Ultramicroscopy 14 (1984) 8996 NorthHolland, Amsterdam
BONDING
89
GEOMETRIES
AT SEMICONDUCTOR
SURFACES
AND INTERFACES
M. SCHLUTER A T&T Bell Laboratories,
Murray
Hill, New Jersey
07974,
USA
Received 11 January 1984; presented at Workshop January 1984
Advances in the theoretical description of electronelectron and electronion interactions allow us to calculate without adjustable parameters atomic geometries for moderately complex systems. The theoretical methods are briefly reviewed and two examples are discussed.
1. Introduction and scope The knowledge of the atomic structure of surfaces and interfaces is regarded as fundamental in understanding the physics of such systems. There are, roughly speaking, three distinct approaches to obtaining the structural information: (i) The mainly experimental approach, using techniques such as, e.g., Xray diffraction [l] or absorption [2], electron microscopy [3] or ion scattering [4]. These approaches yield structural information usually without explicit theoretical calculations. (ii) The combined theoreticalexperimental approach, involving a comparison between structuresensitive experimental data and theoretical calculations of such data as, e.g., photoemission [5] and low energy electron diffraction [6]. (iii) The purely theoretical approach which relies on the calculation of total configurational energies and their minimization with respect to structural parameters. In this paper I shall focus on the last approach and discuss some recent developments in the field. There are a large variety of methods to obtain approximate total configurational energies. As to the choice of Hamiltonian, there are many schemes reaching from empirical to parameterfree schemes [7111. Similarly, there are many different ways to model the surfaceinterface systems: clusters [9], slabs [12], infinite halfspaces [13], etc. Here, I shall discuss a method which is based on the local density functional approximation (LDA) [14,15] to 03043991/84/$03.00 0 Elsevier Science Publishers (NorthHolland Physics Publishing Division)
describe electronelectron interactions and on the firstprinciple pseudopotential approximation [16] to describe electronion interactions. The calculations are done using supercell, i.e., periodically repeated slab geometries [12].
2. Computational
tools
The Density Functional Theory represents an attempt to integrate the lattice ion potential l/exl with the mutual interaction of electrons. The theory is based on the theorem by Hohenberg and Kohn [14] which states that all properties of the many particle ground state are given by the ground state electron density distribution p(r). The energy of a system of interacting electrons in an external nuclear potential can then be written as
HP1 = T,[Pl+~c,,,bl
where T,[p] is the kinetic acting electrons,
energy of the noninter
the usual Coulomb energy (where the electrons are treated as fixed and “ uncorrelated”), V,,, =  Z/r, the nuclear potential, and E,,[p] the exchange B.V.
correlation
energy,
teractions
which globally
contains
due to the Pauli exclusion
all in
principle
and
description
of the ground state.
It is general
practice,
to the otherwise correlated motion of electrons. As shown by Kohn and Sham [15], a variational solu
justification,
tion of eq. (1) can be obtained
by solving a set of
systems where relaxation
oneelectron
equations
is of
Schrodingertype
selfcon
sistently:
energies,
particular the
and
the
last
customarily
where
interpreted
significantly
for extended This
semiconductors cg = cc  c\ be
conduction valence
band state
band
state
is
as the energy gap. Empiri
that cg calculated
too small
[19].
in the LDA
It has recently
is
been
realized [20] that there is an additional contribution to the gap arising from a discontinuity in the
and
exchangecorrelation =
c occupied
The
14+)12.
electrons
states
occupation
Although of
the
states
4,(r)
is done
according to Fermi statistics with the eigenvalue c, serving as the energy for the purpose of determining
for
difference
occupied
cally it is known
p(r)
especially
formal
of eq. (2)
effects are negligible.
importance
eigenvalue
without
the eigenvalues
tween the first unoccupied
W)l G,(r) = c,+,(r)?
[Tt
as quasiparticle
where
although
to interpret
occupancy
manybody
of
problem
a state.
The
solution
is thus reduced
of
the
to the solu
tion of a set of oneparticle equations, not more difficult to solve than the Hartree problem. The difficulty tionals
lies in determining
the unknown
func
Wc[ PI
remedy
to the ThomasFermi
approximation (Local Density and V,, is given by regarding
methods,
an
Functional) to E,, a small neighbor
hood of the electron system as behaving like jellium, i.e., p = const, at the value of the local density. Then
there
is no
yet, and empirical
are still necessary
ad
when gap values are
were originally
introduced
to
simplify electronic structure calculations by eliminating the need to include atomic core states. Considerable
success
the band structure
izedplanewave functions
of
was achieved
and optical
in describing
properties
of semi
(OPW)
method
OPWtype
[23].
