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Theoretical Studies of Silicon Surfaces Using Finite Clusters
Citation
Redondo-Muiño, Antonio
(1977)
Theoretical Studies of Silicon Surfaces Using Finite Clusters.
Dissertation (Ph.D.), California Institute of Technology.
doi:10.7907/8M08-WC34.
Abstract
The objective of this thesis is to study the electronic structure, geometries and chemical binding characteristics of the surfaces of silicon and of the initial form of oxygenated Si. We examined the (111), (100), and (110) surfaces, relaxation on the (111) and (100) surfaces and reconstruction on the (100) surfaces. In addition we examined steps on the OM surfaces. In the oxygenated surface we considered the geometry, excited states and ion states of both 0 and 0_2 bonded to the perfect (111) surface.
These studies indicated that surfaces and chemisorption lead to localized electronic states for which explicit inclusion of electronic correlation (many body) effects is essential. These effects are included through use of generalized valence bond (GVB) and configuration interaction (CI) techniques. These techniques require use of a finite collection of Si atoms to represent the surface. We find that very small clusters lead to reliable results if the model system is properly tied off with SiH bonds (to represent internal Si-Si bonds).
In Chapter 1 we report an effective potential for replacing the ten core electrons in calculations involving the Si atom. The potential is obtained directly from ab initio calculations on the states of the Si atom and no empirical data or adjustable parameters are used. The ab initio effective potential is tested by carrying out Hartree-Fock generalized valence bond and configuration interaction calculations on various molecules. We considered Si, Si_2, SiH_3, Si_2H_6 and H_3S10_2 and calculated excitation energies, ionization potentials, and electron affinities both both using the effective potential and without it (ab initio). In essentially all cases the agreement is to better than 0.1 eV, providing strong evidence that the effective potential adequately represents the Si core. This potential is utilized in all of the calculations reported in subsequent chapters.
In Chapter 2 we consider clean (111), (100) and (110) silicon surfaces. For the (111) surface the relaxation of silicon surface atoms is studied by means of an Si(SiH_3)_3 cluster. We find that the surface state is accurately described as a dangling bond orbital with 93% p character. We determined the .optimum relaxation of the surface layer to be 0.08Å toward the second layer. For the positive ion we find that the surface atom relaxes toward the second layer by an additional 0.30Å. Using an Si_3H_6 cluster we find that the interaction between adjacent dangling bond orbitals indicates that they are very weakly coupled (with a splitting of ~0.01 eV between the singlet and triplet spin couplings.) For the (100) surface we used an Si(SiH_3)_2 cluster. We find a relaxation distance of 0.10Å toward the vacuum. We also considered the 2x1 reconstruction of such surfaces using the results for Si_2H_4 and Si(SiH_3)_2 complexes. It is found that adjacent surface atoms form a bond (1.76 eV bond strength), leading to pairing up. of adjacent silicons with an optimum Si-Si bond length of 2.38Å).
In Chapter 3 we consider the electronic structure of divalent steps on (111) silicon surfaces. We find three localized electronic states separated by less than 0.3 eV. These states have quite different electronic structure and are expected to be reactive toward a large range of chemical species.
In Chapter 4 we study the chemisorption of oxygen upon Si (111) surfaces. For single oxygen atoms we find an optimum Si-0 bond length of 1.63Å. We also find ionization potentials in the range 11-16 eV. Then we consider a model in which an oxygen molecule chemisorbed onto the silicon surface has, an electronic structure corresponding to a peroxy radical. We find ionization potentials in the range 11-18 eV
in agreement with experiment. We find an optimum 0-0 bond length of 1.37Å and a Si-0-0 bond angle of 126° for the chemisorbed peroxy radical.
Item Type:
Thesis (Dissertation (Ph.D.))
Subject Keywords:
(Applied Physics and Chemistry)
Degree Grantor:
California Institute of Technology
Division:
Engineering and Applied Science
Major Option:
Applied Physics
Minor Option:
Chemistry
Thesis Availability:
Public (worldwide access)
Research Advisor(s):
Goddard, William A., III (advisor)
McGill, Thomas C. (advisor)
Thesis Committee:
Unknown, Unknown
Defense Date:
13 December 1976
Record Number:
CaltechTHESIS:11182009-101146638
Persistent URL:
DOI:
10.7907/8M08-WC34
Default Usage Policy:
No commercial reproduction, distribution, display or performance rights in this work are provided.
ID Code:
5380
Collection:
CaltechTHESIS
Deposited By:
Tony Diaz
Deposited On:
18 Nov 2009 23:12
Last Modified:
28 Oct 2024 22:37
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THEORETICAL STUDIES OF SILICON SURFACES USING
FINITE CLUSTERS
Thesis by
Antonio Redondo-Muino
In Partial Fulfillment of the Requirements
For the Degree of
Doctor of Philosophy
California Institute of Technology
Pasadena, California
1977
(Submitted December 13, 1976)
-ji-
Caminante son tus huellas
el camino, y nada mas;
caminante, no hay camino,
se hace camino al andar.
Al andar se hace camino,
y al volver la vista atras
se ve la senda que nunca
se ha de volver a pisar.
Caminante, no hay camino,
sino estelas en la mar.
Antonio Machado
-iii-
ACKNOWLEDGMENTS
I would like to thank’my advisors Bill Goddard and Tom McGi1]
for the many things they have taught me, making it, at the same
time, one of the most enjoyable experiences that I have had. In
addition, I would like to thank the members of our research group,
particularly Barry Olafson and Bill Wadt, for many helpful discus-
sions. I would also like to thank our secretary, Adria McMilian,
for her efficiency in getting the job done.
I gratefully acknowledge the support of the California Insti-
tute of Technology, the International Business Machines Corporation,
the Woodrow Wilson Foundation and the Physics Department of the
Universidad de Los Andes, Venezuela.
Finally, I would like to thank my family in Venezuela for their
constant support and my wife, Shelby, for her understanding and
patience in putting up with the whole thing.
-1lv-
ABSTRACT
The objective of this thesis is to study the electronic struc-
ture, geometries and chemical binding characteristics of the surfaces
of silicon and of the initial form of oxygenated Si. We examined the
(111), (100), and (110) surfaces, relaxation on the (111) and (100)
surfaces and reconstruction on the (100) surfaces. In addition we
examined steps on the (111) surfaces. In the oxygenated surface we
considered the geometry, excited states and ion states of both O and
0, bonded to the perfect (111) surface.
These studies indicated that surfaces and chemisorption lead
to localized electronic states for which explicit inclusion of elec-
tronic correlation (many-body) effects is essential. These effects
are included through use of generalized valence bond (GVB) and con-
figuration interaction (CI) techniques. These techniques require use
of a finite collection of Si atoms to represent the surface. We find
that very small clusters lead to reliable results if the model system is
properly tied off with SiH bonds (to represent internal Si-Si bonds).
- In Chapter 1 we report an effective potential for replacing the
‘ten core electrons in calculations involving the Si atom. The poten-..
tial is - obtained directly fron ab initio calculations on the states
of the Si atom and no empirical data or adjustable parameters are used.
The ab initio effective potential is tested by carrying out Hartree-Fock
generalized valence bond and configuration interaction calculations on
99 3? 6 3910, and
calculated excitation energies, ionization potentials, and electron
various molecules. We considered Si, Si,, SiH SioH and H
~V~
affinities both using the effective potential and without it (ab
initio). In essentially all cases the agreement is to better than
0.1 eV, providing strong evidence that the effective potential ade-
quately represents the Si core. This potential is utilized in all
of the calculations reported in subsequent chapters.
In Chapter 2 we consider clean (111), (100) and (110) silicon
surfaces. For the (111) surface the relaxation of silicon surface
atoms is studied by means of an Si (SiH), cluster. We find that the
surface state is accurately described as a dangling bond orbital with
93% p character. We determined the optimum relaxation of the surface
layer to be 0.088 toward the second layer. For the positive ion we
find that the surface atom relaxes toward the second layer by an ad-
dj tional 0.304. Using an Si3H_ cluster we find that the interaction
between adjacent dangling bond orbitals indicates that they are very
weakly coupled (with a splitting of 10.01. eV between the singlet and trip-
Tet spin couplings. )
For the (100) surface we used an Si(SiH3), cluster. We find a
relaxation distance of 0.10R toward the vacuum. We also considered
the 2x1 reconstruction of such surfaces using the results for ST5H,
and Si(SiH3)5 complexes. It is found that adjacent surface atoms
form a bond (1.76 eV bond strength), leading to pairing up of adjacent
silicons (with an optimum Si-Si bond length of 2.38).
In Chapter 3 we consider the electronic structure of divalent
steps on (111) silicon surfaces. We find three localized electronic
states separated by less than 0.3 eV. These states have quite
-Vi-
different electronic structure and are expected to be reactive toward
a large range of chemical species.
In Chapter 4 we study the chemisorption of oxygen upon Si (111)
surfaces. For single oxygen atoms we find an optimum $i-0 bond length
of 1.632. We also find ionization potentials in the range 11-16 eV.
Then we consider a model in which an oxygen molecule chemisorbed onto
the silicon surface has. an electronic structure corresponding to a
peroxy radical. We find ionization potentials in the range 11-18 eV
in agreement with experiment. We find an optimum 0-0 bond length of
1.372 and a Si-0-0 bond angle of 126° for the chemisorbed peroxy
radical.
-vii-
PREFACE
The subject of this thesis is the theoretical study of silicon
surfaces and chemisorption on such surfaces. The electronic structure
of surfaces is a problem that has been in the minds of scientists for
many years since the pioneering work of Tamm and Shockley in the thir-
ties. However, it has been only in the last few years that theoretical
studies have become prominent in the literature. |
The primary questions that theory should answer with respect to
the study of surfaces concern the nature of the electronic and geomet -
ric structure at the surface; the characteristics of chemisorption,
such as bond energies, geometrical configurations, excited states.
These are questions that are difficult to answer experimentally from
a microscopic point of view. They are quite different from the prob-
lems generally studied by theoretical methods for the bulk of crystal-
line solids. In that case the geometrical structure of the solid is
usually known from x-ray diffraction studies, and the concentration
and characteristics of the electronic states is very different.
In principle, one can obtain all the information required for a
complete characterization of the surface from the solutions of the
Schrodinger equation
Hp = EW .
The exact solution of this problem is not known at the present time,
and different approximations have to be introduced. For bulk solids
the standard approach used in the past has been the introduction of
~Vili-
approximations consistent with the periodicity of crystalline solids,
and the band theory of such solids. For surfaces these techniques
are not immediately applicable due to the lack of three-dimensional
periodicity, the presence of localized states and the lack of expli-
cit electronic correlation for the standard band structure techniques.
We have undertaken the present study with the explicit pur-
pose of including these effects. To do this we have applied quantum
chemical techniques developed for studying molecular systems where
the states are highly localized and thus require inclusion of elec-
tronic correlation for a proper description of the system. Consequently
in our studies of surfaces, we have used generalized valence bond
(GVB) and configuration interaction (C1) wavefunctions.
At present it is not possible to apply such highly correlated
wavefunctions to semi-infinite systems. We have opted, instead, to
apply them to finite clusters of atoms designed to include the most
important interactions for the particular property being studied. In
order to ease these calculations, we have developed an effective
potential to replace the ten core electrons of the silicon atom.
This potential was obtained from ab initio calculations on the sili-
con atom without the introduction of any extemal parameters. The
effective potential accurately reproduces the energies and shapes of
orbitals in molecular calculations.
| To apply these techniques we chose an insulator system of
technical importance where a considerable amount of experimental in-
formation is available to compare with the theoretical calculations.
~ix-
In addition, chemical arguments suggest that nonpolar semiconductors,
like silicon, have electronic states that are rather localized at the
surface, much more, for example, than metals. This has two important
consequences: first, it makes for a conceptually ideal situation in
the application of the GVB and CI wavefunctions, since these were
especially developed for localized systems. Second, microscopic
properties such as geometries and excitation energies are expected to
converge rapidly as a function of the number of atoms used in the cal-
culation: One can then obtain reliable results with a relatively
smal] number of atoms.
The advantages of the present method are: (i) since total
energies are calculated, it is possible to minimize such energies to
find the optimum geometry of the system; (ii) due to the inclusion of
electronic correlation, reliable results are obtained for ground state
and excitation energies and geometries; (iii) low translational sym-
metry configurations like steps and chemisorbed molecules can be
studied without the introduction of ad hoc assumptions as to the geom-
etry of the system; (iv) qualitative interpretation of the results in
terms of localized orbitals makes the analysis of such results easily
tractable. The disadvantages of the present technique are: (i) because
finite clusters must be used, it is not possible to calculate quantities
that depend on the presence of large numbers of atoms, like the inter-
action of surface states with bulk states; (ii) long range effects have
to be introduced by additional assumptions, like the dielectric correc-
tion for ionization potentials. Our results, however, look very
X=
encouraging because many of the properties arising from localized
states do not depend strongly on the size of the system and they agree
with experiment.
We have considered the (111), (100) and (110) clean surfaces of
silicon. For the (111) surface we studied the relaxation of surface
atoms and the interaction of adjacent dangling bonds on the surface.
We found an inward relaxation of 0.08% for the surface atoms. Adjacent
dangling bonds were found to be only weakly coupled.. For the (100)
surface we found a relaxation of 0.108 toward the vacuum. We also con-
sidered the 2x1 reconstruction of this surface and found that a strong
(1.76 eV) bond is formed between adjacent surface silicon atoms, lead-
ing to pairing up of adjacent rows with an optimum Si-Si bond distance
of 2.384.
We have also studied the electronic structure of steps on silicon
(111) surfaces. Experimentally only steps characterized by divalent
atoms (atoms with only two nearest neighbors) are found. For this type
of state we found three localized states separated by less than 0.3 eV.
The electronic structure at these steps is characterized by the divalent
atoms and thus they are quite different from the dangling bonds found
on the clean (111) surface. [They are similar to the states found on
(100) surfaces, which also have divalent surface atoms. ] These states
are expected to be reactive toward a large range of chemical species.
As an example of chemisorption we have studied the oxidized per-
fect (111) surface, with both 0 atoms and 05 molecules. For the single
oxygen atom bound to the surface we have optimized the silicon-oxygen
-Xi-
bond length, finding an Si-0 distance of 1.632. This is consistent
with experimental values on systems having similar bonding between
oxygen and silicon atoms. We have also calculated ionization poten-
tials and excitation energies for this system. We find that the
ionization potentials are in the range 11-16 eV. For 05 molecules
we find that on the first stage of chemisorption, the molecule has an
electronic structure corresponding to a peroxy radical, that is, only
one oxygen atom is bound to a surface silicon. This agrees with re-
cent experimental results in which the two oxygens are found to be
inequivalent. For this system we find ionization potentials in the
range 11-18 eV. The peak at 18 eV arises from an ionization out of
the 0-0 bond and is not present in the corresponding spectrum of
single oxygen atoms chemisorbed onto the surface. For the peroxy radi-
cal we find an optimum 0-0 bond length of 1.37h and a Si-0-0 angle of
126°.
Summarizing, we have applied correlated wavefunctions to the
study of silicon surfaces. We find that the results are very encourag-
ing and agree well with experiment. The methods used in the present
work are expected to be applicable to other semiconductors if care is
taken, through the choice of clusters, to include all of the important
interactions for the problem in question.
The following publications are based on parts of the work des-
cribed in the present thesis:
W. A. Goddard III, A. Redondo and T. C. McGill, "The Peroxy
Radical Model of 0. Chemisorption onto Silicon Surfaces," Solid State
Comm. 28, 981 (1976).
-xii-
A. Redondo, W. A. Goddard III and T. C. McGill], "Ab initio Ef-
fective Potentials for Silicon," Phys. Rev. B 14, Nov. 15, 1976 (in
press).
A. Redondo, W. A. Goddard III, T. C. McGil] and G. T. Surratt,
"Relaxation of (111) Silicon Surface Atoms from Studies on Si gHg
Clusters," Solid State Comm. 20, 733 (1976).
Chapter
Appendix A
Appendix B
Appendix C
-xiii-
TABLE OF CONTENTS
Title
Ab initio Effective Potentials for Silicon
The Clean Surfaces of Silicon
Electronic Structure of Steps on (111)
Silicon Surfaces
Oxygen Chemisorption onto Si (111) Surfaces
The Generalized Valence Bond (GVB) Wavefunctions
Dielectric Corrections
Effect of Correlation, d-Functions and Lattice
Constraints on the SigHe Cluster Model for
(100) Si Surfaces
Page
-)-
Chapter 1
AB INITIO EFFECTIVE POTENTIALS FOR SILICON
I. INTRODUCTION
The idea of using a pseudopotential to replace the core elec-
trons in quantum mechanical calculations of the electronic wavefunc-
tions of atoms, molecules, and solids is now well established. The
first attempts consisted of the work of Hellmann and Gombias! in the
mid-thirties. They realized that these pseudopotentials should incor-
porate the effects of the Pauli principle in order to avoid the
collapse of the valence electrons into the core region. This was put
on a sound basis by Phillips and Kleinman? in 1959. This work,
together with that of Heine and collaborators ,° initiated a vast series
of papers on the applications of pseudopotentials to the electronic
structure of solids.“ These successes also reawakened the interest in
applying this approach to molecules and atoms .>
Although the basic idea in the pseudopotential method is to con-
struct an (simple) operator that reproduces the effect of the core
electrons of a given atom on the valence electron, there are a number
of approaches to determine the specific form of the pseudopotential.
The most common procedure (with many variations) is to select a simple
functional form for the potential and then to adjust the several param-
eters in this potential to fit the experimental energy levels of the
atom or the band structure of the solid while requiring the pseudo-
potential to be weak (leading to orbitals with minimal numbers of
nodes). The alternate approach is to use .only theoretical information
~2~
in determining the potential, requiring the core potential to repro-
duce the results of ab initio calculations. Our approach is of this
657)
latter category (the method of Melius and Goddard 3; we choose the
core potential so as to reproduce the shapes and energies of ab
initio valence orbitals. The resulting core potential is referred to
as the ab initio effective potential, or more simply as the EP. Such
effective potentials have been previously developed for Li, Na, and K
atoms® and for Fe and Ni atoms ® and applied to a number of molecules
containing these atoms. Here we report the effective potential for
the core electrons of Si, which we have applied to all the calcula-
tions to be discussed in the succeeding chapters. We will assess the
accuracy of the effective potential by comparing the results of ab
initio and effective potential calculations on the ground and excited
states of Si SiH3, and SiH.0
2° 372°
The interactions of the valence electrons are handled just as in
ab initio calculations. Appropriate basis functions are placed on the
various centers and the molecular integrals are evaluated. These inte-
grals are then used for self-consistent Hartree-Fock (HF) or generalized
valence bond (GVB) calculations and ultimately in configuration inter-
action (cI) calculations to include various electron-correlation or
many-body effects. A special aspect of our approach is that we cal cu-
late total energies directly so that we can determine the potential
surfaces and geometries for various excited states.
Since the shapes of the valence orbitals are described correctly,
we expect the overlap between orbitals on various centers to behave
properly and hence for bond energies and geometries to be well described
~3-
Excitation energies, ionization potentials, and electron af-
finities are obtained by solving directly for the total energies of
each state and taking the difference. Consequently, one can distin-
guish between the different multiplet eigenstates of the molecule
(usually not possible in standard solid state pseudopotential
methods?) . Another advantage of the present method is that since we
include electron correlation effects explicitly, we can describe
processes that involve bond formation or bond breaking.
I]. COMPARISON OF THE RESULTS OF EFFECTIVE POTENTIAL AND AB INITIO
CALCULATIONS
Before embarking on a description of the calculational details
for the Si effective potential, we will summarize some of the resuits
of the comparisons between the EP and ab initio!® calculations on
molecules. This will put into perspective the procedure and what we
want to obtain from it. We start with self-consistent ab initio
Hartree-Fock calculations of the electronic wavefunctions of Si atoms
(both the ground and an excited state). From this ab initio calcula-
tion (using the method described in the next section) we obtain an ef-
fective potential without the introduction of any experimentally
determined parameters. It is this potential that we have used in the
calculations below.
As summarized in Table I, we carried out both 28-electron ab
initio calculations and 8-electron EP calculations for the electronic
states of the Si, molecule (at the experimental equilibrium geomet ry! !
for the ground state). The low-lying states considered here have
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-5-
either two or four electrons in the 7 orbitals and are denoted as ne
and n'y respectively. The second and third columns show the excita-
tion energies obtained from self-consistent eve 2
calculations per-
formed on the ground state (P25) and the lowest lying excited state
(1s0). The fourth and fifth columns compare the excitation energies
obtained from CI calculations on the low lying excited states of Sig.
In these CI calculations we included all appropriate excitations within
the space spanned by the GVB orbitals for the *ra(n*) or "n") states,
leading to 200 spin eigenfunctions of the proper spatial symmetry for
each CI calculation. In all cases the ab initio and effective poten-
tial calculations lead to excitation energies agreeing within 0.1 eV
for the CI wavefunctions. It is important to note that even for those
states that are close in energy, the ordering is not changed in the
effective potential calculations. Since there are numerous states of
various multiplicities and orbital coupling all within a small range
of energy, we consider Si, to be a stringent test of the adequacy of
our EP.
Next we consider the STH, molecule (using the experimental geom-
etry of silane!3 but with one hydrogen atom omitted). Here we per-
formed Hartree-Fock self consistent field calculations on both the
ground state of the neutral (7A,) and the ground state of the positive
jon (TA,), leading to the ionization potentials of Table II. The EP
calculation leads to an jonization potential within 0.17 eV of the ab
initio value. In order to provide an idea of how similar the wave-
14
functions are, we compare the Mulliken populations ~ in Table II.
Also listed in Table II are the ionization potentials (IP) from
Table JI. Energies for Sil, and sit.
ra
ree
All Energies in eV.
4.
State? Ionization Potential - Mulliken Population per Atom
ab initio effective potential |) ab initio effective potential
. sf on - + Si H
,8i, 7A, | 0.0? 0.0° 3.50 1.17 8.74 «1.09
sin? "A, | 8.637 8.4170 2.92 1,03 3.04 6.99
* The geometry is the same as in silane but without one hydrogen atom; Roy =
2.796 a,, J HSiH = 109°28', The basis sets used are the Si (6s4p) and the Si
(2s2p) of TablelV, and a (2s) contraction of the three gaussian hydrogen bases
of &, Huzinaga, J. Chem, Phys. 42, 1293 (1965).
» otal energy calculated is -290. 56390 h.
© Total enerpy calculated is -5.38986 h.
Gane ten core electrons have been subtracted from the ab initio Si population.
-7-
application of Koopmans theorem. This leads to IP's too high by
0.7 eV for both ab initio and EP.
As a final test, we considered the molecule H3S10, 5
yyw St — 0
H 0
corresponding to the above STH, unit bound to an oxygen molecule (the
new bond lengths are Si0: 1.648, "° and QO: 1.3668, /° while the
>), In these calculations only the Si core
$i00 bond angle is 125°53'
is replaced by an EP. Self-consistent GVB calculations (ab initio and
EP) were carried out on the ground state (2a"), and the CI calculations
were carried out within the orbital spaces spanned by these GVB
orbitals. The range of the excitation energies, as shown in Table III,
is from 0 to 19.6 eV, and in all cases the ab initio and effective
potential calculations lead to the same ordering of states as in the
ab initio calculations.
These results indicate that the excitation energies and ioniza-
tion potentials obtained with the effective potentials are in excel-
Tent agreement with those of the ab initio calculations. Since the
systems compared here are reasonably distinct, we consider these re-
sults to demonstrate the usefulness of our effective potentials.
C. THE AB INITIO EFFECTIVE POTENTIAL
The general forn we use for the EP 16 is
~8-
Table JW. Energies for Various Slates SIH,O, and Sill,O}. All Energies in ev.
State Excitation Energy :
eb Initio effective potential Number of .
cP CI Configurations®
SiH,O, -
24" 0.0° 0.0° 98
7Al 0.692 0. 662 283
*A' a 6. 600 6.617 283
7A” : 4,924 7. 850 98
7A! 8.142 8.116 283
2Ay 15,419 15.453 98
Far 16.174 - 16.172 283
SiH,O;
SA" 11.117 11.173 496
tar 12.210 12,263 316
At 12.582 12,613 340
al 13.985 14.031 ; 340
tA" 15.447 15.459 316
*A’ 15.468 15.493 530
2A" 15.495 15.520 496
SA 15.771 , 15.790 530
3A” 17.366 17.415 496
1A" 18.988 ; 19.048 : 316
tA’ 19.556 19.620 340
ar - 49,619 19.638 530
=the geometry is as follows: The SiH, geometry is the same as in Table II; the O,
bonded as a peroxy-radical to Siatom, eclipsed with one of the hydrogen atoms,
Royo = 3.099 a,; Ro_o = 2.581 a,; 2 O-O-Si = 125°53'. ;
bine Cl was carried out using the SCF orbitals from GVB(2)-SCF calculations of the 7A"
ground state. All double excitations from ground state configuration into the 7 orbitals
“of the O, part were included, These calculations were meant as a test of the effective
potential as compared with the ab initio results and need not represent the most appro-
priate way of describing the excitations within this molecule. :
Crhis is the number of spin eigenfunctions of proper spatial synmetry.
ay otal energy calculated is -440.29763 h.
Crotal energy calculated is -155,15456 h.
