Simple pairing picture of (2 × 1) instability and reconstruction of silicon surfaces

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Simple pairing picture of (2 x 1) instability and reconstruction of silicon surfaces

Mojmír Tomášek

To cite this version:

Mojmír Tomášek. Simple pairing picture of (2 x 1) instability and reconstruction of silicon sur-

faces. Journal de Physique, 1988, 49 (12), pp.2071-2075. �10.1051/jphys:0198800490120207100�. �jpa-

00210888�

Simple pairing picture ^of (2 ^x 1) instability and reconstruction of silicon surfaces

Mojmír ^Tomá0161ek

Ústav fyzíkální ^{chemie a} elektrochemie J. Heyrovského, 010Ceskoslovenská akademie v~d, ^{121 38}

Praha 2, ^Máchova 7, 010Ceskoslovensko

(Reçu ^{le 5} janvier 1988, révisé le 26 juillet, accept,6 le 24 août 1988)

Résumé.

Les reconstructions (2 ^x 1) de trois surfaces typiques du silicium ((111), (110), ^et (100)) ^sont

décrites à l’aide d’un modèle simple d’appariement reliant la saturation mutuelle d’états de surface de Shockley (liaisons pendantes) ^au niveau de Fermi EF ^à l’instabilité géométrique des surfaces idéales (1 1). ^Le

mécanisme mis en jeu fait intervenir le couplage électron-phonon (supposé plus ^{fort en} surface que dans le

volume) ^entre états de surface et les modes de déformation _~. Il est basé sur deux principes importants : ^le

théorème d’appariement reliant les fonctions d’ondes et les énergies des états de surfaces différant d’un vecteur d’onde

et la règle ^{de sélection} qui choisit les modes de reconstruction permis. Le formalisme utilisé

s’applique également à l’interaction des électrons avec d’autres champs de bosons (excitons, plasmons, paramagnons).

Abstract.

The (2 ^x 1) reconstructions of three typical ^silicon surfaces, namely (111), (110) ^and (100), ^are

described within a very simple pairing picture, relating mutual saturation of Shockley ^{surface states} (dangling bonds) from the Fermi energy EF ^{to the} geometrical instability ^{of ideal} (1 ^x 1) surfaces. The mechanism involved is the electron-phonon coupling (assumed ^{to be} stronger ^at the surface than in the bulk) ^between

surface states and the deformation modes ~ and contains two other important tools : the pairing ^theorem relating ^wave functions and energies of surface states differing by ^{a certain wave} vector 03BA, and the selection rule picking ^out reconstruction modes allowed by ^the present theory. ^The formalism utilized is applicable ^also

to the interaction of electrons with other boson fields (excitons, plasmons, paramagnons).

Classification

Physics ^Abstracts

63.20K

73.20

68.20

1. Introduction.

The present very simple ^ideas try ^to approach

surface reconstruction of crystals exhibiting ^{a certain} component of covalent bonding, which have

Shockley ^{surface states} (SS) around the Fermi energy EF. Surfaces of diamond-like semiconductors,

in particular silicon, ^{are classical} examples. ^The

ideas may complement ^{somehow other} investigations

and contribute to the elucidation of the role of SS in surface reconstructions. The importance of the elec-

tron-phonon interaction between SS and lattice deformations is assumed, ^{and as the basic} ingredient

the electron (CDW ^or SDW) and lattice (Peierls ^or pseudo Jahn-Teller like) instability theory [1, 2] ^is

used.

The (2 ^x 1) instability ^{is discussed} by using ^the

« pairing ^{theorem »} [1, 2] ^valid throughout ^the

whole (2 ^x 1) surface Brillouin zone (SBZ) ^{of sur-}

face (and bulk) electronic states of Si (111), (110)

and (100) ^surfaces. The main tool of the analysis ^is

the (Umklapp) electron-phonon ^{matrix element} wkQ

(k ⁺ Q [ w k) ^{which operates as a symmetry}

based selection rule. Here I k) is the Bloch function and w is the electron-phonon (deformation) poten- tial exhibiting ^same symmetry ^{as the} deformation mode _q [3-5]. ^For simplicity ^{reasons, the case of a} single ^SS band, appearing ^{on the Si} (111) ^{1 x 1}

surface is presented ⁱⁿ section 3. New results are

found mainly in sections 2 and 4 showing ^{how the}

selection rule predicts the allowed reconstruction modes, particularly ^{for the not} yet theoretically investigated (110) ^{surface. Central aim is to demon-}

strate how trivially ^{one can describe the}

pairing

of dangling ^{bonds and the} ensuing change ^{of surface} geometry. ^Old well known facts are reworded in a

different language sufficiently ^{flexible to deal with}

some of the presently popular examples ^{of surface}

reconstruction.

