Cluster morphology silicon's nanoparticle
It is proposed to use silicon polyhedral atomic clusters (ACs) in the study of nanoparticles. A scheme of the parametrized density functional theory for calculations of the atomic and electronic structures of these clusters is presented. The accuracy of the method is illustrated by the results of...
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Інститут фізики напівпровідників імені В.Є. Лашкарьова НАН України
2007
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| Цитувати: | Cluster morphology silicon's nanoparticle / V.V Kovalchuk // Semiconductor Physics Quantum Electronics & Optoelectronics. — 2007. — Т. 10, № 4. — С. 81-86. — Бібліогр.: 18 назв. — англ. |
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irk-123456789-1183382017-05-30T03:04:57Z Cluster morphology silicon's nanoparticle Kovalchuk, V.V. It is proposed to use silicon polyhedral atomic clusters (ACs) in the study of nanoparticles. A scheme of the parametrized density functional theory for calculations of the atomic and electronic structures of these clusters is presented. The accuracy of the method is illustrated by the results of calculations for Si-ACs with different sizes. 2007 Article Cluster morphology silicon's nanoparticle / V.V Kovalchuk // Semiconductor Physics Quantum Electronics & Optoelectronics. — 2007. — Т. 10, № 4. — С. 81-86. — Бібліогр.: 18 назв. — англ. 1560-8034 PACS 36.40.+d, 82.20.Wt, 61.50.Lt http://dspace.nbuv.gov.ua/handle/123456789/118338 en Semiconductor Physics Quantum Electronics & Optoelectronics Інститут фізики напівпровідників імені В.Є. Лашкарьова НАН України |
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It is proposed to use silicon polyhedral atomic clusters (ACs) in the study of
nanoparticles. A scheme of the parametrized density functional theory for calculations of
the atomic and electronic structures of these clusters is presented. The accuracy of the
method is illustrated by the results of calculations for Si-ACs with different sizes. |
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Article |
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Kovalchuk, V.V. |
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Kovalchuk, V.V. Cluster morphology silicon's nanoparticle Semiconductor Physics Quantum Electronics & Optoelectronics |
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Kovalchuk, V.V. |
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Kovalchuk, V.V. |
| title |
Cluster morphology silicon's nanoparticle |
| title_short |
Cluster morphology silicon's nanoparticle |
| title_full |
Cluster morphology silicon's nanoparticle |
| title_fullStr |
Cluster morphology silicon's nanoparticle |
| title_full_unstemmed |
Cluster morphology silicon's nanoparticle |
| title_sort |
cluster morphology silicon's nanoparticle |
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Інститут фізики напівпровідників імені В.Є. Лашкарьова НАН України |
| publishDate |
2007 |
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http://dspace.nbuv.gov.ua/handle/123456789/118338 |
| citation_txt |
Cluster morphology silicon's nanoparticle / V.V Kovalchuk // Semiconductor Physics Quantum Electronics & Optoelectronics. — 2007. — Т. 10, № 4. — С. 81-86. — Бібліогр.: 18 назв. — англ. |
| series |
Semiconductor Physics Quantum Electronics & Optoelectronics |
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AT kovalchukvv clustermorphologysiliconsnanoparticle |
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2025-07-08T13:50:23Z |
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2025-07-08T13:50:23Z |
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1837086932523286528 |
| fulltext |
Semiconductor Physics, Quantum Electronics & Optoelectronics, 2007. V. 10, N 4. P. 81-86.
© 2007, V. Lashkaryov Institute of Semiconductor Physics, National Academy of Sciences of Ukraine
81
PACS 36.40.+d, 82.20.Wt, 61.50.Lt
Cluster morphology of silicon nanoparticles
V.V. Kovalchuk
South Ukrainian State University
26, Staroportofrankovskaya str.,65020 Odesa, Ukraine
Phone: 38-048-7384785; e-mail: kovalchuk@edu.pdpu.ua
Abstract. It is proposed to use silicon polyhedral atomic clusters (ACs) in the study of
nanoparticles. A scheme of the parametrized density functional theory for calculations of
the atomic and electronic structures of these clusters is presented. The accuracy of the
method is illustrated by the results of calculations for Si-ACs with different sizes.
Keywords: silicon, polyhedral atomic cluster, atomic and electronic structures,
nanoparticle.
Manuscript received 03.09.07; accepted for publication 19.12.07; published online 13.02.08.
In recent years, the optical properties of silicon
nanostructures (Si-NS) have been the subject of intense
investigations, because the quantum confinement of
electrons and holes leads to linear and non-linear optical
properties much different from those of bulk crystals.
Since Canham’s discovery of visible light emission from
porous Si (see [1]), a great deal of efforts has been
devoted to investigation of the optical properties of Si-
NS. There are the direct experiments on the geometry of
Si-NS [2, 3].
