Surface Science 538 (2003) L495–L499

www.elsevier.com/locate/susc

Surface Science Letters

Vacancy charging on Si(1 1 1)-(7 · 7) investigated bydensity functional theory

Kapil Dev, E.G. Seebauer *

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

Received 6 February 2003; accepted for publication 14 May 2003

Abstract

The structure and energetics of charged vacancies on Si(1 1 1)-(7· 7) are investigated using density functional theory

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

for the unfaulted edge vacancy in the adatom layer, although the )2 state is most stable on real Si(1 1 1) surfaces for

which the Fermi level lies near the middle of the band gap.

� 2003 Elsevier B.V. All rights reserved.

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

1. Introduction

The study of charged defects on semiconductor

surfaces is becoming increasingly important. For

example, studies by this laboratory have uncov-

ered new physics [1,2] for optically influenced dif-

fusion on Si that is governed by charged vacancies

and that could influence device fabrication ac-

complished by optical heating. Such vacancy

charging has received attention in the literature [3–9], but with primary focus on compound semi-

conductors.

We recently extended this focus to include the

elemental semiconductor Si [10], employing den-

sity functional theory (DFT) to determine the

* Corresponding author. Tel.: +1-217-333-4402; fax: +1-217-

333-5052.

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

0039-6028/03/$ - see front matter � 2003 Elsevier B.V. All rights res

doi:10.1016/S0039-6028(03)00734-9

structure and energetics of charged vacancies on

Si(1 0 0)-(2 · 1). The results showed that these va-cancies can support multiple charge states and in

some cases can exhibit ‘‘negative-U’’ behavior, in

which a single ionization event leads immediately

to further ionization. Such effects are well known

for bulk defects but had not been observed previ-

ously for surfaces. The results regarding multiple

charge states offered support for a hypothesis we

have advanced regarding optically influenced dif-fusion [1,2]. However, the experimental results

were obtained on Si(1 1 1), while the DFT treated

Si(1 0 0), so the connection between experiment

and computation was indirect.

The present work examines vacancy charging

on Si(1 1 1)-(7 · 7) by DFT to improve this con-

nection. The calculations also seek to ascertain to

what extent negative-U behavior generalizes tocrystallographic orientations beyond (1 0 0). This

work treats vacancies in so-called ‘‘adatom’’

erved.

L496 K. Dev, E.G. Seebauer / Surface Science 538 (2003) L495–L499SU

RFACE

SCIENCE

LETTERS

positions, which are easier to form than any other

variety on Si(1 1 1)-(7 · 7) [11]. The relative ease of

formation makes such vacancies most likely to

play a role in creating mobile atoms for surface

diffusion [12]. The DFT calculations yield the

identities and energetics of stable charge states at 0K. Methods recently developed in this laboratory

[13] 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. [14] as discussed

elsewhere [10]. In brief, the calculations deter-

mined total electronic energies based on standard

DFT methods using the local density approxima-

tion. The basis functions were plane waves having

an energy cutoff of 11 Ry (150 eV). Above this

cutoff, variations in calculated formation energiesfor the various charge states (relative to the neu-

tral) became insignificant. All calculations were

performed at the gamma point in the Brillouin

zone. Using a finer k point mesh did not affect the

formation energies significantly (�1%).

The surface was modeled as a slab consisting of

a single 7 · 7 unit cell of the Si(1 1 1) in its well

known Takanayagi reconstruction [15]. The slabincluded a so-called ‘‘adatom’’ layer resting atop

four more Si layers. The space over the surface was

treated as a 10 �AA thick vacuum layer, and the

dangling Si bonds at the bottom of the slab were

saturated with hydrogen atoms. The adatom va-

cancy was modeled by simply removing one ad-

atom from the supercell.

The total formation energy EVacðqÞ for acharged vacancy is the sum of the formation en-

ergy of a neutral vacancy and the ionization en-

ergy. If the total formation energy is referenced to

that of a neutral vacancy, EVacðqÞ obeys the rela-

tion [9]:

EVacðqÞ ¼ ½ETotðqÞ � ETotð0Þ� þ qðEVBM þ EFÞ; ð1Þ

where ETot denotes total energy, q the net number

of holes supported by the vacancy, and EF the

Fermi 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.