-9-
vFP(r) = J Uy(r)[s><2 (1)
a=0 *
centered on each atom whose core is being replaced. Here V(r) is a
function of the radius only and
[z><2| = JY [2m is a projection operator onto states of angular momentum % with res- The Vy (r) in (1) are fitted to an analytic expansion of the form nk 2 for ease in evaluating the multicenter integrals required in molecular calculations. Use of two to five such terms allows an excellent fit to -10- 8.0F SI EFFECTIVE POTENTIAL l ee _ Figure 1. Si Effective Potential (EP) Components Vo(r). Curves -]]- we must evaluate matrix elements of the form where the XuA and Xy ¢ are (Gaussian) basis functions centered on the Melius, Kahn, and Goddard®?/ have developed formulae, algorithms, and The EP is obtained as follows. We consider the Hartree-Fock equation for the valence orbital of angular momentum 2, on 5 describing the interaction of oa with the core. The first step consists in replacing the Hartree-Fock orbital 6 CHF goes smoothly to zero as r +0. The reasons for doing this are to ot by the “coreless Hartree-Fock (CHF)" orbital whose amplitude core orbitals of the same 2% with oft -1 -12- CHF (9) 6 n& CHF tains a corresponding Hartree-Fock equation. In it the core and oHF HF . CHF “CHF |. CHF so that ¢ = 0 and the orbital is smooth. [This is analogous placed by a plane wave. ] Once the orbital 9 is determined, one ob- may now overlap the core or- bitals. That is, Eq. (4) becomes oCHF GCHF, CHF _ CHF (h + Voore + Wal) ore = Ene Png ° (4°) now contains a repulsive part (arising from the Pauli principle) which The CHF orbital in (5) is not normalized. After renormalizing, the amplitude of oe at large r differs from that of ae by just the and hence we have modified the CHF orbital as follows, leading to the coreless valence orbital (CVO). The basis set for the HF orbital is partitioned into the core set (those basis functions important for the Is. and 2s core orbitals of Si) and the valence set, -13- where primes denote the core set. The CVO is taken to have the form M' M cvo | where the valence coefficients are exactly the same asin (6). The determine the {ats we set cvo and adjust the other M'-1 degrees of freedom so as to minimize the kinetic energy of the orbital. The net result is gCVv0 gcvo cvo cvo (h + Voore * Val? one = Eng Ong ° (4") In Fig. 2 we compare the HF, CHF, and CVO orbitals for the Si effective potential, vEP(r) (that is, a mere function of r), such that in (4") with an the eigenfunction and eigenvalue of (4") are also the eigenfunction and eigenvalue of (8) gCvO, CVO _ Cvo ; (8) (h + Ve P(r) * Val? Png g one ‘The components of the potential V(r) are obtained by projecting (8) AMPLITUDE [A.U.] AMPLITUDE [A.U,} -]4- 0.2 . 0.0 1.0 2.0 3.0 4.0 5.0 6.0 7.0 _ DISTANCE FROM NUCLEUS [BOHR] 8.0 -15- onto the basis CVO val and adjusting the parameters in (3) to minimize the deviation of (9) that all basis functions are included in the n° of (9)]. After For those angular momenta 2 represented in the core, V(r) with respect to the core orbital). For % not represented in the core, Ve (r) is nearly independent of &. Thus for Si we set and rewrite (1) as vEP(r) = V(r) |s> -16- Table IV. Basis Scts” used in the ab initio and effective potential calculations on the Silicon atom. Set used to obtain Ab initio double zeta EP (484p) EP double zeta -17- FOOTNOTES FOR TABLE IV “The form of a given basis function, x of angular momentum 2% is - pP.q_s . 2 tion (2=0), p= 1, q=s =0 fora p-type function (2=1), etc. bThis basis set is essentially the (11s7p) set of S. Huzinaga ("Approx- added. “Dunning 's double zeta contraction (T. H. Dunning, Jr., private communi- cation) used in all ab initio calculations. this set was used in the EP calculations on the Si atom and the Si, molecule. "This set is equivalent to an ab initio double zeta set. It was used in the EP calculations on STH, and SIH0,. -18- or EP ver = V(r) + Ve_glsess| + Vo-dl P> where Veg") = V(r) * Ver) and where s, p, and d indicate 2% = 0, 1, and 2. D. THE EFFECTIVE POTENTIAL FOR SILICON In Table IV. we list the usual basis for ab initio calculations To determine the d potential, Vae we considered the Vo-d potentials were obtained from the 3. and 3p orbitals of the trip- let ground state of Si In doing this we write S d It p d and solve (9) for V._4 and for Voed" ~19- in Fig. 1. With just three terms each, we were able to obtain devia- tions (sums of the squares) of 1.886 x 10710, 3.05 x 10°78 » and Using the EP, the basis on the Si can be modified to eliminate We compare in Table VI the results of EP and ab initio calcula- perimental triplet-singlet excitation energy is 0.781 eV and hence 0.275 eV below the GVB value. The experimental IP of Si is 8.149 eV 7 18 or 0.769 eV lower than our value. affinity is 1.385 eV ground state (s@p*) of Si for the ab initio and EP calculations. In Table VII we compare the orbital energies for the EP and the ab initio calculations referred to in Table VI. In most cases the difference -20- Table y, Parameters for the Si Atom Effective Potentials. See Eq. (3) for Quantities are in hartree atomic units, # n ms Cc 1 0. 2900090 -0. 07889166 -I 3,2105169 -3. 59100110 -2 1, 8478285 4,08004340 -2 0. 2389864 0. 45326622 -2 0. 9686443 0. 86954814 The effective potential for the core electrons also includes a long-range term of that the nuclear attraction term in h will be ~-4/r. -2|- Table v1, Energies for Various States of Si, sit, and Si’ (Energies in eV). Excitation Energy Type oF Ab Initio | Effective Simplified Electronic State Wavefunction® “Scr Potential ee c clive Si triplet (s’p’) GVB(1) 0.0° 0.04 0. 0° Si doublet (s’p°) GVB(1) 0.684 0.696 1.532 “The basis sets used were the Si (6s4p) and the Si (4s4p) sets of Table IV. Both of them were complemented with one diffuse function for each angular momentum type (Cin = 0.03731 for s and cin 0.2766 for p type). for an explanation of the terms used in this column. See Ref. 12 and the appendix to the present chapter bFor each state we considered the wavefunction using real orbitals and orbital symmetry restrictions. of (2, “Total energy calculated is -288.84378 h. Thus this state is not an eigenstate total energy calculated using the effective potential is -3.67668 h. “Total energy calculated using the simplified effective potential is -3.81514 h. -~C0- Oo SI 3S ORBITAL fe) AMPLITUDE [AU] effective potential calculation ab initio calculation -O.3° 0.2 - SI 3P ORBITAL o) oO -0.2 AMPLITUDE [AU] -0.5 | a ab initio calculation i n L_ l ! ! I 0.0 Figure 3. 1.0 2.0 3.0 4.0 5.0 6.0 7.0 8.0 HF orbitals of Si (Pp) as calculated ab initio and using the EP. a) 3s; b) 3p. Table VII. Comparison of Orbital Energies for Ab Initio (AI) and Effective Potential (EP) Calculations for Various States of Si, Si”, and Si”. All Energies in hartree atomic units. ~23- Orbital State 3s 3p, Spy Si” quartet (s’p°) AI ~0. 3020 -0.0615 ~0.0615 Si triplet (s’p2) Al -0.5544 -0.2958 | -0. 8671 EP -0. 5551 0.2956 ~0, 8725 Si” doublet (s’p*)* AI -0.3187 -0.0586 | -0.0272 7 | (0. 7071)° EP | -0.3199 -0,0593 -0. 0275 Si singlet (s’p’) AI -0.5617 -0.2597 -0. 8747 EP -0.5629 -0.2590 | -0.8804 Si quintet (sp’) AT -0. 7247 -0.3487 | -0.3487 sit doublet (s’p') AI -0. 8661 -0.5817 -1.185 (0. 9819) (-0.1341)4 EP -0. 8664 -0.5801 -1.189 -24- FOOTNOTES FOR TABLE VIT "see footnote b of Table VI. DNumbers in parentheses indicate CI coefficients of GVB correlated pairs. The wavefunctions of these pairs have the form C44 + C55 + C5045 where the C, (Cl coefficients) satisfy )} cf 1. See Ref. 12 and the appendix to the present chapter. “This orbital is correlated with a 3p, orbital having also C, = 0.7071. Ghis orbital and a 3P, orbital like it correlate the 3s orbital in a GVB(1/3) wavefunction. ~25- between the ab initio and EP orbital energies was about 0.001 hartree We have also performed calculations on disilane, H4Si-SiH.. 13 a Si-H bond length ~ of 1.48h with tetrahedral H-Si-H angles were em- ployed. A Si-Si bond potential energy curve was calculated by doing HF 2.53h, respectively) for the "s ground state of the system. From the HF calculations we find an equilibrium bond length of 2.358, The ex- 19 2.3318. We therefore predict a bond length perimental value is the Si-Si stretch is at 0.055 eV, whereas the experimental value 4520 0.054 eV. From the GVB calculations on this molecule and those on ~26- "uaydeyo quaseud ayy 03 Xipuadde ayz pue ZL ‘Jay 9as oyu} SL We sured auz Jo ABuaus uoLzeLauu0d 9y} SL AV f(L = 2 + ‘uted ay} JO SLeZLquO gay ay? yo deluaao AO} ZUALILJJ9OS JQ BYR SL en S629 + oly WAO} DY} SPY ULed paze_sasod e& 4OJ UOLZOUNJoAeEM OLg 2Z pue ZL ‘Sgoy Bag ‘aLed pazejawsod @ JO [eILGO Leunzeu ysaly BYyI SL LeLGO SLUL, ¥600°0 Z0Z0"O - 84200 24.00 $1200 $600°0 av “SHUM o1m0j}e sotjrey UT selszeugq tiv °(°% pz" = UY) FS Jo sazejs 3 2 QU} AO} SUOTFVMOTED [elWa}O aaljoaljq pue OffUl GY 10; sapBZzeugq Te}qQI1O jo uospavdwio) IMA a1qey, *TeyWqs10 patdnos0-ATsutg *Teyiqazo patdnso0-ATqnoq 2 cH 8PSL *O- 6SSL°0- 8PSL “O~ 6SSL‘°0- « G966 “O- 1866 °O- 1087 ‘0- Ze8h “O- Tetquayed aAljoayja OTT ae Tetquajod aatjoayje OTT qe Vv, -HIS ‘y, “HIS Sots1aua [Te}iq10 “SHUN oTuloye sorjaey Uy SeTs1oug Ily “(HIS pue *HIg 10j salZreuq TenqIO “XI ofqeL -28- Table X. Orbital Energies for the 7A” Ground State of SiH,O,.* All Energies in hartree atomic units. ab initio effective potential AE — -0.0397 -0. 0396 ~29~ FOOTNOTES FOR TABLE X athe basis sets used are the Si (6s4p) and the Si (2s2p) of Table IV bThis orbital is singly occupied. ©) NO and 2 NO ind‘cate the natural orbitals of the correlated pair, C1 no * Coos No? C, is the CI coefficient (Cy + CS = 1); the correlation energy of the pair; Sab is the overlap of the GVB AE is orbitals of the pair. See Ref. 12 and the appendix to the present chapter. -30- of Table II we find a bond energy of 3.08 eV or 71.1 kcal. SiH, Experimentally it is found” that this bond strength is 82+4 kcal. excellent agreement with ab initio calculations while eliminating ~3]- Minimum Basis Set Double Zeta u u On the form of a given function is given in footnote a_ to Table IV, bIwo mixed basis sets, MXS1 and MXS2 are obtained from this MBS set by “This set is the same as the (2s2p) set of Table IV. For the DZR set two diffuse functions are added to DZ, of s and Py type, with Syn 003648 the 3-gaussian basis of S. Huzinaga [J. Chem. Phys. 42, 1293 (1965) ] was used. -32- core orbitals and core basis functions. However, although this EP functions of each type (s, Py> Py» p,) on each Si. For use in our studies of large clusters we nave also developed a much cruder, sim- per type per center. In this case we eliminated all core functions Since the orbital is finite at the nucleus, the SEP is much less repul- ab initio calculations for the Si atom. We have also tested the SEP as compared to the EP in calcula- the surface atom, the minimum occurs at 0.153 bohr. Thus the SEP gives a good description of the environment of the core electrons when one —*A aTaeL 0} B aj0ouj00,7 20S, 2289660 ‘0- ZSZ6T ‘0 0 2 SHUN ofMoze sorjTVY UT are SatjTqueny *STetjua}Od aATJIIIA poalyiiduig uloyxyy 1g aj AOJ sxajyaureaegd ‘“IIX LqeL ~34- “W133 UOTJOR.IIIE IvayTonu sy} spnyout payoyd saaano *(2)% squauodwop (das) rermewa aatjoarrg peyridung tg *p aangpy IYHOS] Y OS Ob OF O02 Ol Oo 7 | WILNSLOd SAILOSISS9 GSlAMd Wis IS [av] WILN3LOd -35- than the EP. E. CONCLUSIONS These results are very encouraging. The effective potential pro- costs due to -the smaller number of electrons involved. 2. 10. ~36- REFERENCES FOR CHAPTER 1] in): (a) W. A. Harrison, Pseudopotentials in the Theory of Metals (Benjamin, New York, 1966); (b) M. L. Cohen and V. Heine, in Solid State Physics, ed. by H. Ehrenreich, F. Seitz, and D. Turnbull (Academic Press, New York, 1970), Vol. 24, p. 37. For a review, see (a) J. D. Weeks, A. Hazi, and S. A. Rice, Adv. C. F. Melius and W. A. Goddard III, Phys. Rev. A 10, 1528 (1974). C. F. Melius, B. D. Olafson, and W. A. Goddard III, Chem. Phys. See, for example, M. L. Cohen, M. Schluter, J. R. Chelikowsky, and By ab initio we mean a calculation in which all the electrons of ' the system are taken into account and all energy quantities are calculated exactly. 1. 12. 13. 14. 15. 16. 17. 19. 20. 22. -37- B. Rosen, Selected Constants--Spectroscopic Data Relative to Diatomic Molecules (Pergamon Press, Oxford, 1970). a) W. J. Hunt, P. J. Hay, and W. A. Goddard III, J. Chem. Phys. v,C0) _ -A,(E) r< Ry ~Z/r r> Ry where A, (E) is a constant that depends on the orbital energy E C. Moore, Atomic Energy Levels (NBS Circular 467, Vol. I, 1949). H. Hotop and W. C. Lineberger, J. Phys. Chem. Ref. Data 4, 439 B. Beagley, A. R. Conrad, J. M. Freeman, J. J. Monaghan, B. G. R. Walsh, private communication to W. A. Goddard III. W. J. Hunt, T. H. Dunning, Jr. and W. A. Goddard III, Chem. Phys. —38- Chapter 2 I. INTRODUCTION In this chapter we will consider the (111), (100), and (110) The presence of a surface on a semiconductor can modify the It is the localized states, case (ii), we will be concerned methods to the study of the wavefunctions characteristic of these -39- localized states. Of particular interest in the study of the localized surface The major difficulty in applying the GVB and CI methods to sur- Unless otherwise stated, all calculations described in the Si effective potential (EP) described in Chapter 1] was employed to ~40- replace the core electrons of each silicon atom in the cluster; (ii) a Si-Si bond length of 2.354 (from the Si crystal structure’) termines the motion of the H atom so that the Si-H bonds move as if they were the original Si-Si bonds. II. THE (111) SURFACE OF SILICON A. Introduction 6 On The cleavage faces of silicon crystals are (111) surfaces. Experimentally, freshly cleaved Si (111) surfaces exhibit a atoms might be involved in the 7x7 structure. Although the real cleaved surfaces suffer reconstruction, it is of considerable -4]- OND LAYER Fig. 1. Sketch of (111) Surface of Silicon. Surface atoms are indicated -4?- theoretical value to investigate the electronic structures of ideal For the clean (111) surface we were interested in investigating below) were employed for these studies. B. Surface Relaxation General Description In order to study the relaxation of the surface atoms, we used a potential (IP) and electron affinity (EA) for this complex. Calculational Details The calculations described in the present section were performed orbital (negative ion). This set contains diffuse s and p basis ~43- *souayds Lews Aq suaboupAy ‘sauayds abue_ Aq umoyus sue IS 20v4uNS -44- The geometry chosen is that of a tetrahedral bulk configuration An important consideration is the form of the wavefunctions used. The wavefunction has the form where QL is the antisymmetrizer or determinant operator; o. is a ing bond orbital, $43° In our calculations all 13 orbitals are solved -45- self-consistently, allowing each orbital to delocalize and distort to whatever extent it wishes. The positive ion is a closed-shell singlet and the corresponding Qe LO) yy C1 224) ] ’ where the prime indicates that the optimum orbitals dy to by are not new electron in $73 with spin B , a CO eters 924) $4 3(25) $1 3(26) (25) 8(26) ] : solving consistently for the 13 orbitals. Since two electrons are moving, uncorrelated, in one orbital, bitals ;(1)9.(2) _ 4 (1) 05, (2) + 4, (1), (2) In the Si surface, bss and dip are lobe orbitals localized on the sur- referred to aS in-out correlation). An alternate way of writing the ~46- GVB pair, as described in Appendix A is bj ql1)0;p(2) + 44y(1)054(2) = Cg g(19; (2) + 6% (1), (2) (2) orbitals %ig and oy are referred to as natural orbitals, ig resembles Although the GVB wavefunction includes the dominant correlation $43 and passes perpendicular to the surface (see Fig. 3). We refer to Fige 3. -A7- Sigh, NEGATIVE ION NATURAL ORBITALS Amplitudes for the natural orbitals of the negative ion GVB -48- these as 7 orbitals (inp = +] with respect to the [111] axis) while the dangling bond pairs of the negative ion are described as and the total wavefunction is All 16 orbitals and the coefficients C13 to Cig are solved self- 8,9 . consistently. In the present case, that of the corresponding HF wavefunction. _ Results For the neutral system we find that the surface atom moves towards the For the positive ion the surface atom moves toward the bulk by distance). The new $i-Si bond length is 2.252. The resulting vertical -49- Table I. Summary of Quantities Relating to the SigHyg Cluster Model of the Si(111) Surfaces With dielectric Without dielectric | corrections corrections Relaxation (A) -0.08 -0.38 0.17 -0.36 0.23 Energy (eV) Adiabatic 0.94 5.435 -3.062 7.358 -0.667 Vertical 0.0 5.778 -2.745 7.66] -0. 862 “The relaxation is with respect to the undistorted positions of the sur~- face Si, the positive direction is away from the bulk. The total energy is -20.04811 hartree “Energy necessary to excite the first symmetric C2), vibrational state. tional wavefunctions. The use of the harmonic force constant leads to vibrational frequencies of 0.033, 0.030, 0.028 eV, respectively. The energy at the minimum is 0.024 eV lower than the energy at the tet- -50- a) NEUTRAL Smee : /e b) NEGATIVE ION, Ist N.O. 4.0 -6.0 l Fig. lh. Dangling Bond Orbitals for the Neutral and Negative Ion vor -5]- 10 of 5.6 eV to 5.9 eV. Allowing the ion to relax to its new values For the negative ion we find that the surface atom moves away surface), leading to a new Si-Si bond length of 2.41h. The adiabatic electronic affinity is 3.06 eV. Discussion The relaxation distance of 0.08A reported above was obtained by N relating bond order to bond length was used a formula due to Pauling Ve have used LEED to analyze an impurity sta- Recently Shih et al. In the above calculations the initial (undistorted) geometry was calculations overestimate the Si-Si bond length by 0.028. Assuming such an overestimate for Si-(SiH3)3 we would expect the optimum $i-Si - 52 -~ bond at the surface to be at 2.31A leading to a relaxation of 0.13h two corrections cancel out. Size of Complex: Neutral System To test the convergence of our results with the size of the com- Our conclusion is that the Si-(SiH3)4 complex provides an excel- addressed in Section C of the present chapter. ~53- Table 11. Comparison of Quantities Relative to the Dangling Bond Orbitals of SiH, and ST gH: Si-(SiH3)3 STH, orbital energy - -0.3356 s~functions p,~functions s-functions p,-functions coefficients? 0.2315 0.5197 0.2585 0.5313 ant the undistorted geometry DThese are the expansion coefficients for the appropriate basis func- tions on the “surface” Si atom. -54- Size of the System: Ion Systems For the neutral surface state, the effects of the bulk bonds on and Si-(SiH respectively, the self-consistent calculations yield 3)3 instead developed an approximate procedure as explained in the fol low- ing subsection. The Dielectric Continuum Correction Consider a semi-infinite solid with dielectric constant e¢ , and a positive charge at a height h above the surface as in part a of the ~55- Table III. Comparison of Ionization Potentials for SiH, and Si gly. All energies are in eV. SiH, Si-(SiH,), . “At the undistorted geometry bobtained from the orbital energy of the appropriate dangling bond “Obtained by taking the difference between the energies of the pos- diagram below. oy kh -> A B C The total interaction energy of the charge with the surface is resulting from the polarization induced on the dielectric medium. 138 obtaining h = 0.805R. Using this bond orbital from the surface, the hemisphere of radius ro? due to a positive charge at h (part b of -57- the diagram above). This correction energy jg 3b 2. 2.1/2 which for the undisturbed complex becomes AE = -1.87 eV. Similar corrections were made for each position of the surface Si An ambiguity in our model is the value to use for Yor Is it the The procedure is approximate, of course. One would like to h with a charge density spread over the complex. -58- (4yoq) UOL JOY ZLe°0 gi2°0 - 2vp'o €89°0- 21 °O- = Sip winuLrdo SALZEHSN DALLSOd 52412 e69N geht3}S0d 4 HS" tS »¥0L3402 Sid D14}98L81P UILM 214}03aLp ynoyyim "SaauquRey UL aue saLBuoua LLy ‘aoesuns (LLL) LS @y2 4O Lapow uaqsni9 “AT P1geL -59- FOOTNOTES FOR TABLE IV @Di stance along the [111] direction. Zero corresponds to the unrelaxed buartree-Fock calculations [eqs. (1) and (2)] using DZ basis of Table “Generalized Valence Bond calculations [eq. (3)] using the DZR basis dinterpolated energy from a cubic splines fit. -60- C. Interaction of the Surface Dangling Bonds General Description In order to study the interaction between adjacent dangling Self-consistent solutions of the wavefunctions of this complex occupied, leads to the two wavefunctions (for valence electrons only) Mar(or) = oStnslet eqyo, a £4, (1)o,.(2) #4,(1)6, (2) x fo(1)a(2) + B(1) a(2)]} (5) where -6]- *souauds Lpeuws Aq | IS 3OVsYyNS -62- Sout = [Oy (3de(3) 10, (4)8(4)]---Log(17)a(17) ILog(18)8(18)] (6) Using the same orbitals for both states the energies of (4) and (5) can be written as l4 pSinglet _ Et EX ctriplet _ Eo 7 Ey where S is the overlap, EO = is the energy of the product wavefunction, re and’? Fy ~ 2S<9|h| o> * Ker is the exchange term. Here Jor and Kor are the usual two-electron Coulomb and exchange interactions. In the case that S = 0, we see that EY = KK, > 0, and hence that ar ~6§3- Table VY. Quantities Related to the Interaction of Adjacent Dangling Bonds in the Si He Cluster Model of the Si(111) Surface. All energies are in eV. State Unrelaxed Relaxed? 3a" (er) 0.04 0.07 0.0% 0.0% Ta (or) 0.009 0.014 0.013 0.015 Overlap= 0.004 0.006 0.013 0.006 “The states are defined by the local symmetry, Co. of the Si 3H. complex, fe) "surface" hydrogens were relaxed as if the Si-H bonds were Si-Si “The basis sets are defined in Table XI of Chapter 1; DZ is the most Gr otal energies calculated were -14.12125, -14.47428, -14.54304 and “Overlap of the two dangling bond orbitals, b> and o, » from the respectively. ae) C1] Fig. 6. -64- f “To! x / t -5.5 12.5 [iO] Dangling Bond Orbitals for the | S13H¢. A'(2r) GVB(1) Calculation of Long dashes indicate nodal planes. -6§5- one-electron term in EY is negative and, for larger S (e.g., S > 0.1), Thus, depending upon the sign of EY and the relative magnitude Calculational Details Initially we chose the Si He complex to have a tetrahedral geom- In order to study the effect of basis sets on the wavefunctions, MXS1 (Mixed Set) and DZ (Double Zeta, two basis functions per atomic viates from the MBS only by the presence of an extra P, basis function on each surface center. Excited States The states (4) and (5) are referred to as covalent states since Other excited states of this cluster are as follows: -66- (i) Neutral ionic states: Keeping two electrons distributed over the two dangling bond orbitals, we can construct two states in addition to (4) and (5), namely "yr (2-v?) 2.{9) 4,06, (1)6, (2) -6,(1)6 (2) Ma(1)8(2)] » (7) These states correspond to the ionic or charge transfer states in which an electron is moved from one dangling bond orbital to the other, the local symmetry of the cluster, in this case C.). The Tan (2 2) state is straightforward to obtain from a self- consistent calculation; however, the "ar y2ay?y State 1S more complex (due to a lower state of the same symmetry; note the orthogonalization cribed below. A simple estimate for such charge transfer excitation energies where R is the separation distance of the centers; IP is the ionization potential and EA is the electron affinity. Using the values of Table I -67- for the Sigg complex, and assuming no geometrical relaxation, this leads to It 7.66 eV - (-0.86 eV) - (soe) (27.21 eV/hartree) 4.77 eV which is in very good agreement with the value of 4.82 eV obtained Such states may be significantly stabilized by polarization ef- may lead to an excessively large excitation energy. (ii) Band to surface states: Since the dangling bond orbitals are only partially occupied, one expects low-lying transitions in which an 3 ' (b>s) = ALG El Ooheoeoh Oo )b9% JoBac] ’ (9) and [here alk denotes the wavefunction corresponding to (6) but with the consistent level of description of all states requires a configuration interaction wavefunction. -68- citation energies as compared with the semi-infinite system. (iii) Surface to bulk states: Other transitions involving the surface orbitals consist of excitations into the empty bulk states. Typical excitations have the form 9(s>b*) = QL. ap (oy £o,) M4000] a) and Configuration Interaction Studies In order to explore the effects of electronic correlation in Starting out with the self-consistent wavefunction for the 3 qn bond region (for the 3 A" state two of them are occupied and two unoc- orbitals (denoted as CI). CI should lead to a good description of -69- the covalent and neutral ionic states and to a fairly good description of the b+s and s>b* transitions (within the restrictions of our basis). Hartree-Fock Wavefunctions Combining the on and >. orbitals as bg = (b, + o,)/ CS) leads to two orthogonal orbitals which are symmetry functions for this particular complex. Since we see that wavefunction (5) can be equally well written in the Hartree- Fock form, OTPTEE = A Lo, a Loy (1 g(2) = 6g(1)94(2) Halt )8(2) + 8(1)a(2) I} (5") On the other hand, bodp t Ono) = C7oq0 r gg” ° Coo), u where Cy = C5 for SO. Thus the GVB wavefunction for the singlet Indeed, the best closed-shell MO calculation 9°) = B {5 ypLog(1og2)o(1)3(2)]} leads to energies !® ~ 3 eV above the GVB wavefunction (4). Thus, the -70- HF wavefunction does particularly poorly for singlet coupled orbitals Since Hartree-Fock calculations on the A" (92-12) state should be quite ade- quate, however, the Vat (g? er?) state cannot be well described. Some a HF wavefunction. Results (i) Covalent states - In Table V we show some of the quantities relating to the two covalent states for the Si 3He cluster model of Si Including the surface atoms of the whole surface these results Suggest the use of a Heisenberg Hamiltonian H=E - J Jez Se*Ss with an exchange term between adjacent surface atoms of -/|- J.; = 0.01 eV ij J.. would seem to indicate a ferromagnetic surface for kT << 0.01 eV. ij tortions leading to stabilization of singlet pairing would be expected to produce a lowering of the surface symmetry to at least a 2x1 unit cell (since the singlet paired orbitals would not be translational ly equivalent). (ii) Excited states - Table VI shows self-consistent field re- sults for some of the excited states discussed above. We also compare Section B. The wavefunctions for the positive ion states have the form and *A'(b > vacuum) =QLO).44(8,5, - 66,000] where Oe ulk is similar to (6) but with the , terns omitted. 2q) orbitals while “A! (b + vacuum) is the wavefunction for an ionization Trom one of the bulk orbitals. -]2- Table VI . Comparison of Calculations for the Si3H¢ Cluster Model of (111) Si surfaces using the MBS, MXS1 and DZ Basis Sets.2 All energies are in eV. DZ MXSI MBS athe unrelaxed geometry was used. DT otal energies calculated were -14.54304, ~14.47428, and -14.12125 - /3- It is interesting to note that for the DZ basis the 2 ne state, where the og and dy orbitals in a poor description of the 2 The results of the CI calculations are illustrated in Table VII. In Table VIII we compare the CI results of calculations of the Table VIII we also show the values obtained from the orbital energies of the 3a" (or) SCF calculation. -74- of the Si (111) Surface. 42? All energies are in eV. No. of Spin No. of State® - Energy — Eigenfunctions Determinants “The MXS1 basis set was used with a tetrahedral (unrelaxed) geometry. bin the CI calculations double excitations were allowed within the two occupied and two virtual orbitals corresponding to the dangling “Successive primes on the b's indicate different "bulk" orbitals. Che total energy calculated is -14.48445 hartree. -75- punoub ay} 0} paduapau sue salfuouea ayl "teuqnau ou. JO UOLZeLND1Ld "UOLZELND [CO PLALZ PUSSLsuoOd-J Las (AI) we 3y2 30 ABusua ay. 04 pawresey, 1) aya Jo a2e75 “SqURULWYaTIP ZBL OF LGB JO YS}SUOD SUOLJOUNZaAeM oYL *ALqquOoH (paxeauun) Leupayerray ayy 7e puke Jas siseq LSyw ayy 404, GLL°EL 88p°EL 6LE‘EL 82°21 (unnoea < a OvL*EL 92621 062 °ZL £09° LL (unnoea «® oa OVL*EL 92621 9ev°eL 228 OL suoen«8a),¥, "AQ UL aue salHuaua [Ly ,*seoBsuns (LLL) $S $0 Lapoy 4azsnL9 BY. 4oj S[eL IE 16 “ITIA PL9eL -76- D. Review of Experimental Data One of the most important experimental characteristics of the face of the diamond structure.© This has very important consequences The basic experimental tool in the structural study of surfaces is Low Energy Electron Diffraction. / It is based on principles similar *b of 2 to 10A, thus the electrons collected by the LEED free paths showing a 2x1 pattem, !/ indicating that the unit cell is twice as -7]- large in one direction (compared to the ideal or 1x1 surface). Upon 17 ture. The transition occurs !8 between 350°C and 390°C (depending on the surface roughness). Clearly, some sort of rearrangement has occurred for both Other LEED patterns have been observed in Si (111) surfaces .0°'7 In the early sixties it was speculated that the low energy tail 19 ments was due to surface states. This view was supported later by 20 further photoemission experiments. More recently, photoemission 10a 10b experiments by Eastman and Grobman have and by Wagner and Spicer face. The surface contribution was separated by making a second mea- surement when the surface was contaminated with oxygen (presumably -78- removing the surface states by bonding to oxygen atoms or molecules.) by the surface states. Eastman and Grobman!°8 find, for silicon (111) the Fermi level. From these measurements and from the known width of the valence band, and assuming that the matrix elements for surface and bulk transitions are the same, it was concluded! 2 14 that the surface state density is 8x10 electrons-cm”<. Assuming one electron per each surface silicon atom leads to a surface state density of 7.85 x 10'4 electrons-cm™<. Since there is a band bending of 0.6 eV, Poisson's equation requires | 0.3 x 10!4 surface states per cm*, leading to a total of 8.1 x 1014 occupied surface states per cn, in sion experiments by Rowe?! 4 and Rowe and Tbach®!> also lend support to Other experimental techniques have also successfully identified Their results are in qualitative agreement with the photoemission mea- Surements discussed above. Hagstrum and Becker@? have also reported -~79- ion-neutralization spectroscopy experiments in which the presence It is now possible to detect transitions between surface optical reflection spectroscopy. Chiarotti et al.24 have measured 24 or when heated to obtain the 7x7 structure. Another way of detect- ing surface state transitions is by photoconductance measurements; consider the 2x1 pattern first. This structure is found when cleaving Silicon single crystals at room temperature under ultra-high vacuum!” It is believed” that the rearrangement suffered by the surface does other, thereby healing the crack. The experiment is only applicable to materials that show no plastic flow at the temperature at which the ~80- experiment is performed. This applies to silicon at room temperature .6:28 the lowered rows have dangling bonds with sp hybridization ing thereby raising the atoms). This model was designed to explain the -8]- salient experimental features known at the time; namely, the 2x1 spin density of 0.8 to 2 x 10/4 spins per cm? was found (that is, Our calculations on the Si He cluster (see Section C) indicate 28 and Haneman and Heron?! is that the mechanism proposed by Haneman When a cleaved (111) surface of silicon is heated to at least such patterns requires high temperatures, it is believed? that there is considerable motion of the atoms on the surface. A model for this -~82- structure has been proposed by Lander and Morrison,~~ based on a the present no strong evidence has been found to favor this proposed very weak, suggesting that benzene-like structures are not particu- larly favored. E. Review of Other Theoretical Calculations Theoretical calculations of surface states have been performed 33a and Shockley??? in the thirties. One since the early work of Tamm Cluster calculations on silicon surfaces have been performed by Batra and Ciraci.2” 35 They used the Xa method,~~ which uses a molecular tight-binding formalism, °° the bend orpital inethod>” or pseudo-poten- tial methods . 