Surface Science 572 (2004) 483–489

www.elsevier.com/locate/susc

Vacancy charging on Si(111)-‘‘1 · 1’’ investigatedby density functional theory

Kapil Dev, E.G. Seebauer *

Department of Chemical Engineering, 600 S. Mathews, University of Illinois, Urbana, IL 61801, USA

Received 10 June 2004; accepted for publication 20 September 2004

Available online 5 October 2004

Abstract

The structure and energetics of charged vacancies on Si(111)-‘‘1 · 1’’ are investigated using density functional the-

ory calculations supplemented by estimates of ionization entropy. The calculations predict multiple possible charge

states for surface vacancy. The neutral state of the vacancy dominates up to 1200K, but the �1 state dominates above

this temperature on real Si(111) surfaces for which the Fermi level lies near the middle of the band gap.

� 2004 Published by Elsevier B.V.

Keywords: Silicon; Single crystal surfaces; Surface defects; Adatoms; Density functional calculations; Surface diffusion

1. Introduction

Surface defects such as vacancies play an

important role in mediating surface mass transport

during high temperature processing steps such as

epitaxial film deposition, diffusional smoothing in

reflow, and nanostructure formation in memorydevice fabrication [1]. Recent work in this labora-

tory using density functional theory (DFT) has

provided concrete evidence that vacancies on

0039-6028/$ - see front matter � 2004 Published by Elsevier B.V.

doi:10.1016/j.susc.2004.09.031

* Corresponding author. Tel.: +1 2173334402; fax: +1

2173335052.

E-mail address: eseebaue@uiuc.edu (E.G. Seebauer).

Si(100)-(2 · 1) [2] and Si(111)-(7 · 7) [3] surfaces

can support multiple charge states.

These results offered support for a hypothesis

we have advanced regarding optically influenced

diffusion of Ge on Si(111) [4,5]. The central pre-

mise of this hypothesis is that surface vacancy ion-

ization is primarily responsible for opticallyinfluenced diffusion. One curious aspect of the dif-

fusion behavior is that the influence of illumina-

tion disappears above about 1100K [4,5],

suggesting that the primary vacancy charge state

has changed. The adsorbate-free Si(111) surface

undergoes a phase transition from the (7 · 7)

phase to the so-called ‘‘1 · 1’’ phase in the rough

vicinity of this temperature [6,7], and a related

484 K. Dev, E.G. Seebauer / Surface Science 572 (2004) 483–489

(though broader) transition was observed in the

diffusion experiments with adsorbed Ge.

A related diffusion phenomenon occurs under

the influence of bombardment by low energy no-

ble gas ions [8,9], where the enhancement ob-served below �1030K shifts toward inhibition

above this temperature. Correspondingly, the en-

ergy threshold for the onset of ion influences in-

creases discontinuously from about 15eV to

about 25eV. We proposed [8] that inhibition

arises from the onset of ion-induced knock-in,

which may occur more easily on the high-temper-

ature reconstruction. Knocked-in atoms leave be-hind vacancies that may immobilize adsorbed Ge

atoms by direct annihilation or by formation of

complexes held together by electrostatic [10,11]

forces. The electrostatic forces would of course

change if the primary vacancy charge state

changes.

Thus, the charging behavior of vacancies on

Si(111)-‘‘1 · 1’’ plays a central role in two kindsof non-thermally induced diffusion phenomena.

The ‘‘1 · 1’’ phase does not have the simple bulk-

terminated structure, however, and has received

considerable attention in the literature. Experi-

ments employing second harmonic generation

(SHG) [12], scanning tunneling microscopy

(STM) [13] and reflection high-energy electron dif-

fraction (RHEED) [14,15] have deduced that the‘‘1 · 1’’ surface has a relaxed bulklike structure

with adatom coverage of about 0.25 monolayers

(ML). At the temperatures where the ‘‘1 · 1’’ is ob-

served, the adatoms move quickly and therefore

do not reside in single sites for very long. However,

a useful static model of this dynamic surface is one

in which adatoms are placed in a (2 · 2) periodic-

ity, thereby reproducing the 0.25ML coverage.Quantum calculations by Meade and Vanderbilt

[16] have indicated that this configuration is sec-

ond only to the (7 · 7) in stability at room temper-

ature, and have provided a useful picture of how

this reconstruction behaves.

However, the phenomenology of defects other

than adatoms on this surface remains unknown.

