A model for disorder in vicinal surfaces

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Physica Scripta

A model for disorder in vicinal surfaces

To cite this article: A da Silva Carriço 1991 Phys. Scr. 44 384

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A

Model for Disorder in Vicinal Surfaces

A. da Silva Carrigo

Departamento de Fisica, -Centro de Cihcias Exatas, Universidade Federal do Rio Grande do Norte, Campus Universitario, 59072 Natal-RN Brasil Received March 10, 1991; accepted June 4 , 1991

Abstract

The average spectrum of a disordered f.c.c. vicinal surface is obtained from effective medium Green’s function. The disorder is restricted to the surface plane and the scattering in the surface is treated in the Coherent Potential Approximation. The method applies to site-diagonal disorder and is valid for any crystallographic orientation. An application is made for an s-like Tight- Binding Hamiltonian with substitutional disorder. Depending on the strength of the perturbation potential, resonances or local bands are found in the surface spectrum.

1. Introduction

The increasing interest in the characterization of surfaces and interfaces has driven some experimental and theoretical research effort to the study of disordered surfaces. A great deal of work has been dedicated to the surface electronic properties of metal alloys and also to ternary semiconductors. Controlled disordering of interfaces may lead to special features. It is now accepted, for instance, that adjusting the concentration of selected impurities in semiconductor-oxide interfaces leads to band gap tailoring [l], which is highly desirable for technological reasons.

Disordering in metallic surfaces is also a subject of great interest due to the modifications it may introduce in the surface properties of catalysts.

So far most of the theoretical effort has been dedicated to the study of low Miller indexes metallic surfaces. The Koringa- Kohn-Rostoker theory has been used in conjuction with the Coherent Potential Approximation to describe the photoemission from metallic alloys [2], and also to study alloy effects in a single (unsupported) metal plane [3]. Ternary semiconductor surfaces have been described either by linear chain models [4] or more representative calculations [5], both making use of a configurational average over possible distribution of defects in a set of few planes near the surface, below which is placed a bulk average crystal.

Another approach that has been recently proposed, for semiconductor surfaces, is to represent the disorder by an effective Lloyd Hamiltonian with a Lorentzian distribution of diagonal elements [6].

Disorder at surfaces or interfaces, may incorporate a variety of cases such as the presence of impurities, lattice distortions, missing bonds or atoms etc. In many cases a site disorder model is a good approximation. The possible widths of features in the surface spectrum are expected to be strongly influenced by the concentration and energy-position of the perturbation relative to the spectrum of the host material.

surface plane disorder. Therefore only the surface, or interface, concentration of defects needs to be determined. Two subjects of great interest that falls into this class are, for instance, the effect of partial hydrogenation of semiconductor- oxide interfaces and the adsorption of small molecules at low partial pressures on metallic surfaces.

Vicinal metallic surfaces are of considerable interest because of their special catalytic properties [7]. These surfaces are obtained by cleaving the crystals at a very small angle relative to low Miller indexes (LMI) surface. As a result the surface forms with high Miller indexes (HMI) and its geometrical structure consists of LMI terraces separated by steps often a few atoms in height. For a review of possible defects see [7]. Vicinal surfaces are, therefore, quite open structures in which there is a weak interaction between atoms in the same plane as compared to the interaction between atoms in neighboring planes.

Most of the theorectical models simulate the steps by LMI surfaces, usually (100) from which strips of atoms are periodically taken off [8, 91. The actual dense packing of planes has been considered for clean transition metal surfaces [lo], in this case the calculation is made more complex due to the small distance between planes.

The vibrational spectrum of vicinal surfaces are a valuable tool to characterize relaxation processes. In the last decade a few theoretical models have been proposed to study local phonons in the Pt(332) surface [I 1-14]. This followed from the results of EELS experiments indicating a high frequency phonon localized in the Pt(332) step atom [15].