2° The self-consistent calculations use self-consistent ~83- pseudo-potentials and a local approximation to the exchange energy 22249 Basically, the same conclusions are reached by both Appelbaum and Hamann?2 and Schliter et ato where Vos and Vee are the electrostatic (found by solving Poisson's for the correlation energy of the jellium model. Ve on?) represents the non-electrostatic electron-ion core interactions and was obtained *@ is then introduced into the one-electron Schrédinger equation _ 5 y(F) + v_(F) vw) = EW). (11) Equation (11) is then expanded in the Laue representation”? which Appelbaum and Hamann?" find an ionization potential of 5.3 eV is highly localized on the surface atoms. This band is partially oc- cupied, lying close to the top of the valence band. When relaxation -84- is allowed this band splits into two peaks and additional bands ap- pear. Schluter et a1.9 used a standard solid band structure calcula- tion technique with a slab geometry which is repeated in a direction which is similar to that found in Ref. 39. One characteristic common to all these methods is that for geom- large as the value we have calculated using the Si-(SiH3), cluster (see Table I) and about three times as large as the experimental value 12 of Shih et al. For this reason, since the electronic structures cal- culated using these techniques are highly dependent on what relaxation distances are used, a careful comparison between experiment and theoretical calculations for different geometries would clarify the The techniques discussed in this section use doubly occupied taining the correct electronic structure of the systems in consider- ation. @ -85- II. THE (100) SURFACE OF SILICON A. Introduction The (100) surface of diamond structure (as of crystalline sili- In the unreconstructed (111) surface each surface atom is per surface atom. An important experimental difference between the (100) and the (111) surfaces is the fact that a (100) surface cannot be cleaved. LEED patterns indicating reconstruction of the surface. > <3 Fig. 7. Sketch of (100) Surface of Silicon. -86- ge Es —O Xx — IST LAYER Surface atoms are denoted Second layer atoms are denoted by crosses. Arrows indicate the motion of atoms for one of the reconstruc- -~87- In our approach we again use a finite cluster of atoms. In the Our results can be summarized as follows. For the ideal (100) sur- 45 face by the mechanism proposed by Levine’~ in which two adjacent rows rotate towards each other producing a 2x1 unit cell. B. Basic Electronic Structure General Description For the basic model of the (100) Si surface we have chosen an other silicon atoms but we have substituted them by hydrogen atoms. Here, as before, the hydrogens have the function of decoupling the 99. "back-side-of-the-cluster" electrons from the "surface" electrons. This complex is shown in Fig. 8 for the tetrahedral (ideal) geometry. figuration (shown schematically in Fig. 9): (om): Of the two nonbonded electrons one is in a p-like orbital and Von) respectively. (0°): The other state can be thought of as having both nonbonded orbital pointing below the Si-Si-Si plane, of the form o -A7. These orbitals are spin-paired into a singlet state denoted V2), ~89- "SUOLJELND Le) BdeyuNS COL) 3Y3 UL pasn xXa{dwoy HE 1g *9 Bly -90- *[RLLquo yoea Jo} suoujoaLa JO Uaqunu By} a}zoUapP s}op Suaded ayy Jo auetd ayy $0 }NO BuLwod [ezLquo axL[-d e Sjuasaudau ajduld y *aIeJ4NS (OOL) 243 SuLtepow vazsn1g Hers ayz Jo seqeys ay yo o1qeUIEyDS “6 “BLA TAN —_ wl A2 OO ‘ (49), Bn, -9]- For the (om) states electron correlation effects are not of the ~{o1) and V 42) states to be very close to each other with Vg?) Tower. These states are analogous to the states of the methylene state is generally the 3a) state. / Calculational Details” The geometry variations performed in this system consisted of The calculations for the 3 (57) and Von) states consisted of open-shell Hartree-Fock wavefunctions, in which the o and 7 orbitals -92- were Singly-occupied. Thus the wavefunctions have the form 3(om) =A{d, 1, Co, (1)6,(2) - (16, (2) Mol 1)8(2) + 8(1)a(2) 1} and where Qulk represents the wavefunction of all bond pairs. Appendix A) in which each nonbonded electron is allowed to have its own orbital. The wavefunction has the form (0) Opn pan (sn aml2) + 5 pM Osan l2) J x [a(1)B(2) - B(1)a(2)] . As shown in Appendix C, the closed-shell Hartree-Fock wavefunction for this state Mot )MF -Q fo, alo (1)6,(2)Ia(ta(2)] gives an energy which is approximately 0.54 eV higher than the GVB(1) The basis set used in these calculations consists of the double have tested this for the present system and have confirmed the need for ~93- d-functions on the central (divalent) Si of the cluster (the compari- in Appendix C) . Hence, all calculations include is described as DZd in Table XI of Chapter 1.) Results Table IX and in Fig. 9. As is found’? in SiH,» the ground state of the Since the 3 (on) and Va) states have the same electronic con- central angles because there is only one electron in the nonbonding o orbital. For the V52) state the geometry is quite different. Now two -94. Table IX. Results for the S71 3He Cluster Modes of the (100) Si Surface Vertical Adiabatic Wavefunction® (hartrees) (°) (R) (eV) (eV) *see the text for an explanation of the symbols. obtained by a cubic splines fit to the calculated points (see Appendix C). “Positive values indicate motion toward the vacuum. Zero corresponds -95- electrons occupy overlapping o-like orbitals leading to repulsive in- a system in which the constraints of the lattice are absent (see Appendix B). Tn that case we find optimum angles of 95.2° and 119.9° for the (0%) and 3 (oa) states respectively. Here the vertical and adiabatic excitation energies for the 3 (on) State are 0.49 eV and 0.18 eV respectively. For the V2) state, the two o-orbitals point away from each other, one above the Si-Si-Si plane and the other below. The final 5 0:44 0.53 . : . 49, orbitals is otAmv. These orbitals are plotted in Fig. 10. For the localized at the central Si, and a o-like orbital, also localized at the 49 is 59+20,0.68 0.01 central Si, whose hybridization” j d These orbitals are plotted in Fig. 11. C. Surface Reconstruction Introduction The (100) surface is sketched in Fig. 7. Each surface Si (denoted moving alternate rows towards each other (as shown by the arrows in Fig. -~96- HSi-Si-SiH, *(o7) 3.0 Cog -3.0 Fig. 10. o+Am and o-At GVB Orbitals for the Non-Bonded Electrons of the ores, 2, 2,0 aT FOnq a ee ee ee ee -3,Q Le i Fig. 11. o and mw Orbitals for the Non-Bonded Electrons of the 3 (on) State of the Si 3He Cluster Model of (100) Surfaces. ~98- 7) leading to formation of single or double bonds. In the latter To test this model we have performed calculations utilizing Let uS now consider how the states of this system can be described. Starting with the (o om) and Vom) states of tyvo SiH, units at an infinite Regus; distance we can construct the + *L(o4) (0%) 1. [Recall that for SiH, the order of the states is 48> Detween the Son) 152), 3 (on) and Von)]. Since the difference lower in energy than a Vg2) and a Von), thus the next state should have the form > 3(o,m,)P(o,n,) I. Next a state with Von) must be higher states. As the two SiH, units are brought together to the un- the same with only small changes in the energy. -99- For our calculations we started with the two Si atoms at the As the two silicons move towards each other we expect to find state for the unreconstructed (100) surface geometry, the ground state Yh 2.1 can be accurately described as (og) (o°)1. This is no longer true at the bulk Si, silicon-silicon distance. Here we would have to . 3-1,_2,3 3,3 1,2 state is better described by “Lym )3(o,0,) 1. We thus see that forming a m-1 bond between adjacent silicons produces a considerable each other. For states that do not form a m-1 bond we expect the -100- SIH, R=3.83 A R=2.35A Leon AN PS Fig. 12. Geometry for the Si “fl Cluster Model of Reconstruction on the (100) Surface. (a) R Si-Si = = 3.938 (unreconstructed geometry). -101- 3h 3, 1 Ce? 1te?)] Fig. 13. Schematic of the Different States for the SigH, Model of Recon- "JO[GULS B OZUL parted ave spua ayy uo sLeriquo ayy ySe"Z = PST ES, Et P Y60°E = y toi “[(o ©) ( 4 4), *uOLIe [No “Lb gA90S Vy (2) *seoejuns (O0L)ES Uo UOLJINAZSUOIaY JO Lapoy Aaqsn 9 Weis O47 40 BOURISIG JO UOLIOUNY eB se saBueYD LeILQuO o8 C1103 ce" Od 60¢ ese ((2o Yo), (ets) 3, -103- 45h \ \Gz ae a \ee ~ wan, —l -104- potential energy surface to be repulsive or to have a very shallow minimum. Results Tne results of the SigHy calculations are Sunmarized in Table 1,1 5-3 3 . 3,1 _ 43 1-1 of the surface Si atoms. Both the ~[ (mp7) (o,0,.)] and [ (mom) lower (by 0.13 eV). The bond energy (eneray difference between the ‘ayn (oo) at its minimum and the "Eh (o6) (04) at the unrecon- structed geometry) is 2.4] eV. The SigHy cluster does not include the energy due to the bend- hedral geometry. "AyOg £G2°Z 3e SNLeA OY 02 pauedwod se ja Lp'z JO UIdap [Lam e 02 SpuodsawuOD StUL ‘un {OO SLU} UL pasn s,oOquAS ayy JO UOLZeUeLdxXS Ue “OJ 4xXO] pue EL ‘BLY aS *A® bL*L JO yidap [Lan ® 03 spuodsaus09 StuL, "squlod pazejnoyed ayz 07 S314 sauttds o1qnd Aq pauterag, *SUODLLLS uake] puodas JO Bulpuaq 40 pazoai40d aue sasayjuared UL saneA, (1688 19°6~) 9€0°2 (L6veLe*6-) (66ESLv°6-) (vévlSp'6-) (ZI8Z29°6-) (8S¥Z19°6-) Ad OY “p uOLzoNagsuoday advJans (COL) LS 40 Lapow vazsnig "HLS *sdouTUeY UL due SaLBuaue [Ly -0.05 — ~0,50 — | | | | | 28 5.25 8.28 7.28 Fig. 15 Potential Curves for the SioHy Cluster Model of Reconstruc- energies, dashed lines indicate energies corrected for the -107- To account for the increase in energy due to the change in hy- Since the Si3He calculations were performed by bending the two "surface" The corrected potential curves for the dimer are also shown in Fig. 15. with an optimum Si-Si distance of 2.384. From these calculations one concludes that the Schlier- calculations do not rule out other models. Calculational Details (i) Si,H, cluster For the SigHy cluster the geometry was varied by allowing the states. As in the HSi-Si-Sit, calculations, correlation effects are -108- Table XI. Potential Curve for the Bending of the Central Si Atom in the HSi-Si-SiH, Complex. Angle? Energy 0.0 -14.594074 “This is the angle between the plane formed by the three Si atoms ~109- crucial for the Eh (0%)! (0%) state. For this state we have used a correlated. The wavefunction has the form MEF) (02)] =A fog, ylTo- 28) unpace(errret2dy » (12) where © represents the wavefunction for the Si-H bond pairs (having Si-H "surface (9? 212) = (gern), (9) (cern), 1) Mota) 1) (sera) (2) x{c,fo(9)B(10)0(11)8(12) - 0(9)8(10)8(11)a(12) ~ 8(9)a(10)a(11)8(12)+ 6(9)a(10)8(11)0(12)] + Col2a(9)o(10)8(11)8(12) - B(9)a(10)8(11)a(12) ~ 0(9)8(10)8(11)a(12) - o(9)8(10)0(11)8(12) Correlation is also necessary to properly describe the 37] ; -110- °surface(9e"*+12) = Lo, (90, (10), (1104, (1293 x {c,Ca(9)B(10)a(11)a(12) - 8(9)o(10)0(11)a(12)] + C,[3a(9)a(10)a(11)8(12) - 8(9)o0(10)a(11)a(12) - a(9)8(10)a(11)0(12) - a(9)o(10)8(11)0(12) J} For the quintet i Co CO the wavefunction is also of the form (12) with ®curface(27. 72 12) = Lo, (Ie, O06, (Ve, O2)] x fal9)a(10)a(11)a(12)] . (ii) SisH, cluster For this cluster the geometry variation consisted of "bending" these geometries, only the V2) state was solved at the GYB(1) level. -I11- St; H. X= 45° Fig. 16 Geometry for Si 3H. Calculations in which the Central Silicon -112- D. Review of Experimental Data The experimental situation for the Si (100) surface is, by far, Two basic models have been proposed for the reconstruction of Si Farnsworth>2 and modified by Levine. In this model adjacent rows of they keep the same bond length as in the bulk, but variable bond angles. -113- If only single bonds are formed between adjacent We have previously discussed some of the features of this model concluded that this is a feasible model.>” Recently, experimental the surface atoms rather than by vacancies. Furthermore, using the 45,50 pairing model of Schlier-Farnsworth-Levine one can explain the experimental results of Sakurai and Hagstrum. >? The two stages of H ~-114- chemisorption are explained as follows. Starting with the pairing model, two adjacent atoms are close to each other due to the ToT, bond (i.e., we start out with either the Cr Ce or "E (mga)! (opo,.)] states of Si oH ). This leaves two dangling bonds, or and oe which can be coupled into singlet and triplet spin states with very similar energies. In the first stage of the chemisorption, ducing a 2x2 LEED pattern. This phase has been called by Sakurai and 55 Hagstrum™~ the monohydride phase, since at saturation there is one H atom per surface Si. In the second stage of the chemisorption process additional hydrogen atoms attack the dimer breaking the nr TT. bonds, ve leads to a 1x1 LEED pattern, since the lateral bonds which produced the displacement of adjacent surface rows are no longer present. 55 Sakurai and Hagstrum™~ have termed this phase the dihydride phase. -115- III. THE (110) SURFACE OF SILICON A. Introduction Looking - at an unreconstructed (110) surface of the band, and the third is a second layer atom. This means that each sur- face atom has one electron in a dangling bond. For the tetrahedral As in the case of the (100) surface, the (110) surface cannot allows rather complicated LEED structures to form. -116- COND LAYER Fig. 17 Sketch of the (110) Surface of Silicon. Surface atoms are de- crosses. -117- The basic characteristic we wanted to investigate was the in- To study the basic spin couplings we have chosen three clusters consisting of two, three and four silicons, namely Si SiH. and aigs St 3H. might stabilize the singlet spin coupling of adjacent dangling bonds. B. Summary _ In order to establish the basic nature of the coupling of ad- in the diagram below. / S H Si -118- Here both Si represent surface Si, while both He replace surface Si In the undistorted geometry we find that the triplet state is The results of these calculations are shown in Table XII. For state is the singlet, \(oy09) by 0.19 eV. We also note that the -119- Fig. 18 Geometry for the SigHy Cluster Model of (110) Surfaces. -120 - (UN <9 Fig. 19 Geometry for Three and Four Silicon Clusters Modeling the -I21- Table XII Comparison of Energies for the SigHy Cluster Model of (110) Surfaces. Energies are in hartrees. a . : . . to the ground state "(o405) of the eclipsed geometry. -122- ground state singlet of the eclipsed geometry is lower than the In the (110) surface there are infinite chains of such atoms With three Si (SiH, complex shown in Fig. 19) the low lying states have the form shown in the diagram below “NON Energy ar (c,0y)0,] S = 3/2 LE 0.0 o48,)0,] S= 1/2 | ‘ 0.02 Here a '__...Y indicates singlet pairing of two orbitals, while a be described as the resonant and anti-resonant states or in terms of wavefunctions, as {Lo4(1)o5(2)04(3) #04(1)05(2)o, (3) Hal1)a(2) - 8(1)a(2)Ja(3)} - As expected from the ST5Hy results, the undisturbed surface leads states (for the undisturbed surface) have the form H H 0.10 -124- Note that one electron in each of the four orbitals leads to one For both the Si 3H. and SigHe clusters, correlation effects 56 considerable error. The effect is more pronounced in the case in which two pairs of electrons are singlet coupled, as in "Ll (a,09) Calculational Details Only self-consistent field calculations were performed at the used. The general form of the wavefunction is that of (13) with (405)! Bung = £44 (1)60(2) + 45 (1) 64 (2) Ia 1)8(2) - 8(1)a(2)] For S17 3H. and ST He the geometry used in all the calculations was the unreconstructed tetrahedral one. Two types of calculations were “f (paust Landun) e/61 “UWNLOD SLUQ UL SLOqUAS |Yyz JO UOLJYeUR[dxe Ue “OL 1x32 22S, (2) ang *pasn sem Aujzawoah paxepeuun Leupayeuzey ay. ¢searquey ul aue saibuaua Ly $30. ~126- performed on both clusters. In the first case self-consistent GVB level wavefunctions were utilized. The second kind was a CI wavefunction in which a full CI was performed over the dangling bond space of the high spin state of each cluster CPP (oy5)05] for SiH. and £7 (0405) energy low spin state (2£1(0405)03] for Si3H, and "Eo 405) (o454)] aie The general form of the wavefunctions for the states of the SiH, cluster at the GVB level is eo 5 Ol u1k Purl > (13) where Pulk is the wavefunction for the Si-H and Si-Si bond pairs; ure is the wavefunction for the dangling bond electrons with the form shown below 413; Oo )og]: Binge = 4 (1)b0(2)64(3) aT )a(2)a(3) Zr] . | £3(0 405) (ogo g) Te Og uee = 4118p (2)43(3) 0404) 01) a(2)o(3) 0 4) 3EM (oq) 3(o965)]2 np = 06, (1)84(2)~ ¢4(1)9, (2) ILal1)8(2) - 8(1)a(2)] 9,(3)o,(4)a(3)o(4) “EM (oyog) (ogog)T Supe = £4, (1)9,(2)+ 6, (1), (2) La 1)8(2) - B(1 )a(2) 16, (3), (4)a(3)a(4) (6405) oyeg)]: yg = [04 (1)¢p(2) + o5(1)0, (2) Lal 1)8(2)-8(1)a(2)] x £45(3)q(4) + 64(3)04(4) Ia 3)8(4) - 8(3)0(4)] TEM (oy0g) ogog)Te Bye [4 (1064 (2) + 4 (1044 (2) La )8(2)-8(1)a(2)] x [5(3)03(4) + 65(3)65(4) La 3)8(4) - 8(3)a(4)] The basis set used was the double zeta (DZ) set of Table XI of Chapter For the *L! (a9 )o3] we have also performed SOGVB calculations?” consistent solution of the orbitals. The wavefunction has the form (13) with Seupg 7 (16 (2)63(3) feqLel1)(2) - 8(1)a(2)Ja(3) + + co[2a(1)a(2)8(3) - [a(1)8(2) + 8(1)o(2) Ja(3) Tf D. Review of Experimental Data Several LEED structures have been observed, in general it is agreed that this surface also suffers reconstruction. The different 52a,98 LEED structures include 2x1, 5x1, 7(or 9)x1, 4x5 and 5x2 depend- 52a ing on the annealing process. Jona could not find a recipe for the ~128- preparation of the above structures in a reproducible way. The only mental uncertainty, no models for the reconstruction have been put forth in the literature. 8. 10. -129- See, for example, S, G. Davison and J. D. Levine, Solid State (a) W. J. Hunt, P. J. Hay and W. A. Goddard III, J. Chem. Phys. B. J. Moss and W. A. Goddard III, J. Chem, Phys. 63, 3523 (1975). D. R. Boyd, J. Chem. Phys. 23, 922 (1955). jun) . Haneman in Surface Physics of Phosphors and Semiconductors, C. G. Scott and C. E. Reed, eds. (Academic Press, New York, 1975), (a) J. B. Pendry, Low Energy Electron Diffraction (Academic Press, New York, 1974); (b) C. B. Duke, Adv. Chem. Phys. 27, 1 (1974); Physics of Materials, J. M. Blakeley, ed. (Academic Press, New York, 1975), Vol. I, p. 1. A. Redondo, W. A. Goddard III, T. C. McGill and G. T. Surratt, Solid State Comm., in press. G. T. Surratt, Ph.D. Thesis, California Institute of Technology, (a) D. E. Eastman and W. D. Grobman, Phys. Rev. Lett. 28, 1378 (1972); (b) L. F. Wagner and W. E. Spicer, Phys. Rev. Lett. 28, 1381 (1972); (c) F. G. Allen and W. Gobeli, Phys. Rev. 127, 150 1. 12. 14. 15. 16. -130- L. Pauling, The Nature of the Chemical Bond (Cornell Univ. Press, Ithaca, New York, 1960), p. 239. H. D. Shih, F. Jona, D. W. Jepsen and P. Marcus, to be published. (a) To find h we evaluated > for the dangling bond or- amics of Continuous Media (Addison-Wesley, Reading, Mass., 1960), p. 52. The results for the present case are derived in Appendix 8B; This point is explained in detail in W. A. Goddard III, Lecture In the expression for Ey» h is the one-electron part of the Hamil- In the expression for Eye K,.. is the exchange energy integral, ar ] ys Using the DZ basis for the unrelaxed geometry, this value is (a) F. Meyer and M. J. Sparnaay, in Surface Physics of Phosphors and Semiconductors, C. G. Scott and C. E. Reed, eds. (Academic Press, New York, 1975), p. 321; (b) J. A. Strozier, Jr., D. W. Jepsen and F. Jona, in Surface Physics of Materials, J. M. Blakeley, 18. 19, 21. 22. 25. 27. -131- J. Appl. Phys., Suppl. 2, Pt. 2, 371 (1974) and references cited therein. 30. 31. 32. 34. 36. 37. 38. -132- (a) D. Kaplan, D. Lepine, Y. Petroff and P. Thirry, Phys. Rev. D. Haneman and D. L. Heron in The Structure and Chemistry of Solid Surfaces, G. Somorjai, ed. (Wiley, New York, 1969), p. 24. J. J. Lander and J. Morrison, J. Appl. Phys.34, 1403, 3517 (1963). I, P. Batra and S. Ciraci, Phys. Rev. Lett. 34, 1337 (1975). (a) K. H. Johnson, J. Chem. Phys. 45, 3085 (1966); (b) J. C. (d) J. C. Slater, Adv. Quantum Chem. 6, 1 (1972). (a) K. C. Pandey and J. C. Phillips, Phys. Rev. Lett. 32, 1433 (a) W. A. Harrison, Surf, Sci. 55, 1 (1976); (b) S. Ciraci, I. P. (a) R. 0. Jones, J. Phys. C 5, 1615 (1972); (b) J. A. Appelbaum in Surface Physics of Phosphors and Semiconductors, C. G. Scott and and C. E. Reed, eds. (Academic Press, New York, 1975) p. 95 and references cited therein. 39. 40. 4l. 45, ~133- (a) J. A. Appelbaum and D. R. Hamann, Phys. Rev. B 6, 2166 (a) M. Schluter, J. R. Chelikowsky, S. G. Loufe and M. L. Cohen, 4200 (1975). E. P. Wigner, Phys. Rev. 46, 1002 (1934). J. A. Appelbaum and D. R. Hamann, Phys. Rev. B 8, 1777 (1973). M. von Laue, Phys. Rev. 37, 53 (1931). This is well known for molecular systems and is well documented gas which is assumed to move in the potential created by a uni- J. D. Levine, Surf. Sci. 34, 90 (1973). 46. 47. 48, 49, 50. 52. 53. 54, 55, 57, -134- Here o and w represent orbitals that are symmetric and anti- G. Herzberg, Electronic Spectra and Electronic Structure of Polyatomic Molecules (Van Nostrand-Reinhold, New York, 1966), Table 62, p. 584, (a) I. Dubois, G. Herzberg and R. D. Verma, J. Chem. Phys. 47, (c) H. Burger and R. Eujen, Topics in Current Chem. (Silicon Hybridizations are obtained from Mulliken population analysis [R. S. Mulliken, J. Chem. Phys. 23, 1833, 1841 (1955)]. We did R. E. Schlier and H. E. Farnsworth, J. Chem. Phys. 30, 917 (1959). (a) F. dona, IBM J. Res. Dev. 9, 375 (1965); (b) B. Goldstein, (a) J. J. Lander and J. Morrison, J. Appl. Phys. 34. 1403, 3517 T. Sakurai and H. D. Hagstrum, Phys. Rev. B 14, 1593 (1976). See parts I and II of this chapter for a more complete discussion F. Bobrowicz, Ph.D. Thesis, California Institute of Technology, -135- 59, T. Sakurai and H. D. Hagstrum, J. Vac. Sci. Technol. 13, 807 -136- Chapter 3 A. Introduction An important aspect of real cleaved semiconductor surfaces is As described in Chapter 2, the cleavage face of silicon crystals directions are divalent. Both sets of directions have been -137- distinguished experimental ly !4 by X-ray diffraction. Henzter!4 has In other words, experimentally only the steps with divalent Si atoms The calculations to be described in the present chapter were trivalent silicon atom steps (those toward the [112] directions), by -138- Fig. 1 Divalent Step Geometry. Shaded atoms are those represented in the Si3H, cluster. Fig. 2 Additional Configuration Possibly Present at Divalent Steps the S3H, Cluster. ~140- Fig. 3 Divalent Step Configuration Similar to that in Si (100) Cluster. -14]- recalling the results of our calculations that model the (110) sur- tion as atoms on (110) surfaces.) B. Divalent Steps on Si (111) Surfaces: Basic Step Configuration Summary All the configurations for the step with divalent edge atoms In the (111) divalent step (geometry sketched in Fig. 1) each divalent atom is bonded to a trivalent surface atom leading to a structure of the form -142- A. DIVALENT SURFACE Si TG ora) ae: 1.64 eV de 2 *on) 0.34 eV B. DIVALENT STEP Si AT (Ill) SURFACE Q Nera 2[hem (a “Pom (5)| 0.38 eV (2G) 2[lo*)(H)] [.23eV_ 4, 0.0 eV Fig. 4 Schematic of the Electronic States at Divalent Steps on Si -143- The electronic states expected for this complex are outlined in Fig. 3 (on) (5): Starting with the 3 (on) state of the divalent Si, expected to be lower in energy. Von) (3): There is one additional (o7)(c) state, correspond- denoted 20 on) (5). M625): The 142) ground state of the divalent Si is coupled spin (S = 1/2). This state is denoted 202) (5)]. The results of self consistent field and configuration interac- spin couplings, the quartet is the lowest with a vertical excitation -144- A. H3Si-Si-SiHy Schematic B. 7 orbital of step Si ' “ an ‘ a\/ 2 | © bulk Si SiGC*) © Mae C. & orbital of step Si D. & orbital of surface Si “7.5 Fig. 5 Nodal surfaces are indicated by long dashes, solid lines indi- amplitude values. Contours are drawn every 0.05 atomic unit. "yuLod "qurod awes ay} ze anlea (|) GAD 942 UeYI “dUBlLYy A® 7G°O SL ,8Z.60L 72 3907S SLY} wos AHYoUD 4H MUL g Sjulod paze[no,eo ayz 02 714 saut|ds a1qno e Aq pauteiqg, 9(1)8A9 BSLMABYJO SsSatUN Saaiquey UL sue Salbuaua [Ly “SUOLZeLNOLeQ PLal4y-jUsazsLsuo7 “sdaqg quaLeAid Jo Lapow 4arsnig “yis-1s-1S%H ay, 07 BuLgeLey seLzequend ‘I aiqel *squLod paqe[nojed ay} 02 414 sauLtds DLqno e wou; paurezqg, =i 92Z°L vEG"L O6ELL vOZGE"EL- G6LG6"EL- LLOG6"EL- PvOLvE"EL- L80v6"EL- —[() (*9) ], ~147- energy of 0.23 eV and an adiabatic excitation energy of 0.07 eV. The The optimum Si-Si-Si angles for the “[3(on)(@)}- and *[3(on)(3)] The electronic structure found here for the divalent step sug- chemical species. In the of state the o pair can act as a donor (for 0.2 eV of each other, these step sites should be considerably more reactive than the sites of the perfect (111) surface. As would be ~148- expected for such a reactive site, few molecules have been detected singlet (07) ground state? (with an optimum angle of 92°), Correlation effects As in the 3 (on) state of SiH, (Chapter 2) correlation effects tion, 23 (om) (pe AL pL 0y(1)4(2) al 82) - 8(1)01) 196, (3)0(3)} (GVB) wavefunction, where ooulk represents the wavefunction of the Si-Si and Si-H bond Electron correlation effects are also crucial for the tr (o2)(5)] state, as they were in the of state of the divalent Si. From Table I (footnote b) we see that the HF wavefunction, ~149- Teads to an energy which is 0.54 eV higher than that of the corres- ponding GVB wavefunction, x Co(1)8(2)-8(7 a1) J_(3)a(3)] (4) The HF and GVB wavefunctions for the fre on) (S)] state are identical, @.g.; "Boom (pe = “Com gyp = OG pl Og (16g (2) ~ 65,(1)8, (2) IEa( 1 )8(2)#8(1 a2) J4(3)a(3) } (5) Therefore, had we performed our calculations using HF wavefunctions Comparing Tables I and II we see that for the GVB CI calcula- 9 _ -150- at the GVR level only one of the important spin configurations for this state is included, namely that of eq. (2), e.g., X, = [o(1)8(2) - B(1)a(2)Jo(3) (6) The other important spin coupling is Xy = 2a(1)a(2)8(3) - Lo{1)8(2)+8(1)o(2)]Ja(3) . (7) Because no spin optimization is included at the GVB level, only the the SCF and CI calculations are shown in Figs. 6 and 7, respectively. Calculational details As we have seen in Chapter 2, to obtain a consistent descrip- For the GVB CI calculations we used the valence spaces of both orbitals (not involving the correlated GV8 pair) were not allowed to ENERGY( HARTREE > Fig. . 7} 00 -—— -151- -0.SY -- — 2.722 SiH -0.95 — -0.98 — -}.02 — KS SCF -).04 — -- 0.0¢V 95.0 10/9 tos 9 nog tis a 120.0 6 Potential Surfaces for the SCF Calculations on the Si3H, ENERGYC HARTREE > “0. = pr — 2.72 —0.02eV Fig. 7 Potential Surfaces for the GVB-CI Calculation on the Si.H Cluster 35 -153- have zero occupation number. This leads to wavefunctions having tuals added). C. Divalent Steps on Si (111) Surfaces: Additional Atomic Arrangements _ General description A particularly interesting divalent state for a step configura- This has been previously described in Chapter 2 using an H,Si-Si-SiH complex. For the present case the appropriate complex to use (see Fig. 2) results in a divalent Si atom at the center, bonded to two The central silicon is again a divalent silicon atom with two dangling bonds, Ty and Op» on the adjacent silicons. The low lying states of the system are dominated by the (0) and (om) confiqurations of the central Si (these are outlined in Fig. 8): -154- STEP KINK H,Si-Si-SiH, *TAOVGON gaa eV 5Penyan)) 0.72 eV 3 (on) oe GI 0.36 V % o 6m NS 4 = 3 ie ToGo) o.0 V Fig. 8 Schematic of the Electronic Structure for the SizH, Cluster. ~155- 3 . . . Loos 37. “for)(o,)(o,): Starting out with the (or) state of the leads to an overall triplet (S = 1), denoted SP (on) oo). When we couple the Cy and om orbitals into a triplet, an overall quintet (S = 2) state results, which we denote 563 (on)? (o o)]. The quintet will be lower in energy if the overlap between the Tp and om orbitals is small. "(on) (a, )(0,,)! We now start with the Vion) state of the divalent Si. Two states are possible: if we couple ony and Cy. into "ony! (o,0,)1. For the triplet coupling of o, and g), an overall] triplet (S = 1) is obtained, $E1 (on) %(0,0,)1.- The lowest of the two bitals, Ty and On. 1 2) o)(o,)(o,): Starting with the V6?) ground state of the divalent silicon we can obtain an overall singlet (S = 0),. Mh o® ono). and an overall triplet (S = 1), 301 (6? )3(0,0,)]. One can compare these states with those with (om) configurations and estimate (before any calculations are done) some of the relative oy)! should be lower in energy than the 33(on) (ao) or the "on oo). Similarly, one expects ~156- 3£(07)?(c,0,)] to be lower in energy than 573/ *El(on)3(0,0,)1. on)*(o,0,)] or 1 3 — . oo, . divalent Si, whereas the other dangling bond (0, or o couples into Q) be a combination of this and *! (00, 3(n0,) 1). Results shown in Tables III and IV. The o, m7, oc, and o,, orbitals of the COCKS state are shown in Fig. 9. Figures 10 and 11 show respectively. Note that for 913 (on) 90,,) J instead of o, and Oo, we show the corresponding symmetry orbitals oq and Oy where ay + Oo wv oO, - The ground state of the system is "EN(0?)(oy0,)], with an move together. This can be represented by the following diagram anea (L)GA9 Bujypusdsaisod ayy ueyy waybpy Ad ESO St 82.601 38 9327S SpyQ wo} AbJaua 4H ayi, *anjea (2) GA9 Bulpucdsauuod sy ueyy wayhpY Ae CO'Z SE .BZ.601 72 9323S S}4y} AOs ABJaUa JH lg *sgupod paze[Nd(e> ayy 07 343 Saup_ds 3pqnd e uio4ay paupesqo, (t)@A9 *pajezs aspmiayz0 Ssa_uN saaiquey UL aue Sapbuaua [LY *SUD}ZR NILE} PL Bpd-Fuazs}suaj-jjas *sdazs yualeard JO Lapow 4azsn19 CH bS-1S-4SH OY} 0} UOPJELAY SOpIPWUENY LTE Lge, *KYALOMISMIY BuOW ase Sanirs 49S 841 324} padunoucsd os sp Spy [(*9%o) (40) J, yy JO aSeD ayy UY ‘ALJDI4AOD BF2IS SYR Bqi4dsap 03 AuRSsadaU Sleyiquo [eNqats “4 "34s SBUL[AS Dyqnd e wos, pauter49, - - - LOBSLE*EL- OZEBLETEL~ 2OLOZEEL- OLLece El - - - - - el'l ZOTLLL P2LOLE*EL~ OZ669E"EL- EBSOLEEL- 2HLBSEEL- OLE9SEEL- - - - - 92°0 2E°GUL BBEPSE“CL- LEGLZEEL~ LBOZVC“EL- LeBEREEL~ SBLEBE*EL- - - - - 0°O 66°US ZEVZLPEL~ BGVLZEEL- LPLIVEEL- GeesBE*EL- G2ZzL6E'El* eBSOOP'EL~ ISPCOR'EL- G2zib-Ei- 600607 El~- (ote) (40) | ((7e2) (Fe2), 3, (70%) (293), (A®)Ab4BUZ (6) eAbiauz SLL 1820601 eS0L 056 G8 ol 099 GS *pazeys aspMiayzO SSajuN Saaujuey UE aie sapb4soua Liy *SU0;32{N9 1,2) 739 GAQ ‘Sais YuaLeA|G JO Lapoy 4aqzsni9 Suis ES-bSH ay 02 Bupgeisy sayayquend ‘Al aiqes -159- A, B. Fig. 9 Orbitals for the 5P(on)! (oo) State of SigHg. (a) o, orbital; (b) o, orbital; (c) o orbital (d)norbital. ~160- . . D Fig. 10 Orbitals for the lon) 3(o,0,)] State of SinH,. 4 (a) 0, Orbital ; ~161- £.0 {i oo} a“ ” ] 2,1 -162- Si ft" Note also that the of electrons get out of the way of the Si-Si excited state is the on)! (0,0,.)] at 0.76 eV. The Pon) 3(0,0 and 3010?) 