- Email: [email protected]
H,Cl
104
Surface
Science
115 (1982) lCM-
I16
~o~~-H~l~a~d Pubk&ing Campany
ELECTRONIC STRUCTURES ON Si(ll1) SURFACE I. Si(ll1) 7 X Y/H,Cl
Received
18September
OF CHEMISORPTION
SYSTEMS
I981
Electronic structures of chemisorption on Si( 1I l)/W,Cl are investigated by the first principle DV-Xa cluster method. The calculations are carried out for chemisorption on different sites. based on the Si,,H,, cluster, and the effect of surface vacancy and buckling on the electronic structure is examined in detail. The present calculation shows that the Sit,H,, surface cluster reproduces very well the more sophisticated band calculation for the Si( I I I) surface. It is concluded that the vacancy model with chemisorbed atoms at appropriate sites is reasonable to interpret the observed UPS of Si( I I I) ?X7/H.C1. The charge transfer between the substrate atom and the adatom depends strongly both on the chemisorption sites and on the electronegativity difference.
I. Introchrction Since Schlier and Farnsworth [l] found the long-period reconstruction of the clean Si( 111) 7 X 7 surface in a LEED pattern, many experiments such as RHEED {2], UPS [3-51 and LEELS [6] have been made to investigate the supperlattice structure. Numerous distinct surface structural models [2,7,8] for Si( 111) 7 X 7 have been proposed to explain the experimental observations. In spite of the long history of research, however, no structure model has been established. It is a very interesting problem which still awaits theoretical elucidation. The direct theoretical approach [7,9] of determining the surface structure is to execute the total energy ~i~~tion of the surface system. The method has been carried out for some simpler systems, such as Si(ll1) 2 X 1 or Si(OO1) 2 X 1 surfaces. At present, however, it seems to be very difficult to apply the same method to complex surfaces such as Si( 111) 7 X 7. On the other hand, the intermediate theoretical approach of determining the surface structure is to calculate the electronic structure for the assumed structure models, and to compare them with the UPS and LEELS experimental results. From this point of view, we have already applied the DV-Xcx cluster method [lo] to investigate the electronic structures of the clean Si(l11) 7 X 7 surface and the chemisorption systems Si( 111) 7 X 7/H,Cl. The calculated results for oo39-6ozg/gzjoooo-ooarti%o2,75
0 1982 North-Holland
the electronic structures for different structural models of the clean Si( 111) 7 X 7 surface [ 1I] and the preliminary results for Si( 111) 7 X 7/H,Cl[ 12) were reported. The purpose of the present paper is to supplement the previous paper and to give further detailed theoretical information on Si(ll1) 7 X 7/H,Cl. A comparison of the present calculations with the observed UPS indicates the presence of vacancies on the Si( 111) 7 X 7 surface. We have also calculated the electronic structures of Si( 11l)/Ag to determine the chemisorption site at the high temperature phase. The result will be published in a forthcoming paper. The reason for adopting the DV-Xcu cluster method [lo] stems from several of its aspects. First, the cluster method is adaptable to calculate the electronic structures which are determined principally by the local atomic ~rangement. It is very difficult to apply theoretical band c~culat~ons to a complex system such as the Si(ll1) 7 X 7 surface. Secondly, the technique of the DV-Xa method allows one to take into account the directional properties of the
.I -
Si+H
5.
Sit6kt
-15.
% Htl Fig. I. Structure
of the clusters
Si,,H,S,
Si,,H,,
-1Q Energy(eV)
-5.
and Si,,H,,
Fig. 2. Total density and partial state density without hydrogen in fig. 1. The arrow indicates the highest occupied level.
components
for the three clusters
potential which are well known to be more important in the surface than in the bulk. The DV-Xa method [lo] is already established, as far as one electron energy is concerned. The plan of this paper is as follows. Section2 gives a short account of the DV-Xcr method [lo]. Section 3 presents numerical results for the electronic structure of three kinds of ideal Si(ll1) surface clusters, Si,,H,,, Si,,H,, and Si,,H,, (fig. l), and discusses the effect of size and structure of the cluster on the electronic structure. H atoms are attached so as to appro~mately saturate the dangling bonds artificially produced by the removal of the cluster from the substrate. Sections 4 and 5 describe numerical results of the electronic structures of Si( 111) 7 X 7/H,Cl respectively. The characteristics in the electronic structure, related to surface vacancy and buckling, are clarified in detail. Section6 gives a summary and discussion.
2. DV-Xar method The present calculations are based on the DV-Xa method which is already reported in previous papers [lo]. The effective exchange-correlation potential is given by statistical approximation and is written as
f’ktd = --3#/8~)
~(r)]“~,
where the quantity cxis the only parameter used in the model and is fixed to be 0.7 in all calculations of this paper. The Coulomb potential is calculated self-consistently using the self-consistent charge approximation [lo]. The matrix elements of the Hamiltonian and overlap integrals are calculated by the random sampling method. It enables us to take into account the non-spherical part of the potential. Numerical atomic basis functions ls-3p for Si and Cl, and 1s for H are used to expand the molecular orbitals.
