Structure and dynamics at metal surfaces

Chapter 3

Structure and Dynamics at Metal Surfaces K.P. Bohnen Kemforschungszentrum

Karlsruhe, Institut fiir Nukleare Festkbrperphysik, P.O. Box 3640, D- 76021 Karlsruhe, Germany

and K.M. Ho Ames Laboratory, US Department of Energy and Department of Physics, Iowa State Unicersity, Ames, IA 50011, USA

Contents 1. 2. 3. 4.

Introduction Method of calculation Pseudopotential approach Relaxation of clean metal surfaces 4.1. Simple metals 4.2. Transition metals 4.3. Noble metals 5. Reconstruction of clean metal surfaces 6. Surface phonons 7. Conclusions Acknowledgments References

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Surface Science Reports 19 (1993) 99-120 North-Holland

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surface science reports

Structure and dynamics at metal surfaces K.P.

Bohnen

Kernforschungszentrum Karlsruhe, Institut fiir Nukleare Festktirperphysik, P.O. Box 3640. D-76021 Karlsruhe, Germany

and K.M.

......

.:.‘...‘..::::.~::~.~~.”

Ho

Ames Laboratory, US Department of Energy and Department of Physics, Iowa State Unillersity, Ames, IA 50011. USA

Using first-principles total-energy calculations, lattice relaxation, reconstruction and surface phonons have been determined for various metal surfaces. Use of the Hellmann-Feynman theorem allows for a very efficient determination of equilibrium geometries and interplanar force constants. Results for simple metals as well as for noble metals will be presented and compared with available experimental information from LEED, ion channeling, helium scattering and EELS as well as with other theoretical treatments.

1. Introduction The determination of structure and dynamics of surfaces is a basic question in surface science. Atoms near the surface of a crystal are under the influence of different forces from those in the bulk. This leads in most cases to relaxations or reconstructions. However, the modified forces at the surface do not only influence the static arrangement of atoms at the surface but modify also the vibrational modes of surface atoms. They are involved in many processes on surfaces at ambient or elevated temperatures. A detailed knowledge of the surface-phonon spectrum is essential in studies of surface diffusion, phase transitions on clean and adsorbate-covered surfaces, and desorption processes. It is also indispensable for any quantitative studies of energy transfer and dissipation at surfaces. Furthermore, the frequencies of vibrational modes associated with adsorbed species can yield information on surfaceadsorbate bonding, the geometry of the adsorption sites and the lateral coupling between neighboring atoms or molecules on the surface. The past few years have witnessed a rapid growth in the experimental effort to determine lattice relaxation, reconstruction and also surface vibrations. In addition to the well established technique of LEED [l], ion scattering [2,3], X-ray diffraction [3] and atom beam scattering [3] have yielded valuable information about the geometric structure of the surface. Although different methods yield similar results for simple surfaces, for more complex structures which involve reconstructions only a strong interplay between experiment and theory can help understand the observed atomic arrangements.

K.P. Bohnen, K.M. Ho / Structure and dynamics at metal surfaces

101

For surface vibrations the situation is similar. With the development of high-resolution electron energy-loss spectroscopy [4] and inelastic helium beam scattering experiments [5] we now have the capability of accurately measuring the surface-phonon dispersion curves. However, the data revealed the need for theoretical work to interpret the observed spectra in simple physical terms. On the theoretical side over the past few years also substantial progress has been made in determining lattice relaxation, reconstruction and surface phonons for metals from first principles. All theoretical approaches exploit the fact that the equilibrium geometry is characterized by the total-energy minimum of the system. However, different approaches have been used to calculate the energy of the system. Approximate forms for the energy are used in the jellium model [6], the semi-empirical models (glue model [7], embedded-atom model (EAM)) [8] and the effective-medium theory [9] (EMT). While the jellium model is restricted to close-packed surfaces and systems which can be described by a weak electron-ion interaction, the EMT uses jellium-related quantities to describe the energy of the system. The semi-empirical models usually rely on an extensive fit to measured or calculated bulk properties. However, as recent studies of the multilayer relaxation of Cu(ll0) demonstrate [&lo], they depend critically on the fitting procedure. Both methods (EAM) and (EMT) do not incorporate effects of charge self-consistency. The electronic charge density is constructed from overlapping atomic charge densities. This questions immediately the results obtained for systems which exhibit strong charge rearrangements. Typical examples are the very open fcc(llO> surfaces. These surfaces show either large oscillatory relaxations or reconstructions. A typical example is the Am1 10) surface which is known to reconstruct in the (1 X 2) missing-row structure [ill. Although EAM [121 and EMT 1131 describe this reconstruction qualitatively correct, a detailed comparison with experimental results from LEED [ 141 and ion scattering [15] reveal some discrepancies between experiment and theory. EAM and EMT seem to over- or upder-estimate the relaxation effects and the error seems to by typically of the order of 0.1 A. Furthermore, due to the lack of charge self-consistency, the atomic positions for atoms in the second layer deviate from the observed ones. These deficiencies have drastic consequences for the surface-phonon spectrum [161. Furthermore, EAM and EMT are constructed in such a way that they allow only the calculation of total energies and related quantities. They do not give information about surface states, work function, etc. which should be included to give a complete physical picture of the observed relaxations/ reconstructions. Although the EAM and EMT have the advantage of being easily applicable to a wide variety of systems due to their simplicity, ab initio first-principles calculations which do not rely on input from experiment or assumptions based on jellium calculations are strongly needed. These calculations can be done either by a direct calculation of total energies in the framework of local-density theory [17] or by the dielectric response approach which requires the knowledge of the surface response function [18]. The latter approach, however, has been used so far only for Al and Na surfaces and is restricted (in its present form) to local potentials describing the electron-ion interaction. Among the direct methods, there are the slab or supercell approaches which work best for clean surfaces or adsorbate/substrate systems with a commensurate periodicity [19], the cluster approaches which allow the treatment of adsorbate-substrate systems without restrictions concerning the periodicity [20], but describe the surface only approximately. A third method is the “matrix Green’s function” treatment [21l which is optimum for spatially compact adsorbates on an extended substrate. Since this article is concerned only with clean metal surfaces, we will restrict the discussion in the following to the slab or supercell approaches which are adapted from bulk treatments.

