Moièover, fog smaner ratios J/rtl the gap opeeg îs reduced with respec't to the =lue U relcvant in the atomic limit :5.1224. Probably cx'p'erimeatal tenlnnqquessuch as PFXS elmnot rexlly omtzibute to resolveS'UC.IZ a very mnn.ll gap of the order of 0.1 eV. At room temperatmm the styfaceshould look metallic. The naturazway cotzldbe.to work at very 1owtemperattzres. Unforttmatezy,the 7x7 surface shows a strong suface pbotovoltage shift. whic,hresults iu an tmdeftnH Fermi-levelpositioa in the eedra. TH lowersthe preddon of the low-t'emperatttrePES measurements (5.127, There is aaother compli5.128J catjon for the theoreticaland experimental studio of the Si(11l)7x 7 surface. Many bands appear kl the projectedAlndamentalgap (5.129). Their ene-rr spacing is small resulting i'a problems concerni'agthe peak identfcation in PES. Farthermore, a singlmbandHubbard Hailtonian cannot describe the real situation of strong electron correlation efeds g5.130j. .
'
Excitations II: 6. Elementary Pair and Collective Excitations $.
6.1 Probing
Surfaces by Excitatiolas
6.1.1 Optical Spectroscopies
Optical ypectrdscopies a're emergingms particalarly promising tools to probe surfaces,siace they allow for in sitn, non-destrueive ar.d realrtime monitori'ng tmder challenging conditions as may be encouutered,for instance, dzzring epiteal vowth. For epiteal growth by men.nm of clmrnl'calzeadions, such chemical mpor depositîon IMOCVDI, as, e.g., metal-6rgxnsc optical spectroscopiesprovidethe only possibitity for suchmonitoring. Other advantagesare that the materiaz damagean.dcontxmsnation associatedwith chargedparticle bfoltrnqare avoided. Insulatozs can be studied without the proble,mof chargicg eFects,and buried interfacesare accessible owingto the large penetration depth of the electromagnetic rnrliation. Optical tenbnsques ofer micron lateral spatial resolution a'ad femtosecondtemporal resolution. Howeverj since light peaetration and wavelengt'hare mucbularger than surface titiplcnersses (a few i), sucll tec>nique aze actually poorly snnRitive to surfaces. Some ltrinlm' have to be employedin order to iucreasetheir smface sensitMty. Tite expmrsrnem.tal progress k'athe characterization of s'arfacesusing light hmsbeen splrnmarlzed in a coupleof excellentfeviews ar.d monograpbs(6.1-6.51. rfhcoretical considezationzcan be fouud in review aztides by R. Del Sole (6.6, 6.*4. V The probkg depth of light in a solid, even ia the spectral range of Mghest absorption, is of the order of 10-500 n=. For a charactedstic smface layer of O.5 nm tbânk-ness,the relative surface contribution to the total optical signal only Jtnnotmtsto 10 -10 SeveralapproaGeshave been developed to improve the surface sensitMty. The bazic idea is to mexs'tzre dsFerence signakswhich enhauce tke surface contribution with rcspect to that of the blllk-. Fottr terthniquesare comnnonlytlsed. One is snöace (Jzfererùtïc,l reyectance (SDR)speciroscopy.It is basedon measuring the dsference ia reîectance due to chenkicalmodifeation of the surface,for example,often the Vsorption of oxvgen or hydrogen. The percentagedsFerence is related to the surfacestructare. However,to what exteat it is related to the c'lta-qn or to the c'hetmipsnrbed surfaceand.whether or not it is sensîtive to the spectrnnm of suzfacestates and/orto the atomk structtzre of the smface,is in general dilcult to determîne(6.8). Figure 6.1 shows .
238
6. Elementaz'yExcitatioas 11:Pair and CollectiveBxcstations
6 *
@* o
5
c?-h o
m
4
* * *
.5,
*
W 3 c P 'œ
*
.
X
0
@
@
2
(n œ 1
*
@
*
*
;
*
GWoXOOO 0.4
0.5
0 o
0-6
O
0.7
Photon energy (eV)
6.1- Diferemtial reûectazce of spectra a smgle-domain smface for ligh.t poSi(:.11)2x1 laaqzqdalong tlze tq- 11(711)(open aud v LIgOllq cizcles) (dots)directions. nom (6.91.
Fig.
spectrnlrn that docllrn'ents the breakthot'tgh of SDR spectroscopy because of the lzsc of polnHzed light on Si(111) sampleswith singllz-domai'a2x1 reconstruction (6.9). A:a oxédizedsurface is tusedas a reference.After oxygen chemîsorption,surfacestates are saturated a'ad therefore optical tmnqitions across them occtlr at higher energies,having the DR sigmqlrelated only to the suzfaces'tate.sof the cle-ansurface.Thc measured10O%rmisotropy of the 0.45ev pemkyields s'trongevidencei'a favor of the Pandeycb.ainmodel (see a
4-2.2).
The memsurement of the relative reîectance diference for two orthogm naz ligh.t polarhations (z and p) in the surface pla'ne is cnlled renectance csùsfrop!/ (RA)syedroscoyy (RAS).Since the bulk of cubic materials Jts optically isotropic (atleast, as loag as xni'rvotropies due to the photon wave vedor are negligible), any RA observedfor suck cnrsta.lsmus't be related to the reduced spnmetzy of the surfaceor to another syznmetrsy-inre-q.kl'ng perturbation, for >ample an electric seld. However: RXS is not restricted to surfacœof cubic czystals. For e-x=ple normxl-incideztce spedroscopy phkratle1to the c-azs of a uuia'xial crystal (e.g.: can also be tzsed.The wurtzite) great advantageof the ItA. ten'hnique comparedto the SDR method is that it doesnot involve imdta6nedreferences'arfaces,so that it can indeed be usedto monitor C'O-based epitaxisl growth. The theoreticalinterpretation is also simpler in principle, since impoz'tant atzdpoorly known iuformation) e.g., the atomic structmc of the refereacesurface,is not necessary. The t'wo othe,r methods are ''urlacsphoioabsovption(SPA) whic.hmeasures ambient-inducedchangesiu the npolarized reoectanceat or near the Brewster azzgle,aud ellipsomeiry or spectral :JJ#UIIZ/JeJZ'.!/ whic,h meav (SE)1 slzres the comple.xre:ectanceratio of s- and p-polp.rizeêlight.
6.1 ProbingSurfacesby Excitdtions 11
239
z
.1.
7
#
* '
%
y
X
ds!
c=(f.b), syyltol
Eb(œ) Fig. 6.2. Sckematicconfguration of a rededanceexwriment vith poladzed light. A three-layer system is aumed. Tie propagation clirection of tile light and tâe divisior of zts polnHMationdirectioa with respect to the surfaceare tdicated.
Theoreticaldescriptionsof reâectaaceexperimentsat surfaceswith polmshowni.n Fig. 6.2. ized light often start with the three-layermodel (6.6,6.10) The solid consists of the butk aad of a surface layer with efedive tikicloless ds m'achsmalle,rtibsn thc wavelength,j of the ûght. The bulk is mssalmed The surfaceis describedby a to have all isotzopic dieledric 'hlmctios cbtuJl. gequemzy-dependen.t dielectric tensor with eigenvectorsparallel to z aad p. ?1Yecorrespondingcomple,xdiagonalelementsare a=(aJ) and sw@).The ttpper halfspaceis the vacullm. For normal iccidence L$= 00)the comple,x signal is given by the dslerence betweenrefectance amplitudesat (z = 0Q J1tA. and a = 900, 1-J. azd %,respedkvaly,(6.101 -
.
I
zïiz = zlrridss..@) -
W
y
as
(.;
evv
.y.
(w)
(6 1) .
with ,4f: = f:z Dv and 'F = (irz+ /p)/2.Genernlizationsto a'a nrnbient halfspace,a, non-norma,lligb.t incicteuce,and an nniqotropic substrate cau be i fo'uzd izz g6.111. The standard IRASsetup measkkres the polrm''zation state of the reâected and 1m(z%//). Most eaverimentalists publish spectra ligsht,Le.) Re(.zA///) :éf the real part of the relative variation of the reoectionAmplitudes(6.1) ; = the refectivities J?.z formally related to This quantity is I.i% l .1te(1///). : aud .% = 1/ (2for the two polarizations. This situatiou is schematically ! indicatedin ng. 6.3, though the,irrelative dl.s erence js uot directly measured. Thea -
Re
.4/
-f-
=
1 Z.S
ï
R
(6-2)
6. Elemento Excitatiozts1I: Pxir and CoûectiveExdtations
Rx- Ry
y
9 X
X
Fîg. 6.3. Schemaficreprœentation of the dxerential meamlring tec%nsqtze in a reoectamce-auisotropy spectroscopyexpersmeat.The rlîffbrent light poln.m'zationdirectiomsare indicated.
with M = & % and 2t = (& + G)/2 ms the diFeremceand the mean value of the redectivities for orthogonat polnrszations.Herc the approximate validity of the relation 2Re (,&%*) = has been assqlmed. I/z(2+ IFvI2 Second-orde.r tmrmq rw 1.4/12 are neglected.StacetLe algeaence of two reîectivities is studied in (6.$,RAS is sometimesalso termed rclccfcncz d@àr-
ence
spectroscopy(RDS) Accordsngto (6.7j.
(6.1)the relative Gange of the
polnrlzation-dependent reiectivities Lsgvenas z.a R
=
zlu;ds lm c
s=(u,) swta,l &b(u?)1 -
.
-
(6.a;
The sensîtMt'y o? the reEectanceesotropy to the smface reconstructîon ard stoichiometz'yis demonstratedin Fig. 6.4 for GaAs(100) seaces (6.122In Figs. 2.19and 2.20the reconstructions appenrlng in Fig. 6.4 are relatedto the surfacepreparation conditioas.J.t1 Fig. 2.17 possibleatomic structures are given for the indkated 2D transhtional symmetries. 6.1.2 Light Propagation
in Surfaçes
1n.orclerto detamminereNectanceand transmittaace of a'a electromagnetic wave with Feque'ncwy a7 in a cr.ystal surface,one izwsto solve Maxwell's equations for tEe ezectrk displacementvector DLm, uJ),whic,his related to the electric Eeld ELT, a))by the constitutive relation = a/; w)Jy@/, D0:@, ta)) J') d3a/staptœ, (s).
J7
Heret:s aud J label Cartesiancoordlmatesand eœp(œ, Wb(,J) is the microscopic
dielectric Rnnctionof the vacu4tm-crysta,liutezface. Instead of the longitudinal fh:ncvtionin Sect.5.2,herewe use the generalizationto a tensor in order to accotmt for the transverse character of the light. The dielectric hlrction/tensor
6.1 ProbingSuzfaces by Excitations
Photon energy (eV) 6-4- Refectance antqotropy memsuredfor difrerent reconstnlctions of the sllrfxce. The zr-tT/-la7ds is parallel to E0ï1) The àorizontal linœ ((0111). Ga&s(100) mark the zero level of eacà spectrnlm. From (6.12q.
f'ig.
arbitrazy system accordiagto linear-responsetheActually the temsorcàaracter can alsobe derivedhom the spaceory (6,131 dependent longitudiral response gtndioa in the l5rnl't of small photon wave dielectric of the gequezzey-dependent vectors by relating it to the projection tensor onto the propagation vector. J.nthe case of light propagation ia bulk crysta'ls,thc constitative relation (6.4)greatly simpMes. The electric displacement feld contnsmsionly 10%wavelengtkcomponents.Dgher Fourier components ic the total zaicoscopic spstial homogeadt.yof the system. electzic ûeld cnn be neglectedassttrnn'mg of the dieledric fnlnction. That m/-qanq, Thksresults $nan @-œ/l-dependence one neglectsthe spatial dependence of the electron dersity induced by the atomic struM'are of the crystal, umzallyreferredto as local-feld Gects (6.14, 6.15!Nevezlheless,they play a role i:a m=y crystals, in particalar in the static lirnit (6.16). As an approvlmate resalt tbe dielectric respoase can be œ?;a;). = dccribed by a Fequency-dependeattensor c.4(og) J d3œ/stxygta In cubic crystals, becauseof the symmetrs the dielectric tensoz becomH a, = scaza,rq''p,ntit.g ssta7l witlz s.#(ttJ) sbttbillap. can
be calculated for
a'a
.
.
-
6. ElementaryExcitatioas 11:Pair and Colledive Exdtations
242
In the ceaase of a surface:even when neglectiuglocal-âeldeseds dueto the atomk stractme, the Aniqotropy, the spatial non-loelzi'tya'ad iAomogeneity of the delectdc resporse, i.e., its z- an.d z/-dependeace,have to be takea into accotmt. That makesthe solution of lkfax-well'sequations a non-tri'dal task. 'fhey are eexsilysolvei however,for the rigorously simpved c,ax of an abrupt interface between a snmûln6nite czystal occupying the halfspace takes z < 0 aad the vacuntm foz z > 0. Iû that case, the dielectric Ahmcvtioa the fo=
(6.5)
c@;z)= #(z)+ p(-z)cb@).
.
leadsto the well-knowaFkesThe solution of the light-propagatioa equatsoms 0f comse, in thœe formulasthe microscopic nel formulmsof reâectivity (6.171. feata'resof the surface are lost, since the surface contriimtion i.s completely negleded.Oneway to indude it ksto tzse th.ethree-laye.rmode.ldvribed i.u. ng. 6.2 (6.10,6.11). approvh was taken in 1979by Bagchi, A more general (since microscopic) Barrera and Rajagopal They started 9om the jellblrnmodel for the (6.1S). halfspacez < 0. The tmnncation of the bulk leadsto a moocation of the optical propertiœ in a mtrface region of thinlcness%.BaC.Gon the assllmption. that da 'zs of the ozderof a few bllllc lattice const=ts or less aa.dtkus mtte,h smaller than the light wavelength,% << lj the ligbt propagationeqaations were solved.De,l Sole (6.19) genernllzedtids method to the case of larger ia real crystals a'adobtained a'rl e-xpressionfor the pohm'zatjonAniqotropses dependentstzrfM.n contlbution to the rezetaace ARJ/Rfor a given ligllt polazlation diredion c. The result for s-polxrized Eght aud normal incidence rods as (a = z, y)
zlJu x-zkszm (1&.(x(ta)) It
t
c
cbtazl 1 -
)
.
A11surface featmu are mrnbodiedin the sudacerespoasehmction
dz
(zM=$))
=
dz''
-
dz'
(6.6)
lsaolz,,-/;a;)
dz'''As
(6.7)
,z#;aJlnsaatz'', z'''T t.g) (,z?, (z1z'; tojez-l
with the non-local slprfkce contribution beyond (6,5) = Z;tJ) QpJ(z .z/)c@; z';fzJ) z)lsrx#ta, eopLz, -
The inverse (pllmtit'y z''; &zl'e-ï JJ (z,
-
zJ;?z) is defm.edby ihe i'atega.alrelation cz-z:tz,
z'; w)= J(z z'). fxlsaalz/', -
(6.8)
6.1 Probing Surfaces by Bxcitatîons
243
that atotz, It was show.aby Del Soleand Fioriuo (6.201 F,; (.J) shouldbe the non-local macroscopicdielectzictdosor of the solid-vactulrn interfaceaccounting for a'il macy-body aud iocal-âe'ldefects. The occurrence of oF-diagonn.ltevms sœz of the dielectric tensor (a # z) makestke calculatioa of the,surfacerespoztse(6.2) dz'mcultbecausethe iuversion requires to obtssn sz-l,and becauseof tlze fourfoid iatevation. However: for special surfacesymmetries,suchas for cleavageplaae.sof group-l'v sezaiconductors; these os-cliagonaltez'mqvaaish. Also a careful tuspectionof these 1 a'adGaP(11O)1x1 surfaces(6.21) clid not iadicate terms for GaAs(110)1x a relevant contribution to the refedance. The os-diagonaleontributions a're therefore usually neglected. The remaining fzst tnmn on the rigNt-lzaxtd side by a supercez: o f (6.:1 cata be evaluatedbykephciug the seml'-'ln6nstecrystal large enoughto represent the vactrtm as well as the surfaceand bpllkregions Provided that of the crystal tmde,rbwe-stigation (cf.discussionin Sect.3.4.3). (i)the slabis largeeuougllto pro'perlydescribethe surfaceregion of the czystal, i.e., the surface as well ws s'urf'tïcfg-modc'6eê blllk wave flmdions, aad $) the oF-diagomnl tfarrns of the dielectrictensor are small comparedto the diagonal ones, a simple ex-pressionfor the surfacecontribution to the resectivity in (6.6) can be deriveâ (6.6, The sttrfaceresponse AAnctioa (zM..(a7)) 6.211. witit the half-slabpolarizability has to be replacedby 4zrxhJ.(tJ) +*
1
'Yjs xzlkstwtsgl
+K
dz'
da
=
-X
-X
z'; aJ) &Lz (ssokabtz, z')) -
-
.
(6-10)
The quantity in the square brackets Tapidly appzoacheszea'o for z and/or z' outside the slab. Therefore, the tmrofolâiategral ove,r z aad z? converges bn.q the flsmensionof a ienwth, due to withoat problems.Notice that xbtxt?gpl the twofold ictegration. Iu explicit calculatioas it becomesproportiomn.l to the think-ness % of the slab. The memsurablequautity, the n.nisotropy of the refecvdvit.gof light (6.3) vith two perpendicular pohrlzation dirèctionsz and v iu the mnrfaceplane, z.a R
=
lsrraa Im
x:@) xhgtxl
c
-
sbla7l 1 -
,
(6.11)
:ksl'mnnediatelyobtainedfrom the conwponding half-shb polarizabilities. In for the Hsferentîal re:ectivity c-an be derived simi)ar way an eax-pression iom (6.6). 1.I1this caze one has to subtract the polnmizabity for a pmssivated mtrênne,but for the sltrne polarization direction. a
6.1.3 Electron Enerr
E: .
é ('. .
to probeelementazye'xcitatiou of a surfaceis the memsurement of the energylossesof incident electronswith point chargee aud velodty ,t7. 1'a feld orj i'll or near the stufacereglon the eledrons generateaa extermnl e-lectdc
Ei:.Anothe,rway
'': '
Losses
'
6. ElernentaryExdtatiors 11;Pair and CollectiveBxdtations
244 Z
V
0
/,,
sz.,yy-s. yyy ,
.
of a low-energyeledzon with >xg. 6-5. Iuelastic scattering process (resectioa) velocity 't1 on a hn.lhmace(z< 0).Possibleenergy lossesare due to the polarhable The electzon trajedoryis quazimedium v'itk a bulk dielectz'icfrnction sb/,(zJ). elastic becauseof the smn7lrless of the energy transfcr.
the langaageof macroscopicelectrodynamjcs,a dieledric displacementse-ld (coride'nngthe most important longitudinaâfeld components and, hence, accordiug to negaleaingretardation egec-ts) 1
c
r(z, tt --V=r@ 't7à) 1= 't:zja(m mt c = e2/Iœ1 with w@) To be surfacesensitivea reoedion scatterlng geometzy and prsrnary energies m172/2 < 50 ev are used (Fig. 6.5) The enerr is =
-
=
-
-
.
.
small that the electrons penetrate only a few Mgstromsinto the solid. Hbwever,these eledroas are accompaniedby a lonprauge Codomb fteld halRpace. The whic,his screenedby the electronsazldions in the poln.m'zable tot/ eledrscEeld E nxn be related to the sceened Coulombpotential W' in so
(5,22)or (5.36)in in (6.12).
a
similar way
as
the D Eeld.to the Cotfomb potential
r
The enerr losse.s of the eledrons pemetrating or approachtngthe solid are to the clynamicsof the screeningprocœses. The total emergjrtransfe,r r#ated Qis give; by the c'haugein the eaergy densityof thc Coalombâeld.ji.c refectfon geometry essentiatlyby that outsid.ethe polazizablehalfspacebolmded The totat (time-htegrated) by the imageplace z = 0 (cf.Sect.5.2.3): energy t'rnnsfe.r1om a'n inelastir-qlly reiected eledron to the solid occupying the halfspaccz < Ocan be written az (6.22, 6.23) ::
Q s -
(6.1a) ae/+-dt/daze(z)s(a,t)o(z,t), -X
î
where b icdicata the time dezivative.The totat etectric seld .E@, t) in near the hxlfxpace z < 0 kclades the efect of the pola6zablemedbnrn.
or
6.1 Probing Smfaccsby Bxc-itations
245
Witbi'n the sxrne approvlrnntions whic,hhavebeenuseêfozthe calculation of the displacementfeld (6.12), tke total electzicûeld ca'a be related to tke screenedCoulombpoteatial W' by (5.22, 5.34)
--lva j j d3a/
X@,:)=
d/ctz,
z?; f
f/). Y)p(z?:
-
(6.14)
Since the dezusit.gof the scattered electron is simply g'iven by Dirac's & R:nction
plm,t) 5Lm.?J1)1 =
one
(6.15)
-
Mds 1 --V.
X(m,1)=
+K
.
, dt , W'(œ,.?J'I t , ). ;t
(6.16)
-
&
-x
=J; 1-Y) = r@-a/)J(t-tJ), Without the polarizable medbxrn,i.e., for Wr@, the eeressions (6.12) ar.d (6.16) becomeidemticàl. The spatiazsymmetry of the scattea.ingproblem and the tgmeintegration in (6.13) sugges'tthe tlse of a Fomier representationof the tvo Nelds.f = D, E sirnt-lnr to that msedi:a Sect. 5.2.3,
.f@,t)= with p as trlmsfer
Q
=
+0*
d2Qciop z
(2.,4
cu e-i-zytq,z, -.
ap)
217vedor i.zl the zr-plazze.One obtes
a
(6.1.7) for the total energy
+x
j.. j dp,g
d2QF=#(Q,(a)
(6.18)
with the scattezing probabitit for the traasfe,rof the eaea'gjr&,dand a vector Q parallel to the surface(image plane)
.ëLQ,u)? -
1
-
s,r,s,
wave
-u,). (6.19) utkj*qjz.eLq,zzu,lnL-q,z, c
The Fomier trsmqformsof the âelds caa be easily calcalatedassllrnsrtgthat the electrontrajectorgm' = 'l:tl is ventially that of an elmsticallyscattered electron. The time t = 0 js takea to be the momem.t of reiection at the plane (z = 0).Tlzeu,becauseof the smallaessof the energy surface/image losses (FâzJ it holds rztf < 0) = Iwl = -rz@> 0) for tbe z<< zp,12/2) compone'nt of the ve-locity.For z > 0 one :5.r.*
erwq e-qa , y) , D(Q,z,rs) 2c(-iR (6.ay , + q (aJ Qn4 Qazj 2 1 X(0'7'&J) 1 + c,(Q,();a,) Q.)2jnk,u;)DçQ'z,u'1' + vz ss( (u, -
-
=
-
246
6. ElememtazyBxcitations 1I: Pair and CollectiveExdtations
aud 0; uJ)is defined ia expression(5.37), where the quantity aLQ, represents a 3D vedor. The resulting scattering probability =
x:2
,
is (6.19)
cze,t?y:c.tg)2.V(Q?:zJ), c j(-u? qn4a+
X(Q)tzJ)
(-iiJ1) (6'21)
-
-1
gLQ,u't Im =
a
.
09u7)) + a(Q, &s( Q + (t,J vvlaypjjro,) E1 by tsko factors. The resonance-type is charactem-zed The probabilit.y (6.21) -
prefactor is most important for graziag-incidence-likemeasmemen.t conditions, 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 slzrface plsmeequals the phmsevelodt.y of the surface elementary exdtatiop e.g.) of a surfaceplmsmonor pkononthat is createdi'n.the inelasticscattezizlgprocess. The appearis the slzrfaceloss Aznction (6.24). The second factor é(Q,uJ) ing imaginary part indicates its relation to real ene'rr losses.This factor determl'nesthe spectral stractmw of the loss spectranmof the electrorbsmeatzsuaz.yizt its s'tzredwithszza certnn'nelectronenergylossspedzoscopy(EELS), TH is obdotsswhen the wave-vector form, TOFRELS. high-resolutiox(NQ.) ss giy(tJ). of the bl:lk dielectricfunction rltn be zteglected,sb (q;(gJ) dependeuce the sœcalledsurface loss fnlnction becomes 0; cJ)= 1/sb((zJ) Becatlseof a(Q, Thtls, as ic the cmse of blplk scattering = + sb(uJ)!J (6.251. Im(-1/ E1 .ç(Q,tz) spectralstrucvttzre are expected (6.23, 6.251), processesx, Im (-1/cb@))(see evh-lbitsstrong featmes.They are the rme for energies&aJfor which LnwsLuq as in optical absorption, i.e.; electron-holepairs created by intezband trare mnx-irna in the sitions or trnmsverseoptical phonozls.1tt addition, signiâc-q,mt enerr dependcnceof the scattering probability occm whe,nthe condition -
= -1 Reslattdl
(6.22) give lmpittzlLssmatt.The solutionsof (6.22)
whi:e the dnrnping/xz is A11611ed the eigeeequendesof the surface-relatedcollective excitations of a polazizSects.6.3.3 able lnnlfspace,e.g., surface plasmoztsand tpolarlphonons(see and 6.4.4). measuredin The s'pectral featlzresof the surface eue-rgyloss% (6.21)
reâectioa a're determined by the gequencndependentimage-poteatial-lîke screeningof the externnl chazgeoutside (z > 0) tke polnr'zable halfspace. T:e inelmstic scattezing of electrons witic.h pauetrate the m'ystal (z < 0) is ikside the polarizable dominatedby the sceened Coulombpotential (5.36) halfspace(z < 0).T*e correspoadûlgtoti electdc Seld is governed by the In the ll'=l't of weakwave-vector dependence, quantity a(Q,z zûuJ)(5.37). = e-QIz->?I/sb(uJ)j z'b (,g) this te= generates the bulk loss fzlnction z aLQ, -
-
Im (-1/cs@)). The surface scatteakg menhn.nimm consfderedso faz is restricted to the poLzxinieraction of electronswith a homogeneous imnge-potentiaz-mediated ,xp
6.l Probing Surfacesby Excitatious
247
izable hopace. The details of the atomic aud eledroic structuze of the surface are not tnt-nn into account. In a rough approvimation eledron encrgy lossesdue to elementar,gexcitations eo=ected diredly with the suzface itself can be treated xqgnminga three-layersystem as in Fig. 6.2. .&n.additional suzfacelayer with thîcuess ds and anl'qotropicdielectric teasor with disgon.alelementscxtall La= m,y, # is nmnnnedto l)e embeddH bet'weeu The wave-vector dependence of the vacu.tzm and the bulk czystalwith sb(w). t,hesediagoaalelementsshotlld be reglected in the folloq. and sndhce-layer/vamzum t%t the bulk/slxdhce4ayer one derivesa loss fundion scatteringoccqzrs in the Jz-plan.e (Ql1l-axis1, .
.
'
,
(e,-)
-z-(,+-.-1(e,.))
wb-ic,his sl'nnila.rto that of the two-layer system. The esedive dielectricgpnc.'tion is given by (6.26) slnh s fp(al)cosll.g+ f(tzJ) ce.e(Q, aJ) s@)s@)coshs + c:(aJ) s5.T,s
(6.24)
=
:'
l With a'n.ds = Q% a/o@l/cazttsl (7 = z, r). In s(aJ)= c/v(aJ)c=(a;) EfhcTs=l't of qrnx:l wave vectors, Q% --/ 0, (6.M)becomesce.e(Q, œ)= *72: k:ktwl + Qd.qs##(u') Lwljezz @)jwe note
that tha loss stnction (6.23) ks Muemcedby the norrnxl compone'ntsuLwjof the sllrmce dieledric tensoz Efwhic,h1 asually not directly accessibleto optiW ex-perimcnts. :' 'r A.urwnrnpleof the irnrnediaten'niueace of the surfaceLsgiven in Fig. 6.6. It; J. J 'sùows tbe electrondistributios versus' eneror Bssesfor a Sî(12.1)2x1 surface ylqand In agzeememt loss energies&,z smatler than the bulk indirect gap (6.27). Sect. 4.2.2)a signlca'at Anlqotropy is Efwith the zr-bondedchain model tsee ) pbserqd.Lossesrnn.inly happem for trn.nsferredmue vectors Q parallel to ?J diredion in tke suface BZ, whtle they are s=n.ll for wave vectors l':(:thq t Q paroel to 'J'. The primary electron enerr m/2/2Mueaces the peak l'Ft3 ositioa in ageement witlk the prefaztor5.n(6.21). e'mrnpleof the Anotb.e,z ''lilue'aceof the surfaceYsotropy but wîth Iossesabove tke sandamentatgap ks'shownin 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'lectronicinterbazd trn.nsi'fionsfor trn.nsferredwave vectozs Q parallel(%10J) aud perpendicu.la.r (rf1û!) airnezs riigcrence iö Ethesurface As The rehtive spectmcrn gives addi(6.2$. 'ttonal-'nq-lgb.t into the aature of electro/c transitious-in the suzfacerelon. q 'in' his holds ia partictllxr for the promiment derivative-loe stntd'ttre around ùssenea'gies F= = 2 W wMc,hshouldbe related to interbard traasitions on -
''
.
.
.
.
'
.
;.
.
<
x
248
6. ElemautaryExcitzatiozls 1I: PaJZaad CollectiveRxdtations
Exchanged o. (â-$) 0,04
û.12 3-r6 02o
0.Gs
(a)
vw
.a
2G0 40G 60O 800 1000 023 0.05 0.07 0.()9 0.1ï
=
(b)
L-
W
=
=9
=
2
* Q .J
29O
40G
800
600
.G1 0-02 023
$000
0.04 Q25
(c)
..
203
#
z
e
z
.e *e' .x v
400
%
x
*' %
h. -
600
x %. - ..
800
Energy loss ho
1000
(meV)
Fig- 6.6. ElKtron energy 1= fn:nctîon of tke Si(111)2x1 sarface versus tNe enerpr loss Flt,l memsuredfor rlsFerent azicml. nl argles, Qr1!I10q (solidlineland QIIg11k1 (dashed linel.The azzgleof Ycidenceis Sxed = 700. ne pzimazy electron at .y('t/j (1112) eae'rv mr2/2= 2 ev (a),5 ev (b),and 20 ev (c)is varied. From (6.2:1.
the XJ line ia the surface BZ across electron states iuvolving As atoms of the secondlayer. In tEe lsmt't of small wave vedors, Q% ..+ 0', the emerg.yloss fundion separatesinto two con-butions
J(Q,u8 =
tm
+ :gslm s##@) d@)/E'za(Y) (6.z5) a 1 + cslss;l + g suta7lj -1
-
.
Iu the spedral range of loss enrgies Fla7 for which substrate emergiescau be = negleded (1ms'b(az) 0)7the reEectedelectron can only trnneer enerr Fzu? azd momentum LQ to eacdtationsin the smface layer. Thus
p(Q,*)
Q%
=
7+
z + sb@)1m z 1m(&/u(*)l
cs@l!
-1
sz>@)
-
(6-26)
ln general, the secondterm in (6.25) is dornt'oatedby lossesi'a the sllrfnr;e layer. For metal substrstes and even on semiconductor sarfacesthe factor
6.1 Probing Sarfaces by Excitations 6
(a)
5 o
24â
J)Oj
4
.-.
ld va '>
5 o
-)y(;j
3 2
,.j; ,
1 0 A
0.06
.-d
NV m
:
'
@)
o
'& 0.0a
I
o
l,..'.ru .4 . 0'n J
-0.06 0
1
2
3
4
5
6
7
Energy lossLek7 Fig. 6.T- (a) Bnergy loss intemsity measmed for directions of the momectum trxnKfer Q parallel and perpendicularto the dl'rnezeof a MB&g'rOU As-capped sn'vfpce. The inset showsthe LEED patten acquired at 39 eV. GGs(001)c(4x4) with permsssion (b)The relative Jeerence spectmnm.From (6.28) (copyzight (2003), h'om Elsevier).
exceeds100.The mnsn stnlcture in the loss spectrnlm is therefore des2s@) termined by gLQ, the tblnlk' loss fllnction of the aJ) Qdslm(-1/sza@)J, =
thsmxnlsotropic suzfacclayer weightedby the sma)lprefactor Q%.A qualitative explacation of tb.is e/ec't is givea in (6.254. In the ca-se of loss spedra 1 sudace iu Fig. 6.6, ezzLwj Ls real and nearly a constant for the Si(111)2x as a Ahnction of ul (6.274. Consequently:in this case the meastzredanisotropy is dominatedby the diference-sin the flmt te= cw Ilrz/?gttzJl for 7 = y an.cl p = a. One has to mention that formtllas of the type (6.23)are also tlsed to cvaluate the electzon energ.yloss spectra of reconstructedsemiconductor sarfaces(6.28,6.291 The slab approvsrnationis appEedto calculatcthe tensor of the slzrface clieledric R'nction so(aJ) Lp= z, y, z). .
