- Email: [email protected]
Instability and (2×1) Reconstruction of Silicon Surfaces
Ins tabili ty and \ 2x I) Re c o n s truc Lion of 5i 1 icon S ur-t' ace s ;·10 jnd.r Tomas ek
Institute of Physical Chemistry and Electrochemistry Czechoslovak Academy of Sciences, 121 38 Prague 2., :'lachova 7, Czechoslovakia
1. INTHODUCTION The present very simple ideas try to approach surface reconstruction of crystals exhibiting a certain component of covalent bonding, wn.i ch have Shockley surface states (SS) around the Fermi energy E Sur-t a.oe s of diamond-like semiconductors, F• in particular silicon, are classical exampl.e s , The ideas may complement somehow other investigations and contribute to the elucidation of the role of SS in surface reconstructions. They are based on the author's work /1,2/ published long time ago, which, with various modifications, has been tested recently /2-5/ on surface systems of' current interest. The importance of the electron-phonon interaction between SS and lattice deformations is a s s ume d , and as the basic ingredient the elec tron (cnv or SDl\') and La t tice (Peierl 5 or pseudo Jahn-Teller like) instability theory /1,2/ is used. The (2xl) instability is discussed by using the "pairing theorem" /1,2/ valid throughout the wh o Le (2Xl) surface Brillouin zone (SBZ) of surface (and~) electronic states of' Si (111). (110) and (100) surfaces. As in the pseudo J.-T, theory, main tool of the analysis is the (Umklapp) electron-phonon matrix element w .(k+(III{/k which operates as a symmetry based selection kU r~e. Here Ik) is the Bloch function and W is the electron-phonon (deformation) potential exhibiting same symmetry as the deformation mode, . For simplicity reasons, the case of a single SS band, appearing on the Si( 111) t x t surface .is presented here. rteconstruction of ideal S.i(110) and (100) surfaces, whe r-e two SS bands occur in the gap is also treated easily, but mainly results are mentioned.
=
>
2. PAIRING THEOREM A.ND SELECTION RULE The symmetry based selection rule, wh.i ch by means of 1{ decides whether a reconstruction mode '7Z. is a.Ll owe d or forbidden, is based on the existence of the "pairing theorem". This governs symmetry properties of' the wave functions of Shockl.ey sw'face states (S5). The theorem hol.ds due to the fact that the three investigated surfaces are alternant systems, i.e. each of their atoms falls into one of two interpenetrating eUbljtices A ,nd~B (there are two s.ublattice pairs on the \llQ surface), ror such syaems /1,2/, any t~o states in the
:346
ae.
SS band diff er iug by a cer tain k= (in 0 ur o a se ee ;c{) f' orm pairs of bonding and antibonding (with r-e npe o t to A and TI) states. Since this pairing runs through the who l.e surface Brillouin zone, it signals that local ("chemical") el"fects are at work on the surface. It strongly differs from the usual nesting mechanism occuring on the Fermi surface only. One easily checks that 'k>.:l:~ + (jJ., and I k+(i'Yi= !In hold and hence, WI" can be non-ze:r:o oiily jJ 1.. is, antis)~metric in A,D!. i.e,. \vZ!'A...:-t1D' Here and I'C are the Dlocn f uric t Lo n and the d ef o r-me t Lo n potential, respectively, of th~sublattice C. This selection rule is in apparent analogy to the pseudo J, -T. theory, wher e different parity wave functions can couple only via Wl odd parity deformation - a process stabilizing total electronic energy. Of