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphys:0198800490120207100

2072

2. Pairing theorem and selection rule

The symmetry ^{based selection} rule, ^which by ^means

of w decides whether a reconstruction mode q ^is

allowed or forbidden, ^{is based on the existence of} the pairing theorem. This _governs symmetry proper- ties of the wave functions of Shockley ^{surface states} (SS). ^The theorem holds due to the fact that the three investigated ^{surfaces are alternant} systems, ^i.e.

each of their atoms falls into one of two interpenet- rating sublattices A and B (there ^{are two sublattice} pairs ^{on the} (110) surface). ^{For such} systems [1, 2],

any ^{two states} in the SS band differing by ^{a certain}

k

K (in ^{our case K}

Q) ^form pairs ^of bonding ^and antibonding (with respect ^{to A and} B) ^{states. Since}

this pairing ^runs through the whole surface Brillouin zone, it signals that local (chemical) ^{effects are at}

work on the surface. It strongly ^{differs from the}

usual nesting ^mechanism occurring ^{on the Fermi}

surface only. ^One easily ^checks that I k) = ^{9 A + 9 B}

and Ik ⁺ Q) " 9A - (PB hold and hence, wkQ ^{can be}

non-zero only ^{if w is} antisymmetric ^{in A, B, i.e.}

W -- wA - wB. Here gc and wc ^are the Bloch func- tion and the deformation potentials, respectively, ^of

the sublattice C. This selection rule is in apparent analogy ^{to the} pseudo ^{J. T.} theory, where different

parity ^{wave functions can} couple only ^{via an odd} parity deformation

a process stabilizing ^total

electronic energy. Of ^course, the polarization ^vector

of the antisymmetric reconstruction mode q ^{can be} directed either perpendicularly (buckled mode) ^or parallely ^to the surface (in-plane mode). ^{The latter}

can be either longitudinal ^or transversal with respect

to Q. It is in particular ^{the mode} longitudinal ^with

respect ^to pronounced ^{bonds on} the surface (or having ^an important longitudinal component) ^which

can be expected ^to bring ^a sizeable contribution to the electronic stabilization _energy (see ^also [6]).

In the following, figures 1, ^{2 of} [7] ^{are referred to} display ^the geometry, SBZ and SS of the three ideal Si surfaces. The high symmetry points of the SBZ of

(2 x 1) ^{surfaces are labelled} analogously ^to figure ⁸

of [8] ^and figure ^{2 of} [9]. ^{It is} interesting ^{to notice}

that for the (2 ^x 1) structures of the investigated surfaces, ^a degeneracy ^occurs between the symmetric

and antisymmetric ( folded) SS branches along ^certain

SBZ directions which read : J’ - K - J for the

(111), ^{J’ - K -} leX) ^{for the} (110) ^{and J’ - K for}

the (100) ^surface, respectively. Along these direc- tions, ^a pair ^of equivalent ^SS states I k) :t k ⁺ Q )

can be formed, ^{the wave functions} 9 A and 9 B of which being exclusively ^{localized on} either the A or

B sublattice, respectively (for ^{a recent} experiment apparently confirming ^{this fact in a} particular ^{case :}

see [10, 11]). ^{These states are} non-bonding ^with respect ^to the A-B interaction and show the unsatu- rated character of the corresponding dangling ^bonds [12]. ^Chemical saturation (pairing) of the latter

contributes to the energetics of the reconstruction process.

Let us enumerate those Q ^{of the} investigated

surfaces for which the selection rule is fulfilled, leading ^{to a} (2 x 1) reconstruction of the original (1 ^x 1) surface. For the (111) surface it holds that

Q ^{= 1/2} G(112)

² w /3 a (112), where G is the

reciprocal ^{vector in the} (112) direction of the

(1 ^x 1) surface, ^and similarly ^for (110) ^and (100)

surfaces one has Q ^{= 1/2 G} (001 ) = ’IT/a(001) ^and

Q = 1/2 G (011 ) = 7r la (011 ), respectively. ^{In the}

surface layer ^{to which we limit our} considerations here, ^the Pandey iT-bonded chain model [8] ^{of the} (111) surface reconstruction corresponds ^{to our} longitudinal ^{mode. The same is true for the} (2 ^x 1)

reconstruction of the (110) ^{and the} symmetric ^dimer

of the (100) ^surfaces, respectively. ^The correspond- ing asymmetric ^{dimer is a} combination of the in-

plane longitudinal ^{and the buckled modes. Of}

course, the Pandey ^model [8] ^is typical ^{for its} large

surface deformation amplitude and hence involves subsurface layers ^{into the} game. This is in line with the fact that, contrary ^{to the} remaining ^{two surfaces,}

