Ultramicroscopy 14 (1984) 89-96 North-Holland, Amsterdam
M. SCHLUTER A T&T Bell Laboratories,
Hill, New Jersey
Received 11 January 1984; presented at Workshop January 1984
Advances in the theoretical description of electron-electron and electron-ion 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., X-ray diffraction [l] or absorption , electron microscopy  or ion scattering . These approaches yield structural information usually without explicit theoretical calculations. (ii) The combined theoretical-experimental approach, involving a comparison between structure-sensitive experimental data and theoretical calculations of such data as, e.g., photoemission  and low energy electron diffraction . (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 parameter-free schemes [7-111. Similarly, there are many different ways to model the surface-interface systems: clusters , slabs , infinite half-spaces , etc. Here, I shall discuss a method which is based on the local density functional approximation (LDA) [14,15] to 0304-3991/84/$03.00 0 Elsevier Science Publishers (North-Holland Physics Publishing Division)
describe electron-electron interactions and on the first-principle pseudopotential approximation  to describe electron-ion interactions. The calculations are done using supercell, i.e., periodically repeated slab geometries .
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  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 non-inter-
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.
due to the Pauli exclusion
of the ground state.
It is general
to the otherwise correlated motion of electrons. As shown by Kohn and Sham , a variational solu-
tion of eq. (1) can be obtained
by solving a set of
systems where relaxation
for extended This
semiconductors cg = cc - c\ be-
as the energy gap. Empiri-
that cg calculated
in the LDA
It has recently
realized  that there is an additional contribution to the gap arising from a discontinuity in the
according to Fermi statistics with the eigenvalue c, serving as the energy for the purpose of determining
cally it is known
of eq. (2)
effects are negligible.
tween the first unoccupied
W)l G,(r) = c,+,(r)?
is thus reduced
to the solu-
tion of a set of one-particle equations, not more difficult to solve than the Hartree problem. The difficulty tionals
lies in determining
to the Thomas-Fermi
approximation (Local Density and V,, is given by regarding
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
yet, and empirical
are still necessary
when gap values are
simplify electronic structure calculations by eliminating the need to include atomic core states. Considerable
the band structure
however, a certain problem. The normalized pseudo-wave functions have the correct shape outside the core region, but incorrect is because
of the so-called
hole” puts too much charge inside the core region. The problem tions,
and expressions for the homogeneous electron gas exchange-correlation energies can be used . Local density functional theory is generally very successful
needed . Pseudopotentials
the formal justification for the use of pseudopotentials was initially based upon the orthogonal-
for adding holes or
conductors and simple metals with the use of empirical pseudopotentials . On the other hand,
E,, [ p] and
to the insulating
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 . Improved functionals, going beyond the local approximation, seem to be necessary in some of these cases to obtain a correct
it will cause
in the Coulomb
can, in principle,
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 self-consistent 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 “norm-conserving” 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. , norm-conserving 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 . Combined with the norm-conserving property, the pseudopotentials do thus not lead to any first-order transferability error. For treating heavier atoms, relativistic effects can be absorbed into the core pseudopotential  which enables one to treat heavy atoms in a non-relativistic 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 “full-core” 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.
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 norm-conserving 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
Clean semiconductor surfaces show involved reconstruction patterns . 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  both experimentally and theoretically. For many years, the buckling model, originally introduced in 1961 by Haneman , 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 Jahn-Teller symmetry-breaking 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 1981-82 which clearly showed the inappropriateness of this model for Si(ll1). Originally motivated to understand photoemission data, Pandey proposed a new surface topology, the r-bonded 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
surJaces and interJaces
0 f 0 G
( I x I) SURFACE
MAGNETIC 2 x I SURFACE
-025 Fig. (top) The gling
1. Perspective view of the ideally terminated Si(ll1) surface and the v-bonded chain reconstruction model (bottom). (2 X 1) surface unit cell is indicated and the surface danbonds are emphasized (from ref. ).