Disordering in vicinal surfaces is a subject of great complex- ity due to the large surface unit cell and consequent number of inequivalent atoms near the surface. Our purpose is to present a model calculation which, although not directly connected to a representative metal surface, is valid for a general Tight Binding Hamiltonian, as well as for phonons, and incorportates the effects of disorder in a vicinal surface. We focus on surface disorder in a single plane and obtain the average spectrum treating the multiple scattering in the disordered plane within the Coherent Potential Approximation (CPA) [16]. In section 2 we present the Green’s function for a perfect lattice with a planar defect. Section 3 is devoted to adapt the CPA method for site disorder in a single plane. Following, in section 4, we apply the method to a model (103) f.c.c. surface with substitutional disorder and in the final section we discuss our results and possible future applications. For alloy surfaces an additional difficulty lies in determining

the appropriate concentration profile.

Many physical systems of interest can be studied in a

2. Green’s functions for a planar defect in a perfect vicinal surface

_ _ -

rather neat way if -the leading effect of disorder is fairly localized. In these cases a first approach is to neglect all but

The stacking of a vicinal suface is characterized by large unit cell planes densely packed. In the tight binding description,

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A Model for Disorder in Vicinal Suvfaces 385 which is also valid for phonons with minor modifications, the

Green’s function is defined as the resolvent of a Hamiltonian whose elements are defined in real space (site-energies and hopping intregals, or pair-wise energy of interaction in the case of phonons). The equations of motion are given by

( 2 - h ) g = 1 ( 1 )

where z stands for the energy for the case or electrons of the Mw2 for the case of phonons.

Writing eq. ( 1 ) in a basis of plane waves { n , k } propagating in each plane (n) in a given direction ( k is the momentum parallel to the surface) we obtain an infinite set of coupled equations for the Green’s functions.

(Z

-

hm(k))gnm(k) = 6nm

+ C

h n p ( k ) g p m ( k ) P

(n, m,

P

2 0) (2)

In this set of equations, written for a fixed k-vector, the sub-indexes refer to the planes ( n = 0 being the surface), and all the matrices ( z , h and g) have the size determined by the representation chosen for the Hamiltonian [16]. For the description of surface electronic properties the matrices hnm include the couplings between Bloch waves of the appropriate orbital character propagating in planes n and m in the direction specified by the wave vector k (parallel to the surface). For the description of surface vibrational properties, with force constant-fitting of the bands, the matrices have order 3 [17]. In the following we adopt the terminology appropriate for a tight-binding description of electrons.

In order to have a condensed description the planes are grouped into principal layers, in such a way as to have only adjacent layers coupled [18]. In this case we rewrite eq. (1) in the form

0

(2 - Hnn(k))Gjm(k) = 6nm

+

Hnn+iG,O+im

+

Hnn-iGn-im

(3) where now the matrices include all the coupling between principal layers.

A common feature of vicinal surfaces is that each principal layer contains a big number of planes. For Pt(332), for instance, this number is 6 [14]. This number increases with the Miller indexes and is determined by two factors. The first is the crystalographical orientation of the surface itself, which determine the interplane spacing. The second factor is the range of interaction between atoms. For an electronic hamiltonian this amounts to choose the possible hoppings between site-orbitals (first, second neighbors, etc.). If no relaxation is considered the H matrices are invariant from the 2nd layer onwards. In the following we denote H ; , H,, H I

and H2 the intralayer matrix for the surface and subsequent layers and the interlayer coupling matrices in the directions pointing from and to the surface, respectively.

If N is the number of planes in a principal layer then H; incorporates all the couplings involving the planesj = 0, 1,

.

,

N

-

1, whereas H I incorportates all the couplings involving the sets of planesj = 0,l

. .

.

,

n

-

1 and j = N , N

+

1 ,

.