3. Electronic
structure of ideal Si( 111) surface
To study the effect of size and structure of the cluster, the electronic structures are calculated for the three clusters Si,,H,,, Si,,H,, and Si,,H,,, simulating the ideal Si(ll1) surface. The surface clusters, which have C,, symmetry (fig. l), are removed from the ideal semi-infinite surface. is the smallest one which includes three “sixThe first cluster, Si,,H,,, membered rings of bonds” in the plane parallel to the surface. The third rings cluster, Si,9H2,, is the smallest one which includes three “six-membered of bonds” in the planes parallel and perpendicular to the surface. The importance of six-membered rings of bonds in the diamond lattice is known very well [ 131. The Si-Si and Si-H bond lengths are chosen to be the same as those in the bulk (4.44 au) and in the SiH, molecule (2.80 au), respectively. The
total numbers of electrons are determined from the charge neutrality condition of the respective clusters. Fig. 2 shows the calculated total state density and that without the H components for the three clusters. The overall features of the state densities for Si,,H,, and Si,,H,, are similar to those for thin films as obtained by Schltiter et al. [14]. The valence band width, the appearance and the gross feature of three major peaks in the valence band (table 1) correspond well with the characteristics of the UPS for the Si( 111) surface[3]. The sharp peak around the Fermi level (the highest occupied level) of the state densities for Si,,H,5 and Si,9H2, corresponds to the band D, (D”) of-the dangling bonds (fig. 3a). It is noted that the state density of the dangling bonds is enhanced because of the larger proportion of top Si atoms in the present clusters. The relative weight of the information peculiar to the top layer in the UPS structure considerably changes according to the escape depth of the photoelectron. If the escape depth is deeper than the thickness of several atomic layers, the peak of the dangling bond is not so strong, but manifests itself as only a weak structure, as shown in ref. [3]. It should also be noted that the total state density and the partial state density without H orbital components for Si,3H,, and Si,,H,, are similar to each other. This indicates that the hydrogen atoms used for the bond saturation do not cause any extra peak of the hydrogen states, and that the Si-H bond very much resembles the Si-Si bond. Corresponding to this, the Mu&ken population analysis leads to an almost neutral charge on the hydrogen atoms (table 1). In contrast to these results, the state density of Si,,H,, (fig. 2b) is distorted as compared with the result of Schliiter et al. [ 141: the dangling bond band is much wider and splits into two peaks. The total state density and the partial state density without H orbital components are quite different to each other. We find excess Mulliken charges in the H atoms (table 1) and anomalous lowering of the Fermi energy (fig. 2b). This feature stems from the inclusion of incomplete six-membered rings of Si and consequently from the Table I Valence band widths, energy positions of the three peaks below the Fermi level and Mulliken charges of saturator H atoms for the three kinds of surface clusters Si,,H,,, Si,,H*, and Si,,H,,: the calculated result for thin films is shown for comparison Cluster size
Valence band width
SiJ-L Si16H,, SiI,H21 Thin film ‘)
-I2 -I2 -I3 -13
a) Self-consistent
pseudopotential
Energy
position
1st peak
2nd peak
3rd peak
Mu&ken charge of saturator H
-2.3 - 1.5 -2.5 -3.3
-6.1 -5.6 -6.2 -8.0
- 9.0 -8.X -9.3 - 10.9
1.02 1.19 I.12 _
method
[ 141.
b)
osi
OH
( ) adatom
Fig. 3. Substrate surface cluster, Si,,H,, including Si(l) and B,2H,, without Si(I). D, and 0; denote the dangling bonds perpendicular to the surface. (b) The six kinds of chemisorption site are shown: (I) and (I’) are “one-fold coordinated sites”, while (II), (III) and (III”) are “three-fold coordinated sites”.
appearance of three neighboring SiH, units. The interaction between the H atoms of the SiH, units facing each other (fig. 1) seems to be very strong. A comparable situation is found in hydrogen chemisorption with neighboring H atoms (section4). The presence of only the complete “six-membered rings” in the cluster seems to be an important condition to reproduce an electronic structure similar to that of the infinite surface. This condition is satisfied by the Si,,H,, and Si,,H,, cluster, but not by the Si,,H,, cluster. As shown above, the truncation of the cluster does not introduce any significant artifacts to take into account the for Si,3H,s. The size of this cluster is manageable atomic rearrangement and the self-consistency of the potential. Therefore Si 13H,, is used to investigate the chemisorption systems Si( 111) 7 X 7/H,Cl in the subsequent sections.
4. Hydrogen chemisorption
onto the Si(ll1)
surface
The hydrogen chemisorption is investigated for two typical reconstruction models of the Si( 11 I) 7 X 7 surface, i.e. the vacancy model and the buckled model [7]. The chemisorption bond distance is assumed to be 2.80 au.
T. Hoshino. M. Tsukudu / Electronic structures on Si(lI1).
I
109
First, we discuss the H chemisorption in the vacancy model. For this model, the Si(1) atom is removed and the other atoms remain at their respective positions of the ideal surface. Fig. 4 shows the calculated total and local densities of states for the clusters Si,,H,, and Si,,H,, + H,(I), with and without hydrogen chemisorption. The H atoms (site (I)) sit on top of the Si(2) atoms (fig. 3b). We find the following facts from a comparison between figs. 4a and 4b: (1) The vertical dangling bond peak D, of the top Si(2) atom disappears dramatically. (2) A strong peak (A) of the state density emerges at about -10.5 eV, as the result of the strong Si(2)-H(1) chemisorption bond formation. A Mulliken population analysis for the clusters Si,, H,, (nonrelaxed) and Si,,H,, + H,(I) (table 2) shows that the chemisorptive bond Si(2)-H(1) is slightly ionic and that the charge is transferred from the Si(2) atoms to the H(1) atoms. (3) Peak (B) at about - 12.5 eV is remarkably enhanced due to chemisorption of H(1) atoms. These features are not unchanged even though the displacement around the vacancy is taken into account. The level positions of the Si-H chemisorption bond agree very well with the calculated result for the slab of finite thickness [ 151. We have also calculated the electronic structures of the clusters Si,,H,S + H,(I) + H(II1) and Si,,H,S + H,(I) + H,(III’) where the chemisorbed H atoms exist not only at the site (I), but also at the vacant site, alone or in three-fold, respectively (fig. 5). For the Si,,H,S + H,(I) + H(II1) cluster, the H(II1) atom
-
IO.-____ -.-
(a)
Total.
A
;93 S id-45
‘;,
IO.- -.-
(b)
H(I)
1
c al
5 & I-&+ H& I)
n 5.-
Energy(eV) Fig. 4. Total and local state densities of the vacancy model cluster: (a) without chemisorption, (b) with hydrogen chemisorption at sites (I). The vertical line indicates the highest occupied
and level.
Table 2 (a) Mulliken
charges
for various
clusters
with substrate
Si,,H,,
Adatoms
Si(l) Si(2) Si(3) Si(4) WI) X(1’. II)
(b) Mulliken
None
H,(I)+H(I’)@
cl,(I,+cl(I’)
3.94 3.97 4.01 3.92 _
3.x9 3.96 4.01 3.90 1.00 1.01
3.52 3.69 4.23 4.07 7.14 7.14
charges
for various
clusters
with substrate
a)
CI,(II) 3.71 3.12 4.15 4.04 7.25
Si,,H,,
Adatoms
Si( 2) Si(3) Si(4) X(I) X(III,
III’, III”)
None
He,(I)
H,(I)+
3.93 3.92 4.10 _ -
3.83 3.91 4.11 1.09 _
3.94 3.93 3.96
H(III)
I .os 1.04
‘) Buckled model. The Si(2) atoms are relaxed outward inward by 0.5 au.