It is quite possible that vacancy ionization behav-ior on the ‘‘1 · 1’’ surface differs markedly from

that on the (7 · 7) surface, which could then lead

to changes in observed diffusion behavior. The

present work examines structure and energetics

of charged vacancies on Si(111)-‘‘1 · 1’’ by calcu-

lations using DFT. The calculations yield the iden-

tities and energetics of stable charge states at 0K.

Methods recently developed in this laboratory [17]to estimate ionization entropies then offer a means

to make closer contact with temperatures in real

applications.

2. Computational method

The computations employed commercial CA-STEP software from Accelrys Inc. [18] as discussed

elsewhere [2]. In brief, the calculations determined

total electronic energies based on standard DFT

methods using the local density approximation.

The exchange-correlation term was parameterized

following the approach of Perdew and Zunger

[19]. The basis functions were plane waves having

an energy cutoff of 180eV (�13Ry). Above thiscutoff, variations in calculated formation energies

for the various charge states (relative to the neu-

tral) became insignificant. Troullier-Martins pseu-

dopotentials [20] were employed. All calculations

were performed at 2k points {(1/4,1/4,0) and (1/

4,�1/4,0)} in the Brillouin zone. Formation ener-

gies changed by only �1% with the use of a finer k

point mesh.The surface was modeled as a slab consisting of

4 · 4 supercell of the bulk truncated Si(111) in the

(1 · 1) reconstruction. This bulk truncated slab

comprising of 96 Si atoms was further decorated

with 4 Si ‘‘adatoms’’ in the so-called T4 site (di-

rectly above the second layer Si atoms) with

2 · 2 periodicity. The reasons for choosing this

particular representation of the high temperatureSi(111)-‘‘1 · 1’’ phase will be discussed in the fol-

lowing section. The space over the surface was

treated as a 10A thick vacuum layer, and the dan-

gling Si bonds at the bottom of the slab were sat-

urated with hydrogen atoms. The surface vacancy

was modeled by simply removing one surface atom

from the supercell. Geometric relaxation of the

slab was performed assuming no symmetry con-straint, with the top four Si layers allowed to move

and the bottom two held fixed in the bulk config-

uration. Relaxation proceeded in the conventional

K. Dev, E.G. Seebauer / Surface Science 572 (2004) 483–489 485

way until the Hellman–Feynman forces [21] be-

tween atoms decreased below 0.1eV/A.

The formation energy EVac of a neutral surface

vacancy with respect to the undefected surface can

be calculated as [22]:

EVac ¼ ETot;v þ nEbulk � ETot;p; ð1Þ

where ETot,v and ETot,p denote the total energies of

the defected and perfect surfaces, respectively.

Ebulk represents the total energy of an atom at a

lattice site deep within the bulk, and n denotes

the number of atoms that must be removed to cre-

ate the vacancy. The total formation energy

EVac(q) for a charged vacancy is the sum of the for-mation energy of a neutral vacancy (from Eq. (1))

and the ionization energy. If the total formation

energy is referenced to that of a neutral vacancy,

EVac(q) obeys the relation [23]:

EVacðqÞ ¼ ½ETotðqÞ � ETotð0Þ� þ qðEVBM þ EFÞ;ð2Þ

where ETot denotes total energy, q the net number

of holes supported by the vacancy, and EF theFermi energy. Errors in EVBM due to the finite

supercell were corrected by aligning the vacuum

levels of the defect-containing supercell and the

undefected supercell. These corrections amounted

to roughly 0.15eV for singly charged defects. No

correction was applied for the electrostatic attrac-

tion between the charged defect and the neutraliz-

ing jellium [24]. This correction scales inverselywith the size of the supercell, and in the present case

amounts to the negligible level of roughly 0.06eV.

The results of Eq. (2) are valid only at 0K, how-

ever, as no information about entropy is supplied

by total energy calculations. We extended the re-

sults to non-zero temperatures by accounting for

the entropy of ionization as detailed elsewhere

[17] for vacancies on semiconductor surfaces. Thatwork showed that for Si, the entropy of ionization

DSI(T) can be approximated by electron–hole pair

formation entropy DScv(T), which can be calcu-

lated from the derivative of temperature variation

of the band gap energy DEcv [25]. A consequence

of the correspondence between DScv(T) and DSI(T)

is that, as T increases and DEcv decreases, free

energies referenced to EVBM for vacancy ionization

levels remain at a constant energy below the con-

duction band for negatively charged vacancies

and at a constant energy above the valence band

for positively charged.

