Progress in Surface Science. Vol. 25(14). pp. 147-154. 1987 Printed in the U.S.A.

0079-6816/87 $0.00 + .50 Copyright © 1988 Pergamon Press pic

SIMPLE PAIRING PICTURE OF SURFACE RECONSTRUCTION: (2x1) INSTABILITY OF

SILICON SURFACES

MOJMIR TOMAsEK

Institute of Physical Chemistry and Electrochemistry. Czechoslovak Academy of Sciences, 121 38 Prague 2, Machova 7, Czechoslovakia

Abstraot

The (2xl) reconstructions of three typical silioon surfaces, namely (111),(110) and (100), are described within a very sim­ple pairing picture, relating mutual saturation of Shocltley Surface States (dangling bonds) from the Fermi ener~y EF to the ~eometrical instability of ideal (lXl) surfaces. The mechanism involved is the electron-phonon couplinc (assumed to be stronger at the surface than in the bulk) between surface states and the deformation modes '72.- and contains two other ilIIportant tool.: the pairing theorem relating wave functions and enerGies of 8ur_ :race states differing by a certain wave vector dl!-, and the se­lection rule picking out reconstruction modes allowed by the present theory. It is stressed that the latter two theoretiCal tools can be applied to infinite l_dil!!ensional chains of iden­tical atoms, thus explaining ni£ely the mechanism of the Peierls transition. 1.1so, the zi~-zag (H~) reconstruction of the "(100) surface was treated recently by the present approach.

1. Introduction

The present very simple ideas try to approach surface reconstruc_

tion of crystals exhibiting a certain component of covalent bOnding,

which have Shockley surface states (55) around the Fermi enDr~y ~.

147

148 H. Toma~ek

Surfaces of diamond-l1ke semiconductors, in particular silicon, are

c1assical examples. The ideas may complement somehow other investi­

gations and contribute to the elucidation of the role of 55 in sur­

face reconstructions. They are based on the author's work [1,2J

published long time ago, which, with various modifications, has been

tested recently [2-5] on surface systems of current interest~ The

importance of the eleotron_phonon interaction between SS and lattice

deformations is assumed, and as the basic ingredient the eleotron

(COli or 5DlI) and Lattioe (Peierls or pseudo Jahn-Te1ler 1ike) insta­

bility theory [ 1,2] is used. The (2xl) instability is discussed by using the ·pairing theorem"

[1,2] valid throughout the whole (2Xl.) surface Dri1louin zone (SDZ)

of surface (and~) e1ectronic states of Si (111),(110) and (100)

surfaces. As in pseudo J._T. theory, main tool of the anal.ysis is

the (Umkl.app) electron-phonon matrix element wkQ = (k+QI111 k > 1<hich operates as a symmetry based selection rule. llerelk> is the Bloch

function and 11 is the electron-phonon (deformation) potential exhi_

biting same symmetry as the deformation mode 1L. For simp1icity rea_

sons, the case of a single S5 band, appearing on tha 5i(111) 1Xl. sur­face is presented here. neconstruction of ideal. Si(110) and (100)

surf aces, ~here tuo 55 bands occur in the cap is a1so treated easil.y J

but mai~y results are mentioned.

2, Pairinc Theorem and Selection Rule

The symmetry based se1eotion rule, which by means of 11 decides

whether a reconstruotion mode 1l, is al.10wed or forbidden, is based

on the existence of the ·pairing theorem". This governs symmetry

properties of the wave functions of Shockley surface states (55).

The theorem ho1ds due to the faot that the three investigated 3ur­

faces are alternant systems, i.e. each of their atoms fal.1s into one

of two interpenetrating sub1atices A and 8 (there are t"o sub1attice

pairs on the (110) surface). For such systems [1,2J, any two states

in the SS band differing by a certain k= ae. (in our case de = Q) 1'0=

pairs of bonding and antibonding (1<ith respect to A and 8) states.

Since this pairing runs through the "hole surfaoe Dri1louin zone, it

signal.s that local. ("chemical") effects are at 1<ork on the surface.