, 2N. The same applies for the Green’s functions. The surface spectrum is obtained from G;,(k). The infinte set of eq. (3), involving the layer Green’s functions, can be decoupled using transfer matrices [I71 defined by G;,,,(k) =

T ( k ) G f o ( k ) , satisfying

( Z - HO(W T(k) = K l ( k )

+

HI

(k)

T ( N 2 (4)

T(K) is obtained from an iterative solution of eq. (4) and the surface layer Green’s function is given by G!,(k) = ( 2

-

H,’(k)

-

H , ( k )

T(k))-’

.

Local Green’s functions are obtained from an average of the spectral Green’s functions over the surface Brillouin zone. A useful guide in selecting the surface Brillouin grid of K-points was given by Cunningham, in the past [19].

Next let us consider a planar site-diagonal defect in the surface (first plane of the surface layer). Commonly the leading effect of substitutional disorder is a site-perturbation [20]. Therefore we represent the perturbation potential by

Vgg(R,, R I ) = 6H6ug6R,6R,R, ( 5 ) where Ro is the impurity site, E is the site-energy which

is modified by the impurity and 6H is the strength of the perturbation. The corresponding ( n k ) transform is V,,(k) =

6 H 6,, 6,, and acts only on the first plane subspace of the surface principal layer. The perturbed Green’s function is determined from

G ( k ) = G o ( k )

+

G o ( k ) ( l

-

V ( k ) G o ( k ) ) - ’ G o ( k ) ( 6 )

Now, if we are interested in matrix elements of G ( k ) within the impurity subspace, eq. ( 6 ) is used in the defect space. Therefore, given the form of the planar perturbation, we need only to deal with matrix elements of G ( k ) in the surface plane. As we shall see in the next section for a CPA description of a disordered surface plane, the effective medium consists of a planar defect substitutional problems in which the perturbation is energy dependent.

3. Effective medium to represent the disorder in a single

The Coherent Potential Approximation for a random distribution of impurities in a single plane, consists in representing the disordered plane by a coherent one, with an average self-energy C ( E ) , around which the self-energy fluctuations produce no single site scattering. The idea, as in bulk calcu- lations [ 161, is to define a uniform medium in which all single site scattering is already taken into account. Since the disorder is confined to a single plane, this amounts to redefine the plane self-energy. The renormalyzed self-energy is obtained by requiring that the average single site t-matrix for fluctuations from the average, by zero. Formally we have first to determine the Green’s function for a planar defect: a plane with self-energy C ( E ) . The value of

C(E)

is then chosen to satisfy the CPA condition ( ( t ) = 0). We need to know only the local surface layer Green’s function Goo within the defect space, for the CPA equation reads

plane

c(E

-

C(E))(l

-

G W ( 1 , 1))-’

+

(C - l ) C ( E )

x (1 - G w ( l , 1)(E - C(E))-’ = 0 (7)

where we denoted the impurity plane by number 1 .

Equation ( 7 ) is an algebraic equation to be solved simul- taneously with the equation defining the coherent effective plane Green’s function ( G ) . G is given by

G = Go

+

G O ( l -CGo)-’Go (8)

where Go is the Green’s function for a perfect (clean) surface. Equations (7) and (8) involve local Green’s functions that are obtained from the spectral Green’s functions (for fixed K- values) by averaging over the surface Brillouin zone.

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386 A . da Silva Carrigo

110s 1

I

Fig. 1. The geometry of the (103) f.c.c. surface, showing the primitive vectors and the four inequivalent atoms.

4. Application to an s-like Tight-Binding Hamiltonian on a

We choose to apply the method to an (103) f.c.c. surface and a nearest neighbor s-like Tight-Binding Hamiltonian because, although simple, this system has the necessary ingredients of the discussion. The principal layers contained four planes.

The geometry of the surface is displayed in Fig. 1. The basic translation vectors are given by t , =a(O, 1, 0) and t2 = (a/2)(3, 0, - l), where a is the lattice constant. The corresponding reciprocal lattice is generated by the vectors GI = (2n/a)(O, 1, 0) and G2 = (2n/5a)(3, 0,

-

1).