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First-principles total-energy calculations have been very successful in determining structural properties of a large variety of bulk materials 1221, however, application to surfaces has become feasible only recently due to the rapid development of supercomputers. Among the systems considered, semiconductor surfaces have been preferred since for metals the necessity of accurately representing electronic states near the Fermi level requires sampling more grid points in the surface Brillouin zone. However, a number of clean metal surfaces as well as adsorbate systems have been studied by these first-principles total-energy calculations and calculated relaxations and/or reconstructions agree generally very well with the observed structures. These calculations have contributed substantially to our understanding of relaxation and reconstruction of metal surfaces. In certain cases, a simple physical picture for the underlying driving mechanism could be extracted. For obtaining theoretically surface-phonon dispersion curves the situation is more complicated. The surface dispersion curves can be obtained by solving for the vibrational modes of a slab or a semi-infinite crystal. Early calculations modeled the interatomic interactions by a Lennard-Jones potential [23]. Subsequent studies 124-261 used more realistic force constants deduced from fitting experimental bulk-phonon dispersion curves. Empirical adjustments of the surface force constants were made to reproduce the measured surface mode frequencies. However, it was found that the changes necessary are very model-dependent: two different models reproducing the same bulk-phonon dispersion curves may require very different changes in parameters to reproduce the same surface modes, giving rise to very different physical interpretations 1271. Recently also the EAM, EMT, and the glue model have been used to calculate surface vibrations for a number of metal surfaces [28-311. While these methods avoid part of the aforementioned complications, for the Au(1 lO)( 1 X 2) surface there is a strong disagreement with the experimentally measured surface modes [lb]. It is obvious that a determination of the surface force constants from first principles is very useful for interpreting experimental data and guiding experimental investigations on unmeasured crystals. In the bulk total-energy calculations have been very successful in determining phonon frequencies at high-symmetry points in the Brillouin zone (BZ). These frozen-phonon calculations [32] are unfortunately not directly transferable to the calculation of surface mode frequencies. Frozen-phonon calculations require as input the knowledge of the vibration pattern related to the phonon mode under consideration. For monoatomic systems this is no problem in the bulk at high-symmetry points in the three-dimensional BZ. However, at a surface even at high-symmetry points in the two-dimensional BZ the decay of the mode in the direction perpendicular to the surface is usually not known. This difficulty can be avoided by calculating directly forces and force constants acting on the atoms. Knowledge of the force constants allows us to set up the dynamical matrix and solve for the eigenvalues and eigenvectors of the system, In contrast to the frozen-phonon calculations no assumptions about the eigenvectors have to be made. To reduce the computational effort these calculations are carried out on high-symmetry points on the two-dimensional BZ where not all atomic force constants are needed but only certain linear combinations (interplanar force constants) which are obtained much easier. This limits of course the knowledge of first-principles surface frequencies and surface mode eigenvectors to these high-symmetry points. For systems with short-range interactions a small number of these calculations are sufficient to determine the atomic force constants uniquely and thus allow the calculation of surface phonons on arbitrary points in the BZ. In most cases, however, the range of interaction does not allow this unique determination of atomic force constants yet. In these situations one relies on alternative methods. For W(OOl) [331 and Mo(OO1) 1341 a tight-binding

K.P. Bohnen, K.M. Ho / Structure and dynamics at metal surfaces

103

model has been used where the parameters have been determined from a fit to first-principles density-functional calculations. In calculating the dynamical matrix the long-range part is treated perturbatively within the tight-binding framework while the short-range part is parametrized using Born-von K&man (BvK) force constants. It was possible to show that phonon instabilities for W(OO1) and Mo(001) occur in agreement with the commensurate and incommensurate phases observed. However, these calculations have been restricted to the unreconstructed surfaces. A similar method has been applied to semiconductor surfaces, too [35]. Calculations determining the dynamical matrix for arbitrary points in the two-dimensional BZ directly from the knowledge of the surface response function have been carried out so far only for Al [36] and Na 1501 and cannot very easily be extended to more complicated surfaces or systems containing strongly localized electrons. In the following section, we will describe first the general theory underlying the total-energy expression used for all first-principles calculations of relaxation and reconstruction, followed by a discussion of one of the most widely used implementations, the pseudopotential method. Section 3 is devoted to the discussion of relaxation results in comparison with experiments and results obtained using approximate treatments. A similar study concerning reconstructions is presented in section 4. In section 5 we present surface-phonon results while the concluding section 6 gives a summary and outlook.

2. Method of calculation General theory The first-principles total-energy calculations are based on the Hohenberg-Kohn-Sham theorem (HKS) [17] which states that the ground-state energy of an interacting electron system can be expressed as a unique functional of the electronic charge density n(r). Furthermore, it can be shown that the true physical electronic charge density minimizes this functional. The exact functional form of this energy expression, however, is not known since one is dealing with an interacting many-body system. Thus, Kohn and Sham suggested to express this energy functional in terms of certain known expressions plus a term which contains all the complications of the many-body system which has to be approximated. The most convenient form is

E[n]=T[n]+/n(r)

V(r)

dr+f/n(,;)_:(,;‘)

dr dr’ + E,,[n],

where T[n] is the kinetic energy of a noninteracting system of electrons with density n(r), the second term describes the interaction with an external potential V(r), the third contribution is the Hartree energy and the last term, the so-called exchange and correlation functional contains all the complexity of the many-body system. Minimizing the energy functional (1) with respect to number-conserving density variations is equivalent to solving the following set of Kohn-Sham equations self-consistently [17]: Hqi(

r) = E,1yI( r),

(4

with H=

-(1/2m)V*+

I&(r),

(3)

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where the eigenfunctions ql solve eq. (2) with eigenvalue E,. The density is given by summation over the N lowest eigenvalues. The Hartree potential Vr,artrer(r) and the exchange and correlation potential V,,(r) are given as

and I/x,(r)

= ~&J~l/~~(~~.

(7)

Due to the density dependence of l’,,;lrtrcc and V,,, eqs. (2)-(7) have to be solved self-consistently. These are the basic equations underlying all first-principles total-energy calculations for surfaces. Once the eigenvalues E, and eigenfunctions Y’[ are known, the energy expression (1) can be calculated. For applications, however, an explicit form for the exchange and correlation functional has to be given. The most widely used form is the local-density approximation

(8) where e,,[n(r)} is the exchange and correlation energy per particle. Various forms have been suggested for E,~ [37] based on electron-gas data. Having outlined the general theory, we will concentrate now on applications

3. Pseudopotential

approximate to surfaces.

approach

One widely used approach makes use of the pseudopotential concept [38] which has been applied successfully to a large variety of bulk systems. In this treatment, one exploits the fact that for most applications to the solid state two very distinct regions of space exist. The core region, which is close to the nucleus which contains the tightly bound core electrons which are hardly affected by the presence of neighboring atoms and the remaining volume that contains the valence electrons which are responsible for the bonding properties. This physical fact has led to the frozen-core approximation widely used with the LAPW method [39], where the core electrons are not allowed to relax. The pseudopotential formulation goes one step further and eliminates the core electrons completely by introducing a “pseudopotential” which describes the influence of the core electrons on the valence electrons (orthogonalization, etc.). These potentials are usually non-local in nature. Different potentials are seen by the s, p, d. . . valence electrons and, in addition, these potentials are not unique. The construction of these potentials follows from atomic calculations and has nothing to do with the bulk or the surface. In determining these potentials, one has, however, to take great care to achieve maximum transferability of these potentials. A number of schemes for constructing these potentials are now available [40]. If carefully applied, the resulting pseudopotentials lead to bulk properties which are very insensitive to the actual pseudopotential chosen.