The results of Eq. (1) are valid only at 0 K,

however, as no information about entropy is sup-plied by total energy calculations. We extended the

results to nonzero temperatures by accounting for

the entropy of ionization as detailed elsewhere [13]

for vacancies on semiconductor surfaces. That

work showed that for Si, the entropy of ionization

DSIðT Þ can be approximated by electron–hole pair

formation entropy DScvðT Þ, which can be calcu-

lated from the temperature variation of the bandgap. A consequence of the correspondence between

DScvðT Þ and DSIðT Þ is that, as T increases and DEcv

decreases, free energies referenced to EVBM for va-

cancy ionization levels remain at a constant energy

below the conduction band for negatively charged

vacancies and at a constant energy above the va-

lence band for positively charged.

3. Results

The 7 · 7 reconstruction has four inequivalenttypes of adatoms, referred to as the unfaulted

corner (UFC), unfaulted edge (UFE), faulted

corner (FC) and faulted edge (FE). A structural

model sketching the different adatom types ap-

pears in Ref. [11]. In the unfaulted half of the cell,

the calculations yield formation energies of 1.26

and 1.28 eV for edge and corner adatoms, re-

spectively. Corresponding energies in the faultedhalf are 1.31 and 1.35 eV. Thus, the calculations

indicate that vacancies form more easily in the

unfaulted half of the cell than the faulted, and

form more easily in edge positions than corner. By

contrast, Lim et al. [11] used similar methods to

determine average formation energies near 0.9 eV.

In addition, the trends observed in that work for

faulted vs. unfaulted locations and edge vs. cornerpositions were opposite to those described here.

The disagreement in the formation energies may

originate from the use of different cutoff energies.

The value of this energy used is not mentioned in

Ref. [11], although prior work [16] from the same

laboratory employed cutoff energies no higher

than 8 Ry for this surface.

Fermi Energy (eV)

0.0 0.2 0.4 0.6 0.8 1.0 1.2

-1

0

1

2

3

+2

+1

-2

-1

0

0 KUFE Vacancy

Fo

rmat

ion

En

erg

y (e

V)

Fig. 1. Formation energies as a function of Fermi energy of

various charge states of the UFE vacancies on Si(1 1 1)-(7· 7) at

0 K. The formation energy is referenced to the neutral vacancy,

while the Fermi energy is referenced to the valence band max-

imum.

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, ½V þ� ¼ 2½V 0�.

K. Dev, E.G. Seebauer / Surface Science 538 (2003) L495–L499 L497

SURFA

CESCIEN

CE

LETTERS

The present results, however, correspond clo-

sely to two experimental observations showing

that vacancies form more easily at edge positions

than corner. Tsong et al. [17] used thermal evap-

oration methods to deduce this trend, and assigned

an energy difference of �0.1 eV to the two posi-tions. Uchida et al. [18] employed field evaporation

to obtain similar results, although the energy dif-

ference was calculated to be only �0.01 eV. Our

results yield energy differences in the range 0.02–

0.04 eV, which lies comfortably between the ex-

perimental values.

The present calculations do agree with the con-

clusions of Lim et al. [11] that structural rearrange-ments accompanying formation of the four kinds of

vacancies are small. Atoms originally bonded to the

missing adatom shift less than 0.1 �AA. Neighboring

dimer bonds lengthen, but by less than 1%.

The similarity in structure and energetics of the

various kinds of vacancies examined here offers

reason to believe that the ionization behavior re-

mains largely invariant as well. Hence we investi-gated the ionization of vacancies only in the UFE

position as a typical example. Also, this vacancy

has a (slightly) lower formation energy than all the

others, so this species is somewhat more likely to

participate in surface self-diffusion. We calculated

formation energies according to Eq. (1) for charge

states ranging from )2 to +2. The dominant

charge state is the one with the lowest formationenergy. Fig. 1 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 )2, )1 and 0. Positive states are not stable

for any value of EF.For comparison, the monovacancy in the bulk

Si can support charge states of )2, )1, 0 and +2

[19]. Surface monovacancies on Si(1 0 0)-(2 · 1)

exist in ‘‘lower’’ and ‘‘upper’’ varieties, depending

upon whether an atom was removed from the

lower or upper position of the buckled asymmetric

dimer. The lower monovacancy supports charges

of only 0 and )1, but the upper supports +2, 0, )1and )2 [10]. The absence of the + 1 state for