Due to the dense packing, the equations of motion (2) couple the Green’s function in a given plane with those of eight nearest neighbor planes (four in each direction). Therefore the principal layer consists of four adjacent (103) planes. A principal layer can also be described as a single (103) plane with a basis of four atoms. Atom 1 being the one in the plane itself and the other three corresponding to the remaining planes in the layer. This is illustrated in Fig. 1 where we can also see the (001) terraces of width 3a/2 with four inequivalent atoms limited by (010) steps of height 4 2 . The coordination numbers of atoms 1-4 are 11, 9, 8 and 6. Therefore atom 1 is almost in the same environment as in the bulk, so we expect its spectrum to be rather similar to that of a bulk atom. Local Green’s functions have been obtained from an average over a set of 16 points in the irreducible part of the surface Brillouin zone [19]. The model Hamiltonian was defined by a hopping of 0.5 between nearest neighbors and the energy is measured from the atomic s-level. In this arbitrary energy scale the bulk spectrum ranges from - 2 to 6.

There are four inequivalent positions for substitutional impurities in the surface. They correspond to having a foreign atom substituting either one of the atoms in the terrace (atoms 2 or 3), the atom in the step corner (atom 1) or the atom in the step edge (atom 4). We consider here only the lost possibility. In this case the surface disorder consists on having two kinds of atoms in the step edge. Only the resulting site-disorder is accounted for in the present calculation. Therefore we keep the same hopping matrix elements between impurities or an impurity and a host atom. This is generally a good starting point for metal alloys [20].

Since 6H is confined to the surface plane, the self-consistent determination of the renormalized self-energy C ( E ) involves only the fourth diagonal element of the surface layer Green’s functions: G,(4, 4). Then eqs. (7) and (8) read

(103) f.c.c. lattice

Goo(4, 4) = (Gk(4, 4, k )

+

(1

-

VE)G&(4, 4, k ) ) - ) (9)

c(P - W ) ) ( l -

(P

-

wmGOo(4, 4))-1

l(d)

-10 -5 0 5

lo

ENERGY (arb. units

1

-10 -5 0 5 10

ENERGY (arb.units)

Fig. 2. Local density of states for the clean (103) surface layer. The spectrum of atoms 1 to 4 are shown in pictures (a) to (d).

+

(C

-

l)Z(E)(l

+

C(E)G,(4, 4))-I = 0 (10) In eq. (9) ( ) denotes an average over the surface Brillouin zone.

The self-consistent determination of

Z(E)

requires the simultaneous solution of eqs. (9) and (10). They were numeri- cally solved as a system of four nonlinear equations on the real and imaginary parts of C ( E ) and G,(4, 4). The roots were determined within a relative error estimated to be with an excellent convergence.

5. Discussion of the results

The local density of states (LDOS) in the first four planes of the (1 03) clean surface are shown in Fig. 2, for comparison purposes. No surface band is seen at the surface layer, but only the projection of the bulk spectrum. We observe a rapid evolution from the surface density (atom 4) to the bulk density. The effect of the increasing coordination number from atom 4 to atom 1 is clearly seen from the relative width of the spectra. The fine details of the curves are slightly smeared out due to a small imaginary part (0.02) attached to the energy in order to speed up numerical convergence.

The LDOS for the disordered surface is shown in Fig. 3. Values of the perturbation potential

(6H

=

p)

were chosen in order to examine the surface spectrum in two different regimes. The random distribution of impuritites in the surface may lead to resonances or local modes depending on whether the energy of the impurity states fall on the projection of the bulk spectrum in the surface. We considered two values for

p.

The first is

/?

= 6 and as shown in Figs. 3(a)-(b) corresponds to the split band limit. A smaller perturbation

P

= 2, was also considered. In this case the impurity surface band overlaps the surface projection of the bulk spectrum (Figs. 3(c)-(d)). The results shown in Fig. 3 are the LDOS in the surface plane. We have also examined the local densities in the other planes of the surface layer for the two values of

p.