course, the polarization vector of the antisymllletric reconstruction mode "2. can be directed either perpendicularly (buckled mode) or parallely to the surface (in-plane mode). The latter can be either longitudinal or transversal w Lt.h respect to Q. It istllparticular the mode longitudinal wi t.h 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. In the f o Ll.ow Lng , Figs. 1,2 of /6/ are referred to to display the geometry, SBZ and SS of the three ideal Si surfaces. The high symmetry points of the SBZ of (2xl) surfaces are labeled analogously to Fig. 8 of /7/ and Fig. 2 of /8/. It is interesting to notice that for the (2xl) structures of the investigated surfaces, a degeneracy occurs between the symmetric and antisymmetric (":!:?1-d2d") SS branches ~t2!l&: £ertain SDZ directions ~h.:ich read: J-K-J for the (111), J-K-J(X) for the (110) and J~K for the (100) surface, respectively. Along these directions, a pair of equivalent SS states /k)±lk+Q> can be formed, the wave functions ~\ and B of which being exclusively localized on either the'A or B sublattice, respectively. These states are non-bonding with respect to the A-B interaction and show the unsaturated character of the corresponding dangling bonds /9/. "Chemical" saturation (pairing) of the latter contributes to the energetics of the reconstruction process, Let us enumerate those q=Q of the investigated surfaces for which the selection rule is fulfilled, leading to a (2x1) reconstruction of the original C!.~1) surface. FQJ;: the (111) surface it holds that q=Q 1/2 G(1121_= 2'ir/Ja (112), "here G is the reciprocal vector in the (112) direction of the (lxl) surface, and similarly for (110) and (100) surfaces orie has Q = 1/2 G(OOl) ='ir/a (001) and Q 1/2 G(1,-1,0) :'i//a(I,-I,O), respectively. In the surface layer to which 'we lind t our considera tions here, the Pandey""" -bonded chain model /7/ of the (111) surface reconstruction corresponds to our longitudinal mode. The same is true for the (2xl) reconstruction of the (110) and the symmetric dimer of the (100) surfaces, respectively. The corresponding as~nmetric dimer is a combination of the in-plane longitudinal. and the buckled modes. Of course, the Pandey model /7/ 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
Cf\ -
Y'c
9
=
=
in the bulk as we.l.L, there might exist a s pe qi.n.L friD.fi·e effect Ln chemical bonding (force constants K = W .~) helping the subsurface .layer( s) to adjust its (their) geClmetry more ea.sily to the reconstruction in the surface layer, Notice, that the in-plane longi t udLn a.l, mode (i=21/ /Ja (112) of the (111) sur:face has opposite ph as e in the subsurface layer /7/; i t might be that geometrical stress, steric or mismatch effects are relaxed in this wa y , Of' course, there are t wo SS barid c on (110) and (100) surfaces which cause that ViI (! has interband niat.r-Lx elements, being n01, a 2x2 matrix. ~,
3. FOHMAL MATllE"lATICAL FHA:·!EiiOlU< Let us sketch briefly some simple mathematics behind our ideas, without claiming much accura~cy in the notation. One starts with two coupled hamiltonians, the electron one H and the phonon one H , of the self-consistent (Hartree-Fock 1:1ke, mean-field) aBproach to the electron-phonon interaction
H
e
Hp
= .L k
'k -: c k
= ~ O?, b~
be;
-;..L. ktq
+
~
"k('
"kq
-'.,>
c~+c.~
~
ck
('0(1 'r
ck
(la)
'o~q)
(1'0)
whe r-e c describe electron and '0 phonon operators.