K.P. Bohnen, K.M. Ho / Structure and dynamics at metal surfaces

105

The elimination of the core electrons has the advantage that the pseudodensity is very smooth in the core region. Thus, in many cases, a plane-wave basis is appropriate. Furthermore, since only the pseudodensity enters the calculation, it is possible to apply the Hellmann-Feynman [41] theorem for calculating forces on the ions in the cell. Since forces do not satisfy a minimum principle like the energy, the force calculation requires a very high degree of self-consistency which is difficult to obtain in an all-electron calculation. Although most pseudopotential calculations have been done using a plane-wave basis, this is not necessary. For systems with strongly localized orbitals (e.g. Cu), however, a mixed basis consisting of plane waves and localized orbitals is more appropriate. In this case, the wave function W,(r) e.g. has the form

with plane waves l/m centered at the nucleus 4jtm(k,

ei(k+G)‘r and Bloch sums position r’ = r - R - rj,

r) = l/v’%c

eik.(R+rJ

$,,,(k,

r) of localized

basis

functions

+j,m(r-R-rj)y

R

where R is a lattice vector and rj is a basis vector. Using these basis functions, eq. (2) leads to a matrix equation (H - ES)A = 0 with the Hamiltonian matrix H and the overlap matrix S. A is a vector which contains the expansion coefficients {d”, p”}. The Hamiltonian is given as

” = - wwv2 with the non-local Vpse”dow =

+ I/pseudo(r) + I/Hartree(r) + Kc(r),

pseudopotential

cc / V(r) pr>

being

(10)

angular-momentum-dependent

(11)

where PI is a projection operator on angular momentum 1. For details, the reader is referred to ref. [42]. The solution of the eigenvalue problem gives eigenvalues en(k) and the expansion coefficients {d”(k + Cl}, {p,$Jkl). Having solved for the wave functions W,(r), the energy expression (11 can be evaluated, however with the modifications that the full charge has to be replaced by the pseudo valence charge only since the core electrons have been eliminated. The nuclear potential V(r) has to be replaced by the pseudopotential, thus leading to an energy expression for the electron-ion interaction as

(12) which replaces the second term in eq. (1). For an efficient application of the method, a fast and accurate algorithm for calculating the charge density n(r) and the Hartree and exchange and correlation potentials is needed once the wave functions q:(r) are known. In a pure plane-wave basis, the Hartree potential is easily calculated in Fourier space while the exchange and correlation potential requires a real-space representation of the charge density. However, this can be obtained using fast Fourier transform techniques. Within a mixed-basis description one has, of course, the option to expand the full wave function in a plane-wave basis and to follow now the above outlined procedure. However, this is not very economical for systems with strongly localized orbitals. An alternative is to use a mixed-basis representation not only for the wave function and the solution of the eigenvalue problem, but also for the charge calculation. Details can be found

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IO6

in ref. [43] where an extensive discussion of the latest developments for the mixed-basis pseudopotential method is given. For an extensive review of the pseudopotential method in general, its foundations and formal limitations, see ref. [38]. So far, we have been discussing only the total-energy aspect of the determination of relaxations and reconstructions, however, since usually many atoms are involved in these phenomena, a knowledge of forces on the atoms would be very helpful in determining efficiently the equilibrium geometry for complicated structures. There are two ways to achieve this; one is the calculation of forces numerically via finite energy differences. This can be done by changing the positions of atoms in the unit cell and determination of the corresponding energy changes, however, this has to be carried out separately for all atoms in the cell and is usually very tedious. An alternative is the use of the Hellmann-Feynman theorem which states that only those terms in the energy expression (1) contribute to the force which explicitly depend on the external parameter with respect to which the differentiation has to be done. From eq. (l), it follows immediately that the force on atom i is given as (7, is the shift of atom i away from the equilibrium position) F, = - d E/dri

+ Fion,

( 13)

where the first contribution is the electronic force and the second term describes the nuclear force. In the pseudopotential framework, the second term is the pseudo-ion force while the first term involves the non-local pseudopotential. Explicit expressions can be found in ref. [441. The advantage of using the Hellmann-Feynman theorem lies in the fact that all forces in the unit cell can be calculated easily once the self-consistent solution is known. Application of the above-mentioned formalism to surfaces is being done using slab or supercell geometries. In calculations based on supercells a periodic arrangement of atomic layers separated by a vacuum region is being used. Three-dimensionality is artificially recovered at the price of very large unit cells.

4. Relaxation

of clean metal surfaces

Having outlined in the last section the method of first-principles total-energy calculations and its main implementations for surface studies, we will discuss now results for relaxations of clean metal surfaces. Representative results are summarized in table 1. The three classes of systems studied are simple metals, transition metals and noble metals. 4.1. Simple metals Systems studied in this class are alkali metals and aluminum. These systems can be very well described with pseudopotential theory employing only a plane-wave basis. However, since screening effects in these systems are fairly long-ranged due to the delocalized nature of the electrons involved, the computational effort to deal properly with relaxations in these systems is by no means small. Careful studies have shown that slab thicknesses of 13 to 15 atomic layers are needed to obtain converged results for the interlayer spacings at the surface [451. So far only relaxations of Na and Al surfaces have been investigated using first-principles total-energy studies [45-481. These investigations are supplemented by calculations within the dielectric response formulation [49,501.

KP. Bohnen, K.M. Ho / Structure and dynamics at metal surfaces Table 1 Relaxations/reconstruction

for clean metal surfaces

First-principles total-energy densityfunctional calculations Na(ll0)

Ad,,

Other

theoretical

of interlayer

approaches

Semi-empirical

spacing Experiments

Diel. screen.