The
pseudopotentials
wave have,
however, a certain problem. The normalized pseudowave functions have the correct shape outside the core region, but incorrect is because
neglect
of the socalled
amplitudes.
This
“orthogonality
hole” puts too much charge inside the core region. The problem tions,
and expressions for the homogeneous electron gas exchangecorrelation energies can be used [17]. Local density functional theory is generally very successful
available
needed [21]. Pseudopotentials
groundstate.
is identified,
the formal justification for the use of pseudopotentials was initially based upon the orthogonal
aplr)
In analogy
justments
the problem
for adding holes or
Nparticle
conductors and simple metals with the use of empirical pseudopotentials [22]. On the other hand,
E,, [ p] and
v,c[Pl=
practical
potential
to the insulating
in predicting
structural
properties
of
molecules and solids. There are, however, systems such as 3rd row transition metal atoms, where the approximate form of exchange and correlation is less satisfactory [18]. Improved functionals, going beyond the local approximation, seem to be necessary in some of these cases to obtain a correct
since
potential.
is serious
in selfconsistent
it will cause
The problem
errors
calcula
in the Coulomb
can, in principle,
be over
come by going back to the original OPW scheme, but, the procedure is cumbersome and obviates most of the advantages of using a pseudopotential. The problem of the orthogonality hole, however, is not a necessary consequence of replacing an allelectron potential by a pseudopotential. Several practical schemes have recently been proposed, and in the following the “spirit” of such a scheme is outlined.
IU. Schluter / Bonding geometries at semiconductor surfaces and tnterfaces
We start with a “reference” atom for which we have a selfconsistent local density calculation, including core states (i.e., solutions for eqs. (1) and (2)) at hand. A pseudo wave function for any valence state of the atom need have just two properties to be consistent with our intentions: it should be nodeless, and it should, when normalized, become identical to the true valence wave function beyond some core radius R,. Such a function can be constructed in arbitrarily many ways. For any one pseudo wave function, the radial Schrodinger equation can be inverted to yield a pseudopotential which has this function as its eigenfunction at the correct eigenvalue. By this construction, it is clear that the pseudopotential and full potential are identical beyond R, and that inside R, the pseudopotential correctly simulates the scattering properties of the full potential at the eigenvalue energy and for the particular angular momentum under discussion. To be useful, the ionic (i.e., unscreened) portion of such “normconserving” pseudopotentials must be transferable to situations other than the atomic reference state. Such situations are, e.g., molecules, solids or excited atomic configurations. As shown in ref. [24], normconserving pseudopotentials satisfy an important phase shift egality and show exact transferability to first order in eigenvalue changes. The construction of any pseudopotential is naturally based on the assumption of a “frozen” core charge. It has been shown that the error in the total energy associated with this approximation is of second order in valence charge density changes [25]. Combined with the normconserving property, the pseudopotentials do thus not lead to any firstorder transferability error. For treating heavier atoms, relativistic effects can be absorbed into the core pseudopotential [24] which enables one to treat heavy atoms in a nonrelativistic band structure formalism. A set of these pseudopotentials for all elements in the periodic table (iH through 94Pu) has recently been prepared [24,26]. The potentials contain no adjustable parameters and faithfully reproduce the original “fullcore” atomic spectra over a large energy range. Bulk properties of many semiconductors, among them C, Si, Ge, GaAs, GaP, AlAs and Se have been successfully calculated, within a few percent error.
91
Surface structures of some of these semiconductors have also been studied. Work on insulators like NaCl or CsAu and on metals like Al, Be, W or even magnetic Fe exists. Similarly, there are studies on molecules like Si,, Cr, or MO,, and most of the recent work on surfaces, interfaces and defects is based on the use of normconserving local density pseudopotentials. For references the reader is referred to recent reviews [27,28].
