A note about the effect of surface states on the stability of the Si (110) surface

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A note about the effect of surface states on the stability of the Si (110) surface

M. Tomášek, Š. Pick

To cite this version:

M. Tomášek, Š. Pick. A note about the effect of surface states on the stability of the Si (110) surface.

Journal de Physique, 1988, 49 (1), pp.99-102. �10.1051/jphys:0198800490109900�. �jpa-00210678�

A note about the effect of surface states on the stability ^of

the Si (110) ^surface

M. Tomá0161ek and 0160. Pick

J. Heyrovský Institute of Physical Chemistry ^and Electrochemistry, Czechoslovak Academy ^{of Sciences,}

Máchova 7, ^{121 38} Prague 2, Czechoslovakia

(Reçu ^{le 15} juin 1987, accepté ^{le 29} septembre 1987)

Résumé.

Nous étudions le rôle éventuel des états de surface de Shockley

la stabilité de la surface (110)

du silicium à l’égard ^de plusieurs modes de déformation. La fonction d’onde des liaisons pendantes ^est

construite à l’aide d’une généralisation ^de l’approche simple d’Harrison, tandis que les ^autres liaisons sont considérées grâce ^à des termes élastiques phénoménologiques. ^{Nous trouvons}

^net ramollissement de

phonons pour le mode antisymétrique ^{des atomes} de surface des chaînes individuelles,

analogie apparente

avec

la déformation de Peierls des systèmes quasi-linéaires.

Abstract.

The possible ^{role of} Shockley ^{surface states in the} stability ^{of the Si} (110) surface with respect ^to several deformation modes is investigated. ^The dangling ^bond

function is constructed by

^of

generalization ^{of the} simple ^Harrison approach, whereas the remaining ^bonds

accounted for by phenomenological ^{elastic terms. A} pronounced phonon softening is found for the antisymmetric ^{mode of}

surface atoms of individual chains, ⁱⁿ apparent analogy with the Peierls-like deformation of quasi-linear systems.

Classification

Physics ^Abstracts

73.00

68.00

1. Introduction.

From the theoretical point ^of view, little is known about the geometrical ^structure and reconstructions of the Si (110) ^surface. The existence of quasi-linear zig-zag ^{chains of atoms on the ideal} (1 x 1 ) ^surface [1,2] ^{suggests a sort of} analogy ^{with the} Pandey

bonded chain model [3] ^{of Si} (111) (2 x 1 ) ^{in which}

such chains also appear, exhibiting comparatively large stability. ^An important experimental ^fact [1, 2]

helping ^to guess possible types ^{of the Si} (110) ^surface

reconstruction is the existence of the energy gap

(about 0.2-0.4 eV wide) ^{in the surface state} (SS)

band. Contrary ^{to the} Pandey model, the ideal Si

(110) ^surface displays ^a symmetry ^element (glide plane _ag parallel ^{to the} chains, ^see Fig. 1) ^which

prevents ^{formation of a} gap. Two simple ^deforma-

tions can remove this symmetry operation and open the gap : a) ^the buckling ^mode shifting the alternat-

ing ^atoms of the chains perpendicularly ^{to the} (110)

surface but in opposite direction, ^and b) the Peierls- like deformation in the surface plane, in which the bond to the left of an arbitrary ^{chain atom differs in} length ^{from that on the} right.

Namely, ^the glide plane o-g invokes the appearance of ^a double degeneracy [4-7] along ^{the X-M direction}

Fig. ^1.