In addition, the optical data [4] indicated the
existence of atomic clusters (ACs) in films and were
mediated by the localized electronic states associated
with clusters. In particular, the peculiar Raman spectra in
Si-SiO2 systems, being different from those of bulk c-Si,
a-Si, and microcrystalline Si, but very similar to the
density-of-state spectra of Si33 and Si45 [3], were
observed. The experimental data strongly suggest that
these films are the systems of ACs much smaller than
about 2 nm embedded in SiOx matrices. Although the
size and the size distribution of the clusters are not
known, the introduction of clusters with various sizes
randomly into SiO2 thin films may generate localized
electronic states. If these previous results are taken into
account, it is rather straightforward to attribute the
presently observed ln σ (σ – conductivity) versus T –1/4
(T − temperature) behaviour to the VRH (variable range
hopping) conduction through the localized electronic
states associated with C, Si, and Ge clusters [5, 6]. As
the Si concentration increases, the slopes (B) of ln σ
versus T –1/4 straight lines decrease. Since the increase in
the Si concentration leads to the increases in the size
and/or number of Si clusters, the decrease in B may be
attributed to the increase in the size and/or number of Si
clusters. Thus, the optical data indicate the existence of
clusters in the films, and the present VRH conduction is
thought to be mediated by the localized electronic states
associated with clusters.
On the other hand, the small silicon clusters of
atoms (Si-ACs) are the subjects of the intense study in
the hope for that their properties can provide a new
insight to the physical and chemical behaviour of the
nanodimensional materials [7-11]. Probably, the Si-NS
consist from Si-AC structures. The large surface-to-
volume ratio and the large step and defect density allow
one to expect that small Si-ACs will be highly reactive.
Moreover, the flexibility afforded by the small number
of atoms could give rise to novel structures, which can
possibly lead to the synthesis of artificial materials with
uncommon properties.
In the present paper, a new approach to the study of
Si-NS consisting of small and intermediate-size Si-ACs
is proposed.
Some calculations [7, 12, 13] predict that Si-ACs in
this size range have tetrahedral bonded network
structures whereas others predict more compact
structures. Taking the above-said into consideration, we
based on the following premises. First, for small Si-ACs
(n < 10), it is possible to construct a tree with core of
fourfold-coordinate atoms, similar to bulk Si (for
example, see [7]). Second, for larger Si-ACs
(10 < n < 100), it is possible to construct a network
containing a core of fourfold-coordinate atoms, similar
to bulk Si, which are surrounded by a surface of atoms
with optimal (threefold or fourfold) coordination, similar
to Si surfaces (as [10]). Structures of exceptional
Semiconductor Physics, Quantum Electronics & Optoelectronics, 2007. V. 10, N 4. P. 81-86.
© 2007, V. Lashkaryov Institute of Semiconductor Physics, National Academy of Sciences of Ukraine
82
stability can be obtained when exactly all surface atoms
participate in the surface reconstruction, and any change
in the size (addition or removal of atoms) introduces
“defects” similar to those on c-Si surfaces. To construct
a real Si-NS, our model used: 1) a core of fourfold-
coordinated atoms similar to bulk Si (complete Si-NS) or
2) 5-, 6-, and 7-membered Si-rings and their com-
binations for the construction of surface blocks (empty
Si-NS).
The nature of Si-NS and the reconstruction of Si-
ACs in the range size 1 < n < 100 (n is the number of
atoms in AC) remain open to intense debate too. Si-ACs
are a small piece of the nature in the range of nanometers,
are not molecules, and cannot represent themselves as the
bulk material as well. For example, J.R. Chelikowsky and
J.C. Phillips [12] have suggested that the small Si-ACs are
metallic rather than covalent in nature. This is due to of
their physical size. Clusters of covalently bound non-
metals usually have more open geometries that satisfy the
specific highly directional bonding requirements of silicon
and germanium. Si-ACs form prolate structures up to
n ≈ 25-35 [3, 5], but then they rearrange to more
polyhedral morphologies [8, 9]. In fact, the Si-AC appears
to be more closely related to the high-pressure metallic
phases of bulk silicon than to the diamond structure [7,
12-14]. These works appear to define metallic clusters as
those which have bond angles of 60°.
The absorption spectra from 0.94–5.58 eV are
obtained for gas-phase neutral Si-ACs containing 18-41
atoms [6]. The spectra of all clusters are essentially
identical. The optical signature of these clusters has
much in common with that of c-Si. These spectra are
unexpected: theoretical calculations predict a wide
variation in structure over the size range, but,
experimentally, a Si-AC has numerous strong sharp
absorption lines which do not shift in energy over the Si-
AC size range. Furthermore, the partial spectra of Si-AC
are smaller than Si18 and larger than Si41. The spectral
signature common to Si18−Si41 persists for sizes up to at
least 70 atoms. This spectral similarity is completely
unexpected. Because of the molecular point of view,
these clusters span a sufficiently large size range to have
structural differences which should show up in their
optical spectra.
It follows from these data that Si-ACs in this size
range are expected to undergo rapid structural changes
because of the large surface/volume atom ratio.