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 iaformation about sudaces.Oneovnmple'issecond-hnrmonicgenezation(SHG). For czystaks,suG as diamond-stractm'eczystalswith vazkishing bulk coûtdbutioa, the SHG represents a powerhll tool for surfacestudies.Mother prominect lt iuvolvesthe inelasticscatteeg of phoexnmpleksthe Raman efecf(6.302. torus in tke visible or W raage by elemental'yexdtations of tite system: e.g., surf'acephonoas and plnmons. It thereforeallows some additional inso' t into the vibrational azd electronk properties of sudacesif its suzfacesnmq'ltivit'y is mxlnxnced.This tmhancemeat is substaatîal since izt general only a mnn.ll nttmbe,rof photons is inelastially scattered ia a cel'tain solld angle. For snml'conductorsthe typical Qmmn.nelciezlc'y has been estimateêto be In all inehstic light scattczing process about 10-6- 10-? jterad.c-ml-l (6.31J. aa incident photon with emergyLulj aad mMe eaea'gyks trxnsêerredbet'wee,n vector g$ tmd the sample,resukGg 5.&a scatteredphoton of a dseerezlt enea'r es and mme vedor qs. The aotmt of tmneerred energy correspondsto the eigecenerr Mu(Q)of an demeatazy excitation labeled by the izde.xx, e.g., tàe phononbrrch) azd the wave vector Q. Eaergy conservationyields
fàtsli tJsl :l;#=v(Q).
(6.27)
=
-
The t+' sîgn standsfor those Raman processcsin which a'n. elementanr axh citation is generated.Tkese are called Stokesprocesses. The nnnnslnslntion of an elementanrvdtation correspondsto the t-' sign. Thcy are referredto ms anti-stokesproc-es. A correspor.dingR.q.rn= spectmlm is schematically drawn ia Fig. 6.8versus the lequency shlfk (aq -&Js).Generationaad xnnihilation of thealementaryemxcitations dependozt Mmperatlzre i;a a càaracinm'stic the Stokesscatttm''ngis way. For not too high temperatzzresNT L F?z,;v(Q) more tutertse.ID.analog.yto energyconsezvatiop the qumsi-momot!zm conserution iaw gives a corelation of the componerts of the pbotoa wave vectors )'
2% c
.'
S
;
-040)
0
'
(tk(0)
'
q- rz't
Fig. 6.8. Intensit'y of inelaatically scattezedlight versus lequency shl'ff
i:%--i)1)r) .
(schemat-
6.1 Probing Surfncesby Bxcitahons
2ë
0)o ,zz
.
yzhk;à j/rj-';j
(j:yk
(j(,y;
Fig. 6.9- Sclnrnxtic representationof the mœt resonxnt onmphononKlt-azz pr( 9ess. Stokesscatter.iagis assumed. *
.
.
.1
. .
.
'.'!parallel to '
the sudaceand the 2D wave vedor Q of tha surfaceelementar '
Hdtytion
l 'Q1 '
(6-21
Lq? 4alli &Q=
-
iBecauseof the smaltcessof tie transferredphoton mm vectora practicall p)11 y elementar.yexcitations with waye vedors sear the center r of the stu ,f,1..e BZ are excited. Amqumingtkat vksiblelight is usedto eexcitethe Ramai trans qç>ttexingîn a samplewith rcfractive'index of about 3, the mnr'== terred wave vector IQlis of the order of 10-6 cm-l. This vatueLsabout 1/101 J :6fthe size of the BZ of an Dmeconstmldedsmrface. E2:EA pronnl'ne'nt Rn.rnan scatteeg by optical phonons exampleis (resonance) :.rlthisscattea'c is mediatedby the electroaic system. The photonshterac l . the phoaons are created or xnnilln.ted via the electron. tEvth electrons and ; jj.(. .: Xbnon icteradiota. Six elementaryprocessescontribute to the so-calledone ) Rnmnn scattering. The mœt resonaat prouessis des crsa yc'shoaon (Stokes) 'E'by a Feynma'n diagzam of the type drawn in Fig. 6.9. The correspondia! ,iptàcàttexizzg probabitity ca,a again (cf.seets.5.1.1 aud 5.1.2) be obtained h'on :!' :j s GoldenRlzle (6.32) ,llqFermi '
'
.
.
.'
:
.
..
.
c,,v
a
P@ila7sl -g.Xl lAsol&&l Ysll Jtruai *x(0) &,.zsl,
(6.29
Juotaq, u'sl
(6-30.
=
-
-
'
j' ( '
'
'
(o l/zi-eteq-azill z) (z 1.:-,(0,-(0))1 z-) (z.' 14-(.-,:.,-)1 0). .)....) (&,& (Sz -F70)! EAaps (r.4w .E%)1 -
AJ'
-
-
-
the excitar the izdepemdent-particle picture used in (5.12), tbn of the electzoaicsystem is hezedescribediu terms of electron-holepais i2a tes 1d)with energîes EA. IO)denotes the bzitîal state of tbe scatteriat It is usually identEed with the gzotmdstate of the system, in witicl E>rocess. the electron-photon iater. :/o electron-holepair is e-xcited.Corresptmcllngk, :'' trièfion Hn.mtlton Jytut (c,a7)has beemgenern.4.7ze(j compared with tue siuglo oiitiic)ezepresentation(5.13)The dependenceon the polazization vector e i: àE l'Aplicitlyindicated.The photonwave vector is mssnmedto be negligible.Th
(1nèontrast to
.
, ,
>m''..
.
6. ElementaryBxdtatiozzs11:Pair and CollectlveExcktations
252
ligb.t scatterïg proceedsin three s'teps-In the 6rst step, the incident photou M;L vith polarization ek exdtes the system, e.g.; a s-rn''condttctor surface: into an intermediste state ld) by creati'ag an electron-hole pair. In the secoud step, this eledron-hole pa,tris scattered irto another iatermediate state with vaaisAingwave vector Q ss 0 via the IT) by emittiag a phonon &zx(0) electron-pbononinteraction Hn.rnsltonianX*. Iu the lmststep, the electronradiatively with emission of the scatteredphoton holepair ia JT)recombi'aes after with polarization es. The electrordc subsystemremn.lnq tmc-hanged Fpags the onemhononRmmnxnprocess. Witln'ln the single-particlepfcture three sttrfazeeledrouic bands are bwolved in azl elementary seatt/rimg process. EM% eledron-hole pair izl a:a intermediate state is azsoaiatedwith a conctuction and a valenceband. The electron-phononinteraction cac scatter the electrou band. in anothe,rcocduction (vezlence) t1lolel The most prominent coupling mecltn.nistznsfor bqllk semiconductorsare deformation-potentialscattezing and pola,r Fröblic,h scattering (6.3% 6.321. 80th. zone-cente,r trs.nwerse optîcal(TO)a'acl lon#turlimaloptical (LO) phonozlsshowa deformxtion-potentialeoupliag to the electrons. 1I1systems with pmially ionic boads the longitudinal phononsca'a aksocouple to the electronic syste,m via the accompanying zong-razge electzic feld (Fzlblich dependson the The strength of the Rm.=a,I1Jtmplltude (6.30) m'vtbxnsRm). The Rwtrnazl amplitude depends scsttering geometnr,denoteda.s qitei,eslçs. not only ozt the directiorusof the light polarization ez oz es but also on
the direcdon of the phonon displacemeut.For tEe
(100)surfaceof a, zinc-
()01J
1O41 E0
-
ej
qy
qs es
Fig. 6-10. A possiblepohrs'zation coHg-urationfor backscattering at a face.A.longitudinal phononcnn be obsmwed.
(100)sur-
6.1 Probing Surfacesby Excitations
253
blendeczys'tal lo-symmetryl and a bnnlcscatteeg covguration, tw Ir(ï0()1 '
the created/xommllated b',lk. optîcaalphonozzsare polaœized 11(1OOj, aloz:gthe k010)/(0011 direction (TO)or along the g100J directioa (LO).ln and qs
this coHguration tke T0 phononsare not Rsrnazwactive. The zone-center LO pbonon ca.n be excited via the deformation-potentialmeclmnsqm whec + escsv) (cyvesz # 0 holds for the light polarizations, e.g., whemci 11(010) and es ILjoûlq or ei (1 es IrE011). Suc,ha situation is showain Fig. 6.10. For surfacesthe symmetzy îs reduced..A gene-raldiscussiozz of the symmetqy of the so-calledRxrna'a tezssoz's(6.30) aad the correspondsngselectionrules can be found in articllasby Loudon (6.31, 6.332. The 1bmazt seledion rules for bnztlrscattczingfrom a (110) surface of a zino-blendecrys'tal are completely dlfrereut. ln the blAl1-only TO phonozzs caa contribute 'via the deformn.tion-potentialmeclmnfqmfor polnrssations + figdsvl# 0 Or leiatGz espl + csatc;-,c lEipGz eipl)# 0, for instatlce in a parallel covguration eë 1(es 11 or for perpendicala.rpolarizations (1ï0J aad es Iy(1101. ci 11 For that reason the strongcst peak in the Rnmn.n (001) spectra of a relaxedIA(110)1x1 surfaceizl Fig. 6-11is relatedto blll'lr zoneceater TO phonons (6.344. However,the 'Lwospedra for paralle,land crossed polarizations evhsbit signifca'at dilerenccys. This is due to the.ir surfacesezl= sitivits Thc photon energieshtz;â = 2.96 ev of incident or 3.00ev and Fttz?s scatteredûght are in closere-sonancc qrktitthe electronicsllrface b=d gaps at -
*
R
6s
5
S
S'm *
c
n o
ç-'l
cy o
2
i
f 2 5
: ;
i !
:
*
c: o ..-, =c
(v
E
X ne
.
i ë
!
J -
! ! : i i E :, ë j i : g : i ; : i ; : i
:
E
w
> ...'
i
:
:.
3 ! :,
i
a47 : ;
: :
.
;
: :
'
2,4270
:46
,
; ;
. '
'.
'J 2
-
s
2 :
(j l jj (j .â.
J
-
(110J(1 10)
2 : : : ': : : :
x
i ! i '
30S
100
200
300
Raman shift
2
e I ..j.o s
po-jlg-l lcj 400
(cm-1)
Fig. 6.11. One-phononmamanspectra of the clean 1nP(110)1x1 surfaceobserved în the baclrzcatterlnggeometry for parallel and crossedpoln.rszationcovgurations. The photon energy of the inmdemtlight is huh= 3.00eV. From (6.34J.
6. ElemenàaryExcitationsI1: Pair and Colledive Exeitations
û54
Fig. 5.7 aad Table 5.1).Consequently,the X and X/ in tke sarfaceBZ (see .
generatkon and recombination of electron-kole paiz.s mediating the phonon scattezing happen =n5nly in the locnlization reson of the contributing stu'Fig. 5.22). Rtrthermore,tke surfacesymmetry is faceelectronicstates (see. Table 1.3).Apart Bwered. The poiat group Ls reducedfrom Q to m (see from spectral featmesdaeto multiphonon procves surfacephouortpeAs of and A' symmetry of 69) 146:254, 270, and 347 c'm-1 appear (cf.(6.35-6.371 the discussionin Sec't.6.4).
Pairs: Excitons
6.2 Electron-Hole 6.2.1 Polarization
nAnction
Tlb.eceltzal quaatity ia the calculatioa of a dieledric Ahnction or a polarizabilit'y is the polarizatioa flnnction or irreducible poladzation propagator, P, of the polarhable electroaic system. Accordimgto the dsénstionof the and tite 3D Fourier trxnqformationu dieledric G3nction (5.33) (longitudinal) pohrszability tsusazd d-iezectz'ic the macroscopic dielectric tensor a.#(aJ) ceptibitity) x.p@)e.an be relatedto the correspondinglongitttdl'nal quantity photon wave vectoz in the direction # = g/1çIby depencllngon the vn.niqlnsmg
sçn,u,)
=
)7 tlceco.pta'lt.?p,
(6.31)
o,p
c(o@) %n+ 4rxcg@l =
with
:(4,t(J) =
1
-
55=
q->0
a 'ûLq4d œ
t z d.a= Je-$q(x-=z).p œ (zœ, z ; tz7) (6 aa; .
-
The Fourier tzxnnform 'ïLqj= 4zrc2/(F')ç)2) (F'is the 'volllme of the syshmsbeen i'atroduzedhere. temlof the ba'reCoulombpotential v@)= c2/jm1 I'a contzast to the earlier notation of the polarization fanction in relation the four spatial argametatshdicate that the polsm'zationhtndion (5.33), a/z/; (gJ) 5.s:ia generaz,a four-point response'h'nction i'a P(m,F; uJ)> ?Lxœ, accordrce with its rehtion to the No-particle Green's Al'ncwtioa(6.38-6.401 The horizontal bar oa P indicatesthat, in contrwst to the microscopiccase P, local-feld eseds are now explicitly included in the deterrnimxtion of the polazization Glnctioa of the rnnmoscopic clieledrk h'nction, The combiuaallows the K-draction of the ar.d (6.32) tion of thc two representatiozzs(6.31) complete macroscopic dielectric teanqor.Consequently,the response of the eledronie system to a trrmgveHe pertttrbation suchas light cnn be described density responserepreseuted bz the poby stat'tiug from the (locgitudfnnl) lnn''zatîon fllnction X(a=,a/rt/i aJ). Without Coulomb correlatioa, i.e., witht'n the independentmquasipmicle with approvirnation ttsedin Chap.5,,it holds J'5- Lo (6.22: 6.40)
6.2 Eledron-Hole Px5m:Excitons +x
.!b@lœz,e1 œ'z;Y) -X =
The replacementP
(u
255
,
+ *)J@'z,œz;s). 2rrî t?@l,a?z;f
(6.33)
Ln z:l GG ks often also C'aIIHthe random-pbxqeapHowever,this is somewkatincorred (6.41j Even when provsznatîon(RPA). aad -are negteded,the Koha-sham Tnrniltonia,a (3.46) J quasiparticle eEec'ts llence G contlin efects of both exchaugeand correlation. Ic general, the 'single-partic'leGreen's 'A'nnctioasG for electrons and holesobeythe quasipar'ticle equation (5.21). Thc Coulombcorrelation of aa eledzonand a hole caa öe mcludedonly ic 'an. approximate rnn.nner. 'ne most important long-range mteraction of the two pat't i c les occllrring in P shoald be treated vithl'n the sceenedladdez for This is basedon the GW approvlrnation (5.23) appTovimnAsoa 6.421. (6.40, the setf-energyof atl inclividual particle. The kernelof the most gemerali'nte. gral equation foz P, the so-exlledBethe-salpete,requation (BSE), is given ms a hlnctional dedmtive of the sel-enera with resped to the single-partîcle fllmction. Within the GW approvimation (5.23) the kernelis replaced EGzeen's unde,rthe additional assllmp''ijkthe screenedCoulombinteraction I'F (5.22)1 ïion that the derivativeof the screenedinteradion W' with resped to G ca,n A secondshort-raagecontribationto the k/rnel .%' e'.neglected (6.22,6.40,6.43). or, pf the BSEfor P can be traced to the possibilit,gof particle mccaha'age (6.4% Eihenseamlhsngfor tke mnnrtx%opicdidectaic fnnction (6.3$,to a Coulomb t(' ho restrided to one ''n't ce,n rpjtelatter thteracthon (6.44:6.451. g@)= rtmllsraugrt 'àescribe,s locabûeldegects.For the pobrlqation flmction dependingon only é ($de gequenc'ya closeêBSE follows by negleding the dyunml-csita tlle screenœ?;0)taken at (z2= 0. .: by the static one 'A'@, CQ .; Le., replackv T,F(5.22) Ci'' he BSE then becomu (6,22, 6.42,6.46) =
.
..
.
'.
J'5@zœc, œ/aœ/zif,gl œla(;ts) Jmtœzaa, (6.34) dza œ'za/z:tz?l œ'azaj u')Wr@3,z'a; E.:?:@zœ2, 0)#(auz'3, P @'aa(, 2.:0 @laa,traza; w)'p @t? a?a) zla(;a81 tthe factor 2 incticatesthat the sulet ftmaion is coztsideredazd formany =
-
j Jdz'a -
.
'.
-
.
ry
th'i spin s':rnmation ha,sbeencazriedout. : E ( ;'
'
7d:2 Hnmlltozgau :i.. r 2 'rwo-particle '
*
'
'ètti. e txse o f the C.w approvlmxuou suggets the diagonalapprovlrnxtjon (5.25) for the siagle-particleGreen'sAanction.Thksm/u.r'l that the wave fud (5.27) 'without quaaiparticlee/ects, e.g., l'lttnctions are replved by thorye(#v,@)J E Erttie' solutions of the Kohn-sha.m equation for the surface problem (3.46). r ormoze ' the dîagonruGrœn's 'htnctîon cau 'be replaced by i'ts zvoth'Eltirt,h' ( f....... 'Vdèr (5.27)with respect to the satellite stmzcttzre. Only the eigenvalues ..J ( Ejkiï shrfkedby the quasiparticlecorzections(5.26). A rigorotts proof of this L.IJ. J':' ''
,
.
,
...
'
.
.
. .
.
6. Elemectar.gExciiations t1: Palr a,rd CoEeciiveExdiations
procedtzrecxnrot be Sven. Howevez', it caa be arguedthat tMs approximation js cozzsisten.twith the neglect of the lequency dependenceof T&rLu the BSE Jt has beev shown that the dyzmmicefects of W' in (6.34) laœgely (6.34). comptmqate the stzengthof the satellite stntctures related to the frequeucy of the self-enerr operator (5.31) dependence (6.47) Tke eompleteand orthonormzlszedset of zeroth-orderquasiparticlewave hlndions (%u(z))allows a Bloc,hrepresentation of the polazizabitity ic For the diagonalelementswith respec't to the light polarization direc(6.31). tions one dezivesin the long-waveleugthlimit .
262:2
xpp (=)
-
t
v -
/
)J'b(cw:,'n'c'& ; u') E X''2(Ms-Q(:)pzc,J'v,(R
-
c l z',5c/,v',;'
-ts)1
M(,z,i (k8')23* (ctJ:,'c'c'k'q (:)z4J*v,
with the Bloch matrix elemautsof the velodty operator
'tJ
(6.35) (6.411
tck8l'olvil Mj,(:) sc(k) evqvt -
(6.36)
.
-
The fador 2 h'om spic sïlrnrnationhas beentalceninto account. ln the liznit the matzix elements of the velocity of locaz shgle-particle potentiks (3.48) apa,rt from a operator are identicalto thosecf the momentllm operator (5.13) only hllly factoz given by the Fee-electroamyms. Jn the representation (6.35) occupied valencebands ('n)or empty conductionbands (c) ard interbaud trn.nsitions across tbem are considered.Jntrabandtrn.rqitio:as are discassed in (6.35) we sxl'rn over late,rin the coctext of surfaceplasmons.Conseqpently, pnl'm of electronst'a condudion-baudstates lck)acd holes in valenceband states lwi) that are virtually or physically excited by photozss. The Bloc,hrepresentation of the macoscopic polarization ftmction
P(X1R,œiJ(iL;l (6.37) = l'ala; $b))#z. .P(1:A'z, (aa)#k(z'z)#.k(a() F-1J-.2 @1)#k A7.,Aj, àasàg
with the abbreviation l = vk obeysa BSE whosek-rnel is governedby the m>trix elelents of the Coulembinteraction (6.49)
.F(ltI:, laàû) =
-
d3œ
(=') 0)#Aa@)#); /:k@)#àz(a/)k&r(z,z?; (œ)5(==')#Az@')#lj(œ')1 2/,)@)#Az
d.3z' -
-
.
These do not on:y resonantly couple eledzon-hole pairs cw azld c''v' but contain also antiresonant Mteradions cw and wfc? as we2 as non-particlecoaserving terms in whidz three conductiomband indices or three valenceresonaat terms are takeminto band izldice,socmm Usually only the 1m.q35mg
6.2 Electroa-HolePairs: Bxcitons '
accotmt
detaileê discussion' see
(fora
257
6.43,6.48,6.40)). The omitted (6.38)
efects are only importaut in special casxsesto appronr'h cxtreme accauvy. M exmmpleis the calculation of plmsmonresoaances where the rnlxn'mgof interband trn.nsitions of both positive ar.d negatkvegequenciesmust be irs cluded (6.50) Becauseof the vaaislasmg photon wave vector in (6.35) the :'mlnomogeneous BSE can be substantially simpved. Since,together with the siugle-particle Green's Annc'tion (5.27), the Bloch representation (6.37) of the polarization hlnctioa for inâepemdentquaziparticles (6.33) leads to azz energy denoMnator, the relev=t BSE c,an be transforrned i'nto a'n. iAomogcneous t'wo-parqcmle Sclnrödsnger equation in this represeatatioa26.4% .
c'''z/'k''j &.(ùd PLc''v''k''z/c'#';r,g) + X-.2 'X-.I gzftcpi, Lrnléccwévvz/lsszzj -
c/d qzd?'
1
tày
=
-joc,Jvv,J/u,
(6.3g)
with
.FJ'tcw: + a'(c.r:Ic'v?1')(6.40) , c%'E') gcQP(:) c 'lJ é'cczlvv/tkjp cQP(E)j =
-
.
A small dampicg r of the electron-hole pair c''t?iis introduced in (6.39). The solutions of (6.39) deterrns'nethe inter'baud poinrlrzability (6.3$). The He-rznitian quautity Hlcwv, c?rTV) ca'a be interpreted rzs the Bloc,hrepresen(6.40) tation of a t'wo-partscleHamiltooa'a dacribing the intertal interaction of arz electron-holepair (more precisely: quasielectron-vpnsihole pairlar.dits inte'raction with othe: pairs. Aftez real or virtual eaccitationof eledron-hole palrs with photons, the e-xcitudelectrons and holes do not only interact with the stzrrotmris'ngremnsmingmlemceeledrons resulting in the renornmlization to fwasipacticlœ,sv (&)-> gQ,zP(:) there Ls aksoa direct intcraction (cf.(5.26)); IF (6.38) betwee,nthe excited electrons a'ad holes. This contes the bngrange eledzon-hole attraction deterrnimedby the screenedCoulombinteraction x' -W. The additional jmscreeaed) short-rangeiuteraction ,xz 2/ represents a'a electron-holee-xchacge A diagrnmrnaticrepresenyation of (6.42,6.46). the t'wo contributions is given in Fig. 6.12 (6.49;. For a vxnseîng electron-hole
(b) cT
Cj4LL4LL
ck
vf :
, vP
và v'J,
cI'
Fig. 6-12- Seahernn.tic representation of (a) electron-hole attraction a,nd. (b) eleotron-hole mcchange.The screenedjlnqcreened) Coulomb intezaction is indicated by a duhed (dotted) liue.
6. ElementanrExcitations 11:Pair aud CollectiveExcvitatiors
258
the polarization Alndioa in'(5.3S) follows to give (6.40) + irlj rsjng tbjs P(i) avQP(:)?à(a) PL=k,v/c/i'; a7) -%c'Xn'%k,( gcoc one obtxlnq the well-kcownformu-la(6.514 qllxntity hzthe stusceptibitity(6.35)
interadion .E H 0 ic =
-
-
.
for the iaterband contribution to the dielectzic tensor i'a the GdependentP quasiparticle approxnemation or, with the replacement sQ,, (:) -> s;ztil,in the independent-particleapproYrnation (6.521. 6.2.3 Excitons
BSE (6.39) can be trn.nRformed kto an e'igenvalueprobThe inlnomogeneo'as lem involving the efective two-particle Hnmlltozzia'a (6.40)
Hl=k, c'w'i')ad*'(#').Eu.4rtil 5--2 )((g
(6.41)
=
YTQ?LV
with eigeuvckluesEA and eigenvectors x4T(i).The iudex .t1represents the set of a2 (plnamttl,'naumbers of a.u i'ateracting electroa-holepair in the b=d states c,i and 'ck. Becauseof the attractivc electron-hole interaction both bouad and scattering states are pos
.
z
The eigenvectomallow a spedral represOtation of the polazizstion flmdion = PLC'IIL , 'u'ctk'; (,7)
-
.AI'(:)A$'v''(V) J-) sz + k.s; s,(;.a Z
(6.42)
-
small damping parn.rneter P. The diagonal dielectric susceptibility takes the trMal fo= (6.35)
with
a
2
x=(u')
-
x
282:,2 v
57 J-2MJv(:)-4$**(:) c,,z,:
z
1
EA
-
+ ir) Attd
+
1
EA +
hlul+ ir)
.
(6.43)
The optical propez-tiesare drmsticallymodsfledby excitoaic eseds. TMs holds for the oscillator strengtlls becamsethe siagle-particle trxnKitiozsmatrix elements Mcxh (i) are weightedby the eigenvedorsWIJ(:). In addition, the joint density of states of the two particles is càaugedas indicated by the new pair
6.Q Eleciron-HolePstlr.q: Bxdtons
259
:
Fig. 6.13- Imag'inazypart of the hwudacpdependent dieledric fnnction ibr bulk Si i;it comparison with expezimentaldata (dotled The theoreticalspectra liuel(6.542. :, 'have been calculated at rlsge-rent levels of the iuc-lusioaof many-particle eFeds: appronc'h,and (c) :.(#iudependent-pYicle approach, (b)iudependent-quasiparticale gCouiomb-correlated quasipartkles. From (6.532. ; t
:'
.'
lénergiesSz and theiz dependenceon the quantlnrnntlrnbea'sA In. a s'pec-trtnm pve.ra wide eneror range a,s in Fig. 6.13c the ftrs't Xed, the coupliqg of (11.6ferentclectron-holepaizs, Lsmost important. Near a'a absorptionedge,howEeker: the density-of-statesefects, for exxmple addition.atbolmd-statepenlcs, àrQ dorn:'na'at )j : The 'nBuenze of many-particle efec'tson an optical spectmtm ksdemonstrated 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 stepsof approximstionsare Lfsedfor the i'aclusionof mry-body eiects. For Coulomb-correlated electronbolepairs (c)the fltll Hnmîltonian (6.40) js appDed-Witbin the indepOdentRuasiparticleapproxlnnation (b)(6.411 the electron-holeinteradioa S is nethe single-particle jlbcted.Iu the independent-particleappronrth (a)(6.41J i'a (6.40) are replaced by the Kohn-sham eigemwalues Vergies sv (k). The Eq. eFectssbsfi the spectrtlm to iligher ener#es iu accordaucewith jiùasiparticle tîe openi'ng of the gaps aad trxnsition energies. Tke Coiomb iuteaaction :: kikesrise 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-scribedM'ith the ikitdusioa of E. Due to the èargenklmericalcost, calculations of optical speci 'tzar i'aclucllngquasiparticle shlf'fs .4.(2)and excitonic eseds .F have become :1' ..r. only reccntly for blllk semiconductorsor instzlatorswith about tqro 'ljpéssible tiipmsL'athe elementanr cell (6.55-6.572. ;.,; The physici menmng of the pair exdtatioas defned by the pole,sof the :.: :( jbiàizationA'ndion (6.42) as excitons nnqn emsfwy :.:' be tmderstood neglecting èiiue interactions with other pairs acd consideringa dizectmode,lsemiconduc.
'
.
.
.
.
.
..
'
:
.
,
.s
'C)l.
Excitations II: Pair andCollectiveExcitations s. Elementazy tor with one conductionb=d c an.done valenceband 'v and a frequency-and screening constant a'b. The pair Hnmsltonia'a(6.40) wave-vector-indeperdeat then takes the form
c'Wk') -FJtct,k, =
1
scsôx''u,sku, (scQP(k) -g(k svQP(k)j -
-
ck.
-
kt )
.
(6-44)
For b=ds with s'trong k-vedor dispersioaa 3D Fourier trn.nqformationof the electron-holepair eigenvectoo
z47(k) =
1
daxe
W
-ip..y(m;
equation in real space yields the Schrödimger p sQc
(-iv.)
c.qe
-
(-iv.)
c2
-
Sz4JX'@) *7(œ) ssj-mj =
(6.46)
for tîe internal motion of the consideredelectron-holepair. The center-ofsince vazaisbl'ngphomass motion of this pair does not appear Lu (6.46) ton wsve vectors have bee,naamlrned.Thereforeno momentltm trnnsfcr to the electron-hole pair happemsdltrc'ngthe opticaz excitatiom The quasimomenta of the electron and Eole are F&kazld -Dk, respectively. Witlnin for the two bands,&Qc the esective-mazsapprofmation (EMA) P(k)= (6.58) = and svQP(1) -mkklpmgwith tke ecective ma.sses mc X. + h2k2(2mc and zp,v and tke separating quasiparticle etezgy gap X, the hydrogenatom-like Scblfclinger equation for an telectron' with the reduceêmass mr = merrtnjLmc + rzv)and a reduced tproton' c'hnmgeof ejebappears. Hydrogezslike bound states ./1 = nim (n,= 1, 2, 3...; 0 S ! f zz 1; -! S m :; J)with pair emergiesSaîm = X Rqjn?below the gap ensuc,ha.s Ga-ks,the e-xdtonic Rydberg ergy eacist.For typical semiconductors, -
-
(b)
(a) QOOOQQO n
m -œ'e-*--
m
00100
l
(E) O g' zO C) U''. (:) (3 -. N
11111111 11!1 1111666ï%%I:Ir
o7eooöo p l K
M
Oocoooo
/
*
x
x
%
O O
m :1
=*v
x
C-hsO
Q
é--.
x
-.
:) '
'
y
.
a
O
x
() ----0 O .
...'s
O
m
C) t7
hM . ,
O C) O Oi O
u
m'C m
x
C) O
O O x
C) O
=
m
O
O O m
O O C)
O () Fig. 6.14. Sclmmxticrepresentation of 1he spatial extent of e-xdtons:(a)wealdy botmd Wnnnier-Mott exciton ia a semiconductor;(b) strongly correlated Frenkel exdton in an Soniocrystal. The dashedline fndicatesthe extmlt of the wa've ftmction C)
of $he inteamn.lmotion in
a
fctitio:;s
(here: 2D)czystal.
6.2 Eledron.-HolePairs:Exdtons constant
261
is a factor of abottt 10-3 smazlerthan the Ry.% RHzr?a./tpzszsl =
the dbergcomqtant RH = 13.601ev of tlte hydrogenatom. Consequently,
is a factor of about 102 larger t'hn.n exdtonic Bohr radius câ = anesmjm.z as = 9.529 A of the hydrogen atom. The excitonic Rydberg cons-tant J?)j correspondsto the bindicg energy of the e'xciton hl its grotmd state with quazztlzmzzttmbern = 1 and energy Szuc. and Suchweakly botmd electroa-hole pna'mwith srnxll biadiag ener#es large eledron-hole distances are kcown as Wxnna'er-Motte-xcitons (6.59spatii exte'nt is i'adicatedin Fîg. 6.14a with respect Their chaœacteristic 6.621 to the atomic stractme of the semlconductor. Another lfrnstiag case of e-xcitons appears Ln ionic or molecular czystals.ïn thesesystemsthe k-ctkspersion aze geuerallyweaker. The electron-hole of the band.sas well a.s the sc-reensmg Gteraction is much stronger, and the electron and hole are tightly bovmdto .
(a)
4
R rx
œ r.t
20 ag u
=
2 c
= =
1 D= & X & ()
<
((;)
c &
c :D 3 X
33
****
..*--
.*
F
-a ..4 o 4
8. -12$6
=2
10
o * r
o
9
<
.