3(0,0,)] are quite close in energy at 1.13 and 1.14 eV. ~163~- Potential curves are plotted in Figs. 12 (SCF results) and 13 (GVB CI results). Correlation effects As in the systems previously discussed, correlation effects Hartree-Fock and GVYB wavefunctions are identical, e.g., *E3(on)$(ogo pe = PL°(on)*(045,,) Igys = OM l4o(M9q(2) - HAI C2IIEF, (34g (8) = dg (314, (41 r Yr g x a(1)oa(2)a(3)a(4) } (8) 1- 2 . - os . of electrons. As shown in Table III (footnote b) the HF wavefunction 111662) ogo p) ae = OLS y rel 4(1)6, (2) IEa(1 )8(2) - 8(1)a(2) x fo, (3)6, (4) Jfo(1)8(2) - BC )af2)]} (9) (where 5 an d, + o, is an appropriate symmetric orbital), leads to tion, TH)" (o 90.) Joyg = Attar ota Goa rae(2) * Sg -anl ey 9(2)] x [a(1)8(2) ~8( )al2) Ihe, (3)9, (4) +9) (3), (4) Iba(3)6(4)-2(3)0(4) J, (10) ~164- -0.2— : . Z.1ZeN cluster. Potential surfaces for the SCi calculations on 43 on ~165- 3r 72,3 *S — ~~. -0.38 — \ -6.0— ENERGY( HARTREE 9 ~0.42 — | - 4 I | I 7 I " ANGLEC CEGREES > Fy ar » a a _ . oat + As -166- Similarly, for the 301(0? )F(o,0,.)] state the HF wavefunction 301(04)3 (oo )Iye =OeL% yp 60, (136, (2) Ho 1)8 (2) ~ 8(1)0(2)] « Lay (3), (4) =a (3)¢, (4) IEa(3)804) + 8(3)a(4)I} 5 (11) leads to an energy 0.53 eV higher than that of the GVB wavefunction *1(07)7(o,9,.) Joys = BP lS otan yaya (2) * Sy eam G4 yq(2) I x [a(1)8(2) -8(1)a(2) IE, (34 (4) = 4, (3)¢,, (4)3 x [o(3)8(4) + 8(3)0(4)]} . (12) For the 30100, )%(70)] state appropriate correlation is intro- form 3,1 3 r u 2 r spin + Cy[3a(1)a(2)o(3)8(4) - 6(1)a(2)a(3)a(4) - a(1)8(2)a(3)a(4) - a(1)a(2)6(3)a(4) ] -167- From these results we see that, had we used the standard HF wavefunction, we would have obtained the wrong ground state again. Corrections due to changes in the hybridization of the trivalent If the system we are treating by means of the Si 3H, comp] ex The potential curves obtained in this way are shown in Fig. 14. Table V summarizes the results when this correction is included. total optimum energy of the ground state is raised so all adiabatic excitation energies are decreased by approximately the same amount.) Calculational details For the central Si atom, as discussed in Chapter 2 and in (p. 31). Here, as before, the reason is that the of and cm states ENERGY( HARTREE > Fig. 1l. -0.32— 2. — -0.26 — ~0.40 — 0.2 — it aoa N Sigh, CORRECTED I A 85.0 2.726V 0.0¢cY I | | ANGLEC DEGREES > Potential surfaces for tne 3VB-Ci calculations on the 6131), -169- Table V. Summary of Results Relating to the Si,H, Cluster Model 3.4 'E(o*) (oo )] 0.0° 89.01 $P(on)"(o,0,)1] 0.36 101.96 aSae Chapter 2 for the SizHe calculation. -170- are not correctly described unless d-functions are present at the For the geometry optimizations all silicon atoms were al- In the GVB CI calculations we used the valence spaces of the leading to wavefunctions consisting of 32 to 278 determinants. D. Trivalent Steps on (111) Si Surfaces Although the trivalent edge atoms in steps on Si (111) sur- faces are not observed experimental ly !@ , they should exist in at By looking at a model of trivalent steps! one quickly re- alizes that the geometry for such steps is the same as that of the -171- unreconstructed (110) surface. Each step atom has then one dangling bond and each surface atom in the upper terrace also has one dangling conclusions about the trivalent steps. We first note that, at the etry. E. Review of Experimental Evidence As mentioned in the introduction to this chapter, some experi- experimentally investigated. Henzler!48 has studied the prevalence al.7° have provided evidence that step sites on Si lead to an increase of 0, sticking probabilities by ~ 10°53 this is consistent with the -172- electronic structure at the divalent states we have found. For Somorjai et al. 1¢2 10, For Si steps on (111) surfaces, it has been already mentioned The work of Rowe et al.2> has indicated that surface states charac- . Electron Diffraction (LEED) to establish the relationship between the spectra and the surface structure as the step density was in- found by Henzler~ in germanium. Other theoretical methods |! have included a semi-infinite or a a priori chosen geometrical configuration. These methods use -173- wavefunctions in which the orbitals are doubly-occupied. As shown previously, this leads to the wrong ground state for the localized states require a proper account of correlation in order to be cor- rectly described. ~174- REFERENCES FOR CHAPTER 3 (a) M. Henzler, Surf. Sci. 36, 109 (1973); (b) H. P. Bonzel, in Surface Physics of Materials, J. M. Blakely, ed. (Academic Press, New York, 1975), Vol. II, p. 279; (c) G. A. Somorjai, R. W. Joyner and B. Lang, Proc. Roy. Soc. (London) A331, 335 Hybridizations are obtained from Mulliken population analysis [R. S. Mulliken, J. Chem. Phys. 23, 1833, 1841 (1955)]. We did (a) I. Dubois, G. Herzberg and R. D. Verma, J. Chem. Phys. 47, 4262 (1967); (b) B. Wirsam, Chem. Phys. Lett. 14, 214 (1972); (c) H. Burger and R. Eujen, Topics in Current Chen. 50 (Silicon (a) W. A. Goddard III, T. H. Dunning, Jr., W. J. Hunt and P. J. S. Huzinaga, J. Chem. Phys. 42, 1293 (1965). F. Bobrowicz, Ph.D. Thesis, California Institute of Technology, See, for example, Fig. 6 of Ref. la. (a) M. Henzler, Surf. Sci. 39, 159 (1970); (b) ibid, 22, 12 (1970). (1975). ~175- 10. (a) K. Baron, D. W. Blakely, G. A. Somorjai, Surf. Sci. 41, 45 11. (a) V. T. Rajan and L. M. Falicov, J. Phys. C: Solid State Phys. -176- -177- Chapter 4 A. Introduction In this chapter we will describe the results of calculations in Experimentally, when a clean Si (111) surface is exposed to radical model). As in previous chapters we have used finite clusters in which from crystalline silicon®, i.e., R = 2.358. Si-Si amount of controversy existing in the literature’ as to the actual ~178- manner in which oxygen chemisorbs on silicon. Two basic mechanisms We start our discussion by considering the chemisorption of atomic oxygen on the surface. In Section C we consider the peroxy radical model.” B. Atomic Oxygen Chemisorption General Description We will start by considering the chemisorption of single oxygen The ground state of the oxygen atom 3p, has the configuration 6,7 cules it is found that the Is and 2s electrons do not participate actively in bonding. Thus the active part of the 0 atom can be des- cribed schematically as shown in Fig. Ja. We have four electrons to Fig. -179- (a) ; [Edie 1 Schematic diagrams for (a) oxygen atom (?p); (b) ground state -180- be distributed between the three 2p orbitals, leading to one doubly Now consider bonding this 0 atom to a dangling bond orbital of oxygen, a configuration we will denote as (py)! (p,)° (1a) also has the same energy so that the ground state of the chemisorbed citations will involve . (2) Considering now the positive ion states associated with the 0, direct jionizations from the ground state lead to configurations of the form? Px74'a7 + (5i0)°(Op)' (op, (3) -181- way? 2 (810)! (op,)" (Op, )° where in each case the symmetry designation is in terms of the Si0 For our calculations we have used two clusters, one having only Results The results of Hartree-Fock (HF) calculations on OSiH, are shown in Table I. The HF wavefunction [eq. (6) below] was utilized In Table II we show the excitation energies for the GVB-CI cal- the two components of the “Il state, (1), are separated by 0.1 to 0.2 eV. This result obtains because we solved for one component, (la), sélf- consistently and then carried CI calculations based on these orbitals. -182- Table I. Hartree-Fock Energies for the Ground State of the O-SiH3 Cluster.* All energies are in hartrees. Si-O Distance Energy “The geometry is that of silane for the SiH, part. The DZd basis set of Table XI, Chapter 1, was used for Si. “Experimental values of Si0 bond lengths are: Roig = 1.633K from -183- "YEO" L SBM BOURISIP O- LS SUL ‘pako| dua J) JO LAA@L 3y2 JO UOLYdLuoSap e “UWINLOD SLUZ UL OF PatUaJou SLEPLqUO dYyz JO OLyewsyoS e uO} | “HL4 aas "90UIMeY /GIQ6°P6- SEM payzeiNoleo ABuaua LejoL ZL *8u, "9a4JUPY E/BEEOB- SEM pazeLNd,ed ABuaua |eqoL, AOL 4X9} 84} 89S, L6'L 6L°2 By°Z 622 gatgnop—ssaZ dzo + ols -184- This leads to a slight bias, 0.1 to 0.2 eV, in favor of (la) with Comparing the Sid + O2p,, excited state to the experimental |! O2p, > 02p, transition of the free oxygen atom (see also Fig. la). This transi- surfaces, '4 but the interpretation of these spectra is made very diffi- In Table III we compare the (Koopmans' theorem) ionization poten- potentials (from Koopmans' theorem) are not very sensitive to the size -185- Table III] Comparison of Ionization Potentials for the O0-SiH, “Using a GYB(1) wavefunction of the form (2). The Si-0 bond length is 1.63h. Orbital energy of the first natural orbital of the GVB pair. This energy does not correspond to the Koopmans' Theorem energy, but should be close to it. ~186- In Table IV we show the ionization potentials obtained fron The comparison with the experimental values for a free oxygen 1] atom follows the same lines as that made for the neutral. As in the atom, it is easiest to ionize out of the doubly occupied (02p,) orbi- two states have been referred to 3 1 zy and 'a7: (si0)2(op_)'(op_)! axis, we obtain two states again, 3 ] nm and |n (810) (02p,,)'(02p,.)° Ionizations out of the singly-occupied 02, orbital leave all orbitals doubly occupied. Two possibilities exist, 1,4 1 AY and 's* (si0)“(o2p,)° ~187- Table IV GVB-CI Calculations for the Ion States of O-SiH3 and 0-Si(SiH,) Clusters.* All energies are in eV. State? Ionization Potential O2p, vacuum triplet 12.27 11.91 13.62 Sid vacuum Singlet 16.30 15.88 18.64 {see the text for a description of the level of CI employed. The Si-0 Dsee Fig. 1 for a schematic of the orbitals referred to in this column “Referred to the corresponding energy of the neutral ground state. qi oni zations involving simultaneous excitation of an electron are omitted as are ionizations involving Si-Si and Si-H bonds. Polariza- CRef. 11. -188- of which Table IV shows only the lowest one. These ionization potentials range between 11.9 and 16.3 eV. In Two types of errors are present in these calculations: (i) a Calculational Details For the OSiH3 system we have carried out calculations to optim- The ground state was solved self-consistently, using the DZd basis of ~189- Table XI, Chapter 1 (p. 31) for the Si atom and the double zeta (4s2p) contraction of Dunning! for the 0 atom. A Hartree-Fock wavefunction of the form Mur {putk®siol)sio(@) SB(Z) 0925 (3) 9p (4) oO « 0 3)B(4) $995 (5)a(5)} ( was used. Here O ulk represents the wavefunction of the SiH bond Once the optimum Si-O distance was determined we performed GVB Si atoms, and Dunning's DZ basis for oxygen. !2 The GVB(2) wavefunc- tion has the form x Moap (9) tozp) * Fox, S)Poap ) He)8a) - B(3)a(4) Hoa, (S)a(5)} (7) For the CI calculations we used the space spanned by the orbitals (denoted 02p,)- For the neutral system the configurations were generated -190- by doing all single excitations from the basic configurations shown figuration. Thus, if we write the CI wavefunction as (o; is called a configuration, cs is the CI coefficient), then for configuration 1 of Table V, ©. has the form = *ourePozs(%o25'2)#si0*si0l4 tozp (5) 029, (#029, (7 (9) For the ionic system, all single excitations were performed represented by Souik were kept doubly occupied. C. Molecular Oxygen Chemisorption General Description tronic structure, new features appear in the Spectrum. Also because ~191- Table -V Basic Configurations for CI on the Neutral Systems of OSiH, and 0Si (SiH) All SiH and SiS? bond orbitals 3 3° Configuration Orbital® ] 2 2 0 2 0 ] 0 11 ] 0 2 2 0 2 0 “Number shown represents the occupation of the orbital. bThis is a virtual orbital constructed by the orthogonalization to all other orbitals of (2). ~192- Table VI Basic Configurations for CI on the Ion Systems of OSIH, and OSi (SiH3) 4. All SiH and SiSi bond orbitals were kept doubly occupied. Configuration Orbital? @Number shown represents the occupation of the orbital. DThis is a virtual orbital constructed by orthogonalization to all other orbitals of (2). -193- of the added degrees of freedom, more than one vibrational frequency First we consider the electronic structure of an isolated molecule leads to Fig. 3b as the ground state?’, Here two singly- occupied 02p orbitals are paired up into a singlet to form the 0-0 pairing these orbitals®?/, Fig. 2c, leads to excited singlet states (13 and the other component of | q)* Soiving self-consistently for doubly-occupied pi orbitats |? can delocalize onto the other center for Consider now bonding the 0, to a dangling bond orbital of the However, this orbital is perpendicular to the Si00 plane and hence parallel to the surface so that it cannot (for the perfect surface) ~194~- eurf, oy ue Fig. 2 Orbital diagrams for (a) oxygen atom (3), {b) ground state 0,5 ~195- bond to another surface atom to form a bridged structure. That is, the chemisorbed 05 is a peroxy radical. On the other hand, the excited state of 055 Fig. 2c, leads to Fig. 2e upon bonding to the surface. In this case it is possible for the unpaired orbital of the silicon to form a bridged 0, bond. The vertical excitation energy from the ground (Fig. 2d) to the ex- cited (Fig. 2e) states of the bound 05 should be similar to that The basic ionizations in this system occur out of the Py or- singly-occupied orbital; the Py doubly-occupied orbital the Si0; and 00 bonds. Results To test the ideas discussed above and to allow detailed com- ized the 0-0 bond length and the $i0-0 angle, leading to Ry »= 1.366R and an Si-0-0 angle of 125.9°. These compare with 0-0 bond lengths 10 ] of 1.214 for the oxygen molecule!?, 1.28A in ozone 03), 1.23 and cal !9, to 110° to 120° for C00. compounds”, to 135°-137° for Fed complexes. We thus see that the geometry we have obtained for this peroxy radical falls within the values observed in other systems in ~196- which the oxygen molecule has the same type of bonding with other In Fig. 3 we show the orbitals for the ground state of 05Si13He5 (and found for other oxygen systems°?’), the 02s orbitals are only Slightly changed (Fig. 3c). The other orbitals are much as pictured Table IX summarizes the results for the peroxy radical. The 2 r ‘Apoarpoadsal ‘spuog OO pur Q's au} ysnosy} durssed pur ouryd QOIS au} 0} JepNatpuadied soured ur syojd aysoduioo yora oe (3) puke (J) ‘aurjd OOIS oy) 00 ols V6 00 YE O!s os- Z 2 QNOd O 0 (9) GNOS O !S (9) ~198- ue uol 2 Bly aag *Auqeawwss ‘un Lod Stud UL pasn stoquAs jo uoLqeueydxa ui 40 ,Y aAey Sazeqys dy2 OS pasn SL uazsn{d ayy $0 Arjouwss 55 [edo SUL, pl29y'vSl- ss: OSBL ‘HSL - cObLybSL- LOE PSi- UNNOBA + 0-0 Vp sabusue LLY »“4eIsn{g HEs-°9 yz soy uoLzezpwLidg Arqawoag au} 0f SALQeLay seL649uz 19-aA9 TIA PLqeL -|99- Table VIII Optimum Geometries for the GVB-CI Calculations on the all angles are in degrees State Roo 85409 Excitations 2A" ground state 1.366 125.88 “A Si0 > py 1.459 124.78 3 py > vacuum 1.303 135.22 The local C, symmetry of the cluster is used so the states have A' or A" symmetry. See Fig. 2 for an explanation of symbols used -200- Table IX. Excitation Energies and Ionization Potentials for the . Experi- Corresponding 17.42 17.54, 18.02, 18.06 *For the states of the neutral system, the two entries correspond to surface orbital with the singly-occupied orbital of the 0 5° For the infinite solid, one expects a narrow band of states corresponding to each of these pairs of states. For the ion states there are gener- ally two doublet states and one quartet for each configuration. Here SURF indicates the surface dangling bond orbital. “Ref. 19¢ “Ref. 13 Ref. 15, assignments are tentative. -201- GVB-CI Calculations of the 05-Si 3H. Cluster.* All energies State Frergy Spin Excitations b . -202- Table X - Continued State Energy Spin Si0 > vacuum 15.001 doublet 15.280 doublet 15.874 doublet S10 > vacuum + py > Py 15.975 quartet 17.543 doublet 0-0 + vacuum 18.021 quartet 0-0+vacuum+ p. > p 18.707 doublet At the optimum geometry for the 0,-Si-H, cluster. The orbital space used was obtained from a GVB(3) calculation for the 2q! ground state. Silicons, and the double zeta contraction of Dunning (Ref. 12) for Total energy calculated was -164.25595 hartree. ~203- absorption. This transition has the same origin as the *x + om absorption band of oxygen !3 at 8.6 eV and should result in disso- the singlet state. tion potentials in very good agreement with the experimental values 15 of Ibach and Rowe. The calculated excitation energies and ioniza- tion potentials are close to what would be predicted using our model and the known values for 0, or HO). The main changes occur in (i) the singly-occupied oxygen orbital (p, ) whose ionization poten- and (ii) the doubly-occupied orbital (Py) which is a bonding orbital Replacing the Si3H¢ unit with an SiH, unit leads to small 2] dix B. For the present system we estimate” them to be of the order of 0.2-0.4 eV. Combining excitation energies from calculations with experi- 22 mental thermochemical data™” for silicon compounds with bonding PHOTOEMISSION INTENSITY Lt ep] -204- Si(it1) +O. x2 CLEAN @=0.26 ®™=0.52 | i a I I -20 «+5 = -10 “5 0 “Tan Vane - a 7 . — boon im - . . — nm Lt peroxy racical model and the experimental spectrun (after tef. -205- configurations similar to those found in oxygen chemisorption, our con surface is at -6.56 eV. -206- Table XI Total Energies Relative to the Energetics Involved on the Peroxy Radical Model.? 0-SiH, -80.31098 ~80.33873 0 -74.79884 - 0, -149.60125 -149.68480 *Values obtained from SCF GVB calculations. -207- Calculational Details The geometry of the clusters used in these calculations is For 0,-SiH, we optimized the 0-0 bond length and the Si-0-0 the HF method is not adequate for these studies. (For 0, the HF wavefunction only accounts’ for 0.9 eV of the 5.2 eV experimental bond energy. Including a configuration interaction over just the GVB orbitals (GVB-CI) accounts’ of 0,.) for 4.9 eV of the 5.2 eV bond energy The basis set used was the double zeta basis of Table XI, Chapter 1, for the Si part in O5-SiH3. The oxygen double zeta con- traction of Dunning! 2 was used. In O5-Si3H¢ we used the MXS2 basis -208- For the GVB-CI calculations on O5-S7 3He the space spanned by . r x Vinty All double excitations were allowed orbitals (p and Py within the 0-0 and Si-0 bonds for the configurations shown in Table ception that there was no surface dangling bond orbital. D. Chemisorption Models and Review of Experimental Data The oxidation of silicon surfaces has been studied experimen- 14,15 ,23-33 tally since the early sixties. Two basic models have been proposed; in one case the oxygen molecule is assumed to fissicn upon 14,29 chemisorption. The other model proposes that initially the the 0-0 and Si-0 Orbitals tations within Double Exci ming for Perfor 10ns gurat rT Table XII Basic Conf for 0,-Si 3H. virt vi Py Pre 0-0 0-0 02s, Py surf 02s, Neutral -209- Oo Oo Tons continued Table virt virt Pxr 0-0 0-0 02s, pyr surf 02s, Tons =210= -211- molecule chemisorbs without fissioning. °° Several possibilities Considerable controversy has existed in the literature concerning these two basic models. 4 Two vers ions of the fissioned 05 chemisorption have been pro- The nonfiss ioned 05 chemisorption models propose that the 05 Green and Maxwet 123 proposed that a peroxide bridge is formed between is not dissociatively chemisorbed but only one 0 is bonded to a surface The basic experimental features to be explained are as follows: and Rowe 9 have also observed transition energies at 3.5, 5.0, 7.2, 11 and 23 eV. These are due to electronic excitations in the oxygen 25 surface system. Ibach et al.” have observed three different vibra- tional frequencies with a perpendicular component due to the oxygen at -212- the surface. Chiaradia and Nannarone*| showed that the oxygen 26 have showed that two surface silicons. In addition, Rowe et al. The most important objection that can be raised against the 25 14 vibrational modes observed by Ibach et al. Ludeke and Koma have suggested that two vibrational peaks (at 0.125 eV and 0.090 eV) yt and Ih states of Si0. This they are similar to those of the Ibach et ale’, thus the vibrational frequencies of an excited elec- tronic state (like the \ Il of Si0) are irrelevant; second, bonding the Si0 to the other Si atoms in the surface leads to great changes leads to large changes in the vibrational frequencies; third, the a state of Si0 has the electronic configuration where oSi0 is an Si0 bonding orbital along the axis of the molecule; Si3s orbitals are singlet coupled. Hence, to excite the surface Si-0 to the Il configuration requires breaking all three Si-Si bonds. -213- E. Conclusions We have performed calculations on clusters that model the For single oxygen atoms chemisorbed to the surface we see that inequivalent types of oxygen atoms are present in the first stage of chemisorption. 10. -214- REFERENCES FOR CHAPTER 4 H. Ibach, K. Horn, R. Dorn and H. Luth, Surf. Sci. 38, 433 (1973) M. E. Straumanis and E. Z. Aka, d. Appl. Phys. 23, 330 (1952). See for example (a) F. Meyer and J. J. Vrakking, Surf. Sci. 46, W. A. Goddard III, Lecture Notes for Chem. 120, California Insti- tute of Technology, 1975 (unpublished). (a) W. A. Goddard III, T. H. Dunning, Jr., W. Jd. Hunt and P. Jd. Hay, Accts. Chem. Res. 6, 368 (1973); (b) B. J. Moss, F. J. Bobrowicz The Jahn-Teller theorem would remove this degeneracy but the strong Cy: ‘The angular momentum, 4 , about the axis of rotation deter- mines the types of symmetries that the wavefunction can take. For example (a) Interatomic Distances, L. E. Sutton, ed., The Chemical Society (London), Special Publication flo. 11 (1958); (b) Interatomic Dis- tances, Supplement, ibid, Special Publication No. 18 (1965). V1. 12. 14, 16. 17. 18. 19. 20. 2}. -215- C. E. Moore, Atomic Energy Levels, Nat. Bur. Standards (U.S.) Circular No. 467 (U.S. Government Printing Office, Washington, (a) G. Herzberg, Spectra of Diatomic Molecules (Van Nostrand, N.Y., 1950); (b) D. W. Turner, Molecular Photoelectron Spectros- copy (Wiley, N.Y., 1970). 10, 710 (1974). was taken to be the Si-Si bond length for the bulk, 2.352. o orbitals are symmetric with respect to any mirror plane that J. P. Collman, R. R. Gagne, C. A. Reed, W. T. Robinson and G. A. (a) D. H. Liskow, H. F. Schaefer and C. F. Bender, J. Am. Chem. We assumed h to be between 1.63 and 2.43A corresponding to the smallest and the largest perpendicular distances to the surface 22. 23. 27. -216- from the peroxy radical part. Taking ry as 2.354 leads to the figures quoted. D. Wagman, W. H. Evans, V. B. Parker, I. Halow, S. M. Bailey and R. H. Schumn, Selected Values of Chemical Thermodynamic Proper- Government Printing Office, Washington, D.C., 1968). M. Green and K. H. Maxwell, J. Phys. Chem. Solids 13, 145 (1960), H. Ibach, K. Horn, R. Dorn and H. Liith, Surf. Sci. 38, 433 (1973). R. Dorn, H. Liith and H. Ibach, Surf. Sci. 42, 583 (1974). J. E. Rowe, H. Ibach and H. Froitzheim, Surf. Sci. 48, 44 (1975). J. Archer and G. W. Gobeli, J. Phys. Chem. Solids 26, 343 (1965). B. A. Joyce and J. H. Neave, Surf. Sci. 27, 499 (1971). (a) R. E. Kirby and D. Lichtman, Surf. Sci. 41, 447 (1974); (b) R. E. Kirby and J. W. Dieball, Surf. Sci. 41, 467 (1974). ~21|]- APPENDIX A The Generalized Valence Bond (GVB) Wavefunctions In order to clarify the discussions of wavefunctions in this Consider a two-electron system such as Ho or the He atom. The Hartree-Fock wavefunction has the form oF 12) = QALo(1)9(2)0(1)8(2)] $(1)o(2)[o(1)8(2) - a(1)8(2)] (A.1) HAF, = e¢ (A.2) where HF H =ht Ug > h is the one-electron Hamiltonian, and Jy is the Coulomb potential for an electron in orbital 9 . In the GVB wavefunction for two electrons one allows each elec- tron to have its own orbital, thus oVB(1 2) =a fo,(1)8, (2)La(1)@(2) - (1) a2) 1} (A.3) = [9,(1)9,(2) + 6 (2)6,(1)]£0(1)8(2) - B(1)a(2)] ~218- where d, and o, are generally nonorthogonal. Applying the variational principle to the GVB wavefunction leads to the equations GB, __, which must be solved for the optimum orbitals. In solving for these of (A.3) to the natural orbital (NO) form NEO (2) 4 G62] = c19,(1)8 (2) + C,05(1)45(2) » (A.5) where aa (A.6) and N, D> and De are Suitable normalization constants. Since 4 and Replacing a HF pair such as in (A.1) by a GVB pair such as in body effects involving these electrons. From the form in (A.5) one can -219- Say that in the GVB method we solve for the occupied and correlating For a molecular system the correlated orbitals generally local- in the ab initio wavefunction. for the x4 state of Sis» we might cor- Such a wavefunction would have the form * Cr 6b 1615) (Cy 7917918 * ahighi 7808-08} (A-7) [ten doubly-occupied core orbitals and four GVB correlated pairs The programs for calculating the GVB orbitals can also allow ad- ditional natural orbitals in an expansion of the form in (A.5), and occasionally there are cases where such additional correlations (CHS). Such a wavefunction for two electrons is denoted as GVB(1/4) -220- indicating 4 NO's for describing one electron pair. For a many-elec- of some doubly~-occupied orbital. ~221- In this appendix we describe in detail how the polarization The geometry is shown in Fig. B.1. en Vac lil Fig. B.1. Geometry for Dielectric Corrections The change in free energy is given by! > > oH =-7(P-8 dr (B1) é-5 , (82) Y is the unit vector in the radial direction, P is the polarization vector, given by! -] _|~ where ce is the dielectric constant of the semi-infinite continuum. We now make the change r > rh, so that from Eqs. (B2) and (B3) we obtain x fF. ys aa 1 _ 1) yen] ] where 6 is the angle between r' and hi‘. Substituting (B4) back ( dr' dddor' sin 9 : T These integrations lead to éG= - (Sh i lim [s 1og(A2+ h?)]- 4 log(rs +h?) \ > +] ro Now ~223- [24 ne A ‘ctl’ h roth ~ roth I} REFERENCES FOR APPENDIX 8B 1. L. D. Landau and E. M. Lifshitz, Electrodynamics of Continuous Media (Addison-Wesley, Reading, MA, 1960), p. 52. 2. For a derivation of a similar problem in a spherical cavity see -224- Appendix C - EFFECT OF CORRELATION, d-FUNCTIONS AND LATTICE CONSTRAINTS ON THE Si 3He CLUSTER MODEL FOR (100) Si SURFACES As reported in the text, we have studied the effect of corre- We have carried out calculations on the Si 3H, cluster for the straints of the lattice (i.e., letting the central Si atom move only in the [100] direction), the 3 (on) state is still the ground state, -225- Table C.I Summary of Results for Si 3H. Clusters Modelling the Opt imum® Optimum Relaxation? Vertical® Adiabatic® With lattice constraints DZ basis set “State: |(o7)GvB -14.555173 103.72 0.150 0.44 0.28 Von)¢—-14.542376 123.01 - 1.84 1.45 -226- Footnotes for Table C.I “Ob tained by a cubic-splines fit to the calculated points. The values trons, since these have been replaced by the effective potential (EP). Positive values indicate motion toward the vacuum, away from the tion. Che minimum obtained for this state falls outside the range of calcu- lated points and as such represents an extrapolation. Table C.II - Potential Curves for the Si -227- Lattice Constraints? H- Cluster with the 36 95° 105° 109°28' 115° DZ basis set V06%) eve -14.547280 -14.555010 -14.551941 -14.543080 "62 ) eve -14.579739 -14.594074 -14.592091 -14.583152 fATy energies in hartree atomic units. Energies do not include the contribution of the core electrons of silicon atoms (an effective potential has been used for these). -228- with a vertical excitation energy for V6?) of 0.44 eV. The adia- intermediate values of the $i-Si-Si angle (between 100° and 115°). This is due to the onset of other effects that become more important at small -229- Table C.III. Effect of d-Functions and Correlation on the Si 3H¢ Constraints).* All energies are in Ev. Correlation V2yeva (0? )HE Energy? (on) 4011 energies are referenced with respect to the 162) eve DZd mini- The correlation energy is the difference between the 1062) eve and “This is the difference produced by the inclusion of d-functions. "e[bue 1S-LS-Ls 3432 JO UOLJOUNy e O° Ct O°Stl oot o°SG1 O° cot C°S5 A20°O — ~230~ — S2't- — hS'O- ~ae —c5'0- -231- siseg Gza & Buisn uarsniy [H®is |yz 4og seoesun§ [elqueyog 2°9 “BLY "(9919421 BY2 JO SJULeUZSUOD 342 YILM) ,ATTUATIO YTTOM GQ — 08° 0- — eS" O- — 95°0- — ¢S°O- ~232- and large angles. The most important of these other effects is The effect of d-functions can be evaluated by looking at each The above discussion concerns calculations we performed for the (including d-functions and correlation) with an optimum bond angle of -233- Table C.IV.. Potential Curves for the SiH. Cluster without the Lattice 36 95° 100° 109°28!' 120° 201] energies in hartree atomic units. Energies do not include the contribution of the core electrons of silicon atoms (an effective potential has been used for these). ~234- Table C.V Effect of d-Functions and Correlation on the Si3H¢ Clusters Modelling the (100) Surface (without the lattice constraints)°. All energies are in eV. 1,2 1,2 Correlation? 3 any energies are referenced with respect to the V(0% yep DZd minimum The correlation energy is the difference between the V2) eve and "(8919921 3Yy} yO szuLeszsuod ayy yNoYI LM) CS3ae030 DA TENG ocr} Onstt oor o*sat anno: or O68 | | ! 4 = c9"0- iw ~235~ “m™ — Ss’ O- — 0S°0- -236- . * (2079 92T oy FO SyUTeIMSti0O eA qnoUZTH) — G3" O- | — 85°0- “] pena HAH iy li A. a) oo eee nS‘ J — OS" O- ~237- 95.2°. The 3 (on) state is at 0.49 eV vertical excitation energy and of 119.9°). REFERENCE FOR APPENDIX C 1. The correlation energy is the difference between the GVB and HF values.
pect to the center of interest. As described below, the Vy) are
obtained from ab initio calculations on various states of the atom; no
readjustments are made to fit the molecular systems. Rather, we have
in mind that the potential (1) describes the interactions of the atomic
core of interest with orbitals on al] centers of the system. With
this effective potential we then completely eliminate the core orbitals
from the system. Consequently, no basis functions for describing the
core orbitals are required, considerably simplifying ab initio calcu-
lations. We do not require that other orbitals be orthogonal to the
core being replaced and hence Vo contains components representing the
effects of the Pauli principle. As a result, for Si the Vy for & = 0
and & = 1 are highly repulsive in the core region, as can be seen from
Fig. 1.
Vo(r) = b Cyr exp(-¢,r ) ’ (3)
3 4.0F
ne i: Vs
< Z
kK
2 -4.0+
0.0 {.O 2.0 3.0 4.0 5.0
R [BOHR]
‘plotted include the nuclear attraction term.
the ab initio atomic wavefunctions.
In calculations of the wavefunctions of the molecules (or solids)
various nuclei (A,B,C) of the molecule. For terms of the form (3)
computer programs allowing rapid evaluation of the various one-, two-,
and three-centered integrals.
CHF CHF HF _ HF
(h + Veore * Val) One = Ene Ong , (4)
where OF ee is the operator (involving Coulomb and exchange operators )
a)
avoid singularities in the resulting local potential Ve(r) and to
minimize the number of basis functions required to describe the valence
orbitals. The CHF orbital is obtained by simply mixing Hartree-Fock
CHE HF Oo" HF
rn oy Cee (5)
to the procedure used in OPW formalism where the CHF orbital is re-
n&
valence operators Vine and Val are replaced by new operators Voore
and Vial which reflect the fact that dng
Note that the orbital energy is stil] the same while the operator rr
serves to prevent the collapse cf the valence orbitals inte the inner
shells.
. normalization factor. This means that overlaps and other interaction
quantities between orbitals on different centers will be modified by
this same (small) factor. We want the transformation from HF orbitals
to coreless orbitals to leave intermolecular interactions invariant,
M' M
HF
¢ = y Cl y' + ) C ux ; (6)
nz wl pH “ue weM +] uk
go = ax, + Cx , (7)
n& ey yur vette yw pd
conditions on the a, of (7) are: (i) 6° as written in (7) must be
normalized and (ii) the CVO goes smoothly to zero as r goes to zero. To
. ne
lim (r) = 0
r>0 r
atom.
. CVO
The next step is to replace the operator Voore
S1] 3S ORBITALS
O.1 4H
0.0 AN
Ne gcve
3s
-O.1 4
“ HF
? 35
-0.2 —_
-0.3 L ! I me ! 1 |
0.2
| SI-3P ORBITALS
Ol
CHF
$3,
0.0
cvo
$s,
-0.!