Pandey  and by Nielsen, Martin, Chadi and Kunc , all within the computational frame discussed
in section 2. The results (some are shown
in fig. 2) confirmed models
fact, buckling polar
did not stabilize surface.
on this homoby Haneman
would cause rehybridization
atomic orbitals centered at the outermost atoms which, in turn, should cause charge transfer from the
that this charge
Fig. 2. Calculated total energy variations for various reconstruction models for the Si(ll1) (2 X 1) surface (from ref. ).
as such was resisted
calculations was strongly
a common If buckling
feature of covalently is introduced
this energy gain Nevertheless,
may suspect that certain metastable configurations can exist. Among the “standard” models, not containing topological changes, a non-buckled, insulating surface with a (2 X 1) antiferromagnetic spin alignment
is found to be most stable . 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 intra-atomic 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 rr-bonding between surface atoms. The change in topology, proposed by Pandey, arranges surface atoms into zig-zag 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 a-bonding rather than neighbors as in the ideal of buckled
next nearest surfaces (see
fig. lb). The energy gain is rather large, i.e., calculated to be - 0.3 eV for a non-optimized chain  (fig. 2) and up to - 0.37 eV for an optimized chain configuration . 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 (2-5) while establishing a new bond (l-5). It has been calculated by Northrup and Cohen  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  or chlorine . Although competing reconstruction patterns such as, e.g., the r-bonded molecule  are proposed, the r-bonded chain model seems to gain overwhelming experimental support . 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
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 band-structure calculations  using specific structure-related spectroscopic fingerprints to deduce chemisorption geometries. The initial studies involved qualitative comparisons between polarization-weighted density-ofstates spectra. The band-structure calculations used semiempirical pseudopotentials and were carried out for Cl on a Si(lll)-(1 x 1) substrate in either
surfaces and interfaces
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 polarization-dependent 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  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  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  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 norm-conserving pseudopotentials have then been carried out on these systems [41,42]. The Si and Ge(ll1) surfaces were approximated by six-layer 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
M. Schluter / Bonding geometries at semiconductor
surfaces and interfaces
a rigid substrate,
be 42 and 40 meV for Cl-Si(ll1) respectively.
will be accordingly
and Cl-Ge( motion,
63 and 49 meV, respectively.
If one as-
case of isolated
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
to test the predictions.
The calculated equilibrium bond lengths of 2.02 A for Cl-Si(lll) and 2.09 A for Cl-Ge(ll1) are in excellent
data. A comparison
A for Cl-S1
to be considerably
and 2.21 A for Cl-Ge) shorter.
of these bond lengths
+ 0.02 A). Pauling
in the calculations
in the bond and
in part to some additional r-bonding, “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
(2.02 + 0.02 A) and GeCl,
part to the strong ionic character
for Cl chemisorbed
on Si( lll)-
and Ge(lll)-(1 X 1) agree quantitatively the SEXAFS data taken on annealed Cl-
surfaces. they also confirm the original spectroscopic identification of unannealed Cl on cleaved Si(lll)-(2
1) and annealed
The question on cleaved
scopic fingerprint To
Cl on Si(lll)-(7
then arises: Why does unannealed
1) not show any spectro-
for atop bonding?
calculations, originally only done for Si, were extended to Ge . The comparison shows a much broader and less pronounced fingerprint for Ge
and the hcp hollow
the fee hollow
are shown in a (110) plane intersecting three
in the atop site (top),
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 atop-bonded chlorine is present on Ge(ll1) in sufficient quantity, it should be detectable by ppolarized photoemission. A recent analysis of new photoemission data  from annealed Cl-covered 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 Cl-covered 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 low-energy-electron 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 (non-atop) state on the cleaved surface which would convert to the stable-atop 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, non-equilibrium 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 r-bonded 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
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.
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M. Schluter / Bondrng geometries at semiconductor
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surfaces und interfuces
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