H,(I)+H,(III’)
cI,(I)+cI(III”)
3.94 3.63 3.91 I .05 1.35
3.78 4.10 3.96 7.18 7.20
by 0.5 au, while the Si( I) atom is relaxed
sits just on the vacant site. On the other hand, for the Si,,H,, + H,(I) + H,(III’) cluster, the bond length Si(3)-H(II1’) is assumed to be 2.80 au (fig. 3b). It should be noted that peak (B) decreases with increasing H chemisorption on the vacant site. We’ find from table 2 that in the cluster Si,,H,, + H&I) + HJIII’), the bonding among the three neighboring H(II1’ ) atoms is very strong and a lot of charge is transferred from the Si(3) atoms to the H(II1’) atoms. The bonding states among H(II1’) atoms are found around the valence band bottom (fig. 5b). Next we discuss the calculated results for the H chemisorption in the buckled model [7]. The present calculation takes only account of the ring-like buckling of surface atoms: the Si(1) and Si(2) atoms are relaxed outward and inward by 0.5 au, respectively. Fig. 6 shows the calculated total and local densities of states for the clusters Si,,H,S and Si,,H,, + H,(I)+ H(I’), with and without H chemisorption. The level positions of the Si-H chemisorption bond are almost the same as those of the vacancy model. Comparison between figs. 6a and 6b clearly demonstrates the disappearane of the vertical dangling bonds D, and D“ of the top Si atoms (fig. 3a), and the formation of the
I”-
_..._
H(l) f+(m)
VI
5.
Fig. 5. Total and local state densities of the vacancy model cluster: (a) with hydrogen c&misorption at sites (I) and (III), and (b) with hydrogen chemisorption at sites (I) and (III’). The vertical line indicates the highest occupied level.
r -Total . . Shf ____
S&?J
%3&5
-.-“-
H(I) H(r)
n
(b)
Sii$!# f-kJtff)+ Hfl’)
Fig. 6. Total and local state densities of the buckled model cluster: (a) without chemkorption, and (b) with hydrogen &em&or&on at sites (I) and (I’). The vertical line indicates the highest occupied IeveI.
-15.
-10. Energy (eV)
-5.
0
Fig. 7. Total and local state densities of the ideal cluster Si,,H,,: (a) without chemisorption, and (b) with hydrogen chemi~~tion at sites (I) and (I’). The vertical line indicates the highest occupied level.
chemisorption bond at about - 10.5 eV. It is very similar to that of the vacancy model although the chemisorbed H(1) atom is neutral compared with that in the vacancy model Si,,H,, + H,(I) (table2). However, we should notice that the peak (B) is split into a double peak, which is quite different from the sharp intense peak at - 12.5 eV in the cluster Si,,H,5 + H,(I). These features are insensitive to the amplitude of surface buckling and remain unchanged even without buckling (fig. 7).
5. Chlorine chemisorption
onto the Si( 111) 7 X 7 surface
We discuss the electronic structure of the Si( 111) 7 X 7 surface with chlorine chemisorption on different sites. First, we discuss the chlorine chemisorption in the vacancy model. Fig. 8a shows the state density for the cluster Si,,H,, + Cl,(I) + Cl(II1”). In this model six Cl atoms (Cl(I)) are chemisorbed on the Si(2) with bond length 3.80 au and a single Cl atom (Cl(II1”)) sits at 2.0 au, directly above the vacant site. There is no displacement around the vacancy. In this case, the distance (S.&j au) of the Cl(II1”)-Si(3) bond is slightly longer than the sum of the Si + and Cl _ ionic radii, 5.30 au. Peak (C) in the figure consists mainly of the 3p,, 3p,
Density
of
states
Table 3 Value of electronegativity
for H, Si and Cl atoms by Pauling
H
Si
Cl
2.1
1.8
3.0
[21]
chemisorbed at 3.80 au directly above the top Si atoms. As shown in fig. 8b, the two sharp peaks are induced by the chlorine chemisorption: peak (C) is made up of the Cl(I) 3p,, 3py and Cl(1’) 3p,, 39,. orbitals, and peak (B) of the Cl(I)-Si(2) and Cl(I’)-Si(1) bond orbitals. This agrees very well with the calculated result by Schltiter et al. [14] for the covalent site chemisorption model Si( 111) 1 X l/Cl(I) with a self-consistent pseudopotential method. We have also calculated the electronic structure of the cluster Si,jH,, + Cl,(II) where the three Cl(I1) atoms exist at 2.00 au above the center of the triangle formed by the Si(1) and the two Si(2) atoms (fig. 3b). We find a quite different peak structure (fig. 8c) as compared with those in figs. 8a and 8b. There is no reminiscence of the double peak (B) and (C). The calculated result is also very similar to that for the ionic chemisorption model Si( 111) 1 X l/Cl(II) by Schltiter et al. [16]. On the basis of the present numerical results, it is shown that only the vacancy model with chemisorbed Cl atoms both on sites (I) and (III”) can explain the polarization dependence of the UPS spectra of Si( 111) 7 X 7/Cl obtained by Rowe et al. [5] (see section6). In closing this section it should be noted that the Cl atoms are always more negative than the chemisorbed H atoms (table 2). It is a natural result, judging from the electronegativity of the H, Si and Cl atoms (table3). Further, we notice from table2 that the Cl(I1) and Cl(II1”) atoms in the three-fold coordinate (ionic) sites are more negatively ionic than the Cl(I) and Cl(1’) atoms in the one-fold coordinated (covalent) sites. It should be remarked that the Si atoms near the chemisorbed atoms are also positively charged.