h'hen mo r-e than one (say n) 55 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 /10/) with the key quanti ties (like the gap :function 0k l ) ) changed from scalars to nxn matrices. Obviously, the exac-f"solution of' the phonon problem is trivial since the latter represents the displaced harmonic oscillator problem. By puttiDG the expectation values o:f the commutators <[b. ,H and H J) equal to zero, one gets the phonon sRi:f¥s -~ p
< >= '0q
<'0+ -q
>
=
<['0+,.,
J>
-~ k
(2 )
which appear because terms linear in '0 are contained in (1'0), Completing the squares in that equation (1. e. diagonalizing
(1'0)) gives H
""'Wq ('0+q -<'0+») ('0 - < b » ) P = ~ q q q
-
H
=
po
1
<'0:'><
wL th the st~bilizing "polaron energy" H I 4;:'~, 'or)' resulting :from the displacement o:f ions ¥8 the n~w structure~ equilibriwn positions, What determines the final physical picture is the model in which H is diagonalized. Our qualitative considerations above suggestethe CD1V and Peierls t.r-aris L tion model in the sense of /1,2/, i.e. a single-mode (q=Q) model. In this model, c:. o Ok'> is evaluated from (la) exactly following /1/ and the~resuTt is
«
<
+
<,
c k+ Q c k / =
6k Q£(2.) «n Cx k ,..(-1)
1)
k
-
(2 ) n k
)
(3 )
:34H
n(~), n(~)
whe n occupation numbers (Fermi functions) new energy levels /1/
of the
(4)
:
+
('2
°kQ)
1/2J
are also included to allow for temperature effects. The selfconsistency cycle is closed by denoting
t
:
kQ
"ikC,
<
be:
+
b~Q
>'
( .5 )
substituting (2) in (.5) and inserting (3) in the re~ult. A homogeneous system of linear equations arises in Q which kQ has a non-zero solution when
1
(2) - n k" (2 )
+
~k"
] =
a
(6)
holds. The quantity in square brackets of this BCS like "gap" equation can be called the phonon self-energy (polarization operator) IT(e~,w:o). (6) is the instability condition from which the phase transition temperature T is in principle to be determined. Total energy (adiabaticalCpotential) E t is approximated as the sum over occupied energy levels (~1 stabilized by H l ' plus the repulsive phonon part.dOne finds the "softening" 8P (,J, (at T:O) b)' evaluating the 2 n derivative of E t t with respsct to the ion displacements, at the equilibrl&n positions of the original lattice. Alternatively, by expanding the electronic part in small displacements ~ , using (for illustrative purposes only!) similar approximations /11/ as in BCS theory and assuming that wkO~ Wo holds, one can calculate the reconstruction free ene~gy cllange
AT:
1/2 [K -
4ns(EF)''i~
tn(1.14W/k BT
)J t 2 ,
(7)
where n (E) is the 55 density of states, l.U is the cut::~ff energx, Rnd k B is the Boltzmarm constant. By writing We : :(~6r/i~2)~=o one immediately obtains the renormaliz~d f'resuency Go " Equating r.h.s. of' (7) to zero, one could get T~. Howe ve r , ~he approximations made do no t grasp correc tly the important contribution coming from degenerate ::is branches.- The improved resul t w LLl. appear e Lsewhe r-e /12/. The emel)ing pairing mechanism of (2x 1) silicon surf ace reconstructions is related to other pairing phenomena in solids, in particular to the small polaron (negative U or bipolaron) problem /11~/. It can be checked that
5
>
into account properly /3/, the present selection rule can also be applied to the problems of ordered adsorption, adsorption induced reconstruction, epitaxial growth or surface atom scattering, where instead of the electron an "external" atom in interaction with surface phonons is considered. il.
DISCUSSION
The generally accepted (2x1) reconstructions of the surface layer of SiC 111) and (100) surfaces belong to reconstruction models allowed by the presently introduced. selection rule. As usual recently, buckled modes are excluded by arguinJ1fihat in LCAO total energy calculations they result from an artifact of the method which exaggerates charge transfer between neighbouring surface atoms. The (2xl) 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 which causes a double degeneracy along the X-M direction ofgthe (lx1) SBZ /13/. A selection rule analogous to the present one allows a longitUdinal Peierls-1Lke distortion in separate chains which splits the degeneracy, leading to two separate SS bands. Notice, that analogous transversal displacements of alternant chain atoms are also allow~d.z.. however, they do not remove They can help to shift the X-M degenerate states to EF and t5 optimize the energy gain. The (2x1) reconstruction follows then from the selection rule of Section 2 and occurs perpendicularly to the chains. One can imagine that it arises from opposite-phase location of the Peierls distortion in neighbouring chains. A very simple total energy calculation of several reconstruction modes of the Si(110) surface supporting this picture, has already been done /5/.