the upper monovacancy corresponds to so-called

‘‘negative-U’’ behavior, in which the removal of

one electron from the neutral defect leads imme-

diately to the removal of a second. No such be-

havior is indicated for Si(1 1 1)-(7 · 7). Clearly,

there is only modest correspondence in the number

and type of stable charge states among the bulk

and various surface crystallographic orientations.The calculations showed that structural relax-

ation effects due to vacancy charging were negligi-

ble for Si(1 1 1)-(7 · 7). This behavior corresponds

well to that of negatively charged vacancies on

Si(1 0 0)-(2 · 1) [10]. However, that surface exhib-

ited significant rearrangements for the +2 state,

and the bulk monovacancy exhibits significant

Jahn-Teller distortion of the nearest neighboratoms for all charge states [20,21]. Once again, there

is only modest correspondence among the bulk

and various surface crystallographic orientations.

Fig. 1 shows that the formation energies of the 0

and )1 states on Si(1 1 1)-(7 · 7) at 0 K equal each

other at EF ¼ 0:15 eV. When EF rests at this so-

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

two charge states are equal to within a spin de-generacy factor of two [22]. 1 The corresponding

T (K)0 400 800 1200 1600

-0.4

0.0

0.4

0.8

1.2EC

-1/0

-2/-1

EV

EF

UFE Vacancy

Fre

e E

ner

gy

(eV

)

Fig. 2. Variation with temperature of the vacancy ionization

levels.

L498 K. Dev, E.G. Seebauer / Surface Science 538 (2003) L495–L499SU

RFACE

SCIENCE

LETTERS

()1/)2) level appears at EF ¼ 0:58 eV. As men-

tioned previously, the temperature variation of

acceptor ionization levels (as referenced to thevalence band) mimics the variation of the con-

duction band edge. The latter variation can be

obtained from the empirical Varshni relation [23]:

EcðT Þ ¼ Ecð0Þ � aT 2=ðT þ bÞ; ð2Þwhere T denotes temperature in Kelvin. Fig. 2

shows the ionization levels as a function of tem-

perature based on Eq. (2). At about 640 K, 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

There is no evidence of negative-U behavior for

the vacancy investigated here. The structural re-

arrangements that accompany such behavior both

on Si(1 0 0) [10] and in the bulk [24,25] evidently donot stabilize higher charge states over lower.

The dominant charge state at any given tem-

perature depends on the Fermi level position at the

real surface. Si(1 1 1)-(7 · 7) has a high density of

surface states that sets the Fermi level near midgap

[26]. Himpsel et al. [27] reported EF � EVBM to be

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

temperature. There is in principle a dependence ofthis number on both doping level and temperature.

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

independent of substrate doping [28], making this

independence plausible for 7 · 7. We are not aware

of any studies on the temperature dependence of

Fermi level position for Si(1 1 1). For Si(1 0 0)-

(2 · 1), however, EF appears to remains constant to

within about 0.1 eV from 0 to 1200 K [10]. Thisconstancy is therefore also plausible for Si(1 1 1).

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

mains 0.63 eV above the valence band irrespective

of the doping level and temperature. Fig. 2 then

suggests that the )2 state of the vacancy dominates

under virtually all conditions of temperature and

doping.

As mentioned in Section 1, nonthermal effectsof optical illumination on surface diffusion appear

to be mediated by charged surface vacancies whose

population statistics vary in response to the gen-

eration of photogenerated charge carriers [1,2].

Experiments detailing this phenomenon employed

a symmetric sequence of group III, IV and V ele-

ments on Si(1 1 1). The hypothesized mechanism

relies on the ability of surface vacancies to supportmore than one charge state. Measurements of

purely thermal surface diffusion have offered evi-

dence for this hypothesized ability [29], and anal-

ogies with vacancy charging effects in the bulk give

additional support. DFT calculations for Si(1 0 0)

offered much stronger evidence that vacancies on

Si surfaces can exist in different charge states

whose relative populations depend on the positionof the surface Fermi level [10]. The present work

offers still more evidence for Si(1 1 1). Direct

comparisons with experiments are still not possi-

ble, however, because the experiments employed

adsorbates capable of substituting into the top

layer of the Si substrate at near-monolayer con-

centrations. Such surfaces do not necessarily have

(7 · 7) symmetry, and offer many permutations forforming vacancies surrounded by mixed chemical

compositions.

Acknowledgements

This work was partially supported by NSF

(CTS 02-03237). Computations were performedwith support from the National Computational

Supercomputing Alliance at UIUC.

