ELSEVIER Applied Surface Science 123/124 (1998) 131-135

applied surface science

Electron correlation effects in the Si (111)-5 × 5 and-7 × 7 surfaces

Jos~ Ortega *, Alfredo Levy Yeyati, Fernando Flores Departamento de Ffsica Tedrica de la Materia Condensada C-V, Facultad de Ciencias, Uniuersidad Autdnoma de Madrid, E-28049

Madrid, Spain

Abstract

LDA-calculations have been performed for determining the electronic properties and the surface band structure of the Si (111)-5 × 5 surface. The six narrow surface bands that appear around the Fermi energy are essentially related with the six adatoms dangling bonds, and contain 3 electrons. From these calculations, a two-dimensional model is introduced to describe the electron correlation properties of those narrow bands. These results are compared with the ones published elsewhere for the Si (111)-7 x 7 surface. Our analysis shows that the Si (111)-5 x 5 surface has a semiconducting character while the Si (111)-7 X 7 reconstruction presents a metallic behaviour. Both surfaces exhibit important correlation effects that arise from the localization of electrons on some elementary structures having a hexagonal ring geometry. © 1998 Elsevier Science B.V.

PACS: 73.20. r; 73.20.At; 71.27.+ a

1. Introduction

The S i ( l l l ) - s u r f a c e is probably the most exten- sively studied semiconductor surface. Under specific conditions [1], it exhibits different reconstructions: in this paper, we shall discuss the electronic properties of the 5 X 5 [2,3] and 7 X 7 reconstructions. Fig. 1 shows the geometry of the currently accepted struc- tural model for the S i ( 1 1 1 ) - 5 X 5 surface: the d imer - ada tom-s t ack ing fault (DAS) model [4,5]. The 7 X 7 reconstructed surface has been analyzed elsewhere [6]; in this reference we have shown that the adatom dangling-bonds control the electron den- sity of states at the Fermi energy. Our analysis has shown that 5 electrons per unit cell fill these 12

* Corresponding author.

adatoms dangling-bonds, in such a way that 3 elec- trons get localized in the hexagonal ring structures appearing around the comer-hole (see Fig. 2b). Then, the other 2 electrons fill the adatom dangling bonds forming the dimers depicted in Fig. 2b, and con- tribute to the metallic character of the Si(l 11)-7 × 7 reconstruction.

The aim of this paper is to extend the arguments already presented for the S i ( l l l ) - 7 × 7-surface to the 5 X 5 reconstruction. In a first step, we use a local density approximation (LDA) and discuss the adatom dangling-bond surface bands and their occu- pancies. In a second step, we analyze the electron correlation effects associated with these bands by introducing a 2-dimensional many-body Hamiltonian that embodies the main properties obtained in our LDA calculation. In a final step, we show that a simple picture of the electronic properties of the

0169-4332/98/$19.00 © 1998 Elsevier Science B.V. All rights reserved. PII S01 6 9 - 4 3 3 2 ( 9 7 ) 0 0 5 1 6-3

132 J. Ortega et al. / Applied Surface Science 123 / 124 (1998) 13 I - 135

O

O

Fig. l. Unit cell of the Si (111 )-5 × 5 surface: top view and side view. The six adatoms are represented by black dots and the two rest atoms

by the dark-grey dots. The comer-hole atom is the one at the corners of the unit cell.

5 × 5-surface is obtained by assuming that the elec- trons filling the adatom dangling-bonds are localized in the hexagonal rings depicted in Fig. 2a. We conclude that, at variance with the 7 X 7-surface, the 5 × 5-reconstruction has a semiconducting character.

2. LDA-rcsults: surface bands and dangling-bonds occupancies

We have applied a local orbital self-consistent LDA-method [7] to calculate the relaxed geometry and the electronic ground state of the DAS-model for the 5 X 5-surface. Fig. 3 shows the 6 LDA-bands we have found around EF; these bands are essentially derived from the 6 adatom dangling-bonds and are filled by only 3 electrons. This occupancy can be easily predicted by using a simple 'electron counting rule ' , if we assume - as our calculations suggest - that the dangling-bonds associated with the rest and comer-hole atoms are doubly occupied (the dan- gling-bond states associated with the corner-hole and rest atoms are ~ 0.8 eV below EF). Notice that, in the 5 X 5-surface, there are 6 adatoms, 2 rest-atoms

and 1 corner-hole atom. The doubly occupancy of the rest and corner-hole atoms leaves 3 electrons filling the surface bands associated with the adatoms

As shown in Fig. 3, the 6 adatom surface bands are split into a bunch of three bands whose width is around 150 meV, which accommodate the 3 elec- trons, and another bunch of three bands located around 200 meV above the first ones. Fig. 3 also shows the fitting we have obtained to the LDA-bands by introducing the following Hamiltonian:

H = EE,~, . , , + E t,?~.~,'~ .... ( l ) i,¢r i,jcr

where E i is the energy level corresponding to each adatom dangling-bond wave-function, and tls defines the different hopping integrals. In our fitting we have used two levels E r and E, , associated with the faulted and unfaulted parts of the unit cell, and two hopping integrals, T and s, shown in Fig. 2a. Our

Out of the 9 electrons provided by the 9 dangling-bonds in the unit cell.