3. Results and discussion In this section I shall discuss the results of total configurational energy studies on two different systems to exemplify the general approach. 3.1. The clean Si(l1 I) (2
X I)
surface
Clean semiconductor surfaces show involved reconstruction patterns [29]. Apart from the question as to what physical mechanism drives these reconstructions, the determination of the geometrical structure itself represents a challenging problem. The metastable Si(ll1) (2 X 1) surface which is obtained by cleaving has been actively studied [29] both experimentally and theoretically. For many years, the buckling model, originally introduced in 1961 by Haneman [30], has been the favorite model to explain this reconstruction. The buckling model is obtained when alternating rows of surface atoms are raised and lowered (e.g., in fig. la atom row 1 up and atom row 4 down). The buckling model is very intuitive since it can be viewed as a Peierls or JahnTeller symmetrybreaking distortion of the metallic (111) surface, rendering it semiconducting or at least decreasing the density of states at the Fermi level. Although there persisted doubts about the detailed interpretation within the buckling model of experiments (notably photoemission data), it was the work by Pandey [31,32] in 198182 which clearly showed the inappropriateness of this model for Si(ll1). Originally motivated to understand photoemission data, Pandey proposed a new surface topology, the rbonded chain (see fig. lb). With this new idea, total energy calculations were carried out by Northrup and Cohen [33,34], by
M. Schluter / Bonding geometries at semiconductor
92
surJaces and interJaces
BONDLENGTH
CONTRACTION
(ATOMIC
UNITS1
005
0 f 0 G
005
RELAXED
( I x I) SURFACE
t
2
/
MAGNETIC 2 x I SURFACE
‘1
020
____

:\_:___________ \
cHA,N&,/
025 Fig. (top) The gling
1. Perspective view of the ideally terminated Si(ll1) surface and the vbonded chain reconstruction model (bottom). (2 X 1) surface unit cell is indicated and the surface danbonds are emphasized (from ref. [35]).
0
30
0
351
,,./‘/’
MODEL
! I
I
0.6
Pandey [32] and by Nielsen, Martin, Chadi and Kunc [35], all within the computational frame discussed
in section 2. The results (some are shown
in fig. 2) confirmed models
fact, buckling polar
Pandey’s
did not stabilize surface.
[30], buckling
suspicions.
Si(ll1)
As originally
stated
In
on this homoby Haneman
would cause rehybridization
of the
atomic orbitals centered at the outermost atoms which, in turn, should cause charge transfer from the
receding
atoms. showed
The
atoms
to
the
outwardly
new selfconsistent
that this charge
transfer
4
0
RELAXATION
I
, 2
(AUTOMIC
I a
I 02
UNITS)
Fig. 2. Calculated total energy variations for various reconstruction models for the Si(ll1) (2 X 1) surface (from ref. [32]).
Buckling
significantly.
as such was resisted
0 SURFACE
I
displaced
calculations was strongly
[32] re
eV/atom, systems.
a common If buckling
is approximately
feature of covalently is introduced
offset
again.
bonded
this energy gain Nevertheless,
one
may suspect that certain metastable configurations can exist. Among the “standard” models, not containing topological changes, a nonbuckled, insulating surface with a (2 X 1) antiferromagnetic spin alignment
is found to be most stable [33]. One
should note, however, that the structural properties of this surface have essentially (1 X 1) symmetry and should therefore not produce a strong
sisted in Si by intraatomic Coulomb repulsions which, in turn, destabilized buckling. Heteropolar semiconductors, such as GaAs, are different in this regard, since charge transfer dissociated with buckling can reduce the ionic character of atoms in these compounds. Fig. 2 summarizes the calculated energetics of various Si(ll1) 2 x 1 reconstruction models. A
(2 X 1) LEED pattern. The chain model adds extra stability due to some additional rrbonding between surface atoms. The change in topology, proposed by Pandey, arranges surface atoms into zigzag chains such that dangling bonds are nearest neighbors promot
uniform inward relaxation of the outermost layer by  0.5 a.u. lowers the surface energy by  0.15
ing some abonding rather than neighbors as in the ideal of buckled
next nearest surfaces (see
M. Schluter
/
Bonding
geometries
fig. lb). The energy gain is rather large, i.e., calculated to be  0.3 eV for a nonoptimized chain [32] (fig. 2) and up to  0.37 eV for an optimized chain configuration [34]. Interestingly, the change in topology can be obtained rather easily from the ideal (111) structure by simultaneously depressing a surface atom (No. 1 in fig. 1) into the second layer and by raising a second layer atom (No. 2 in fig. 1) into the surface layer. This breaks a subsurface bond (25) while establishing a new bond (l5). It has been calculated by Northrup and Cohen [34] that the barrier for this process (2 0.03 ev) is significantly lower than a typical Si bond strength ( 2.3 eV). This makes creation of this surface during the cleavage process quite likely and also explains its ready transformation into a simple (1 X 1) pattern upon chemisorption of, e.g., hydrogen [36] or chlorine [37]. Although competing reconstruction patterns such as, e.g., the rbonded molecule [35] are proposed, the rbonded chain model seems to gain overwhelming experimental support [38]. The birth of this model demonstrates how the interplay between a new idea and theoretical total energy studies has drastically altered our understanding of this surface system. 3.2. Chlorine chemisorption