^Left, the chain formed by ^{Si surface atoms} (open circles)

^{the Si} (110) surface is shown, together

with their nearest subsurface neighbours (full circles). ^The

atoms belonging ^to

and b sublattices are labelled and the

numbering ^{of bonds} b, ( j = 1-3 )

^well

^the glide plane ag

^also displayed. Right, ^{the modes} A1 ^and AZ together

with the mode N suggested ⁱⁿ [2]

depicted.

of the surface Brillouin zone of the Si (110) ^surface,

the wave functions of the degenerate ^branches being Oa and Ob, ^where cPa and Ob ^{are Bloch functions}

localized on a and b sublattices (Fig. 1), respectively [7]. ^{The wave} functions show that with respect ^to a-b interactions, ^{the two} SS levels exhibit the unsaturated character of dangling ^bonds [8]. ^This

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphys:0198800490109900

100

effect arises from symmetry ^reasons, since lateral interactions in the chain cancel mutually. ^The

«

chemical

saturation between neighbouring ^chains

is unlikely because of long ^{distance between them} (third nearest-neighbours). ^Hence, saturation can occur by ^either charge disproportionation ^{between a}

and b sublattices (the buckling mode), ^or by ^dimeri-

zation (the Peierls-like distortion). The latter mechanism takes place ^{for the} A2 distortion (Fig. 1), antisymmetric ^with respect ^to a- g’ for which the

electron-phonon ^matrix element lfJa I wi lfJb) ^is

non-zero along ^{ff-R as a} result of suitable symmetry

(a ^{sort of a} selection rule [9], where w is the electron-

phonon (deformation) potential [10], exhibiting

same symmetry ^{as the} deformation mode). ^The

above reasoning explains ^the suggestion ^{of the} degeneracy splitting pointed ^{out a} long ^time ago [5].

Naturally, ^{the real Si} (110) ^surface displays [1, 2,11] ^a number of modulated structures (e.g.

(4 x 5 ), (2 x 1 ), (5 ^x 1)), ^a successful theoretical

explanation ^{of which} by ^{means of our} simple dangl- ing ^{bond based} approach ^{is hard to expect since even}

in realistic calculations [4-6], the SS band shows ^a small dispersion ^{in the direction} perpendicular ^{to the}

chain. However, ^{one can} hope that modes important

in our local chemical approach ^{will be} similarly important ^{in more} complicated ^{modulated structures}

where the interaction with subsurface layers, ^the geometrical strain, ^steric and mismatch effects etc.

have to be inherently ^included.

2. The generalized Harrison model.

Contrary ^{to our recent} qualitative approach [10] ^to

surface reconstruction, ^based mainly ^on the role of surface states from the vicinity ^of EF in the electron-

phonon coupling, ^the present paper emphasizes

more the chemical point of view and tries to get

qualitative conclusions on total energy differences for particular surface deformation modes or on

phonon softenings. Naturally, ^main emphasis ^is again put ^{on surface} states, half-filled on the ideal surface in agreement ^with simple ^SS theory. ^To get SS energies ^{and wave} functions in simplest possible

way, ^a generalized ^{Harrison model} [12] ^{is used as the} starting point. ^This approach ^{is based on the be-}

haviour of the dangling ^bonds during the defor- mation. The possibility ^{of a} charge ^transfer (appear-

ance of ^a surface ionic structure) ^{is a} priori ^excluded.

Consider a surface atom on the Si (110) ^surface (Fig. 1). It is attached ^to its three nearest neighbours by ^bonds bj, j

^{1-3. According to Harrison [12], a}

«

bond-orbital »

is associated with bj, where the p-orbital Ipj) ^is

directed along bi. ^To these three orbitals, ^the

dangling ^bond orbital I 0) ^{of an} analogous ^{form is}

added. The a j ^{as well as} the orientation of

[ 0 ) ^are determined by ^the orthonormalization condi- tions (i I j)

5 ij, ^{i, j}

0-3. The orbitals change during the deformation and therefore, their proper

use helps ^to discover the surface instability.

The matrix elements between nearest-neighbour dangling bond orbitals are computed using ^the

Harrison parametrization [12], including ^{the d- 2} dependence ^{on the} distance. The orbitals 0) ^are

not directed along ^the nearest-neighbour ^surface bonds, however, their interaction is responsible ^for

the SS band formation and has to be considered. Let

us denote as vl ^and V2 the matrix elements between the dangling bond orbitals connected by ^bonds bI ^and b2, respectively, ^and further let us put

zl

vl - v2, ^D

via ⁺ v2, eo

(0 I H 0) . ^{For the}

SS bandwidth one obtains - 2 D, the width of the gap which appears when A =A ⁰ being 2 I L11.