r:s:
e ,
1. M. Tomasek: Physica 36, 420 (1967); ibid. 39, 21 (1968); Physics Lett.~6 A, 374 (1968); Int71.Quant. Chern. ~, 849 (1969) - 2. M. Tomasek: In Phonon Physics, Proc. 2nd Int, Conf'. Phonon Physics, Budapest 1985, eds. J. Kollar, N. Kroo, N. Merryhar-d , T. Siklos (1{orld Scientific, Singapore, 1985) p. 673 3. M. Tomasek, S. Pick: J. de Physique (Paris) i2, Colloque C5-125 (1984); Surf.Sci. 140, L 279(198 11); TCM Techn. Rept. (15 January 1985), Cavendish Lab., Cambridge Dniv, , England; Physica ~ B, 79 (1985); Czech. J. Phys. B 35, 768 (1985); ibid. B 37, No. 12 (1987) 4. M. Tomasek: Theor. Chim. Acta, to appear 5. M. Tomasek, S. Pick: J. de Physique (Paris), submitted 6. I. Ivanov I A. Mazur, J. Pollmann: Surface Sci. 92, 365 (1980) 7. K.C. Pandey: Physica ill + 11§.. B, 761 (1983) 8. M.A. Olmstead, D.J. Chadi: Phys. Rev. B 33, 8402 (1986) 9. M. Tomasek, J. Ko u t e ck y r Int. J. Quant. Chern. 3, 249 (1969) --
:)50
10. H. Balian, N.H. \;'erthmner: Ph y s , Hev. ill, 1553 (1963) 11. Ch , Kd t t.e l : Quantum Theory of Solids (J. \iiley, New York 1963) Chapt. 8 12. M. Tomasek: to be pubLished 13. V. Heine: Proc, Roy. Soc. London A 331, 307 (1972) lLf. P.li. ,\nderson: Ph y s , Hev. Lett. J4, 953 (1975) APPLNDTS
To get (6) in a from more familiar from 13CS theory, one writes
I)~l)
-
~2)=
1 _
2nl~;2)=
-tgh
tk/;~kBT
an.d assumes exact pairing Here the notation
used. Then the analogy of the finite temperature dependent DCS result f ol Low s immediately.
NOTE paper was originally prepared for presentation at the 2 n International Conference on the Structure of Surfaces (ICSOS II, Amsterd~l, Jw~e 22_25,1987), but not presented there f inall s-
Th~s
Institute of Physical Chemistry and Electrochemistry Czechoslovak Academy of Sciences, 121 38 Prague 2., :'lachova 7, Czechoslovakia
1. INTHODUCTION The present very simple ideas try to approach surface reconstruction of crystals exhibiting a certain component of covalent bonding, wn.i ch have Shockley surface states (SS) around the Fermi energy E Sur-t a.oe s of diamond-like semiconductors, F• in particular silicon, are classical exampl.e s , The ideas may complement somehow other investigations and contribute to the elucidation of the role of SS in surface reconstructions. They are based on the author's work /1,2/ published long time ago, which, with various modifications, has been tested recently /2-5/ on surface systems of' current interest. The importance of the electron-phonon interaction between SS and lattice deformations is a s s ume d , and as the basic ingredient the elec tron (cnv or SDl\') and La t tice (Peierl 5 or pseudo Jahn-Teller like) instability theory /1,2/ is used. The (2xl) instability is discussed by using the "pairing theorem" /1,2/ valid throughout the wh o Le (2Xl) surface Brillouin zone (SBZ) of surface (and~) electronic states of' Si (111). (110) and (100) surfaces. As in the pseudo J.-T, theory, main tool of the analysis is the (Umklapp) electron-phonon matrix element w .(k+(III{/k which operates as a symmetry based selection kU r~e. Here Ik) is the Bloch function and W is the electron-phonon (deformation) potential exhibiting same symmetry as the deformation mode, . For simplicity reasons, the case of a single SS band, appearing on the Si( 111) t x t surface .is presented here. rteconstruction of ideal S.i(110) and (100) surfaces, whe r-e two SS bands occur in the gap is also treated easily, but mainly results are mentioned.