-8 ...4 0 4
3 12 IB
mo- Eg(R))
'hro-Eg (R))
Fkg. 6.7.5. Optical absorption with (solidline)azd Mthout (dmshed line)electronhole attraction in the WxnnleaMott l''m5t. (a)Schematicrepresentation without lifettme broadening.The absorptiox by botmdpa,ir statœ is representedby vertical exciton of bttlk Ga.A.s line. (b)SpeotmlmlzzEMA. for parameters of the heayy-hole v'ith JG = 4.7 mev and r = 0.25;z.(c)Spec'tr'amm the true two-dîmensional limit with the parameters of ('b)but no modifcation due to the reduced smface sceening. The pa,ir deztsit.yof states withotzt interaction showsa sqx'pte-root (3D) or stemûke(2D)variation with FIts fg. -
Elementarylxcitations 1I: Pair and ColledivcExcitatiozus
26;
p.lu+ othe.rwitbin the same or nearest-neighbor ttnc't cells.rnœe exdtons are lœownas bnrenke,l exdto:as (6.61, They are Rhematically indicated in 6.621. Eig. 6.14b.We mention that near the band edgeE the absorptiou spedrum ! can easily be calculatedusing (6.43) and the Wnnnnler-Mottappronmh(6.46). 1n.the 3D case a hydrogen series occurs for photon energies&zgbelow Es, while a rather structmelesscontinu:nrn appeazsfor F= > X i'a the regioa of the scattnrlng states (6.6:4. The drastic chamgœ of the optical absorptiond'tze to the electron-hole attradioa are hdicated i.n Figs. 6.15a and b. .&n analytical solution for the absozptionspectmlm can alrsnbe fotmd in the 2D >se (6.64) Fig. 6.15c). I'a the extreme two-dim-nsional llmn't of (see i.a whic,hth: pairs are located i!l a sheet at z = 0 (which may be (6.46), = identifed with a surfacelj the pair e'xcitatioa enertesare (,/1 rzzrl; n = 1, 2, ...; 0 S l'mlS zs 1)(6.65) -
E om
=
f Ssur g
-
JLj
81, .
a sarf
2 rz
1 -
2
(j
where tize gap enerr Egsk're and the dielectric constant ssurf relevaat to tze mlrfncm sheet are introduced. With csurf r? @b + 1)/2 Jk$ sb/2,exprvion predids a sigvezmt increaseof the exdton binding energy by about (6.47) i.n compm'qon' a faztor 16 vith resped to the bulk val'ue .%. Consequeatly, to the bulk situation stronger excitoaic efeds Ootlld appear for suzfaces. 6.2.4 Surface Exciton Botmd States
It shouldbe possibleto s'tudybotmd states of smface e'xcitonsfor the lowest lmrface-state Aansitiozss if these are energetie-qllywell separated from bulk stlrRztc (see s'tates. This ts the cmse for the Si(111)2x 1 and Ge(111)2x1 The sudace bandsDdowntemptyl an.dDup toccuSect. 4.2.2 and F1.5.25). arising from tEe daïzgling-bondstatœ of the buckled ûr-bondedclmsmq pied.) lie essentiallykï the balk ftmdameat/ gap. The surface-iaducedoptical absorption has an onset energy EgsurîLd.Table 5,1)that is smaller th= the
i'adirec't bulk flmdamental gap. As a consequencea well-pronotmcedpeak is observedic the dsFerentialzefedaace (see Fig. 6.1)for the altowedoptical traztsitio:aswith the transition operator paralle,lto the cimsns.A ssmllaz peak J7 l surface(6.66). is observedi'a Fig.6.16 in the N A s:1)4*:*,4-17= of t'he Si(111)2x rne broad peak possves a mnvsmltm at photon energiœ of about &gJ= 0.48 eV. Its Mgll-energytail becomessmall closeto the bulk iudired gap of about 1.1 eV, Sl'mslarvalues of F= = 0.45 W azisefzomch'fvential reoectMty specand phototimmmxldeiection (6.67). On the other hand, 1om a :: troscopy (6.9) combinatioa of dired and iuverse photoern:xqionspectroscopy,a surfacegap E calcttlationsgive :: of about 0.75ev Lsdeterrnl'nedtsee Table5.1). Quuiparticle The dxerence betveen the mn.='mn. :1 a ml 'nimtlm dired gap of 0.69 W (6.68). J in opticaz spectroscopiesazldthe deriyed gaps of about 9.2-0.3 ev can be in.- z.1 tezweted as an iudication of the large bindiûg energy of the surfaceexciton. The dtuation is slm51arfor the Ge(111)2x1 surfaze (6.69). 4,
Exdtons 6.2 Electron-HoleP%1rR:
26:
4.5
4%
4.O
.
@ * @
3.5 *
en b 3.0
.
.
X
@
* @
we
ch 2.5 1'& < w œ
.
% *
2.0
*
.
*
t
**
1.5
* @* **
*
1O
*
*
@
*
@ *
*
().61 ().0 '
().:$ ().:1 0.5
0.6 Q.7 ().19 0-9 1.0
photon energy
:
(eV)
x'ig. 6.16. rlze reqectancexnisotropy spectmlm of a single-domain sitzzklzxl temperatme aud norrnnl inddence. The two poln.rirzationsaa'f :.sllrfmce ibr room : iosen paralle,land perpemdicular to the zr-bondednlnnsnq.The LBBD pattern o: the surface is given in the hsset.n'om (6.66).
.
.:
!'
The physical ptcttu'e has been c-hrsfled (6.43, 6.68jby careftf tpnmeric,as and the t'wo-particlt ttalctûations basedon the solution of the BSE (6.39) :. Hnmsltozf.an(6.40) for the eledronic subsystea of the t'wo staface ban.df the results are presentedin Fig. 6.17. làowzli'a Fig. 5.25. For Si(11l)2x1 E: The forbidden trn.nsitions for ligb.t pohrizatioa perpendicular to the clmlnç. ':Eià: without intezest for the exciton problem. The spedznlm of the alloweé : it Jpt ical transitîons for light polarizatîon paralle,lto the nbxims shows the dra. ir ) tlàtic ''nnuence of the bound exciton states. Thesegeneratethe bz'oadppm.Q:rti''èapee-rentsaz resectMky spectmzmbelowthe single-quwsiparticleabsorptior Egs''rt= 0.69 eV. Above the st:rlhce QPgap, the dlferential re:ectivity tyèike LtL'j ,'..,.:jj muc,h reduced.. Due to the electron-bole interaction, spectri att d osc juw E:'E: strengths iJr are redistributed to smaller energies.Thc =n.6n reason is th ,, .
'
.
.
.
264
6. ElementaryExcitations II: Pair amdCollediveExciiations
8 o No
> >
(a) *
4
*
= O
X
= œ =
= c œ
*
* @
# 0
l
*
*
2
e x
.
o
=
x
x s
7 -@ z
--
x
E suf g
*
z
x
N'
N
w
x.
M'
elchain
2
0 0.0
x
o
z .-*
z
l
(b)
m
D
eIIchain
6
' -
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 sudacecalculated for norrnnl inddence. The tclusion (nerslect) of the ezectron-holeinteraction is ldicated by solid (dashed) curves. The two lijht polarizatîons are chosenparallel (a)and perpendicular (b)to the zr-bondedchams..&nartiûcial broadrasngof r?,r= 0.05 ev Lsmcluded. The dots denoteexperimental data by Clniaradiaet a1. (6.g;. From (6.684.
destrudive couplingof tmcorrelatedpair oscillators c'ri Mritit diFerent wave vedors by the eigenvectorsXJ (i) (cf.(6.43)). Bciow the seace QPgap, a n4lrnberof discreteexeiton states are formed. The optical oscillatoz streng'th is, however,nearly completelyconcecttateê in the lowest-energyexciton at 0.43 eV. The dominant sph-siuglet exciton at 0.43ev possessesa,n exciton bintiimg eneror of 0.26 eV. This is more than one order of magnitude larger thaa the value of pbout 15 meV kl bulk Si. About a factor 4 may be due to the reduccd screensngin the surfacereglonwith au esective dielectric constant of csus = The othe,rmairt reason (sb+1)72(cf.the ctiscussionia Sect. 5.2.3an.d(6.47)). for this icczeased bindlng is the spatii conBnement of b0th the eledron and the hole at the stzrf'sce(see also (6.47)). For the direction perpendëcularto thc suface this is already dear hom the 217model (6.47). Becauseof the localization of the surface s'ta'tes derived mns'nly 9om llze Ddownand Dgp orbitals (see Fig.4.19 or 5.25), the b=ds get a partial 2D character with the strongest dispersion parallel to the chains. TMs ID character may further Mcreasethe e-xcitonbizlding.
6.2 Electron-HolePairs: Bxcitors
kqN---hble
'% @
o
2 c -2 W -6
o
:
c,n w
.
X
c
-15
-10
-5
o
0
5
10
so .
15
Distancefrom the hole (# Eig . 6.18. The electror-hole space for a flxed hole position
fht'nction of the lowest-energy exciton in real For detn.17R see text. From (6.685. (irecated). wave
A gencral visualization of the eledron-hole conelation at the sarface2lustrates the electron-holewave hznction dependingon b0th the electron coordinate œe axzdthe hole coordinatezh. It is plotted ia Fig. 6.18for a flxed hole position œjz slightly above one of the ttp atoms in a r-bonded chatu, position whezc the amplitude of the Dup hole state twhicllcontributes strongly to the excitoa) is very high. The contom plot in Fig. 6.18, which shows the distribution of the excited electron relative to the ftxed plane perpendic'ala,rto the Pandey nhn.l'n5k, can hole, m = tre zh, in a (011) i.e., at
a
-
therefore be interpreted pzsa visualization of the wave 'hlmctionof thc internnq motion desczibedby, (6.46). The Arnplitude of the electron is ve.r.glarge on the sxrne Pandey chna'nwhe'rethe hole is located. Ou the neighboring Pandey cAnsnqto the left and to the right, the ampûtudeis mue,hweaker.On the second-aeighbor Pandey clzltlrtK,the xmplitude is already closeto zero. As a eonsequenceof the qllnuqi-lDband structttre with strong dkspersioain clirecthe chain direction the exdton showsFrnnkel-like behaviorizzthe g2îîq tion and, hence, a large binrls'ngenergy. The situation LsdsFerent alongthe dmims (notshown in Fig. 6.18)The probability distribution of fmdm'g the eledron in the (01Iq direction is more extended.The mean square distanvce in the closn direction of 40 J't may be interpreted as a'a indication of the Wannier-Mott characterof thc exdton in this direction. .
6.2.5 Surface-Morln-Ged B111'k'Excitons The interpretatîon of smface optical spedza for photon energiesabovetlb.e butk hlndnrnentatgap is much more complicated. B111lc and surfaceoptical trn.nRitionsi'aterrnsv.Botmd states of one absorption edgeoccur izl the contizmllm of scattering states of other absorption edges.Ic the presence of a resonaace iateraction mediatcdby E this mteraction may give rise $o a Fazo lineshapenear the bolmd states of e-xcitons(6.702. In orde,rto avoid conhtsion
266
6. Elementary Excitations F1:Pair and CollectâveExcitations
Fîg. 6.19. Uppermostatomic layers of t/e hyclrogen-covered Si(110)lx
wé conside,ra, model stuface, such as the hydrogetssaturatedSi(1 surfacerepresentedin Fig. 6.19. Becauseof the bonclingto hydrq the surface states are removed h'om the eaea'or region of the f '
tal bulk gap. The optical reQectancen.niKotropy (6.3)ca'a only trnmqitioas betweensttrfaco-modi6edbulk states in the surfacereg'id cussedin the fozowiug.Accordingto ciculations tkis rezion is ret lessthaa 30 atoznic layers. The conwpoudiug experimental RA sp The spectmlm is rat Fig. 6.20 (6.71) can easily be reproduced(6.72j. sitive to the strtzc'tttral and clmrnicazdetnslqof the smface passiva,t
0.01
'
V2
E1
Experiment rzl
<
Theory 2
4 5 3 Photon energy (eV)
Fig. 6.20. Measua'ed'R A spectmtm of the Si(ll0)1x1-H sudace E6.71). and (6.39) for pared with tbe spectrnlrn calctûated according to (6.11) slab (6.721.
6.2 Electron-Hole Pn.5m:Bxdtozls
267
!It hmstherefore becomea calibration standard for RAS ,apparatus and a tex-tbook evnmple for surface optical propertiœ (6.32j. The mcmsmeda'ad calculated spectra in Fig. 6.20 showtwo stzong posiThe tive RAS feattzresnear the Eï and & blllk critical-point ener#es (6.321. àtrongpolarization n.nl'sotropyof optical trsmqitions nea,r bl:lk-like mitical pohts Lsrelated to the symmetzy reduction to the relevant point grou.pmIn detail it is mnu5nlydue to the diserent deformationsof tke 3D Bloclwlike wave hlnctions in the smface region for the z-direction ((1ï0)) and v-diredioa Amplitude, phasej oscl'llxtion width., a'addecay into the vacunlm may (y001)). by the tznlncationof the matezial and the orienV i'nAuenced(cf.Fig. 5.22) i'ation of the Si-si an.d Si-H bonds (d. Fig. 6.19). The botmdary conditions 6f the blllk-like wave fllnctiorts at the surfacedependon z aztdy. Tke Si-TI bonds lie in the pz-plre aud: henee,have a veater ln6uence ou the vztriartion of the hlnctions with &. As a qonsequemce, the trxnqition-matrix elaments for the excitation of one qumsipartideeledron-bolepaiz a-swell a-s the (6.36) Ecotllombinteraction of electron-hole pairs (6.38) are modioed. This men.nq Enotonly the optical matrix elementsMm$ (i) but aksotheir tterference mediaze cilanged.The isotropy with ated by the pair eigenvedorsAI'(i) in (6.43) resped to the z an.dy directions is destroyed.The strong opticz rmsqotropies in Fig. 6.20 are directly related to the shb polarizabilities for light polnHzaon the iion parazle,lto (110) or (001q. Iu Fig. 6.21 their signlcant dependence johvizationdirection is demonstratedfor a lz-layer slab,in particu)ar ic the lz and E2 spectrat regiozus.Figttre 6.21 also indicate,sthat the dip on tke
,
60
Ec E4
> *'-
r
40
...
------' .
.'
.tJ
.' .
'K.1 Ix *'%--
;
4
? '.O
...
(n.
.
rn r
.
:
nn = VJ
X
..
'.
,;,
%'
1*
*
1 % * .i,k w
Nw... nw
.e e#
E.--.. . .
xw
-w..
J
O 2 i
e a.K' MM*J
a?
4 5 3 Photon energy (eV)
6
calculated from (6.31) Fig. 6.21. Tmn.giuarypazt of the slab polnHzabili'ty(6:10) bbra lllayer slab. Tûe light po7xrizationis pazalle'lto E1ï01 (solidlinelor (Q01q ('dotted linel.From (6.72).
6. BlementaryKxcitations H: Pair and CollectiveBxcitations
268
low-energyside of the Ej peak in the RX spectrum (see Fig. 6.20) is more ralated to the Meshapeof 1m(1/hb@)1;) as a consequeuceof exprassbn -
(6.11).
Becauseof the iazgeslab witb. 24 atoms per supercell, t'wo conduction and two valencebandsper atom a'adl40 i-pointsin the surfaceBZ, about 35O000 pair s'tates are involved in a nl:merical calmtlation. To avoid the diagonalization bottleneck (6.41), a novel time-evolutiontenbnsqueis applied to solvethe BSE (6.70j. I'n.partinnllnm,the remarkablearkisotropy near Eï is due to e-xcitonic efects. This 5s demonstratedin Fig. 6.22. Hcludîrg the quassparticle corrections (5.30) the IRA.spedznlmis slniftcdto higher energies.The short-range electron-hole e'xchaageiateraction =d, hence, the locaz-âelddfeds harfy in6uence the spectmnrn.The attractive electron-bole interadion in (6.38) stronglycouplesezectron-hole patrs with quasiparticleenergiesnear the b7:7kEï and .F'z critical pomts. The constructive interferenceincreases thc strengthof the Ek feature. The opposite eFect happensfor tke Eg peak. Thas, the .E': peak remmîns almost l:n' Ahl'ffedwher- the excitonic E2 feam tm'e is. slightly shifted to lower energies.Coztsequently, one may conclude that e-xcitonic efects aze mop important near E1, at leut i'a the RAS of 1-H. Sî($10)1x
'
(0.01
Ez
Et
(d)
kX
-
(c) E2
(b) E1
-
(a)2
3 4 5 Photon energy (eV)
ptHnce cpzlcuhted Eig. 6.22. RAS spedm'm of the Si(110)1x1-1I for a lFlayezrslab: (a)independent-pattcleapprovsrnxtion; (b)hdependett-quasiparticleapproximam tion; (c)quaslpartîcleapproxsrnxtionwith electron-holeexcAnnge/local-feld esects; and (d)fully Coulomycorrelatedelecwtron and hole quasiparticles.&om (6.72J.
6.3 Plasmons
269
6.3 Plumons 6.3.1 Intrabaud Excitations
describingthe optizal propemtiesof uonmetals, For the evesluationof (6.35) completely fzlled @)or empty (c)bauds have been azsnlmed.Iu metaksor l'tighly doped semiconductorsalso paztiatly fzlled bands u appeaz. 1n.such are systems the sm'eenkzgis rather complete. Efects related to W' (5.22) negligible. 1lz a &st appronz'lnthe local-feld esects .'w f should aksobe n.ewitbl''n gle#ble,and the polarization AnmctionP can be replacedby qh (6.33) the independent-quaskpartideapprofmation, whic,h is eaentially identical to the i'adependent-particleapprov'rnation becauseof the GW approvimafor the self-emergjroperator and the Almost vanishingscreened tion (,5.23) potential W- in systems witlz fzee carziers. Combn'ning(5.25), (5.27),(6.32), wave aud (6.33), assllmsnglow temperatres, a'n.dstill consideriugvzmislnsng becomœ vectors, the intratfandcoatfbution to the dielectzic Aqncwtion (6.32) f
1...'2s(q)
1u--(#,,s)
q->o
y'l s
*
-&-*+t.')) O ç''b- f-t*) (6.48) (&>' + Lr') svtk+ g) svlk)+ 56u1 -
-
-
with the Fermi energ,g e's of the electronic s'ystem. A 3D b'xllr system is
consideaedas indicated by the replacementk -> k. The ktraband contzibu.tion to the dielectric glnctions of metals rtn.n also be calculate by ab initio In the limit of pa-rabolic eledzortic-stmctme methods (forcopper see (6.734). describesfor eadzpartially fllled baad v bau.ddispezsion,ex-pression(6,48) the watl-kaow.nLindhard dieledric A:nctgon(6.74J. the intraband In the ll'=l't of small wave vectors (or large h'equencies) contribution cau be approvlmately treated. With the elementsof the ttmqor of the inverse elective mmss
p2sv(k) gpzv-1(k)q., ksok. ,k# 1
=
(6.49)
of the dielectrk flmdion takesthe Drade fo= the intraband pazt (6.48)
sintrat*) (s;
=
1
-
/p21op
(6.75)
(6.5û)
jtx + g (y (uJ iF) o4
with the aaisotropic plmsmaFeqttenc'yof the ealectronicsystem 2
8xv.6 yzlto 1(k)q a J? :7 e (aF' cv(k))qzrz-v P =
-
.
(6.51)
zsir
band u svith isotropic azld parabolic kIn the case of one partially SIIC'CI dispersiot characterizedby the efedive rnnAs zn*, the plazma gequencybecomes
6. ElementaryExcitatsoms1I: Pair and CollectiveExdtations Ya p
4rc2% =
(6.51)'
m.
with the homogeneouselectzoudensit.y n
=
2
p Etayp
-
c.(k))
(6.53)
.
For 1ee electrons ia the jelliammodel we have rn, = m-. Such a'a appcox-: imation allows tEe calckllation of tke intraband clielectric Alnction also foq non-vcmishingwave vectors. One Snds a wave-vedomdependentplcma 1t.4E + 3cp/(5'rrJ*) queucy wp E1 Lq/iupllj (6.76j ,
6.3.2 Pluma
('
Oscillations
For vanishing damping the zesulting isotropic intraband clielectric A'nctio/ TMs men'nq that an eletq sintrwtu,) (6.5c)uas a zero at the frequency so (6.52). troa gash:u an 'm6nttelylazgeresponseto ields appliedwith this frlzqaency. k 1.n.other words, there exist self-sustnsnceng chazgeoscillations of the systeù; E ï' These are jas'ttbe long-wavelemgth plasma osdllations of the eledzordcsuV , the use of the terrn : plasma frequencvy is Jus tjsed 1::) system. Consequentlyj general, zeros in the dielectric flmdion correspondto Gcited states of tV system. They give the eigenenergiesof (longitudsnxl) cozedive excitatiol.' In norrnxl metals the (anergyFztoof thtaseexdtations) the plmsmonsat vanish: i'ng wave vedor, is typically severaèeV. The valuesfor bulk alknlt metals ai/ Fw)p= 8.03 (Li),5.90 (Na),and 4.36 ev (K) (6.76). For dopeâ semicondùèu: tors the plasma gequendesaze muc.h smaller becauseof the srnnller carrier densities involved and the presence of the fmite electronic interba'ad polarizh abili'ty (6.35), mpinly (cb 1)/4x,at such small gequencies,which tscreed'E the plasmafawuency.Moreover,the modïcation of the eecvtiveban.dmais '
.
,
.
'
'
-
has to (6.52)
be takezti'ato consideration. Physically the plasma oscillations correspondto solmdlik-e compzession waves in the electron gas. However, becatlse of the long-range natme of the Coiomb potential, whic,h sustnanKt he osc illations ; their frequencydoeq not approac,hzero at long wavelengthsbut approachesthe 6nlte plmsma::1quency. BG plazma oscillations or bulk phsmozls may be excited icumeta.ts electrons at tlnir fozs. Electrons i'ateract strongly witi by flring Mgh-enea.gy the plasmamodesa'adthe caharacterlstic by stadyhg eaergy lossis obseawable the trsnqmitted elecqzonbtoltrn(e,f. the discussionill Sed. 6.1.3). Another pom sibility Lsthe obsemtion of the satellite stzucvtttresin photoelecxtzonspectfi (seeFig. 5.9) '
,
6.3.3 Surface MoalGcations
The presemzeof a smfacemorlmes the lossesobservedby EELS i'a a reâedits geometl'y (see or the satellite stzudares in PES (see Figs.5.8 and 5:01 (6.21))
6.3 Phsmons
271
ty tnqncation a'ad image-potentialeseds. One type of thesemodl'6cations gives zise to plazma oscillations locxlized in the surface region arotmd z = O (6.771. For the displacement of electrors in sach a sheet during plasma ioscillations the driving force is reducedin compn.rllonwith the 3D cafe due to the restriction of the density fuctuatiors to 2D space. This gives zise to a fedttction of the phsma gequeccy. The eigenlequency of a surfaceplasmpn, als, followsfrom (6.22) using (6.50) a'adneglectMgthe interbandcontribution to the dielectric înnction at this lequency as (6.35) ws
=
wp/v'it
(6.54)
Therefore, the energy of the surface plunfon at vazzis%ingwave vector Q it is diredly related appeal'sto be a pzopert.y of the blllk'. Becauseof (6.52) to the bll'k electzondensity zz. The wave-vecvtor dispersionof the surface ljlasmon enerr is, howevea',completely diserent from that of the b771> (see Sect.6.3.1). Jnsteadof the eue'rgy increasing with the square of the wave ''qedor, a aegative stzrface-plasmon dispersionhas been predicted (6.781. More precisely, a linea.r dispersionrelation a;s (1 Ic1 IQr)occurs for small 2D wave ( -
.
kedozs.
>
= v)
= 1-c)
Qx
.
S
o .1
9.163 A
2.6meV
Q=
' o . c 16 A'@
#'
e
ae
O
1
2
t3
4
Energy Ioss (eV)
fiik. 6.23.
Electron energy lossspedza (measklred in the reèectioa from geometzy) > jiliclc K metal flm grown onto A1(lll) for tsvodiferent Mues of tke momezttzzm transfer and primaz'y electron enezgyof 12 eV. From (6.79J.
272
6. ElementaryExcitations J1)Paizand CollectiveExcitations
2.7$ 2.74 ,,-h
>
2.72
*
2.70
d. f,rl
..1
m
x
2.68
=
2.66
P X
%
ku
2.64 2.62
Y s & N s N % % N % s N N % A' A
n Ai
0.0
0.1
0.2
0-3
Q $1) as a fn'nctîon of IQ1 Eig. 6.24. Memstzredctispersîonrelation of the,surfaceplasmon for K metat The dmshedline correspondsto a tkeoretlcal prediction for the littear From (6.794. te= (6.781.
A soace-sensitiveelectzonenergy loss spedrctm for au alka,limetal is The cozresportdingmeaslzreddispersion relation of shown in Fîg. 6.23 (6.79). betveen the sarfaceplasmonis given in Fig. 6.24.Thereaze two discrepancies theozymentionedabove. The &st is that the expeaimentand the An'rnplifying valu.eof als (6.54) deducedfrom Fig. 6.24is 2.74 ev while the auticipated value for the densityof the 11metalwould have beenabou.t3.0 eV. This djscrepancy may be relatedto the fad that the bulk K metal is not an ideallellium.Bandstructme efects haveto be taken into accout. The seconddiscrepancythe positive dispezsionfor large wave vectors, iudicates tkat Mgher-orderterms ia IQ1becomeimportant. However,also the treatment of the liuear te= has to be improved by the inclusion of exeange and correlatson esects (6.80). We note that the observatîonof surface piasmons oa surfacesof doped A semicondudorsby N'RRELSis more diëcult than for metals (6.81, 6.82j. dead layer of surfaeeplssmonsseems to exist. Moreover,their energiescome and coupled sarface plasmonwitidn the Tange of optical phonon eneargies the sarfaceplmsmonsin sezniconductors phonon lossesappear. Consequently, èrenot sensitiv!to the details of the atomic geometzyatd eledroaic stmzcture of the Srst atomic layers in the smface.An example for sach coapledmodes smface with a bulk eledron is #venin Fig. 6.25for a cleavedInSb(110)1x1 concentration of n = 1.5x 1017cm-S and a'n LO phoron entargjrof 24 mev. The double-ppxkstrudtu'e relatedto coupled suzfacemodesazd its mriation prefactor (6.21) are deazly visîble. due to the wave-vector-dependent
6.4 Pkonons
(a)
273
'
K :3
Eo
.'
.
(b)
L
0
-90 -60 -30
O 30 60 90 120 Energy Ioss (meV)
1 surfacewiG t'wo Fig. 6.25. EELS spectra measuzedozz a deaved HSb(l10)1x and (b) 5 eV. A doped 4:::.yst9with a hiz,h electron p 'nnrnaryener#es(a) 20 ev cm-3 coscentration .p,= 1.5x 1017 is studied. Fkom(6,82i.
6.4 Phonons 6.4.1 I'Iarmonic Lattice Dyzmmics
I'a order to dœcribethe lattice dyzomics n.ear surfaces,the same attempts at modelicg should be used ms i'a the cmse of the electronic-stmzcztttre and are the slab and total-energ.gstudies in Sect. 3.4,3. Promlnem.tRvnmnples scatteriug-theoretical methods. The repeated-slabmethod or the Green's fnrnction method togetb.er with a special mnrface ssnrllntio:a aze commonly The instantaueous position of an atomic core (atom, applied (6.83!. ion)is given by -
m
.
=
m. +
w
(A),
where R.t = R + 'rs (Bravais lattice vector R and atomic basis vector z%) gives the eqlliln'brinrrnposition of the particle and w(A) the time-depemdent displacemeut.Tllese displacementsa're the ceatral quantitîes of the lattice
dynanlics. Withsmthe hltrmonic approvimation the equatioz:sof motion of lattice particles
are
274
6. ElementazyExcitations 1I: Pair azzdCollectiveExcitations
d2 Mkyg uJ.I.Rt) =
sj
t , R J7 C.PLR, lzb4(S, t)
-
(6:5q)
z?zo',J ;'
M:. In general: tNe hteratoznic force constaùtà
witll the atoznic rnn.qses i Ci.;) of ià'e' (X,R ?) are dm6 ne d a,s the secondderivatives02Ej@R%.=0.%p) of the consideredsystem iu agreementwith the de6ritotal energy (3.$5) for vauishing forces (3.39) tion of tke cvpilibz-blrnatomic structure (.Rs.) A nlamberof relations between the force constaats foltow from the beha'fu ior of the totat eztea'gya'ad the atomic forces unde,r rigid-body transhtionà J aad rotations (6.84j. Oae of these relations 1(,)a*to aa expression for ï'lli 'self-iateraction' force constants .
'i
-'s R) cutA, .&) C;=çR, =
.
=
l
-
',i Jt ?). C%œ;?IR,
zz4,:
The prlrne indicatesthat ltra-atomic terms are not taken into the s7lmmae tioa.
Becauseof the 2D trnmlational i'avarianceof the s'ystem, the force ccfy!r = JU stants depend only on the aiFeavnce (A .Jr),C'U RJ4 (A X$!) (A, œ;? .# Their tramqlationalpropezty impliesthat the norrnxl mode solutionsto (6.5d) have the form of 2D Bloch waves -
-
u#(Jz,z)
1
=
Jg
)(Te .J(())et(q(z?+.ï)-?.,,(e)tq
where N denotestke n'lmber of xlmit ctallq. J.nthe cmse of bnlllcsystmrnsràhd.. 'a in the repeated-slabapprornnation snrnliltr formu)a,sare valid witll 3D q((: >!l tities. Withi'n the repeated-slabappror'rnationj however,the modesstZ (Vz pend maizslyon the 2D wave vector Q 5.nthe surfaceplane since the dépetïu dence on the third wavewector component is negligible for su/dently laq4 i'ato (6.56) leads to the eigenvaluepToblem slabs. Substitution of (6.58) .
.
.
.
.
JT-IZE/J œp(Q)? s#(Q) dCQ)e'(Q) -
.
''
' .
(6.(59) :
lna
,f.J-3
with tlze Hermitian dynamçcalmcfrâ'r .j
D.,LQ)
=
1
c'it x)e-w(a+r:-r,1 .p( 'X'I u : syj .
(6kt't) 1
x
eigenvalueproblem (6.59): In the generalcase of S atoms pe.'rsupercelkthe pps! 2 ë 35 solutions (as(Q) (/1= 1, 2, .-., 3S)for s' at eachpoint Q in the lst'afà4.dlj: Briliontsn zone, which can be interpreteê as the brauc'he.sof a multiokttd' Eaehnormalmode nQwith frequenc'yw(Q) and dgenviètbz!! 'hlmctiontsztol. ei a excitatioa of the vibratiug lattice of all atomd.: & (Q)desczibes particular is called a pltonon. A quaattlm of the lattice vibrations with energy hu)nLql ..
...
6.4 Phonons
Consequentlyj the relationsexpressed by the equatinnsa7 =
V
Itonorbdfapersïtmrdations.