6 . Summary and discussion By using the self-consistent DV-Xa cluster calculation, we have elucidated the electronic structures of. Si( 11 l)/H,Cl. It has been shown that the ideal surface cluster including only thirteen silicon atoms gives a good agreement with the band calculation by Schltiter et al. [ 141, and that the cluster method is very powerful to investigate the feature of a chemisorptive bond peculiar to surface vacancy and buckling. As for the hydrogen chemisorption, the vacancy model of Si,,H,S + H,(I) explains the observed peaks in the UPS of Si( 111) 7 X 7/H by Sakurai and Hagstrum [4]. A strong peak (A) (- 10.5 eV) and a subsidiary peak (B) (- 12.5
eV) correspond to those at - 10.0 and - 12.0 eV in the UPS result [4]. The enhancement of the subsidiary peak (B) (- 12.5 ev), however, is not observed for the vacancy models Si,,H,, + H,(I) + H(II1) and Si,,H,S + H,(I) + H,(III’) with the excess H adatoms on the vacant site (fig. 5). The calculated densities of states for such model clusters are not similar to the UPS spectra. Also, peak (B) is not enhanced in the buckled model. It should be remarked that the buckled model has a decrease of the state density in the valley region (- 11.5 eV) between two che~so~tion peaks (- 10.5 and - 12.5 eV). Such a decrease has never been found in the observed UPS [4]. The numerical results for the chlorine chemisorption as well as the hydrogen chemisorption may support the vacancy model, peak (C) consists mainly of p,, p, states of Cl(I) and peak (B) is made up of three parts, B,, I$ and B,. Peaks B, and B, consist of p,-like bonding states between Cl(I) and Si(2) atoms with a considerable admixture of Si(2)-Si(3). In contrast to this, peak B, consists of px, py-like bonding states between Cl(II1”) and Si(3) atoms. These features correspond to the polarization dependent UPS results of Si( 111) 7 X 7/Cl by Rowe et al. 151.On the other hand, for the buckled model peak (B) consists of only the orbital of Cl(I)-Si(2). It corresponds to the UPS results of Si(ll1) 2 X l/Cl [5]. This is easily understood because the buckling model of the Si( 111) 7 X 7 surface has a local atomic ~r~gement very similar to that of the Si( 111) 2 X 1 buckling model by Haneman [ 171. Thus the vacancy model which has chemisorption on the appropriate sites shows satisfactory agreement with the observed UPS results of Si( 111) 7 X 7/H,Cl. The feature of UPS data for Ge(l1 l)/Cl [5] is very similar to the calculated result for Si( 11 l)/Cl(II) which is characteristic for chemisorption on the three-fold coordinated site (II) (section5). This may show that for Ge( 11l)/Cl, the Cl atom is chemisorbed at the three-fold coordinated (ionic) site rather than at the one-fold coordinated (covalent) site. For Si(l1 l)/Ag, it has been suggested from experiments with both the ISS (ion scattering spectroscopy) [IS] and the LEED/CMTA (low. energy electron diffraction constant momentum transfer averaging) technique 1191, that the Ag atoms are embedded in the first double layer of Si. A comparison between our calculated results for the. electronic structure and the UPS data 1201supports the experimental proposal. This will be published in.2 subsequent paper. In order to give the theoretical justification of the chemisorption geometries for various systems, we must perform the total energy minimization. Such a calculation cannot be done at the present stage. However, we may expect that, as the strength of the covalent bond between the substrate atoms becomes weaker and the electronegativity difference between the substrate atom and the adatom is larger, the adatom tends to chemisorb at the ionic site rather than at the covalent site. Finally, we must discuss the possibility of the existence of vacancies on the Si( 111) surface. Both the present and the previous calculations obviously give a justification of the vacancy model. If that is the case, one may well have the
simple question why the vacancies are stable on the Si( 111) surface. To answer this problem, we are now calculating the formation energy of surface vacancies, based on the chemical pseudopotential method 191. Our preliminary result shows that the energy loss to form a single vacancy is not so large if the
displacement around the vacancy is taken into account. The interaction among the vacancies on the three sites arranged in a regular triangular form [2] may stabilize vacancies on the Si( 111) surface.
We would like to thank Professor S. Sugano, Professor K. Nakamura and Dr. S. Ohnishi for valuable discussions. The numerical calculations were performed on M-180 and M-200H systems at the IMS Computer Centre. We acknowledge a Grant-in-Aid from the Ministry of Education, Science and Culture. References (11 R.E. Schher and H.E. Farnsworth, J. Chem. Phys. 30 ( 1959) 917. [2] S. Ino, Japan. 3. Appl. Phys. 19 (1980) 1277. [3] J.E. Rowe and H. Ibach, Phys. Rev. Letters 32 (1975) 421: DE. Eastman, F.J. Himpsel. J.A. Knapp and K.C. Pandey, in: Proc. 14th Intern. Conf. on the Physics of Semiconductors, 1978, p. 1059. DE. Eastman, J. Vacuum Sci. Technol. 17 (1980) 492. [4] T. Sakurai and H.D. Hagstrum, Phys. Rev. 12 (1975) 5349. [5] J.E. Rowe. G. Margaritondo and S.B. Christman, Phys. Rev. Bl6 (1977) 1581. [h] J.E. Rowe and H. Ibach, Phys. Rev. Letters 31 (1973) 102: J.E. Rowe. H. Ibach and H. Froitzheim, Surface Sci. 4X (1975) 44. [7] D.J. Chadi, R.S. Bauer. R.H. Williams, G.V. Hansson. R.Z. Bachrach, J.C. Millkelsson. F. Houzay. G.M. tiuichar, R. Pinchaux and Y. Petroff, Phys. Rev. Letters 44 (1980) 799. [X] J.J. Lander and J. Morrison, J. Appl. Phys. 34 (1963) 1403. [9] K. Suzuki and T. Hoshino, J. Phys. Sot. Japan 49 Suppl. A (1980) 1055. [IO] H. Adachi, M. Tsukada and C. Satoko, J. Phys. Sot. Japan 45 (197X) X75: C. Satoko. M. Tsukada and H. Adachi, J. Phys. Sot. Japan 45 (197X) 1333: M. Tsukada, C. Satoko and H. Adachi, J. Phys. Sot. Japan 4X (1980) 200. [I I] K. Nakamura, T. Hoshino, M. Tsukada S. Ohm&i and S. Sugano, J. Phys. Cl4 (1981) 2165. [ 121 T. Hoshino and M. Tsukada, Solid State Commun. 38 (1981) 23 I ; M. Tsukada and T. Ho&no, Intern. J. Quantum Chem., to be published. [I31 J.D. Joannopoulos and F. Yndurain, Phys. Rev. BIO (1974) 5164; V.T. Rajan and F. Yndurain, Solid State Commun. 20 (1976) 309. [I41 M. Schltiter, J.R. Chelikowsky, S.G. Louie and M.L. Cohen, Phys. Rev. Bl2 (1975) 4200. [IS] KC. Pandey, J. Vacuum Sci. Technol. I5 (1978) 440. [ 161 M. Schltiter, J.E. Rowe, G. Margaritondo, T. M. Ho and M.L. Cohen, Phys. Rev. Letters 37 ( 1976) 632. [17] D. Iianeman, Phys. Rev. 121 (1961) 1093. [18] M. Saitoh, F. Shoji, K. Oura and T. Hanawa, Japan. J. Appl. Phys. 19 (1980) L42. [19] Y. Terada, T. Yoshiaawa, K. Oura and T. Hanawa, Japan. J. Appl. Phys. 20 (1081) L333. [20] G.V. Hansson, R.Z. Bachrach and R.S. Bauer, J. Phys. Sot. Japan 49 Suppl. A (1980) 1043. 1211 L. Pauhng, The Nature of the Chemical Bond (Cornell Univ. Press, New York, 1960) ch. 3.