=
>
2. PAIRING THEOREM A.ND SELECTION RULE The symmetry based selection rule, wh.i ch by means of 1{ decides whether a reconstruction mode '7Z. is a.Ll owe d or forbidden, is based on the existence of the "pairing theorem". This governs symmetry properties of' the wave functions of Shockl.ey sw'face states (S5). The theorem hol.ds due to the fact that the three investigated surfaces are alternant systems, i.e. each of their atoms falls into one of two interpenetrating eUbljtices A ,nd~B (there are two s.ublattice pairs on the \llQ surface), ror such syaems /1,2/, any t~o states in the
:346
ae.
SS band diff er iug by a cer tain k= (in 0 ur o a se ee ;c{) f' orm pairs of bonding and antibonding (with r-e npe o t to A and TI) states. Since this pairing runs through the who l.e surface Brillouin zone, it signals that local ("chemical") el"fects are at work on the surface. It strongly differs from the usual nesting mechanism occuring on the Fermi surface only. One easily checks that 'k>.:l:~ + (jJ., and I k+(i'Yi= !In hold and hence, WI" can be non-ze:r:o oiily jJ 1.. is, antis)~metric in A,D!. i.e,. \vZ!'A...:-t1D' Here and I'C are the Dlocn f uric t Lo n and the d ef o r-me t Lo n potential, respectively, of th~sublattice C. This selection rule is in apparent analogy to the pseudo J, -T. theory, wher e different parity wave functions can couple only via Wl odd parity deformation - a process stabilizing total electronic energy. Of course, the polarization vector of the antisymllletric reconstruction mode "2. can be directed either perpendicularly (buckled mode) or parallely to the surface (in-plane mode). The latter can be either longitudinal or transversal w Lt.h respect to Q. It istllparticular the mode longitudinal wi t.h 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. In the f o Ll.ow Lng , Figs. 1,2 of /6/ are referred to to display the geometry, SBZ and SS of the three ideal Si surfaces. The high symmetry points of the SBZ of (2xl) surfaces are labeled analogously to Fig. 8 of /7/ and Fig. 2 of /8/. It is interesting to notice that for the (2xl) structures of the investigated surfaces, a degeneracy occurs between the symmetric and antisymmetric (":!:?1-d2d") SS branches ~t2!l&: £ertain SDZ directions ~h.:ich read: J-K-J for the (111), J-K-J(X) for the (110) and J~K for the (100) surface, respectively. Along these directions, a pair of equivalent SS states /k)±lk+Q> can be formed, the wave functions ~\ and B of which being exclusively localized on either the'A or B sublattice, respectively. These states are non-bonding with respect to the A-B interaction and show the unsaturated character of the corresponding dangling bonds /9/. "Chemical" saturation (pairing) of the latter contributes to the energetics of the reconstruction process, Let us enumerate those q=Q of the investigated surfaces for which the selection rule is fulfilled, leading to a (2x1) reconstruction of the original C!.~1) surface. FQJ;: the (111) surface it holds that q=Q 1/2 G(1121_= 2'ir/Ja (112), "here G is the reciprocal vector in the (112) direction of the (lxl) surface, and similarly for (110) and (100) surfaces orie has Q = 1/2 G(OOl) ='ir/a (001) and Q 1/2 G(1,-1,0) :'i//a(I,-I,O), respectively. In the surface layer to which 'we lind t our considera tions here, the Pandey""" -bonded chain model /7/ of the (111) surface reconstruction corresponds to our longitudinal mode. The same is true for the (2xl) reconstruction of the (110) and the symmetric dimer of the (100) surfaces, respectively. The corresponding as~nmetric dimer is a combination of the in-plane longitudinal. and the buckled modes. Of course, the Pandey model /7/ 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
Cf\ -
Y'c
9
=
=
in the bulk as we.l.L, there might exist a s pe qi.n.L friD.fi·e effect Ln chemical bonding (force constants K = W .~) helping the subsurface .layer( s) to adjust its (their) geClmetry more ea.sily to the reconstruction in the surface layer, Notice, that the in-plane longi t udLn a.l, mode (i=21/ /Ja (112) of the (111) sur:face has opposite ph as e in the subsurface layer /7/; i t might be that geometrical stress, steric or mismatch effects are relaxed in this wa y , Of' course, there are t wo SS barid c on (110) and (100) surfaces which cause that ViI (! has interband niat.r-Lx elements, being n01, a 2x2 matrix. ~,
3. FOHMAL MATllE"lATICAL FHA:·!EiiOlU< Let us sketch briefly some simple mathematics behind our ideas, without claiming much accura~cy in the notation. One starts with two coupled hamiltonians, the electron one H and the phonon one H , of the self-consistent (Hartree-Fock 1:1ke, mean-field) aBproach to the electron-phonon interaction
H
e
Hp
= .L k
'k -: c k
= ~ O?, b~
be;
-;..L. ktq
+
~
"k('
"kq
-'.,>
c~+c.~
~
('0(1 'r
ck
(la)
'o~q)
(1'0)
whe r-e c describe electron and '0 phonon operators.