HF
$3,
-0.2
-0.3
-0.4
-05 J I { L | i I
Figure 2. HF, CHF and CVO orbitals for the Si atom.
a) 3s type; b) 3p type.
from zero. [More precisely we require that the square of (9) summed
over all basis functions is minimized.] In (9) one uses the normal
basis functions for an atom plus additional basis functions represent-
ing important regions of function space for which V(r) is signifi-
cant. In particular, it is important to add diffuse basis functions
to the basis in order to ensure that fitting (9) will lead to the cor-
rect long-range behavior of Ve(r). The basis used to solve (9) is
included in Table IV [we solve the HF equations for the new basis so
obtaining the effective potential, all basis functions required only
for the core can be eliminated (along with the functions added only
for fitting the EP).
contains a large repulsive component representing the effect of the
Pauli principle (the orthogonality of the ab initio valence orbital
Vo (r) = V gtr) | for 2£>2
the EP Eq. he “(6s4p) basis sete basis set (2s2p) basis sct
t Sun B un H Con u un Bu “un
5 26740.0 1 1,0 1 0.002583
8 4076.0 2 1.0 1 0.019237
5 953.3 3 “1.0 1 0.093843
Ss 274.6 4 1.0 1 0.341235
8 90.68 5 1.0 1 0.641675
s 90.68 ~-- 2 0.121439
8 33,53 6 1.0 2 0.653143
8 13,46 1 1.0 2 0. 277624
8 4.051 8 1.0 3 1.0 1 1.0 1 0.043662
8 1.484 9 1.0 4 1.0 2 1.0 1 ~0, 274872
8 0.2704 10 1.0 5 1.0 3 1.0 1 0.653119
s 0.2704 --- --- on 2 -0, 200408
s 0.09932 11 1.0 6 1.0 4 1.0 2 0.424753
s 0.03731 12 1.0 --- --- ---
‘s 0.01401 13 1.0 --- --- ---
P, 163.7. 14 1.0 1 0.011498 -+- ---
Px 38.85 15 1.0 1 0.077726 --- ---
Py 12.02 16 1.0 7 0. 263595 --- ---
Px 4.185 17 1.0 1 0.758262 5 1.0 3 | -0,004717
P, 4.185 --- 8 -1,173045 --- ---
Fy 1.483 18 1.0 8 1.438335 6 1.0 3 -0. 036542
Py 0.3350 19 1.0 9 1.0 ” 1.0 3 0. 345438
Py 0.3350 --- --- 4 -0,030736
Py 0.09699 20 1.0 10 1.0 8 1.0 4 0.144725
Py 0.02766 21 1.0
Py 0.007890 22 1.0
a, 2.973 23 1.0
a, 0. 7966 24 1.0
dy 0.2863 25 1.0
dy 0.1154 26 1.0
ay 0.04998 27 1,0
ay 0.01789 28 1.0
ay 0.007211 29 1.0
XN, L On XYZ exp(-o yr”)
Ny is the normalization coefficient; p = q = s = 0 for an s-type func-
imate Atomic Functions. II.", Report from the Department of Chemistry,
The University of Alberta, unpublished) with diffuse and d functions
Vi-glt) = V(r) - Vglr) ,
on Si and the additional functions used in (9) for determining the poten-
tial.
(1s)*(2s)*(2p)°(3s)! (3p)? (3a)!
quintet state of Si, solving (4) for the o3q orbital. The Vo-d and
(1s)*(2s)*(2p)°(3s)? (3p)?
The resulting potentials are listed in Table V and are plotted
8.469 x 10° in the least squares fit to (9) for the Vas Vege and
Vo-d potentials, respectively (for the large basis set of Table IV).
the functions required for describing the core orbitals. This reduces
the double zeta valence basis from 18 to eight functions as indicated
in Table IV.
tions on various states of Si, Si, and si®. Here we Tind errors of
the order of 0.01 to 0.06 eV, quite satisfactory for our purposes.
Bear in mind that the EP was determined from fitting of the d orbital
of a quintet state and of the s and p orbitals to a triplet state. No
further adjustments were made and hence the good agreement here is
already evidence that the potential adequately represents the core
electrons. At the HF and GVB level, the lack of complete electron
correlation leads to errors in the excitation energies. Thus the ex-
or 0.864 eV higher than the GVB value, and the experimental electron
For comparison in Fig. 3 we show the 3s and 3Py orbitals of the
the definitions of n, ¢, and C,andEq,(10) for the form of the total potential.
Vy 0 0. 0991736 * .0,01189620
Ve-a 0 3,5641009 30. 31756200
-2 0. 1570854 0. 24891789
Vo-d 0 4,0620237 36.5855 7100
+10/r. There is also a -14/r term in the h of (4), corresponding to the nuclear
attraction. We have deleted the +10/r term from the table with the under standing
Si” quartet (s’p*) HF ~0.616 -0. 623 ~0. 030
Si singlet (6*p’) GVB(J) 1.056 1,078 1.107
Si quintet (sp’) HF 2.893 2. 836 2.476
si? doublet (s’p’) GVB(1/3) 1,285 7.276 7.192
effective potential calculation
DISTANCE FROM NUCLEUS [BOHR]
EP -0.3028 -0.0620 | -0.0620
| (0; 9904)? (-0. 1380)
(0. 9900) | (-0. 1409)
(0. 70771)
(0. 9904) | (-0. 1382)
(0. 9900) (-0. 1412)
EP -0. 7298 -0.3489 | -0.3489
(0. 9811) (-0. 1370)
= 0.037 eV. This difference is as high as 0.005 hartree only for
some of the GVB correlated pairs. In Tables VIII, IX, and X we show
the orbital energies for the Sigs SiH, and S1H0, SCF calculations,
respectively. Here we note that the differences between the ab initio
and EP values are larger than they were in the atomic case. This is
to be expected since the effective potential was constructed from the
atomic SCF calculations. In the molecular orbital energies most of
the differences between the ab initio and EP values are below 0.020
hartree =~ 0.3 eV except for some of the GVB correlated pairs in which
it is as high as 0.030 hartree. We note, however, that the correlation
energies for most of those same GVB pairs (given in Tables VIII-X)
agree to better than 0.005 hartree = 0.015 eV.
The double zeta (2s2p) basis set of Table IV was used for silicon and
and GVB calculations at four different points (2.22, 2.32, 2.43 and
which is 0.028 too long. By using a Numerov numerical integration of
the HF potential curve we find that the first vibrational level for
Ly) teqiquo jeunqeu puosas 343
PEP9 "0 8279 ‘0 $4790 6ezs0 | * 186L°0 LL6L “0 a5
SLE ‘O- 0662 '0- PLLS “O7 8862 0- LETT 'O- 8IIt*O- ve)
ayed Sut (upue%y) ed (8 pun By ayed puoq gS ?HHuenb
~puoquou 0 spuog 2 | Surpuoquou o spuog 4 ayed GAD ‘@
,9L6% °0- ch62 °0~ 9TTS‘0- FOTE O- 3
T9LbO~ poPLh'0- ZELP“O- 8ZLPO~ 9
bd ° na. . Bp
GL99 °O- 28799 0- ese "0" 2vlee'0- >
$539 "0- 6289 ‘0- 8L99 “0- $199 °0- 9
. solstoua
TeHqIO 'V
Teyuajz0d oATpaIIa OTU Qe TeTquajod oaqqoas30 OTTUT qe 7
3 3
+S, Se
+S, OH pus Pz.
esee*o- | eseeo-
Z08h °0- Sz8h'0-
OTTL ‘0- LETL'O-
A. Orbital Energies
Ent ~20. 6740 -20.6777
Ea! ~20.6181 -20. 6148
Ea! -1.3737 ~1.3771
€,/ -1.0588 ~1.0638
Ey! -0. 7313 ~0. 7293
Ea! -0.5940 ~0.5972
€,! -0. 5099 -0. 5084
eqn? -0.6172 — ~0,6207
B. For GVB Pairs©
a. Si-O bond
E1NO | -0.7721 -0. 7691
C, | -0.0852 0.0881
Sab 0.8424 0. 8376
AE -0. 0142 -0. 0147
b. O-O bond |
EiNO ~0. 7871 -0. 7909
CG -0.1639 -0.1637 —
Sb | 0.7151 0. 7154
and the H (2s) and 0 (4s2p) of T. H. Dunning, Jr., J. Chem. Phys.
53, 2823 (1970) and S. Huzinaga, ibid. 42, 1293 (1965).
2 2. («; . . os 2 2. 4).
21
For reference in future chapters we have listed in Table XI
all the different basis sets used in the calculations to be described
in them. Two basic types of basis sets have been used: first a
minimum basis set (MBS) which consists of the same number of functions
per symmetry type as there are orbitals in the atom in question. This
basis set is the one that has the least amount of flexibility of all of
the types described here. A second type of basis set is the double
zeta (DZ) in which the number of functions is twice that of orbitals
in the atom. We have also used in some of the calculations basis
sets which consist of modifications of the two mentioned above. One
such type of basis set is what we have termed as a mixed basis set
(MXS1 and MXS2) which for some parts of the molecule consists of a
MBS set while the "surface" part has DZ character.
The set MXS1 has only been used in calculations on the SiH. cluster
of Chapter 2. The set MXS2 has been used in the silicon part of the
Si,H,.0, calculations of Chapter 4. Two other types of sets have been
used; one is a double zeta set with additional d-functions (denoted
DZd) which has been used in various calculations in Chapters 2 and 3.
The last set consists of a double zeta part with diffuse s and p func-
tions (denoted DZR). This set was used in the Si gHg calculations of
Chapter 2.
We find that the EP obtained using the above ideas leads to
Table XI. Si Atom Basis Sets* Used in Calculations Discussed in
Chapters 2 to 4
4 Gin (MBs) (pz)°
S 4.051 1 0.043662 1 0.043662
s 1.484 1 -0.274872 1 + 0.274872
S 0.2704 1 0.452711 1 0.653119
5 0.2704 - 2 -0.200408
5 0.09932 1 0.424753 2 0.424753
Pp, 4.185 | 2 -0.004717 3 -0.004717
P, 1.483 2 ~0.036542 3 0.036542
Pp, 0.3350 2 0.314702 3 0.345438
p, 0.3350 - 4 -0.030736
p, 0.09699 2 0.144725 4 0.144725
together with a meaning of the symbols used in this table.
adding the function with yp = 4 of the DZ set (for MXS1) and functions
with yp = 2 and u = 4 of the DZ set (for MXS2). In all calculations in
which hydrogen atoms are present the 3-gaussian basis of S. Huzinaga
[J. Chem. Phys. 42, 1293 (1965)] was used.
and b,7 0.02808 , respectively. For the DZd basis set, a set of
d-functions with Sine 0.3247 was added to the DZ basis set. In al]
calculations in which hydrogen atoms are present a DZ contraction of
leads to great computational gains, it still requires four basis
plified EP (SEP) adjusted so as to require only two basis functions
from (7) leading to a smooth valence orbital with Finite amplitude at
the nucleus. The parameters for the SEP are listed in Table XII.
sive at small r than the EP. A plot of this potential is shown in Fig.
4. Table VI compares values of energies obtained with this SEP and
tions in SigHg complexes. Here the complex models a Si(111) surface
with one Si surface atom bound to three "second layer" Si atoms. These
in turn are bound to nine hydrogens, used to decouple what otherwise
would be "second layer dangling bonds". These second layer electrons
are then localized in Si-H bonds and do not couple with the surface
dangling bond. In the calculations (described in more detail in Chap-
ter 2) the surface atom is allowed to move along the [111] direction,
keeping all other atoms fixed. When we use the EP to substitute the
core electrons of all four Si atoms the minimum energy is found at
0.152 bohr from the tetrahedral geometry, toward the bulk. When in
the three second layer silicons we use the SEP, with the EP only on
IPOL8 °S LOOTE'T 0 p-d,
PrOS6T “0 -. gzesz"o 0
ShIL PT $9106 °T 0 PrSy
es
OLLOOT6S *S- 69TSOTZ’S T-
99168910 ‘0~ 0600062 ‘0 I-
02968110 °0- 98L1660°0 0 PA
3 2 u
uses it in atoms which do not determine the principal characteristics
of the electronic states of the cluster.
For band calculations on solids using plane wave expansions, an
overriding consideration is the reduction of the number of plane wave
components. Thus, for such studies the SEP is likely to be more useful
duces wavefunctions of ab initio quality, as well as very good agreement
in the energy quantities of molecular and atomic systems. It must also
be noted that since the wavefunctions obtained with the effective poten-
tial are smooth at the core, we can reduce considerably the number of
primitive functions on the basis sets employed. This produces an appre-
ciable reduction of cost in the calculation of the integrals. A further
reduction in cost (for large complexes ) is obtained when one uses the
simplified effective potential for atoms that are not actively involved
in the calculation (e.g., "bulk" atoms) when calculating surface proper-
ties. Besides this, one gets the corresponding reduction in the SCF
3.
(a) H. Hellmann, J. Chem. Phys. 3, 61 (1935); (b) P. Gombds, Z.
Phys. 94, 473 (1935).
J. C. Phillips and L. Kleinman, Phys. Rev. 116, 287 (1959).
(a) M. H. Cohen and V. Heine, ibid. 122, 1821 (1961); (b) B. J.
Austin, V. Heine, and L. J. Sham, ibid. 127, 276 (1961).
For a review, see the following (and the references cited there-
Chem. Phys. 16, 283 (1969); (b) J. N. Bardsley, Case Studies in
Atomic Physics 4, 299 (1974).
Earlier work on related approaches is summarized in (a) L. R. Kahn
and W. A. Goddard III, J. Chem. Phys. 56, 2685 (1972); (b) idem,
Chem. Phys. Lett. 2, 667 (1968); (c) L. R. Kahn, Ph.D. Thesis,
California Institute of Technology, 1971.
Lett. 28, 457 (1974).
5. G. Louie, Phys. Rev. B 12, 5375 (1975).
18.
21.
57, 738 (1972); b) W. A. Goddard III, T. H. Dunning, Jr., W. J.
Hunt, and P.. J. Hay, Accts. Chem. Res. 6, 368 (1973).
D. R. Boyd, J. Chem. Phys. 23, 922 (1955).
R. S. Mulliken, J. Chem. Phys. 23, 1833, 1841 (1955).
W. A. Goddard III, A. Redondo, and T. C. McGill, Solid State
Comm. 18, 981 (1976); see also Chapter 4.
A similar form for the effective potential was used by V. Heine
and I. Abarenkov [Phil]. Mag. 9, 451 (1964)]. They used
of the eigenstates of the jon core of charge Z. Ru is the core
radius.
(1975).
Norton and G. C. Holywell, J. Mol. Structure 11, 371 (1972).
G. Bethke and M. K. Wilson, J. Chem. Phys. 26, 1107 (1957).
Lett. 3, 606 (1969).
THE CLEAN SURFACES OF SILICON
surfaces of silicon. In particular, we will be concerned with the
electronic structures for clean surfaces (i.e., in the absence of im-
purities) and some of the consequences the electronic structure has
on the geometrical configurations of such surfaces.
electronic structure in two ways: | (i) The interruption of the long
range periodicity will modify the properties and characteristics of
the bulk states; such effects should not be very sensitive to the
specific surface; (ii) there will generally also be localized elec-
tronic states associated with the unsaturated valences of the surface
atoms and hence quite sensitive to the specific atomic arrangement at
the surface. Case (i) can be studied using the techniques of bulk band
structure calculations to treat the two-dimensional region. parallel to
the surface and matching layers in the direction perpendicular to the
surface using appropriate boundary conditions.
with herein. For such localized states it is essential to account
properly for electronic correlation or many body effects, and hence
the usual band techniques are inadequate. Consequently we will apply
the generalized valence bond@(avB) and configuration interaction? (CI)
states is the determination of the displacement of the surface atoms
from their respective locations in the bulk. We will distinguish
two such distortions: (i) relaxation, which we define as a uniform
motion of the surface atoms along the direction perpendicular to the
surface, either toward or away from the free surface; (ii) reconstruc-
tion, that is, nonuniform displacement of atoms, either laterally or
perpendicularly to the surface, or movement of the atoms to entirely
new positions. Here we will consider both relaxation and some types
of reconstruction.
faces is that we do not yet know how to include such correlation ef-
fects for infinite systems. As a result we must use a finite cluster
of silicon atoms. In order to obtain rapid convergence of the surface
states as_a function of cluster size, it is very important to ensure
that all Si atoms included in the cluster have the same coordination
as in the semi-infinite system. Thus visualizing the formation of a
cluster by cutting it away from the semi-infinite solid, we maintain
the proper coordination numbers by replacing any broken Si-Si bond
with a Si-H bond. This procedure has been applied to the clusters
discussed herein.
present chapter have the following characteristics on common: (7) The
and a Si-H bond length of 1.48h (from silane? SiH; ) were used. In
studies of relaxation and rearrangement effects involving motion of
a Si atom attached to a hydrogen, the H atom was moved so that all
bond angles were the same as for the semi-infinite system. That is,
the virtual position of the (stationary) Si represented by the H de-
this plane each surface Si atom is bonded to three Si atoms on the
plane below. (this is sketched in Fig. 1) and one electron is left in
a nonbonding or dangling bond orbital, pointing away from the surface.
Retaining the full symmetry of the surface(1x1 unit cell), we find
(vide infra) that the surface Si atoms relax 0.08h toward the bulk
positions (this is 10% of the bulk interlayer spacing of 0.788).
2x1 unit ce11® in the low energy electron diffraction’ (LEED) pat-
terns, indicating some degree of reconstruction. Further treatment
(usually thermal) leads to additional rearrangement and a more stable
7x7 unit ce11.° This suggests that considerable motion of the surface
Ist LAYER
e x a x ©
x 6 xX ©
@ x 6 x ©
x @ Xx 6
@ x @ X @
X @ X @
@ X r) X L*)
by filled circles. Second layer atoms are indicated by crosses.
(tetrahedral geometry) and relaxed surfaces.
two basic problems: (i) the relaxation of the surface Si atoms; and
(ii) the interaction between dangling bond orbitals on different
neighboring surface atoms. Two different types of clusters (described
cluster consisting of one surface Si atom, its three Si neighbors on
the next layer plus three H atoms bonded to each of these second layer
Si's, leading to a Si 4Hg complex as shown in Fig. 2. To investigate
the effect of relaxation we allowed the surface silicon to move along
the [111] direction. Similar studies were also carried out for the
positive and negative ion systems in order to determine the sonization
using the double zeta (DZ) basis set of Table XI, Chapter 1 (p.31). (Double
zeta means that two basis functions are included for each orbital
present in the atom). Since a negative ion generally leads to more dif-
fuse orbitals the DZR basis set (Table XI, Chapter 1) was used for the
case in which a second electron had been added to the dangling bond
suoye UODLLLS ‘aaysnt9 ILA QYy1 AOJ UOLPeUNBLyuoZ [edOLuyawoseg °Z “BHL4
BCR HIS) -1S
functions on the surface Si to allow a greater flexibility in the
variational calculation of such an orbital.
except that the position of the surface atom was varied along the [111]
direction (threefold axis). The range of the variation was from -0.8
to 0.6 bohr. (Positive displacement implies motion away from the sur-
face. )
The neutral complex is a doublet state and we carried out fully self-
consistent open-shell Hartree-Fock (HF) calculations for this state.
= 8 yp (1se++ 924), 5(25) a(25)]} (1)
spatial orbital, a and 6 are the up and down one-electron spin func-
tions. (Note, spatial and spin functions are always ordered with
sequence of increasing electron number unless directed otherwise.) On
the right hand side Sourk ele? 924) denotes the wavefunction of the 24
nondangling~bond electrons. (Recall that the Si 1s, 2s, and 2p elec-
trons are included in the EP). $4 3(25) and «(25) correspond to the
spatial and spin functions of the dangling bond electron. We thus have
a total of 12 doubly-occupied bonding valence orbitals plus the danal-
closed-shell HF calculations were performed. In this case the wavefunc-
tion is
the same as (although very similar to) those of (1).
For the negative ion state the HF description is to place the
this description of the negative ion should lead to too low an electron
affinity. To be consistent with (1) we allowed these two electrons to
be correlated. In the GVB wavefunction such correlation is introduced
by replacing a doubly-occupied orbital with a pair of overlapping or-
face atom but $5 3 is more compact while dip is more diffuse, thereby
allowing for radial correlation of the motion of the electrons (this is
where
%ig = ($5, + o:,)/D,
fu = (¢,. - 55) /D.
Sg 2 its
GH. 14 sé
S = <5 41 05h
(D, and D, are appropriate normalization constants). The orthogonal
a localized Hartree-Fock orbital (see Fig. 3); os is denoted as the
correlating orbital; generally Ce = 1.0 and C, = 0.1.
term, there are smaller terms that are important in properly describ-
ing negative ions. In general, the most significant correlating
orbitals are those that have one more nodal surface than the orbital
being correlated. Thus the two important correlations in addition to
those in (2) involve correlating orbitals whose nodal plane bisects
a} ist NO.
(1/4) SijHg Caleulation., a) First Natural Orbital, 073; b)
Second Natural Orbital, O14), (in-out correlating orbital); c)
Third Natural Orbital, )5 (angular correlating orbital).
Solid lines indicate positive amplitude values, short dashes
indicate negative amplitudes, long dashes indicate nodsl sur-
faces. Contours are drawn every 0.0% atomic units. Atoms are
denoved by an asterisk. Distances are in atomic units.
orbitals of (2) are referred to as o orbitals (m, = 0 with respect to
the [111] axis). The correlations effected by 7 orbitals are called
angular correlations. Including all these correlations together the
the coefficients Cia to Cie
have values between 0.04 and 0.08, whereas C13 has a value of 0.995.
The energy obtained with the wavefunction (3) is 0.35 eV lower than
Our results for the SigHy complex are summarized in Table I.
bulk 0.08h (10% of the interplanar distance), leading to a new Si-Si
bond length of 2.33 (compared to 2.35A in the bulk). The resulting
dangling bond orbital, $43: is shown in Fig. 4. It is localized in
the region of the surface atom (93.1%) and is mainly p-like (92.9%).
an additional 0.308 (a total relaxation of 48% of the interplanar
. Positive Negative Positive Negative
Neutral Ion Ion Ion Ton
. Q a .
Res gy (A) 2.33 2.25 2.41 2.26 2.44
Excitation
Aw (ev) © 0.036 0.030 0.030
This was calculated by solving numerically for the lowest two vibra-
rahedral undistorted geometry.
SIgHg DANGLING BOND ORBITALS »
0, | i
2 ae " r ‘ a as ~1 “3 3 ad » : ~
the Si), ely Uluster. (a) Dangling ond Orbital vor the Joutral
Goaslex. (b) Dominant Natural Oroital for the van, flin::-Bond
slectron Pair or the Negative Ion.
jonization potential is 5.78 eV which is to be compared with experimental
equilibrium position leads to an energy decrease of 0.34 eV giving an
adiabatic ionization potential of 5.44 eV.
from the surface by 0.252 (from the location of the optimum neutral
optimizing the total energy of the system. This is, to our knowledge,
the first time this has been done for a silicon surface. In the past,
to estimate relaxation distances. 34-40 This leads to a relaxation dis-
tance that is 4 times as large as our calculated value.
bilized Si(111) 1x1 structure. They find excellent agreement between
the observed and calculated spectra when the first layer relaxes 0.124
toward the bulk (15% of the interlayer spacing). This is in very good
agreement with our results.
based on the experimental geometry of the solid’ (Rei gj = 2.354). As
shown in Chapter 1, the results on H3Si-SiH, demonstrate that similar
rather than 0.088. On the other hand, in our relaxation calculations
we allowed only the one surface Si to relax. Repeating the calcula-
tions and letting the six hydrogen atoms representing other Si surface
atoms also relax a proportional amount, produces a calculated Si-Si
bond length of 2.348 and the surface relaxation changes from 0.084 to
0.032. Combining both corrections leads to a corrected $i-Si bond
length of 2.33K and hence a surface relaxation of 0.08, that is, the
plex, we also carried out calculations on SiH35 certainly an extremely
small complex for modelling the surface. The orbital coefficients for
the dangling bond orbital for SiH, and Si-(SiH3), are compared in Table
II. Here we see that the dangling bond orbitals are very similar. In
addition, the orbital energy (which by Koopmans' theorem is the ioniza-
tion potential for the case where the other orbitais are not allowed
to readjust) for the dangling bond orbital changes only by 0.5% between
these two cases.
lent model for the dangling bond state and its interactions with the
bulk bonds.. The remaining question concerning the interaction of sur-
face Si atoms with each other (through their dangling bonds) is
(hartrees ) 0.3340
0.0992 0.5558
0.1239 0.5263
the surface orbital are basically related to the overlap of the local-
ized bond pairs with the localized dangling bond orbital. Hence the
effects should decrease exponentially with the distance and satis fac-
tory results are expected with a small complex. On the other hand for
states with a net charge at the surface long range effects are ex-
pected. Thus, upon ionizing the electron from the dangling bond
orbital, the resulting positive charge leads to effects that fall off
as rl, In the semi-infinite system this results in polarizations
extending over a large region of the crystal surrounding the surface
charge. For example, in Table III we see that although the Koopmans'
jonization potential is nearly the same, 9.13 eV and 9.09 eV, for STH,
smaller and much different values, 8.47 eV and 7.81] eV, respectively.
These differences are due to polarization of the bonds in the complex
in response to the positive charge at the surface. Our estimate is
that it would require a complex having a radius of v 652 to treat cor-
rectly all polarization effects to within 0.1 eV (~ 20A for 0.3 eV).
Such large size complexes are not currently practicable, and we have
Koopmans ' theoren® 9.13 9.09
Self-consistent ioniza-
tion potential’ 8.47 7.81
orbital
itive jon and the ground state self-consistent calculations
Considering our finite complex as a hemisphere of radius ro? as in
part c of the diagram above, we include the polarizations within this
hemisphere but ignore the polarization effects in the balance of the
semi-infinite system (part b in the diagram). We have estimated this
additional correction as follows. From the wavefunctions of the
Si-(SiH3), complex we evaluated the average position h of the dangling
value of h, the self-consistent energies for the Si gly complex were
corrected using the polarization energy of the semi-infinite slab minus
Wendy ph (ro +h’)
AE = - gay) nine p> lost rt h I}
leading to the results in Tables I and IV.
radius to the nearest Si atom (ro= 2.358), or to the midpoint of the
SiH bonds (ros 3.098), or to the H atoms (ro 3.83K)? (The numbers
used in parentheses are for the undisturbed complex.) Since the Si-H
bond is much less polarizable than the Si-Si bond, the value of Yo
should be smaller than the value to the midpoint of the Si-H bond but
not smaller than the Si-Si bond length. ‘In our calculations we took
r, to be the distance to the nearest Si atom (ro= 2.358 for the undis-
turbed geometry). Changing Yo by 40.12 leads to a change of 0.1 eV
in the correction energy. Similarly changing h by+ 0.32 changes the
correction energy by 0.3 eV. Thus we estimate that our corrections
are probably good to +0.4 eV.
carry out such corrections self-consistently, replacing the charge at
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uo] Udt uo] Uo] LeuqnaN (y) (ayoq)
SUOL}DS4N09 SU0 1398409
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XI of Chapter 1 {p. 31). °
of Table XI of Chapter 1. From G. T. Surratt, Ph.D. Thesis, Cali-
fornia Institute of Technology, 1975, Table VI.4.
bond orbitals on the unrelaxed-unreconstructed surface, we used the
Si He cluster illustrated in Fig. 5, the smallest complex suitable
for this study. It consists of three silicon atoms, two of which are
on the surface, and the third is bonded to both surface Si atoms. The
six hydrogen atoms replace the bulk Si-Si bonds broken in cutting the
complex from the semi-infinite solid.
lead to a ground state possessing a singly occupied dangling bond or~
bital on the end of the two surface Si atoms. These two orbitals will
be denoted as d» and pn Coupling these orbitals together leads to
both a singlet state and a triplet state. Treating all Si-H and Si-Si
bond pairs as doubly-occupied (as in the Hartree-Fock description) but
allowing the dangling bond orbitals, oy and dye to each be singly-
x [a(1)e(2) - (1)a(2)I} » (4)
and
Sar(gy) = otPiPlet -Qya Lo, (1)6,(2) - 6,(1)4, (2)]
suaboupAy *‘seuayds abuet Aq paqzuasaudau aue suodLiLis ‘aoeJuns
(LLL) 243 BuLpapow uagsn 9 9H 1s BY} MOF UOLJeUNHLjJuoZ Ledtuzawosyg “¢ “BLY
contains all bond pairs. Solving for these wavefunctions self-consistently
leads to the results in Table V. . Here we see that the triplet state is
consistently 0.01 eV below the singlet. Although both states were solved
self-consistently, the orbitals of wavefunctions (4) and (5) are nearly
identical so that use of the psinglet orbitals in the ptriptet wavefunc-
tion increases the energy by only 0.002 eV. The dangling bond orbitals of
(4) are shown in Fig. 6.
148°
1-s¢
the triplet state is the ground state (just as in Hund's rule). The
MBs © mxs1° pz° DZ
see (4) and (5) for the wavefunctions.
bThe relaxation distance of the surface Si atoms was 0.08A. The
bonds.
flexible. |
~14.54255 hartrees.
Th (2r) GVB calculation.
Si,He A ORBITALS |
or ~~ ry
i ¢ 7 ~ A
t ,f soa ‘
+.
amplitudes by short dashes.
Amplitude contours are drawn every 0.05 atomic units.
Positive amplitudes are denoted by solid lines, negative
this term dominates over Kor leading to ELS 0 and hence a singlet
ground state.
of EY and kT (thermal energy) the surface could be diamagnetic, para-
magnetic or ferromagnetic.
etry, as in an unrelaxed-unreconstructed (111) Si surface. A second
calculation was performed at the relaxed geometry, obtained from the
calculations reported in Section B.. [The surface silicons were relaxed
by 0.088 towards the bulk along the [111] direction, keeping the other
silicon fixed and rotating the Si-H bonds as if they were Si-Si bonds
with the (virtual) second layer silicons fixed. ]
we carried out similar calculations using three different basis sets:
MBS (Minimum Basis Sets, one basis function per atomic orbital),
Orbital), as described in Table XI, Chapter 1 (p. 31). The MXS1 de-
each atom is bonded as expected from its neutral atomic configuration.
"ar (ner?) Af 4, [0,(1)4, (2) +6,(1)6,(2)
- (4, (1)8,(2) #9,(1)6, (2)) a 1a(2)]} (8)
leading to a positive ion on one surface Si and a negative ion on the
other. For the actual calculation it is computationally more expedient
to recombine these states to yield symmetry functions (with respect to
parameter A ). Both states were studied with CI calculations, as des-
is
1]
° e)
from self-consistent calculations (Table YI).
fects around each charge center and hence the use of a finite complex
electron is excited out of a "bulk" orbital, say dy» into a dangling
bond orbital (o, or o,)- The simplest description of such states is
"(b> s) =ALO, 1, (4990 29,0 p09% 0808] 5 (10)
oy terms deleted], leading to A’ and A" symmetries for both spins.
Although certain of these states can be solved self-consistently, a
Of course, the use of a finite cluster with an abbreviated
description of the bulk states should have a great effect in the ex-
Such transitions should require a more complete basis (diffuse s- and
p-functions in addition to d-functions) as well as a large complex.
leading to A' and A“ symmetries for both spins.
the bulk orbitals upon the covalent states and to obtain a consistent
description of the other excited states, we carried out configuration
interaction calculations as follows.
ground state and using the MXS1] basis we allowed al] double excitations
within the space spanned by the four orbitals describing the dangling
cupied) simultaneous with all single excitations out of the "bulk"
by = (b) ~ O)/ VATS)
- _ ee
bug 7 Pqhu = (Sp > Oyo) M18)
cannot be expressed as a simple molecular orbital (MO) wavefunction.
having a smal] overlap. Since most band structure techniques are
based on the closed-shell HF formalism we expect such approaches to
yield erroneous results for such surface states despite the use of a
large complex or a semi-infinite system.