Surface
Science
115 (1982) lCM-
I16
~o~~-H~l~a~d Pubk&ing Campany
ELECTRONIC STRUCTURES ON Si(ll1) SURFACE I. Si(ll1) 7 X Y/H,Cl
Received
18September
OF CHEMISORPTION
SYSTEMS
I981
Electronic structures of chemisorption on Si( 1I l)/W,Cl are investigated by the first principle DV-Xa cluster method. The calculations are carried out for chemisorption on different sites. based on the Si,,H,, cluster, and the effect of surface vacancy and buckling on the electronic structure is examined in detail. The present calculation shows that the Sit,H,, surface cluster reproduces very well the more sophisticated band calculation for the Si( I I I) surface. It is concluded that the vacancy model with chemisorbed atoms at appropriate sites is reasonable to interpret the observed UPS of Si( I I I) ?X7/H.C1. The charge transfer between the substrate atom and the adatom depends strongly both on the chemisorption sites and on the electronegativity difference.
I. Introchrction Since Schlier and Farnsworth [l] found the long-period reconstruction of the clean Si( 111) 7 X 7 surface in a LEED pattern, many experiments such as RHEED {2], UPS [3-51 and LEELS [6] have been made to investigate the supperlattice structure. Numerous distinct surface structural models [2,7,8] for Si( 111) 7 X 7 have been proposed to explain the experimental observations. In spite of the long history of research, however, no structure model has been established. It is a very interesting problem which still awaits theoretical elucidation. The direct theoretical approach [7,9] of determining the surface structure is to execute the total energy ~i~~tion of the surface system. The method has been carried out for some simpler systems, such as Si(ll1) 2 X 1 or Si(OO1) 2 X 1 surfaces. At present, however, it seems to be very difficult to apply the same method to complex surfaces such as Si( 111) 7 X 7. On the other hand, the intermediate theoretical approach of determining the surface structure is to calculate the electronic structure for the assumed structure models, and to compare them with the UPS and LEELS experimental results. From this point of view, we have already applied the DV-Xcx cluster method [lo] to investigate the electronic structures of the clean Si(l11) 7 X 7 surface and the chemisorption systems Si( 111) 7 X 7/H,Cl. The calculated results for oo39-6ozg/gzjoooo-ooarti%o2,75
0 1982 North-Holland
the electronic structures for different structural models of the clean Si( 111) 7 X 7 surface [ 1I] and the preliminary results for Si( 111) 7 X 7/H,Cl[ 12) were reported. The purpose of the present paper is to supplement the previous paper and to give further detailed theoretical information on Si(ll1) 7 X 7/H,Cl. A comparison of the present calculations with the observed UPS indicates the presence of vacancies on the Si( 111) 7 X 7 surface. We have also calculated the electronic structures of Si( 11l)/Ag to determine the chemisorption site at the high temperature phase. The result will be published in a forthcoming paper. The reason for adopting the DV-Xcu cluster method [lo] stems from several of its aspects. First, the cluster method is adaptable to calculate the electronic structures which are determined principally by the local atomic ~rangement. It is very difficult to apply theoretical band c~culat~ons to a complex system such as the Si(ll1) 7 X 7 surface. Secondly, the technique of the DV-Xa method allows one to take into account the directional properties of the
.I -
Si+H
5.
Sit6kt
-15.
% Htl Fig. I. Structure
of the clusters
Si,,H,S,
Si,,H,,
-1Q Energy(eV)
-5.
and Si,,H,,
Fig. 2. Total density and partial state density without hydrogen in fig. 1. The arrow indicates the highest occupied level.
components
for the three clusters
potential which are well known to be more important in the surface than in the bulk. The DV-Xa method [lo] is already established, as far as one electron energy is concerned. The plan of this paper is as follows. Section2 gives a short account of the DV-Xcr method [lo]. Section 3 presents numerical results for the electronic structure of three kinds of ideal Si(ll1) surface clusters, Si,,H,,, Si,,H,, and Si,,H,, (fig. l), and discusses the effect of size and structure of the cluster on the electronic structure. H atoms are attached so as to appro~mately saturate the dangling bonds artificially produced by the removal of the cluster from the substrate. Sections 4 and 5 describe numerical results of the electronic structures of Si( 111) 7 X 7/H,Cl respectively. The characteristics in the electronic structure, related to surface vacancy and buckling, are clarified in detail. Section6 gives a summary and discussion.
2. DV-Xar method The present calculations are based on the DV-Xa method which is already reported in previous papers [lo]. The effective exchange-correlation potential is given by statistical approximation and is written as
f’ktd = --3#/8~)
~(r)]“~,
where the quantity cxis the only parameter used in the model and is fixed to be 0.7 in all calculations of this paper. The Coulomb potential is calculated self-consistently using the self-consistent charge approximation [lo]. The matrix elements of the Hamiltonian and overlap integrals are calculated by the random sampling method. It enables us to take into account the non-spherical part of the potential. Numerical atomic basis functions ls-3p for Si and Cl, and 1s for H are used to expand the molecular orbitals.