J. Ortega et al./Applied Su,rface Science 123 / 124 (1998) 131-135 133

a)

b)

Fig. 2. Schematic representation of the adatoms interactions: (a) the Si (111)-5 × 5 surface, and (b) the Si (111)-7 × 7 surface. The adatoms are shown as black dots. The thick lines connecting adatoms represent the big hopping-interactions, T, and the thin lines are the small hopping-interactions, s (for clarity, only a few s-interactions are represented). The unit-cell border is also shown.

results yield E r - E . = - 1 8 meV, T = - 7 0 meV and s = 21 meV.

It is worth comparing T and s with similar hopping integrals calculated for the S i ( l l l ) - 7 X 7- surface (see Fig. 2b). For this case we have obtained the following values: T(7 × 7) = - 85 meV and s(7 X 7 ) = 25 meV. The agreement between these pa-

rameters for both reconstructions is encouraging. Notice that the distances between the dangling-bonds are similar for both surfaces, suggesting that both cases should yield similar hopping integrals as our calculations show.

Hamiltonian (Eq. (1)) is a one-electron approxi- mation to the LDA-bands. Many-body effects, asso- ciated with the different Coulomb interactions be- tween the adatoms dangling-bonds, play an impor- tant role in these surfaces. We describe these effects

by means of the following 2-dimensional Hamilto- nian:

i,o- i,jcr

I Jiinl,f n i~,, (2) i4~j,o-o-'

where U, Jii are the intrasite and mtersite Coulomb interactions between adatom dangling-bonds, ni, ~ = (ni, ~ - n 0) defines the electron charge fluctuations with respect to the mean occupation number per spin, n o = 1 / 4 (3 electrons in 6 orbitals).

3. Correlation effects in the surface bands

Different Coulomb terms, U and Ji.i' have been analyzed by considering how much the levels of the effective Hamiltonian (Eq. (1)), E i, are modified by introducing some charge transfer between different surface states. This can be achieved by changing the filling factor of each surface state; in practice, this amounts to modifying the occupancies, n i, of the orbitals associated with the dangling-bonds of the adatoms localized either in the faulted, nr, or in the unfaulted, n u, side of the unit cell. From this calcula- tion we can obtain the parameter:

a[E,-E.] ue't- a [ n , - n . ] ' ( s )

defining how the levels change as a function of the orbital occupancies.

Eq. (3) defines an effective Hubbard term that is related both to U and Jii" Our calculations yield Ucr r = 0.7 eV. This procedure does not yield all the different Coulomb terms appearing in the Hamilto- nian (Eq. (2)), because we do not have as many independent charge transfer processes between sur- face bands as independent interactions we have in the problem. In order to determine completely all the different interactions of Hamiltonian (Eq. (2)), we have calculated all the Coulomb interactions, Jii, by using a point charge approximat ion for the dangling-bond states, and replacing the Si-crystal by a semi-infinite crystal. This model is justified by the long distances between different dangling-bonds

134 J. Ortega et al. / Applied Surface Science 123/124 (1998) 131-135

~D

0.4

0.3

0.2

0.1

-0.1

; i

p

F" Q

o o . . . . - - - - - ' J o <~

o o o o o o o

_ J o - - - .

o o j .

. . . . - . . . . . . . J - o

o o o ~ o o o

i i

F Q P F

Fig. 3. Surface band-structure along the F-Q, Q-P and P-F directions of the two-dimensional BZ (see inset). The dashed lines show our fitting to the LDA-eigenvalues indicated by diamonds. The zero of the energy-scale corresponds to the Fermi energy.

(more than 6 riO, and has been used successfully for the S i ( l l l ) -7 × 7 surface.

Knowing U elf and Jii, w e can determine the Coulomb interaction U. This also allows us to calcu- late the levels ef and eo, defined for the case for which nf = n, = 1/4. Our results yield: U = 1.5 eV and e l - e, = - 108 meV.

Hamiltonian (Eq. (2)) is defined by these quanti- ties, the Coulomb interactions Jii (the nearest neigh- bour interaction is 0.386 eV) and the hopping inte- grals defined above, T = - 7 0 meV and s = 21 meV.