h'hen mo r-e than one (say n) 55 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 /10/) with the key quanti ties (like the gap :function 0k l ) ) changed from scalars to nxn matrices. Obviously, the exac-f"solution of' the phonon problem is trivial since the latter represents the displaced harmonic oscillator problem. By puttiDG the expectation values o:f the commutators <[b. ,H and H J) equal to zero, one gets the phonon sRi:f¥s -~ p
< >= '0q
<'0+ -q
>
=
<['0+,.,
J>
-~ k
(2 )
which appear because terms linear in '0 are contained in (1'0), Completing the squares in that equation (1. e. diagonalizing
(1'0)) gives H
""'Wq ('0+q -<'0+») ('0 - < b » ) P = ~ q q q
-
H
=
po
1
<'0:'><
wL th the st~bilizing "polaron energy" H I 4;:'~, 'or)' resulting :from the displacement o:f ions ¥8 the n~w structure~ equilibriwn positions, What determines the final physical picture is the model in which H is diagonalized. Our qualitative considerations above suggestethe CD1V and Peierls t.r-aris L tion model in the sense of /1,2/, i.e. a single-mode (q=Q) model. In this model, c:. o Ok'> is evaluated from (la) exactly following /1/ and the~resuTt is
«
<
+
<,
c k+ Q c k / =
6k Q£(2.) «n Cx k ,..(-1)
1)
k
-
(2 ) n k
)
(3 )
:34H
n(~), n(~)
whe n occupation numbers (Fermi functions) new energy levels /1/
of the
(4)
:
+
('2
°kQ)
1/2J
are also included to allow for temperature effects. The selfconsistency cycle is closed by denoting
t
:
kQ
"ikC,
<
be:
+
b~Q
>'
( .5 )
substituting (2) in (.5) and inserting (3) in the re~ult. A homogeneous system of linear equations arises in Q which kQ has a non-zero solution when
1
(2) - n k" (2 )
+
~k"
] =
a
(6)
holds. The quantity in square brackets of this BCS like "gap" equation can be called the phonon self-energy (polarization operator) IT(e~,w:o). (6) is the instability condition from which the phase transition temperature T is in principle to be determined. Total energy (adiabaticalCpotential) E t is approximated as the sum over occupied energy levels (~1 stabilized by H l ' plus the repulsive phonon part.dOne finds the "softening" 8P (,J, (at T:O) b)' evaluating the 2 n derivative of E t t with respsct to the ion displacements, at the equilibrl&n positions of the original lattice. Alternatively, by expanding the electronic part in small displacements ~ , using (for illustrative purposes only!) similar approximations /11/ as in BCS theory and assuming that wkO~ Wo holds, one can calculate the reconstruction free ene~gy cllange
AT:
1/2 [K -
4ns(EF)''i~
tn(1.14W/k BT
)J t 2 ,
(7)
where n (E) is the 55 density of states, l.U is the cut::~ff energx, Rnd k B is the Boltzmarm constant. By writing We : :(~6r/i~2)~=o one immediately obtains the renormaliz~d f'resuency Go " Equating r.h.s. of' (7) to zero, one could get T~. Howe ve r , ~he approximations made do no t grasp correc tly the important contribution coming from degenerate ::is branches.- The improved resul t w LLl. appear e Lsewhe r-e /12/. The emel)ing pairing mechanism of (2x 1) silicon surf ace reconstructions is related to other pairing phenomena in solids, in particular to the small polaron (negative U or bipolaron) problem /11~/. It can be checked that
5
>
into account properly /3/, the present selection rule can also be applied to the problems of ordered adsorption, adsorption induced reconstruction, epitaxial growth or surface atom scattering, where instead of the electron an "external" atom in interaction with surface phonons is considered. il.