3. Electronic
structure of ideal Si( 111) surface
To study the effect of size and structure of the cluster, the electronic structures are calculated for the three clusters Si,,H,,, Si,,H,, and Si,,H,,, simulating the ideal Si(ll1) surface. The surface clusters, which have C,, symmetry (fig. l), are removed from the ideal semi-infinite surface. is the smallest one which includes three “sixThe first cluster, Si,,H,,, membered rings of bonds” in the plane parallel to the surface. The third rings cluster, Si,9H2,, is the smallest one which includes three “six-membered of bonds” in the planes parallel and perpendicular to the surface. The importance of six-membered rings of bonds in the diamond lattice is known very well [ 131. The Si-Si and Si-H bond lengths are chosen to be the same as those in the bulk (4.44 au) and in the SiH, molecule (2.80 au), respectively. The
total numbers of electrons are determined from the charge neutrality condition of the respective clusters. Fig. 2 shows the calculated total state density and that without the H components for the three clusters. The overall features of the state densities for Si,,H,, and Si,,H,, are similar to those for thin films as obtained by Schltiter et al. [14]. The valence band width, the appearance and the gross feature of three major peaks in the valence band (table 1) correspond well with the characteristics of the UPS for the Si( 111) surface[3]. The sharp peak around the Fermi level (the highest occupied level) of the state densities for Si,,H,5 and Si,9H2, corresponds to the band D, (D”) of-the dangling bonds (fig. 3a). It is noted that the state density of the dangling bonds is enhanced because of the larger proportion of top Si atoms in the present clusters. The relative weight of the information peculiar to the top layer in the UPS structure considerably changes according to the escape depth of the photoelectron. If the escape depth is deeper than the thickness of several atomic layers, the peak of the dangling bond is not so strong, but manifests itself as only a weak structure, as shown in ref. [3]. It should also be noted that the total state density and the partial state density without H orbital components for Si,3H,, and Si,,H,, are similar to each other. This indicates that the hydrogen atoms used for the bond saturation do not cause any extra peak of the hydrogen states, and that the Si-H bond very much resembles the Si-Si bond. Corresponding to this, the Mu&ken population analysis leads to an almost neutral charge on the hydrogen atoms (table 1). In contrast to these results, the state density of Si,,H,, (fig. 2b) is distorted as compared with the result of Schliiter et al. [ 141: the dangling bond band is much wider and splits into two peaks. The total state density and the partial state density without H orbital components are quite different to each other. We find excess Mulliken charges in the H atoms (table 1) and anomalous lowering of the Fermi energy (fig. 2b). This feature stems from the inclusion of incomplete six-membered rings of Si and consequently from the Table I Valence band widths, energy positions of the three peaks below the Fermi level and Mulliken charges of saturator H atoms for the three kinds of surface clusters Si,,H,,, Si,,H*, and Si,,H,,: the calculated result for thin films is shown for comparison Cluster size
Valence band width
SiJ-L Si16H,, SiI,H21 Thin film ‘)
-I2 -I2 -I3 -13
a) Self-consistent
pseudopotential
Energy
position
1st peak
2nd peak
3rd peak
Mu&ken charge of saturator H
-2.3 - 1.5 -2.5 -3.3
-6.1 -5.6 -6.2 -8.0
- 9.0 -8.X -9.3 - 10.9
1.02 1.19 I.12 _
method
[ 141.
b)
osi
OH
( ) adatom
Fig. 3. Substrate surface cluster, Si,,H,, including Si(l) and B,2H,, without Si(I). D, and 0; denote the dangling bonds perpendicular to the surface. (b) The six kinds of chemisorption site are shown: (I) and (I’) are “one-fold coordinated sites”, while (II), (III) and (III”) are “three-fold coordinated sites”.
appearance of three neighboring SiH, units. The interaction between the H atoms of the SiH, units facing each other (fig. 1) seems to be very strong. A comparable situation is found in hydrogen chemisorption with neighboring H atoms (section4). The presence of only the complete “six-membered rings” in the cluster seems to be an important condition to reproduce an electronic structure similar to that of the infinite surface. This condition is satisfied by the Si,,H,, and Si,,H,, cluster, but not by the Si,,H,, cluster. As shown above, the truncation of the cluster does not introduce any significant artifacts to take into account the for Si,3H,s. The size of this cluster is manageable atomic rearrangement and the self-consistency of the potential. Therefore Si 13H,, is used to investigate the chemisorption systems Si( 111) 7 X 7/H,Cl in the subsequent sections.
4. Hydrogen chemisorption
onto the Si(ll1)
surface
The hydrogen chemisorption is investigated for two typical reconstruction models of the Si( 11 I) 7 X 7 surface, i.e. the vacancy model and the buckled model [7]. The chemisorption bond distance is assumed to be 2.80 au.
T. Hoshino. M. Tsukudu / Electronic structures on Si(lI1).
I
109
First, we discuss the H chemisorption in the vacancy model. For this model, the Si(1) atom is removed and the other atoms remain at their respective positions of the ideal surface. Fig. 4 shows the calculated total and local densities of states for the clusters Si,,H,, and Si,,H,, + H,(I), with and without hydrogen chemisorption. The H atoms (site (I)) sit on top of the Si(2) atoms (fig. 3b). We find the following facts from a comparison between figs. 4a and 4b: (1) The vertical dangling bond peak D, of the top Si(2) atom disappears dramatically. (2) A strong peak (A) of the state density emerges at about -10.5 eV, as the result of the strong Si(2)-H(1) chemisorption bond formation. A Mulliken population analysis for the clusters Si,, H,, (nonrelaxed) and Si,,H,, + H,(I) (table 2) shows that the chemisorptive bond Si(2)-H(1) is slightly ionic and that the charge is transferred from the Si(2) atoms to the H(1) atoms. (3) Peak (B) at about - 12.5 eV is remarkably enhanced due to chemisorption of H(1) atoms. These features are not unchanged even though the displacement around the vacancy is taken into account. The level positions of the Si-H chemisorption bond agree very well with the calculated result for the slab of finite thickness [ 151. We have also calculated the electronic structures of the clusters Si,,H,S + H,(I) + H(II1) and Si,,H,S + H,(I) + H,(III’) where the chemisorbed H atoms exist not only at the site (I), but also at the vacant site, alone or in three-fold, respectively (fig. 5). For the Si,,H,S + H,(I) + H(II1) cluster, the H(II1) atom
-
IO.-____ -.-
(a)
Total.
A
;93 S id-45
‘;,
IO.- -.-
(b)
H(I)
1
c al
5 & I-&+ H& I)
n 5.-
Energy(eV) Fig. 4. Total and local state densities of the vacancy model cluster: (a) without chemisorption, (b) with hydrogen chemisorption at sites (I). The vertical line indicates the highest occupied
and level.
Table 2 (a) Mulliken
charges
for various
clusters
with substrate
Si,,H,,
Adatoms
Si(l) Si(2) Si(3) Si(4) WI) X(1’. II)
(b) Mulliken
None
H,(I)+H(I’)@
cl,(I,+cl(I’)
3.94 3.97 4.01 3.92 _
3.x9 3.96 4.01 3.90 1.00 1.01
3.52 3.69 4.23 4.07 7.14 7.14
charges
for various
clusters
with substrate
a)
CI,(II) 3.71 3.12 4.15 4.04 7.25
Si,,H,,
Adatoms
Si( 2) Si(3) Si(4) X(I) X(III,
III’, III”)
None
He,(I)
H,(I)+
3.93 3.92 4.10 _ -
3.83 3.91 4.11 1.09 _
3.94 3.93 3.96
H(III)
I .os 1.04
‘) Buckled model. The Si(2) atoms are relaxed outward inward by 0.5 au.