on Si(lll)
and Ge(ll I)
As mentioned above, chemisorption on semiconductor surfaces can drastically alter the original reconstruction pattern. In fact, when Cl is chemisorbed on Si(ll1) and Ge(ll1) which after cleavage exhibit (2 X 1) reconstruction patterns, a simple (1 x 1) structure is usually obtained. The interest then focusses mainly on the question: where do the adsorbate atoms chemisorb? Before total energy studies, such as discussed here, were feasible, the systems Cl on Si(ll1) and Ge(ll1) were investigated by comparing UPS photoemission data with model bandstructure calculations [37] using specific structurerelated spectroscopic fingerprints to deduce chemisorption geometries. The initial studies involved qualitative comparisons between polarizationweighted densityofstates spectra. The bandstructure calculations used semiempirical pseudopotentials and were carried out for Cl on a Si(lll)(1 x 1) substrate in either
at semiconductor
surfaces and interfaces
93
the onefold atop or the threefold hollow (fee) bridging site. Bond lengths were not calculated but were assumed to be given by the sum of covalent and ionic radii for the respective sites. The photoemission measurements for Cl on both cleaved Si(lll)(2 X 1) and annealed Si(lll)(7 X 7) surfaces showed, a polarizationdependent peak which, on the basis of the calculations, was interpreted as arising from a a, bond along the surface normal, appropriate for atop but not bridging bonding. The absence of such a strong spectroscopic fingerprint in the data for Cl on cleaved Ge(lll)(2 x 1) led to the conclusion [37] that Cl does not bind in the atop site but most likely binds in the threefold hollow site. First doubts about this arose from empirical extended Hiickel calculations [39] on Si and Ge clusters which confirmed the atop site for Si but challenged the conclusion for Ge, suggesting instead the same atop site. Later, SEXAFS measurements [40] have determined accurate bond lengths and have established that Cl does indeed bind in the atop site for both Si(ll1) and Ge(ll1) surfaces. Such direct knowledge of the bonding geometries thus allows not only new calculational methods to be tested on an absolute basis but also allows further theoretical insight to be gained about the electronic spectra responsible for the original assignments. Local density functional calculations in connection with normconserving pseudopotentials have then been carried out on these systems [41,42]. The Si and Ge(ll1) surfaces were approximated by sixlayer slabs of substrate and two overlayers. An ideal lateral (1 x 1) geometry was assumed, which is a good approximation in view of the small surface relaxation energies ( 0.2 eV/atom) compared with the calculated average chemisorption site energy differences ( 1 eV/atom). Three sites, the onefold atop, the threefold fee hollow, and the threefold hcp hollow, were investigated (see fig. 3). For each site the relative binding energies, the equilibrium bond lengths, and the force constants were evaluated. It became clear from these calculations that the atop site is energetically much more favorable than the bridging sites for both Si(ll1) and Ge(ll1). From the calculated stretching force constants, vibrational frequencies can be deduced. Assuming
94
M. Schluter / Bonding geometries at semiconductor
surfaces and interfaces
a rigid substrate,
the frequencies
are predicted
be 42 and 40 meV for ClSi(ll1) respectively.
Allowing
frequencies sumes
substrate
will be accordingly
the limiting
or GeCl
for
and ClGe( motion,
higher.
diatomic
the calculated
frequencies
63 and 49 meV, respectively.
the
If one as
case of isolated
molecules,
to
11 I),
These numbers
Sic1 are are in
excellent agreement with data available on gaseous molecules, i.e., 66 meV for Sic1 and 51 meV for GeCl. Electron energy loss, absorption or inelastic atom scattering experiments should be carried out on these surface
systems
to test the predictions.
The calculated equilibrium bond lengths of 2.02 A for ClSi(lll) and 2.09 A for ClGe(ll1) are in excellent
agreement
SEXAFS
with
the
data. A comparison
with
the
(2.16
A for ClS1
corresponding
them
to be considerably
ever,
almost
molecular
sums
of
covalent
and 2.21 A for ClGe) shorter.
identical
SiCl,
experimental
of these bond lengths
to
the
They bond
+ 0.02 A). Pauling
[43] attributes
in the calculations
in
(2.08
this anomaly
in
in the bond and
in part to some additional rbonding, “partial double bond”. The inclusion
leading to a of d orbitals
which must be involved in such
is clearly important.