K. Dev, E.G. Seebauer / Surface Science 538 (2003) L495–L499 L499

SURFA

CESCIEN

CE

LETTERS

References

[1] R. Ditchfield, D. Llera-Rodriguez, E.G. Seebauer, Phys.

Rev. Lett. 81 (1998) 1259.

[2] R. Ditchfield, D. Llera-Rodriguez, E.G. Seebauer, Phys.

Rev. B 61 (2000) 13710.

[3] Ph. Ebert, X. Chen, M. Heinrich, M. Simon, K. Urban,

M.G. Lagally, Phys. Rev. Lett. 76 (1996) 2089.

[4] Ph. Ebert, K. Urban, M.G. Lagally, Phys. Rev. Lett. 72

(1994) 840.

[5] G. Lengel, R. Wilkins, G. Brown, M. Weimer, J. Gryko,

R.E. Allen, Phys. Rev. Lett. 72 (1994) 836.

[6] Ph. Ebert, Surf. Sci. Rept. 33 (1999) 121.

[7] G.W. Brown, H. Grube, M.E. Hawley, S.R. Schofield, N.J.

Cursons, M.Y. Simmons, R.G. Clark, J. Appl. Phys. 92

(2002) 820.

[8] H. Kim, J.R. Chelikowsky, Phys. Rev. Lett. 77 (1996)

1063.

[9] S.B. Zhang, A. Zunger, Phys. Rev. Lett. 77 (1996) 119.

[10] H.Y.H. Chan, K. Dev, E.G. Seebauer, Phys. Rev. B 67

(2003) 035311.

[11] H. Lim, K. Cho, R.B. Capaz, J.D. Joannopoulos, Phys.

Rev. B 53 (1996) 15421.

[12] For a detailed review of such self-diffusion phenomena, see

E.G. Seebauer, M.Y.L. Jung, Surface diffusion of adsor-

bates on metals, alloys, oxides and semiconductors, in: H.P.

Bonzel (Ed.), Landolt–B€oornstein Numerical Data and

Functional Relationships: Adsorbed Layers on Surfaces,

Springer-Verlag, New York, 2001, vol. III/42A, Chapter 11.

[13] K. Dev, E.G. Seebauer, Phys. Rev. B 67 (2003) 035312.

[14] V. Milman, B. Winkler, J.A. White, C.J. Pickard, M.C.

Payne, E.V. Akhmatskaya, R.H. Nobes, Int. J. Quant.

Chem. 77 (2000) 895.

[15] K. Takanayagi, Y. Tanishiro, M. Takahashi, S. Takahashi,

J. Vac. Sci. Technol. A 3 (1985) 1502.

[16] K.D. Brommer, M. Needels, B.E. Larson, J.D. Joanno-

poulos, Phys. Rev. Lett. 68 (1992) 1355.

[17] T.T. Tsong, R.-L. Lo, T.-C. Chang, C. Chen, Surf. Rev.

Lett. 1 (1994) 197.

[18] H. Uchida, D. Huang, F. Grey, M. Aono, Phys. Rev. Lett.

70 (1993) 2040.

[19] J.A. Van Vechten, Phys. Rev. B 38 (1988) 9913.

[20] O. Sugino, A. Oshiyama, Phys. Rev. Lett. 68 (1992) 1858.

[21] S. Ogut, H. Kim, J.R. Chelikowsky, Phys. Rev. B 56 (1997)

11353.

[22] See J.A. Van Vechten, Phys. Rev. B 33 (1986) 2674.

[23] P. Lautenschlager, M. Garriga, L. Vina, M. Cardona,

Phys. Rev. B 36 (1987) 4821.

[24] O. Sugino, A. Oshiyama, Phys. Rev. Lett. 68 (1992) 1858.

[25] S. Ogut, H. Kim, J.R. Chelikowsky, Phys. Rev. B 56 (1997)

11353.

[26] M. McEllistrem, G. Haase, D. Chen, R.J. Hamers, Phys.

Rev. Lett. 70 (1993) 2471.

[27] F.J. Himpsel, G. Hollinger, R.A. Pollak, Phys. Rev. B 28

(1983) 7014.

[28] F.G. Allen, G.W. Gobeli, Phys. Rev. 127 (1962) 150.

[29] C.E. Allen, R. Ditchfield, E.G. Seebauer, J. Vac. Sci.

Technol. A14 (1996) 22.