H,(I)+H,(III’)
cI,(I)+cI(III”)
3.94 3.63 3.91 I .05 1.35
3.78 4.10 3.96 7.18 7.20
by 0.5 au, while the Si( I) atom is relaxed
sits just on the vacant site. On the other hand, for the Si,,H,, + H,(I) + H,(III’) cluster, the bond length Si(3)-H(II1’) is assumed to be 2.80 au (fig. 3b). It should be noted that peak (B) decreases with increasing H chemisorption on the vacant site. We’ find from table 2 that in the cluster Si,,H,, + H&I) + HJIII’), the bonding among the three neighboring H(II1’ ) atoms is very strong and a lot of charge is transferred from the Si(3) atoms to the H(II1’) atoms. The bonding states among H(II1’) atoms are found around the valence band bottom (fig. 5b). Next we discuss the calculated results for the H chemisorption in the buckled model [7]. The present calculation takes only account of the ring-like buckling of surface atoms: the Si(1) and Si(2) atoms are relaxed outward and inward by 0.5 au, respectively. Fig. 6 shows the calculated total and local densities of states for the clusters Si,,H,S and Si,,H,, + H,(I)+ H(I’), with and without H chemisorption. The level positions of the Si-H chemisorption bond are almost the same as those of the vacancy model. Comparison between figs. 6a and 6b clearly demonstrates the disappearane of the vertical dangling bonds D, and D“ of the top Si atoms (fig. 3a), and the formation of the
I”-
_..._
H(l) f+(m)
VI
5.
Fig. 5. Total and local state densities of the vacancy model cluster: (a) with hydrogen c&misorption at sites (I) and (III), and (b) with hydrogen chemisorption at sites (I) and (III’). The vertical line indicates the highest occupied level.
r -Total . . Shf ____
S&?J
%3&5
-.-“-
H(I) H(r)
n
(b)
Sii$!# f-kJtff)+ Hfl’)
Fig. 6. Total and local state densities of the buckled model cluster: (a) without chemkorption, and (b) with hydrogen &em&or&on at sites (I) and (I’). The vertical line indicates the highest occupied IeveI.
-15.
-10. Energy (eV)
-5.
0
Fig. 7. Total and local state densities of the ideal cluster Si,,H,,: (a) without chemisorption, and (b) with hydrogen chemi~~tion at sites (I) and (I’). The vertical line indicates the highest occupied level.
chemisorption bond at about - 10.5 eV. It is very similar to that of the vacancy model although the chemisorbed H(1) atom is neutral compared with that in the vacancy model Si,,H,, + H,(I) (table2). However, we should notice that the peak (B) is split into a double peak, which is quite different from the sharp intense peak at - 12.5 eV in the cluster Si,,H,5 + H,(I). These features are insensitive to the amplitude of surface buckling and remain unchanged even without buckling (fig. 7).
5. Chlorine chemisorption
onto the Si( 111) 7 X 7 surface
We discuss the electronic structure of the Si( 111) 7 X 7 surface with chlorine chemisorption on different sites. First, we discuss the chlorine chemisorption in the vacancy model. Fig. 8a shows the state density for the cluster Si,,H,, + Cl,(I) + Cl(II1”). In this model six Cl atoms (Cl(I)) are chemisorbed on the Si(2) with bond length 3.80 au and a single Cl atom (Cl(II1”)) sits at 2.0 au, directly above the vacant site. There is no displacement around the vacancy. In this case, the distance (S.&j au) of the Cl(II1”)-Si(3) bond is slightly longer than the sum of the Si + and Cl _ ionic radii, 5.30 au. Peak (C) in the figure consists mainly of the 3p,, 3p,
Density
of
states
Table 3 Value of electronegativity
for H, Si and Cl atoms by Pauling
H
Si
Cl
2.1
1.8
3.0
[21]
chemisorbed at 3.80 au directly above the top Si atoms. As shown in fig. 8b, the two sharp peaks are induced by the chlorine chemisorption: peak (C) is made up of the Cl(I) 3p,, 3py and Cl(1’) 3p,, 39,. orbitals, and peak (B) of the Cl(I)-Si(2) and Cl(I’)-Si(1) bond orbitals. This agrees very well with the calculated result by Schltiter et al. [14] for the covalent site chemisorption model Si( 111) 1 X l/Cl(I) with a self-consistent pseudopotential method. We have also calculated the electronic structure of the cluster Si,jH,, + Cl,(II) where the three Cl(I1) atoms exist at 2.00 au above the center of the triangle formed by the Si(1) and the two Si(2) atoms (fig. 3b). We find a quite different peak structure (fig. 8c) as compared with those in figs. 8a and 8b. There is no reminiscence of the double peak (B) and (C). The calculated result is also very similar to that for the ionic chemisorption model Si( 111) 1 X l/Cl(II) by Schltiter et al. [16]. On the basis of the present numerical results, it is shown that only the vacancy model with chemisorbed Cl atoms both on sites (I) and (III”) can explain the polarization dependence of the UPS spectra of Si( 111) 7 X 7/Cl obtained by Rowe et al. [5] (see section6). In closing this section it should be noted that the Cl atoms are always more negative than the chemisorbed H atoms (table 2). It is a natural result, judging from the electronegativity of the H, Si and Cl atoms (table3). Further, we notice from table2 that the Cl(I1) and Cl(II1”) atoms in the three-fold coordinate (ionic) sites are more negatively ionic than the Cl(I) and Cl(1’) atoms in the one-fold coordinated (covalent) sites. It should be remarked that the Si atoms near the chemisorbed atoms are also positively charged.