SS on the (111) ^{surface are} highly delocalized into the bulk. Since the pairing ^theorem operates ^{in the} bulk as well, ^there might ^{exist a} special fringe ^effect

in chemical bonding (force constants K = w Q) ^help-

ing ^the subsurface layer(s) ^to adjust ^its (their)

geometry ^more easily ^{to the} reconstruction in the surface layer (cf. ^also [6]). Notice, ^{that the} in-plane longitudinal ^mode Q

² rr /3 a (112 ) ^{of the} (111)

surface has opposite phase ^{in the} subsurface layer [8] ; ^it might ^{be that} geometrical ^{stress, steric or}

mismatch effects are relaxed in this way. Naturally,

there are two SS bands on (110) ^and (100) ^surfaces

which cause that wkQ ^{has interband matrix} elements, being ^{now a 2 x 2 matrix.}

Recently, ^the theory of this and the following

section was successfully used for the W(001 ) ^surface [2]. ^The pairing ^theorem and the selection rule

picked ^{out the} experimentally ^found zig-zag (Ms :

Q ^{= 1/2} G(110) ^" (110 )) reconstruction mode

a

there. In principle, ^the alternative X3 (Q

1/2 G(lOO) = ’IT la(100» ^{mode with similar local}

environment would be also possible. However, ^band

energy estimates based on simple considerations

using ^{the 4th moment of the local} density ^{of states} preferred ^the R5 ^{mode. As} with the silicon surfaces,

the unsaturated (non-bonding) character of the

degenerate ^{SS branches occurs here as well.}

The theorem and the rule (partly corroborated by [10, 11]) represent ^also adequate ^{tools to deal with}

the Peierls transition in infinite linear chains since

these can be formally divided into two subchains A

and B, being typical ^alternant systems. Hence, ^one

finds a rewording of the Peierls transition theory.

Originally ^{of course,} both tools have been introduced

to deal with genuinely 3-dimensional alternant sys- tems. Due to its Peierls-like character, ^{our recon-}

struction model can exhibit a whole spectrum ^of soliton (anti-phase boundary ^{or domain} wall) ^effects

on the investigated ^surfaces, this time in two dimen- sions.

3. Formal mathematical framework

Let us sketch briefly ^some simple mathematics behind our ideas, ^without claiming much accuracy in the notation. One starts with two coupled ^Hamilto- nians, the electron one He ^{and the} phonon ^one

Hp of the self-consistent (Hartree-Fock ^{like, mean-} field) approach ^{to the} electron-phonon interaction

where C describe electron and b phonon operators.

When more than one (say n) ^{SS bands are} present, the summation over k is to be complemented by ^that

over the band index i. (1a) ^then corresponds ^{to a 2 n-} component theory (analogous ^to [13]) ^{with the} key quantities (like the correlator (3)) changed ^from

scalars to n x n matrices.

Obviously, ^{the exact} solution of the phonon problem ^{is trivial} since the latter represents ^the displaced harmonic oscillator problem. By putting

the expectation ^{values of the} commutators

[bq, Hp]) and [b: q’ Hp]> ^{equal to zero, one gets}

the phonon ^shifts

which appear because ^terms linear in b are contained in (1b). Completing ^{the squares in that} equation (i.e.

diagonalizing (1b)) gives

with the stabilizing «polaron energy» Hpol

L Cùq (bq) (bq) ^{resulting from the} displacement ^of

q

ions to the new structure equilibrium positions.

What determines the final physical picture ^{is the}

model in which He ^is diagonalized. ^Our qualitative

considerations above suggest ^{the CDW} and Peierls transition model in the sense of [1, 2], ^{i.e. a} single-

mode (q

Q) model. In this model, C t + Q Ck) ^is

evaluated from (la) exactly following [1] ^{and the}

result is

when occupation ^numbers (Fermi functions) nk(1), nk2) ^{of the new} energy levels [1]

are also included to allow for temperature ^effects.

The self-consistency cycle ^{is closed} by denoting

substituting (2) ⁱⁿ (5) ^and inserting (3) in the result.

A homogeneous system ^{of linear} equations ^{arises in} 8 kQ ^{which has a non-zero solution when}

holds. The quantity ^{in square} brackets of this BCS like gap equation ^can be called the phonon ^self-

energy (polarization operator) 1T (Q, (ù = 0). To ^get (6) ^{in a form more familiar from BCS} theory, ^one

writes

and assumes exact pairing

Here the notation

and

has been used. Then, ^the analogy ^{of the final} temperature-dependent ^BCS result follows im-

mediately.