The calculations (1 X with
are, howlengths
(2.02 + 0.02 A) and GeCl,
part to the strong ionic character
bonding
radii shows
for Cl chemisorbed
on Si( lll)
and Ge(lll)(1 X 1) agree quantitatively the SEXAFS data taken on annealed Cl
1)
covered
Si(lll)(7
7)
X
and
Ge(lll)(2
8)
X
surfaces. they also confirm the original spectroscopic identification of unannealed Cl on cleaved Si(lll)(2
1) and annealed
x
The question on cleaved
Ge(lll)(2
scopic fingerprint To
Cl on Si(lll)(7
then arises: Why does unannealed
gain insight
x
7). Cl
1) not show any spectro
characteristic into
X
for atop bonding?
this question,
spectroscopic
calculations, originally only done for Si, were extended to Ge [41]. The comparison shows a much broader and less pronounced fingerprint for Ge
Fig.
3.
Calculated
ClGe(ll1)
(pseudo)
chemisorbed
site (middle),
charge
and the hcp hollow
substrate
(from
ref. [42]).
contours
for
the fee hollow
site (bottom).
are shown in a (110) plane intersecting three
density
in the atop site (top),
the surface
layers (Le.. half the slab modeling
The contours Cl layer and the surface)
M. Schluter / Bonding geometries at semiconductor surfaces and interfaces
than for Si, making its detection by photoemission more difficult and less reliable. Nevertheless, if atopbonded chlorine is present on Ge(ll1) in sufficient quantity, it should be detectable by ppolarized photoemission. A recent analysis of new photoemission data [22] from annealed Clcovered Ge(lll)(2 X 8) (giving Ge(lll)(1 X l)Cl), in fact, shows evidence for such a weak a,like structure. this result is in contrast to the unannealed Clcovered cleaved Ge(lll)(2 x 1) data (also giving Ge(lll)(1 X 1)Cl) where no such feature was apparent. This suggests that despite the similar lowenergyelectron diffraction pattern observed after Cl absorption, the preparation of the Ge substrate surface has some influence on the chemisorption process of chlorine. For example, Cl could form some metastable precursor (nonatop) state on the cleaved surface which would convert to the stableatop geometry only after annealing. This example shows how total energy studies in connection with independent experimental results help deciphering and understanding surface bonding geometries. While clarifying some fundamental aspects, the studies also demonstrate the limitations of structural studies involving the interpretation of spectroscopic data. Furthermore, the results obtained ask for refined interpretations, considering, for example, nonequilibrium precursor events.
4. Conclusions This short review was necessarily rather selective. Among a variety of approaches to the theoretical determination of surface and interface structures, the pseudopotential density functional approach was singled out. This particular method is currently very actively used. The two selected examples illustrate the power of the approach. In the case of the clean Si(lll)(2 X 1) surface the calculations gave a convincing basis for abandoning the buckling model in favor of the new rbonded chain model. It is, of course, clear that this new model has not been found as an automatic result of calculations, but rather originated from a novel idea based on physical arguments. In the second case of chlorine chemisorption on Si and Ge
95
surfaces, the calculations gave quantitative explanations of independent experimental measurements (SEXAFS). They also illustrated the limitations and dangers associated with the approach of extracting structural data from spectroscopic measurements. In addition to these two examples, many different calculations exist to date, and more and more studies appear on systems with increasing complexity. Complete studies on systems with a large number of degrees of freedom, such as interfaces between two solids, are becoming feasible.