^The dispersion ^relation for the SS band energy reads

The one-electron theory gives the covalent energy

gain (E -- 0) [13]

associated with the dangling bond band. Let us show in more detail that the value _Eo does not play any essential role in the total energy, supposing ^{that the}

local charge neutrality is conserved. With 4 valence electrons per ^atom and with bonding ^{states filled} (the orbital Ij) half-filled, j

0-3), ^one has nj

¹

for the corresponding occupation numbers. On the other hand, in the first order perturbation theory, ^a change 5 Ej ^{of diagonal matrix} elements E

(j I HI j) ^{leads to the} change ^of the total energy 6E

Xnj 5 Ej

⁵ (-v Ej). ^Since ’VEi

^{es + 3 ep}

holds by definition of the orbitals ij), ^{one has}

6E

0. In other words, during the deformation the valence state electronic configuration of the surface atom remains 3s1p3, irrespectively ^{of the amount of}

s, p character contained in each orbital. Therefore, any deformation changes the total electronic energy

only ^{as much as it leads to the} broadening ^{of the SS}

band via the interaction between the resulting hyb-

rids. In the original buckling ^model [12, 14], ^the

violation of local charge neutrality by ^the charge

transfer (no

^{1 ±} An) is crucial for the stabilization.

According ^{to more recent} analysis [3, 15], ^however,

this stabilization is counterbalanced by ^Coulomb

effects arising from ionic configurations.

As a matter of fact, the SS band is modified due to the interaction with the bulk states (especially ^{at the}

lower edge) [6]. Nevertheless, ^{it is} correct to use the

«

bare » bandwidth - 2 D when evaluating (2) ^and (3). Namely, ^{let us} write the Hamiltonian of the

occupied ^{states in the form}

where Hd is the bare dangling bond Hamiltonian based on equation (2), Hb ^describes bonding ^bulk

states and Vbd ^{is an} interaction term. Ignoring ^the

inessential constant term XEj ^{= £S +} 3 Ep ^{per atom}

discussed above, ^the energy is given by ^{the trace Tr} Hocc ^of equation (4). ^Since below, ^Tr Hb ^is replaced by ^a phenomenological ^elastic Hamiltonian and since Tr V bd

⁰ (a matrix with zero diagonal elements), the evaluation of the energy of the

dangling ^bond band is based on the bare parameters of Ha, ^{i.e. on} equation (3).

The integral (3) is transformed into the complete elliptic integral ^of the 2nd kind E(K) ^{and is} easily

evaluated by ^the expansion in the small parameter

K’2 - 1- K2 = (4/D )2 [16, 17]. ^{Note that for}

A =A 0, ^the expansion ^{contains a} particular ^term

~

K 2InK’, which, ⁱⁿ principle, destabilizes the surface for any values of the remaining parameters.

Nevertheless, it appears that the respective ^deforma-

tion is quite negligible if it is not enhanced by physically significant ^effects.

Instead of computing ^the contribution from the

orbitals I j) , j ^{= 1-3, in a} way similar ^to equation (3),

the deformation of bonds bi is taken into account by

an empirical elastic Hamiltonian [12, 14]. Namely,

the contributions 1/2 CO(5bjlbj )2 ^{and 1/2 C 1 5} 09?.

describe the change ^of length 5bj ^{and the change of}

angle 8 eij ^between particular bonds, respectively.

Whereas a reasonable guess should be obtained in this way for contributions quadratic ⁱⁿ deformations,

the description ^{of an} important feature is missing :

the energies associated with bi contain linear terms for fully symmetric ^modes. Here, ^{we have those}

linear terms [18] ^{in mind,} which cancel in the bulk due to a higher symmetry. ^{Because of lower coordi-} nation on the surface, ^{one can} expect that these linear terms will cause a certain shrinking ^{of the}

surface bonds (cf. [3]). ^{Hence, in the} present ap-

proach ^we expect ^an underestimation of the surface relaxation as well ^as the fact that one can not rely ^on

the prediction of the surface bond bl, 2 lengthening (see below).