3. Results

3.1. Structure of Si(111)-‘‘1 · 1’’

As indicated earlier, the ‘‘1 · 1’’ phase of

Si(111) comprises a relaxed (1 · 1) lattice with mo-

bile adatoms at 0.25ML coverage [12–15]. The sta-

tic (2 · 2) reconstruction offers a useful model of

this structure. There exist two non-equivalent sites

that the adatoms can occupy: T4 sites directly

above the second layer Si atoms, and H3 sites di-rectly above the fourth layer Si atoms. Experi-

ments indicate that adatoms preferentially

occupy the T4 sites [12,14] and are situated 1.3–

1.4A above the first layer atoms [14,15].

For comparison with previous work, we com-

puted the adatom-induced surface energy gain

per (1 · 1) unit cell relative to the surface with

no adatoms. This energy change is given by [16]:

DEsurf;ad ¼ Esurf ;ad � Esurf ;na

¼ fðEtotal;ad � Etotal;naÞ � N adEbulkg=16;ð3Þ

where Esurf,ad, Etotal,na and Ebulk, respectively rep-

resent the calculated energies for the adatom cov-

ered surface slab, the no-adatom surface slab,

and the bulk slab. Nad denotes the number of ada-

toms in the (4 · 4) supercell. The 0.25ML adatom

coverage leads to maximum energy gain of 0.27eV,

which matches exactly the value obtained by

Meade and Vanderbilt [16].In terms of structure, the first layer atoms di-

rectly backbonded to the adatom relax closer to

one another by 0.18A compared to the ideal bulk.

The second layer atom directly below the adatom

moves downward by 0.38A. The rest atom (first

layer surface atom not bonded to the adatom)

experiences upward relaxation by 0.45A. Vertical

(plane normal to the surface) separation betweenthe adatom and the rest atom after relaxation is

0.67A. The directions and magnitudes of atomic

486 K. Dev, E.G. Seebauer / Surface Science 572 (2004) 483–489

displacements obtained in these calculations agree

closely with those previously reported by Meade

and Vanderbilt [16].

3.2. Neutral surface vacancy

Fig. 1a shows the geometry of the ‘‘1 · 1’’ sur-

face together with the vacancy examined here.

The vacancy we chose corresponded to the ‘‘rest

atom’’ in the first full atomic layer of the surface.

The rest atom site is the only one in the first layer

unit cell that is not bonded to an overlying ada-

tom. We chose the rest atom vacancy to examinein order to make the results most directly compa-

rable with previous work on vacancies in the

(111)-(7 · 7) and (100)-(2 · 1) reconstructions.

Fig. 1. Diagram of the Si(111)-‘‘1 · 1’’ slab including a

vacancy (a) before relaxation and (b) after relaxation. Only

the first two surface layers and the adatom layer are shown for

clarity. Atoms in the first layer are shown with a white dot in

the center. The arrows in (b) show the direction of relaxation.

During diffusion under illumination or ion bom-

bardment, surface vacancy formation functions

to create additional mobile species. Removal of

an adatom on the ‘‘1 · 1’’ surface corresponds to

removing mobile species. This considerationmakes the rest atom vacancy the most appropriate

one to consider.

The structural rearrangements induced by the

vacancy formation are large but are limited to

the three nearest-neighbor atoms in the second-

layer. Fig. 1b shows the geometry; the second layer

atoms move toward one another by 0.34A into the

free space left behind by the missing atom. Thisbehavior contrasts with Si(111)-(7 · 7), where

structural rearrangements accompanying adatom

vacancy formation are minimal [3].

Although the second layer atoms relax towards

one another upon vacancy formation, no new

bond formation occurs. At first glance, this lack

of rebonding might seem surprising, since removal

of the adatom creates two additional danglingbonds. However, geometrical considerations show

that formation of a bond between two second

layer atoms would require angle distortions in

the pre-existing bonds on the order of 20� and cor-

responding stretching near 0.25A. The newly

formed bond would be similarly non-ideal. For

comparison, surface reconstructions of Si such as

the (111)-(7 · 7) and the (100)-(2 · 1) supportsimilar bond angle deviations, but the bond

stretching is only on the order of 0.05A. Thus,

the strain associated with bond stretching prevents

rebonding between the second layer atoms upon

vacancy formation.

Eq. (1) yielded a formation energy of 1.60eV for

this vacancy in the neutral state. This value lies

close to the average adatom vacancy formation en-ergy (�1.30eV) on Si(111)-(7 · 7) [3]. This agree-

ment is probably fortuitous. Vacancy formation

from rest atoms on the ‘‘1 · 1’’ surface increases

the number of dangling bonds in a surface unit cell

by 25%, whereas vacancy formation from adatoms

on the (7 · 7) surface increases this number by

only 10%. However, structural relaxations due to

‘‘1 · 1’’ vacancies have no counterpart on the(7 · 7). The free energy gained by these relaxations

apparently happens to cancel the free energy lost

by creating additional dangling bonds.