It strongly differ. from the usual nesting mechanism occuring on the

Fermi surface only. One easily checks thatlk)~ rA + g'8 and

Pairing Picture of Reconstruction 149

I k+Q)::::jl.. - 'tn ho~d and hence, "l<Q can be non-zero onl.y ir II is

antisymmetric in A,n, i.e. I,::::: l'A- liB' Here re and lYe are the n~och ~unction and the d~ormation potential, respectively, o~ the 8ub­

lattice e. This selection r~e is in apparent analogy to the pseudo

J.-T. theory, where dir~erent parity wave :functiona can coup~e onl.y

via an odd parity de~ormation - a process stabi~izing total e~ectro_

nic energy. O:f course, the polarization vector o:f the antisymmetric

reconatruction mode ~ can be directed either perpendic~arly (buok­

led mode) or paralloly to the sur:face (in_Plane mode). The latter

oan be either longitudinal or transversal with respect to Q. It is

in partic~ar the mode longitudinal with respect to pronounced

bonds on tho sur:face (or having an important longitudinal component)

which can be expected to bring a sizeab~e contribution to the elec­

tronic stabilization energy.

In the :following, Figs. ~,2 o:f [6 J are r~erred to to display

the geometry, SHZ and SS o:f the three ideal Si sur~aces. The high l

symmetry points o~ the SBZ o~ (2x~) sur:faces are ~abel/ed analogous_

ly to Fig. 8 o~ [7Jand Fig. 2 o~ [8J. It is interesting to notice

that ~or the (2Xl) struotures ~ the investigated sur:faces, a dege­

neracy ocours between the symmetric and antieymmetric ("f'o1ded") 5S -,- -branches along certain SBZ direotions which read: J_K_J :for tho

(~~~), j~K_J(X) :for the (~~O) and j~K :for the (100) sur:face, respec_

t1ve1y. A10ng these directions, a pair of' equivalent SS states

I k)tlk+Q> can be :formed, the wave ~unotions rA and <fB o~ whioh

being exc1usive1y 100alized on either the A or D sublattice, respec­

tive~y. These states are non-bonding with respect to the A_B inte­

raction and show the unaaturated charaoter o:f the corresponding

dang~ing bonds [9J. "Chemical" saturation (pairing) o:f the ~Qtt"r contributea to tho energetics o~ the reco~truction prooess.

Let us enumerate those q=Q o:f the invosti8ated sur:faces :for which

the 8e~ection r~e is ~~~il~od, leading to a (2x~) reconstruotion

o:f the oricin~ (LX1) sur:face. For the (~~~) sur:face it holds that

qEQ = ~/z G(iiZ) = 27'( /3a (ii2), where G is the reoiproc~ veotor

in the (ii2) direction o:f the (~X1) sur:face, and simi~arly ~or (~~O) and (100) sur:faces one has Q = ~/2 G(OO~) = 'ir /a (OO~) and Q = = 1/2 G(l,-l,O) =1r/a(l,-l,O), respectively. In the sur:face ~ayer to which we ~imi t our considerations here, the Pandey 11 - bonded

chain mode~ [7] o~ the (~~~) sur:face reconstruction corresponds to

our longitudinal mode. The same is true :for the (2x~) reconstruction

150 H. Tomasek

of the (110) and the symmetric dimer of the (100) surfaces, respecti­

vely. The corresponding asymmetric dimer is a combination o~ the 10-

plane longitudinal and the buckled modes. Of course, the Pandey mo­

del [7) is typioal for its large surface defoX'll1ation amplitude and

hence involves subsurface layers into the game. This is in line with

the :fact that, contrary to the remaining two surf'aces, SS on the

(Ill) surface are highly delooalized into the bulk. Since the "pai­

ring theorem" operates in the bulk as well, there might exist a spe-1-

oial fringe effeot in ohemioal bonding (force constants K a ~G)

helping the subsurfaoe layer(s) to adjust its (their) geometry more

easily to the reoonstruction in the surfaoe layer. Notice, that the

in-plane longitudinal mode Q=2~/Ja (112) of the (Ill) surface has

OPPOSite phase in the subsurface layer [7J; it might be that geometri­

oal stress, steric or mismat~ch effects are relaxed in this way. Of

course, there are two SS bands on (110) and (100) surfaces which

oause that w1c.Q has interband matrix e1ements, being now a 212 matrU. Recently. the theory of this and the following Section "as success_

fully used for the U(lOO) surface [2J . The pairing theorem and the

selection rule picked out the experimentally found zig-zag (MS) ro_

cOn3truction mode there. The theorem and the rule represent also ade­

qua te tools to deal t<i th tho Peierls trans ti tion in in!' ini te linear

chains since these can be formally divided into two subchains A and

D, being typica.1 al. ternant systems. IJenco, one 1'inds a rewordinc of

the Peierls transition theory.