- 1.6 [47] 0.0 0.6

0.33 [911

AK1 10)

- 6.8 [45] 3.5 - 2.0 1.6

- 10.5 [lo] 3.6 - 2.9 -1.5

AK 100)

1.2 [46] 0.2 -0.1

-4.9 [lo] 2.2 2.3

Al(331)

- 11.3 [481 -6.3 10.1 - 4.4 - 1.8 4.8

CldllO)

- 9.3 [581 2.8 - 1.1

Cu(100)

Ag(ll0)

in percentage

107

-5.4 [18] 1.2 -3.0

0.7 1181 0.2 0.4

- 8.5 [531 5.5 2.4

- 8.6 [52] 5.0 - 1.6

0.0 [92]

- 11.7 [931 -4.1 10.3 -4.8 - 2.4 0.3 -8.7 [lOI 1.6 -1.2

- 4.9 [Sl 0.2

- 8.5 [941 2.3

- 5.3 [951 3.3

- 3.0 [581 0.1 -0.2

-3.8 [lo] -0.5 0.0

- 1.44 [8] -0.3

- 1.2 [971 0.9

- 2.4 1981 1.0

- 1.3 [581 0.6 - 0.3

- 2.5 [lo] -0.0 0.0

- 1.4 [8] 0.1

- 0.7 [991

- 7.0 [581 2.8 -0.2

-6.9 [lo] 2.2 - 1.0

- 5.1 [8] 0.3

- 5.7 [loo] 2.2 - 3.5

- 9.5 [loll 6.0 -3.5

- 3.0 [lOI 0.0 0.0

- 1.9 [8] 0.0

- 1.9 [lOI 0.1 0.0

- 1.3 [8] 0.0

- 18.0 [15] 4.0

- 20.0 [141 2.0 2.0 8.0

A&00)

6.2[591 2.1

AgUll)

- 0.4 [581 -0.2 0.2

Au(ll0) (1x1)

-9.2[11] 7.7 0.7

- 12.1 [lOI 3.6 - 1.6

- 15.3 [8] 2.1

Au(ll0) (1x2)

- 16.0 [ll] 2.0 3.0 7.0

- 27.0 [7] -5.0 -2.2 13.0

- 15.0 [121 -5.0 - 1.0 4.0

0.05 A

- 0.29 A

- 0.07 J&

7.0 0.07 A

- 7.5 [96] 2.5

Table

I (continued)

First-principles tot&energy densityfunctional c2tlculations

Other

theoretical

Semi-empirical

approaches

Experiments

Diel. acreen.

W(OOl)

Arl,? It/,, A&,

- 5.7 [561 2.4 I.2

- 7.0 [ lO2]

Mo(OOl)

Ad,, AC/,,

- IO.7 [X4] 2.7 0.3

-Y.5 [lO3]

G

-

I I.5 [IO-ll

This class of systems has, of course, been studied extensively with jellium or jellium-related models which should work best for these systems since the pseudopotentials are fairly weak. However, even for these systems, these models give a good description only for the most densely packed faces [9,30,51,521. The more interesting (because of stronger relaxation effects) open faces ((110) and (100) for fee lattices) cannot be dealt with within perturbation theory. This can easily be seen by comparing surface energies for AI(l10) and Al(100) from jcllium calculations with low-order perturbation theory [52] to those obtained from first-principles studies [45,46]. Results clearly show that for AhlOO) and for AN1 lo), low-order perturbation theory does not give results close to the converged ones. High-order perturbation theory has no advantage over the exact treatment since all the simplifications due to the jellium approximation enter only the low-order perturbation terms. A second shortcoming of these jellium and jellium-related calculations is due to the fact that all these studies employ a very simple local pseudopotential. Thus, even an exact treatment would be bound to give only qualitative results due to the errors involved in the pseudopotential. Unfortunately, this is also the major shortcoming in studies based on the response function approach which would otherwise offer a nice alternative to the total-energy approach. The surface which has been studied most extensively is AN1 10). This is due to the fact that experimentalists found in this system for the first time a very pronounced multilayer relaxation which is oscillatory [53]. The origin for this effect lies in the electrostatic interaction of the ions and is related to the stacking sequence of layers ABAB.. . This effect is large so that the screening effects of the electrons cannot completely counteract the oscillatory behavior. The Al(100) surface which also exhibits the ABAB . . . stacking differs in that the interlayer distance is much larger. Thus the bare ion effect is much smaller such that finally the screening determines the outcome. Due to this, no substantial oscillatory behavior was found for the Al(100) surface [46]. The phenomenon of oscillatory multilayer relaxation is not restricted to Al(110) but has been seen for all fcc(l10) surfaces studied so far. In view of the fact that the driving mechanism lies in the electrostatic interactions, this is not surprising. Recently, even a stepped surface Ah33 1) has been studied and results are in excellent agreement with experiment [48]. Summarizing all studies for surfaces of simple metals, one can state that the densely packed fcc(100) and (111) as well as the bcc(ll0) and (111) faces show only small relaxation effects of the order of 1% or less in terms of the interlayer spacing. The fcc(l1 I) and beet 110) surfaces can be described fairly well with approximate methods. For the open fcc(l10) and bcc(100) faces, these methods give only a qualitative description. The exact first-principles methods have to be applied to obtain quantitative agreement between theory and experiment. These methods also provide detailed information on other ground-state properties like work function and surface energies.

K.P. Bohnen,

4.2.

KM. Ho / Structure and

dynamics

at metal surfaces

to9

Transition metals

In these systems, most of the physical properties are dominated by the fairly localized d-electrons. Thus, they have been dealt with very often in an approximate way using a tight-binding picture neglecting the influence of the s-p electrons completely [54]. These treatments are, of course, not accurate enough to treat such subtle questions like lattice relaxation. Besides these model calculations, a great number of self-consistent first-principles calculations have been carried out (without trying to give a complete list, some typical examples are given in ref. [55]). However, these studies have been concerned with surface states and surface magnetism without studying the questions of equilibrium geometry at the surface. Usually the ideal unrelaxed surface has been used in these studies. The equilibrium geometry at the surface has been studied using the first-principles methods only for W(OO1) [56] and Mo(001) [34]. Although these surfaces have gained a lot of interest due to their interesting reconstruction patterns, the unreconstructed surfaces are also interesting since they exhibit again multilayer-relaxation effects. In contrast to the simple metals here, the screening effects are fairly short-ranged due to the fact that the localized d-electrons dominate. Thus, the calculations for these systems require much thinner slabs than those for the simple metals. Usually slabs of 7 atomic layers are sufficient to obtain converged results for the interlayer spacings. For both systems, W and MO, the first interlayer spacing is substantially reduced from the bulk value. The bulk lattice constant in these systems results from an interplay between d- and s-p-electrons. The d-electrons alone would favor a smaller lattice constant, however, due to the increase in kinetic energy in the s-p-electrons with compression, this cannot be achieved. At the surface, however, a shorter atomic distance can be supported since the kinetic energy of the delocalized electrons can be lowered by extending the wave function into the vacuum. For the 4d transition-metal series a systematic study of surface relaxation has been carried out for all low-index faces [57]. Unfortunately, this study has been restricted to the first-layer effects only, thus no detailed comparison with experimental data for the open fcc(l10) surfaces is possible. In general, first-principles results are in good agreement with available experimental data. As has already been seen in studies of the simple metals, the relaxation energies are usually only a few percent of the surface energies and work functions are also not very sensitive to relaxation effects. 4.3. Noble metals The most extensive relaxation study using first-principles total-energy calculations has been carried out for this class of systems. All low-index faces of Cu [58,591 and Ag [58] as well as the unreconstructed Au(ll0) 1111 have been investigated recently. The results, which are given in table 1, can be summarized as follows: the most densely packed (111) faces show the smallest effects. The relaxations are of the order of 1% of the interlayer spacing or smaller. The more open (1001 faces show effects of the order of l%-3%. The largest effects are again seen for the very open (110) surfaces. For the outermost layer, the effect varies between 9% for Au, 7% for Ag and 9% [58] or 6% [59] for Cu. At present it is not known why the two calculations for Cu differ by 3% for the first interlayer spacing. Although the second-layer effect is smaller, it is still substantial, 2%-3% for Ag and Cu. For Au, it is even larger. Again, the effect is oscillatory. The outermost layer is always moving inward while the second layer moves outward compared to the bulk interlayer spacing. Screening effects are much shorterranged than for the simple metals due to the presence of the d-electrons. However, in contrast to the transition metals, they do not dominate all effects since the d-bands are full

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and only the s-p-bands lie close to the Fermi surface. The relaxation is again the consequence of the competition between the localized d-electrons and the delocalized s-p-electrons.