275
w(Q)aze lcowa
There are many phenomenologicalmodelsto determsnethe force conEétants XJ (A,A/)in (6.56)(6.32j. Contrary to phenomenologicalmodels, ab ktitio caalcalationsrequire an accmate ard parametergeeMowledgeof the microscopic zespoaseto lozen-in lattice vibrations. The basic ide,ato all irst-prindples methods is to detemninetbe inteaatomic force cortstants Nia the total enea'gy(3.55) of the s'ystem with gozen core .coordjnates.ln the list decade,mnany theoretical advrces have beea made toward the appm 'cation of these concepts to lattice dynamlcs.There are two cornmonlyused àpproazhesfor this knd of caLculation:the direct lozen-phonoa method and fhe pemtmbattveapproacz.Thesemethods aze essemtiallybasedon the den.'2 àztrg:nctional theol'y presentedin Sect.3.4.1. Withl''n the hnzen-pltonovt ûpprocch (6.85, one considersthe propa6.86) cation of a phonon wave of a ftxedwave vector whic,his comnnenstlrate with a :r E(,/etikproc,al lattice 'vector. Tbis causes the atomb to vibrate witiz a d.e6nitedis7 ,.ylacement pattfacn. At a given moment the syste,mwill cocespondto a new E' al strtz dure corrcxspondingto the Aozea' vibrationi mode. For suc,ha ',tjtllst jhononmode the total energyand the atomic forcesare calclzlatedas a, fhnrdcof atomic displacemeatsin a supercellappropriate for the new crystal ttickn dtrudare. Using the energy chfkre'ace betweenthe distozled and tmdistorted E E:btructt'trE's or equivalently from the atomic forcesiu the supercellgeometzy ibneconstruc'ts the force constant matrix. Meanwhile,this method hasbeen 'mânedto the placaz force constant method (6.87). Tite atomic planes perto a given wave vector are regardeêa,s rigid bodies, and the lattice E/çndicala.r i'2' D nrnscs can be treated in the spirit of a linear ckta,inmodel. The most Mportant pexurbative approae comsiders the linea.rrezpozxse is?the electronic system to the displacements.Basedon the DFT it is called 'the density Jvrlcfïorlipectncbationtheorl (DFPT) meih'od(6.88,6.891. For a of tNe elecwtrondensit'y jjveu'lattice distortion the resulting chauges.zAn,(z) and 1v'(œ)of the total Kohn-siam potential (3.48) thus needto be (3.47) qvaluated.Thesechangesare Oectly related to the electroGc contributio:ato hn.rmonicforceconstrts. The ionic contribution ca'a be straightforwardly Etffe i uated 9om the Ewald snlnnmationmethod (6.841. :'y: Polar czystals, suc.has semicondudorswith partially ionic bonds and ktjzuc czystals,needaddaionalconsiderations. Irzstzclzsystems,wheredieerent ::' are involved, the long-ranger-haracter of the Coulomb iljrjesof atoms/ions .!. g'ivesrîse to macroscopicelectl'ic îelds for loagitudinaloptical phonons Yrcqs ''trt'the long-wavelengthlz'mx't.As a consequence, the dynamicalmatrbc is aot EValjrtic at q = 0 (ia3D).There is an additional non-analytic contribution the dpormicaLmatzix (6.60) : Etd It %smtke geaezalfo= (6.00) ..
,.
q...l'.
'
.
.
CJ -> D=o Lq
0)
=
z-t ph (n( a-!f k 3 h.=
l 4cr: V' MLM.f -
-
.
z- ,*)
-*7= %)Jç q êx q -
.
-
-
.
(6.61)
6. ElementaryExcitations 1I: Paiz and CoEective Excitatio>
276
The tensors êx and k*ç denotethe ielectzic and the efective dyzomical ion chargetensors, respectively.Thereby,the index x stands for the (Born) purely static eledronic contribution to the screensng.The tensor êco ca'a be calculatedtusing (6.J1)for vanishy'ngfrequency ul = 0. J.'athe cubic llml't acd without the lattice conebution, the scala,rsoo correspondsto the btTlk ,
cozlstant sb discussedabove.A new abbreviation is 'ttsfafltl, since another &.-
electric constrt so hcluding the static contribution of the lattice will late,r be introduced. The qu=tities X* ç an d êx can also be obteed in the framework of the DFPT i'a termn of derivativesof the dielectdcpolnm'zationfeld of the s'ystemwith resped to the dispacementsa'adthe electdc âeld (6.88,6.90) However,at least for cubic systems it has been shown that the screened Born Garges Z;/ sx caa also be derived directly wlthsma supercezcalculation (6-91, 6.92). .
6.4.2 Surface and Bulk Mpdes
The poln.rszaticnvectors eix(Q) of the lattice displacementsiu (6.59) have 35 components, nxmely three cï .(Q) for each pazticle at 'rï izzthe l'nst cell. 5 They indicate how the atom ï vzbratesizl the paztioTlltr mode sQ. Sincethe dpmrnl'calmlttrtx $sHerxnitian,the polarizationvcdozs satisfy the orthonormalit'y condition
et*.(Q)4-.(Q)Jxxz 5-2 'lta
(6-62)
-
and the clomtrerelation
X'le'(Q),f* MJ(Q)
-
nœ
kiè'œp
(6.63)
N
edx(Q)eL.(-Qtand au(Q) *s(-Q)
For ins'tance,the poTv'zation vedoz's of a givec mod.eare normalized to 'nn7't.yove.r the whole thicAmessof a slab. This enablesas to determine, by izzspectîonof the variation of the ei NG (Q)for that mode over the variation of the atomic positions vz parallel to the sttrfacenormal withn'n the thielenessof the slab, the locxll'zation charmode acte,r of the vibrational mode,i..e., whether it is a HJ: mode,a s'urfac.e mode. TMs is quite sre=l'ln.rto what we have lenrmt or a m'ized(rc-s/zwzzcc.l about the zectroaic state of a surfacesystem in Sect. 5.3.3, Consequotly, the identlcation of tkevdsferentmode Garacters r.a.zl also be basedon the of the uu(Q)of the surface system with the b:llk phonon discompazqsozz J.nthe b:711c persioa relations projected onto the suzfaceBZ (see Sckct.1.3.3). case there are three tconstic p/ztmtmbraachesand 3(S 1) opticai Jl?lozltm branches with S ms the Irlrnber of atoms i'a the prlrnstive n'n''t cell of tke crystal. Mong high-symmetzydirectionsin the bphl'lrBZ, suc,haz the (100) directionsin cabic c'nrstals, the phonons can be c-lassifedas transand (111) nerse or longitndino.taccorclingto whethertheir displacements o:r polarization
=
=
.
-
6.4 Phonons vedors e,)' Lq)are perpendic'alazor paratle,lto the directionof the 3D wave vector q. The projectionof the relations ulxLqj onto the surface BZ yields gap resoas,pocketsor stomach gaps iu which sT:rfnne phonon modestsu(Q) c,ac appear. Another possible lequeacy region for surfacemodes concerns that above the mmcimllm frequeacytxmax allowedin the bulk cmse. Modes localigeda,t the smface appear izl thks region if lighter atoms are adsorbeclor force constants aze increasedby sulface reconstruction/relaxation. Projected bulk phonon branches a'adpossiblesurfaceor resonrce modes are indicated in Fig. 6.26. We note that also the wave vector may Muence the locn.lîzation of a phonon mode at a surface. The locmuh'zation is geaerally stronger for shorter This holds in particular for so-calledmaccoscopéc surfacemodes wavelen#hs. The attenuation of the pazticleamplitude away from the sarfaceis in (6.831 some way proportional to the reciprocal wavelength. Therefore, for long waveleagthsthesemodesextendover consiêerabledistance.s i'ato the czystal. Since at lon'g wavelengthzthe atomic cnrstal stntctme (butnot its anisotropy) is lqnl'nqpordant,suc.hmodes can be found jn thc frameworkof e'lutic a'addieledric continut'lm theozy A.aexample is the Rayleigh mode (see Sect. 6.4.3). The microscopic modes are chn.racterizedby the fact that their penetration depth into the mys'tal extendsover oaly a.few interplanaz distancc.efor all wavelengtlzs.Most sulfacemodes that are aucolmteredin practice are microscopic modes,whic,hare locn.lszed in the stttfaqelayezs. .
t
i
! !
*a
l i
i i !
E
! E
0
l
0
QBz Q
Pig. 6.26. Schematicrepresentation of tîe projeeed bulk phonon bzanches
(hatched re#on)and smface phonon brazees (solidlines)or smface resonxnce phonon branches (dashed linel.'rlze boundary Qsz of the smdaceBZ and the maxim=
Wbrational frequenc,yaJ=
i'a the 'bulk czys'talare indicated.
6. ElementmyExcitations 11;Pair and CollectiveExcitazions
278
-
n
.
Q shear horizonll
longitud-lnal
transverse
.'.'''.'''.'''
:'j' :'.'
Fig. 6.27. Tite three mx'''n types of pollm'zations of surfacephonons'with res/èctlE ;: ;' to tke satttal plaue (Q,'rz). .'
. ... ..
.
.. '
.
.g .
k(! j :1t ('. .
':. '
.
.
.
''
helpful in simtj .
khzsurfaceproblemssymmetry considerationsaze fa.r1=
)
:
'
thV j oi' the solationsof the eigenvalaeproblem (6.59) plifying and clmssifyi'ag bu)k problems.Nevertheless,some simpMcations due to the remnsnimgS#iV'E è metry are still possible.A centr al ro l e is played by the 4J#ifOJplaae devv' 1 surface n Fig. normal by the phonoa wave vectoz Q az.dthe (cf. 6.27). Tvi, t'êi mode polarizations aze customn'm,1y referred to this plane. When it coktciè' wlth a re:edion plazleof tke slab, the dyzmrnlcalmatrix can be reducedjj.j jj t'wo blocks. One contnsnR t'wo thirds of the modes, of wlkic,hthe are ellipse,sin the sagittal plane. Thesemodesare labded SP (sa#ttal-/liùé' )y 'V modo. Titey may be classled into two typest transvez'se and longitudluak it is aksoindicated in Fig. 6.27. The other bloek contnsnqthe rernnlniug ozx normal to the sagittz third of the phononmodes,which are lineearlypob.znzed > : Note only theq! that modes. plane. Thesewill be labeledshear-horizontal(SH) polarhation normal to the sagittal plane (siti! SH modeshavea well-defu'aed 17 modesare attti.r.E Fi f- 6 27)) whereasthe so-calledtrn.nsvezse and longiturismnl .( atly coherent rnlx-tm'es of b0th polnvlzations, the name indâcatingwhie,h hl 5: the lazgestrelative amplitude. .
.
'
'
.
polarizafv/ . .. .
'.'
--
,.
.
-
6.4-3 Myleigh
Waves :' :';':':'
T he f o rrnnlism describedin Sect.6.4.1 glves complete iaformation about thètiq: y ,.jg lattice dyzmrnl'csof a surfaces'ystem. However, for ceztna'mfrequeney aù2,), to apply certxirl approsnrnate descrtpttùriïy wave-vector regiorusit ks pozssible that give deepet physicazinsight. TMS holds in partichlln.r for rnnztroscolft' vibrational modes.We lmow from practical exercisesi'a solid state physics(i.èï that izl the lonpwaveleng'tklimt't the equations of motion (6.5à)l e.g. (6.93j) over into the wave equations of an elutic contintplm for a linear chnnneahauge .: ..7i of motiid of elasticii'y the equations of the linea'r theory Witlnsnthe frnmework of an elastic medlum are (6.94) ..
..
..
.
,
'
' ' '' '
' *' '
.
..
o,
pazzralx,t)
=
(-9.., (w,tj )C ozp p
La,p =
z,
v, z),
..:.
(6.6i''
6.4 Phonons
2$9
@ere p is tEe mass
densityof the medium,.u@, of the tj is tite displacmment zzfèililll'n at tbe point m and time 1, aad &#t t) îs the stress tensor. The latte'r
1kgiven by Cœp=
Hooke's law
c.o.,o' -''tG' Dzp'
J'')
ue/,J3/
khere the (tQ,.,,r)
-
eobofqeo, J-.z
(6.65)
,
the elementsof the elastiqsbfr-neôstensor (atconjtint macroscopic electric Eeld), and (etz.#)are the elements of the piezochargetensor. The l'nnsuenceof the macroscopic electric âeld X i'a èidric li 'C. tftemedblm is take,ninto account to allow aso the treatment of piezoeledric tikàteials The tensor of the pieaoelectriccharge,sis symmetric Cfn the second rkiji.oficdices.The elemeatsof the tensor Coboorn' are symmetric j.nh a aad ;, 1ik;q?'aud p' a'adia the paizs an and a'p'. This makesit possibleto exprerss trvmeqtzivalemtiyin a two-subscript'notation, Cœpœ,pr -> q.;, according to stbemez:r = $, yy = 2, zz = 3, yz = zy = 4, zz = zz = 5, n = yz = 6, t11'è tkudto iutroduce the elementsof symmetrizedstress an.dstram tensors. The elpstic mectb:rnLsconsideredto occupy the lowe,rhnlRpace z < 0. The surfàce z = 0 ùs azsumed to be stresseee, i.e., cazlzxo = 0 (withcuz in the dkmmetrized form).The electric Neldsf1116l1the botmdaryconditionsof elccaze
.
tvàtatics.
ln order to simplif.g tbe considerationswe study an Lsotvoptcmedbnm 4k'i'h ou t the piezoelearic Kect. There are 'two iudeperdeat ehstic moduli tî:, cza, au.dcu with cu = c11 czz. with Laml's mod.uliit = lz(c11 cza) ttktdà = clc, (6.64) becomes(6.93,6.944 J ,,.
-
:2 = 'tztm, t) &2
njl
-
graddivuta,Lt -
v:2
carl cuzlxztm, tj,
(6.66)
.wrllrethe longituainal and trxnwerse soundvelocitie,s ',(a= 'nt
=
L2g+ à)/p, y/p
(6.67)
1'eltzoduzed.
Equation (6.66) suggets splitthg up the displacamentâeld ''QL# 'tz! +w icto a turbulence-free contribution tcurlw= .7.L, 0)and a source-free '
'.
.
jz't ldivut 0).In the btllk', tEe resulting dzflbrezttialwave eqttatiozlscan be ikved independentlyfor b0th contrsbutioas,giving longituan'nalsotmdwaves l éïi;(jp, aad trxnsvezse (sheaz) sound waves 'tsutz: t) propagatingwith the ( t) .
.
=
'
: ,...
'
sotiid velodties /t& or ,2 t) = 'tzvttm, (9$2
vt.
Eac,hof these soandwaves obeys a
wave
equation
'tf/taltrlt,l/ttm, t).
In the cmse of the elastic halfspace (z < 0) we only seeksolutiozzsof the equations (6.68) 'Lqtpke witic,hrepresent surfacewaves, i.e'., which are characViiked by = ex-ponentialdecay iuto the crystal. With a wave vector Q in i dùrfaceplaue (z= 0)one has
28û
6. ElementazyFzxcitations 11:Pair and CollediveFxcktatioz)s
'tz:/tlz, t)
=
e O2-bJ2/WpJ el/te5(Q=.-a7t)
(6.69)
for Q > Wqyt.The special fo= of the positive decay constant is a consethe conditions quence of the wave'equation. The polarization vectors F7r1611 et
.
e) x
(Q:r1
K%/ï =
Qv,-i
A-t
0, AA= 0, =
Q2 cJz/rtzs -
.
aze called Rayleigh waves (6.952. Ffowevez,its Wavcsof the type (6.69) dividual waves (6.69) cnmmotfttlfll! the botmd.a'cyconditions of a stress-lee stTrfnre. This is only possibleby linear combitmtions('t1: + 'tttl'with a mixed chxracter. Moreover,the bolmdar.yconditionscFmnot longitudlmal-transverse be satisled for shear-hozizontal modes.ln a'a isotropic e-lmsticmedî''m only vector = of thc sagittal-planeE'ayleighwaves are. Vowed. The displacemem.t surfacewave lies iu the plaue spannedby Q and the smface norrnnl zz. = 0 The botmdav conditions,cuzqmc (a= 1, &, # , lead to an eigenvalue equation. Slml'larto the bulk case a linea'r dispezsionrelation
uzawlol rswkol, =
rRw
=
'pt
'
(6.71)
ï
Ls obtained fov the Rayleir,hwaves. The phase velocity luw is #venby the solution ( (0< # < 1)of the polyaomial eqaation
ï
6
-
12
2
16 1 84 + 84 3 2.-.% p 'tl 4
a
-
-
'r
-
-.tz %J :
=
0.
The quactity (' dependson the ratio of the sound velocities. The ratio (14/q)2 VHe.S between 0 and 0.5 for the various mateziaks.Tllis corrœponds to a xariatiozzof #'iu the interval 0.955 0.874. For Gaà.sa value ( = 0.92 is obtaiaed (6.974. J.nany case thc velodty of the Rsyleigh wave zxw is srnnller tha'a the tzansvezsesolmdvelocià rt.. ThksSsalso tmzefor cubk mystals with three hdependent elastic constants (6.96j. Consequently,the Rayleigh mode should be belowthe projeeedacoustic branches,at lemstfor wave vectors in higkssympetry directioms. The rn5vedlongitudical-t:ansversechrader ks seen from the direction of the dîsplacementswhic.h are paztially parallel acd pactially norm/ to the to be parallel to the r-aMs. At the surface, propagatioa direction msslzmed z = 0, the polazization vector is given by -
2
eaw =
1
-
-
($2 2 2
('pt/'t&) ('
-
2 1
-
$2,0, -i$ 2
.
(6.73)
The dispersionrelation of a Rayleighsuzfacephonon Lsshown5.1Fig. 6.28. surface by men.nq TEe dispersionrelation Ls measuredfor a clea,zz Ni(100)
6.4 Phonons
5
.
R
150
R
S .E,
%o
k
k
# 1Oo
O u
ö' @ = g' v:z 2 o
X
% n4 80
->
c:
281
8
3
O c
a u
2
a u L
50
1
ul
n-
0 0.0
D = <.
c
c a
=
>x
'
0.5
F
1.0
O c C =
œ
()
X
Q1
Q (K )
Fiy. 6.28.
Wave-vectorclispezsiozz of the Rayleighphonon on tke Ni(100) surface. It ls memsured by menx!q of EBLS usitg t*e primav emergiœ180W (squares) and 322 ev tcirclesl. After (6.982,
of M-RMELS26.9% Near the bouad.azyof tile slzrênre BZ êeWatioasfrom the resdt of the continutlm occtzr as in the bullt case. The phnse velocity exhibits a wave-vector dispemion.For wave vedors at the zone botmdary the localization of the Rayleigh phonon near the surfaceiacex'xse,s according to t.he decayconstauts, e.g., Q 1 ê2.J.nsach a Rayleighwave only a few atomic layers beneath the surfaceWbrate.Consequently:Rayleighwaves with short wavelogtkts may also be used to detect efects of bond changes aad reconstruction. One drastic example is the sharp (iip kl the surfacephonon disperskonof W(110) whic,happears 0e,r hydrogensatttration of the stlzface .
-
(6.99!.
6.4.4 Fuchs-Kliewer
Phonons
The surface iHuence onwoptical pbonons can also be studied i'a the longwavelecgth tsud,hence,quui-contiauum) lirnst ) simn'lar to what we have learnt for acortic phonons. We consider the simple case where a bulk Eikactive czystal has tvo atoms in the uait cell with mmsses 3f+ aad M-. Such a crystal could again be a polar smmicondudor or azt ionic c'rystal.The bonds
6. Elementary Exdtations II: Pair and Coiedive Excitations
282
shouldbe partiatly ionic, ard the pair of atoms conwpondsto cation and anlou witlz sm-'ned dynnrnscalion charga e* and -c* with e. = ezijsfc (see force Taking iuto accotmt only the efedive nexet-nearest-neighbor (6.61)). for the disphcements'tz.y and .tzconstant .f, the equationsof motion (6.56)
read ms
M+ M-
:2
z*E,
'tt+
=
-2.f('u+ 'tz-)+
'tz-
=
-2f ('tz- 'tz+) e*B.
d@2 d2
-
-
(6.74)
-
(uz I'a the consideredloug-wavelength l''m'lt the displacementsdo not depend oa the position R of the Atnit cell. 1r. the continutlm limit, however: one may discussthe dependenceon the space coordinatez. Becaaseof the effectîve càarges of the cation aad anlon additional restoring e-lectricforces occuz. Tàe total electric feld E ac'ts at the position of the respective core. It is therefore inBuencedby Bcal-âeld esects wizich wilk however, not be discussedhere (6.90, 6.93).Skce we are interested in lonpwaveleugth opticat phozmnsjthere is only a need for stad>g the relative disphcement 'u = '?z,+ 'u- of the cation-anion pair, whic,hdbrate.swith the reduced mass Mr = M+M-/(M++ M- ) Then -
.
Mr
d.2 dzz
'tz
=
(6.75)
-2.fu + 6*E.
with frequencytzl one obtnl'nq J.11 the ll'ml't of a harmoztictime dependence c*
n,
=
1
V;
u savo
.
ts
-
(6.76)
g.E
with tEe zone-center frequency u'l'o
=
ag'ro
(0)=
2.VMr
of the bulk TO phonons azsumed to be kmdamped. The displaced catioms and anions fo= dipoles with moment henceja dielectric polnrization âeld puaz=
N. F
..e
N
+2
e
xhztfz;) V Mr ww;
ands
(6.78*)
w,.
and tNe relation P lat Becauseof (6.77) susceptibility a,s =
c''tt
1 -
u' a
.
=
x
lat
.s , (6 zr8) cu%es jue (&J) .
latticèE
(6.79)
In additionoatsothe electrons contribute to the total polarization âeld byL .PO: = x e.lE. Fbr frequenciesbelow the absorption edge of the electrdiiv '
6.4 Phonozus
283
Fystemthe electroak polarhability xd = (:x 1)/41ex.n be replacedby the static polarizability with ex ms the higisfreqttenc,yalectcotdcdielectric constant. From the deAmitionof the dielectric dksplacementîeld -
D
E + 4zr,FDM + 4rr.Pd H
=
LbLu)4E
(6.80)
the lcquenca--dependen.t blllk dselectricAlnction Q
sbtapl cx =
1 + Ltoz
u'To
-
-
wz 'ro LJz
(6.81)
k derivedfor lequencie,sbelow the htndamentalgap in the elèctronic band s'tmzctare.He'rethe (sceeaed) ionic plasmalequency 2
2
2
DJLO û7TO = -
4re* N E'xafrF
(6.82)
;i: fntroduced.This dm6nstiong'uaranteesthat the zero of the dielectz'icflcncof the zone-center LO jion,cbt(al= 0, eqz:xlslthe SequencywLo = 1t?z,0(0) With the statie dielectric constant sz = sb(0)j lshonon. that also accotmts fPr thc static lattice pobqrtzabilit'y the Eyddane-sn.nlnK-r-reller , i.e., &a > sx, = tèl>tion tu&o/u&olz ca/cxholds (6.100). Sincemacroscopicallythe crysta,li,seledric-allyneutraz,one 3àkksshw
dîvr with
or
=
cau
0
apply the '
(6-83)
eqttivazently (6.80)
= 0. skhtslltlivf
(6-84)
Nèglectingretardation efects, in addition the Maxwez'sequatioa .' ;' ''';.''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''d
r'E'
curl E
=
O
1i6l&.Equation (6.84) is f1116:1ed when eitheerdâvf
(6.85)
= 0. The 0 oz sstoJl qt case ctivx = 0 implit's that tlze electric âeld E and with (6.76) also i1k4relatkvedisplacemezefeld '?z.kave to be trxnmvezse,div'u = 0..The diJtqqtric A'nction sb(uJ) h.asa resonxnce at ts = tga'o which is assocîatedwith trxiivez'se lattice Wbrations.Thus) az'ro Lscatled the transverse rcsoncp'ce ,y = yknertcy.In the secondcase, div E # 0 (butcuzl.F = 0)and sb(aJ) 0, iiè :eledric Eeld E of the excited waves is longitadinal, Tke chvadetistîc ièquenc'y is EO:E.II at w = fztoo the lonnitudinal rssonance Jrazuerzcr. Shce k''w:r/ss(w),E kscot necessxrlly zero even for D = 0. The displacement iéïd.satisses cuzlvz= 0. Fpr Fequenciestzx-o < ;(J < wuot the lattice polarization bnn simtztane=
,
, l'
:. .
.
.
-
.
ottslfto
An161!
284
6. ElementaryBxdtatîons F1:Pair and Colledive Bxcitations
div Jva,t= 01
cml-rfat
=
(6.86)
û1
zesulting ia slmslar relations for the relative displacementfeld 'tz. These conditions have consequènces for possible displacement felds in a polarizable haltspace(z< 0)with smface(z= û) The felds ia such a system are cilarazterizedby a scalarpotential /@) with E = -grad/. Becauseof (6.83) and the potemtialmust Gal6ll a Laplaceequation (5.85) .
Zm/@)=
(6.87)
0
a'ad.the standardbound.ar.y conditions
/(z)Iz=-0= /@)Iz=+c, cb(Y)s/(=)Iz=-o yr/@)1z-+0
(6.S8)
=
of elec-tros-tatîcs.
.
Waves propagating paraltel to the sudace along a 29 wave vedor Q IL g-azds,whic,hdecayinto the vacuhtrn as well as into the crystal, are described by the fozowing solutions of (6.87)
/(œ)= /(œ) =
yuiozeom BeîQne-Q%
fozz < 0, forz > 0.
(6.89)
The 2D wave vector Q deterrnl'nu the attenuation of the electric îeld and the displacementâeld into the blllk. The boundarjr conditions#ve a1.= B and the eigenvalueequation (6.22) for the dieledric smfaceexc-itatioas.With = -1, leads to the gequency of the (in the eigenvalueequation, sb(uJ) (6.87) the pre-sentapproxlmation dispersioaless) Fizc/ls-fflïelccr phonon (6.1O1q tz'.p'lc= u:sto
+ 1 , soo + l co
whic,h indeed lies ill the intezvezltppc?o<
(6.90)
< Yco. The accompanying displacementfeld '?zin the polarizable hnlfxpaceLsparallel to the vector ($0, 1). u:
Therefore,the Fhchs-lcewe,r phonon also repreelts a sagittal-plre modve, but with a ftxed rnl'ved longitudinal-trnnsveasec'haactez. Sincethe trxnsfen'ed wave vector has no in-pla'n.ecomporent, ushg the backscattering geometa Fuchs-Kliewer phonons cazmot be excited ia a one-phononRxrnan experiment.
The Fuchs-newe,r phonons are nlnn.racteristiccollective elementary e-xcitations of the vibrating atoms in sarfacesof crystals with partially ioaic derived for thesesurfacephononsis generallyvalid. bonds.The formula (6.S0) This ha.sbeen shownby inspection of the material parameters ugpx, tzro, cc, a'ad scn for many compotmd smrniconductors (6.102jFttehs-Kliewersurface Typiphonons were flzst detected by HREELS with Zn.O surfaces (6.103) cally tke Fkchs-cewer phonons gi've rise to stron.g (even mtlltiple)lossesin .
.
6.4 Phonons
285
EnergyIoss hro (meV) -J.0
-K
O
20
40
0
-:-
aa: 1 XSCO
=
D
.d %R
rëp ; .S S
17o
I
-28: 1
-!7c
xqa
l
Q
-J.DO
-2tt
O
200
KC
Energy Ioss 'hro(cm-1) Fig. 6.29. N-RFVLSspeetmlm of a GaAs(110)1y l surfaceobservedin the specalar geometry (.Ppoiut)for a prinzazy enerr 5 eV. Fkom (6.1O4q. '
the HRFELS spectra of polar crystals obseaved i'a spccularscattering geometry. Oneexample is presectedin Fig. 6.29 (6.104q. The spectmlrn of a cleaved 1 stzrfaceusing a semi-insulating substratemainly showsthe GaAs(:.1O)1x Fuchs-lfliewer phonon at nso.x = 35.8meV (289 cm-l).A second,extremely weaker (byordersof magnitude) featuzein Fig. 6.29appears at 21.1 mev (170 cm-:)amd.is identïed a,s another surfacephonon neaz 1-h. 6.4.5 Tmfluence of Relaxation and Reconstruction
Accordiag to the abovediscussions,the smface 'dbrations may be usedto probe the surface atomic structttre, since the restoring forcesdeviate â'om their bulk values. This holds for the micaoscopic modes but, in pzinciple, also for the macroscopic modes.For instaace, the dispersionof the Rayleigh modeis Sn6uenced by the surfacedetailsfor wave vectors aear the smfaceBZ boundaaor. Soaces with an atomic s'tructure not too digerent from the bulk one are relaxed surfaces.The most intensîvely s'tudieârelaxed smfacesare the (110) cleasragefacesof EU-V compotmds crystallizing in the zinc-bltande structme.
The (110) smfacephonon dispersionof one of thE'secompotmds, Inpj is shown in Fig. 6.30. Calculateddispezzionrelations (6.36j are comparedwith measlzredvalues (6.34, II).P îs consideredto be a, prototypical material. 6.1054. Sincethe masse,s of the anion aad cation are vezy dxerent, a large gequemcy gap betweenbulk optical ard acoustic branchesexists that allows for szzrfaceirduced gap modes.Tlle theozy,of course, predicts more brancahes of surfaee
286
6. Elementanr Excitdtions 11)Pair and CollectiveExcitatiozus
Wavevedor Q
surface obtnt'nedby difFîg. 6.30. Surfacephonon deersionfor a.rt 1nP(11:)1x1 solid lines, 'FFRIVLSE6.1O5!: ferent methods.Ab initio D.FPT cakulations (6.36): The shnrled.areas desczibethe prœ squaro, au.d Rnrnan spedroscopy: dots (6.34J. jectedbulk phonor branches.From (6.341.
modesor surfaceresonxnce modesthn.n observedspectroscopicallyfor difFerent remsons, suc,has scattering geometzies,matrix elements, selection rules, locnllzation, etc. The mayleighmode appears as the lowest smfacephonon 1 surfaces:typibraach, at least along tke PX' dtrection. For 11I-V(11O):tx the Myleigh Sed. 4.2.1) cmlly, it is found that upon sarfacerelaxation (see. wave modesat the zone bolmdary points R, X', aud a az'e shified upwards feat'aresfor sarfacemodcs and surface Clnnzvtezîstsc by up to 2 meV (6.106j. cm-l, resonance modesappear forïaptllollxl at lequenciesof about 60/82 cm-l, a'ad 347 cm-l tlsing the Rmman signature's (see 146 cm-l ) 254/270 Fig. 6.11). ln these frequenc.yregiozlsmore or less weaMy clisperskvephonon branclzesare also observedby TTRRELS(6.1051 to At the center of the surface BZ, lh as we2 as a2. ong ?X' (parallel the smrfacevibzational modes c-q,n be claasifed according to tke trreg001)), duciblerepresentations of the poiu.t group m (orQv or G) of the redangnllar direcsmfacellnst cell vith a rnirror plane that inclades the (0O1q IE-V(110) i.e., Accordingly,atoznic vibrations along (110), tion (see Tables 1.4 and 1.6). along the III-V zig-zag abasn direction are represented as AM modes, aad vibrations perpendiculaz to the chain directionu are represented az AI. The A'' modes represent shear-horizontal'divations, whereas the A' modes are .
6.4 Phonons
287
sagittal-planemodes.Suc,ha clear dassfcation is not possible (transverse) along the symmetzy directions PX (parallel and Pa (parallel to g1ï0J) to deânectby the surfacenorrnnl and the (1ï1)).The sagittal plane (Fig.6.27)
phonon wave vedor is not a reoectionplane. Consequently,along these &redions modesshow a rni'vklzre of sheaohozlzontaland sagittal polarizations, i.e., tbe atomic disphcementsllave nonzero components in all Cartesiandiredions. The Rs.rnan feattu'es at 69-82, 146, 254, and 347 cm-l (Pig. 6.11) can be mssignedto smface phonon modesof A' symmetzy The t'wo surface phonon modesat about 254 aud 270 cm-l (6.:?.44 are locatedin tEe gap between acoustjcatld optical bu?uk brarches.Their eigenvectoo are irecated in Fîg. 6.31. They =M5n!y conwpond to nearly opposiug motioas of the Ast-layer ar.d second-layeranions, with the most sîgnifcant contributions from the furst layea.The chaœacte,r of the 17.n7,,=peak near 347 çm-l is not complete'ly cleaz. A surfacephonon mode at the 15point, that is predided at 353 cm-h (slightly above the LO phononfrequemcy) by DFPT cm-l aud at 349 Ecalculations (6.36) by sezai-empirical calculations (6.37j, hmsbeen assignedto matc,hthe Fuchs-cewer pbononwith A' sym-petr-?. %.Fig. 6.82 the isplacement pattern of the correspondingsurfaceF opti,Vlphonon of GizAs(11O)1x1 at 286 cm-l (6.35q is showm For TnP(110)1x1 cm-l for tize Rtchs-cewer pitonon é:XEELS 1 however,yields a value of 342 chna'n Ebèiil.g in bewreecthe TO and LO phonon energiesof I'IIP (6.105J c The . zitodeof the relaxed(110)1x l s'arfaces for wave vectors along 1nX' possesses ))/ symmetzy a'adcorrespondsto the opposing motîon of the catiozzsand anionsin one atomic layer parazlelto the zig-zag-chna'ndirections (1101. The disill the Ast-atomic layer are larger. This is shown for hN(11O)1x placeznemts 1 For the I'CNsurfacetMs mode lies i'a the uppe,r(L.a,If tàhd1-hin Fig. 6.33 (6.107q. = 55.6 mev. However, $î the bulk acoustic-opticalgap at Hchautol for othe,r = 35.0 VI-Vcompounds,it is also a gap mode, e.g., for A-lsbwith *01t::$,:(0)
lev (6.10$. r
102 E1
10E1 E1
)
0:(I OE
Sng-6.a1. Atoznic dksplacementy on J.nP(llO)-1x1 whicz #ve rise to (eigewectors) Fîoizatkr:ed gap phononmodesat r. 'ne :engths of the arrows are takenfrom DFPT kalculatiozus open t5J)edl smbols denote In (P)atozas. (6.362.