In both cases, for different concentration of impurities, practically only the surface iteself is affected by the disorder. This effect is probably an artificial feature of the nearest neighbor approximation. = 6 a very slow surface band appears. This band is centered at the energy E = 6.6 and it is strongly localized in

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A Model for Disorder in Vicinal Surfaces ₃₈₇

cn

W l- t-

cn

LL 0

>-

m

z

w

a

k

n

0 )

b)

-10 - 5 0 5 10 ENERGY (arb.units) -10 -5 0 5 10

E NE RGY (arb. units)

Fig. 3. Local densities of states in the surface plane for two values of the

perturbation (p) and two values of the impurity concentration ( e ) .

the surface plane (atom 4), with a small leaking to other planes in the surface layer. For low concentration of impurities (c = 0.1) practically only the surface plane is affected. For #? = 2 a surface resonance band appears at the energy of

E = 2.6, predominantly localized in the surface plane but with considerable weight within the surface layer planes if the concentration is high (c = 0.6). It is interesting to notice that the transfer of weight of the projection of the bulk spectrum in the surface to either the surface band or surface resonance band is correctly described in all cases. This is a valuable information to deal with the scattering which involve the states near the surface.

of random surfaces and can be extended to representative models of metal or semiconductors. Although it is based in site diagonal disorder it can be applied to these systems with minor modifications. It is known, for instance [17], that the electronic properties of Si-Silica interfaces is well represented by a modification of the Si site-energy. A study of hydrogenation of the Si-Si02 interface is now in progress and will be reported in the future.

Another possible application of the ideas presented here is the study of the initial stages of adsorption of small molecules on vicinal surfaces. We hope this will prove to be useful in interpreting the photoemission data from disbrdered vicinal surfaces. References 1. 2. 3. 4. 5 . 6. 7. 8. 9. 10. 11. 12. 13. 14. 15. 16. 17. 18. 19. Our method can be applied to describe surface vibrations 20.

Perfetti, P., Quaresima, C., Coluzza, C., Fortunato, C. and Mar- igotondo, G . Phys. Rev. Lett. 54, (1986).

Durham, P. J., J. Phys. F Metal Phys. 11, 24475 (1981).

Wille, L. T. and Durham, P. J., J. Phys. F Metal Phys. 13, L117 (1983). Bryant, G. W., Phys. Rev. B31, (1983).

Bryant, G . W., Phys. Rev. Lett. 55, 1786 (1985).

Rodrigues, D. E. and Weiz, F. 1. Phys. Rev. B39, 1662 (1989).

Somorjai, G., Surf. Sci. 89, 496 (1979).

Tersoff, J. and Falicov, L. M., Phys. Rev. B24, 754 (1981). Salzberg, J. B. and Falicov, L. M., Phys. Rev. B15, 5320 (1977). GonGalves da Silva, C. E. T., Phys. Rev. B22, 5945 (1980). Allan, G., Surf. Sci. 85, 37 (1979).

Mostoller, M. and Landmann, U., Phys. Rev. B20, 1755 (1979). Knipp, P., Phys. Rev. B40, 7993 (1989).

Nogueira Jr., E. and Carrico, A. S. to be published.

Ibach, H. and Bruchmann, D., Phys. Rev. Lett. 41, 958 (1978). Elliott, R. J. et al., Rev. Mod. Phys. 46, 465 (1974).

See for instance CarriCo, A. S., Elliott, R. J. and Barrio, R. in Phys. Rev. B34, 872 (1986) for the case of surface electronic properties of Si or reference 14 for the case of Pt (332) phonons.

Lee, D. H. and Joannopoulos, J. D., Phys. Rev. B32,4988 (1981). Cunningham, S. L., Phys. Rev. B10, 1988 (1974).

A model for disorder in vicinal surfaces

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