We have checked that Hamiltonian (Eq. (2)) yields a good description of the LDA-bands, by solving it within an LD-approximation [8]. This amounts to substituting the many-body terms E; Un;~ n;+ + ½ ~ i , i .... , Jij~ti,r tti,r, for the following contributions:

[ Uf i ' a+ . j , i ~ ' ~ J i i h J ] n; '~+V×c(ni 'r)n; '~" (4)

The first term is the Hartree contribution, and the second term represents the exchange-correlation po- tential that we approximate by

dEx~(ni ,~) , ( 5 3

dni,~

where E ~ ( n ; , ~) = - ( ~ / 2 ) n;~(1 - ni~). In this Eq.

(5), f is the effective Coulomb interaction between the charge n;~ and its hole (1 - n;,,).

Correlation effects in the surface bands can be obtained by analyzing the full Hamiltonian (Eq. (2)). This task is outside the scope of this paper: we shall limit our discussion to analyzing the simple case that is obtained by taking s = 0. This is suggested by the values of T and s given above, with T much larger than s.

For s = 0, Hamiltonian (Eq. (2)) is decoupled into different Hamiltonians associated with the hexagonal rings shown in Fig. 2a. For this case, we have the following problem:

E r Lr j + i , cr i , j o - i

+ '2 E n j , (6) i 4 - j , o ' c r '

with 3 electrons. Hamiltonian (Eq. (6)) can be solved exactly, and using the conventional Green-function techniques we can calculate the density of states associated with electrons and holes. Fig. 4 shows this DOS around the Fermi energy, with 3 electrons and 3 holes below and above E F, respectively. Six more holes are located above this spectrum by an energy of U, but they are not shown in this figure.

J. Ortega et al./Applied Surjitce Science 123 / 124 (1998) 131-135 135

,]'!

-d.3 £.2 -o., o:1 0.2 o'.3 (eV)

Fig. 4. Shows the density of states for the hexagonal ring struc- tures (s = 0). The origin of energies is the Fermi level.

the responsible of the semiconducting character of this surface.

It is worth comparing these results with the ones obtained for the S i ( l l l ) -7 × 7 reconstruction. For this case, 3 electrons are also localized in the similar hexagonal structure, while two other electrons filling the adatom surface bands contribute to the metallic character of this surface. We suggest, in conclusion, that in the geometries of the different Si(l 11)-(2 n + 1) × (2n + 1) reconstructions (n > 2) [9,10], 3 elec- trons are filling the hexagonal ring structures located around the corner-hole; the metallic or semiconduct- ing character of these surfaces is associated to the bahaviour of the other electrons filling the remaining adatoms surface bands.

The crucial points to realize are the following: (i) firstly, the large value of U splits the spectrum into two regions, separated by U. (ii) The spectrum lo- cated around E v is symmetric with respect to this level, and has a density of states similar to the one found for the same ring structure appearing in the S i ( l l l ) -7 × 7. (iii) Finally, we note the insulating character of the Si(111)-5 × 5 surface. Although the full solution has to be obtained by introducing the hopping s, neglected in the previous solution, we do not expect this small hopping to modify the insulat- ing character of the DOS shown in Fig. 4.

4. Conclusions

In conclusion, we have shown how to calculate an effective two-dimensional Hamiltonian, in order to describe the correlation effects associated with the electron gas of the Si(111)-5 × 5 surface states. From this Hamiltonian, we have deduced that the 3 elec- trons filling the adatoms dangling-bonds are essen- tially localized in the hexagonal ring structures ap- pearing around the corner-hole. This localization is

Acknowledgements

This work was funded by the Spanish CICYT under contracts No. PB92-0168C and PB93-0260, and the EC (CHRX-C793-0134).

References

[I] C.B. Duke, Chem. Rcv. 96 (1996) 1237. [2] R.I.G. Uhrberg, E. Landdemark, L.S.O. Johansson, Phys.

Rev. B 39 (1989) 13525. [3] G.B. Adams, O.F. Sankey, Phys. Rev. Lett. 67 (1991) 867. [4] K. Takayanagi, Y. Tanishiro, M. Takahashi, S. Takahashi,

Surf. Sci. 164 (1985) 367. [5] K. Takayanagi. Y. Tanishiro, M. Takahashi. S. Takahashi, J.

Vac. Sci. Technol. A 3 (1985) 1502. [6] F. Flores, A. Levy Yeyati, J. Ortega, Surf. Rev. Lett.. in

press; J. Ortega, F. Flores~ A. Levy Yeyati, to be published. [7] A.A. Demkov, J. Ortega, O.F. Sankey, M.P. Grumbach,

Phys. Rev. B 52 (1995) 1618. [8] F.J. Garcfa-Vidal, J. Merino, R. P~rez, R. Rinc6n, J. Ortega,

F. Flores, Phys. Rev. B 50 (1994) 10537. [9] R.S. Becket. J.A. Golovchenko. G.S. Higashi, B.S.

Swartzentruber, Phys. Rev. Lett. 57 (1986) 1020. [10] R.S. Becket, B.S. Swartzentruber, J.S. Vickers, T. Klitsner,

Phys. Rev. B 39 (1989) 1633.