_ 2
boty * by8g = (8% > Spell 1-8
of the b>+s and s>b* transitions can also be adequately described with
(111) surfaces. For both the unrelaxed and relaxed calculations the
overlap. S$ = <9] 0,> is very smal] (S = 0.01). This means that there
is only a relatively small amount of interaction between adjacent dangl-
ing bonds. This leads to a 0.01 eV splitting between the triplet ground
state and the singlet state.
07 355) TS
For kT (thermal energy) large compared to Jaze the surface dangling
bonds would act as individual paramagnets, but for kT small compared
with Jag we could expect cooperative magnetic behavior. The calculated
However, slight displacements of the surface atoms can cause energy ef-
fects much larger than 0.01 eV and indeed could stabilize singlet pair-
ing of adjacent dangling bonds. Because of the Pauli principle, dis-
the effect of the basis set on the energies. The ionization potential
of 8.63 eV is about 3 eV higher than the experimental value. |! This
discrepancy is due mainly to polarization of the solid, ‘as discussed in
(s + vacuum) represents an ionization out of one of the dangling bond
State Basis Set Basis Set Basis Set
3A" (er) 0.0° 0.0° 0.0°
"at(ar) 0.013 0.014 0.009
"aH (g2_r?y 4.816 5.064 6.793
3a'(b+s) 4.856 4.824 6.415
3a (5 +b*) 7.252 11.172 10.871
2A"(s > vacuum) 8.630 8.440 7.830
4A (b> vacuum) 9.496 9.236 8.711
hartree, respectively.
Ta" (22-2) state is lower in energy than the 34" (b+s) state. For
the MSX1 basis the 3A" (b+ 5) state is relatively well described be-
cause the surface part of the basis has the same flexibility as the
DZ set (for the p,~functions). On the other hand, MSX1 is the same
as the MBS set for the "bulk" part of the complex. This is reflected
are more delocalized over this region.
The separation between the singlet and triplet covalent states is still
very smal] (~ 0.02 eV) but the total energy of the ground state is
0.26 eV lower than the corresponding SCF value. The order of the
Van (ger?) and 3A! (bos) is reversed with respect to the SCF value for
the same basis set. The reason for this is that the CI description for
the ger? state is better than that of the b>s state (some configura-
tions important for the b>+s state are not present in CI).
positive ion, Si gles with Koopmans ' theorem values for the ionization
potentials. The CI configurations were obtained by doing all single
excitations out of a basic set of 10 configurations formed by omitting
one electron at a time from each occupied orbital. This latter set of
basic configuration was used for the Koopmans' theorem calculation. In
Table VII. Results of CI Calculations for the Si He Cluster Model
34" (21) 0.04 824 1036
"a (ar) 0.019 614 1632
"an (22-2) 4.286 612 -
3A (b +s) 4,293 814 1018
3a" (b+s) 4.492 824 1036
"ar (rtp?) 4.561 614 1632
Tan (pt +s) 4.607 612 -
"A'(b>s) 4.629 614 1632
3A" (b' 55) 6.392 824 1036
3A" (b' $s) 6.407 814 1018
Ta" (b> 5) 6.487 612 .
Tat(bt +s) 6.492 614 1632
3A'(b" +s) 7.407 814 1018
bond. Simultaneously, all single excitations were allowed from all
“bulk” orbitals to all virtual orbitals of the basis.
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(111) surface of silicon single crystals is that it is the cleavage
on the experimental determination and study of silicon surfaces. Any
other surface has to be cut or prepared and annealed before one can
study it. This can modify the ideal environment of the surfaces in
ways that are difficult to predict. On the other hand, a freshly
cleaved surface corresponds more closely to the idea that one has of
what an undisturbed surface is. The cleaving can be performed under
conditions of ultra-high vacuum, so that for good cleaves the surface, be-
Sides being undisturbed, is not contaminated by other atomic or mole-
cular species.
to those of X-ray diffraction techniques for crystalline solids. In
LEED, the probe consists of electrons ejected by a gun at energies
ranging from 10 eV to 1000 eV. This means that their de Broglie wave-
lengths lie between 3.878 and 0.398, that is, of the same order of
magnitude as the typical interatomic dimensions in a solid. On the
other hand, low energy (10 to 500 eV) electrons exhibit inelastic mean
apparatus come from the first few layers of the surface.
A freshly cleaved silicon (111) surface leads to a structure
heating this 2x1 structure changes irreversibly into a 7x7 struc-
structures. The temperature behavior indicates that the degree of
rearrangement involves greater motion for the 7x7 structure than for
the 2x1 structure. Since the 7x 7 structure cannot be transformed
back into the 2x1 structure, the 7x7 pattern most likely is the more
stable one.
However, since they are not easily reproduced and since some of them
seem to be due to the presence of small amounts of impurities !/2, we
will ignore these cases.
in the total electron yield vs photon energy in photoemission experi-
shown the presence of surface states on silicon and germanium (111)
surfaces. They used synchrotron radiation with photon energies in the
range 7 to 30 eV. The energy distribution curves of the emitted elec-
trons were measured for fixed incident photon energies. The emitted
electrons contain contributions from both the bulk solid and the sur-
The photoemission curves of the oxidized surface would then only have
the bulk contribution, together with photoemission from the oxygen-
surface complex. By studying the difference spectra between the clean
and contaminated surfaces, one can observe peaks assumed to be produced
surfaces, a band of states centered at 0.75 eV below the Fermi energy,
or 0.43 eV below the valence band edge. Wagner and Spicer! obtained
similar results, with a peak at 1.1 eV and a shoulder at 0.5 eV below
good agreement with the photoemission estimate. Additional photoemis-
the idea of characteristic surface states.
the presence of surface states on silicon (111) surfaces. An example
is shown by the field emission experiments of Lewis and Fischer. °2
of surface states is made evident.
States. One of the methods used in this context is total internal
a transition peaked at 0.5 eV, which occurs in 2x1 structures of
Si(111) surfaces but disappears when the surface is exposed to oxygen
Muller and Mnch2? have shown that a surface sensitive shoulder ap-
pears in the bulk photoconductance curves for cleaved Si(111) sur-
faces. The shoulder disappears when the surface is exposed to oxygen.
Further experimental evidence for surface state transitions is
obtained from ellipsometry experiments“° and eneray-loss experiments .°/
We now return to a more detailed analysis of the LEED patterns
for (111) Si surfaces and their interpretations in terms of theoreti-
cal models. Two basic patterns are found: a 2x1 and a 7x7. We will
not involve large migrations of surface atoms. This is confirmed by
mating experiments® in which two surfaces are created by brittle
cracking in ultra-high vacuum. They are then replaced on top of each
under mild fracture stress. Evidence has been obtained® that almost
perfect atom-on-atom replacement occurs when the two surfaces are
mechanically fused together. One of the important aspects of this
experiment is that the two surfaces are not allowed to fully separ-
ate; this insures that precise replacement is possible. Large atomic
rearrangements are practically ruled out by this experiment, suggest-
ing that the 2x1 structure is originated by small displacements of
the surface atoms (possibly more than one layer) about the positions
that they occupied on the solid. (Note that LEED cannot be used in
this technique to determine what pattern is present in the crack;
the assumption is that the 2x] pattern is present since the experi-
ment is performed in ultra-high vacuum and at room temperature. )
Unfortunately, these experiments cannot be performed at high tempera-
tures (to observe the transition to the 7x7 pattern) because the
necessary electrical contacts to the Si crystal are lost upon heating.
Haneman?® has proposed a model to account for the observed
2x1 structure on Si (111) cleaved surfaces. This widely discussed
model assumes that the surface undergoes buckling, so that alternate
rows of surface atoms are raised and lowered, producing the observed
LEED pattern. The mechanism for the buckling is based on the follow-
(tending ° to produce a planar configuration, hence the lowering of
the row), while the raised rows have pure s character for the dangl-
ing bonds (producing 90° angles for the bends to the second layer,
structure observed in LEED experiments and the Electron Paramaanetic
Resonance (EPR) experiments by Haneman and coworkers“? in which a
roughly one spin per four to ten surface atoms). It is now known that
single crystal (111) surfaces of silicon have no detectable EPR sig-
nal°2 that can be ascribed to a surface spin density.
that the coupling between adjacent dangling bond orbitals is too small
to produce the distortions proposed by Haneman.28 Of course, slight
motions might account for such reconstruction, but it is our opinion
not entirely correct (i.e., we do not find the hybridization schemes
proposed in Ref. 31). It is possible that charge transfer states might
account for such a reconstruction. This is suggested by our calcula-
tions on Sigg clusters (see Table I), where we found that the positive
jon state relaxes inward (toward the bulk) whereas the negative jon
state relaxes outward (toward the vacuum). Further calculations, using
larger clusters, to test this model are warranted.
400°C, the 7x7 LEED pattern appears. All the fractional order spots
are present in this pattern, indicating that the basic structure of the
surface has a truly 7x7 unit cell and it is not the result of a smaller
unit cell with an overall 7x7 pattern. Also, since the formation of
32
series of vacancies of the surface atoms so that benzene-type rings
are formed, presumably stabilizing the surface structure. Up to
structure. Our studies of Si-Si bonds indicate that the m bond is
can partition the techniques into those utilizing finite clusters and
those utilizing semi-infinite systems.
orbital (doubly-occupied orbital) wavefunction. Their results seem
to be in general accordance with those obtained using semi-infinite
systems. They find a dangling bond orbital with a high degree of P,
character whose orbital energy is in the neighborhood of 7 to 8 eV.
Most of the theoretical calculations in the literature use a
semi-infinite solid. These calculations can be divided into two
classes: (i) calculations utilizing an effective hamiltonian (non-
self-consistent); and (ii) calculations involving siome sort of self-
consistent procedure. Non-self-consistent calculations have used the
The method of Appelbaum and Hamann??? assumes a potential of the
form
> > > >
Ver) = Vag) t VO) # Vig gO) >
equation) and exchange potentials. Vel?) is a local approximation
to the exchange energy which uses the Wigner interpolation form 4
by fitting to bulk band structure calculations. The potential V5 (r)
assumes two-dimensional periodicity parallel to the surface. This
results in a one-dimensional set of coupled differential equations
that have to be solved numerically.°22
for the ideal (111) surface, which is insensitive to small normal dis-
placements. For all geometries a dangling bond state is found which
normal to the slab. They find an ionization potential of 14.0 eV.
They find a surface band structure near the top of the valence band
etry variations all these methods use a priori chosen values of the
relaxation distances. The reason for this is that the techniques
cannot be used to calculate total energies, and therefore cannot opti-
mize geometries. One of the methods used to estimate relaxaticn dis-
tances?/~70 uses a formula due to Pauling! | in which bond lengths are
related to bond order. In the case of the unreconstructed Si (111)
surface this predicts a relaxation of 0.34K. This is four times as
37,39,40
Situation.
orbitals (i.e., explicit electronic correlation is not included),
leading to systematic errors which can be of great importance in ob-
con) presents a completely different geometry from that of tne (111)
surface. Whereas the surface atoms of the (111) surface form a hexag-
cnal pattern, with nearest (surface) neighbor distances of 3.83h, the
(100) surface presents a square lattice structure (shown in Fig. 7),
also with nearest neighbor distances of 3.83K,
bonded to three second layer nearest neighbors. This leads to one
dangling bond electron per surface atom. In the (100) surface each
surface atom is bonded to two second layer nearest neighbors (for the
ideal unreconstructed surface). This leaves two nonbonding electrons
Instead one has to prepare the crystals by first cutting and then sub-
mitting them to extensive treatment to clean and heal the disruption
caused by the cutting process. This may eliminate simple metastable
structures similar to the 2x1 pattern of the (111) surface. The treat-
ment to which the crystal faces are subjected is capable of producing
the most stable structure, even if this means very considerable rear-
rangement of the surface atoms. It is not surprising then, to find
x X
e> eo
x x
- eo
x X
Bo —e
by filled circles.
tion models (discussed in Section C).
present case we were interested in investigating two types of problems:
(i) the basic electronic structure of an ideal surface, as well as the
relaxation distances for the different states; and (ii) the feasibility
of one of the proposed reconstruction mechanisms for the (100) surface.
face we find that the electronic structure is determined by the divalent
character of the surface silicons. There are two "surface" electrons
per surface silicon atom with two basic low-lying states. For the
ground state the relaxation distance is 0.108 towards the vacuum, while
for the first excited state the surface relaxes inward by 0.058. We
also find that bond pairing of adjacent surface atoms leads to a bond
whose strength is 1.74 eV. (Each Si atom moves so that the Si-Si bond
length is 2.384). This makes plausible the reconstruction of the sur-
Si3H¢ cluster (quite different from the SigHe complex used in the (111)
surface), in which only one silicon atom is a surface atom, bound to
two nearest neighbors corresponding to second layer atoms in the ideal
unreconstructed surface. These, in turn, should each be bound to three
Since each surface Si is bound to two other silicons, two nonbonded
electrons are left. Thus the complex can be schematically represented
as HSi-Si-SiH3, where the "bulk" (second layer) Si are fixed at the
normal tetrahedral positions, but the surface Si (divalent) are allowed
to relax. Two basic low-lying states result from this particular con-
perpendicular to the Si-Si-Si plane (this is denoted by m and indicated
in Fig. 9 by a circle; visualize an orbital sticking out of the paper)
and the other is in an sp-hybrid orbital located mainly in the plane of
the three silicons (this is denoted as o and indicated in Fig. 9 by a
lobe). Two different spin couplings are possible, a triplet and a
singlet. Since the two orbitals are orthogonal, a favorable exchange
integral predicts the triplet to be the lowest of the two (as in Hund's
rule). The triplet and singlet states are denoted in Fig. 9 by 3 (on)
valence electrons in ao orbital. In our GVB calculations this o pair
is correlated, leading to one orbital pointing above the Si-Si-Si plane
with a shape of the form’? o+ An and the other electron is in an
UL [2ZLquo © B squasaudau * Cd*aqo] e Suaded ayy Jo aueld ayz
(30) ZZ oN
A2 bS'0 |
og Z XY a
A2 791
IS 4OVSHNS LNA WAIC
major importance because each orbital is singly occupied. For the
(o7) configuration a Hartree-Fock description leads to large corre-
lation errors and as a result obtains the wrong ordering of the
states. Introducing normal correlation (GVB wavefunction), we find
molecule’! (CH, ) and its derivatives. Few such systems are known for
silicon; an example is SiH,» which has a singlet Vg2) ground state’®
(with an optimum angle of 92°). For the carbon systems the ground
keeping the “second layer" silicons fixed at the tetrahedral positions
and letting the "surface" Si relax along the [100] direction. The
actual calculations were performed for four different values of the
Si-Si-Si angle; namely 95°, 105°, 109°28' (tetrahedral geometry) and
115°, corresponding to 0.402, 0.116, 0.0 and -0.135%, respectively,
for the relaxation distance along the [100] direction (positive values
indicate motion away from the bulk, toward the vacuum). For each of
these points the Vege, 3 (6x) and Vow) states were each solved self-
consistently. .
"(om) = Q14,,7406,(1)6,(2) + ¢,(1)8,(2) a(t )a(2) -8(1)a(2)]
For the Vy state a GVB(1) wavefunction was utilized (see
result, therefore predicting the wrong ground state.
zeta (DZ) basis on the bulk Si and H atoms (see Table XI of Chapter 1).
However, from studies of methylene systems, it is known that d-functions
are essential for a consistent description of (0) and (om) states. We
son between basis sets with and without d-functions is discussed
d-functions on the central Si atom of the Si git complex. (This basis
The results for the relaxation calculations are summarized in
system is the V62) state. The optimum Si-Si-Si angle is 105.4°,
corresponding to a displacement (along the [100] direction) of 0.108
toward the vacuum (with respect to the unreconstructed, unrelaxed
tetrahedral geometry). The first excited state is 3 (oa), with a verti-
cal excitation energy of 0.34 eY. The optimum bond angle for this state
is 111.5°, corresponding to a displacement of the surface atom by 0.0528
toward the bulk from the unrelaxed tetrahedral geometry. The Von)
state, as expected, is higher in energy, with a vertical excitation
energy of 1.64 eV. The optimum angle for this state is 111.7°, also
corresponding to a displacement of 0.054 toward the bulk. Geometric
relaxation effects account for a further drop of 0.16 eV in the energy
of these states, giving 0.18 and 1.48 eV for the adiabatic excitation
energies of the Son) and Vos) states, respectively.
figuration, their geometries are very similar. They both lead to larger
Optimum Relaxation Excitation Excitation
State and Optimum Energy Angle Distance® Energy Energy
V052) eve (1) -14.594092 105.43 0.104 0.0 0.0
3 (on) HE -14.587655 111.49 -0.050 0.34 0.18
Von) HE -14.539713 111.67 --0.055 1.64 1.48
to the tetrahedral (unrelaxed) geometry.
teractions (because of the Pauli principle) with the adjacent Si-Si
bonds. This leads to the surface Si moving toward the vacuum and hence
smaller angles at the central Si. These effects are more drastic for
hybridizations for these orbitals” i p , where the shape of the
3 (ox) state the two (triplet coupled) electrons occupy a m-like orbital,
by a filled circle) is bonded to two bulk Si (denoted by x) leaving two
non-bonded surface electrons. It has been suggested (by Schlier and
Farnsworth? and modified by Levine**) that these surface Si pair up by
-2.8 (i00] 3.2
102) State of the Si 3He Cluster Model of (100) Surfaces.
~2.8 CIOOJ 5.2.
case it is assumed that the optimum dimer distance should then be
very close to the optimum distance observed in the gas spectra?! of
the molecule Sis of 2.252. If single bonds are more appropriate, it
is expected”? that the dimer distance is very similar to the Si-Si
distance in the bulk, namely 2.358.
an SigHy cluster consisting of two separate SiH, units.
2) 3
states by considering the different combinations as sketched in Fig.
12. Four different singlets (S = 0), four triplets (S = 1) and one
quintet are possible. At Roa_gq = © the ground state should consist
of two Vg2y, The next state should have the form Sof Pom)
and Von) of SiH, is about 1.4 eV, one expects that two 3 (a) is
considered, namely "Eh (02) Mo) + "Ch (oqm,) (02) 1. Similarly for
reconstructed Si (100) nearest surface neighbor distance, we expect
the shapes and spin couplings of the orbitals to remain basically
positions they would have in an unreconstructed (100) surface. We
then allowed them to rotate as if the SiH bonds were Si-Si bonds,
letting the two Si atoms get closer. In Fig. 12 we show the geom-
etry for this complex for two different Si-Si distances (at the un-
reconstructed (100) surface and the bulk Si-Si distances for the two
silicons).
more drastic changes in the nature of the orbitals and the spin
couplings. The total energies also change noticeably as sketched
in Fig. 13 (for that figure we have used the calculated shapes of
the lowest singlet, lowest triplet and the quintet states; all other
curves constitute guesses). In Fig. 14 we show how the orbitals
change as a function of distance for the singlet ground state and
the lowest triplet state. We notice that for the orbitals of the singlet
describe this state as "Tn my) (o,0,)1- Similarly, the triplet
state can be well described by ~[ (of) (0 .,.)] + [ (om) (oJ
for the unreconstructed geometry. At the bulk silicon distance, that
lowering in the energy, particularly as the silicons get closer to
(b) Ros gy = 2.35R (bulk Si-Si bond length).
if (7G)
' ('G, x) (a, a3}
struction for (100) Surfaces. Small horizontal lines denote
values obtained from actual calculations. Values at infinity
are estimates based on Ref. 48b. All other features are approx-
imate. Orbitals connected by a solid line are singlet coupled;
orbitals connected by a wavy line are triplet coupled.
40, pue wane = ESTES, Ago) (4,09
(VI
oe”
X. Figure 15 shows the potential curves obtained for the
Emm) (ogo )Is Engr)" (o,0,)] and °[3(n,0,)3(1,0,)] states. The
ground state, at the unreconstructed geometry, is the £102) '(02)1.
This corresponds to two, noninteracting Vg?) states localized on each
"(o49,)1 States are very close in energy, with the singlet state being
ing of the Si-Si bonds at the second layer (i.e., the Si-H bonds of
the SigH, cluster do not account for hybridization changes). We have
accounted for the change in the hybridization at the second layer Si
atoms by doing a calculation on the HSi~Si-SiH, cluster. Here we
bend the central silicon off the plane formed by the two second layer
Si atoms and the divalent silicon (at the tetrahedral geometry) as
shown in Fig. 16. Here one expects the eneray due to the change in
hybridization increases as the bending angle departs from the tetra-
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Si- Si UISTANCELBOHR]
tion on Si (100) Surfaces. Solid lines indicate uncorrected
bending of second layer silicons.
bridization of the second layer atoms, we must add to the Si5Hy results
the potential curve obtained from the Si3H¢ bending calculations. The
energies obtained in the SiH. calculations are shown in Table XI.
hydrogens as well as the central silicon, they already include the poten-
tial barrier that we should add for both sides of the dimer (SinH,)-
When these corrections are included we obtain a bond energy of 1,74 eV
Farnsworth-Levine model is energetically feasible. Of course, these
two STH, groups to rotate as if the Si-H bonds were Si-Si bonds (as
if the second layer Si atoms were fixed at the tetrahedral unrecon-
stituted geometry). The actual calculations were performed for Si-Si
distances of 3.84, 3.09, 2.46, 2.35 and 2.254 (corresponding to bending
angles of 0.0, 15.91, 33.24 and 55.31 degrees). The basis set con-
sisted of a double zeta basis with d-functions added to each silicon
(DZd set of Table XI, Chapter 1, p. 31). We have shown in Appendix C
that this is necessary for a proper description of the o@ and
(hartrees }
15.0 -14.589742
30.0 ~14.575700
45.0 -14.549754
and the vertical plane that contains the Si atoms and the two
"surface" hydrogens for the tetrahedral geometry.
SOGVB wavefunction?” in which both pairs of "surface" electrons are
the standard closed-shell Hartree-Fock form); and where
+ 28(9)B(10)a(11)a(12) - 8(9)a(10)o(11)8(12)]}
L (1,7,,)°(0,0,)] state. The wavefunction has the form of (12), with
+ Col2a(9)o(10)8(11)a(12) - «(9)8(10)a(11)a(12) - 8(9)0(10)a(11)a(12) J
the central Si atom off the vertical plane, indicated in Fig.16 . Al]
the calculations were performed for an Si-Si-Si angle of 105°. The
bending angles (determined by the three silicon atoms and the verti-
cal plane; a = 0 for an unreconstructed surface) for which the cal-
culations were perforned were 0, 15, 20 and 45 degrees. At each of
is bent off the tetrahedral geometry. (a) No bending (tetra-
hedral geometry). (b) Bending angle equal to 45° (the angle
between the plane of the 3 silicons and the vertical plane is
45°),
much less clear than that of the (111) surface. The reason for this
is that these surfaces are cleaned usually by ion bombardment fol lowed
by annealing. © Several LEED patterns have been observed, but differ-
ences among them suggest that impurities might be of crucial importance
in obtaining such patterns.© The involvement of impurities has not
been exhaustively studied? and no conclusive results are as yet avail-
able. The most commonly known structure is a 2x2 pattern. 20794 A
4x4 structure has also been observed. °° The 2x2 and 4x4 LEED patterns
can be explained by 2x1 and 4x2 structures (unit cells with basis vec-
tors that are twice or four times as long as those of the unrecon-
structed tetrahedral geometry). The observed LEED patterns are obtained
from the 2x1 and 4x2 structures by orienting these in orthogonal direc-
tions in different domains of the surface of the crystal.
(100) surfaces. Lander and Morrison? 24 have proposed a model for the
4x2 structure in which atoms in the first two layers are drawn together
in pairs, leading to the appropriate reconstruction. The other model
is for the 2x1 reconstruction, initially proposed by Schlier and
Si surface atoms form bonds via their dangling bonds, producing double
rows. The surface atoms are constrained to move in such a way that
surface atoms, it is expected that the dimer distance is approximately
that of the bulk interatomic spacing, 2.358. This leads to 2 displace-
ment of +0. 75K in the horizontal plane (for the surface Si atoms), and
a vertical displacement of 0.238 in such a way that the surface sili-
cons are still at 2.358 from their nearest neighbors in the second
layer. This is equivalent to a rotation of +34° of the surface atoms
as measured from the [100] direction to the [010] direction. This
model leads to a unit cell with a rectangular shape with sides a/ [2
and 2a/f2, where a is the buik unit cell side, 5.41734. The atoms
are arranged in "caves" (grooves) and "pedestals" (formed by the two
rows drawn together).
in relation to our calculations on SigHy complexes. At that time we
evidence has been obtained that suggests that the 2x2 LEED pattern is
due to the 2x1 pairing (dimer) model. The experiments” consist of a
study of the chemisorption of hydrogen on Si (100) surfaces. Two dif-
ferent phases have been found, one with a 2x2 LEED pattern, the other
with a 1x1 LEED pattern. Each phase can be obtained from the other one
by controlling the temperature and the time of exposure to atomic hydro-
gen. This suggests that the reconstruction is produced by motion of
hydrogen atoms can bind to each of these dangling bonds still pro-
with the result that two hydrogens bind to one surface silicon. This
diamond structure from above, the surface atoms appear arranged in
bands with a rectangular unit cell. Each band consists of atoms in
a zig-zag fashion, lying on the (110) plane and with tetrahedral bond
angles of 109°28' (see Fig.17.). The nearest neighbor distance for
the atoms within one of these zig-zag bands is 2.358 (for the unrecon-
structed geometry). The length of the short side of the rectangular
unit cell is 3.83A, while the distance between two adjacent zig-zag
bands is equal to the side of the cubic unit cell of the bulk, namely
5. 41h. Each surface atom is trivalent, i.e., it is bound to three
nearest neighbors , two of which are surface atoms in the same zig-zag
geometry this orbital makes an angle of about 36° with the normal
to the plane of the surface.
be cleaved, and therefore has to be prepared by other methods, usually
cutting and cleaning with ion bombardment followed by annealing at
high temperature. This means that considerable rearrangement of the
surface atoms is possible, so that the surface attains its most stable
structure before any kind of experiment can be performed on it. This
Ist LAYER
/-
& xXx € X @
2 xX = xX e
te X @ x €
& xX «eX &
& xX & x @
@ xX OX
Ss xXx € X @ |
noted by filled circles, second layer atoms are denoted by
teraction between dangling bond orbitals on adjacent atoms. In par-
ticular we wanted to determine the basic spin states when a number of
Si atoms are present in the zig-zag configuration. Since the orbitals
initially point towards opposite sides of the zig-zag band and at an
angle of approximately 36° (for the tetrahedral unrelaxed geometry)
from the normal to the surface, triplet and singlet pairing of these
orbitals are nearly equivalent.
SigHe- From these calculations the most important conclusion is that
for the undistorted geometry the ground state has high spin coupling
(all dangling bonds having the same spin). Of course distortions
jacent Si dangling bond orbitals we considered the SigHy model shown
Ss H
ar
and both Hy replace bulk Si. There are two important electronic
states *(0405) and "(049). depending upon the coupling of the dangI-
ing bond orbitals.
the ground state, with the singlet state 0.1 eV higher. This is a
small excitation energy, one that might be removed upon relaxation
of the surface Si atoms. In order to examine the effect of such
distortions we allowed the hydrogens to rotate about the Si-Si bond,
so that the two pairs of hydrogens end up in front of each other
(eclipsed geometry) as shown in Fig.18. In this case the two dangl-
ing bonds are almost parallel to each other. (This would be the op-
timum configuration for pairing). For this system we also expect two
low lying states: a singlet, "(o459) and a triplet, *(a405). For
the real reconstructed Si (110) surface one would not expect the
eclipsed geometry to be a very strong possibility, since this would
introduce strains in the lattice, raising the energy. An intermediate
geometry is more likely in which the Si atoms would rotate slightly
so that the dangling bonds point in directions that are closer to
being parallel than the directions in which they point at the unre-
constructed tetrahedral geometry.
the tetrahedral. geometry the ground state is the triplet. The
singlet state is up by 0.10 eV. For the eclipsed geometry the ground
(a) Unreconstructed geometry. (b) Eclipsed geometry.
(110) Surface. (a) Si3H,. (b) Sige.
Geometry
Dangling Bands Point- . Dangling Bands Point-
State and ing in Opposite ing in Parallel
Wavefunction Directions (tetrahedral) Directions (eclipsed)
"(o,05)6VB(1) ~9659599 -9. 668258
; (0.23) (0.0)
(0,05) HF -9.663312 -9.661324
(0.13) (0.19)
Values in parentheses are the energies in eV measured with respect
triplet *(6495) of the tetrahedral geometry by 0.13 eV. This means
that singlet coupling of adjacent dangling bonds on the Si (110)
surface is at least possible for a non-tetrahedral geometry. This
can be a cause for some of the rearrangement that the surface suf-
fers.
and in order to study the effects of coupling more than two together,
we considered complexes with three and four silicons.
*[Mo405)03] S= 1/2 toy 4 0.01
Paanat indicates triplet pairing. Note that two different doub-
let states can be obtained from three electron systems, which can
to a high spin ground state.
For four Si (Si,He complex shown in Fig. 19) the low lying
H A A
Energy Average
V H
H H
5a, oyry)] 2 famt Taal 0.0 0.0
3. 'aj0,)3(o,04)] s=1 ! Tanna! $ 0.05
3M o,0,)'(o04)] $=) Teh} ot 0.14 0.13
3(3(o,0,)"o,9,)] $=} fet tut 0.21
‘eyo, Noxoy)] $= 0 y f ff 0.10
'Eo,04)"to,03)] s= 0 t tii 4 0.10
quintet (S = 2), three triplet (S = 1) and two singlet. (S = 0)
states. Again, the ground state is the high spin state, but the
excitation energies are smaller than for the 3 and 2 silicon cases.
These results are also shown in Table XIII.
are important for all states that have at least one pair of dangling
bond orbitals coupled into a singlet. In this case a closed-shell
Hartree-Fock wavefunction would put both electrons in the same orbital
Spread over both centers which, even for adjacent centers, leads to
(o0,)].
GVB level for the SigHy complex. The two geometries chosen are shown
in Fig. 18. The double zeta (DZ) basis of Table XI of Chapter I was
$0405)! Mange = [4 (1)49(2) ~ 05(1)4, (2) Ilalt)9(2) + 8(1)a(2)) «
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19 498 UOLIOUNLAARN, 49S 19 498 pUOLqounsaney 49S
pue a1eis pue a1e45
ABuauz Abuauz .
FHV ts SHE 1g
“sadesuns 1S (OLL) 40 Lapoy uszsnig [Hig pue “Hers oug 07 Burqelay solGuauz “ITTY aLqes
%(o494)1 for Sigg) augmented by the dangling bond space of the lowest
for Si,H,).
[i loyoy og]: Ong = [04 (1)b5(2) + 65(1) 64 (2) Ha )8(2)
- B(2)a(1)] 64(3)a(3)
For the Su cluster the GVB level wavefunction has the form
(13), with Sure having the form shown below for the different states:
yl
1.
in which the spin coupling is optimized concurrently with the self-
where cf + c =].
general statement that could be made was that the 5x1 structure was
likely to appear at high temperatures (T> 1000°C). The high tempera-
tures used in the cleaning and annealing process introduce the possi-
bility of impurities at the surface. These can possibly come® from
the bulk by diffusion or from the ambient. Sakurai and Hagstrum= have
recently determined by hydrogen chemisorption, that the 5x1 recon-
struction is likely to be due to relaxation or slight motion of the
surface Si atoms, rather than due to surface vacancies. (They con-
cluded this from the observation that when the 5x1 structure is
exposed to H atoms at 350°C the pattern is replaced by a 1x] pattern).
The experimental situation for this surface, other than what
was mentioned above, is in a relatively poor status, as compared with
the (111) surface and even with the (100) surface. Due to the experi -
REFERENCES FOR CHAPTER 2
Phys. 25, 1 (1970).
57, 738 (1972); (b) W. A. Goddard III, T. H. Dunning, Jr., W. J.
Hunt and P. J. Hay, Accts. Chem. Res. 6, 368 (1973); see also
Appendix A.
M. E. Straumanis and E. Z. Aka, J. Appl. Phys. 23, 330 (1952).
p. 1, and references cited therein.
Many excellent reviews on LEED are available, we only list a few;
(c) J. A. Strotzier, Jr., D. W. Jepsen and F. Jona, in Surface
1975 (unpublished).
(1962).
13.
bital.
(b) See, for example, L. D. Landau and E. M. Lifshitz, Electrodyn-
see also Ref. 9, p. 169.