5. Reconstruction

of clean metal surfaces

So far, only relaxation effects have been discussed, however it is well known that many surfaces reconstruct showing a lattice periodicity in the outermost layer which differs from that of the underlying bulk structure. This surface structure can be commensurate or incommensurate with the bulk. Some experimentally well studied reconstructed surfaces are the W(OO1) [60] and Mo(001) [61] surfaces, the (110) surfaces of Ir, Pt and Au [14,1.5,62], the (111) surface of Au as well as the (100) and (111) surfaces of Ir, Pt and Au 1631. These reconstructions can be divided into two classes, those which have unit cells which are small enough to be handled completely form first principles (W(OO1) and the (110) surfaces of Ir, Pt and Au) and those which can be dealt with only in terms of hybrid methods where models are used to describe the reconstructions, the model parameters, however, being determined from first principles, the Mo(001) being a typical example due to the incommensurate structure. Reconstruction studies were originally carried out for monolayer systems. The alkali, alkaline earth, transition metals [641 as well as Hg [65] and Ag [66] monolayers have been studied by total-energy methods. Most of these systems, however, have not been fully studied with respect to possible reconstructions since very often it was assumed that they formed a hexagonal close-packed structure with the bulk lattice constant. The studies for Cs [67], Ag [66] and Au [66] confirmed indeed that the hexagonal close-packed structure is the most stable one, however, the optimum lattice constants deviate significantly from the bulk values. This, of course, is significant for answering questions about the structure of monolayer adsorbates, especially for systems with weak adsorbate-substrate interactions. Furthermore, this study indicates clearly the presence of competing effects in the reconstructions of Ir, Pt, and Au(ll1) and A~(1001 where the topmost layer forms a denser phase than the substrate. The energy gain due to a uniform contraction in the topmost layer is bigger than the energy loss due to the fact that surface and substrate are no longer in registry. These effects have been studied successfully recently for Au(lll) [68] and Au(100) [66] by using a two-dimensional Frenkel-Kontorova model. The model parameters have been determined from firstprinciples total-energy calculations. The minimum-energy configurations for these models arc in excellent agreement with the observed reconstructions for Au(lllI and AutlOO). Among the bee materials, the reconstruction of WtOOlI has also been studied completely by first-principles methods [69]. At low temperatures, these surfaces exhibit a ~(2 X 2) superstructure. However, for a long time it was not clear whether this structure involved an M,-phonon distortion (fig. 1) (surface atoms alternately moving up and down) or an MS-phonon distortion (fig. 2) which is directed along the (11) direction (Debe-King model). The ab initio total-energy methods clearly favored the MS-distortion and it was found that for W the MS-distortion is restricted to the last layer and does not involve deeper layers. This system has also been studied by a hybrid method where first-principles results for the unreconstructed surface are used to determine the parameters for a tight-binding description of the surface vibrations [33]. It could be shown that the reconstruction is a consequence of Fermi-surface nesting which leads to an instability in the susceptibility ~(4 ,,I for a certain two-dimensional wave vector q ,,. The instability occurs for W(OO1) for a q,, -value which corresponds to the MS-phonon model. For Mo(OOl), the experimentally observed reconstruction is an incommensurate

K.P. Bohnen, KM Ho / Structure and dynamics at metal surfaces

111

M, (up/down)

Fig. 1. M,-phonon distortion

~(2.2 x 2.2) structure. This has been dealt with again using the above-mentioned hybrid method [34]. It could be shown that indeed an instability in the susceptibility occurred for a wave vector corresponding to a ~(2.2 X 2.2) structure. The difference in behavior between MO and W is due to the smaller value of the Fermi wave vector for W which is a consequence of the stronger relativistic effects. Among all reconstructed clean metal surfaces, the Au(llO)(l X 2) missing-row structure is by far the most thoroughly studied one. Total-energy calculations for the missing-row structure gave detailed information about the relaxations which accompany this reconstruction ill]. Besides a large inward relaxation for the outermost layer, a pairing of atoms in the second layer and a buckling of atoms in the third layer have been calculated. The structure of the fully relaxed (1 x 2) geometry is shown in fig. 3. While qualitatively similar effects are found within the framework of the effective-medium theory, the embedded-atom theory or the glue model, the actual numbers for the relaxation effects vary up to 10% compared to experiment and first-principles results [ll-131. The most striking difference being the fact that the pairing of atoms in the second layer is of opposite sign in the approximate treatments compared to experiment and first-principles calculation results. Although it seems at first to be only a small effect, it indicates that certain forces are not described properly which has dramatic consequences for experiments probing just these forces. Surface-phonon measurements are probing these forces and as recent experimental results indicate, the failure to describe the forces properly in these semi-empirical models is related to the existence of spurious modes above the bulk-phonon band edge [751. The first-principles total-energy methods do not have this artifact [16]. The failure to describe the pairing of atoms in the second layer correctly is related to the fact that these simplified methods do not treat charge rearrangements properly. The driving forces for the reconstruction can easily be seen from fig.

M, (zig-zag

lateral shift)

Fig. 2. MS-phonon distortion.

II?

Fig. 3. Schematic

drawing

of the geometry

of the missing-row

structure

for the (1 x 7) reconstructed

Au( 110) surface.