(6) ^{is the} instability condition from which the

phase transition temperature Tc ^{is in} principle ^{to be}

determined. Total energy (adiabatical 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 _{w Q} (at

T

0) by evaluating ^{the 2nd} derivative of Etot ^with

respect ^{to the ion} displacements, ^{at the} equilibrium positions ^{of the} original ^lattice. Alternatively, by expanding ^the electronic part ^{in small} displacements

q, using (for illustrative _purposes only !) ^similar

approximations [14] ^{as in BCS} theory and assuming

2074

that wkQ = wQ holds, ^{one can calculate the recon-} struction free energy change

where ns (E ) ^{is the SS} density ^{of states, w is the cut-}

off _energy and kB ^{is the Boltzmann constant.} By writing io-2 Q

^{( a2 å[f /ð’TJ} 2),a

^{0 one} immediately ^ob-

tains the renormalized frequency W Q. ^Equating

r.h.s. of (7) ^{to zero, one could get} T,,. ^{However, the} approximations ^{made do not} grasp correctly ^the important contributions coming ^{from both the} peak

in ns (E ) ^caused by degenerate SS branches and the

logarithmic (saddle-point) ^{van Hove} singularity ^in-

voked by ^the two-dimensional character of the

problem. ^{It is worth} noticing, ^{that this} singularity

resembles the one found recently ^{in the} density ^of

states of high-Tc ^ceramic superconductors (see e.g.

[16]). ^{Here of course,} only electron-phonon ^matrix

elements of the Umklapp type ^are dealt with so that in agreement ^with experiment, structural instability

is preferred ^to superconductivity.

To get ^an idea how the above two peaks ^can

influence the determination of Tc, ^{let us mention} shortly the results obtained under simplifying ^as- sumptions. ^Full discussion will be given ^elsewhere [15]. ^Let us assume that the (flat) degenerate ^SS

branches contribute into ns(E) a 5-function like

peak ^of amplitude S ^{and that} the width of the

logarithmic singularity is R. Let us further assume

that both peaks ^{lie at} EF. ^{Then, the} expression ^to replace the second term in the square brackets of (7)

will consist of two terms, each from its own peak

The emerging pairing mechanism of (2 x 1) ^silicon

surface reconstructions is related to other pairing phenomena ^{in solids, in} particular ^{to the small} polaron (negative ^{U or} bipolaron) problem [17]. ^It

can be checked that bq> ^and Hpol ^{give correct}

results in the small polaron ^{limit. Also,} by consulting [1] ^with 8k,j ^{given by} s kQ ^{of (5), one} finds that in the

present picture, strong electron-phonon coupling

leads to the localization of charge ^{on one of the}

sublattices (another ^electron pairing effect, ^{of the}

sort known form the negative ^U problem [17]). ^This

effect might be of relevance for the existence of

buckled modes. Especially if local surface symmetry is taken into account properly [3], ^the present selection rule can also be applied ^{to the} problems ^of

ordered adsorption, adsorption ^induced reconstruc-

tion, epitaxial growth ^{or surface atom} scattering,

’ where instead of the electron an external atom in interaction with surface phonons is considered.

It should not be overlooked that the formalism of this section is equally applicable ^{to other kinds of}

bosons (e.g. ^excitons [1], plasmons, paramagnons)

in interaction with the electrons, ^{the last case} leading

to a sort of Heitler-London-Heisenberg coupling.

We expect ^{to be able to} exploit ^these possibilities

elsewhere [15].

4. Discussion.

The generally accepted (2 ^x 1) reconstruction of the surface layer ^{of Si} (111) ^and (100) ^surfaces belong ^to

reconstruction models allowed by ^the presently

introduced selection rule [4, ^{10, 11,} 18]. ^{As usual} recently, buckled modes are excluded by arguing [8]

that in LCAO total energy calculations they ^result

from an artifact of the method which exaggerates charge transfer between neighbouring ^{surface atoms} (however, ^see [4, 18]).

The (2 ^x 1) reconstruction of the Si (110) ^surface

which has not yet been treated in the literature,

deserves attention. The surface layer of the ideal

(110) surface consists of parallel chains of atoms, each chain exhibiting ^a glide plane symmetry o-g ^{which causes a double} degeneracy along ^the X M direction of the (1 ^x 1) ^SBZ [19]. ^{A selection}

rule analogous ^{to the} present ^{one allows a} longitudi-

nal Peierls-like distortion in separate chains which

splits ^the degeneracy, leading ^{to two} separate ^SS bands. Notice, ^that analogous transversal displace-

ments of altemant chain atoms are also allowed, however, they ^{do not remove} _og. They ^can help ^to

shift the X-M degenerate ^{states to} EF ^{and to} optimize the energy gain. ^The (2 ^x 1) reconstruction follows then from the selection rule of section 2 and

occurs perpendicularly ^to the chains. One can im-

agine that it arises from opposite-phase location of the Peierls distortion in neighbouring ^{chains. A} very

simple total energy calculation of several reconstruc-

tion modes of the Si (110) ^surface supporting ^this

picture, ^has already ^{been done} [5].

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