J. Formal l-fat..'1ernatical Framel~ork

Let us sketoh briefly some simple mathemntios behind our ideas,

without claiming muoh acouracy in the notation. One starts with two

ooupled hamiltonians, the electron one H and the phonon one II , or e p tile self-consistent (Hartree-Fock like, mean-field) approaoh to the

elootron-phonon interaotion

II L Ck + L "kq < bq + b+ > o~+q (la) .. c k c k

+ Ok e k k,q -q

11 • z:: W b+ bq +~ Wkq<C~+q Ck)(bq + b+ ) (lb)

p q q k,q -q q

PaIrIng PIcture of ReconstructIon 151

where c describe electron end b phonon operators. When more than one

(say n) 55 bands are present, the summation over k is to be comple­mented by that over the band index i. (la) then corresponds to a

2 n-component theory (analogous to [lOJ) lfith the key quantities

(like the gap function ~Q) changed from scalar to nxn matrices. ObViously, the exact solution of the phonon problem is trivial

since the latter represents the displaced harmonic oscillator prob­

lem. Dy putting the expectation values of the commutators <[ bq ,lIp]) end<[b:q ,lIp]) equal to zero, one gets the phonon shifts

-L (2) k

which appear because terms linear in b are contained in (lb). Com­

pleting the squares in that equation (i.e. diagonalizing (lb» gives

with the stabilizing "polaron energy" Hpol .. L;Wq<b:><bq), resul­ting from the displacement of ions to the new structure equilibriUIII

positions. 1o'hat determines the final physical picture is the model in which

II is diB80nalized..Our qualitative considerations above sUGgest the C;'" end Peierla transition model in the sense of [1,2], i.e. a

single-mode ('1.=0) model. In this model, <,c~+Q Ck:>is evaluated from (la) oxactly follolfing [1] end the result is

.( c~+Q ck > .. ~Q (~l) _ .{2»

e~)- 2:) when occupation numbers (Fermi functions) ~l~ ~2)Of gy levels [1 ]

..

(3)

the new ener_

(4)

ore a150 included to allow for temperature effects. The self-consis_

tency cyclo is olosed by denotinc

152 H. Toma~ek

+

substituting (2) in (5) and inserting (3) ous system of linear equations arises in

solution when

(5)

result. A homocene­

which has non-zero

(6)

holds. The quantity in square brackets of this DeS like "t;ap" equa_

tion can be called the phonon self-energy (polarization operator)

1f (Q,W=O). To get (6) in a form more fBl:liliar from DeS theory, one writes

~l)_ ~2)" 1 _ 2~2)" -tgh E.k/2kDT

and a~sum.ea exact pa:l.ring

= Here the notation

~l,2) ..

haa been used. Then, the analogy or the rinal temperature-dependent

DeS resul t follows iDDDediately.

(6) is the instability oondition from which the phase transition

temperature T is in principle to be determined. Total energy (adie_ o batioal potential) Etot is approximated as the sum over occupied

energy levels (4) stabilized by Hpol ' plus the repulsive phonon

part. One finds the "softening" of GU Q (at T:O) by evaluating the

2nd derivative of Etot with respect to the ion displacements, at

the equilibrium positions of the original lattice. Alternatively,

by expandint; the electronic part in small displaoements 't, using

(for illustrative purposes only!) similar appr_oximations [11] as

in DeS theory and asswnint; that wkQ~wQ holds, one oan oaloulate the reconstruction ~ree energy change

(7)