DISCUSSION
The generally accepted (2x1) reconstructions of the surface layer of SiC 111) and (100) surfaces belong to reconstruction models allowed by the presently introduced. selection rule. As usual recently, buckled modes are excluded by arguinJ1fihat in LCAO total energy calculations they result from an artifact of the method which exaggerates charge transfer between neighbouring surface atoms. The (2xl) 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 which causes a double degeneracy along the X-M direction ofgthe (lx1) SBZ /13/. A selection rule analogous to the present one allows a longitUdinal Peierls-1Lke distortion in separate chains which splits the degeneracy, leading to two separate SS bands. Notice, that analogous transversal displacements of alternant chain atoms are also allow~d.z.. however, they do not remove They can help to shift the X-M degenerate states to EF and t5 optimize the energy gain. The (2x1) reconstruction follows then from the selection rule of Section 2 and occurs perpendicularly to the chains. One can imagine that it arises from opposite-phase location of the Peierls distortion in neighbouring chains. A very simple total energy calculation of several reconstruction modes of the Si(110) surface supporting this picture, has already been done /5/.
r:s:
e ,
1. M. Tomasek: Physica 36, 420 (1967); ibid. 39, 21 (1968); Physics Lett.~6 A, 374 (1968); Int71.Quant. Chern. ~, 849 (1969) - 2. M. Tomasek: In Phonon Physics, Proc. 2nd Int, Conf'. Phonon Physics, Budapest 1985, eds. J. Kollar, N. Kroo, N. Merryhar-d , T. Siklos (1{orld Scientific, Singapore, 1985) p. 673 3. M. Tomasek, S. Pick: J. de Physique (Paris) i2, Colloque C5-125 (1984); Surf.Sci. 140, L 279(198 11); TCM Techn. Rept. (15 January 1985), Cavendish Lab., Cambridge Dniv, , England; Physica ~ B, 79 (1985); Czech. J. Phys. B 35, 768 (1985); ibid. B 37, No. 12 (1987) 4. M. Tomasek: Theor. Chim. Acta, to appear 5. M. Tomasek, S. Pick: J. de Physique (Paris), submitted 6. I. Ivanov I A. Mazur, J. Pollmann: Surface Sci. 92, 365 (1980) 7. K.C. Pandey: Physica ill + 11§.. B, 761 (1983) 8. M.A. Olmstead, D.J. Chadi: Phys. Rev. B 33, 8402 (1986) 9. M. Tomasek, J. Ko u t e ck y r Int. J. Quant. Chern. 3, 249 (1969) --
:)50
10. H. Balian, N.H. \;'erthmner: Ph y s , Hev. ill, 1553 (1963) 11. Ch , Kd t t.e l : Quantum Theory of Solids (J. \iiley, New York 1963) Chapt. 8 12. M. Tomasek: to be pubLished 13. V. Heine: Proc, Roy. Soc. London A 331, 307 (1972) lLf. P.li. ,\nderson: Ph y s , Hev. Lett. J4, 953 (1975) APPLNDTS
To get (6) in a from more familiar from 13CS theory, one writes
I)~l)
-
~2)=
1 _
2nl~;2)=
-tgh
tk/;~kBT
an.d assumes exact pairing Here the notation
used. Then the analogy of the finite temperature dependent DCS result f ol Low s immediately.
NOTE paper was originally prepared for presentation at the 2 n International Conference on the Structure of Surfaces (ICSOS II, Amsterd~l, Jw~e 22_25,1987), but not presented there f inall s-
Th~s