288
6. BlememiaryBxcstations11;PG and CollectiveExcitations
10(1 E1
%
E00 11 Fig- 6.32. Atomic displacementsof the .? p/onon with energy 35.5 meV oq the
surface.The arrows GaAs(l1O)1x1 nniorus,
Filled drcles:
give the Jtrnplitudesof the eigenvectorsd(0). empty circles: cations. Fkom (6.351.
rrhe phonon modes measm'edor c/calated for 1H-V(1$0)1x 1 surfaces show rather clear clmrnlca,ltrends. This is demonstrated in Fig. 6.34 for the Fequendesof the Rmbq-Kliewe,rmode (Fjg. 6.32)acd the sttrfacolayerzigthe h'eThe xlmost linear vadation of Flau(0)c,c, zag clmt'nmode (Fig. 6.33). quenc'ymaltiplied by the blll1zlattice constant, versus T/ M: indicatesthat or J(&0 the efèctive force constant f of the type in (6.77) + 1)//x + 1)izl
EtïOq
lt >
1 surface (top Fig. 6-33. T:e qhizinphonon mode of A'' symmetry ozz a (110)1x Lsshown. It 11%in the upper Bn.1fof the view).Here the In modeof IzzN(1:0)1x1 = 55.6 mev. After bulk acoustic-optical gap at ntzgcluuatol (6.10e(.
6.4 Phonons
28g
400 InN M
35o
o:F j;& 300
AM K Alsb
X é'
(7
X
253
m
G
Ga/ks
:-) 200 G
Gasb
lnsb
M
K
D
150
:a
100 0.1
0.16
O.2
0.25
0.3
l/qo-r (a.u.) Fig. 6.34. Product of phozton enerr Nd lattice éonstant versas the reciprocal squaze-root of the reducedmpA.q for two F'sn'vfnmemodesof 1II-V(:10)1x1 surfaces. Fachs-Kliewermode: ftlled sqaares; surface-layerzig-zag ckaiu modet open squares. From g6.107, 6.10$.
varies nearly ms the reciprocal (6.90)
bond length.
square of the lattice constant
or
blllk
The Si(111)2x1 surface can be considered a,s a prototypical sllrfnzte to study the l'nsuence of surfacereconstruction.It is chvaderized by the formation of tilted zc-bondedctlnnlnqiong tize Eï10J direction (see Sect. 4.2.2). Also this surface sbovrs a Rxyleigh mode. Along 13.1, i.e., along the chnsns, there Lsexcellentagreement for the dispersionof the Rayleighmode between ab initio theory (6.10% a'udmeasuledresults obtained from Ee-atom energy loss spectroscopy (6.110j. At the zone botmdarg the RW mode is Yredly Oected by the reconstnzction. Tize Rayleigh mod.e at the J' point lies at = 10 mev and is poladzed perpendicular to thc n'approxlrnately F/z.uxw(J) bondedchatus. The additional dispezsioeessresonant mode detectedat 10 meV aloag ?.1g6.110j has not beenveMed by the DFPT calculations(6.1094: but hasbeeareproducedin adiabaticbond-cZargemode.lcalcuhtions L6.111). Ab izkitio latticedpmnnical calculations (6.109) also predict several 6.112) the blllk localizcdar.d resonant modesin the energy range 45-68 mev arou'n.d optical zone-ceaterphononwith F/zs'ro= Flamo= 65 mev. The displacement patterns of three characteristic .I9modes being more or lesé resonant with blllk optical phononbTanchesa're shownin Fig. 6.35.Thesesurfaceresonance modespossessenertesof 57 meV (after improvement of the i-poin.t sxmpling
6. ElementanrBxoitatiomsII: Pair aud CollectiveExdtations
290
1H
D
ET1 0)
l I
!
1Z E1
112 E1
$)
St
kk
(@JI2
>xg. 6.35. Disphcement patterns of selectedresonant surfacepkonon modes at ? 1 sudace.ARe.T(6.112q. of the &(111)2x
However,they ap'ç ita the calcalatioa) (D),55 meV (r) and 50 mev (J??). strongiy localized i'a the St'st atomic layers. The D mode corresponds t3. longitudinal-opticatvibrations along the chnsnsin the &st atomic layer. TV two other modœ11 au.dJ& also contain contributiozssfrom the secoadatomiç layer. These modes are essentiaxypolarized pazallel to the surface normk. The D modewit jyliy vith a smaz (121) component perpendiclzlarto the c-halms. is accompaniedby a lafgè: or 59 meV (6.111) a'a encrgy of 57 meV (6.112) polxrization âeld. lt may therefore be identled with the strongly dipol/: the optical-phononzegion at 56 meV K taivemode detec'tedby HR.EELS5.zt .
or 57.5meV (6.114q. (6.7134
6.5 Elementary' Excitations
for Reduced Dimension
'Fhe mnmpbodytheory of intenading particles,espedallyeledrons, in solidà predic'ts the existence of mazty dementary exdtations. Because of the,ir mtl i taal interactiou or their inte-raction with other particles they are renorrnniz ized and therefore named qltn.qipal-ticles;whiclkmay show a one-to-one corzr respondencewitk non-interacting electrons tseeSect. 5.$. In the case 6f metals the correspondjngphysics can be describedwithin the Femni-liquv % (6761. The properties of electromsbecomemore and more exotiè &PPrORn ;L1 world into lower dnrnenskonà. as oae progrcsses from the three-dpmensional sttrprising phenons: In a two-zirnertsionateledron gmsone a'treadyobsez'ves ena) suc,ha,s fractional nbxrge aad statkstscsia the regime of the gactiona? quaatltm Ha.IJeFect.The corelated motion of electrons and magnetic vof. Efects sternming frozi tices generates theseuzzasualphenomena(6.115,6.116). the jncreasedeledron corelation aear smfaceshave bee,nOcussed in Sedi: electron gas are even morr 5.4 and 6.2. Predîeions for a one-dsrnensional exotic. This applies ic particular to one-dn'rnensionalmetallic clmimq.Suçfq low-dimensionalsystems nxbibit a variety of novel physical phenomena)suci .
.
.
6.5 EzemerzaryExdtations for ReducedDimension
291
J: charge-densitywaves
Peierksirtstabilities, or the formation of (CDWs), rlcm-Fezrfrti-liqdd-Degrotmd states (6.761 6.117-6.1212. structtkres are t'ypical for rn>myreconstractedsemiQuasi-onew-fisrnezasiortn:
jiönductorsurfaces.Important Rvmples are the nhnan structmes discussecl in jéc't 4 2 However,more interesting iu the many-bodycontex't are adsorbate:E induced modp6catîonsof semicoaductorspqrfaces.Self-orgn.nt'zed adsorbatemodi6cation of semicondudorsurfacesis a powerful tenhnique for tnduced fàbzicati'ng suc,hlow-dimensionaznnnoscalequaatlxrn structtzres. Jmportant :: jxxmples are one-dMettsionalquantl:nn chnanqof metazatoms, sue,has In or lu.oc surfacez.The nrlqozption of irtdblm inducesthe formation of Si(111) :': on a 4x 1- or 8xz-reconstructed Si,(111) tmaai-one-Hnmenqional cktyunsl surface Arrays of monatomic cllnsnsaze also observedfor Si(2.11)5x26.1231. (6k422, ia,or stepped Si(557)-Au smfaces(6.12*6.126j. ((111)5x1-Au) The arrays of quasi-ono-dzemezusipnal chnsnqinduced by metals on the suzfaceexhibit a valiety of iuteresting electronicstates. Theze are E!i(1'11) of correlation eects dastroyixtg the metallicity of suzfaceswith a,!z ,bè>oz'ts J'édelectroncotmt per llrp't cell (6.127-6.1292, of anomaloussurfacecormlgaEiiàn eitser by cows E6.180, or by large atomic displacements(6.132q, .7 6.1:12 .8?E,metxll5c nanowi'res (6.131, 6.133,6-1341) of sudacesof rnl-vedri'-rnfmKional;.:''j;' azd of spitschargeseparationin a Luttinge,rliquid (6.1351. #y1(6.).24j) E.1n a Luttiuger liqtzid (6.76, the elecvtronlosesits identity and 6.119-6.121) into t'wo q'tlmipazticles,a spénonthat carries spin without charge Erçsarates Vd a hoio%that carrîes the positive charge of a, hole vfthout it.s spih. T'h.e ieon for the positive chargeof the holonis not peculiarto a one-dinnensional 'Blid. It is simply related to the fact that one cxnnot probe the e'nergy and iömentnlm of a:a electron in a solid without ejectkgit, for exztrnpleas a phoiioeledron tn a photoeznissionexperirnent accompn.nsed by a photohole(see jèèt 5 1 2).This leeavesthe solid iu a positively charged,excited state. The t*o asgerent quasipmicles, spinon a'ad holonhhave rlsFerent grou.p velocitteirand rkm away fxomeac,hother. T*e proble,mof spin-chazgesepvation îs jj..;: ianxlyzedn'lrnerically in the met>llsc phmse41s3.r.g a one-ba'nd Eubbard Hn.rnik tözïia.u i'a one dimension (6.1214. The spedral Annctiozzof the (cf.Sect.5.4.2) iigle-particle Greea'sAlnction (5.20) with spin can be decomposed icto t'wo peskq that changetheir energiescvztk)licearly with the (wwsiktfwsipazticle ùzùmentx'rn Flk. Thesetwo pen,k-qcoincideat the Ffarrn''level ss ozfy, where Efh' e sphon and holon lines intersect. The spinon hmsa laTgergroup ve-locity '8ii (k)/d(Fzk) th= the holon by nlrnost a factor of two. e.
.
*
.
.
--
.
.
-
''
.
-
-
.
.
.
. .
-'
.
'
'
<'
The ùzcreasizlgamolmt of cozrelation bet-wto.e'n eledrons in lower dirnenjions can be rationldized by a simple, clazsicalpictttre whe-reeledrons behave ik é billiard 1m1lq(6.136j. They are forcedi'ato head-on collisions i'ctone dnrnen!siàn beca use they cannot escapehom each other on a true one-dirnensional ti-k.. QpnnmtT:nn menhn.nicallythei.rwave packetshaveto penetrate eachother àt some point în time a'adthereby geneaptemnv'nnllm ovLrlap.In t'wo or three lzifnensions sucz a situation is impzobable.In one rlsrnension,the IFe'T:Gsuo
6. ElemertaryBxcitations1I: Pair and Collechtive Excitations
202
face' 6v wave
consistsozf.yof t'wo points at kp aud -kv (withthe Fermi szyatkl vector kp).The i'aterestingphysicazproq-es happenarotmdthe sFfA=l' =
surface'. Consequemtly,the-reis no such thing as a single electron i:a one dimension. When exciti/g it datrimgthe meazurement process one necessarily generatesa chain reaction that exdtes other electrons.The rewstlltis more like a cozectiveexcitation. Consequemtly, one may visun.lszea holon in a Imt'tinge.r liqttid as a CDW acd a spiuon as a spin-densitywave (SDW) Kxotic phenomeaa,sucb ms sph-charge separation, havenmn'nly been discussedfor metallic systtarns.They allow l'n6mstesMallysmall excitations and are sensitive to tiny pertrbations. Recently,signatmesof spin and c,harge collective modesj.n onmdirnensionalmetallic chnsnqhave beenbelievedto be obsmed withim photoemismion ex-persments(6.135). Aa array of monatomic .
Au dmins on
a
snghtlymisorientedSi(111) surface(Si(557)) h.asbeenstudied.
Afte.r xnnenls'ngof
'
0.2 monolayer-thick overlayer a Si(111)5x 1-Au sprrface appearswith parallel chaimscf Au atoms i'n an intrachain distanceof 3.83i W hic,hare separatedby about 20 i twhie,h correspondsto the tezrace widthlf Photommission seems to indicate a metallic characterwith a parlially ftlle'd band. The authors have no indications suggestiveof CDW and claz'mthe absence of a Peierls trausition (6.118) for such a, system. Fbr one-dMensional systems: however,the vezy efstence of a metal is in question. Accordsng to the Peierls theore,m(6.1181 a one-rlsrnension.al c,hn5nof atoms is kmstable with rcspec't to a pairing of atoms, which creates a'a eae'rgygap in the band stracttzre at the Farnnllevel. In a trae ID system of idcntici atoms such a reconstruction is the ozfy possibility for lowering the total energy. The enthe occupiedstates at the bottom of the gap exceeds eror geed by lowem'mg the strlu''n energy that is necessary for the displacement. Tm.fact, it is diëcult to Snd one-dsmensional systems that a're metalc. hdeed, for rzhnsns on a Peie'rlsgap is obsezved(6.1241. However, for metaltic sysSi(111)5x2-Au tems suc,has Si(557)-Au a Luttinger Iiqt'dd witll a spGon-holonsplitticg htzs beenrulcd out and, moreover, details of the atomic structkzre,suc,has broken bonds at the steps, have to be taken into coMideration (6.125, 6.126j. One of the signlcant featuzes of qumsi-one-dimcnsional sysIVSi(111) tems is a reversiblephasetrn.nqitîon, TEe room-temperature Si(111)4x 1-1'a surfaceundergoesa tr>m.qitionat about 100 K to a new phmsewith a 4)4$2' zeconstructiongwhic.h is ddven by a ID CDW or, equivalently, a Peierls instabitity alonythe Iu chnsnq (6.122, It is specttlated that the 6.123,6.1371. trae low-temperatme grotmd state mig'h.tbe a well-ordered 8x2 phase with the CDWSlockedin pbmse(6.131j. Electronically the room-temperature 4x 1 phwseevhibits the Ferrni-iquid-iike behavior of a strongly anisotropic twoHsrnensional metal (6.131, whereastke 8x2 phaxseevlnibits feature,sof 6.134), an insulati'ag system. Recent studilas (6.123, show, however,that the 6.137j simple SD-CDW formation camnot be the dri'ving force for the phasetransition. a
7. Defects
7-1 Realistic ard Ideal Smfaces picture of what a solid sttrfaceloolcslike at the atomic level wms fairrly idealized.Atomic reazrangeaents and mon-stoiœometric distributiors of cations aud atfons were only allowedto zeslzltic relaxedor reconstruc'ted sarfaceswith a deMed 2D trauslatioaal symmetzy In realit'y, invaziablys'arfacesaze found to consist of a mbttt'tre of Qat regions,the so-exl7edterraces bat surrotmdedby Ene defects,e.g., with a cel'tain reconstmction/relaxation Figs. 1.1 ard 2.1).The tersteps, kn'nk-s,and modled by point defects (see races Tepresentpoztions of low-inde.xsttrfaces.Only extendedterraces without po i n t defeds represent realizations 'of the idvealsmfacesused az physical objects=d, hence,are dlscussedcornrnonly. On the other hand, the presence of a large ntlrnber of defeds in bulk czystals acd on slufacesis a nat'ural occurrence. This hoids in pazticalar for aative dcfeds, which on surfacescan be categorizedas point defects (vasteps, 'irsnks, and their complexcs.Steps are natural for cancie. ackisjtes), Sect.2.3).For a given temperature native point defects viciual surfaces(see suc,has vacanciasshould occtzr with a certain probabitity i'a thermodrmmic Despite the tTHV conditiomsatoms in equilibritlm as in the bulk case (7.1). the restga.smay give rise to contxmsrations. Suchadatoms9om the rœtgas represent impuritîcs (atlemst,msi'azthe lazguageof bulk semiconductors). Onemay classif.gsurfacedefeds accoreg to their dsrnensionn.ll'ty.Threedimensionaldefectsdue to the mosic structure of a crystat or due to'straiu shottldnot be discussedhere. The same holdsessentiallyfor two-rlimensional defectssuc,has stacldug faults or dornna'nbotmdaries.Only one evn.rnpleis studed i'a Sect. 7.4. Importaat one-dimensionalor line defeds a're steps in Figs. 1.1 and 2.1)separat% t'wo terraces lom eachother. which the ledge (c,f, The step hdghts dependon the surfaceozientation, the elcctrostaticbehavior of the atomic layers, and the polmorph of th.e czystal.Jn many cases steps of singleatomic Eeight prevail.However,for polar surfacesdouble-layersteps are smatl more Dkelybccauseof chargeneutrality aad, hence,the accompauying surfacesof hexagonal a'n.d(0001) electrostatic forces. I'a the caqe of (0001) 611(4H)compolmdczystalssuchaz 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 to the sarfaceorientation.
Otherto
our
.'
294
('
7. Defects
There is go ltnivocal deBnstionof the step keight. Here, it is expressed in terms of single atomic layers with an orsentationparallel to the normnl of the low-indexsllrface represemtingthe adjacentterraces. Accordz'ngly, the
denotation ms a monatomic (shgle-layer) or a biatomk (double-laye'r) step is ased.Wke,na step is charactfm'zedby a bilayer, usplltlly a'a electricaxy neutral combiuation of two atomic layersis discTpsmed, e.p, a combhation of a cation and an xnlon layer in the (100q or (111) direction of a ziuc-blezdccrysti or ic the (0001) direction of a he-xagonalpolytype of a compolmd sach as SiC. Zero-dinnensional or poht defects(Fig. 7.1)hvolve impurities (adatoms, è ledge adatozns), i'aterstitials (adstoms or substrate atoms at a non-lattice Other surfate r site),ldnkq, vacaacies,and antisites tinsurfacesof compokmds) defeds aze related to dislocations,wllic.hzepresentone-dimemsional defeds hE the blll'k. An edge dislocationpenetrating icto the smface with the Btuge,p q5 vec tor ozieated parallel to the surface gives rise to a pomt defect at thi : surface.Step dislocations Mtting a surfacealso cause poi'nt defectswllic,h aréE:: uslpn.lqysottrces of step lines. The local eations related to defec'tsmay cause changtastn all jmportani sl:rface propelies, sac,hms bondzngbehavior,atomic coordlnation, electrordd(: s'tates, lattice vibrations, etc. A thorough study of suc.hdefects is therefore important for a hmdmental tmderstaadlngof their role iu growth nttcleu: surface cliRlsion, adsorption, and desorptëonlçrE ation/epitaxy (evsporation, smface chemicalreadions (e.g.the slprface catallic eëciency), and eleis tro/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-blendesemiconductors(Sect. 4.2.1)are ideal objec'ts to study vacancies.Thesesurfacesare non-polar, i.e., equal num-, bers of cations and aniozusoccur izzone atoznic layer i'a the absenceof defects;E They are only relaxed,i.e.) the bulk trxnqlatioual spnmetry is coasezved 1: two Hlrnemsiozus, azldthek surfacegeometries, at least tlze lateral coordmatcigE: E are sirnslar to the bulk ones. The defect densitjr direc'tly after deawage Lssmall?:. which makes f 110)sttvL.ces ideal for investigationsof isolaied defects.TllçE sabstratesare AE compouuds,i.e., 50th 1-.0e,5of vacancies,Wx acd Ws (see E .ï l c..%'n be stuctied.Thereby a ç-fold càargestate T'ig. 7.1)) Lq= ..., +1., 0, -1, ...)E may occm. Suchvacanciesare generatedduring the cleavageprocess. IIl genz eral, the dezzsityof atomic vacandes is consideredto be cleavage-depeadeui .!:. but for A.svacancies on (110) surfac%of p-GaA.ssamplesit is estimated 'to be roughly 5 x 1011cm-z ) or one per thousand surface anions (7.24. The ssg,rj aad the magnitude of the clzargeq of the vacanciesis ''n6uencedby the clo/i iag level of the bulk czystal and, hemce,band bendingnear the smface. Ah :. exampleof an STM lmnge of defeds occuzring on the 1nP(110) surfaceof 'a , '
.
.
..
o
.
.
'
'
r.
7.2 Point Defects
295
Ideal surfaces o o *
S
A atom B atom
.
Impurityatom
Substitutional impurities SA
:
.
Vacancies
V
VA
VB
(y' .
(Eigen-llnterstitials I '
!
IA
IB
O
O
.
O
:
iikerstitial impurîty '
l..y ),)r.(-.
/. q
?lkrltiïiitf,is AB
BA
S
Pig. 7.1. Elustration of impoztant point defect,sia two-dsrnensionalelememtaland ttimpoundmystals, i.e., conwpoading sarfacelayezs. '
2S6
7. Defects
Pig. 7.2. STM image (snmple voltage -2.2 eV)of a (110) smface of a Zn-doped J.'aPcrystal (7.3) by the Arnerkm.nPhysical Society) . (copyrigùt (2003)
Zn-dopedI=P czystal is show.ain Fig. 7.2. (copyright by the Ameri(2003) nltn Physici Sodety) I'a addition to pEosphoruswacancies(V)appexrlng ms blnztkdepressiousp white elemtions suzroundingZn dopaat atoms (Zn)and smaller blac.kdepressiozzs relatedto vacsncy-impurit.ycomplexes(V/Zn)ca,n be observed.The depressionsand elewations aze relatedto nlnnmged states azd are thereforeinfuenced by t:e band bending, The occuzrence of such a dafect,e.g., an acion vacaccy, strongly depends on the surrounds'ngatomic geometzy acd its eHrge state (7.4, 7.5).The c%Aractezistic quantity that governs the probabslityof tba appearancein th-rrnodrmrnl'c equilibrium is the formation energ.vvf%. Jt can be fbrmulated izza m'mslarway to the sarfaceenerr Rs (2.45) using the forrnnm'mof Zhang and Northm:p (7.61 as = f)f (Ws)
SIXA)Nn
-
1, q/
-
ELNXt NB)+ #B + qlsr+ fvsM).
1 snrfn.ceu The total energies of the syssem 'Gt,h an ideal relaxed (110)1x of the surfaceon wikicha B atom Lsmissing, ELNh, NB-1, q), A%l,-an.d ELNA., For q < 0 (ç > 0) i:l additional may be calculated Mcording to (3.55). electrons (Yoles) aze considered5.:1 the system which aze compensatedby a backgrokmdas in tàe jellbnrnmodel. However,vithin the positive (negative) numericaltreatment using the (repeated) slab approxn'mationthe-reis a complication due to the local missing of such a compensating backgroundin the vrunm regîon. The ch-rnl'calpotential JZBof B atoms ân a ceztainz-rvoir is detmrrnined az described:in Sec't.2.5.3. The Lastterm in (7.1) accotmts %r the energ.yof l.t2k electrons Lq> 0)oz holesLq< 0)ia the eledronic reservoir.
7.2 Point Defects
da
297
db
dc
Eo01)
s
p'$oj
representation of ar anioravacancy on a III-V sornlcondudor Fig. 7.3. Schnrnn.tic (110)stkrface.Three dacgling bonds (Q, dN?and dcjare created dtlring the formation of a vacancy. Small circles correspond to atoms in the secondhyer. Filled smbols represen.t cations tanionsl. (open)
The Fermi level sv (thes=n.11temperatttre-induzedcliFereace to the chemiis measlzred1om the mlen.ce-band cal potential of the electronsJsnegleded) mnavlrnam evnu. Usuallyan alignmentproceduzeof the energ.gscalesin the Ia exideal system aud the system with a chargedvacancy is applied (7.71. plicit calculations often a shift of the valence-baudmaOmllm due to the cbvging of the dafec'tLqconsidered.lt may be calcuhted by the dlference + &k@)(3.48, of the spatially averagedeledrostatic potentials Montœl 5.41) for the càarged and neutral situations (7.7, 7.81. The charge state of the anion vacaucy i'n the surface layer detnrml'nes the atomic relaxatioa around the dmfec't(in contrast to the bulkl and the energeticsof its formatioa. The three daugljng bonds of an anion 'vacrtzjr VGs(see Fîg. 7.3)are occupiedby (3 qt electrons.Consequentlythe chazged vacancies Vs- azd V+s prefer rebonded geometric whereasthe aeutral N'acar.cy Wsexhibits a weakertemdencyfor rebondtng.The positively charged aaion vaca'aclesa're dominated by an inwazdrelaxation of the neighboring cations (7.8-7.101. However,the symmetrically non-rebondedand the nonsymmetrically rebondedcovgwations are almost deguerate i'n eneror. ln the non-bonded geometry the catiomsa aad b (Fig.7.3)relax inward. In the Sed. 4.3)with the c rebotded geometu, the a or b cation forms a dimer (see cation. In 170th.cases a single empty dangling-bond-derivedstate appears ixt the 'h:ndamentalgap. -
298
7. Delhects
Fcg. 7.4. Formation ener#escalculated for dl'lerently charged P vacalcies öy!. mlrfaces tmder P-rich conditions verstus the Ferrnl'energy. Adapted fibm 1nP4110)
(7.:3-:1.) -
Accozdîngto the formnt,iolk eneror (7.1)tEe most favorableeahazge statq deper.dson the doping level. TMS is klltzstratedin Fig. 7.4 for dlFere/ttk chargedP varnncies on 1nP(110) surfacesprepm'edtmdez plmsphoruscrich bunc conditions, Jzp = pzp On surfaces of pdoped substrates a singlt (7.11J positively cttargedP vacancy JCis stablej wheremsa singlenegatively chargèd P vacancy V- hmsthe lowest formation energg for zsdoped substrates.Fck tmdoped TnP the neutzal P varancy F'eo be the most stable one. Tlzev may Mdsmgsare in agreement with m=y ex-perlmentalobservationsfor Gavj! 1nP, acd GaP(110) surfaces g7.12J. The positions of the Fnrml' level sp at wllich the formatioa enfzrorJ)f 8f a defect iu two cllFe-ren.tchargestates q and q + 1 become.s equal in Fig. z.iï or (7.1) deft'aesthe éonémiion Jewelor charge-transition J6VeJ vitil sLq+ 1jt2) respect to the VBM. According to (7.1) one %n.qg7.7, 7.81 '
.
:(ç + l1ç) ELNX,NU1, qt EINX,NB 1,ç + 1) =
-
-
-
-
Sv'aM
disregardingfor a momest tke alignment procedure. For the acion vacacéîr!q there aœe t'wo gap levels,s(+I0)and s(0E-), as demonstratedin Fig. 7.4 fùt 5'pvon TnP(11O)1x1. The Srst (second) one is related to a more acceptötlg like (donoz-)
state. The euerg.gc(+I0)may be interpreted as the bindiik energy of a hole to the neutral vacancy, while X c(0I-)gives the e'nerj/ o f a'a electron bonded to the neutral vacancjr. Uaforttmately, neither STV'' a'n.dphotovoltage memsurementsnoz ftrst-principlescalculationsgive a uzziqttèE -
pictzlre for the t'wo energy levelsin the gap g7.12). Even for the mode,lsystelE
7.2 Point Defec'ts
(a)
(b)
-):( >
:-01 f
299
y
Q
y j yj
Fig. 7.5. Stmzctuzalmodels (topview)for the C-type ddec.ton Si(10O). Dànglt'ag ibùnds rezatedto rniAsingatoms and located at the furst-hyeraad tbhd-layer atoms ;' E:. o aàeindicated..Bu of the z'mers Ls allowed aa kzdicatedby the dlflkrent shes 6f the circ'les representing Srst-layeratoms. (a)A two half-rl':rner model (7.15, 7.16J, (b)m''qsixgatom in the secondlayer E7.17). '
.
.
v
.
ö?a
sl:rfbnrtethe leve,lpositiozls single .ks vacancy at a cltoltnGaAs(110)1x1 lait'4 the intezwetation of the chaapder of the levels va'cy (7.8-7.10, 7.13). bnx recently remaruble progress ia the calculations been aœeved .k 9wever, t.'LL?Jncerns'ng states (7.112. convergenceand treatmemt of the c-harged Vacaudes are also discussedfor sttrfaces of covalentsmmiconductors, for k 'Li acce the Si(111)2x1 surface(7.14) and Si(100) suzfaces(7.15-7.171 One pvn.nnpleis the so-enlledC-defect on Si(109) wllich is always obsetwed (7.15J i/ room-temperatttre STM lmltges. At positive snmplebias it appears as a Vghprotrusion adjacent to a depressionin the surfacelayer. Two interpreVtionstseeFig. 7.5)have beengiven for these STM images.Izzone case they ake interpreted to inclicate a dimer ddect in. which t'wo adjacent Si atoms kong the dsrner row are rnr'qsiztg,i.e., like t'wo 'hn.lf-fssrners(7.15,7.162. Conscthis defectshould represent a mcaucy pair in thc fkrst atomic layer ktzently? Ciculations (7.171 haveshowatha,t the optirnizedstructtzre start(Fig. .LE 7.5a). ,k#jFomthe two lnnlf-dimezcougttration is llnnble to rèproducethe observed j''FCMimages. Tnqtramzla C-type defect structme izl whic.h only one Si atom i'a ,,
'
.
.
.
3OO
7. Defeds
Fig.. 'J.6. Lltustration
(topview)of a mlqsjng-dimerstractuzej a sty-nn.lledtype.4 defed, on a Si(l0O) smface; (a)idea,ldimer row; (b)cll'mer zow wiih a dimer vacaztc'y.Large (small) atoms. open circles represent Srst-layer (second-layer) the secondlayer is mlqskzg,a,s shownin yaig.7.5b, IhAAbeen proposed.rIn1np's correspondsto an isolated vacancy in the atomic layer bene-aththe dimer layer. Many other defeds disturbing the dirner rows along (OI1j havebeen discussedfor Si(1O0) surfac%.lmportant ones are related to missing dimers, i.e., to patt's of vacancies izz the 6mt atomic layer. A dsrner-mcmmcymodel i'n whic,ht'wo adjacent Si atoms along a (O11j direction are absent is shown in Fig. 1.6. suc,b. a model has beeaproposedby Pandey (7.18) to elurpln.in the type-A defect obsemrecl on Si('1O0) (7.15, 7.16j.Other defect complexes consisting of t'tvo oz more missing dsrnersare observed.) e.g.,the so-calledtype-.s defed (7.15,7.16j, an.dgive rise to confgurations with not too lmgc formation energiesF.19). 7.2.2 Impurities
The abil!ty to kcorporate reprodudbly dopant atoms, in particulaz shallow impurîties as acceptorsand donors,with precisely controllcdconcentrations and.spatial distriimtions is essentialin vadous de
7.2 Poht Defeds
301
conducti'ng materials. The geaea'allyaccepteclview ks that the chargeof a shallow impurity is smvencyd by the charge cn.rrsez'sixl tlze doped semicoxductor, restllting in a lonpracge repulsive screenedCoalombinteraction betweert dop=ts. Suc.ha repulsion ia turn leads to a rather homogeneo!zsdistribuSurfacesdisturb such a tion of the dopant atoms in a bulk somsconductor. repdsive interadion and, as a consequence,the bomogeadt.gof the spatial distribution of nlnnrgeddopant atoms is destroyed(7.20J Scnnnsngtntnneling microscopy is particutarly well smted for investigations of near-surface dopant atoms and their defectcomplexc, becauseSTM rnxlces it possible to identify mdMdual atomic defects and to explore thek III-V semiconductor properties. TMs holds in particular for (llol-oriented surfacc witk their simple atomic relavxtion and their reproduciblepreparaI'a bulk crys'tals A prototypical impttrity ksSi in Gal.s 27.214. tion by cleavage. slcoa on a Ga site, Sica, is a perfed examplefoz a shallow donor with hydrogenic s'tates witb. a small electronbindiug energy of 6 meV sn'rnilarto that On an. A.s site, Sipks,it of the Wxnnier-Mott exciton i'a this material (6.46). The is still a shallow imptzrity with a hole bindiag enera of 35 meV (7.22). formation energy of such an impurit.y atom X on a certain lattice site nxlled B also remarltablydeperds oa the Fermi lcve.lsv. Then .