Finally, ^{for a} deformation of magnitude 6f, ^one

defines the

frequency » w by ^the equation

where m is the mass of the Si atom and AE is the energy change per surface ^atom with respect ^{to the} equilibrium energy, evaluated for small 5f -

Although generally ^{is not an} eigenfrequency, ^{it is} helpful ⁱⁿ discovering surface mode softening.

3. Results and discussion.

Let us describe the numerical results obtained. The model predicts ^a relaxation and an A1 mode defor-

mation, giving ^the bl,2 ^bond lengthening (sbl,2

0.05 d ) ^{and a small} b3 shortening (- ^0.01 d), ^where

d is the bulk nearest-neighbour distance d

2.35 A.

The energy gain ^{is 0.05} eV per surface atom. We believe that the bond lengthening ^{is an} artifact of the method discussed above. (The A mode ^{causes also a}

considerable s - p transfer for the 1 ) and 2)

orbitals, ^{which is} probably ^an energetically ^unfavour-

able effect not described correctly by ^{the elastic} term.) ^{In the new} equilibrium, the surface is stable with respect ^{to the} A2 mode, although ^the frequency

w

is not too high :

= 7.3 THz. For comparison,

w

(A2 ) evaluated with bulk elastic constants only [12] amounts to 9.7 THz.

If we freeze the geometry ^with respect ^{to the} A, ^{mode, we still} get ^a mild relaxation 5b3 -

-

0.02 d accompanied by ^a small energy gain

AE

0.01 eV. The surface (as ^{well as} the unrelax- ed one) ^allows only ^a negligible A2 deformation

6&i 2

^{± 10-3} d, åE - - 10-3 ^eV. Although ^this

can be an unphysical ^effect having ^its origin ^{in the} logarithmic ^{term of} equation (3) (see ^the respective

discussion in Sect. 2), ^{there is} undoubtedly ^{a soften-} ing ^{of the} A2 mode : the evaluation of equation (5)

for 5 = 0.01 d gives w

3.6 THz. Further soften-

ing is obtained when ^a shortening ^of 612 by ^the A, mode is allowed. Note that the energies ^{of order} 10- 2 eV decide here about the stability ^or instability

of the surface with respect ^{to the} A2 ^{mode - a} quantity ^{which can not be} reliably grasped by ^our approximate ^model. Nevertheless, ^{small deforma-}

tion (5bt,2 - :i: ^{0.03 d) is} sufficient to open the

experimentally observed gap of width 0.2 eV [2], leading ^{to no essential s}

p charge transfer in the individual orbitals (a ^more favourable dangling ^bond

orientation takes place). All these facts show that the A2 mode deserves attention for its possible participation ^{in the} (110) ^surface reconstruction or at

least, ^for giving ^{rise to a} low-frequency ^surface phonon.

There is ^a difference with respect ^{to the Si} (111) (2 ^x 1) surface, where the gap in the SS band opens due to a small inequality in the effective charges ^on adjacent ^atoms in the chain (as ^{well as on subsurface} atoms) [3]. The latter effect comes from the necessity

to match the wave function of the surface and subsurface layers ^with differing geometrical ^struc-

ture. The fact that the A2 ^mode might ^{be less} important ^{on the Si} (111) (2 ^x 1) ^surface originates

from the absence of the o-g glide plane ^there.

The recently suggested ^model [2] ^{of the Si} (110)

102

(2 ^x 1) reconstruction (denoted by ^{lV in} Fig. 1) ^does

not seem to be realistic for ^two reasons : a) ^it

assumes a non-negligible interaction between dangl- ing ^{bonds on} adjacent chains, ^the distance between

which is rather large ^{for a} direct chemical interaction, b) it preserves the glide plane o-g ^which prevents ^the opening ^{of the} experimentally ^{observed SS band}

gap.

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Poverkhn. Fiz. Khim. Mekh. No 8 (1986) p. 81

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