4 where the original charge density resulting from cutting the bulk is compared to the smoothed one as obtained by the Smoluchowski argument. This smoothed charge density immediately forces atoms to move in the direction as seen in the experiment. This smoothing effect which comes about the self-consistency requirement is missing in the approximate methods. For this Au(llO> surface, a thorough study of the stability and the driving mechanism has been carried out [70]. The gain in kinetic energy due to the delocalized electrons is favoring the (1 X 2) compared to the (1 x 1) structure. This is true for the ideal (1 X 2) and (1 X 1) surface as well as for the fully relaxed ones. The relaxation energy stabilizes the (1 X 2) even more. Studies for the (1 x 3) have shown that the ideal structure is again lower in energy and this would be the case also for the (1 X 4). . . (1 X n). The kinetic energy is favoring the (1 X n 1 structure for large n. If one takes into account the relaxation effects, the energy difference between (1 x 2), (1 x 3), and (1 x 4) is substantially reduced and within the limits of accuracy of our calculations [71]. The small energy difference is suggesting that in real systems the actual structures seen might be defect-stabilized. Besides explaining why Au(llO> shows the (1 X 2) structure, these calculations have also been successful in explaining the difference between A~(1101 and Ag(ll0) where the Ag(ll0) surfaces do not reconstruct. An important role in determining the difference is played by the d-electrons. In these systems, a substantial part of the bonding forces between the atoms comes from the d-s hybridization, the contribution of which is bigger for Au than for Ag. This leads to a larger gain in kinetic energy for Au than for Ag. The origin of the larger hybridization results from the strong relativistic effects in the Sd metals which move the

. Fig. 4. Schematic situation

before

.

representation charge

.

. of charge

rearrangement,

full

. distribution

.

.

.

at the (I X 2) missing-row

line indicates

the final smoothed

indicate the atomic positions before relaxation.

surface.

Dashed

charge distribution.

line indicates Heavy

the

dots (0)

K.P. Bohnen, KM. Ho / Structure and dynamics at metal surfaces

s-band closer to the d-band. In addition, there is the compared to the 4d ones. For Ag(llO), however, the reconstruction can be adsorption of alkali metals. All these effects could be total-energy calculations in excellent agreement with

larger

d-d

overlap

113

for the 5d electrons

induced by an electric field [721 or the consistently described by first-principles experimental results.

6. Surface phonons Having discussed so far theoretical treatments for lattice relaxation and lattice reconstruction we will now concentrate on methods to calculate the surface phonon modes from first-principles total-energy studies. The simplified approaches which have been used in the past have already been discussed in the introduction. Results in comparison with experiment can be found in ref. [115]. Consider the vibrational modes of a slab consisting of N atomic layers (N = 50-200). Because of translational periodicity parallel to the surface, the modes can be characterized by their wave vectors (q,,) in the surface Brillouin zone (SBZ). For a given q,, , the displacement of atoms in the nth atomic layer is given by u(R, t) = U, ei’q~~‘R-““. The frequencies and polarization vectors for all the modes in the slab with surface wave vector q,, can then be found by diagonalization of a 3N X 3N matrix equation:

C

Diajp(

4

[i)"jp

=

--“iW2Uia)

(14)

where (Y and p indicate the x, y and z directions and Dii(q,,) is a 3 X 3 matrix representing the interplanar force constants coupling the motions of the ith and jth atomic layers. In the bulk region of the slab, these interlayer coupling constants can be obtained from first-principles bulk-phonon calculations or from a simple Born von KQrman analysis of the experimentally measured bulk-phonon dispersion curves in the direction perpendicular to the slab. The calculation of the interlayer force constant matrices for the surface layers will be discussed below. Advances in computer speed and computational techniques in the past decade have enabled us to calculate total energies and forces at crystalline surfaces with a high degree of accuracy from first-principles self-consistent calculations. Detailed information on the equilibrium surface geometry can be obtained from precise first-principles self-consistent total-energy calculations of surfaces using the density-functional approach as has been shown in the preceding sections. For selected q ,,‘s in the surface Brillouin zone, the distortions induced by the surface phonons are commensurate with the surface unit cell. In such cases, the distorted system still has surface periodicity and we can calculate the change in total energy as well as the atomic forces induced by the distortion. Of course, such calculations often involve surface unit cells which are larger than the undistorted case. Since the computational effort increases rapidly with the size of the unit cell, it is obvious that such an approach can be used only at special wave vectors e.g. at the zone center or high-symmetry points at the zone boundary. To calculate the surface force constants, we start with the zero-force (equilibrium) geometry (table 1) and distort the surface layer slightly by small atomic displacements corresponding to the surface wave vector q,, under consideration. The self-consistent electronic structure for each of the off-equilibrium geometries is calculated and the forces exerted on the atomic layers in the slab evaluated using the Hellmann-Feynman theorem [41]. From this we obtain the interplanar force constants coupling the surface with the subsurface layers <.a( R) =

CDia,,Li~lB eiqli’R. P

(15)

113

Tutorials on Selected Topics in Modem

Surface Science

By moving the surface layer in three orthogonal directions, the interlayer force constant matrices coupling the top layer with all other layers can be determined. These surface force constants are then used together with the bulk interlayer force constants to construct the full dynamical matrix of the slab and to obtain all the vibrational modes of the slab at the wave vector q,, . It should be emphasized that this approach does not require the knowledge of the eigenvector as input for the calculation, in contrast to the widely used “frozen phonon” method [73]. Such ab-initio calculations of interplanar surface force constants have been carried out for a number of metal surfaces including Na [47], AI [74,46], Cu [58,75-771, Ag [75,78] and Au 1161. The frequencies for surface modes are listed in table 2. For comparison also results from other theoretical treatments as well as available experimental data are included, For detailed information on eigenvectors and surface resonances the reader is referred to the original literature. In many cases, a simple comparison of the surface-phonon frequencies with the experimental data is not sufficient to discriminate between different theoretical models. This is because of two reasons. Firstly, not all the surface modes are contained in the experimental data. For example, in both EELS and He scattering experiments, it is usual to have the incident beam scattered in a plane perpendicular to the surface. With such a geometry, the horizontal shear mode with vibrations perpendicular to the incident plane cannot be observed. It is interesting to note that these modes are usually very localized at the surface and thus provide the most sensitive probe of changes in interatomic interactions at the surface [74,76]. Only very recently it has been possible to measure these shear horizontal modes for Ni(ll0) [82]. Secondly, the scattering matrix elements of different modes can change with different scattering conditions. This can cause shifts in the observed peak positions when a surface mode is close to another surface mode or exists in resonance with a bulk-phonon background. Comparison with experimental data in such cases requires a detailed calculation of the scattering cross-sections using the polarization vectors of the various normal modes. For EELS spectra the combination of first-principles phonon calculations with a full multiple-scattering calculation of the cross-section has been carried out recently for a number of systems very successfully [79,81]. Due to the interplay between experiment and theory surface modes close to bulk edges as well as surface resonances could be clearly identified. Concerning data obtained with inelastic helium scattering the agreement between experiment and theory is not as perfect. While there are no problems with true surface modes there seems to be a difficulty to explain the observed intensities for resonance modes, the best known example being the Cu(ll1) surface where the surface resonance shows up in the spectra at least as strong as the Rayleigh wave [83]. Recently it has been suggested that this is a dynamical effect which results from the charge-density modulations induced by the phonon [84]. Again first-principles methods discussed in this article offer a unique way to test these suggestions since they allow the calculation of charge-density modulations due to the phonons. Using the Norskov formula [85] which connects charge density and interaction potential these charge modulations lead to modulations of the scattering potential which might explain the observed helium scattering intensities. The major disadvantage of the first-principles total-energy method is the fact that it can only be used at selected wave vectors. In some cases, for example, to study surface-state nesting effects, it is necessary to have a scheme which can interpolate between the first-principles results to obtain the entire surface phonon dispersion in the SBZ. A complementary approach to the method described above uses perturbation theory to compute the interlayer force constants. Although first-principles calculations using the perturbative approach exist