= Of (XBV) A%) lzx+JzB +ç@F' +svBM)('DZ) E(NhtNB 1, X, Q E I'ZVA, -
-
-
NB 1, X, q)of a system with aa X atom izt the |
This value should approae,h spatial ex-temsionof 2.5 = i'a STM images (7.23). with iucreasingdepth of the Bohr radius of the shallowbllllr impurity (7.221 the lattice site below the sttrface. The STM measurements allow tus to cotmt the Si atozns in all defectsacd to detemm'me the surface concentration. Jn the absenceof any d-lm:qion at room temperatmc the surface concentrations reoectthe bulk concentrations. Garader ca'a or acceptor (SiAs) The n7lmberof Si atoms with doaor (Sios) let 1zs eocpecta strong increase also be deterrnsned.&71k-calculations (7.24) of the concentrstions of donors a'ad acceptors with Si dopmg. This is not suzface,sof substrateswith obselwedusing the STM data of the GaAs(l1.0)
3O2
7. Defects
diFerent Si incorporations. Foz smnller Si conceatrationsy Si atoms are initially incorporatedonly on Ga sites as donon. With incremsingrstype dopiug accompanying the incremsingSi concentrations, however,the formation enof Sixs acceptorsdecremses a'ndSi is incremsHglyLncorporateê er> (7.3) (7.25J on A,s sites as acceptors. Cozlsequently; compenqation eFectshappen.I'a addition, tlb.eattractive i'ateraztion of Sioa and Sihsshallow impurlties suppoz'ts the formation of Si clustea's.Suc.ha view e-xplnsnm the obsenation of nearly constau.t donor and acceptor concentratior (7.21J itl contrut to theoretici predieionsj whie,husllxlly study uornînally Ssolateddefects and do not také the defect-defectiateractioashto account. Nevertheless, the electrical 5nAuL ence of the other defects in the system is reasonablymodeledby tqb.e chenlical potential of the eledrous. 7.2.3' Antisites
Mtîsite ddbcts
are
not identïed ia Fîg. 7.7. However, the
iccotrpozatipyjEéf
arseaic at low vowth temperatures leadsto the forrnyktionof a hij concen tration of defects.Cross-sectional STM showsdiretztly that the Yfé/' il'' fo=ed by the incorporation of Acess arsenic are mostly isolated Ge2i: antisite (Asoa) defects,aud that tllese antisite defec'tsgive zise to an iatènbytj , bazd of midgap states (7.26,7.271. One of the most stm'kl'ngfeatttresrevéalèà b: the STM imagesare the spatially extendedlocal DOS of the defeds (i.jd)r. u ë Basedon the symmetr.g aud the intensit,y of the local DOS as imagedby.STV) Asca defeds in dllerent sabsurfacelayet'swere distiagaished. The syznmetzyand intezlsft.yof the Iocal DOS also allow the ideutifcatld:( surfaces éf of ardon antlite defects ën the topraost surface layer of (110) excess
,
'
Fig. T.ï. STM ''mxgesof occupied (upper aud empty (lower elecpanels) panels) tronic states of the majordefects on Si-dopedGn.Aq(1l0) surfaces. (a1)a'ad (a2): Ga vacancy; (b1)and (b2): Sioa donor; (c1)and (c2):Sixs acceptor; (d1)and (4i2): Sioa Voa complex;and (e1)and (e2): intersedion line of a plxnn.r Si cluster (perpendictzlarto the surface). Tke tnmnelingvoltagesare -2.4 (al),-2.0 (b1,cl, d1); -2.2 (e1), 1.8 (V), 1.4 (b2jc2, d2),aad 1.5 V (e2).Adapted from (7.21) (copyright lmerican 'by the Physical (2003) Sodety). -
7.3 Line Defects: Steps
a-doped
303
surface. GaP(110)
Four Poa antisite defects by t:e (copyrigît(2Oô3)
GaAs: GaP, azd J.IIPusing STM, in pazticular, by compariqon with results Sf DFT calculations (7.28J. The aaion antisite defect in the surfacelaye,rof smfacesb.asa vezy localized DOS, i'a contrast to the extendedDOS E(110) As an mvnmplethe STM image of a'a zz$f the sabsurfacea'atksites(7.26). cleavczge soace with four Poa antisites is shown i'a Fig. 7.8. Ediped GaP(110) (The constaut-cunent STM image eebits rows of occupieddanglhg boads with E:locnlszed above the P atoms of the lx1 surfacestracture (compare In addition to the regularly spaceddangling bouds, one obsmes k:Fig.4.9). by arrosvs in Pig. 7.8), which i additional mnorima betweenthe rows (indicated may also be identifted with completely Slled P danr;ingbonds =d, Eenze, with Pca aatisite ddects. The lack of any apparen.tEeigkt chnangeor long-rangevoltage-depezden.t contras't around the defect featmes in large-scaleSTM images indëcatethat the kmdealyingdefect type Ls electhcally uachazgedon a11investigaiedztdoped substrates. Consequently,the surfaceaaion antisîte defects aze elect21 ically iaactive a'addo not induce a Fermi-level pinning 'lnlike bullc aatssite ddects. This holds not ozktyfor GaAs but essentiallyalso for Gap and Tnp. .
'
7.3 Line Defects:
Steps
7.3.1 Geometry and Notation
The most impoztant ckmssof liue defectsoa surfacœ are steps i'a whic,h the ledge separates t'wo terraces from eac,hother. Steps'are impoztaut i'a the i.e., surfaceswhic,h are formation of vicinal surfaces(high-inde,x mmhces), with respectto a low-indexsarface.Suc.h orientedat a sma;llangle I (i 100)
3O4
1. Defeds
vicinal surfacesare formedby small Eow-inde,x terraces and a 1nltg. h deusi'ty of regnzla,r steps (see Fig. 7.9). Thetr surfaceenergy shouldbe srnnller thaa that of the high-inde,x surfacewith a certsu'n atomic structure (see Sect. 2.4.1). As an evn.rnple,a 2D kut of a simple cubic stzuctttre with a suzfacenorma,l slightly misorientedwith respec'tto the (901) direction is sbownin Fig. 7.9. The peziodicsuccessjonof terraces and s'teps on a vici'nal surface c.ark be specfed by their correspondingMille.r irdice.s,for i'astance(5,57) for the situation of an fcc crystal (without atomic basîsl in Fig. 7.10 with a tilt angle 0 = 9.40 of the surface aormal with respect to the (111) direction. However, this notation is not ve,zyconvenientsince it doesnot indicate,at Sz'stsight, the tmze geometrical stractme. Such rstaircEtse'stzazcttlresms shoqm i'a Fig. 7.10 are more conveHentlyreferredto tzsi'agthe fmicrofacet'notation (7.2%
p(AkI)x Lhtkllî),
(7.4)
in wizic,hh'kk and Wk'l' are, respectively, the M511e.rindices of tbe terraces and of the ledges, an.dp gives the wzmber of atomic rows in the terrace paralle,lto tlve edge. Hence:an fcctlll) surface may alternatively be labelled 64100) x (111), i.e., a series of six-atom wide (100) terraces separated by (100) x (111) place may be referred to as steps. Sinzla,rly,a'a fcc(331) x (111), i.e., three-atomwide (111) terraces sepazatedby (111)x 3(111) (111) x (001) steps.The suzfacein Fig. 7.10is1thusj a 6(11:) steppedsurfaceof azl fcc czystalwith a lattice cozsstautco. The geometzy of the (111) ar.d (001) planœ of aa fcc crystal are describedin Fig. 1.2. The steps a'e propagating along the (EE21 clirection. The coordinstesof the lattice site,sare given bl Table 1.1. Periodicitrin this dizectionhappens for L = 17cc/(2W) with a distance c = Wcc/W betvee,nt'qro adjacent atomfc rows fa a tevace, the step height is d = (U/W,and the horizontal shifk bet-ween t'wo consecutive terraccxs nmouts
to g
=
co/A/d.
We mention that, contrary to low-index surfaces,the sign a'adthe order of Miller indices in the notatio:a pLhkljx (Wktl')are important ('D30j. For example in a bcc lattice p(110) x (1ï0)is dsFerent 9om p(110) x (011). I'ndeed,it is easy to see that the edge of the Srst step is parallel to the
Eig. 7-9- 2D cat of a simple cubk crystal, showingterrace atd ledge atozns in proftle. The nominal (014) suzfacedecaysinto (001) terrace and (010) steps. The 1. incline angle8 to the low-inde.x (001) plane is gfven by tan8 = 4 .
7.3 Line Defeds:Steps
5Oq E1
E-l ï21
(EI,.f 00:)
periodicltyL ;
-11.1 (5541 E1
(b)
)'h'jl *---
,; a
d
9 L
Fig. 7.10*. Vicinal (557) smfaceof an fcc czystal. (a)Positionsof lattice point.sin a s'teppedsurface. The characteristic lengths and heights are indicated x (001) 6(111) in (b).
diredion., in which the suzcessiveatoms are second-nearestneighbors, (001j whtle for the second step it is pazalle,lto (ï11Jin whic,h the successive ,
atoms are flmt-nearest neighbors. Motker problem occuzs for orientations in the centrat part of the stereographctriaugle (Fig, 2.6).For example,at = x (111) x (100) it sllotf.d be noted that steps and ter2(111) (311)*2(100) pole abng the jo:ïj races becomeindistinguishable.Traveling from the (111) pole, the size of the terraces decreasesand (eveniuatly) direction to tke (100) is referred to a,s terraces becomeslarge'r.Therefore,(311) the size of the (100) the t'lrning pokt of the zonç.The stereographictriaagle may also #veother wsually cor&om the edges) îzlformation. Points witbsn the triangle (away containing regular non-linear respondto k-inkedhigh-inde,xplaues (sïtrênces However,recent studieshave shown that high-inde,xsurfaceson aa steps). edge of the s'te-reographictrsn.ngle,e-g., Si(311) ) or Mrltbs'nthe stereo(7.31j 5 11)(7.3241 can be s'tableagn.inqt decayinto graphic triangle, e.g., GaAs(2 terraces of low-index surfacesseparatedby steps. For realistic cabic czystals, for example for dinmond-stnzcturecrystals, and (331) vichlal surfaceshaa beendiscussedby the stntcture of (211), (311), One hms partimzlarly for steps propagatiag along(îI21and g11l). Chadi (7.33),
7. Defects
306
to mention that for viainxl surfacesof suchcprstals sometimesaksoa deviating nomenclatme is ased (7.34). =d. Si(2(111)(ïï2))suzfaces Sig3(111) (112)q corzis't of regtzlarlyspacedterraces tkat are three/t'wo Si atoms wide iong the or g1E2) directiop separatedby siugle boyer steps. The nomenclatare E11I) is also simplifed aûd s'ac,hstnzdttres aze zeferredto as (11R acd (1ï2j(or more genea'ally(11i)aud (ïï2))steps. -
7.3.2 Steps
-
Si(100)Surfaces
on
We coasidersteps on Si sa'rface.s as prototypical objects. Siliconis availableizi the fo= of the most perfect single crjrstals. lt is used as one of the most common substrates for homo- a'ad heteroepitax'y. There are severi remsons for s'tudying the geometric and electronic structure of steps on atomically cleaziE vicinal Si(1O0) stzrfaces(perhaps after n.nnen.lingat high temperature for sktfa scient time)ms well ms of steps on cleavedSi(l11) surfaces.Fit-st, dedactioy of the azomic s'tructare of the stepsis izztrsmm'rm.lly interestiug. Second,an ux/ derstanding of the step s'tmzct'azeleads naturally to possibleinsights iato the structure of high-indexsmfaces,whic,hare themselvesperiodic arrays of Low ( i'ad e.x steps (7.33). Third , the heteroepitaxial growth of other snmiconductozp suc.haz GGs or Gap ozl silicon appears to dependon the step stnzd'ure 6f the sllrfnne. surfacesthe atomic layerson the terraces arq ln tlle case of vicinal Si(1O0) at equal distances of co/4(d. Sed. 1.2.2 and Fîg. 1.6).At least siugle-layer (S)or mozlatomic aud double-laye,r(D)or biatomic steps shouldbe possiblqt 'lnwodistinc't tgpes (.4,B4may be distinguished accordingto the orientatio/ of the dirners on the adjacen.t (100)terraces. The steps are labeled as Sz, Ss, Dz, and Ds accordi'agto the notation of Kroemer (7.351 The three moàt '
'
.
SA HH-H e
-
-
-
-
-
ezzzmzz
-
-
-
e
-
-ZZTTZTZ ezzzzzT
-
H---
-
**
-
-
-,z,,,,
-
JZ zyz, - ezzzzzz
De
ss
TTTTTTT zzzz zzz
e-
2222222
zzzzzzz
-
-
221r122
ZZTCTTT
-
-
e-
-
-
-
Fig. 7.11- The three most impoz'tant single-laye,r (Sz,Ss)and double-layer(Dsk steps on Wcinal S6(1O0) surfaces (sche-matfc) The step height and the orientation of the dimers oa tlte terrres a'e iadkated..The terraces show 2x 1 ar.d 1x2 rlsrnéf reconstructions. .
7.3 Line Defects;Steps
307
1*
Fîg. 7.12- Top view of tlsmer-vacancy strudares used to model Sx, Ss, azd Ds iarfacesteps of Si(100), The size of the circle.sw.1-1e.s with the depth of the three comsideredatomk layers beneath tke surface. Anom Mdlcate lime,r Sup' atoms: dotted lines the resulting surface uzkit cell, and dashed lines approm'rnxtely the steps. After (7.362. .
*
importaat ones are sczematicallyshownitt Fig. 7.)1. The âguzemakes.evident that the subscripts .4 and B denote ,whethe,r the dimerization d/ection on azf upper tezrace near a step is normal (A)aad pazallelLBtto the step edge. Possibleatomic stmzctures kzsedto model the steps Sx, Sxv and D.s oa 'a sarfacejat least for a high dtmqityof steps, aze shownin Fig. 7.12 Si(100) The vicinal Si(100) surfacesare assllmedto be tûted about g011q and, (7.36!. consequently,the steps appear along the g0î1j diredion. Monatoxnicsteps separate 1x2 an.d2x1 domainsof dimerhation, therefore Sx aa.dSs steps 'bltmz-nate. The Sx s'teps do not require the formation of new or the breaking of existiug bonds. The Ss steps, however, aze more complicated aud nviqt izl three mriatiozts. M titree types az'e observedin snAnning txlnnelLngzaicroscopy,even tllough the rebondedversion has the lowest formntion The comrnonly observedbiatomic steps are of type Ds and anergy (7.37j. rebonded,Tile neighboring terraces show a 2x 1 reconstrudion (forrot too largeterraces and not too low temperatures) '
.
308
7. Defect,s
Experimentatly the occmrence of a ceztnl'n step and hence the step height dependon the temperatttre ar.dtkc misorientation. Surfacesteps have monatomic height below a miscut angle J of 1-20. For larger miscuts.more and more biatomic ste/sare observed.The steps aze zlmost exclusivelybiatomic for 9 e-xceedl'ng6-80 (7.37) J.nthe gn.mework of the tight-bindl'ng approach the formation energy 'yf = qslwith of step) Stidealsttrfacelj/fzs a cerwin step pez llnit step Iength Ls haBbeenciculated (7.384 The values are yf (Sx)= 0.01 ev/fgs, %(Ss)G 0.1: eM(Ls, (Dx)= 0.54 ev/fs, artd 'g/f yf (Ds)= 0.05ev/fs,whereLs = cc/uf = 3.85A Lsthe lx 1 sarfacelattice constart. 111reaâty thesevczluesdepezdon the step-step intezaction, i.e., on .
-
.
the periodicity L of the large surfacelnnit cell usedirt the cazculations(7.394. Thc tendcrc'yis boweve,r obviousfrom the above-mentionedvalues.StepsSx, s:lrfaces to a certna'mextect in Ss, a'ad.Ds sholzldoccur on the tilted Si(100) tlmvranody.cLnml'c eqllilr'brilIrn. However, their occunence is also l'n6uencedby sllrfaces misorientedtoward (O1lJ the smfacepreparation..For (100) oz (011) the eaergy situation is completely chamgedin compadson with the :at sur-' facesprepared by a repeated c'ycleof sputtering a'n:dnannen.lsng h'om a (100) Si wafez.For a tilted surfaceit is fmpossibleto haveonly Sz steps. If such a step doesoccur then a Ss step kstmavoidableacross some botmdar.vbetween terraces. For vazkishing spadng of Sx and Ss steps, howeve'r,a biatomic step of type Ds is created (see Fig. 7.11). Stepss'nflttence many surfaceproperties and, hence, are measurableiu many mzrfacemeaslzrementtenhnsquessuc,has LEED, STM, RAS, etc. The dslezenïcombinations of 2x 1 and 1x2 dimer reconstrudions and the elec)I
!
i
g g 1
l l'
: :
1 1
Tr
1
L' h i ter
l
/
o.co2
12
I
;.n
q
uq Gh-o c
-
S
t :
2
3
:,
t : :r k Tz t
4j :
'
: :,
'1
@)
:1
:
.
: l ; ; : 1 l
(s) (y; j (d)
ll %
w 4
5
6
Photonenergy(eV)
Fig. 7.13. Refectaztce n:n''qotropyspedra measuredfor clean Si(100) sarfacesby a'-merent Dmshedlines mark , and (d) (7-43). (a)(7.402: (b) (7.412, (c)(7.421 poups: the posîtlon of the bulk matical pokn.tenergesEï and F'z. After (7.361.
7.3 Line Defects:Steps
'B c .P m
K
1. s
ï
-
QG (:: (:;:p *
-6o
l : l 2 :
l
d .'
.
*.w.
.. *6 q...
.p
...w...m
RWW''
..
%%<
xe
ex w- e*
we
41
s
1
.. * ... -.An?.c?.
-P ..
1-
A'w
>...
1
1 I
, ---.v.
--
w.
1
1 '
#
ê
>. .A
T
I
()j:, ;j
I I
1 1
p
j . $ 1
1 E1
'
3 Photon ener
oE
N I 'j
,' ,;
t
2
(a)
@ l
>*
t
0.001
Ss
:
:8?î
zp
sA
:
f-
1
t:p .-ztI.
&;
l : :
)
Q= -1 0 .2 o
:
: l
1.
(1(1.()()2
l
309
E.2 5
4
(e
opiical Jmsqotropy (a)RAS calcuFk. 7.14. Step'-hduced Re((FIoïz: Fyzsl/rol: -
lated for Sx, Ss, and Ds steps in Fig. 7.12 (K36J; (b)RAS memsuredfor cllokrent miscut angle,s9 (7.41q. Terracecontributions havebeensubtzacted. troaic strttdme of the stepsthcselves sholfd particulazly iHueace the sarface anisotropy wlzich can be deteded using optichl reectance aaisotropy spectzoscopydescribedin Sect. 6.1.2. In. Fig. 7.13 we present the reûectance surfaces of fo'tm dferently prepved Si(10O) aaisotropy (see Sect. 6.1.1) A1l experimentalspedra show comrnon featmes:sach as a max(7.40-7.431. im= at or close to the & critîcal-point eneror ard negative auisotropiœ around the Ez enezgy.The spectra (c)an.d (d)obtainedfor 'vinsnn.lumples havea rninn'mttm around3 ev denotedby S in. Fig. 7.13. The spedra (a)and (b)of tEe higitly orientedsamples,instead,are charadezizedby msnima atc and 8.6 eAC(7*3). 1.6 (:71), 3.1 (S*2), The featmu T1, T%and T3 can be relatedto smfRœslate-relatedoptical surfres with essentiuy c(4x 2)reconstructrausitio:nsof :at, sizlgle-domain tion (d. Sed. 4.3.2). They are Muenced by the interMtion of tbe dirnerz parallel aad perpendictllar to the dimer rom (7.361. T'he RA feature S ks cleady relatedto steps. This is demonstratedin Fig. 7.14. J.nordez to extraswt the pttre step contribution alsofor Sx and Ds steps) for wïc,h the contribueac,hother, the norvnxln'zed tions of tlte upper ard lowerterraces do not omnce,l spedra of singlodomainsurfaceshavebem subtracted.Giventhe lsznitations in particxllnr with respect to the modeledstep density, of the caèettlations(a)s studied experimentally, the compariwhic,his far higberthan at the sample.s son witit the experlmemtal data (b)is gratifging. The surfacesteps, especiuy belov the Ss an.dDs, #vc rise to a broad negative Jtnlqotropy (Sfeature)
310
7. Defects
a minirnam at about 3 eV. This agreeswitk the expprt'men.% energy w1t21
tal ftndiazgs,whic,hshow increasing AnsRotropywith increasing miscut angle 1:1.Experimentatly a positive acisotropy between3.5 aud 4.0 e'V is observed for zniscut anglesj9Ihrger than 4O.A s5m57= featme appears iu the IRA spectrum calculatedfor Ds steps. 7.3.3 Steps on
Si(111)Surfaces
If siticon Rxmplesare ealeaved dn situ along the
or (711j dizections,2xl(2114
reconstruded terraces occur. Theseterraces are botmdedby steps. LEED aûd STM studies obxrve predomlnantly gzlîj-oriente s'tel)s with a step Eeight
s
(c)
-j
4
c@a
y
4 6z '
2 o
()
20 zo 40 scan distancealocg (g1t'z (&
lc
6:
Fig. T-15. STM Lmageof a step on Si(111); aquizedat samplevoltage +1.2 V and a comst=t tttannelingcurren.t of l nA.. (a)perspectiveview; (b)top view of tEe same step! an.d (e)crosmsedionalcut alorg the line indicated 1. (b).The step edgeis identifed by tick maz'lcsat the border of srnsgein (b).After 17.45).
7.3 Liae Defeds: Steps
(a)
..
,
(2-1-(
-
.
.
(b)
.
.
)17*
' Nh .,
f....:; .!
,
'.
,.
.ji?
Fig.. 7.16. Stde<ew of two po%ible con4gurations for
(0l 1)
311
a
m'.
);
,
'.
,.Li:
step lnmning iong (2-1-)1
.
of
az/s/z 3.14 .t
correspoadingto a'a atomic bilayer i'a thd (111q direction A typicat STM hnage of sucha step on a Si(111) surface(7.45) is (7.44,7.45). shown in Fig. 7.15. Two cegions of the step edge are apparent in the uppe,r and lower parts of tke fLgure.80th regions have tlnst pedodicity along the step edge, the (O1Ej direction. The lower region consists of a row of mn.='ma in the image-s,with one maxkzplrn per llnst cell. This row splits into two in the ttpper pazt of the image. The cornzgatsoaalongthese two rows is wealc A cross-sectionthrouglz this uppe,rregion of the image is shown in Fig. 7.1:c. The 'kworows observedalongthe step edgegive rise to Fmxn'rnn. in the STM contotzr, with the rows separatedlateratly by 4.5 + 0.5 A. The STM images indicate that there are two possibleclassesof steps along (01î)and a bilayer height (7.33, Conceivablemodels are shown 7.461. in Fig. 7.16. The outermost stom at the step edge f->tn eithe,rbe tvofoldcoordinatedas sitowain panel (a))or threefold-coordlnated as slmwuin panel =
ng. 7.17- Two possible models of step reconstruction (sideview) a , iztcludz'ng r-bonded cknn'n(ândicated by arrows) reconstrtzctiohof the tercaces. Afte,r a sup ges-tionin
(7.45) .
312
7. Defects
the flrst class,with only six-memberzingsalso occms on high-inde,xsurfacessar,hws (211) and (311) , which may be inter(7.33q preted as s'teppedsoaces with shozt terraces. The daagling bonds of the outerrnos't edgeatom form a drabbît-earttypeazaangement,ideatical to that of the Si(100) surface (Fig.3.11, Sect. 4.3.1). Consequently,a rlsrnerization along the s'tep edgeis expeded. Howeverj it leads to a doubleor quadruple periodicit.galong the edge,in coatradiction to the STM obsezwations. The seconâclass of steps in Fig. 7.16b is Garacterized by a geometry where the closestzizzgbn.q lost onc atom resulting in a sve-memberring a,t the step edge. Suc.hstructures are also discussedfor (211)suzfaces(7.33,
(b).The idealstructtlrej
They 7.464.
l'rnrnediately be combined with a 2x 1 reconstruction withs'n the r-bonded nhnsn model (Sect. terraces. Two clangling 4.2.2)of the (111) bonds remai'a at the two step-edgeatoms ar.d may be completelySlled or empt'y after recoastnzction. A11other danglingbondsare satttrated during the formation of tEe new tezrace topology with Nve-memberand seven-membe'r The dl'lerences T>. Two possiblestractmes are shomt i:rt F'ig.7.17 (7.45J izl the fmrö modelsare due to the topology of the rin.gswith whic,hthe upper and lower terraces approachthe s'tep. Tite step extents in these models of 5.1 at.d 3.6 i come close to the value of the lateral sepazation of rows on rl:'Reren t tezraces of (4' .5 ::E; 0.5)i. However,it sllould be mentionedthat also more abnpt steps havebeen observedby STM (7.454. cltn
.
7.4 Planar Defects: Stacldng Faults 7.4.1 Defect: Reconstruction
.
Element
or
Bulk Propez-ty?
Stanlrs'ngfaults are one of the most cornrnon types of planar defectsin crystalline dinmond-typeazd zinc-blende-typesemiconductors,suc,hms Si, Ge, a'ad GaAs,as well as ia many fcc metals. They have vezy low formation energies (ofthe order of 20-70 mJ/m2 and a're created when chacges (7.471) of the atoznic plane stackingsequence in the perfect czystattake place arong the E111) direction, without breltking bonds. For evnmple, dislocations can dissociatef'at?partials and create stacklmg fatzlts. Two types, the intrinsic and ex-trinqic s'tackqng faults, are indicated in Fig. 7.18. They conwpond to one missing or one ex4ra bilayer, respectively,in a'a otherwise pefect crystal with a stanlrs'ngsequence ABC. The correspondinglayer sequencesare .ABCA ICABC. (1SF) or For zinc-blead.eABCAICEBCAB. (ESF). semiconductorsin att attmmaxti've way one cazs view the (dixmond-lstractttre stanking fault a,s consisting of one (>0) layem of tetram intrinsic (exetrinsic) hedra twisted by 180D.The periodic arrangement of such stacldng fatllts gives new htexagonal polytgpes,211(wuztzite) for ISF, 411for ESF, and 611 for triple stacking faots (notshown in Fig. 7.18)as Mdicated in Fig. 7.19. For compouadsemiconductorswith stroager ionic bonds sttchpolytypes re . . .
. .
. .
.
. .
.
,
.
. .
7.4 PlanarDefects:StaddngFattlts
ideal (3C)
ISF
+ + + + + + +
+ + +
+
+
3l2
ESF + + +
+ +
+ + +
+ A BC A B CA B C
A B CA R C A : C
A B C A B CA B C
Fsg. 7.18. Staclcingsequence for an ideal fcc structtzre tleftpanel), fcc withuan htrhsic-stacldng fault (JSF)(middle and fcc 44th an ex-trlndc stacldng panel), fazklt IBSFI(rightq=ell. A, B, C zepresen.tthe three inequivalent positions withc'n a (110) plar.e with'p an irreducible crystal slab (d. Sect. 1.:.2).In the case of semîconductorsa dot repraents a pair of atoms with the connecting bond parallel to
(111).
e'aergeticallymore stable tha'a the 3C (ziac-blende) one, for evnmple 2H for ZnS, GaN, and .A.INas well as 41-1acd 6H for SiC. (wartzite) Stazlrs'ngfaults also occur on mIrfaces. They are not ac-tually a tfault' fault but a,'a importaat reconstrudion element of the d'lrnemadatom-stpcklng model of the Si(111)7x7 surface(see Sect.4.4.3), in contrast to stanMng faults in blll.k Si czystals.In the case of halogen-terrnsnated smfacesthe Si(111)1x1 introduction of a staûlrimgfaalt aloag a (111) step edgeatlowsthose bilayer steps rin the two czystallovaphic directions (ïï2) and (111) to have the sn.rne atomic structure 7.4-2 sî
on
g7.48).
'
si(111)xdx,vY-B
m:rface whic,h Interestingly, Si homoepitaxial vowth on a Si(111)WxVFB includesj monolaye,rof boron showsa tendenc'yfor rotation by 1801of the Si tetrahedra i.'athe interface, i.e., for the generation of the flrst paz't of a stackiug fatllt (see Fig. 7.18), a so-calledtwin boundar,g(7.49, At 1owtempera7.501. ture, the surfacereconstmzctionis partly prcxserved, buried unde,ran epilayer, ar.d the homopitanial Si layer gwws rotated by 1801with rcsped to the suN strate. This situation is indicated in Fig. 7.20b. The rotation is in coatzmst to all other cazes of bomoepitaxy.Usually the epilaye,ris crystatlographically aligned with the substrate, irrespective of the surface reconstrucion, imput'it.g segregation, oz other Gects at the substzatesurface.Tlle origsnnlsmface reconstntctiox is alwaysreorderediuto an thmreeonstructed iaterfacebetweea the substrateand.61m, since epitak'y requires a snTmcientlylligh temperature for surfacedifhlsion to occttr. That means, in prindplc, there is the possibilit.g of growi'n.ghexagonalSi polytypes (7.51) on a Si(111)WxW-B surface,
314
7. Defects
2H
4H
6H +
+
+
+ +
+
+ + +
+ + +
+ ABCABCABC
A B C A B C A B C
A B CA B CA B C
Fig. 7.19. StnrNsogsowemcesin hexagonalpclytypes 2H, 411,and 6H.
becausethey are strttdmw incladiag twin botmdariesperioically along the direction tsee Fig. 7.19). (111)
shows that A quatitative analysis of the system Si(I11) (WxW)A3O12B the boron atom pe,r surface lnnit cell occapies a, subsurfacesubsitutioaal S5 adsorption site tsee Sec't.4.4.1) becauseof the smmll covalemtrnrh'uzof boron. would lead to B-Si bondsmuch shorter Adsorption in thé T4 site (Fig. 4.35) than the substrate bonds, giving rise to considerable strnan'n(7.52-7.55). Tkè right panel of Fig. 7.20makesevide'nttkat the mcontn.msnatedsurfaceservej is as a template upon whic.hthe new orientatioa of the twi'a crystal (layer)
(b)
(a)
p ,11;
)
Eif1.lq
substratei Fkg. 7.29. Interfacestracture of a homoepitaxialSi layer and a Si(111) (a)witkout B coverage;(b)after ) monolayerboron coverage. The Wx W reconu struc tion of the interface (dashed lm'e) is introduced by occupying every tlnird site izz a single atomic layer.The bond-stnrllng directions are indicated by thsn solid line. Opentfzlledl mrcles:Si (B)atoms. Mer (7.491.