T (K)0 250 500 750 1000 1250 1500

Fre

e E

ner

gy

(eV

)

0.0

0.4

0.8

1.2

1.6

EC

-1/0

EV

0/+1

EF

Surface Vacancy

Fig. 3. Variation with temperature of the vacancy ionization

levels. For a Fermi energy of 0.63eV characteristic of real

Si(111), the primary ionization state changes from 0 to �1 at

1200K.

K. Dev, E.G. Seebauer / Surface Science 572 (2004) 483–489 487

3.3. Charged surface vacancy

Formation energies for the charged surface va-

cancy were calculated according to Eq. (2) for

charge states ranging from �2 to +2. The domi-nant charge state is the one with the lowest forma-

tion energy. Fig. 2 plots these energies as a

function of EF. As EF increases from the valence

band edge, the formation energies of positive

charge states increase, while those of negative

states decrease. The calculations show that,

depending upon the value of EF, vacancies can

have dominant charge states of �1, 0 or +1. Forcomparison, the adatom vacancy on the Si(111)-

(7 · 7) can stably support charge states of �2,

�1 or 0 [3]. The calculations indicated that struc-

tural relaxation effects due to vacancy charging

were negligible for Si(111)-‘‘1 · 1’’. This behavior

corresponds closely to that of charged vacancies

on Si(111)-(7 · 7) [3], but contrasts with the

Si(100)-(2 · 1) surface, where +2 state of themonovacancy exhibits ‘‘negative-U’’ behavior

leading to large structural rearrangements [2].

Fig. 3 shows that the formation energies of the

+1 and 0 states on Si(111)-‘‘1 · 1’’ at 0K equal each

other at EF = 0.14eV. When EF rests at this so-

called ‘‘ionization level,’’ the populations of the

two charge states are equal to within a spin degen-

Fermi Energy (eV)0.0 0.2 0.4 0.6 0.8 1.0 1.2

Fo

rmat

ion

En

erg

y (e

V)

-1

0

1

2

3

+2

+1

-2

-1

0

0 K

Surface Vacancy

Fig. 2. Formation energies of the various vacancy charge states

as a function of Fermi energy on Si(111)-‘‘1 · 1’’ at 0K. The

formation energy is referenced to the neutral vacancy, while the

Fermi energy is referenced to the valence band maximum.

eracy factor of two. 1 The corresponding (�1/0)

level appears at EF = 1.00eV. As mentioned previ-

ously, the temperature variation of acceptor ioniza-

tion levels (as referenced to the valence band)

mimics the temperature variation of the conduction

band edge, so that the ionization levels remain ata fixed energy spacing from that edge. The energy

Ec of the conduction band edge (referenced to the

valence band edge) was obtained from the empirical

Varshni relation for the band gap of Si [27]:

EcðT Þ ¼ DEcvðT Þ ¼ Ecð0Þ � aT 2=ðT þ bÞ; ð4Þwhere T denotes temperature in Kelvin, and Ec(0),

a and b are constants obtained from the experi-

mental work of Ref. [27]. The donor ionization

levels stay constant with temperature because they

track the valence band edge, which represents akey reference energy in this work.

Note that the ionization levels calculated here

are thermodynamic quantities based on formation

energies referenced to the top of the valence band,

not eigenvalues of the Schrodinger equation. No-

where does the calculation use eigenvalues to ob-

tain Ec, and there was no need to look for

1 The �1 and +1 states each have an unpaired electron with

possible up and down spin states. The �2, 0, and +2 states have

no such degeneracy. Thus at the 0/+1 ionization level, for

example, [V+] = 2[V0]. See Ref. [26].

488 K. Dev, E.G. Seebauer / Surface Science 572 (2004) 483–489

specific surface or bulk electronic states. The band

gap can be interpreted as a thermodynamic free

energy [28], so we simply used the experimental

value from Ref. [25]. Also, this treatment considers

only electronic contributions to the ionization lev-els due to ionization entropy. Other factors can af-

fect the temperature dependence of the ionization

levels as well, at least in principle. For example,

there are also vibrational contributions to the ion-

ization levels due to vibrational entropy. Struc-

tural changes resulting from defect ionization can

lead to charge-dependent differences in the vibra-

tional entropy, which in turn leads to changes inthe temperature variation of the ionization levels.