K.P. Bohnen, K.M. Ho / Structure and dynamics at metal surfaces Table 2 Surface phonon

frequencies

for clean metal surfaces

First-principles total-energy density-functional calculations

Other

at high-symmetry

theoretical

points (frequencies

approaches

Semi-empirical

115

are given in THz) Experiments

Dielectric

screening

-

Na(l10)

S

Y

AltllO)

X

Y

S AhlOO)

Y

M Cu(ll0)

X

Y

s

cm 100)

X

M

Cu(ll1)

M K

2.4 [47] 2.8 3.9 0.8 2.5 3.5 4.0 4.2 [741 4.0 5.5 6.1 6.5 7.7 3.4 2.5 3.6 6.2 7.9 3.6 [46] 3.0 8.1 4.9

4.6 [36]

3.6 [107]

3.3 3.1

3.3 2.3

3.6 [lo81

3.2 [76] 3.2 4.4 5.2 5.9 1.7 2.9 3.1 4.9 2.8 3.1 5.7

2.9 [29] 3.2 3.8

3.3 1301

3.1 [109,1101

5.6 1.7 2.5 2.9 4.5

5.9 2.3

5.8 1.7

3.0 4.8

2.9

2.3 [75,81] 3.1 3.4 6.2 4.3 4.9 5.0

2.1 [281 3.0

3.3 [77,80] 6.6 3.4 5.5

6.1 4.0 6.5

1.9 11141 2.9 3.8 5.8 3.9 5.0 5.1

3.2 [111,112] 6.1 4.1 4.9

3.2 [83,112] 6.3 3.4

Tutoriuls on Selected Topics in MO&W Surface S&me

I I6

Table

2 (continued)

First-principles

total-energy

Other theoretical

density-functional

Semi-empirical

calculations

Ag( I IO)

X

approaches

Experiments

Dielectric

screening

2.0 [lo51 2.0 3.4 3.7 4.0

Y

1.1

I .2 [I I.31

1.7 I.0

I.‘)

2.5 3.3 s

3.1

I.9 2.1 3.U

Agt 100)

X

M

AgflllJ

M

I.4 [75,7Y,Xll

1.6 [I

7. I

2.1

4.1

4.0

2.5

2.8

3.2

3.5

3.4

3.1

IAl 2.1 [7Y,811

2.4

2.2 [X0. I Oh]

2.Ofr [X31

4.6 AutllOJ

K

3.7

I_

1.6 [161

0.8 [I21

1.X

2.4

(I X2)

X

I.4 [IO]

3.5

4. I ‘I’

3.7

4.9 ,”

I.8

I.6

I .7

4.0

4.1 “1

4.2-4.4

4.7

45 ‘I’

3.6-4.2

2.x 3.6

,‘J These modes are true surface modes above the bulk-phonon glue model. bulk-phonon

This is an artefact maximum

of the model which predicts

measured with neutron

maximum

according to the calculation

a hulk-phonon

maximum

based on the

17% lower than the true

diffraction.

[361, adopted schemes represent the total energy of the crystal as the sum of a band-structure term and a term from short-ranged pairwise interaction [86,87]. The surface electronic structure contribution enters through the band-structure term which is just the sum of occupied electronic eigenvalues. The electronic structure of the system is described by empirical tight-binding schemes obtained from fitting to bulk and surface electronic structures. Such tight-binding schemes describe well the bulk-phonon dispersion curves of bulk semiconductors as well as transition metals and transition-metal compounds [86,87]. Surfacephonon calculations for both semiconductor surfaces [88] as well as transition-metal surfaces [33,34] have been carried out with this method. The strong point of the method is that it is easy to calculate full surface-phonon dispersion curves for general q,, and to examine the effect of the surface electronic structure on the surface vibrations of the system. For example,

K.P. Bohnen, K.M. Ho / Structure and dynamics at metal surfaces

117

recent calculations on the surface-phonon anomalies on the W(OO1) [331 and Mo(001) [341 surfaces reaffirmed the importance of surface-state nesting in causing the reconstruction on these surfaces.

7. Conclusions Although the number of first-principles calculations of surface relaxations, reconstructions and surface phonons is still fairly small all calculations so far have given results in good agreement with experimental findings. These studies are, however, not restricted to clean metal surfaces. A number of adsorbate systems have been studied already. In some cases structures have been proposed which later have been experimentally verified [89]. Recently, these methods have also been applied to study growth mechanisms 1901. Concerning the surface-phonon aspect the most promising future development lies in the simulation of EELS - as well as He - spectra from first principles. This allows the determination of surface resonances and thus leads to a much more detailed understanding of the interaction potential at surfaces. All these calculations are very involved, however, due to the rapid development of supercomputers as well as new algorithms in the future more complicated systems can be treated by these first-principles methods.

Acknowledgments

We would like to thank W. Weber, B.N. Harmon, C.T. Chan, C.L. Fu, X.W. Wang, Th. Rodach, Y. Chen, S.Y. Tong, C. Stassis and Ch. Wijll for collaborations and helpful discussions. Ames Laboratory is operated for the USDOE by Iowa State University under contract No. W-7405 Eng-82. This work was supported by the Director for Energy Research, Office of Basic Energy Sciences; including a grant of computer time on the CRAY-XMP computer at the Lawrence Livermore Laboratory and by NATO grant No. RG(85/0516).

References [l] G. Ertl and J. Kiippers, Low Energy Electrons and Surface Chemistry, 2nd ed. (VCH, Weinheim, 1985) p. 201. [2] C. Varlas, in: Topics in Current Physics, Vol. 41, Eds. W. Schommers and P. von Blankenhagen (Springer, New York, 1986) p. 111. [3] K.H. Rieder, in: Topics in Current Physics, Vol. 41. Eds. W. Schommers and P. von Blankenhagen (Springer, New York, 1986) p. 17; J. Als-Nielsen, in: Topics in Current Physics, Vol. 43, Eds. W. Schommers and P. van Blankenhagen (Springer, New York, 1987) p. 181. [4] H. Ibach and D.L. Mills, Electron Energy Loss Spectroscopy and Surface Vibrations (Academic Press, New York, 1982); H. Ibach and T.S. Rahman, in: Chemistry and Physics of Solid Surfaces, Eds. R. Vanselow and R. Howe (Springer, New York, 1984). [5] J.P. Toennies, J. Vat. Sci. Technol. A 5 (1987) 440; in: Proc. Solvay Conf. on Surface Science, Ed. F. de Wette (Springer, Heidelberg, 1988). [6] J.R. Perdew and R. Monnier, J. Phys. F 10 (1980) L287; U. Landmann, R.N. Hill and M. Mostoller, Phys. Rev. B 21 (1980) 448; R.N. Barnett, U. Landmann and C.L. Cleveland, Phys. Rev. B 27 (1983) 6534; B 28 (1983) 1685.