Notes for Chem, 120, California Institute of Technology, 1975 (un-
published).
tonian, including the kinetic and potential energy terms. Jor is
the Coulomb energy integral, given by
= ea. i
Jon = <5 (TO (2) FI Og (1) 6,2 )>
Kon =
2.948 eV, while for the relaxed geometry it is 2.978 eV.
23.
24.
26.
28,
29.
ed. (Academic Press, New York, 1975), Vol. 1, p. 1, and references
cited therein.
(a) F. Bauerle, W. Monch and M. Henzler, J. Appl. Phys. 43, 3917
(1972); (b) M. Erbudak and T. E. Fischer, Phys. Rev. Lett. 29, 732
(1972).
J. J. Scheer and J. van Laar, Solid State Comm. 3, 189 (1965).
R. C. Eden, Proc. 10th Int. Conf. on Semiconductors, Cambridge, Mass.
p. 22 (1970), quoted in Ref. 6.
(a) J. E. Rowe, Phys. Lett. A46, 400 (1974); (b) J. E. Rowe and
H. Ibach, Phys. Rev. Lett. 32, 421 (1974).
B. F. Lewis and T. E. Fischer, Surf. Sci. 40, 371 (1974).
H. D. Hagstrum and G. E. Becker, Phys. Rev. B 8, 1580, 1592 (1973).
(a) G. Chiarotti, G. Del Signore and S. Nannarone, Phys. Rev. B 4,
3398 (1971); (b) G. Chiarotti, P. Chiaradia and S. Nannarone, Surf.
Sci. 49, 315 (1975); (c) G. Chiarotti, G. Del Signore and S.
Nannarone, Phys. Rev. Lett, 21, 1170 (1968); (d) P. Chiaradia and
S. Nannarone, Surf. Sci. 54, 547 (1976).
W. Muller and W. Monch, Phys. Rev. Lett. 27, 250 (1971).
F. Meyer, Phys. Rev. B 9, 3622 (1974); see also Ref, 17a, where
further references are quoted.
J. E. Rowe and H. Ibach, Phys. Rev. Lett. 31, 102 (1973).
D. Haneman, Phys. Rev. 121, 1093 (1961); see also Ref. 6.
(a) D. Haneman, Phys. Rev. 170, 705 (1968); (b) M. F. Chung and
D. Haneman, J. Appl. Phys. 37, 1879 (1966); (c) D. Haneman, Jap.
33.
35.
Lett. 35, 1376 (1975); (b) B. P. Lemke and D. Haneman, ibid 35,
1379 (1975); (c) D. Haneman, talk presented at the American
Chemical Society Centennial Meeting, San Francisco, California,
August 30, 1976.
(a) I. Tamm, Phys. Z. Sowj. Un. 1, 733 (1932); (b) W. Shockley,
Phys. Rev. 56, 317 (1939).
Slater and K. H. Johnson, Phys. Rev. B 5, 844 (1972); (c) K. H.
Johnson and F. C. Smith, Jr., Phys. Rev. B 5, 831 (1972);
(1974); (b) V. Bortolani, C. Calandra and M. J. Kelly, J. Phys. C
6, L349 (1973).
Batra and W. A. Tiller, Phys. Rev. B 12, 5811 (1975); (c) S.
Ciraci and I. P. Batra, Solid State Comm, 18, 1149 (1976).
and D. R. Hamann, Phys. Rev. Lett. 31, 106 (1973); (c) R. 0. Jones
42.
43.
44,
(1972); (b) idem, Phys. Rev. Lett. 32, 225 (1974); (c) idem,
Phys. Rev. B 12, 1410 (1975); (d) idem, Rev. Mod. Phys. 48, 479
(1976).
Phys. Rev. Lett. 34, 1385 (1975); (b) idem, Phys. Rev. B 12,
in the literature (see Refs. 2 and 3, for example). For infinite
or semi-infinite systems there are few examples in which proper
account of electron correlation has been taken. One such system
is the jellium model of electrons in a metal (a free electron
form background of positive charge, adjusted so that the system
has zero net charge). Wigner has shown (Ref. 41) that the effect
of correlation can have a magnitude of several eV in this system.
This is large enough to considerably affect the resulting theore-
tical electronic structures; this is particularly acute in the
case of wavefunctions using double occupancy of the orbitals [see
for example, R. P. Messmer and D, R. Salahub, J. Chem, Phys. 65,
779 (1976)].
56.
symmetric, respectively, relative to the Si-Si-Si plane.
4262 (1967); (b) B. Wirsam, Chem. Phys. Lett. 14, 214 (1972);
Chemistry I) 50, 1 (1974).
not renormalize and hence the sum of the s* p¥d7 exponent (xtytz)
indicates the percentage character on the surface atom.
R. D. Verma and P. A. Warsop, Can. J. Phys. 41, 152 (1963).
Surf. Sci. 35, 227 (1973).
(1963); (b) R. E. Weber and W. T. Peria, Surf. Sci. 14, 13 (1969).
LEED calculations for this model do not seem to agree with the
experimental spectra; M. van Hove, private communication.
of these effects.
1973 (unpublished).
58. G. 0. Krause, Phys. Stat. Sol. 35, K59 (1969).
(1976).
ELECTRONIC STRUCTURE OF STEPS ON (111) SILICON SURFACES
the presence of steps. Such steps lead to significant modifications
of the electronic and mechanical properties and the chemical reac-
tivity!, as compared with the behavior expected of an ideal cleaved
surface. Unfortunately, due to significant experimental and theor-
etical difficulties, little is known about the microscopic structure
at steps. Experimentally, major difficulties are reproducibility in
forming the steps and in separating the properties due to the steps
from those of the perfect surface and the bulk. Major theoretical
problems are: (i) lack of symmetry (even lower than that of a perfect
surface) and (ii) the importance of electron correlation effects in
the resulting localized electronic states. As in the calculations
described in the previous chapter, our approach has been to include
the electronic correlation explicitly but to replace the semi-infinite
solid with a finite cluster of atoms.
is the (111) surface. For (111) surfaces two types of steps are pos-
sible: those in which the edges consist of atoms with three nearest
neighbors (trivalent) and those whose edges consist of atoms with two
nearest neighbors (divalent). The edges toward the three [112] direc-
tions consist of trivalent atoms, whereas the edges toward the [112]
determined that, upon cleaving, only one of these types is preferred.
This type has been found to have edges toward the [112] direction.
at the edges are observed (this is shown in Fig. 1). For this reason
we have concentrated our efforts on calculations that model steps with
divalent silicon atoms.
performed on Sigh, and SiH, clusters. The Si3H5 cluster models the
basic divalent step in which the edge atom is divalent (central atom
of the complex). It is bound to a "bulk" atom and to a trivalent,
upper terrace atom with a dangling bond. This complex can then be de-
noted by H)Si-Si-SiH3, where Hpi represents the upper terrace atoms
(the hydrogens substitute other stlicons as in previous chapters),
and ~STH, represents the "bulk" atoms on the lower terrace (again hy-
drogens replace silicons as before). This complex is indicated in Fia.
1 by a cross hatching of the appropriate atoms. By binding other
silicon atoms between two adjacent divalent edge atoms in a step we
can also produce "kinks" on the steps. These are modelled by
HpSi-Si-SiHly clusters as shown in Fig. 2. A similar structure to that
of the (100) surface can be constructed on the edges of the divalent
steps. These are shown in Fig. 3. We will use the results on Si3H¢
clusters discussed in Chapter 2 to describe the electronic structure
of such atomic configurations. Finally, we will briefly consider the
on Si (111) Surfaces. Shaded atoms are those represented in
Surfaces. Shaded atoms are those represented in the SigHe
face. (The steps with trivalent edge atoms have the same configura-
or any of the ombinations obtained by binding new Si atoms bridging
two divalent step atoms have an electronic structure dominated by
the divalent atom. As discussed in Chapter 2, a divalent Si on (100)
surfaces (the same basic configuration as in the present case) leads
to two low lying electronic configurations (see Fig. 4a): a V2)
ground state (optimum bond angle of 105.4°) and a 3 (om) excited state
with 0.34 eV vertical excitation eneray and 0.18 eV adiabatic excita-
tion energy (optimum bond angle of 111.5°).
si si (0-2) 0.0 ev
[Mam (a)] 0.23 eV
(111) Surfaces. (a) Basic divalent configuration, SiH
cluster. (b) Divalent step, Si He cluster. A circle repre-
sents a p-like orbital coming out of the paper; a lobe
represents ao orbital in the plane of the paper; dots rep-
resent the number of electrons for each orbital. Energies are
not drawn to scale.
4b. Three low lying states result:
we couple the o dangling bond orbital of the upper terrace surface Si
in two different ways, leading to quartet (S = 3/2) and doublet
(S = 1/2) states, denoted 43 (on) (SB) and 273 (on) (S)], respectively.
If the overlap between the o and o orbitals is small, the quartet is
ing to the Von) states of the divalent silicon. The new dangling
bond, o, necessarily couples as a doublet with the Von) electrons
leading to an overall doublet spin coupling (S = 1/2). This state is
to the dangling bond orbital, o, of the surface with an overall doublet
tion (CI) calculations are shown in Tables I and II. In Fig. 5 we
plotted the o, and o orbitals of the "03 (on) (S)] state. We see that
the ground state of the system is the 202) (5)] (which is to be ex-
pected since 1042) is the ground state of the divalent Si), For the
3 (sn) (B) states the overlap of the o and o orbitals on the step and
surface atoms is quite small (
a 9.0
[iz] eo OT
: i ‘ c “os ‘, ‘ ‘
, tt
step Si OZ 50
surface Si “5.0 Toole 50
0 o J \ &
o / \
/ | 4
> Ao“
[172] an,
QL mos
-2.0 [0.0
cate positive amplitude values, snort dashes indicate negative
Atoms are denoted by an asterisk.
owes 9yz2 7e SNLeA (L)GAQ ButLpuodseruod sayz uey} uayBLy Ad y9°L SE ,8Z.60L 22 82e3S SLY} UOJ ABUBUa 4H aul,
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UOLZEPLOXZ UOLzeyLOXZ aa Bbuy uinwizdg
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UOLFERLOXZ «= UL }eRLOXQ = Huy = wnwLadg
2LyeqGe Lpy Leotquan = wnwiL3do SaLbBuauq LeyOL
*payzeqs AaSLMaYyZO SSaLUN SaduqueY UL due SBLHuaus [|y ‘*‘SUOLZe{Nd[e9
1D GAD *sdays YUaLeALq 4O Lepow uaqsnl9 Eu ES-IS-1S H ay} 02 Bulqelsy sarziquend “IT alge
203 (on) (5)] state is only 0.15 eV higher than the *(3( on) (S)Jand thus
its vertical excitation energy is 0.38 eV with 0.26 eV adiabatic exci-
tation energy. Finally, 2 lon) (5), as expected, is higher in
energy with 1.93 eV vertical excitation energy and 1.73 eV adiabatic
excitation energy.
states are very close, 112.1° and 111.8° respectively, corresponding
to the step Si moving 0.098 toward the bulk position. The hybridiza-
tions of the o and @ orbitals are 50+ 16,0.74,-0.01 and 59+ 08,0.85,0.03
respectively; the m orbital has pure p-character. For the ground
state the presence of the less rigid surface atom allows the divalent
Si to distert so as to decrease its bond angle to 102.7, while allow-
ing the surface atom to attain its preferred, less-pyramidal configura-
tion (the step Si moves 0.254 away from the bulk position).
gests that this step site is particularly reactive for a range of
molecules with an appropriate acceptor site), while the empty 7 orbital
can act as an acceptor (for molecules with a donor pair). On the other
hand, the presence of the two low lying 3 (on) (3) states provides two
radical orbitals for effective coupling with molecules having radical
sites (singly-occupied orbitals). Having both of and on states within
containing divalent silicons; an example is SiH, which leads to a
are of minor importance for *13 (on) (S)]. For the 273 (on) (5) J state,
correlation between the o and o orbitals is very important since
these orbitals are localized on different centers. From Table I
(footnote c) we see that in this case the Hartree-Fock (HF) wavefunc-
(1)
leads to an energy 1.64 eV higher than the Generalized Valence Bond“
2P(on) (Joys = AUP y lol M45(2) + 41), (2)]
x [o(1)8(2) - 8(1)a(2) I, (3)0(3)} (2)
pairs.
“(07 yp [ALM ype (1), (2) 01 )8(2) - 8(1)a(2)T4(3)0(3)},
(3)
2T(o2)(FJoyn = AL 4 C45 (1). (2) + oy (1645 (2)]
GVB bulk*‘to+At O-ATr ; O-AT Oo+AT
only, we would have found the ground state to be *13 (on) (5)] with
the 2 ty] state at 10.32 eV and. the 213 (on) (5)] state at
v2.16 eV, giving the wrong gfound state.
tion there is almost no lowering of the total energy for the quartet
state. This is also true for the 20 (02) (8)] state. The reason for
this is that the basis space used for the CI calculation involved
valence orbitals of the corresponding state only (with no virtuals
added), thus the basic correlations are already included at the GVR
level. The largest change occurs for the 213 (on) (5)] state, where
solution with spin coupling (6) is solved. The potential curves for
tion of the of and om states it is necessary to include d-functions.
We have done so for the step and surface Si atoms (i.e., these atoms
have the DZd basis of Table XI of Chapter 1, p. 31).. For the "bulk"
Si the DZ basis of Table XI, Chapter 1 was used; for the hydrogens a
double zeta contraction of the three gaussian hydrogen bases of
Huzinaga” was used.
the 463 (on) (S)] and 20h 2) (5)] states. In both cases a total of 10
valence orbitals was used, allowing double excitations out of the 10
Orbitals, with the restriction that all initially doubly-occupied
*[(o7\(F) ]
ANGLEC DEGREES )
Cluster
S135 ve meee eepneeeett
-9.96 — “LU ore\ a)
-0.28 — _
“(on )(T\}
4.00— “|
~~ eee
“me Alea)
GVB-Cl
1.01 —
[ { | | |
95.0 . 100.0 105.0 - 110.0 115.0 120.0
ANGLEC DEGREES 3
from 44 to 142 determinants. We have compared this (at the tetra-
hedral geometry only) with the same calculation doing all double ex-
citations with no restrictions, with the result that all corres pond-
ing energies agreed within 0.03 eV. Thus, we believe that the
restriction imposed on the selection of configurations leads to a
good description of states within the valence orbital space (no vir-
tion is obtained by binding a silicon atom to two adjacent Si atoms
of the row of divalent Si's. The geometry for this system is shown
in Fig. 2. Note that completing a whole row in this way leads to the
configuration observed in a (100) surface and described in Fig. 3.
trivalent silicons, leading to an H)Si-Si-Si, complex. One can think
of this system as describing kinks on the step edge.
(/ Wg *(o\ag)
A =
= _
divalent Si we can construct several states. according to how we
couple the other dangling bonds. Coupling Go and O. into a singlet
xX
a singlet, an overall singlet (S = 0) state results, denoted
will be determined by the overlap between the two dangling bond or-
energy orderings. For example, knowing that the o@ state is lower
in energy than the on states for the divalent Si, one estimates that
the TW g2y lio
L055) (1t0,.): Finally, we consider a state in which one 07
the dangling bonds (o, or 0.) couples into a singlet with o of the
a triplet with 7. The overall spin state is a triplet (S = 1),
denoted >t" (004) %(n0,) J. (The true wavefunction of this state will
The results for self-consistent field and CI calculations are
the appropriate orbitals for *CF(on)3(o,0,,)] and "Eh(o*)!(ogo,)1,
Og O% op °
u g~ Or
optimum angle of 61.0°. Here the three Si atoms are almost in an
equilateral triangle position. This is due to the fact that ory and
oy, can form a relatively strong bond as the two trivalent Si atoms
GL'Z .S2*vOl B99LEC'EL~ ZiPLZe"El- LELOEE*EL- Osglectel- gLzeze"el- = “- - - > ((4e%o), (ue) 3,
aAsos
bE’ —ODOLL. © PLPLOCEL- GLPO9E*EL- ESEL9E'El- EOKESEEL- - - - - - Jon) (bor
[(*cu), (Poe) J,
~ 4H
es OL"L —- LOTLLL. «© 9OSOLETEL~ POLESE*EL- OLEOLE'EL- OZSBdErEL- SEL9SE'EI- - - - - t(to%), (40),9
te leds
' , 5(i)aaa
SO"L 12°COL =: LEOZZETEi- 29109E'EL- HOELIE'EL- LBLOLE"EL- SBoze'el- - - - - (toto) (69)
gta rie
(1) aa9
O8'O © vL*Z0L ««GLLGE*EL- 9LLSLE*EL- 9z96LE"EL- 9zO1BE"CL- zOvGLE'EL- - - - - (40%). (uo). J
, fee
. q(2) a9
OO 19°09 LSLOLy"EL- SBZOLE*EL- EG90BE"EL- JLLBE"EL- OOO96E*EL- OLLGGE'EL- GS9LO¥"EL- OLE6OEL- 99BLOP“EL- [(45%) (_c)}
rages
(,e)AB18u3 (5) kBau3 eit 1820601 oS0L 056 068 oS 099 SS
UO;WeR19xQ albuy dinury3dg
3}32q2 yy wnd;3do SayO4aug Lezon
Sy} UP2zUOI Jou Op UOLZe{NILed [9 GAN ayy 4OJF pasn Sareds [eILquo ayy ashedaq anleA 49S Gutpuodsaiiod ayy UeY, WayBLY Jue saibsaua asa:
oeey Z2°GLL EePeSEel~ GivdSEEL- OOPSSE’EL- G2ecSEEl- 666LbE"EL- - - - -
is Sev PS°OLL PLOCSE EL GELLSESL~ ([e929E"El- OEZIGEEL- - ; - - 7 -
re .
t .
pl’ 6B°COL EBSOLE"EL- GISSSEEL- ZLSOSE°EI- OESS9E°EL- 62999 °EL~ - - - -
[("o), (200) ie
qh CO), 00 i,
[(70%), (40), 1,
[(“o%) (40), ],
UO,PRRLSXZ aL buy unu,3dg
2h32GR py whwy3do $a, G6u4augqZ peyryoy
or Or
PAIRED
C, 5-0 D, 5.0
N oy A
~~ ao
(10d) pono nns CO) ee
YO ree a Woe
-5.0 \ = x . “5.0
-7.0 [ot 7} 7.0 -5.0 Lice] 5.0
6.0 C : §.0
© | |
bree] ae (fen) Ee]
Or. - 10 ee
| —
aan four) 7.0 rar Lieo 5.0
(b) O, Orbital; (c) o orbital; (d) 7 orbital.
Ud
fe] fe) .. oy
Fige 11. Orbitals for the [( a 2%) J state of Si3zil,. (a) Je
orbital, (b) c. orbital, (¢) o+Ar orbital, (d)a~ Arorbital.
<< oe
SIiG— CP 91,
\ ‘
bonds by buckling out. All the other states have geometries with
Si-Si-Si angles more in accord with those of the Sigh, complex. For
this reason we only quote adiabatic excitation energies. The first
3,3
[~( )]
(The vertical excitation energy of the quintet is larger). We then
‘find the?["(o0,)(10,.)] state at 1.35 eV. The "E (oo)! (10) ] state
is only 0.12 eV higher at 1.47 eV adiabatic excitation energy.
| The optimum Si-Si-Si angle for the ground state, as discussed
above, is 67.0°. For the *P (on) (o,0,)] state there are two compet-
ing tendencies: the (on) configuration at the divalent silicon favors
large central angles, the "(o,0,) spin coupling favors small angles
(by trying to form a bond as in the ground state). The resulting op-
timum angle is 100.3°. The 7010? )(o,0,)] is dominated by the V2)
configuration, leading to an optimum angle of 100.9°. The 513 (on)
3(6,0,)] state is dominated by the 3 (sn) configuration, leading to an
optimum angle of 111.1°. The optimum angles for the 3. (09, 30)
and "No0,)! (no) states are very similar at 110.5° and 110.2°.
are of minor importance in the *[3(on)§(a,0,.)] state, where the
For the WV igty lig 0) J state correlation is important for both pairs
g g
g Yr
an energy 2.03 eV higher than that of the corresponding GVB wavefunc-
r r
30° (ors TS, Co .
Ne.
3 \ a
0. | PLR (on) (o,0,.)] ‘. Ny. ME (on) (ono,)]
~ ee NN XN .
~~ — “ON,
Oo ie ~
~ An ‘ ss
~~ 0.55 — . a se a S yee. Ley
wW ss “
Fs 3 3, ( )] ee Ne a |
re, [~(on) (ago, ~ ain
= -o.33— 3 Te
= 8 13 Moog) (a, )d ~--5
oO
io
to
tas
— O.4O— y
-0.u2 — - 5.0 aN
I i | { I t I
$s.0 65.0 75.0 85.0 95.0 165.0 115.0
ANGLEC DEGREES 2
Pig. 12.
. NS
AH — 3-1] \
(' (oo) 3(n0,.)] \ Ss
NN ™“~
. \ IN m. tS 4
~0.35 — . sy ou!) _—
. N\ se ~~. 2%
Se aR. . oa
lig,0.)]
35.0 65.0 75.6 85.0 95.0 103.9 115.0
figs 13. Potential surfaces for the GVB-CI calculations on tie Si3Hj,
cluster.
r r
duced by considering a wavefunction in which the spin coupling is
optimized concurrently with the spatial orbitals. We have chosen a strong-
ly orthogonal Generalized Valence Bond? (SOGVB) wavefunction of the
E (004) (70,.) Isoeve Sour K tol 8g (2) op) (AAC (1-4)
(13)
where *o ~\ 7 5 and by ~\ b5 ~ by are symmetry orbitals. Here
the spin function xX is defined by
Si_atoms
were part of a real crystal (i.e., when the semi-infinite solid is
included), one expects that the bending of the trivalent silicons at
the ends of the Hp Si-Si-SiH, complex would increase the total energy
of the system because of changes in the hybridization. In the present
case we have substituted Si-Si bonds by Si-H bonds; thus this change
is not taken into account. We can include this effect by adding to
the energies of Table IV the results of the Si3H¢ calculations of
Section II-B of Chapter 2, where we calculated the effect of the bend-
ing of the central Si off the plane formed by the three atoms (see
Fig.16 of Chapter 2, p.111).
We notice that except for a change in the optimum angle of the ground
state no significant changes are produced by this correction. (The
Appendix C, we have used the O2Zd basis set of Table “I, Chanter |
55.0 65.0 75.0
25.0 105.0 115.0
cluster corrected for the hybridization changes Gue to the
bending on "bulk atoms".
of Divalent Steps when Bending Corrections for Si 3H. Are
Taken into Account.®
Adiabatic Excitation Optimum Angle
State Energy (eV) °
3-723, 4
[(o*) (o,9,)] 0.73 101.63
*C1(c0,)3(10,,)] 0.94 110.46
*E°(on)3(o40,)] 0.72 110.94
Tr] ]
[ (00, ) (10) J 1.05 110.12
bTotal energy is -13.397155 hartree.
divalent atom.
lowed to rotate so that they would allow the system to reach its
minimum in energy. The Si-H bonds rotated as if they were Si-Si
bonds with the "bulk" (virtual) Si atoms kept fixed.
"E(o?) opo,)1, F£3(on)!(o,0,)1 and °[3(on)3(o,0,)] states. In all
three cases, the total valence space was used (10 orbitals). AT1
double excitations were allowed between the 10 valence orbitals
least one edge. This can be seen from the fact that although most
of the observed steps have edges toward the [112] directions, since
the actual crystals used are finite in size, some edge must point
toward the [112] direction. This means that toward those directions
trivalent edge atoms exist (or atoms in configurations that result
from the rearrangement of such trivalent steps). It is, therefore,
useful to consider the possible electronic states arising from such
configurations.
bond. We can use the results on our calculations on the SigHys 13H,
and SigHe clusters of Chapter 2, modelling the (110) surface to draw
tetrahedral geometry, the overlap between adjacent dangling bonds
(about 0.07) is too small to allow singlet pairing as compared to the
high spin coupling states. Thus, for this geometry one expects the
high spin states to be the lower energy states. However, as we have
seen for the SioH, calculations, small rotations might produce a ground
state in which some or all of the adjacent pairs of dangling bonds are
paired up into singlets. This would produce a relaxation of the
tetrahedral geometry leading to a new ground state at a relaxed geom-
mental difficulties exist that preclude the complete and unique deter-
mination of effects due to steps. However, some properties can be
of steps on silicon and germanium. The presence of steps in the sur-
face, even in small concentrations, can lead to considerable effects
in the electronic and chemical properties of the surface.” Ibach et
either the of or om states, the presence of the o radical orbital
on the surface Si adjacent to the step Si provides even greater
flexibility for scission of molecular bonds. Comprehensive studies
of the increased reactivity of Pt due to steps have been proviced by
that the preferred steps are those that have divalent edge atoms.
teristic of steps have a different nature from those of the clean
surface dangling bonds. They made a study in which Ultraviolet
Photoelectron Spectroscopy (UPS) was used, together with Low Energy
creased. Their main conclusions are that the Fermi level shifted by
0.25 eV toward the top of the valence band for a high density of
steps surface. The one~half-order LEED features of the 2x1 structure
were not present on the highest step-density surfaces [» (16+3)%].
These spots were observed only for step densities of less than
(12+2)%. This means that the 2x1 reconstruction occurs only on ter-
races which have widths larger than 40-508. Similar results have been
slab geometry in which an infinite number of atoms are oresent in an
states present at the step configuration. We believe that such
(1972). |
not renormalize and thus the sum of the s*p¥d* exponents (xty+z)
represents the percentage character on the appropriate atom.
Chemistry I), 1 (1974).
Hay, Accts. Chem. Res. 6, 368 (1973); (b) W. J. Hunt, P. J. Hay
and W. A. Goddard III, J. Chem. Phys. 57, 738 (1972).
1973 (unpublished).
(a) H. Ibach, K. Horn, R. Dorn and H. Luth, Surf. Sci. 38, 433 (1973);
(vb) J. E. Rowe, S.8. Christian and H. Ibach, Phys. Mev. Lotr. ag, 27s
(1974); (b) M. A. Chester and G. A. Somorjai, Surf. Sci. 52, 21
(1975).
9, 2533 (1976); (b) M. Schltiter, K. M. Ho and M. L. Cohen, Phys.
Rev. B 14, 550 (1976); see also J. A. Appelbaum, G. A. Baraff and
D. R. Hamann, Phys. Rev. B li, 3822 (1976). |
OXYGEN CHEMISORPTION ONTO SILICON (111) SURFACES
which oxygen is chemisorbed onto the (111) surface of silicon.
oxygen, the chemisorption process occurs in two phases!: First there
is rapid adsorption until a saturation level is reached (at 10°*-1072
torr-sec oxygen exposures); this is followed by a slow formation of
an oxide layer. We will consider here only the early stages of the
first phase. We conclude that in the initial chemisorption of 05 on
(111) silicon surfaces the 0, is bound as a peroxy radical, in which
the oxygen molecule binds to only one dangling bond on the surface.
This model is consistent with all current experimental data! > 19-23-32
(much of which was obtained subsequent to the proposal of the peroxy
the core electrons of all the silicon atoms have been replaced by the
effective potential (EP) of Chapter 1. Hydrogen atoms have been
utilized in a manner similar to that of Chapters 1 to 3, with a Si-H
bond length of 1.48f (from silane’). The Si-Si bond length is taken
The motivation for these studies arises from the considerable
have been postulated: in one the oxygen fissions as it chemisorbs; in
the other the 0, stays basically intact and bridges between two surface
Si. By performing calculations with both single oxygen atoms and 0,
molecules chemisorbed on the surface and comparing them with the ex-
perimental data, we expect to shed some light on the actual mechanism
of chemisorption. This, of course, is a problem of considerable tech-
nical interest. Also, an understanding of chemisorption in silicon
can be of considerable value in the understanding of oxidation of
other systems.
atoms on the clean (111) surface of silicon. This surface has been
described in Chapter 2, where we found that its basic electronic struc-
ture consists of one dangling bond electron per surface Si.
(1s)*(2s)*(2p)*. From various studies of oxygen containing mole-
VR
COOK) aM
()
“2p
of chemisorbed oxygen atom. Is and 2s orbitals have been ig-
nered and p orbitals parallel and perpendicular to the plane
of the paper are indicated by » and 0, respectively. Dots
indicate how many electrons are in each orbital.
occupied orbital (2p, in Fig. la, denoted by two dots), and two singly-
occupied orbitals (2p, and 2p, in Fig. Ta).
the (111) surface, as in Fig. Ib. A strong Si-O bond is formed by
pairing one of the oxygen singly-occupied orbitals, say 2p, > with the
Singly-occupied dangling bond of the surface. This leaves one singly-
occupied orbital (2p) and a doubly-occupied orbital (2p,) on the
The state
2 1 (1b)
(py) (P,)
0 atom is a doubly degenerate doublet state. Classifying the states
in terms of the Si-0 axis? the ground state becomes en. Low lying ex-
band 0
Si-O + 0
j Py
(si0)* (op, )?
way 2 1
3, Iq (Si0)” ( Py) (Op) (5)
axis only. (SiO) represents the orbital localized mainly in the
region of the silicon-oxygen bond, (Op) and (Op) represent p-like
Orbitals.
one Si atom, O-SiH3, and the other having four silicons, O-Si(SiH3) 3,
with the aaditional silicons corresponding to second layer atoms.
to solve for the (py) (py)? ground state. The optimum bond length for
the Si-O bond is 1.63A which compares well to the values 1.638 to 1.642
found experimentally in systems with Si0 single bonds. '9
culation for the neutral OSiH, and 0-Si(SiH3), complexes. Note that
2]
1.528 -80. 355637
1.617 ~80. 364242.
1.634 -80. 364254
1.740 ~80. 359823
Minimum?
1.626 -80. 364295
bobtained from a cubic splines fit to the calculated points.
(S4itg),0, Reig = 1.64% from (SiC1)205 Reso 7 1.638 from (CH)
Si-0-Si(CH5)4 and Rein = 1.638 from Si[0Si(CH) 4], (see Ref. 10).
The calculated force constant for O-SiH, is 0.362 atomic units,
corresponding to a vibrational energy of 0.096 eV.
LL “48%,
0°0 - 02°0 LL‘O. = 9LGnop—saZziaLss fazo « “dzo
0'0 0°0 20°0 5070 «= waignop——s| 2 93815 punoug
hy %q
,lequauiedxy ypazetnoie9 "(Fyis)is-o — FHs-0 uLds 0 “o o1s SUO 112 L9X3
woe O woxe Q
UOLPRUNG LJUO0D
"AD UL Bue SAaLbuaus LLY
p saarsnig ©(fy1s)is-o pue FHis-o go sazerzs LeuqneN yz 405 suoLgeiNd1e) T)-n9 IT aLqeL
respect to (1b).
and calculated!2 spectra for the free oxygen atom, we see that the ex-
citation energy of 2.3-2.5 eV indicates that this transition is very
much like the
tion is characteristic of the oxygen atom and is not present in the 0,
molecute!3, It should be mentioned that a peak in the range 2.39-3.9 eV
is found in the energy loss spectra of oxygen chemisorbed on Si (111)
cult because of the presence of many peaks due to bulk and surface
phenomena not associated with the oxygen. Ibach and R wel? observe a
similar peak at 2.0 eV, but they associate it with the intrinsic states
of the surface because its intensity decreases as the coverage of oxygen
increases.
tials for corresponding orbitals using the OSiH, and OSi(SiH3) 4 clusters.
The oxygen Is orbitals do not change, as expected from chemical arguments.
The ionization potential of the 02s orbitals increases by 0.9%, indicat-
ing a slight effect due to charge readjustment for the larger cluster.
The largest changes occur in the p-like valence orbitals, and here the
most sizable change is only 4.1%. This shows that these ionization
of the cluster.
and 0-Si(SiH,), Cluster Calculations?
Koopmans' Theorem Ionization Potential %
Orbital fev] Change
O-SiH, 0-Si(SiH3),
Ols 560.53 560. 38 0.027
02s 318. 36 321.09 0.85
02py 17.50 16.8] A.