In the present case, however, vacancy structure re-

mains largely independent of charge state, so such

effects should be negligible. As another example,

thermal vibrations induce moment-to-moment

fluctuations in defect structure. Since ionization

levels depend upon structure [29], the ionization

levels fluctuate in a corresponding way. This paperconcerns itself with the time-averaged levels, how-

ever, so no fluctuation dynamics are considered.

Fig. 3 shows the variation of the acceptor and

donor ionization levels as a function of tempera-

ture. At about 1200K, the (�1/0) level intersects

the valence band. Therefore, the neutral state of

the vacancy is not stable above this temperature

for any value of Fermi energy.

4. Discussion

The vacancy investigated here supports stable

charge states of �1, 0 and +1. The dominant

charge state at a given temperature depends upon

the Fermi level position on a real Si(111)-‘‘1 · 1’’surface. This position can be estimated as follows.

Si(111)-(7 · 7) has a high density of surface states

that sets the Fermi level near midgap [30]. Indeed,

Himpsel et al. [31] reported EF � EVBM to be

0.63eV for the 7 · 7 surface of undoped Si at room

temperature (compared to a bandgap energy of

1.1eV). There is in principle a dependence of the

surface Fermi level position on both doping leveland temperature. The doping dependence for

Si(111)-‘‘1 · 1’’ is not known. However, for

Si(111)-(2 · 1), the position of EF remains inde-

pendent of substrate doping [32], making this inde-

pendence plausible for the other reconstructions.

As for the temperature dependence, a core-level

photoemission study of Si(111) [33] from 300K

to 1673K indicated a shift of only about 0.2eVin the binding energies. However, this shift was

not attributed to changes in the surface Fermi level

but to surface rearrangements such as the transfor-

mation from (7 · 7) to ‘‘1 · 1.’’

In short, it is reasonable to believe that EF re-

mains 0.63eV above the valence band irrespective

of the doping level, temperature, and reconstruction.

Fig. 3 then suggests that the neutral state of thevacancy dominates below 1200K, while above

this temperature the primary charge state shifts

to �1.

As mentioned in Section 1, non-thermal effects

of optical illumination on surface diffusion appear

to be mediated by charged surface vacancies whose

population statistics vary in response to the gener-

ation of photogenerated charge carriers [4,5].Vacancy charging effects may also govern the

switch from ion-induced diffusion enhancement

to inhibition. For Ge on Si(111), both non-ther-

mal diffusion phenomena change substantially in

character in rough correspondence with the phase

transformation from (7 · 7) to ‘‘1 · 1’’ that takes

place between 1060 and 1110K [4] for Ge-ad-

sorbed surfaces. Differences in vacancy chargingbehavior for the two reconstructions may account

for these effects.

The present results show that on real ‘‘1 · 1’’

surfaces with EF � EV = 0.63eV, either the neutral

or the �1 vacancy dominates, with a switch from

neutral to �1 occurring near 1200K. On real

(7 · 7) surfaces, however, the �2 state dominates

for all temperatures, of interest here [3]. Thus,some change in charge state occurs in concert with

the (7 · 7) to ‘‘1 · 1’’ phase transition. DFT calcu-

lations coupled with the theory for temperature-

dependent charging employed here are not

sufficiently accurate to say definitively that the

dominant ‘‘1 · 1’’ charge state upon phase transi-

tion near 1100K is neutral. The position of the

(�1/0) ionization level is too uncertain for such adetermination. However, the conclusion that the

�2 state is never stable on the ‘‘1 · 1’’ is more

robust. Thus, the basic premise of the paper (that

K. Dev, E.G. Seebauer / Surface Science 572 (2004) 483–489 489

changes in vacancy charge state occur upon recon-

struction) is strongly supported.

More direct comparisons with experiments are

still difficult, however, because the present compu-

tations concerned an adsorbate-free surface, whilethe experiments employed a Ge adsorbate. Ge is

capable of substituting into the top layer of the

Si substrate, offering many permutations for form-

ing vacancies surrounded by mixed chemical com-

positions. These various compositions could

change the formation energies and therefore popu-

lation statistics of the charge states.

5. Conclusion

The results presented here suggest that ioniza-

tion behavior of vacancies varies with different

reconstructions of the Si surface. Vacancies on

Si(111)-‘‘1 · 1’’ surface tend to be more neutral

than the vacancies on Si(111)-(7 · 7). This lendsfurther qualitative support to our hypothesis that

non-thermally influenced diffusion is driven by va-

cancy ionization. Calculations of vacancy charging

on Ge covered Si surfaces are required for a more

quantitative assessment.

Acknowledgment

This work was partially supported by NSF

(CTS 02-03237). Computations were performed

with support from the National Computational

Supercomputing Alliance at UIUC.

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