6 . Summary and discussion By using the self-consistent DV-Xa cluster calculation, we have elucidated the electronic structures of. Si( 11 l)/H,Cl. It has been shown that the ideal surface cluster including only thirteen silicon atoms gives a good agreement with the band calculation by Schltiter et al. [ 141, and that the cluster method is very powerful to investigate the feature of a chemisorptive bond peculiar to surface vacancy and buckling. As for the hydrogen chemisorption, the vacancy model of Si,,H,S + H,(I) explains the observed peaks in the UPS of Si( 111) 7 X 7/H by Sakurai and Hagstrum [4]. A strong peak (A) (- 10.5 eV) and a subsidiary peak (B) (- 12.5
eV) correspond to those at - 10.0 and - 12.0 eV in the UPS result [4]. The enhancement of the subsidiary peak (B) (- 12.5 ev), however, is not observed for the vacancy models Si,,H,, + H,(I) + H(II1) and Si,,H,S + H,(I) + H,(III’) with the excess H adatoms on the vacant site (fig. 5). The calculated densities of states for such model clusters are not similar to the UPS spectra. Also, peak (B) is not enhanced in the buckled model. It should be remarked that the buckled model has a decrease of the state density in the valley region (- 11.5 eV) between two che~so~tion peaks (- 10.5 and - 12.5 eV). Such a decrease has never been found in the observed UPS [4]. The numerical results for the chlorine chemisorption as well as the hydrogen chemisorption may support the vacancy model, peak (C) consists mainly of p,, p, states of Cl(I) and peak (B) is made up of three parts, B,, I$ and B,. Peaks B, and B, consist of p,-like bonding states between Cl(I) and Si(2) atoms with a considerable admixture of Si(2)-Si(3). In contrast to this, peak B, consists of px, py-like bonding states between Cl(II1”) and Si(3) atoms. These features correspond to the polarization dependent UPS results of Si( 111) 7 X 7/Cl by Rowe et al. 151.On the other hand, for the buckled model peak (B) consists of only the orbital of Cl(I)-Si(2). It corresponds to the UPS results of Si(ll1) 2 X l/Cl [5]. This is easily understood because the buckling model of the Si( 111) 7 X 7 surface has a local atomic ~r~gement very similar to that of the Si( 111) 2 X 1 buckling model by Haneman [ 171. Thus the vacancy model which has chemisorption on the appropriate sites shows satisfactory agreement with the observed UPS results of Si( 111) 7 X 7/H,Cl. The feature of UPS data for Ge(l1 l)/Cl [5] is very similar to the calculated result for Si( 11 l)/Cl(II) which is characteristic for chemisorption on the three-fold coordinated site (II) (section5). This may show that for Ge( 11l)/Cl, the Cl atom is chemisorbed at the three-fold coordinated (ionic) site rather than at the one-fold coordinated (covalent) site. For Si(l1 l)/Ag, it has been suggested from experiments with both the ISS (ion scattering spectroscopy) [IS] and the LEED/CMTA (low. energy electron diffraction constant momentum transfer averaging) technique 1191, that the Ag atoms are embedded in the first double layer of Si. A comparison between our calculated results for the. electronic structure and the UPS data 1201supports the experimental proposal. This will be published in.2 subsequent paper. In order to give the theoretical justification of the chemisorption geometries for various systems, we must perform the total energy minimization. Such a calculation cannot be done at the present stage. However, we may expect that, as the strength of the covalent bond between the substrate atoms becomes weaker and the electronegativity difference between the substrate atom and the adatom is larger, the adatom tends to chemisorb at the ionic site rather than at the covalent site. Finally, we must discuss the possibility of the existence of vacancies on the Si( 111) surface. Both the present and the previous calculations obviously give a justification of the vacancy model. If that is the case, one may well have the
simple question why the vacancies are stable on the Si( 111) surface. To answer this problem, we are now calculating the formation energy of surface vacancies, based on the chemical pseudopotential method 191. Our preliminary result shows that the energy loss to form a single vacancy is not so large if the
displacement around the vacancy is taken into account. The interaction among the vacancies on the three sites arranged in a regular triangular form [2] may stabilize vacancies on the Si( 111) surface.
We would like to thank Professor S. Sugano, Professor K. Nakamura and Dr. S. Ohnishi for valuable discussions. The numerical calculations were performed on M-180 and M-200H systems at the IMS Computer Centre. We acknowledge a Grant-in-Aid from the Ministry of Education, Science and Culture. References (11 R.E. Schher and H.E. Farnsworth, J. Chem. Phys. 30 ( 1959) 917. [2] S. Ino, Japan. 3. Appl. Phys. 19 (1980) 1277. [3] J.E. Rowe and H. Ibach, Phys. Rev. Letters 32 (1975) 421: DE. Eastman, F.J. Himpsel. J.A. Knapp and K.C. Pandey, in: Proc. 14th Intern. Conf. on the Physics of Semiconductors, 1978, p. 1059. DE. Eastman, J. Vacuum Sci. Technol. 17 (1980) 492. [4] T. Sakurai and H.D. Hagstrum, Phys. Rev. 12 (1975) 5349. [5] J.E. Rowe. G. Margaritondo and S.B. Christman, Phys. Rev. Bl6 (1977) 1581. [h] J.E. Rowe and H. Ibach, Phys. Rev. Letters 31 (1973) 102: J.E. Rowe. H. Ibach and H. Froitzheim, Surface Sci. 4X (1975) 44. [7] D.J. Chadi, R.S. Bauer. R.H. Williams, G.V. Hansson. R.Z. Bachrach, J.C. Millkelsson. F. Houzay. G.M. tiuichar, R. Pinchaux and Y. Petroff, Phys. Rev. Letters 44 (1980) 799. [X] J.J. Lander and J. Morrison, J. Appl. Phys. 34 (1963) 1403. [9] K. Suzuki and T. Hoshino, J. Phys. Sot. Japan 49 Suppl. A (1980) 1055. [IO] H. Adachi, M. Tsukada and C. Satoko, J. Phys. Sot. Japan 45 (197X) X75: C. Satoko. M. Tsukada and H. Adachi, J. Phys. Sot. Japan 45 (197X) 1333: M. Tsukada, C. Satoko and H. Adachi, J. Phys. Sot. Japan 4X (1980) 200. [I I] K. Nakamura, T. Hoshino, M. Tsukada S. Ohm&i and S. Sugano, J. Phys. Cl4 (1981) 2165. [ 121 T. Hoshino and M. Tsukada, Solid State Commun. 38 (1981) 23 I ; M. Tsukada and T. Ho&no, Intern. J. Quantum Chem., to be published. [I31 J.D. Joannopoulos and F. Yndurain, Phys. Rev. BIO (1974) 5164; V.T. Rajan and F. Yndurain, Solid State Commun. 20 (1976) 309. [I41 M. Schltiter, J.R. Chelikowsky, S.G. Louie and M.L. Cohen, Phys. Rev. Bl2 (1975) 4200. [IS] KC. Pandey, J. Vacuum Sci. Technol. I5 (1978) 440. [ 161 M. Schltiter, J.E. Rowe, G. Margaritondo, T. M. Ho and M.L. Cohen, Phys. Rev. Letters 37 ( 1976) 632. [17] D. Iianeman, Phys. Rev. 121 (1961) 1093. [18] M. Saitoh, F. Shoji, K. Oura and T. Hanawa, Japan. J. Appl. Phys. 19 (1980) L42. [19] Y. Terada, T. Yoshiaawa, K. Oura and T. Hanawa, Japan. J. Appl. Phys. 20 (1081) L333. [20] G.V. Hansson, R.Z. Bachrach and R.S. Bauer, J. Phys. Sot. Japan 49 Suppl. A (1980) 1043. 1211 L. Pauhng, The Nature of the Chemical Bond (Cornell Univ. Press, New York, 1960) ch. 3.