02p, 15. 86 15.48 2.5
si-o? 20.18 19.41 4.0
GVB-CI calculations for the 0-SiH, and 0-Si(SiH,), clusters. We first
notice that even though these are not Koopmans’ theorem ionization
potentials (since each system is allowed to readjust its charge density)
the comparison between the two clusters shows that the ionization poten-
tials are not very sensitive to the size of the complex.
tal. We are then left with two singly occupied orthogonal orbitals
(02p,, and 02p,)- This leads to two states: a triplet and a singlet.
The triplet is lower due to a favorable exchange interaction. These
y X
Ionizing out of the Si-O bond (or 02p, for the free 0 atom) we are
left with two singly-occupied orbitals again (the Si-O and 02p.)- Since
Si-O has o symmetry and O2p, has 7 symmetry with respect to the Si-0
d 0 atom @
Ionizations Spin 0-SiH, 0-Si(SiH,)3 Experimental
Sid vacuum triplet 13.87 13.52 16.94
02p, vacuum Singlet 14.60 14.07 16.94
02p, vacuum Singlet 14.63 14.24 -
distance was 1.632. Ionization potentials are calculated by solving
for the appropriate state and subtracting the energy from that of
the ground state.
tion corrections are estimated to be between 0.2 and 0.4 eV.
comparing these values with the experimental data of Ibach and Rowe!®
we see that they find peaks at 11.9, 15.1 and 18.4 eV (see Table VII
and Fig. 4). Although the first two are within the range of values
shown in Table IV, the 18.4 eV peak is not. We shall see later that
this peak can be tentatively assigned to an ionization out of the 0-0
bond and thus one does not expect to see this peak in the spectrum of
chemisorbed single oxygen atoms.
positive charge on the surface leads to long range polarization effects
that cannot be treated by a small finite cluster. Using the dielectric
corrections discussed in Chapter 2 and Appendix B, we have estimated! ®
these errors to be between 0.2 and 0.4 eV. (ii) Because of the use of
Koopmans’ theorem in Table III, the positive ion states are described
in terms of orbitals suitable for the ground state. This prevents the
orbitals from contracting about the charged centers resulting in too
large an ionization potential. A compensating error is that through
the use of a HF ground state, the extra correlation error tends to
lead to too small an ionization potential.
ize the Si0 bond distance. This is done by letting the 0 atom move in
a direction perpendicular to the surface (along the [111] direction).
“Y
pairs.
and GVB-CI calculations on both the OSiN, and OSi(SiH,) 4 clusters. The
basis set in this case was the DZ basis of Table XI, Chapter 1 for the
®pulk represents the SiH bond pairs for OSiH. and the SiH and SiSi bond
pairs for 0-Si(SiH,),. Here both the Si-O bond pair and the 02,
doubly occupied orbital are correlated.
of (7) plus one additional (virtual) orbital in the p. direction
in Table .V. All orbitals corresponding to Po ulk were kept doubly
occupied and no excitations were allowed out of them (thus we do
not include them in Table V). In Table V, under each orbital we
show the occupation number of that orbital for a particular con-
Yor = ) c; o, (8)
(where X is an appropriate spin function). Note that when the oc-
cupation number is 2 the orbital appears twice (two electrons in
that orbital), when it is 1 the orbital appears once and when the
occupation is zero the orbital is omitted.
from the basic configurations shown in Table VI. Again all orbitals
We shall now consider the chemisorption of oxygen molecules
to Si (111) surfaces. Due to the greater complexity of the elec-
were kept doubly occupied.
02s Sid. = S70 O2p, O02p, 02p Oop,
2 2 1 ] 2 0 ] 0
3 2 0 2 2 0 ] 0
4 2 2 0) ] 0 2 0
5 2 ] ] 1 0 2 0
6 2 0 2 ] 0 2 0
7 2 ] 0 2 0 2 0
8 2 0 ] 2 0 2 )
9 1 2 0 2 9) 0
10 ] 1 ] 2 0
02s Sid 330 02>, Op. o2ap dap”
Xx y y
] 2 2 0 2 0 0 0
2 2 ] ] 2 0 0 0
3 2 0 2 2 0 0 0
4 2 2 0 ] 0 ] 0
5 2 1 1 1 0 1 0
6 2 0 2 ] 0 ] 0
7 2 2 0 0 0 2 0
8 2 ] ] 0 0 2 0
9 2 0 2 0 0 2 0
10 2 ] 0 2 0 ] 0
1] 2 0 ] 2 0 1 0
12 2 ] 0 ] 0 2 8)
13 2 0 ] ] ¢) 2 0
14 1 2 0 2 0 1 0
15 ] ] ] 2 0 ] 0
16 ] 0 2 2 0 1 0
17 ] 2 0 ] 0 2 0
18 ] 7} ] ] 0 2 0
19 ] 0 2 1 0 2 0
is possible.
oxygen molecule. We will start with a qualitative description of the
system. The oxygen atom can be pictured as in Fig. 2a (or la) as
discussed in Section B. Pairing two such oxygen atoms to form the 0
bond. This leaves two other singly-occupied orthogonal orbitals lead-
ing to triplet cm) and singlet (140) states of which the triplet is
lower (due to a favorable exchange interaction). The other way of
the orbitals of 0, leads to significant delocalization of the orbitals
but the qualitative picture still applies. The basic reason why the
configuration of Fig. 2b is better than that of Fig. 2c is that the
Fig. 2b (where there is a singly-occupied orbital on this center) but
not for Fig. 2c (where there is another doubly-occupied orbital on this
center).
(111) surface, as in Fig. 2d. After bonding one end of the 0, to a
Surface silicon atom, there is a remaining unpaired orbital on the 05.
({c) excited state 0, (d) ground state of chemisorbed On»
(e) excited state of chemisorbed 05. Is and 2s orbitals have
been ignored and p orbitals parallel and perpendicular to the
plane of the paper are denoted by « and o, respectively. Dots
indicate how many electrons are in each orbital. Orbitals com-
bined into a singlet pair are indicated by a connecting line.
found in the free molecule for the corresponding excitation, about TeV.
bital (refer to Fig. 2d), which is a doubly occupied orbital; the Py
parison with experiment, we carried out theoretical calculations on
0,SiH3 and 0557 3H¢ clusters. For the O,-STH, complex we have optim-
1.264 for iron dioxygen complexes '8 (FeO); 1.28 to 1.48% for various
cobalt dioxygen compounds“? (Co0,) and 1.34 for hydroperoxyl radical !9
(HO,). The bond angles range from 102° to 105° for hydroperoxyl radi-
18
ligands. Our results are shown in Tables VII and VIII. Table VII
shows the GVB-CI energies for the different geometries of some of
the relevant states of the 0,-Si-H, complex. These are indicated
by the most important excitations (refer to Fig. 2 for the meaning
of the orbitals). In Table VIII we show the optimum values for the
0-0 bond length and the Si-0-0 angle obtained by fitting a parabola
to the appropriate points of Table VII.
at the optimum geometry of the 05S1H, ground state (i.e., for
Rog = 1.366A and 853_9-9 = 125.9°). The band orbitals and the Ols-like
orbitals are not shown. Just as assumed in the previous discussion
6 7)
in Fig. 2d, supporting the peroxy radical model.
calculated states are also shown in Table X where more information
is given. Here we see that the neutral states of the peroxy radical
can be interpreted on the basis of the excitations observed in the
oxygen molecule. Thus the transition at 0.92-0.95 eV, in which an
electron is excited from the doubly-occupied Py orbital to the
singly-occupied Px orbital, follows the pattern of the 0.98 eV tran-
Sition of 05> and in fact can be interpreted in exactly the same way.
The transition at 6.2 eV (p, > Py ) is characteristic of peroxy radi-
cals. It has a strong (oscillator strength 0.1) broad (~ 2 eV)
Ul poyqoyd are (9)—(v) syeg ‘N° $Q'O S} sINdJUOD UdEMJ9q UOTIVIndos oY} ‘souRId [eEpOUa}esIpUT soysep SUG] “uMOYS
JOU 91k S}EIIGIO OATL-STO PUE speyquo purg “(+ Aq payeorpul ase suo!Isod woje oYy}) soRJINS 94] O} JepNo[puadiad pue
SWIO}E DOEJINS OA\} YSNOIY) Surssvd ouesd ou} ul st 27Q ayy arom ‘xajduroo *QXxzEIS ay} JO speyqio GAH oyL ‘f “Bly
g'9-
N\ .
/ NY
Oo
3NO OML Sb
rx
Yivd“ “do (4)
Gol zZ zZ
' / c
aed / vA /
a 7 f | pa
re rm / \
! VA oN
\.
SA
OML -~ OML OML
\ ~,
\ \ \
\ \ ~ { (7
~ Vs \E
yivd]
(Gaur),
3NO 3NO | |ano /
LO6SS*bSL- ««6SPHSPSL- —»«s«GGRBS*HSL- —sLELSS*PSL- BSS “SL UNNDBA
LO9LS"VSL- 2029" PSL- LLL6S*PSL- «- 898LS°PSL- 20089 "HSI wnnoeA « Ye
O€989'PSL- BELBO"HSL- = GZELO SL L6069°PSL- -LLE69"HSI- wnnoen + ay,
gy20L"PS1- L66L9°VSL- 80989°PSL- —»«s-2B8EN*PSL- =» 09969 PSL wnnoen « Me aa,
LEGEL*DSL= LOL LL" PSL \822e‘9SL- LOLEL*vGL- -- SO98L “HSL wnnoe A Mb Me
SUOL}ZEZLUOT
zesze"ySl- «€6S28"PSL- «OL99B"PSL- —»«ZBEZV"PSL- = PZESL“PSI- xa. Me WN
89/28 "YSL- L60v8"bSi- -«GP89B"PSL- —»«O89EB“HSL- ~—S>PLOZL “BSI Xd £0-0 WY,
L9LL8°PSL- -«« E69BB"PSL- —««6Z9ZSVSL- ss BVEBBPSL- —=«-L00ZR“PSIL- a +015 WY,
BEBZL"SSL- -«SENSOTSSI- —»«sTSLEL"SGL- —=»s99E2L “SSL LSe60°SSL- «4a ay,
PEvL'SSL- —»«sPOESL*SGL- —»«HRZSL*SSL- _—«OODLSL*SSL- —=BLEZL“SSI- “97835 punosB ,y,
SUO L229 LOXJ
oStl = 0 wSLL = 6 o0fl = 9 o0E'L = 9 o0EL = 6
yet = Oy pert = My spe = Oy yet = My year = My 2723S
¢ *SooAqsey UL ade
0,-SiH, Cluster. All bond lengths are in angstroms and
nly +p 1.381 135.54
Yr Xp
2a 0-0 > Dy 1.495 110.45
2 q" Py, * Oe 1.410 123.35
Tonizations
Van Py. > vacuum 1.306 135.64
Vy >. > vacuum 1.279 129.73
3a! >, > vacuum 1.392 130.11
3," S40» vacuum 1.407 136.00
‘a" si0 > vacuum 1.402 136.45
3a" 9-0 + vacuum 1.756 128.04
in this column. .
Peroxy-Radical Model. TIonizations involving simultane-
ous excitation of an electron are omitted, as are ioni-
zations involving the SiH bonds. All energies are in eV.
State Calculated Energy (GVB-—CI)* mental valucs
Results? for 02° for 102°
Excitation SiH3;02 SizH.O02 OQ, on Si
Ground State 0.00 0.00, 0.007 0.00 0.00 ~ 0.00:
Pyr> Par 0.84 0.92, 0.95 0.98 0:93
Pst > Pr 6.65 6.19, 6.19 (8.6 6.40
O00 py, 6.51 6.54, 6.55 _ 7.54
SURF? = p,, ~ 768 =
SiO > p,, 7.96 7.90, 8.17 ~
Pyp> SURF — 9.98 -
Ionization
Dyp7? Vac 10.81, 11.82 11.00, 11.08, 11.99 Jars 16.7
Par vac 1.51 11.37 12.3
Pat Vac 14.4 - 14.23, 14.23 151 16.7
SiOo > vac 14.73, 14.78 15.00, 15.04, 15.28 ° ~
O00 > vac 18.4 18.2
the triplet and singlet states resulting from the coupling of the
Table X. Excitation Energies and Ionization Potentials for the
are in eV.
ground state 0.0 triplet
0.007 singlet
0.922 Singlet
Py * Py
r r 0.951 triplet
p>) 6.191 triplet
xy Xp
6.193 Singlet
0-0 > Py 6.542 triplet
r 6.548 Singlet
surf > p 7.684 Singlet
Sid > Py 7.904 triplet
8.17] singlet
p +> surf 9.978 triplet
Yr
Tonizations
surf +> vacuum 8.712 doublet
Surf =vacuum + py * Py 10.265 doublet
r r
p> vacuum 11.004 quartet
Yr 11.084 doublet
11.990 doublet
P, > vacuum 11.366 doublet
Py > vacuum py > Px 13.378 doublet
P, > vacuum 14.225 quartet
. 14.231 doublet
surf + vacuum + p > Py 14,684 doublet
15.039 quartet
rr 16.605 doublet
18.062 doub Jet
Yr Xp 18.857 quartet
Tne basis was the MXS2 set of Table XI, Chapter 1 for the two surface
the oxygen atoms.
ciation of the 05 bond, leading to a chemisorbed atom plus a free
oxygen atom, with a significant probability of the 0 atom being in
As indicated in Table IX and shown in Fig. 4 we find ioniza-
tial increases from 12.3 eV (for 05) to 15 eV upon bonding to the Si
in 05 (with an ionization potential of 16.7 eV) but which must de-
localize onto the oxygen on the right, thereby decreasing its ioniza-
tion potential to v11.5 eV.
changes, indicating that these excitation energies and ionization
potentials are not greatly affected by changes in the size of the
cluster. Corrections due to the polarizability of the semi-infinite
solid can be estimated in the manner discussed in Chapter 2 and Appen-
cs Lg 0 :!
hu=21.2 ev
Tx?
ENERGY (ev)
vLomperison or the caleulaued Lonization potentials for the
15). Calevlated values are indicated b3 sal lines. @ is
the fevactional O» coverage. Curves 2-1 are ‘ference specura
between the surface exposed to Op and tue clean surface.
best estimates of the energetics in this case are as follows: refer-
encing all the bond energies to an 05 molecule and a free silicon
surface, the 0, bonded to the silicon surface (in the peroxy radical
form) is at -2.2 to -2.5 eV; the dissociative adsorption of the 05,
leaving a single oxygen bonded to the surface and the other oxygen
free, is at +0.5 to +0.3 eV. The state with the 0, bond broken and
the two oxygen atoms independently bonded to two separate sites on
the silicon surface is at -4.1 to -4.5 eV; however, the barrier sep-
arating this state from the ground state of the peroxy radical is
expected to be large. The excited state of the 05 is at +1.0 eV.
The state with the 0, in an excited state and bonded to the silicon
surface in the peroxy radical form is at -1.3 to -1.5 eV. The bar-
rier between this state and that with two oxygen atoms separately
bonded to the surface should be rather much smaller than the compar-
able barrier for the ground state of the 0, bonded to the surface.
In Table XI we show some of the energies for GYB and GVB-CI
calculations on complexes related to 05 chemisorption. Referencing
all the bond energies to an 0, molecule and a free silicon surface,
we estimate the energetics (based on the GVB calculations) as fol-
lows: The 0, bonded to the silicon surface (in the peroxy radical
form) is at -3.87 eV; the dissociative chemisorption of the 0, Teav-
ing a single oxygen bonded to the surface and the other oxygen free
is at -3.23 eV.. The state with the 0, bond broken and the two
oxygen atoms independently bonded to two separate sites on the sili-
GVB
Complex Total Energy GVB-CI
(hartree)
0,-SiH, -155. 13326 -155.17888
SiH, -5.38986 -
as follows: for O5-SiH, an oxygen molecule is placed on top of an
SiH, unit with one of the oxygen atoms bonded to the Si atom, as
in Fig. 2. For 0,-Si3H¢ 5 an 0, molecule is placed on top of one
of the surface silicons of a (111) Si surface modelled by an Si zHe
cluster as in Chapter 2. Here the three silicon atoms are placed
at the positions of a silicon lattice, as in Fig. 2. Hydrogens re-
place broken Si-Si bonds. The Si-O bond length used was 1.63A for
all cases considered.
angle (see Tables VII and VIII). This was done by (i) choosing a
fixed angle (8c; _9-0 = 130°) and minimizing the energy for
in the 0-0 bond length and (ii) choosing a fixed 0-0 bond length
(Rog = 1.348) and minimizing the energy for variations of the Si-0-0
angle. These calculations were performed for GVB and CI wavefunc-
tions. The choice of wavefunction in this case is important because
of Table XI, Chapter 1 with the bunning!2 double zeta basis for
oxygen.
the GVB orbitals was used. Two virtual orbitals were added, local-
ized mainly in the Px. and Px directions. The orbitals were divided
into two sets. These localized on Si-Si and Si-H bonds were kept
doubly-occupied for all configurations. The other orbitals consist
of two 02s orbitals, a doubly occupied Py orbital, a surface dangli-
ing bond (surf), two 0-0 bond natural orbitals, two Si-O natural or- —
bitals, two oxygen orbitals (p, and Py ) and two oxygen m virtual
virt
XII. In this way 30 configurations are generated for the neutral and
68 for the ionic system. To these we added all single excitations
from the 30 (for the neutral) and 68 (for ions) configurations des-—
cribed above. Thus the wavefunctions actually consisted of 646 to
1004 determinants for the cases considered here. For O5-SiH, the
same procedure was used to generate the configurations with the ex-
xr
Xk
Co ao
AE ON
NE Od
mom ON
Co Oo
Oo ©
N aw
ma WN
bor
AL ON
los aN
xr
Xz
for the structure of the chemisorbed 05 are possible in this case.
posed. Meyer and Vrakking@? proposed a model in which the oxygen
atoms are inserted into Si-Si bonds forming a complicated structure.
Ludeke and Koma! 4 proposed a model in which the oxygen atoms form a
monoxide structure with double bonds between the oxygen atoms and the
surface silicons.
molecule can still be identified as such after it has chemisorbed.
adjacent Si atoms by the two oxygens. Our calculations led to the peroxy
radical model discussed in the previous section in which the molecule
Si.
the photoemission results (of which those presented by Ibach and Rowe!®
are representative) indicate a large peak at 111.9 eV with smaller peaks
at 15.1 eV and 18.4 eV, also a shoulder is observed at 8.3 eV. Ibach
chemisorbs without greatly affecting the dangling bonds in nearby
geometrically inequivalent types of oxygen atoms must occur at the
oxidized surface. The peroxy radical model is consistent and agrees well
(where comparison is possible) with these experimental results.
fissioned 05 models is that they cannot explain the three localized
can be explained by the double bond monoxide model by assuming that
suggestion is untenable on several grounds: first, only vibrational
frequencies of the ground state are observed in the experiments of
in the other orbitals (e.g., the Si0 bond length increases from 1.518
for the Si0 molecule to 1.632 of the chemisorbed 0 atom) and surely
(0810) (nSi0)*(O0, )°(Si3p,)! (si3s)!
ssid is a py like bonding orbital and the electrons in the S1SP and
chemisorption of single oxygen atoms as well as molecular 0... From
them we determined that, for the initial step of the chemisorption,
the optimum geometry for chemisorbed 0, is that of a peroxy radical,
with one oxygen bound to only one surface silicon and the other oxy-
gen in such a way that the Si-0-0 angle is 126°. The optimum 0-0
bond length is 1.3668. These values fall within the range of values
observed experimentally for other peroxy systems. The predicted
spectrum for the peroxy radical agrees with experimental results, and
the vibrational modes expected for such a system are consistent with
experimental observations.
one of the peaks in the ionization spectrum (that one corresponding
to an ionization out of the 0-0 bond for the peroxy radical) is not
present, contrary to experimental results. In addition this system
cannot account for the three vibrational modes that have a component
perpendicular to the surface or with the recent observation that two
D. R. Boyd, J. Chem. Phys. 23, 922 (1955).
287 (1974); (b) R. Dorn, H. Luth and H. Ibach, ibid 46, 290 (1974).
W. A. Goddard III, A Redondo and T. C. McGill, Solid State Commun.
18, 981 (1976).
and W. A. Goddard III, J. Chem. Phys. 63, 4632 (1975).
Si-O bond suggests that only small distortions occur in this case.
The point group is that of a heteronuclear diatomic molecule,
r+A=0
—>A= 1]
Arr =2
13.
15.
D.C., 1949).
T. H. Dunning, Jr., J. Chem. Phys. 53, 2823 (1970).
R. Ludeke and A. Koma, Phys. Rev. Lett. 34, 1170 (1975).
(a) H. Ibach and J. E. Rowe, Phys. Rev. B 9, 1951 (1974); (b) ibid
We assumed h to be 1.63% corresponding to the Si0 bond length. ro
contains the molecular axis; 7 orbitals are anti-symmetric with
respect to mirror planes containing the axis.
Rodley, Proc. Nat. Acad. Sci. USA 71, 1326 (1974).
Soc. 93,6734 (1971); (b) H. E. Hunzinker and H. R. Wendt, VII
International Conference of Photochemistry, Jerusalem, Israel,
August 1973 [IBM Res. Report RJ1373 (#21044), March 28, 1974];
(c) W. A. Goddard III and T. H. Dunning, unpublished calculations.
(a) R. S. Gall and W. P. Schaefer, to be published; (b) A. G.
Sykes and J. A. Weil, Prog. Inorg. Chem. 13, 1 (1970).
24.
25.
26.
28.
29.
30.
31.
32.
33.
ties, Nat. Bur. Standards (U.S.) Technical Note 270-3 (U.S.
G. A. Bootsma, Surf. Sci. 15, 340 (1969).
J. E. Rowe, G. Margaritondo, H. Ibach and H. Froitzheim, Solid
State Comm. 20, 277 (1976).
F. Meyer and J. J. Vrakking, Surf. Sci. 38, 275 (1973).
P. Chiaradia and S. Nannarone, Surf. Sci. 54, 547 (1976).
report we include in this appendix a brief summary of the GVB wavefunc-
tions. For more details see Refs. 12 and 22 of Chapter 1.
where OL is the antisymmetrizer. The optimum orbitals are obtained by
solving the resulting variational equation
Ha 45 E aa
(A.4)
GVB _
Hy = &bop
orbitals it is generally more convenient to transform the spatial part
el
®2_1-s
C, 145
S =
2 2
Cy + Cs = |
5 are orthogonal, the corresponding variational equations are simpler
to solve.
(A.3) or (A.5) accounts for the dominant electron correlation of many-
wavefunction simultaneously and self-consistently.
ize on or around one or two atoms leading to what can aptly be
interpreted as a bond pair, a core pair, a nonbonding pair, etc. It
is possible to selectively correlate only certain pairs; for example,
relate all (four) pairs involving the valence orbitals (3s and 3p)
but not correlate the Si core orbitals (1s, 2s, and 2p, ten altogether).
(9265-480 (Cy 94 ab95 # Cynb45y 4) (Cy adaadyg + Ca aby 9b7 9) (Credrnd
V2 1O."11711712 12712711 13°13"14 “V4 14° 1345715"°15°16
(expressed in the (A.3) form)]. This wavefunction would be denoted as
GVB(4) indicating that four pairs are correlated.
are important; for example, a nonbonding orbital in a negative ion
tron wavefunction the form (A.8) would just replace the doubly-occupied
orbital of a HF wavefunction. Physically not more than four to five
NO's in an expansion of the form (A.8) can be expected to be important,
corresponding, for example, to in-out (1) and angular correlations (3)
Appendix B - DIELECTRIC CORRECTIONS
corrections for the ST gHg complex were performed. We started by
assuming that we had correctly described the polarization effects
in the cluster, which in this case we substituted by a hemisphere
of radius ro (determined by the size of the complex). We also
assumed that a charge of one electron was located at a height h
above the center of the hemisphere on the surface. The location h
is determined by the particular characteristics of the complex. In
order to estimate the effect of the polarization we calculated the
change in free energy for the semi-infinite (continuous) dielectric
with the hemispherical cavity when the charge is placed above it.
QP =
(eh (B3)
|“
into Eq. (Bl) we get
‘a
] - 2r'h cos 8)
+] - Jim [log{\+h)] Ht log(r +h) }
lim (1 Yog(x2+h2) - log(Ath)} = 0
> © 2
leading to the final result?
pH 2-1 ely ly _ yogi 2
G. T. Surratt, Ph.D. Thesis, California Institute of Technology,
1975, p. 169.
lation (or the lack of it), the inclusion of .d-functions in the basis
set and the effect of the lattice constraints on the calculations for
the SiH. cluster model of the Si (100) surface. Let us start
with the effect of correlation; we will then examine the changes that
occur when d-functions are included in the basis set. We will finally
consider what happens when the constraints of the lattice are relaxed.
V9?) and 3 (or) states at the self-consistent field level. The effects
of correlation are most prominent for the V2) state in which, for the
Hartree-Fock (HF) wavefunction (no correlation), two electrons occupy
the o orbital. For 3 (om) this does not occur because the two elec-
trons occupy two different open shell orbitals, o and w . In order to
introduce the most important effects of correlation, we have also
performed calculations for the V2) state using a Generalized Valence
Bond (GVB) wavefunction. The results are summarized in Tables C.I and
C.IIl. For a double zeta basis (DZ) set, and keeping the constraints
of the lattice at the HF level, the 3 (on) state is 0.75 eV below the
Vg2) state (vertical excitation energy). When we allow for geometri-
cal relaxation, an adiabatic excitation energy of 0.61 eV is found. If
we include correlation for the same basis set and including the con-
(100) Surface
Energy Angle Distance Excitation Excitation
(hartree) (°) (A) Energy(eV) Energy(eV)
———Ng2)HF -14.542985 104.18 0.138 0.75 0.61
Sion) 14565318 111.74 -0.056 0.0 0.0
DZd basis set
State: '(o*)@vB -14.594092 105.43 0.104 0.0 0.0
Vg2)HF © -14.574004 105.35 0.106 0.55 0.55
Sion) 14587655 111.49 -0.050 0.34 0.18
Vion) -14.539713 111.67 -0.055 1.64 1.48
Without lattice constraints
DZ basis set
State: }(o%)@vB -14.553597 100.57 - - 0.41
Vg@)HE — -14.541581 101.52 - - 0.74
S(on)4—-14.568625 130.84 - - 0.0
DZd basis set
State: M(o7)avB -14.595822 95.21 - 0.0 0.0
V(o2)ye¢ -14.575881 94.66 — - 0.54 0.54
Ston) 1.599082 119.87 - 0.49 0.18
shown do not include the contribution due to the silicon core elec-
solid. Zero corresponds to the tetrahedral, unrelaxed geometry.
“Measured with respect to the ground state of thecorresponding calcula-
102 uF ~14.53434] -14.542915 -14.540308 -14.532118
3 (on) -~14.534275 ~-14.560361 ~14.564764 -14.564172
DZd basis set
V6?) HE -14.560135 -14.573992 -14.571966 ~-14.563180
3 (on) -14.552411 -14.582328 -14.587146 -14.586129
Von) -14.505636 ~-14.534315 -14.539129 -14.538382
batic excitation energy of this state is found to be 0.23 eV. The
potential curves for the double zeta basis set are shown in Fig. C.L
The effects of correlation for a basis set including d-functions are of
crucial importance for Vg2y, This was investigated by doing the same
calculations over.with the difference that to the double zeta basis set
we added a set of d-functions to the central silicon atom (this basis
set is called DZd in Chapter 1). In this case we found that without
correlation the ground state is 3 (on) with the V( 42) state at 0.50 eV
vertical excitation energy (in this case the adiabatic excitation energy
is 0.37 eV for the Vg2) state). When correlation is included we find a
Vig2y ground state with the 3 (on) State at 0.34 eV vertical excitation
energy, as discussed in Chapter 2. Thus we see that unless both d-
functions and correlation are included in the calculations we obtain the
_ wrong ground state for the system. In terms of the potential curves
(shown in Fig. €.2 and summarized in Table C.III for the DZd basis) we see
that the effect of correlating the doubly-occupied pair for the Vg)
state there is a relative uniform correlation energy! of v 0.32 eV for
the double zeta basis and ~ 0.54 eV for the DZd basis. The reason for
this difference of v 0.22 eV is due to the fact that d-functions can be
effectively used in correlating the thn and o-Aw orbitals of Vg);
this is in spite of the fact that very small amounts of d-character are
present in such orbitals. The effect is slightly more pronounced for
Clusters Modelling the (100) Surface (with the Lattice
Central angle 95°
DZ basis 1.27 1.63 0.35 1.63
DZd basis 0.39 0.92 0.53 1.13
Difference 0.88 0.71 ~0.18 0.50
Central angle .105°
DZ basis 1.06 1.39 0.33 0.92
DZd basis 0.0 0.55 0.55 0.32
Difference’ 1.06 0.84 ~0.22 0.60
Central angle 109°28'
DZ basis 1.15 1.46 0.32 0.80
DZd basis 0.05 0.60 0.55 0.19
Difference’ 1.10 0.86 -0.23 0.61
Central angle 115°
DZ basis 1.39 1.69 0.30 0.81
DZd basis - 0.30 0.84 0.54 0.22
Difference’ 1.09 0.85 -0.24 0.59
mum (-14.594092 hartree).
1052) HE energies.
se pazzyoid si ABuaug "(89137R] 842 JO sqULeuqzsuod 2Yy} YLLM)
Sis®g 7q BP HuLsn uaqsni9 Er ¢ SY} UOJ SAdBJuNS [PL UaIOY
(SASeo50 SIONS
“O21 ost or ort or sor oon o°ss o"
} {
| i | |
P ZO a |
— ~ EAD (y0)
~ to Vy
en nt
a “
Pauli principle repulsions with other pairs of electrons, which
dominate over correlation when the geometry forces the of orbital
to interact with the Si-Si bonds.
of the states individually. This is shown in Table C.1IT. We see that
for central angles between 105° and 115° the effect is relatively
constant, with a drop in value for small angles (~ 95°). This is due
to the fact that for 95° the geometry is such that the central atom
is pulled 0.40 ay (toward vacuum) with respect to the tetrahedral
geometry and the electrons in the central Si make less use of d-
functions. As mentioned above, the inclusion of d-functions is nec-
essary to obtain the correct ground state. This is also true for the
system in which the constraints of the lattice are not included, as
we shall see below.
system which models the (100) surface. When we eliminate the con-
straints of the lattice (i.e., when we let the angle of the central Si
vary without changing the length) the results are basically the
same. The main difference occurs in the values of the Si-Si-Si angles
at the optimum geometries, but this is only a consequence of the
elimination of the lattice constraints. The results are summarized
in Tables C.I, C.WV andC.V and the appropriate potential curves are
Shown in Figs. C.3 and C.4. In this case we obtain a of ground state
Constraints®
State
DZ basis set
1052) evp -14.552850 -14.553585 -14.551941 -14.546634.
V5?) HE -14.540590 -14.541531 -14.540308 -14.535557
3(on) -14.554080 -14.558811 -14.564764 -14.567866
DZd basis set
16?) eve ~14. 595821 - -14.592091 -14.584552
Vote ~14.575879 - -14.571966 -14.564420
3 (or) -14.578006 - -14.587146 -14.589082
Von) -14.528481 - -14.539129 -14.542215
(0°) GVB (0° )HF Energy (om)
Central angle 95°
DZ basis 1.17 1.50 0.33 1.14
DZd basis 0.0 0.54 0.54 0.49
Difference’ 1.17 0.96 -0.21 0.65
Central angle 100°
DZ basis }.15 1.48 0.33 J.01
Central angle 109°28'
DZ basis 1.19 1.51 0.32 0.85
DZd basis 0.10 0.65 0.55 0.24
Di fference® 1.09 0.86 -0.23 0.61
Central angle 115°
DZ basis 1.34 1.64 0.30 0.76
DZd basis 0.3] 0.85 0.55 0.18
Difference’ 1.03 0.79 -0.25 0.58
(-14.595822 hartree).
V5? )HE energies.
This is the difference produced by the inclusion of d-functions.
siseg zg e Bursn aarsniy %y®1s aya 40g sadesuns Lelquaiog ¢°9 "Bi4
ZQ
om
ee
Yah
—oS°O-
STSPC PAG B SUTSN TOISNTN CSTE aya Jor seovrang TeTmueqog ee Star
A \ i ee
‘a
Yw
ate
mal
mee —nsit-
0.18 eV adiabatic excitation energy (with an optimum Si-Si-Si angle