7.4 Plnnn.tDelectmSkackiugFaults
315
enugetically prefeaa'edj at least during the nudeon stageof the 61m gwwth near 400 QC (7.49, Eighe,rgrowth temperatau'œ of 800 OCor nnnealillg 7.501. st 1000OCmakethe B atoms mobile and #verkqeto a normnl st of the Si bonds and tetrahea- The third-nearestneighbor atmmsof the tetœahedra abovethe contxmînxtedlayer te'adto ocmpy a dte directly abovethe B atonxs j.ûSs sites. TMSis only possible by a twist of the Si tetrahedra by 1800.Two :factsmay stabilizesuch a situation. The smallerB atom reducesthe repulsive forcesbetweenthe third neighboz's. The eledron trnnqfer bemeenboroa and Si atoms may give zise to an attradive Coulombhteraction betweenB and ji atoms abovean S5site. '
-
References
Chapter 1 1.1 J. BardeemlPhys.Rev. 711î15 (1947) 1.2 E. Bechstedt, R. Enderleitu Semicondnator zAr/'oce,.s and fnterlace.v Berlin 1988) (AknHemie-verlag, 1.3 R. Kubo, T- Nagnml'yacSolz z6'tctePlzysics(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.HLWeinberg)C--M- Ckau: LomœnergySleclmrzD6pctden: Ezjmrimont: neozwpand ArftzceStructnrn Dcfgzwzïrzcdtm (SprWerVerlag, 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/ l L162 (1978) 1.16 H. Lûth: Solid Syrlaces, and Thin Fdrzus4th ed. (Spzinger-verlag, Jzztczk'coe,.v Berlin 2001)
Further Reading and Jnzerfcce,s BechsteaF., EnderleinR.: SemkonductorS'ur/'cccs (AkademieVerlag,BezH 1988)
LGth H.: Solid s'tface-:,Interfaces cné T/z1/Flrus, 4th ed. tspringer-verlag, Berlsn2001)
Wydcof KW.G.: CrystajSfrucfuras2nd 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: Conceptaïn S'ttrfaaAhv.dc.s(Spzinger, Berlin 1996), Chaptezs2 and 6 2.3 J.W. Gibbs: ne SdenttscPgers, Vol. 1 (Dover, New York 1961) 2.4 A. Zangwilt: Physic.acf Suvfaces UzkiveDiï;yPress, Cvnbhdge (Cmmbridge Chapte.r1 1988), 2.5 H. Ibvht Surf. Sd. Repoz'ts29) 193 (1997) 2.6 R. Shuttlewozth:Proc. Phys. Soc.A.63, 445 (1950) M. Mn.nqâeld: 2.7 R.J. Needs,M.J. Godfrey 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. MolkM. 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 Us6o.faasir.al Mcrmsczefc Cortcepuïn the Svrfcze-fnerr ed. R. Gomes,C.S. Fmblcrzhin: Structure and Fropeïe.a of Soàd lnterface-h of ChicagoPreas,Chicago1953), Smith (Uaivezsity p. 5 2.17 T. BMineau, D. Iofe; Y. Velenilc J. Mathemxt. Phys. 41, 1033(2000) Press, Oeord 1965), 2.18 L.D. Lardau: Collede.d Pcyer.s(Pergnrnon p. 540 2.19 A.A. Basld, S.C. Erwizt, L.J. Widtrnn.n'.Surf. Sci. 392, 69 (1997) A.E. White, L.C. Feldman,N. Moriya, D.C. Jacobson:Pbys. 2.20 B.J. Eagleshn.m, 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) College,Philadel2.28 N.W. Ashcroft, N.D. Mmvmio:SotidStafePhysics (Saundem pMa 1976) 2.29 N.D. Langl W. Kohm Phys.Rev. Bl) 4555 (1970) 2.30 C. Hezz-mgn Phys. Rev. 82, 87 (1951) Sci=tisc, Singapore 2.31 W.A. Harrixon: Exntentary SlcctrozzfcStrv,ctk= (Wor1d. '
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-rMerve: Proc. R. Soc.London, Ser.A198, 2G;(1949) 119) 277 (19Q6) 2.36 M. Volme, A. Weben Z. Phys. Chem. (Leipzig) 2.37 I.N. StrxnKk, L. Krastanow: Sitrmlngsber.Akad. Wkss. Wien., Math.Nattttwiss. Klasse146, 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, R Chen, 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:Materials SaiencqFortrn 353-356, 247 (2001) j.42 M. Jacobi,L. Geelltaar,J.M. Mârque'z:Appl. Phys. A1'5, 113 (2002) :0.43K. 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) Plwsics ()./Matyals, VoL 1, ed. J,M. :7!48J.M. Blakely, J.C. Shelton: ir. Scr/tzce Press,NewYork 1975) Blakely (Academic 'j.:9 M.J. Spwmaay;Surf. SCLRep- 4, 101 (1984) J.-P. Bozel, A. Châtrzxin:Smf. Sd. 1.56)572 (1985) i''8o ) l.t1 K. Reuter, M. Scheser:Phys. Rev. 865, 035406(2001) .2..52A- KJey:PE.D. Th'ym'sTechnscalUn'w' v'ty, Berlin (1997) Berlin 1982),Vol. 2.53K.-H. Hellwege (Ed.):Landolt-Bsrnstein (Springer, '
HI/1'Ja+b
'2.54 G.-X. Qi= R.M. Martin, D.J. Chadi:Pllys. Rev. Lett. 60, 1962(l98E); Phys. D Rev. B38, 1649(1988) W.G. Snbmidt:Appl- Phys. A75, 89 (2002) '.$55 lj Tlj56 F. Reif: Fundamentalao.fzshfkqtfc'al New and T/zcrmclPhysics (McGraw-'Rh'll: York 1965) 'j.57 C. Kittel: Introduction bo Snlïd S'tae Phgsïc.a(J.Wiley & Sons,Hc., New York 1966) $t.àU. 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 New W.A. HrLson.; Blcctronic Sfruct'tz'reand thc Properjïe.soi Solïd.s(Dover? York 1989) /:61 A. Zywietz, J. Parthmiiller, Ft Beestedt: Phys. Rev. B59, 15166 (1999) 22620. Kubaachewskk C.B. Akock Metailnrgtcal'Tlscrwzeemxs'trp'(Pergxmnn Press,Oxford 1979) 2:63 D.R. Lide (Ed.):CRC ZandbookJJ Chemistry and npa-scz, 77th ed- (CRC Press, Boca matozt1996) K.C. Pandey;Phys. Rev. 835, 2.5t E. Krsxiras,Y. BK-Ya. J.D. Jonnnopotklos, %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 77drlFftm Pvocenes tcambridge ' L.* Press, Cambddge 2000) UzkivmsEty ,j';jf, L. Dâweritz, R, %ey:Smf. Sci. 236, 15 (1990) '..j'cd8 V.P. LaBellayz. 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 1 g.g tyggg; :2t70 U. Grossner)W.G. Szthrnsdt) F. Behqtedt: Phys. Rev. 856, 6719 (1997) .
.
'
'
.
Refezences
320
ntrther
Reading
M.-C., Sprjaard D.: Conceptsïrz SnrfaceJqrsïcs, 2nd ed. Desjonqueres B'erlin 1996) (Sprhzgewverlag,
Zangwzll'A.: Physics at
1988)
Uztiversity Press, Cambridge zskr/tzcas (Cambridge
Chapter 3 3.1 W.A. Hnv'lzon: Slezpzrtcrp flectrtrrsïcStrnctnre (WorldSdentec, Singapore
1999)
3.2 K. Ohno, K. Esfarjari, Y. Kawaaoe: Compntaiional Mctcricls Sciczzce
Berlin 1999) (Springer-verlag,
3.3 R. HoFmnnn: J. Chem. Phys. 39, 1.397(1963) 3.4 J.C. Slater G.f'. Koster: Phys. Rev. 94, 1498 (1954) 1 3.5 P.O. Löwdm: J. Chem.Phys. 18, 365 (1950) New York 1980) 3.6 W.A. Harrkson:Electvonic StrucHre o.fi'sblids(Dover, 3.'F D.J. Chadi, M.L. Cohen: phys. stat. sol. (b) 68, 4O5(1975) J. Vac. Sd. Technol.16, 1349(1979) 3.8 S.T. Pamtelides,J. Pol7mnmn: 3.9 S.G. Louie: Phys. Rev. 822, 1933(1980) 3.10 P. Vogl, H.P. Hjalmarson, J.D. Dow: J. Phys. Chem. Solidsz$4,365 (1983) 3.11 R.R. Shnmna:Pbys. Rev. B1.9,2813(1979) 3.12 K. Hirabayazèti:J. Phys. Soc.Japau 27, 1475 (1969) 3,13 A. Schohe,W,G. Szthmidt,F. Bechsiedt:Phys. Rev. 853, 13725(1995) 3.14 Y. Li, P.J. Lin-chlmg: Phys.Rev. B2T, 3465(lS83) 3.15 J.A. Appelba4lm, C..A.Baraf, D.R. Hamann: PTNr. Rev. B11, 3822 (1975) 3.16 P. Kriigez, J. Poplrnxnn:Prog. Surf. Sd. 35, 3 (1991) 3.17 M. Born, R. Oppenheimer:A'nn. Phys. 84j 457 (1S27) Press, 3.18 M. Born, K. Huang: DynaméœzT'Jleo'rg o.fCrps'tttlLattices (Clarendon Orord 1988) W.G. 3.19 W.G. Srlnmsdt,B. Wenzien,F. Becbstedt:Phys. Rev. 849, 4731 (1994)9 Sclnrnl'dt,F. Bertheedt, G.P. Sdvxxstava: Surf. Sci. Rep. 25, 141 (1996) Leipzig 1937) irz d6eQuczztczzclzeznée 3.20 H. Hellmxnn: Einflihcun.q (Deutickec 3.21. R.P. Feynma'n:Phys. Rev. 56, 340 (1939) 3.22 G.R. Darling, S. Hozoway:Rep. Prog. Phys. 58, 1995 (1995) 3.23 S. Glasstone, K.J. Laldler, H. Eyrilg: ne Theorp o.f Jlate Pvocessing New York 1941) (McGww-Hi')l, Comlmtatçon,ed. Ik.B. Christofer3.24 C.H. Bennett: in Alio6thrrts .J/r Chemïco,! ChemicalSociets Wmshington1977)) Chapte,r4 son (zrnezicaa ed. an'dS'ur/cce Scte Conmttmts 3.25 ().. Wnxnerömt jn RatnJkvtzdonxs D'tkbsio'rh Press, New York 1990): by V. Bortolnn:', N.A. March, M.P. Tosi lpletutm Chapter 16 3.26 G. Bhrlicl:, R. Stolt: Am'n. Rev. Phys. Chem.31: 6O3(19S0) 3.27 E. Zeitler: in D@usionct Surfack: Xitrm,cceïc Conczpth ed. G. Brtl Berlin 198$, Chapter 11 (Springer-verlag, 3.28 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. SeinozW.G. S'nhnnidt,F. Beckstedt, J. Bernholc: Appl. Surf. Sci. 190s 264
(2002)
Chapter3
321
3.31 C. KumptL.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. 'KoimtPhys. 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. Pllysique33, 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 ElectvonécJtmcftzreo)S/li& *1, ed. P. ZiesGe, H. E Berlin 1991), p. 11 (Akademie-verlag, 3.4.4A.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) New ï'?ztâ,e Tlteo'r'yo.fMqtal's (Benjnrnl'n, 3.50 W.A. Harrison: Pscmdopotential.s 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, Press,New York 1971) Supplement3 (Academic S.G.Louie,M.L. Cohen:Phys. Rev. Bl2, 4200 3.54 M. Scltlûter,J.R. Chelikowslcy,
(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) Phys. Rev. 835, 9625 (1987) 3.57 B. KaxtrasjY. Baz-Yhm,j.b. Jon.nnopoulos) 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) 845, 6074 (1992) 3.60 N. Chetty, R.M. Martixu Phys. Rev. 844, 5568 (1991)) 3.61 A.T. S'Flkrebtii,N. Esser, W. Ricbter, W.G. Schrrddt,F. Bechstedt,'B.o.Fimlazd, A. Mey, R. De1Sole:Phys. Rev. Ldt. 81, 72l (1998) 3.62 I.P. Batrw S. Ctraci: Phys. Rev. Lett. 36, 17O (1976) 3.63 W.A. Goddard1r1,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) and Jhïer/bce,s 3.66 F. Beckstedt, R. Enderlein: Szmzçcondnctor S'tzrlcces .
Berlin :988) (Akademie-verlag,
3.67 J. Pollmpnn, R Kriiger: in Handbxk ().J SurjaceScience,Vol. 2, ed. by 1(. Amsterdam 2000), Chapte.r2 Horn, M. ScheEer (Ezsevier, Solid State Commtm.17, 3.68 F. Yndtzrair, L. Falicov:J. Phys. C8, 157 (1975); 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) 3.7,5F. 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: SoudState Phv.We.a HiQ New York 1969) (McGraw 3.79 D. HanemamPhys, Rev. 170, 705 (1968) ntrther
Reading
Hnrrison W.A.: Electrœic s'àzvftsrcof Solftfs (Dove'r, New York
1980)
SdSrivastavaG.P.: Tlteoretoal Mcdelën,p ofsemiconductor Stzr/cce.s (World entiâc, Shgapore1999)
Chapter 4 4.1 S. Andersson:in Snrlace ScMce, Vol. 1 llnternnt. AtoMc Enerv Agency Vienna 1975), p. 11 in Handbooko/srtzrfac,e Vol. 2, Electwaic 4.2 E. W''mvnea', A-J. Fkeemx.n: Scïenceo ed- K, Hozm,M. ScheEe,r(Elsevier, Arns-terdam2000), p. 1 Ssrtscïurq 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. OckozD. 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. NeedslPhys.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. NeugebauerlPhys. 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. SCL377-379, 943 (1997) <19 X.-G. W=g W. Weiss,S.K. Shxikbutdlnmr,M. ltitter, M. Petersen,F. Wagne-r, R. Sclzl8gll M. FscheRea'; Pbys. Rev. Lett. 81, 1038 (1998) 4.20 P.R. Watson, J. Miscllenlto111:Phys. Rav. 842, 3415 (1990) 4.21 Y. Ynnmrnoto:Phys.Rev.B50, 8534 (1994)
32
Ckaptzr 4
4.22 N.W. Jepps,T.F. Page:Progz.Crysi. Growth Cbprac't.T, 259 (1S83) Universit 4.23 L. Panling: Tlte Nszzure o.ft/za ClscmieniStlnt$3zd.e(L (Cornell Preu, Iihaca 1960), pp. 2:1-264 Prœs, Ne< 4.24 J.C. Phd'llips:Bands 4z)3Bon, fn Semicotductors(Academic York 1973) Slemenfs (Sàrgent-Welcb 4.25 Sargent-Wele:rasfe olperiodicPzpperlïesofLhe Skolde1980) 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.SrivhstavxxRep.Prog. Phys. 60, 561 (199:1 4.30 D. Hanernn.n:Rap. Prog. Phys. 50, 1045 (1987) 3rd'ed. (Spriager-vezlag 4.31 W. Möncz: Somicondnctov Surïacvc'n,dInterfacea, BerM 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$63 (1.991) 4.35 M.Y. Chou,M.L. Cohep S.G.Ouie: Phys. Rev. 832, 7979 (1985) J. nlrk'hmiiller, F. Becbstedt:Phys. Rev. 858, 1326 (1998) .'4.36 P. K>''n1ce11, 4.37 K.C. P=dey; Phys.Rev.Lett. 49) 223 (1S82) J.S, Vcoers, T, Klitsner: Pkys. Rev.B39: 4.38 R.S. Becker,B.S. Swartzemtruberj 1633 (1989) A. Selloï, E. Tcsat-ti:Phys. Rev. Lett. 69, 648 (1992) 4.39 N. Talceuclzi) 4.40 C. Kressj M. F'iedler,W-G- Scbrnidt,F. Bechstedt:Phys. Rev. Bsû, 17697 .: ''
-
'
.
'
..
.
.
(1994)
.
4.41 J.L.A. Mves, J. Heberstreit, M. Scheler: Phys. Rev. 844, 6188 (1991) 14, 883 (1977) 4.42 W.A. Wxrrison: J. Vac. Sci. Tenlmol.I6, 1492(197$: 4.43 J.ELNozornlp, 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)
Pressl OAord 1955) neozr.!/of SC,Jï/ (Clarendon 24.46R-ELPeierls: Qnantum Phys.Rav. B19, 2074 (1979) 147D J Chadi: Phys.Rev. Lett. +3, 43 (1979)9 D-J. ChHi: Phys-Rev.B21, 4592 (1980) 4.48 J. Dzm, M.L. Cohe,nv 4 49 M.T. Yin M.L. Cohen:Phys.Rev. 824, 2303 (1981) Phys. Rev. B12, 1623(1976) ( k.50 J.A. Appelbqalm,G.A- BaraRD.R. Hamxn'n; Phys. Rev. B40, 19481(1û8S) M.D. Pashley; L;4.51 4.52 D.J. Chadi; m nc zs%'ruclv.re HL ed, S.Y. Tong, M.A. V= Hove, o!S'uzfoce.s Beerlin1991), K. T@kny. anagi, X.D. Xie (Springemverhg, p. 532 Psys. Rev. Lett. 6%2697 (1991) 4.53 M.D, PasMey,K.W. Habevrern: 2:.4.54 H.H. FarrekC.J. PnlmAtrlml J, VR. Sd. Tehnol, B8, 903 (19S0) 4.55 W.G. Scbrnidt:Appl- Pbys.A7'5, 89 (20û2) Jpn. J. Appl. Phys. 26, r 4.56 K. M=da, 1. TpBeayasu, T. Mh-nxto, M. Kawaslnp-mx: zjlaz6 (1qa,r) R.J. Dalby: J, Vac. Sci. Tenbnol.B6, 769 4.67 H.J. Cornelisser,D.A. Cltrnrnaclc? '
.
.
,.
(:gs8)
G. 4.58 Y.S. Wu, C.R. Beclcer, A. Waag, M.M. Krauw R.N. Blnlcn/alr-Tassjtks, Landwev Pkys. Rev. BZIA8904(1991) .': W. Hanke; Phys. Rev. 859, 15261 4.59 S. Gundel, A. Fles-zar,W. Pmscllinger, '
(1999)
:4.60 C.H. Pazk)D.J. Chai: Phjrs.Rev. 849, :.6467(1994)
324
References
4.61 U.M. Dassnrxyake, o1.B11: 1467(1903) W. Chen,A. Knbn: J. Vac.Sci.Tecbn, 4.62 A.U. Mxztlue, G.W. Gobeliti:a Snmicondnctorsand Sezrzïr/zetqls, Vol. 2, ed. R.K, Willardson, A.C. Beer (Academic Preas,New York 1966), p. 115 4.63 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) J. Fkzthmiiller, F. Bec%tedt: Phys.Rev. 858, R1722(1998) 4.67 U. Grossner, 4.68 K. KA.hn:Surf. ScL Rep. 3, l93 (1983) Pmpcrlics @.f New 4.69 W.A. Harrison:Eleziwnic Strucfureand f,??,c Soiids(Dover, 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. OrwidwR. De1Sole:Phys. Rev. Lett. 85, 5440
J, Fttrthmiiller, P. Kekell:lphys.Rev. Let't. 4.76 F. Beckstedt,A.A. Stekolnskov, 897016103(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) M. Lobrneier, H.A. van der Vegt, J.F. Peters,S.A. de Vries, 4.88 W.J. H 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) A. gabol'zz,F4.91 W.G. PveAmldt, 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) Scé.enc.en Vol. 2, ed. K- Horn, 4.95 J. Pollmnnn,P. Krùger: i.n Handbooko.fsur/'t?,cz M. Pvnlmme.rllsevier, Amqterdam2000), Chapter2 4.96 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,r4
325
4.99 H-G. Busamnn, S. Lauer, LV. HertG W. Zsmmermxnn-Bdling,H.-JG'-nntherodt,T. Frauzm%eim, P. Blaudeck,D. PorezaglSurf. SCi.295, 340
(1993)
4.100 W.A. HA.v'soa:Eiementaryflectrtmic StmtctureAorld Scientïc, Singapore
1999)
4.101 W.G. Snhrnldt,S. Mlrbt, F. Bechstedt: Phys. Rev. 862: 8087 (2000) F. Bxhqtedt: Phys.Rev. 854, 16742(1995) 4.102 W.G. Schmt'dt? orth: J. Chem.Pkys, 30) 917 (1959) 4.103 R.E. gcb7le.r,H.E. Jena (l9S6) 4.104 A. ScEolze:Diplomx Thesis)Friedriclt-scitiller-unieut, 4.105 V.M. Bermudc: pkys. stat. sol. (b) 202, 447 (1997) 45 (1996); 4.106 P. Kâckelk J. Fkzthmiiller, F. Bechstedt: Appl. Suzf.Sd. 104/105, Smf. Sci. 3322354, 55 (1996) Press, 4.107 N.N. GraenwoodoA. Emrnqhaw:Chemistry o.fElcmnnts (Pergamon New York 1984)9 p. 643 4.108 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 816541(1996) 4-110 A. 'Rxme-''u!. G. Brocks, P.J. Kelly; Phys. Rev. B51, 14504(1995) 4.111 K5. Hnmers2R.M. Aomp, J.E. Demuth: Pko. Rev. 8341 $343(1f86) 4.112 J. DabrowskbM. 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.TfvthsltxraoT. AmnkntswM..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. SbigekawacPhys. 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. BechstedtlW. 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.rcPhys. Rev. Bs4, 10304(1996) Phys. 4.127 J.M. Powel's,A. Wamder,P.J. Rous,M.A. V= Hove, G,A. Somorjai: Rev. B44, 11159 (1991) Phys. Rev. Let't, 76, 991 (1996) 4.128 J.P. Long, V.M. Bermudez,D.EL Rnmn.ktAr: J. Kitamura, S. Hara, K. Tono, 1.Matsuda, B.S. 4.129 H.W. Yeom,M. Sht'momtzra, 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. AdolpkK. 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) K. Pfnnn-tglza'as, B4.136 J- nlrkllmiiller ! P. Kn-nkfo, F. Beheet, A. 'naSG Schröter,W. Rachtezc J. ElectrozsicMate-rn.yq 2z , azja(:ggs; 4.137 J.ELNortlmnp,J. NeugebauerPhys. Rev. 852, Rl7001 (1905) 4.138 M. SabisckP. 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) F. 0wm=, P. MMensson: Sttd. Sci. Let't. 360, L483 (19S6) 4.140 L.I. JohmnKson, 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) J. Vac. Sd. TeGr 4.144 K. Tnonynnztgi:Y. Tazkishiro)M. TAkmhashijS. Tn.lcsyNKSM; zzol.A3, 1502(1985); Surf. Sci.164, 367 (1985) 4.145 G.-X QiaxbD.J. Chacti:Phys.Rev. 835, 1288 (1987) Phys. Rev. B55, 15444 4.146 M. Koike, Y. Einaga, H. Hirayama, K. 'TnDnyxos%: .
.
'
(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. TeiakllrmPhys. Rev. 862, 15319(2000) 4.149 R.J. Hnmen, R.M. Tromp, J.E. Demuth: Smf. Sci. 181) 346 (1987) ïzl Scbii Data cn# Fuzlctixd.l lïezationxnl6iits 4.150 G. Chiarottî (Ed.):in N New Seriu Group 111,Voi. 24d ezzce and recànniopg,Landolt-Böznzsteic, Berlin, 1994), p. 3?9 (Sprhger-verhg, 4.151 T. Stimpel, J. Scxtzze,H.E. Hoster,1. Eisele)H. Battmgârtner: Appl. Sart. û Sci. 162-163, 384 (2000) 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) J.C. Burl, L.C. Felzlrnaw E.G. Mcltaaj I.K. Robinson;Phys. 4.156 H.J. Gossmztnrh, Rev. >tt. 55, 1106 (1985) and Jszerlcce.a 4.157 F. Becbstedt, R. lnderlein: Semiconductor s'ur//c.e,.s BerH 1988) tAknrliAml'e-'Ve.r1ag, 4.158 L.C. Snyder:Sud. Sci. 140 101 (1984) 4.159 M. Aono, R. SoudaoC. Osh:=n,Y. Lshizawa:Phys. Rev. Lett. 5lj 80l (1983) F'.J. Himpselv1.R Baea: J. Vac. 4.160 F.J. Himpsel: Phr. Rev- 827) 120 (:t983)1 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) R.D. Bringans, J.D. Northrup, L.E. Swaztz:Phys. Rev.Lett. 4.163 D.K. Biegelsem; 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) K. 4.167 U. Stadqe,J. Scharctt,J. Bernhardt, M. Frnnle, K. Reuterl H. Wcxdle-rt Hekkz,J. Furtbrniiller, P. KxsrlreliaF. Becbs*dt: Phys. Rsev.Lett. 80, 758 '
.
.
-
(1998)
Chapte.r5
327
4.168T. A.n,M. Yosblmure.k 1. Ono,1f. Ued.aPhys.Rev. B6i, 3006(2000)
4.169 W. Kx.nk'e: Phys. Rev. B41, 5243 (1990) 4k170J, Dabrowsk, H.-J. Mûssig,G. Wof: Phys.Rev, Lett. 'r3$ 1660(1994) 4.171 W.B. PankxrdoJ.E. Dcw: Phys. Rev. 855, 15643(1997) '4.172 M.A. Kulakov, G. Hezm,B. Bullemer: Sud. Sd. 346, 49 (1996)
Pkcther IRemdlng
à'eryeodt F'.,EndezleinR.: Seméconductor and fnfezïccea s'vrlcee,s tàk-MdAmsekerlag,Berlia 1088) -
#
MönchW.: Semicnnductor tmd Intnvjqces, 3rd ed..(Springer-verlag, Sur/cce: Berlin 2001)
Chapter 5 5.1 G. Bl'n'nlg,H. Rohrer:Helv- Phys. Acta 55, 726 (1982) àk2J. Bardeen: Pkys- Rev. Lett. 6, 57 (1961) à3 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. mchtmnPhys.Itev. B62, 11.046(2Kû) 5.9 R. mesendangerhH.-J. Gfmtherodt ('9Mq.): Scannqng TunnelïngMicvoscoby 1, II, 11TSpringerSer. Surf. Sci., Vols. 20, 28, 29 (Springemverlag, Berlm '' 1994, 1995, 1996) à.10J,A. Stroscio,R.M. Feenstra,A.P. Fein:Phys.Rev. Lett. 5,62579(1986) 5.11 S. Hiiher: Phoialzctron Silcctrtucom,SprsngerSeriesin Sohd-stateSdences, Vol. 82 (Springer-verlag, Berïin1H5) 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.FwlleyyM.A- Van Hove:Phys-Rev. tett. %9,4681 .'
'
(1::7) 116 C.N. Berjltmd, W.L è :
'
Spicer:Phys. Rev. 136, A1030 (1964) 5.17 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, TeclmicazUniveDity, 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. De1Sole;Phys,Rev. Lett. 78, 1528
(19S7)
-
5.27 P. Minnhagen:Phys.Lett. 56A, 327 (1976) 5.28 F. Becksedt, M. Fiedler, C. Krem, R. De1Sole: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) iawan, 0. Gxlnnarsson: Rep. Prog. Phys. 6l, 237 (1998) 5.33 P. Phys. Rev. 5.34 M.S. Hybertsen,S.G. Louie: Phys, Rev. Izett. 55, 1418 (1985); B34, 5390 (1986) 5.35 0. Pulci, F. Gchstedt, G. Onidw R. De1Sole,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) Solid Svte Phys. 5d, 1 (2000) 5.39 W.G. Aulbtm L. Jönsson,J.W. W''1Vr-ns: 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) Vol. 8 (Pergamon, 5.43 L.D. L=dau, B.M. Lehitz: Electrodynamia QJColstin'uav 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. Eguiltzz1M. Heinn'chsmeier,A. Fleszar, W. Hxnke: Phys. Rev. Izett. 68, 1359 (1992) 5.47 N.D. Langj W. Kohn: Phys. Rev. B1, 4555 (1âT0) Collegej 5.48 N.W. Ashtrofi N.D. Merrnl'n: Sol6dxtxaàrPhysica tsaunders Philadelpha 19L6) f 5.49 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) Jena (1995) 5.52 A. Scimlze:Diplomx Tllesis, n'icdricll-scla-lller-uGveauitât, 5.53 C.-O. Alrnbhdhl U. von Barthi Phys-Rsxv.B31, 3231(1985) Miinster 5.54 M. RohlGng: PKD. Titœks, Wœti-alischeWiRLektus-l/nivezvitât,
'
(1996)
'
5.55 F. BMhqtedt. R. De1Sole:Solid State Cornrnun-C4, 41 (199Q) 5.56 W. MönckSomiconductrSîtrfc,cez and Jnteryaceyl 3rd ed. (Springer-verlag, BerDn 2001) 5.57 U. Groiner, J. Fkrtbrniiller: F. Becb-stedt:Phys. Rev. BS8, R1'C22(1998) Pkys. Rev. B38, 7710(1988) 5.,58 F. Beqbeedt,R. De) So1($: 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) Press, 5.62 S.G.Daeon, M. S'MDCH;Ba-* neprp oj s'ldlce Staiea(Clarendon
Ozbrd 1992) 5.63 V. Heine:Sad. Sci. 2, 1 (1964) 5.64 V. Heine: Pbys. Rev. 138, A1689 (1965) cnd Intrdacp 5.65 F. Bechstedt, R. Bnderlein: SzmzconductorSur/c,ce..s Berlin 1988) (Altademie-vezlag,
6.66 M.S. Hybertsen.,S.G.Louie: Phys. Rev. 838, 4033 (1988)
Chapter5
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) 'onal 5.70 C. Kress,M. Fiedler, F. RecbKtedt:in Pvoceedingso/fhcFourtlt I ed- B. Lengeler, tm fze Fovrnaton of Senzïernducttrr Jnterfcms, Con,j'erence H. Lûth: W. Mönch, J. Pollrnann (World Sdentéc, Singapore1994), p. 19 5.71 C. 'Kwtquq.M. nedler, F. BeckstedtcRnrophys.Lett. 28, 433 (1994) 5.72 C. Kzus, M. ncxcller,W.G. Snhmldt,F. Beckstedt:Phys. Rev. B30, 17697 Surf. Sci. 331-333, 1152 (1995) (1994); Phys. Rev. 852, 1905 (1995) 5.73 M. RolsBng,P. Kriiger, J. Pol7rnn.nn: 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); pziveztecomrnllnscation 5.76 M. Rohl6ng, S,G.Louie: Phys. Rev. Lett. 83, S56 (1999) 5.77 M. Ro R. Del Sole:Phys. Rev.Lett. 87 944.0 , M- Pabtmmo, Q. Onidap
(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'xehnoLA11, 1048 (1993) 5.81 G. FrancztP. Kn.n'tl!.l G..Ganther,H. Stupp, P. Oelhafen:phys. stat. sol. (b) 1s4, 91 (1â96) Matter 1, 75 (1989) 5.82 F. Beehstedt, R. De1Sole,F. Manghi: J. Phys.: Condrenn. 5.83 0. Pulci, M. Pal:tmmo,R. De,lSole:in prepazation 5.84 L. Sorba,V. Hinkel, U,U. M-iddmlrnn.n'n, K. Horm Pkys.Rev. 832, 8075(1987) Sm'f.Sd. Lett. 243, L31 (1991) 5.85 Z.M.Nicholls, K.O, Magnusson,B. Re11:t12 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. FloclströmcPhys. Rev. 827, 2594 (1983); J.M. Niclmlls, GX. N'xnRqon, U-O- Karltnanl RJ.GUhrberg: 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. RohremJ.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 '.: Phys. Rixv. Leu. 55, 991 (1985) A.R. Wilhxmq: W. Watlauer, T. Fauster: 5.100 U. Höfe-r,I.L. Shnmay,C. Reus) U. 'Chomrb.nn: Sdence277, 1480 (1997)
330
Referenca
5.101 F. P=ek, M. Donath Phys. Rev. Lett. 69, 1101 (1992) 5.102 M.V. ColeuM.H. CohemPbys. Rev. Lett. 23) 1238 (1969) Srmoondnctor P/zvsïc,s 5.103 R. EknderleipN.J.M- Horinj:FundamentauO.J and De't)ice.sAorld ScientfqSmgapore1997) MezLmic-sVoL 3 tpergamon Pre>, 5.104 L.D. Landau, E.M. T,!'Eqb1tz: Quantum
OAord 1958) , 5.105 D. Straub,F.J. NsmpsekPhr. 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'robeao! Juzpcased. P. Halevi Science,ArnKterdam1995), Chapter$ (Elsevier 5-110 A. Golamlmn,V. Dose, G. Borste.hPhys. Rev. 832: 1971 (1985) 5.111 J.B. Northrap; J. Neugebauer:P/ys.Rev. B5T, 11,:1230 (1998) 5.112 J. Farihmiiller, F. BechstedtvH. 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.LJohxnAqon,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. IzanglaiwH. Be7lrh-lr,J.-M. Debever; E'urophys. Lett. 39, 61 (1997) Mt'inqter: 5.117 C. BeuesciuPL.D. Thesis, WMfn-llqclzeWizlrgnhs-Tlzûvcsitâtp
(1998)
5.118 V. RlbmMltl'.ndr=, R.M. FeezustraPhys. Rev. Lett. 82? 1000 (1999) M. Riehi-cimdoba,B. Sckröter,W. Riczte-r:Eutophys.Lett. 5.119 V.A. Gmsparov, 51, 527 (2000) 5.120 N. Sieber,T. Seyller,L. Ley, M- PolcmD. James,J.D. Ridley,R.C.G.Leckey: Materiab ScienceForltrn 389-393, 713 (2002) 283., 401 (1964) 5.1.21J. HubbardtPrx. K Soc.LondonM76, 238 (1963); New York 5.122 W. Jones, N.H. March: TAerrctic,tzlSotid State Pvaïcd(Dover, '
1S&5)
5.123 W.A. Hxrrlson: Phys. Rev. B31, 2121 (1985) 5.1M W.A- Harrison: flemerer!/
1999)
flectmrlïc Structure
(WorldSdentœc,Singapore
5.125 N.F. Mott: Proc. R. Soc. London A62, 416 (1949) W. Nolting; Phys. Rev. Lett. 86, 3847 (2001) 5.126 R. Sn%sller, 5.127 KLG. Uhrberg,T. lfaum'lx,Y.-C. Clun: Phys. Rev. B58, R1730(1998) 5.128 RJ.G, Uhrberg: private cornrnunicatîon 5.129 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: StTrfBzv. 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/ïnearand Xcrlà'n/nl Szectmscoyvo.fs'wrlc.c'e,s an'd Jxferïccas Berliz ear œfrzr' (Spzingez-verlag,
1995)
6-4 W. Richter, J.T. Zettler: Appl. Surf. Sci. 101, 465 (1996)
Chapte.r6
&3:
'
6.5 D.E. Aspnes:SolidState Commtm.191, 85 (1997) ed. P. Halevi tELsev/.r 6.6 R. Del Sole:in Photonic T'r/he.so! Suzfacas Science Amsterdxm 1995), Chapter 4 6.7 P. Cbiaradia,R. De1Sole:Surf. Rev. Lett. 6, 5l7 (1999) J. Groenem, M- Cardona,W.G. Schmt'dt,F. Becitstedt 6.8 P.V. SaatcGN. Erxse-ra 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)
.