IIX

Tutoriuls on Sekctrd

Topics in MO&WI Surf&c

Science

[7] F. Ercolessi, E. Tosatti and M. Parinello, Phys. Rev. Lett. 57 (1YXh) 719; F. Ercolessi, M. Parinello and E. Tosatti, Surf. Sci. 177 (1986) 314. [Xl S.M. Foiles, M.J. Baskes and M.S. Daw, Phys, Rev. B 33 (lY86) 7Y83. [Y] K.W. Jacobsen. J.K. Norskov and M. Puska. Phys. Rev. B 35 (19X7) 7423. [IO] T. Ning, 0. Yu and Y. Ye, Surf. Sci. 201 (1988) LXS7. [I I] K.M. Ho and K.P. Bohnen, Europhys. Lett. 4 (1987) 345; J.W. Davenport and M. Weinert. Phys. Rev. Lett. 58 (lYX7) 13X2. [12] SM. Foiles. Surf. Sci. IYl (19x7) L77Y. [I31 K.W. Jacobsen and J.K. Norskov. in: Proc. 2nd Int. Conf. on the Structure of Surfaces. The Structure ot Surfaces II. Eds. J.F. van der Veen and M.H. Van Hove. Springer Series in Surface Sciences (Springer, New York, 1988). [14] W. Moritz and D. Wolf, Surf. Sci. 163 (1985) LhSS. [IS] M. Cope1 and T. Gustafsson, Phys. Rev. Lett. 57 (19%) 723. [16] A.M. Lahee. J.P. Toennies. C. Wiill, K.P. Bohnen and K.M. Ho. Europhys. Lett. IO (lY89) 261; B. Voiytlander, S. Lehwald. H. Ibach, K.P. Bohnen and K.M. Ho. Phys. Rev. B 40 (19X9) 8068. [17] P. Hohenherg and W. Kohn, Phys. Rev. B 136 (1964) 864; W. Kohn and L.J. Sham, Phys. Rev. A I40 (1965) 1133. [IX] A.G. Eguiluz, Phys. Rev. B 35 (1987) 5473. [IY] See, for example, K.M. Ho and K.P. Bohnen, Phya. Rev. B 32 (19X5) 3446, and references given therein. [20] A. Beeler, M. Scheffler. 0. Jepsen and 0. Gunnarsson. Phyx. Rev. Lett. 54 (1985) 2525. [2l] P.J. Feibelman, Phys. Rev. B 35 (1987) 2626. [22] V.L. Moruzzi. J.F. Janak and A.R. Williams, Calculated Electronic Properties of Metals (Pergamon, New York. 197X). [23] R.E. Allen, G.P. Alldredge and F.W. de Wette. Phys. Rev. B 4 (lY71) lh48, 1661. [24] G. Benedek, G.P. Brivio. L. Miglio and V.R. Velasco, Phys. Rev. B 20 (1982) 497. [2S] V. Bortolani, A. Franchini and G. Santoro, in: Electronic Structure. Dynamics, and Quantum Structural Properties of Condensed Matter. Eds. J.T. Devreese and P. Van Camp (Plenum. New York, 19%). [26] T.S. Rahman. D.L. Mills and J.E. Black. Phys. Rev. B 27 (lYX3) 4059; J.E. Black, T.S. Rahman and D.L. Mills, Phys. Rev. B 27 (19X3) 4072. [27] J.E. Black and R.F. Wallis. Phys. Rev. B 29 (1984) 6972. [2X] J.S. Nelson. E.C. Sowa and MS. Daw, Phys. Rev. Lett. 61 (IYXX) lY77: Luo Ningsheng. Xu Wenlan and SC. Shen, Solid State Commun. 67 (19X8) X37. [29] L. Yang and T.S. Rahman, Phys. Rev. Lett. 67 (1991) 2327. [30] P. Ditlevsen and J.K. Norskov. Surf. Sci. 254 (1YYl) 261. [3l] X.0. Wang, G.L. Chiarotti. F. Ercolessi and E. Tosatti. Phys. Rev. B 3X (1988) X131. (321 See K.M. Ho, C.L. Fu and B.N. Harmon, Phys. Rev. B 2Y (1984) 1575. and references therein. [33] X.W. Wang and W. Wcber. Phys. Rev. Lett. 58 (1987) 1452. [34] X.W. Wang. C.T. Ghan. K.M. Ilo and W. Weber, Phys. Rev. Lctt. h0 (IYXX) 2066. [35] O.L. Alerhand and E.J. Mele, Phys. Rev. B 37 (198X) 2536. [36] A.G. Eguiluz. A.A. Maradudin and R.F. Wallis. Phys. Rev. Lett. 60 ( IYXX) 309. [37] D. Pines, Solid State Phys. I (IYSS) 367: L. Hedin and B.J. Lundqvist, J. Phys. C 4 (1971) 2064: S.H. Vosko. L. Wilk and M. Nusair. Can. J. Phys. 5X (1980) 1200: L. Wilk and S.H. Vosko. J. Phys. C IS (1982) 213X: J.P. Perdew and A. Zunger, Phys. Rev. B 23 (lYX1) 5048. [3X] W.F. Pickett, Comput. Phys. Rep. Y (1989) I IS. [3Y] H. Krakauer, M. Posternak and A.J. Freeman, Phys. Rev. B 19 (197’)) 1706: E. Wimmer, H. Krakauer, M. Weinert and A.J. Freeman. Phys. Rev. B 24 (19x1) 864. [40] G.B. Bachelet, D.R. Hamann and M. Schluter. Phys. Rev. B 26 (1982) 41YY; G.P. Kerker. J. Phys. C I3 (IYXO) LIX9: B.M. Bylander and L. Kleinman. Phys. Rev. B 20 (10x4) 2774. [41] H. Hellmann, Einfiihrung in die Quantentheorie (Deutickc. Leipzig, lY37) p. 2X5: R.P. Feynman. Phys. Rev. 56 (1939) 340. [42] S.G. Louie. K.H. Ilo and M.L. Cohen. Phys. Rev. B 19 (1979) 1774. [43] C. Elsiisser, N. Takeuchi, K.M. Ho. C.T. Chan. P. Braun and M. Fahnle. J. Phys.: Condens. Matter 2 (lY90) 3371. [44] K.M. Ho, C. Elsasser, C.T. Chan and M. F?ihnle, J. Phys.: Condens. Matter 4 (1992) 5189.

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