References
111I.K. Robinson, Phys. Rev. Letters 50 (1983) 1145. 121P.A. Lee, P.H. Citrin, P. Eisenberger and B.M. Kincaid, Rev. Mod. Phys. 53 (1981) 769. 131 W. Krakow, Thin Solid Films 93 (1982) 243. Critical Rev. Solid State Mater. Sci. 10 [41 L.C. Feldman, (1979) 143. and the PI W. Gudat and D.E. Eastman, in: Photoemission Electronic Properties of Surfaces, Eds. B. Feuerbacher, B. Fitton and R.F. Willis (Wiley, New York, 1978) p. 315. WI F. Jona, J.A. Strozier, Jr. and W.S. Yang, Rept. Progr. Phys. 45 (1982) 527. [71 D.J. Chadi, Phys. Rev. Letters 41 (1978) 1062. PI J.A. Pople, in: Modem Theoretical Chemistry, Vol. 4, Ed. H.F. Schaefer (Plenum, New York, 1977). 191 A. Redondo and A. Goddard III, J. Vacuum Sci. Technol. 21 (19820 344. [lOI J. Ihm, A. Zunger and M.L. Cohen, J. Phys. Cl2 (1979) 4409. [ill G.B. Bachelet, H.S. Greenside, G.A. Baraff and M. Schluter, Phys. Rev. B24 (1981) 4745. S.G. Louie and M.L. WI M. Schluter, J.R. Chelikowsky, Cohen, Phys. Rev. B12 (1975) 4200. P31 J.A. Appelbaum and D.R. Hamann, Phys. Rev. B12 (1975) 1410. w41 P. Hohenberg and W. Kohn, Phys. Rev. 136 (1964) B864. WI W. Kohn and L.J. Sham, Phys. Rev. 140 (1965) A1133. M. Schhuter and C. Chiang, Phys. Rev. F51 D.R. Hamann, Letters 43 (1970) 1494. u71 D.M. Ceperley and B.J. Alder, Phys. Rev. Letters 45 (1980) 566. 1181 0. Gunnarsson and R.O. Jones, Phys. Scripta 21 (1980) 394. [19] D.R. Hamann, Phys. Rev. Letters 42 (1979) 662. [20] L.J. Sham and M. Schluter, Phys. Rev. Letters 51 (1983) 1888; J.P. Perdew and M. Levy, Phys. Rev.Letters 51 (1983) 1884. [21] G.A. Baraff and M. Schluter, Inst. Phys. Conf. Ser. 59 (1981) 287.
96
M. Schluter / Bondrng geometries at semiconductor
[22] M.L. Cohen and V. Heine, in: Solid State Physics, Vol. 24, Eds. H.E. Ehrenreich, F. Seitz and D. Turnbull (Academic Press, New York, 1970) p. 38. [23] C. Herring, Phys. Rev. 52 (1940) 1169. [24] G.B. Bachelet, D.R. Hamann and M. Schluter, Phys. Rev. B26 (1982) 4199, and references therein. (251 U. van Barth and C.D. Gelatt, Phys. Rev. B21 (1980) 2222. 1261 H.S. Greenside and M. Schluter, Phys. Rev. B28 (1983) 535. [27] M.L. Cohen, in: Proc. Enrico Fermi Summer School. Varenna, 1983. [28] M. Schluter, in: Proc. Enrico Fermi Summer School, Varenna. 1983. [29] For a review see, e.g., D.E. Eastman, J. Vacuum Sci. Technol. 17 (1980) 492. 1301 D. Haneman, Phys. Rev. 121 (1961) 1093. [31] K.C. Pandey, Phys. Rev. Letters 47 (1981) 1913. 1321 K.C. Pandey, Phys. Rev. Letters 49 (1982) 223. [33] J.E. Northrup and M.L. Cohen, J. Vacuum Sci. Technol. 21 (1982) 333. [34] J.E. Northrup and M.L. Cohen, Phys. Rev. Letters 49 (1982) 1349.
surfaces und interfuces
[3S] O.H. Nielsen, R.M. Martin, D.J. Chadi and K. Kunc, J. Vacuum Sci. Technol. Bl (1983) 714. [36] H. Ibach and J.E. Rowe, Surface Sci. 43 (1974) 481. 1371 M. Schluter, J.E. Rowe, G. Margaritondo, K.M. Ho and M.L. Cohen, Phys. Rev. Letters 37 (1976) 1632. [38] R.I.G. Uhrberg, G.V. Hansson, J.M. Nicholls and S.A. Flodstrom, Phys. Rev. Letters 48 (1982) 1032; R.M. Tromp. L. Smit and J.F. van der Veen. Phys. Rev. Letters 51 (1983) 1672; M.A. Olmstead and N.M. Amer, preprint. [39] Zhang KaiMing and Y. Ling, J. Vacuum Sci. Technol. 19 (1981) 628. [40] H. Citrin, J.E. Rowe and P. Eisenberger, Phys. Rev. 828 (1983) 2299. [41] G.B. Bachelet and M. Schluter, Phys. Rev. B28 (1983) 2302. 142) G.B. Bachelet and M. Schluter. J. Vacuum Sci. Technol. Bl (1983) 726. [43] L. Pauling, The Nature of the Chemical Bond (Cornell University Press, Ithaca, NY, 1980) p. 310.