-
N 6.12 D. Pnlhllce,M. Aremsj N- Esser, W. Xchtèr: Surf. Sd. 352, 66 (1996)) Bsser:private communicatioa 6.13 H. EhremreiG:ElectromagneticTrnm.qportin Solids:Optic.alPropertiesani Phsma Esects, i'a The OgtfcczProperties OJSOD&,Proc. Mternat. Seool o: Press,New York 1966) PhysicsC>).Femni',ed. J. Tauc (Azozlemic .
6.14 S. Acller: Phys. Rmr.126, k13(1962) 6.15 N. Wj=: Phys. Rev. 129, 62 (1963) F. Becitstedt:Phys. Rev. 855, 4343 (1997) 6.16 V.I. Gavrilezzko, Press: 6.17 L.D. Landau,E.M. Lifschitz: Electrodynamia J./Conbinut (Pergmmon OAord 1959) Phys. Rev. B20) 4824 (1979) 6.18 A. Bagclti, R.G. Barrera, A.K. Rajagopal; 6.19 R. +1 Sole:SolidState Commum37, 537 (1981) C 6.20 K Del Sole,E. Fiorino: PNys.Rev. 829, 4631 (1.984) 6.21 Y. Manghi, R.De,lSole,A. Selloni,B. Molsnxm':Phys.Rev. B4lj 9935(1990) fzzd6e Ffeîezet-trtlp.enfzzsode d6vA'riztclle (Akademie6.22 H. Stolz: Einshmtng Verlag, BerM 1974) 6.23 H. Ibach) D.L. Mills: Elsctron Ensrgy .!,tu.6Speciroscnycnd StdaceWàro Press, New York 1S82) tiez?.slAe-qriemic 6.24 B.N.J. Perssoa Phys, Rev. Let't. ö0, 1989(1983) Berlin and T/ùn Films (Sprhger-verhg, 6.25 H. Lûth: Solid S'tsrltzccs, Interjace.s :
.
2001)
6.26 A. Selloni,R. De1Sole: 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:SOIJZState Commuxu60, 337 (1986) 6.28 A. Balzm'ottij M. Fstrfoni,'F. Patella, F. Arcipretej E. Placidi, G. Oaidal R. De1Sole; 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. DezSole;Phys. Rev. 8581 R1013S
(1998)
6-30 N. Fgsser,J. Gettrt's:N-eimxn Spcvtroscopy,ia Oyhcal C'//zzmcferlcïilm, o: Bpitaziul Sczpâconducl/rLayors,ed. G. Bauer, W. Ricizter (Spriager-verlaz, Berlin 19S6) 4.2C5,218 (1963) ,,6.3: R. LoudozuProc. Roy. Soc. (London) 6.32 P.Y. Yu, M. Cardona: F'andavrœnials ()JS6micondnctors(Springer-verlag, '
'
Berlin 2996) n-.tnm ibid. lzl, 621 (1965) 6.33 R. Loudom Adw Phys. 1a, 423 (1964); 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. TûtiincikG.R SrivmstavaPhys. Rev. 853, 15675 (1996) 6.38 G. 0011a,L. Reiuing, A. Rubio: Rev. Mod. Pkys. 74, 601 (2002)
;J .
332
Reference
6.39 L. Hedin, S. Ltmdqvist:Solid Stzte Phys.23, l (1969) 11, 1 (1958) 6.40 G. Strinlti: Ea Rivista de,iNuovo Cimen't.o
6.41 B. Adolph, V.I. Gavrilertko; K. Tenelsen, Lbn. Bechstedt?R. De1Sole:Phys.
Rev. 853, 979.7(1996) T.M. Wce:Phys. Rev. 1.44, 708 (1966) 6.4.2L.J. S1t%.m, 6.43 M- Robl6ng,S.G.Lottie: Phys. Rev. B62, 4927 (2000) 6.44 R. Del Sole,B. Fiorino; Phys,Rev. 829, 4631(1984) 6.45 P. Habn: Dlploma Thesis:Friedrich-snhl 'ller-universitât?Jena (2001) Pkys. Rev. 821, 4656 6.46 W. Hn.rtke,L.J. Sham:Phys.Rev. Le'tt. 43, 387 (197$; '
(1980)
6.47 F. Becltstedt,K. Tenelsen,B. Adolph, R. De1Sole:Phys.Rev. Lett. 78, 1528
(1997)
6.48 S. Mbrecbt: Ph.D. Tke-ris: Vole Polytebnique, Pn='u (1999) 6.49 F. BeclzstedtjW.G. Stthmtdt,P. Hlthn; phys.stat. sol. (a)188, 1383(2001) 6.50 V. Olevano,L. Re'tning:Phys. Rev. Lett. 86, 59œ (2001) 6.51 F. Bassxnl',G. Pastoz'iParraddni! Electronic Sfcle.eavtdO>$çcclTransitions f'?zJolida,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) F. Becbstedt:Bxcitonic and Local-Field Efects in 6.53 W.G. SnBmt'dt,P. Hsxh'n, Optical SpectrafxomReal-spaceTime-Domnc'nCalmzlations,itz Ifiglz Per/brBerlin, rzcnc: ComputingïzèSdencqcnd Engin6erin.g'tM,: (Springer-verlag, W.G. ScAm''dt,S. Glutsch, P. Htthn, F. Becitstedt: Phys. Rev. 867, 2003); 085307(:003) 6.54 D.B. Aspnes, A.A. StudnmPhys. Rev. B2C, 985 (1983) 6.55 S. Albreht, E. Relnlng, R. De1Sole,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.EottielPhys. Rev. Lett. 81I :37.2(199E) 6.58 R. Erderlem, N.J.M. Horing: Fnndameniaisol Jer/zfemdllztorPàysir.scnd Devices(World Sciemtïc, Singapore1997) 6.59 G.H. Wxtnoier:Phys. Rev. 529 191 (1937) 6.60 N.F. Moti; Trnm. FaradaySoc.34, 500 (1938) 6.61 R.S. Knox: Tlzemy of E=*n% Solid Svte Pbysics,Supplement 5, ed. FPress,Ncw York 1963) Seaz, D. Ttzrnbull (Academic New York 1966) 6.62 D.L. Dexter, R.S.Knox: Escjion.s (Wiley, 6.63 R.J. Elot: Phys.Rev. 108, 1384(195*4 6.64 M. Shinada,S. Sugano:J. Pkys. Soc.Japa.n21, 1936 (1966) 6.65 S. Fliigge, H. M=snhp.ll: Rchenmethode'n46'r Q'ulmtnnihaorés tspringe't-
.
Problem 24 Veriag, Berltn 1952), 6.66 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) G. Lucovsky:J. Vac. Sci. 6.71 T. Yasuda:D.E. Aspnes,D.R. Lee,C.H. Bjorkmnan, 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. DanskeVdensk. Selsk.Mat.-fy's. Medd. 28, 8 (1954) Naw York 1980) 6.75 W.A. Hxrrlsonl Solid Staie Thearg (Dover,
Chapter6
333
Press,New York 1990) 6.76 G.D. Mnuhnr:Many-pnrticle Php.dc.slple.nntrn 6.77 E.A. Stern, R.A. Ferrell: Phys. Rev. 120, l30 (1960) Phys. Rev. B9, 'O'N (1974) 6-78 P-J. F'eibelmxn;Phys. Rev. Lett. 3ô, 975 (1973); 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) Berlin Phymsra (Springer-verlag, 6.83 W. Krcs: F.W. deWette (Eds.): Surface
1991)
6.84 A.A. Maractudsn,B.W. Montroll, G.H. Weiss, LP. lpatova: SoDd State Press,New York 1971) Physiœ,Suppl=ent 3 (AcM>m''c 6.85 K. Ktmc, R.M- Markin: Ph>- Rzv'. B24, 2311 (1981) 6.86 M.T. Y1, M.L. CohemPhys. Rev.B26, 3259(1.982) Phys. Rev. Lett. 42, 06-649(1981)4 6.87 K. Kunc, R.M. Martin: J. Phys. (ParLs) 4s, 406 (1982)
.
6.88 'P. Gixnnozzi, S. de Gironcoli, P. Pavone, S. Baroni: Phys. Rev. 843, 7231
twgz;
Rev. Mod. Pllys. T33 6.89 S. Baroni, S. de Qironcoli, A. DhzlCorso, P. Glpnrhozzi:
515 (2001) Uzzive=ity 6.90 M. Born, K. Huang: Dynanxicalne'orv ojtCrg:tal Jaz>ce.s(Oxford Prœs,Oxsord19-04) 8.91G. 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: Jatmdtzcàïtmto SolqdStatePhysia llob.nWiley, New York 1966) Press, 6.94 L.D. Landau, B.M. LifsMtz: T/zcor,yof Slcsdcïjp, Vol. 7 (Pergamon London 1959) 6.55 Lord Rayleigh: Pzoc. London Math. Soc.17, 4 (1887) Scpleilh-Nat)e Theor.çand Xppzicaen, 6.56 E.A. Ash, E.G.S. Palge (Eds.): BerH 1985) SpringerSez.WavePhoom., Vol. 2 tspringer-verlag, 6.97 M.G. Cottn.m? D.R- Tilley:Inbroduaion fzl S'tdccearltf S'tqwwrlattiafzcfzaïfmn.s(Cambndge Univeat.y Prœs,Cnrnbridge198$,SeG-?.4 5.98 S. Lehwald, J.M, Szeftel:H. l'bnmb,T.S. *.$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) Be'rlin and Jntr/ac> (Springer-verlag, 6.102 W. Mönclu SqmizondnctorScrlcce.s 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) J. Phys. Chem. Solids 58, 685 (1997) 6.106 H,M. Tùtiimcii, G.P. Srivmstava: 6.107 H.M. Tiititncîi, R- Miotto) G.P. Srivastam,J.S.Tse:Pbys. Rev. 866, 115304 (200$)G.P.Szivastavœprivate commtmication 6.108 H.M. Tutiincû, G.P. SdvastawuPhys. Rev. 859, 4925 (lS99) R, Honke,P. Pavone,U. Sceöder: Surf. Sd. 377, 108(1997) 6.109 M. Zitrlsperger, 6.110 U. Haaten,J.P. Toernfes, Ch. Wö'tl: Pbys. Rev. Lett. %T,2947 (1986) Phys. Rev. Lett. 62, 3070(1989) 6.111 L. Miglio, P. Santlni)G. Benedekc 6.11.2F. Ancilot-to, W. .Aneexm.',A. Selloni, R- Cap M. Pxrrsmel1o:Phys. RrvLett. 6s, 3148 (1990) 6.113 H. Ibaciu Phys. Rev. Lett. 2*6û53 (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'merRev. 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.yof sblïtf.s(COendon Press,Oxford 1964) 6.419 J.M. Luttinger: J. Math. Pby;. (N.Y.)4, 1154 (1963) : 6.1.20J. Voit: Rep. Prog. Phys.18, 997 (1995) 6.121 M.G. Zachea'jB. 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. NielsezbKL. JobnqonjR. 1f. Bechgaard:Phys.Rev. Lett. 85, 4916 (2000) Feidenhxnq'l, 6.1,241t. 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. JohnRnajJ. Chen, N.J. DiNaœdo,K. KemN. Phys. Rev. Izett. ;8, 1331(1997) 6.1.29VlGmnzGazkclran, 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. HasegawatPhys.Rsv. 1,e1..82, '
4898(1999) 6.132 A. Mxem.raque,J. Avilw J. Alvemez,M-C. Arhtarmio, S. Ferrer, E.G. Micheli Phys. 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'11t A.B. Mnluaan:Phys. Rev. Lett. 82, 2155 (1999) 6.135 R ScgommD. 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 PhusicstlobmWiley, New York 1966) 7.2 G. Lengel, R. Wilk'snx, G. Browa M. WdflrneryJ. GUICO,R.E. Men: Pbysk Rev. Lett. 72, 836 (1994) M.B'. 7.3 Ph. Ebert, M. l'IeinrsdbM. Simoa,C- Domkp, K. Urlwm, C.K. Sh1'%,
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-claeUmversitât,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) ed. D.J. 7.18 K.C. Paûdey:Froc. 11th Ini. Con.ffm th6 Phvdic.:ofs6micondnctorh New York 1985), Chadi, W.A. Hmison (Springer-verlagj 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. UrbamP/ys- Rev. Lett. 83, 757 (1999) 7.21 C. Domke, Ph. Ebert, M. Heindch)K. Urbam Pitys. Rev. B541 10288(1996) Landolt-Bövrtstein.AkmersctzlDafc 7.22 0. Madeltmg,M. Sckulz,H. Wekss (Bds.)) cnd Tcchnology,New Series,Vol. l7, and Jtmctïtmt?tRetationsltips ïn Scdcrtca Berltn 1982) Smrdconductors tspringer-verlag, 7.23 K. Sauthos,M. Wenderot)hJ.K. Garlef, T.C.G. Reusck,R.G. Ulbricht: Proc. 26th Jnf. Cons()1 ihe Fhs/sïc..s o.fSemïc/ndvct/rd,F'dlnbttrgh2002 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. M. Sclm/er; Appl. Phys. Lett. 79l 2877(2001) Neugebauer, Smf.Sm.30, 440 (1972) 7.29 B. Lang,KW. Joyaer,G-A. Somorjai: D. Spaujaard: Conzepts ïn Sur/accPhysics (Spdnger7.30 M.-C. Desjonquêr%, Verlag, Berlin 1998) 7.31 J. Dabrowski)H.-7. Mikssig,G. Wof: Phys.Rev. Lett. C3, 1660 (1994) 7-32 1f. Jacobi, L- GeelhnnryJ.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) 045322(2001) : 7.36 W.G. Schmidt, F. Becllstedt) J. Bernlnolc:Phys. Rev. 863, and Fomwtïtm o! Jrztcr/ocas t.37 J. DabrowskkH.-J- M'iissignSûkconzgllriace.a :.' (WorldScientdc,Singapore2000) : :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. miiller, F. Bechstedt;Phys. Rev. 866, 075201(2002) , J. Fkrtln. 7.52 R-L. HeadrickoLK. 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-bondedchaânmodel 24?139,153 k-resolvedinverse photoomlssion spectroscopy IKRIPBSI154,195 X. approvsmatton 111
Bound surfacestate 218 Bravais indke,s 3 Bravais lattice 4, 7 cubk 4, 5 -
he=gonal
-
Adatom 3) 78 Hs site 170 Ss geometzy 17â JL site 170 Adatom model 174 Adiabatic approvlmation l96 Adsorption 46 104 enev groumltl etemeats 171
-
-
-
-
-
rectang'ular 14 scuare 166 surface 20 three-dl'mensional(3D) 9 twcxx'rnezusional (2D) 7, 19, 30 Brielp'nr group 169
-
-
-
-
-
Ar(100)
zmagestates
-
229
Au(110) -
Bzillouin zone (BZ) high-syzmnetryline 36 bigh-szmmefaypoint 36 irreducible 42 -
Angle resolvedphotoernlAsionspeotroscopy IARPBSI154,194 Antisite 302 Arrhenius behaWor I07 Atomic position 16 -
oblique 14 plahe 14 quadratfc 14
-
-
6, 14
driving force for rcconstructîon 51 mlqsing row recoztstruction 135
Bnr.k botd S8,126 B=d structttre 96 prolected 100 Band-strudttre enea'pr 112,123,134 Bethe-salpeterequation (BSE) 255 Blocà ftmciion 82 Bloe xnnT'n 85, 120 Block theorem 82 Btmd coeent 83 heteropolar 83 polaz'it 8tlj 136 Bond-contraeion reluation model 144 Bonâ-rotation rtalnvxtionmodet 144 Born-rppenhehne.r approxqmation :03 -
-
projectiozt37
-
threcxasmenssonal (3D) 37 twmclirnensional(2D) 34 BuaMing 51,59, 128 amplitude 147,152 gap opensng 154
-
-
-
C(10O) -
-
-
C(111) -
-
-
-
,-
-
223
anglœrœolvedphotoelectron spura 197 hxnd structare 154 electrostaticpotential 213 surfacesiates 155 TDB surface 157 trimer recoMraction $81 Chxrge-asymmetnrcoe/cient 137, 146,147 (Xlargo.fo.nm'tion level 298 Chemicalpotential 47, 66,69
-
'
hand stauctttre 160 dlnae,r 15: qumsiparticlebazd structure wave flmctions l61
338
Iudc'x
71 compoud s-lconductor ezem=t 71 chornd'c,al trend strudural pazameters l48 Chenlical'vapordepodtion (CVD) 7% 157 Cleavage 2, 138 Clustc method 119 Cohcsiveenergy C1,IM lattice 19 CoMcidezme CmmplaxKqnd stracture 220 Condactance aieerential 1:2 relatîve 193 mode l88 Constant-carrent Comtant-hetght mode 18*, 191 Comer imle 176 Correlation 203,209 strong 225,232 Crys'talslab Heducible 9, 10
rnl-ved 167
-
-
-
-
molemée levels 159
-
symmctl'ic 140
-
-
-
-
-
9 Crystal sjrstem plane 14 primstive
-
-
Cu(100) -
lrnxge statœ 220 231 two-photon pkotpamsp-qson
Dangling bond 9'tt 100,109,126,140, 158,163,176,183,232 de Broglie 'wavelength ?4, 105 ûeld (1SCF') Delta-self-consistemt method 234 Density fnncdonal pertttrbation theory (DFPT) 275 Densîtjr hnnetional theory (DFT) Denst'tyof siyates(DOS)134 electronic 193 local 190 Desorption zlG105 Diezedzicfxpnditm balk Si 259 intraband contribation 269 inveDe 207 longitudinal 207, 240,254 rnxrrroscopk 254 D1*' mion 46, 105 blrm*er 46 107 -
-
-
-
-
Dimer -
,51
asymmetric 163 burlalng t63 fippùy 165 formatzon 158
-
t -z1 tm g
141
twisfmg 163 faatt Dime-ruzlxtommode) 176,178 Dimer-row domru'mwall l76 Dimerization 142 nhJu'nboncls l52
-
'
(DAS)
-
Domain
23 Dynamicalmair:ix 274 screened Coulomb Dyunrnlc.ally
poteatial 203,208 Dysonequation 122,202,2OS
approvlnnrttioa(BMA)(' EEective-mass 227, 260
195 Binste.in1Bizzsteinrelation 106 Electron non'-ty 214 ElHron cotmt'mg rule (ECR) 14$, l83 Blectrondensity 110,202,211 Electron emergyloss 243 Electron energy loss spectroscopy (EELS)246 Electroc trn.nscfer 129 Blectron-holeinteractiozt 197,25:, : 263 Blectron-hole pair Hsxnqtonian 257 stai,e 25l Ezectron-phononMteraction 251 Electron-photoninteavtion 195,251 Electronegativity 136 Eledzostatic energy 112, 122,l35 211 Electrostatic potctial Empirical tigxht-bindingmethod (ETBM)83,88)115 B'knerprd.ensitjr%m'nA.l'lqm119 BpitaMlzlg'rowth 2, 45 crystal shape (ECS) 52,:. Ekujlibrî'tzm 57,59 Ewald construction 31 :) Ewald energy 113 'Rvabnnge 111)255,257 Bxchange-correlation eme'rgy 111 Exdtation 187,237 Fwxdton 258 binding energy 261,264 -
-
.'
'
-
-
F'renkel 265 surface 262 'kwo-dlrnensional case
262
Index
Wxnnier-Mott 261,265
-
Facet 58,65 59 Futtinz Fermi's GoldenRule 18S,195,25l Fock operator 2O2 Fbrceconstaztt model 122 Formatioa enera 296 S'renlceludton 262 approack y75 Frozezephonon bnucbs-lcewerphonon 284 augmented Fkll-potential line--m'zed plane wave (FLAPW)l16 Ftmdamentat gap 06,9â,150,21$
cmxqtjlfï) -
M trimers
182
GaAs(10O) ((4x2) 108,162
-
hnna s'tructure
-
-
-
-
162
EELS spedra 247 LEBD 34 phmsectiagram 76 potertii eneag sudace 1û8 refectaaceAnmotropy 241 slab 118 surfacestructare 69 topmost .&sasmers l43 wave Alnctions l63
Groundstate 187' Growth 76 Growth mode 'hnrxnv-v.,m der Merve 61, 76 Strausld-Krastaaov 61 Volmer-Weber 61 GW approvlmxtion 203,255
-
-
Hûcke,ltheory 124 extended 83 Half-siabpolarizability
-
243 Hnmoaic apprnvlrnxtioa 105,273 SartreepotentM 91, 110,202 Heatof formxtion 71 Hellmnnn-F'eynma.u force 104,113 Hellmxnn-Feynmantheorem 104 Hd/mholtz1ee enerr 47, 66 Heterod'trner 166 Hohenbezg-Mohn tEeorem 110 Eubbard paramete,r 184,233 Hybrid 91 -
-
-
total enera surface 103 HRMELSspectrttm 285
GaN(111) -
I
daagling 97, 126:158 Hydrogeaproblem 228,260
I1R-N(11O) -
STM image
168, 192
Ge(100) -
bands 163
Ge(111)
adatom model 173 b=d structme 154, 175 cleavage 139 isomer 154 swface suates 175 Generxliqedgradient approvirnn.tion -
-
(GGA) ll1
Gib% adsorptionGpation 67 Gibbs 9c.e enthipy 46, 66) 69 GibbspbMe rate 72 Gibbs-Duhe,mequatson 47 Gram's flmction
band stractme 149 ionization energy',electrouxmnity 216
-
stnlctaral parameters 146
L11-V(110)
combhed photoemt'skqlonaad inverre photoemissionspectra 198 phonon enera 289 strudu'm.l parameters 146 Imagecharge 20g Tmngeplane 208,226 Tmngepotemtial 210,246 Trnxgestate 226 lmpm'ity 3, 300 Icdependentpmicles 110,187 Appzovsrnation Odependemt-pxrtscle 19$,251,zeoy259 Independent-qumsiparticle appzova'rnnEion 254,258,259 -
electrostaticpotential 212
Ga-P(100) -
.+ 94 syz 9:3 sp% 9l: 59
-
GxAq(11o) -
120, 202)205
-
fzrs'titeration 206
-
perfect crystaâ 121
339
-
1nP(100) -
-
-
msved-dimermodel 167 pàasecliagram 76 STM imaje 16E sarface stnzcttzre 69
Tn'p(ll()) -
deposition of As atoms
78
340 -
-
-
hdex
qloxswarde-le baad skrudure 224 Rxmxn spectra 253 STM 145 surfnz'ophonondkspeon 286
Mb(l1O) -
18 Massxctlon law 72 Matrix notation 20 Metal surface 18 excbxnge-correlationpotentx 210 image states 23l reconstraction 134 relaxation 133 Metal-inducedgap states ImGSI 221 Metal-orgec chemîcalmpor deposition IMOCVDI237 Mg met,al spectntm 2O0 pkotoelectron Mllk-stool model 18O Miller indices 3, 4, 53 Molemzlaœ beam epitav (MBE) 2, 45, 70 Mott-Hubbard msulator 172, 184:234 enera Madelunj
-
EELSsyectra 273
-
Interatom)c force constant 274 Interdifusion 46 Intemmlenergy 47, 66,68 Hterstitial 294 Iuversephotoemlmqion speciroscopy
-
(œF-3)194
lon bombardmentand n.nmep7lng (1BA) 2, 45 lonization enerv 214 semiconductozs 216 Islrd 46: 64 Isomer 154
-
-
Neaarewneigkbor interaction
Jxhn-Teller dbplacement 17, 18t l63 theorem 17,126 JAMAV theozem 123 Jellb:m model 208,2ll
87>95,
151
-
Nearly-freGelectron (N5V)approvlmtion 219
-
Ni(100) -
K/M(111)
-
image s'tates 229 Myleigh phoaon 2.80
271
eleeron enerr lossspectra Kiak 3, 294 -
Optical absorption bulk Si 259 vith electron-holeattruion Overhp integral 83, 124
Koha-askam(KS)eigenXues 110 Kohn-sham (KS)stata 1l2 Koha-shxm equation 110 Kohn-sham poteutial 110,204 Kramer's grand poiential 4%66
-
26l
Pardal pressure '74 Peierls eAec't 141,291 Peripherazatozaic state Löwdin theorem 83 Pkase diagram 74,80 Lattice plane 2, 3) 6, 8 Phonon Lattice vibration 68, 274 Layer-orbital zepreentation 12ô,l22 eigenvector 287 macroscopic mode 2?7 Lebmam.arepresentation 202 mëcoscopicmode 27T LifetMe 187 surfacemode 276 Liuea.r combinationof atomic orbitG Photonmlxqionspectroscopy (PES) (LCAO) 82,85 194 Linearizedmuen-tin orbîtal (LMTO) Piasmon method 116 losses 199 trocal dtansrr't.g approvlmafion(LDA) skake-tzp 199 71, 109 scface 271 Local spin density approzmation Pocket 100,228 II.,SDAI111,235 Local-ield efects 208,M1, 255,268 Polnt defect 295 Point group 42 Lone paiz 140 laow-ene'raeledron doaction ILERRI kolohedri 14,25 plaae l4, 15 29, 33 Polarity 9 mewsmement 146 .
-
-
-
'
-
-
-
-
-
lnde,x Pohrization flmction 258
207, 208, 2.54,
Bloch representation 256 macroscopic 254 Polytype 136: 312 Potentia,lenergy surfaceIPF,,SI103 vr'nlmxlm 1O4 Potential-enm'> sudace(PES) saddlepoht l07 Prlm''tive bmsisvec-tor 7, 29, 33 Prlml'tive lattice vector 4, 8 Prindple of detailed ba7xnce 46 Pseudohydrogen 118 Pseudopotential 110 Pseudopotentialpla'ae wave (PPPW) method 1l6
-
-
241
Rehybridization 126,140 ltelaxation 16, 17, 19, 128,135,146 outward
-
133
Repeated-slabmethod 116,274 Rest atom 173 Ring structttre 139 Roughen''mg 59 R''rnpliag 18, 135
-
Pyrxrnldal-clustermode) l80
Sagittal plane 278 Satellitestnzdme 199,206 Scaxmingtllnneling microscopy :S8 -
(STM)
Ge nanocrystal 63 Tn As pyr a 'mz d 63 Scattering-tkeoreticalapproach 120 equation Sclzrödirtger eoccitoas 260 l'mxge states 227 onmcllrnensional 219 siazgle-pardcle81 two-particle 257 Screening 207 Seiwatzmodel 157
-
-
Quazielectron196 Quasihole 196 Quasipaztide -
-
-
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 204 equation 201 Quaasiparticle
-
250 by optical phonons 251 selection rttk 252 Randomphase approvimn,tion IR,PAI 203, 255 wave 280 ltaqleigh Remprocf lattice 4, 29, 32 Reconstructîtm 16, 19, 24 driving force 135 missing row 17 pairing 17 pzinciples 138 vacancy 140 ReEelztacceanisotropy (RA) 243 Refectance azdsotropy spectroscopy (RA.S)238 Refec-tancediference spectroscopy (1tDS)240 Resectiott Mgh-eue'rgyelectron dOaction I'R.N%EDI 76 Rxman scattering -
Self-enera exchango-correlation 202,2O4 GW approvlznn.tion 203 -
-
surfaces 210 Self-hteraction 274 -
Shuttleworth equation 50
Si(1O0) -
1tA spedzum
-
-
-
-
-
-
-
-
-
-
-
baud structure 99, l54 buclding model 128 cleavage l39 a''eerential resectacce spectrum 238, 264 electrone'nergylosshmction 248 photoelectronspectra l99 quasipazticle b=d stmzcture 223 1kA.spectzum 262 relative conductance l93 step 310 sTM l'mxge 1zs smfacephonon modes 290 surface states 178 '
SiC(000l) -
LEED
35
SiC(111) -
308
266
Si(111) -
-
''
bud structure 99 reiectance anisotropy spedra
Si(110) -
-
'
near
adatom geometry
172
342
Hdex
b=d strudare l72 electrostaticpotential 212 pllue diavam 185 slab 117 tetzamer 183 t'wisted Si J'rlrxyer l83 Singledangling-bond(SDB) surface 150,156, 180 Slabmethod 115 Shter-Koster pazamete.r 89 srnxll poinz voup 42 Spaceroup 10j 25 -
-
-
-
plane
25
Spectralellipomekry (SE) 238 Spectralwelght 2O5 ftmction 187,190) jpectralt-weight) 196,201,2O6 core-hole 201 Si 206 Stnnklng fault 139,176,312 st,ariangvector 8, 9 Star of m.'ve vector 42 Step 2, 46, 52,303
-
biaemic 306 Eeigsht 293 Dicrofacet aotation 304 monatomic 3O6 Steawgraphctrin.nrle 54 Strain 51 Sudda approvimation 196 Superlattzœ 19 Supersatuzation 76 Surface 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 49 Surfacedieeautial reiectance (SDR) 237 Slxrlhceenergy 67,74J 128 snbonded metals a7 jelliummodel 56 metals (table) 56 sexniconductors(table) 56 Surfaceloss ftmction 246 pkn.- 74 SlxrTsrm Surfacephotoabsorptioa (SPA) 238 -
-
Smfazeplaae 13 Sudacereconstraction 16 Snlrfnceresonxnce state 218 Surface rougkue% 3 Sudace-state gap 225 Symmetry point 13 roiational 15 txrazslakiona: 16, 82
-
-
Terrace 3, 46, 52
Tersol-Hxrnxnnapproach191
Tetrakedron dfrect:on 91 Tetramer 183 Tetramer-xdlnyermodel 184 Three-layermodel 239,247 Tilzee-stepmodel 195,201 Tight-binding method 83, 123,151, 160,193 Total energy 68, 103,109,1.12 Traasfer-matrkxmethod 120 Trnndtion-state theozy (TST) 106 'Trxrmlationalg'roap 13, 19 rrrirner 181 Triple daugting-bond(TDB)surface 156,181 z'nlnnelingcurreat 190 'nlnneling miœoscope ideal 190 Twin boundazy 313 -
Ultrahigh vacuztm IUHVI 2, 45 Ultraviolct photoemissionspectrcvpy
(WS)
-
194
-
-
-
vv-xncy 3, 294 Vacuum level 2l4 Vapor pltn.te epitaxy (WE) Vi 'cma1 surface 52, 304 Virtual gap states (ViGS)221
-
-
-
W(100)
reconstruction 134 relnx-xtion 133 Wlmnser-Mott exciton 261 Wi>er-seitz cell 25)33 Wood notation 20 Work ftmcdon 214 typioal metaks 215 WUIScomstzuction 57 WUISplot 51 -
-
-