Pairing Picture of Reconstruction 153

where n (8) is the S5 density of 5tates, 6J is the cut-off energy and S W""2 . 02 '7'/" 2) kB is the Boltzmann constant. By"Writing Q. ( ~/0'1/.. = 0

one immediate~y obtains the renormalized frequency UUQ_ Equatinc

r.h.s. of (7) to zero, one could get TC. lio"ever, the approximations

made do not grasp correctly the important contribution coming from

degenerate 55 branches. The improved result will appear elsewhere

[lZJ • The emergir.g pairing !;1echanism of (Zxl) silicon surface reconstruc­

tions is related to oiher pairing phenomena in solids, in particu­

lar to the small polaron (negative U or bipolaron) problem [14] •

It can be checked that <b > and lip 1 give correot results in the q 0 ('Ii j

small polaron limit. Also, by consulting [1] with Ok' given by

8k\l of (5), one finds that in the present picture, strong electron­

phonon coupling lends to the localization of charge on one of the

sublattices (another electron pairing effeot, of the sort known from

the negative U problem [14J ). Thi. effect micht be of relevance

~or the existenoe of buc~ed modes. Especin1~y if ~oca1 surface

symmetry is taken into account properly [ :3 J. the present Bo~eoti.on rule can a1so be app~ied to the problems of ordered adsorption,

adsorption induoed reoonstruotion, epitaxi~ growth or surfaoe atom

scatteril16', where instead of' the electron an tl externa1" atom in

interaction with surface phonons is considered.

4. Discussion

The generally acoepted (2xl) reconstruction of the surface lay­

er of 5i(111) and (100) surfaces belong to reconstruction models

allowed by the presently introduced selection rule. As usual recen­

tly, buckled modes are excluded by arGuing [ 7] that in LCAO total

energy calculations they resul t from an artifaot of the method which

exaggerates charge transfer between neighbouring sur.faoe atoms.

The (2xl) reconstruction of the 5i(110) surface which has not yet

been treated in the literature, deserves attention. The surraoe

layer of the ideal (110) surface consists of parallel chains of atoms,

each chain exhibiting a glide plane symmetry €I which causes a -- g

double degeneracy along the X_M direction or the (lXl) SBZ [13] •

A seleotion rule analogous to the present one allows a longitudinal

Peierls-like distortion in separate chains "hich splits the degene_

154 M. Tomasek

racy, leadine to two separate SS bands. Notice, that analogous

transversal displacements o~ alternant chain atoms are ~so allowed,

hOliever, they do not remove ~. They can help to shi1't the X_M de-g

generate states to EF and to optimize the energy gain. The (2xl) re_

construction rollows then rrom the seleotion rule or Section 2 and

occurs perpendicularly to the chains. One can imagine that it ari­

ses from opposite-phaBe location of the Peierls distortion in neigh­

bouring chains. A very simple total energy calculation of several

reconstruction modes or the Si(110) surface supporting this pioture,

has already been done [ 5].

References

1. H. Toma.aek, PhYsica 22., 420 (1967); ibid. 12., 21 (1968);

Physics Lett. ~, 374 (1968); Int.J.Quant.Chem. ~, 849 (1969).

2. M. Toma§ek, in Phonon PhYsiCS, Proc. 2nd Int.conr. Phonon Phy­

cjcs, Budapest 1983, J. Kollar, N. Kroo, N. Menyhard, T. S~os

eds., 110rld Scienti1'ic, Singapore, (1985), p. 673.

3. H. TomaSek, ~. Pick, J. de Physique (Paris) !!:l" Col1oque C5-125

(1984); Surf .Sci. !!!.2" L 279 (1984); TCH Techn.Rept. (15 Janu_

ary 1985), Cavendish Lab., Cambridge Univ., England; Physica

~ B, 79 (1985); Czech.J.Phys. ~, 768 (1985); ibid. B 21, No.12 (1987).

4. M. Tomasek, Theor.Chim.Acta, to appear.

5. M. Tomahk, il. Pick, J. de Physique (Paris), submitted.

6. I. Ivanov, A. Mazur, J. Pollmann, Surface Sci. ~, 365 (1980).

7. K.C. Pandey, Physica 117 + 118 n, 761 (1983).

8. M.A. Olmstead, D.J. Chadi, Phys.Rev. !!...2l, 8402 (1986).

9. M. Tomasek, J. Koutecky, Int.J.Quant.Chem. l,249 (1969). 10. R. Balian"N.R. llerthamer, Phys.Rev. 1;)1, 553 (1963).

11. Ch. Kittel: Quantum Theory or Solids, J. Wiley, New York (1963),

Chapt. 8.

12. M. Tomasek, to be published.

13. V. lIeine, Proc. Roy. Soc. (London) !Llll., 307 (1972).

14. P.W. Anderson, Phys.Rev.Lett. li, 953 (1975).