Nonetheless, major changes occur in the geometric
structures of Si-ACs as a function of size below
50 atoms. Such changes are mirrored in the photo-
electron spectra measured for Si-AC anions containing
3-12 atoms [8, 9]. Structural differences of Si-ACs
(similar size range) for positively charged ACs have also
been inferred from mobility and chemical reactivity data
[4]. Thus, the electronic spectra of neutral Si-ACs larger
than Si18 do not show any evidence for any structural
changes. This allows us to conclude that all of these
small silicon clusters share one or more common
structural entities. One possibility is that a small Si-AC
shares a common bonding network which persists and
extends as the cluster grows in size.
Yet, the absorption spectrum of crystalline silicon
has much in common with the silicon cluster spectra.
Strong absorption features at energies above 3 eV
dominate in both Si-AC and c-Si spectra. The 800-K
bulk spectrum, which is red shifted by the thermal
expansion of the lattice, more closely corresponds to the
cluster spectrum. Nonetheless, a number of discre-
pancies between the bulk and cluster spectra should still
be explained. For instance, the cluster spectrum has six
peaks between 3.0 and 5.6 eV, whereas the bulk
spectrum contains only three features. Furthermore, the
comparison with the 800-K bulk spectrum neglects the
effects of lattice contraction and quantum confinement
which are known for Si-NS, and these characteristics do
not change as a function of the AC size range.
Given that Si-ACs containing only few tens of
atoms are too small to have either band structures or
bulk excitations, the similarity of their optical signature
to crystalline silicon is unexpected. Moreover, these
spectra are far more similar to the spectrum of the most
stable crystalline form of Si than to the spectra of other
Si forms.
Furthermore, optical spectra of small InP-ACs
exhibit the absorption analogous to that of a-InP [13].
Other forms of more compact Si such as the β-tin and
primitive hexagonal metallic phases [14] are also
candidates for comparison in the light of theoretical
calculations, which predict high coordination numbers in
small Si-ACs. But none of these structures have optical
properties which are as closely related to Si-AC spectra
as c-Si. Moreover, Si-AC spectra cannot be easily
compared to Si surface spectra [6].
In addition, it is reasonable to concentrate on
models for experimentally observed sizes of exceptional
stability [the so-called magic numbers (MNs) [15]]. A
MN is observed and reflects differences in Si-ACs
[11, 12]. The MN can be understood basing on structures
predicted theoretically and observed experimentally. Si-
ACs of somewhat larger size (for example, 10 < n < 100)
show the dramatic changes in their chemical reactivity,
depending on the number of atoms in the cluster.
Using the thermodynamic force field, the
equilibrium structures of Si-ACs (n = 11-25) were
calculated in [12]. The following interesting results
about the structure of these clusters were obtained. Since
a Si-AC containing 7 atoms (and n = 13 or n = 19),
clusters have a pentagonal pattern (or icosahedral plus
the associated face capping). These results are fully
consistent with MNs found in the reaction rates for the
addition of first C2H4 molecules to Sin
+ clusters.
These experimental and theoretical data showing
the spectral regularity of Si-ACs challenge all known
theoretical models which are used for the construction of
space structures in the size range 10 < n < 100. This
leads to the intriguing conclusion that silicon clusters
containing tens of atoms are derived from a common
structural entity. We do not know the nature of this
Semiconductor Physics, Quantum Electronics & Optoelectronics, 2007. V. 10, N 4. P. 81-86.
© 2007, V. Lashkaryov Institute of Semiconductor Physics, National Academy of Sciences of Ukraine
83
entity, but its spectral similarities to the bulk crystalline
spectrum provide a firm ground for further theoretical
investigations. However, this is not the optimal practical
procedure for clusters with larger sizes: for a system
with many degrees of freedom, the problem of
identifying the lowest-energy configuration becomes
computationally difficult and depends on the sizes of
clusters. A Si-AC with size range 10 < n < 100 requires
to search for the possible new calculation schemes. In
addition to this, in order to elucidate the properties of
ACs, the development of different specific models has
great importance.
These facts became the motivation of our computer
researches. To attack this problem, we investigated the
electronic and geometric structures of Si-ACs
theoretically using the parametrized density-functional
theory (PDFT).
To study the dynamics of clusters, we offer an
approximate calculation scheme. This is the density-
functional theory (DFT) in the realization of Kohn and
Sham (KS), using a few empirical parameters. This
method, which is named parametrized DFT-PDFT, is
based on the Hartree-Fock scheme plus a proper treatment
of the electron correlation [16-18]. The use of only a few
parameters minimizes the effort for the determination of
the parameters, and it yields a close relation to full ab
initio DFT schemes (for example, GAMESS [16]). This is
a guarantee of the good “transferability” of the
parameters, while going from one system to another. On
the other hand, the use of some approximations in
connection with a few empirical parameters makes the
scheme of computations to be extremely fast. PDFT
allows also the study of dynamical processes through the
coupling with molecular dynamics (MD).
The method is based on the LCAO ansatz for the
KS wave functions:
( )jRrCr
rrr
−ς=Ψ ∑
µ
µµ)( . (1)
Here, µς − the atomic orbital which will be
utilized as basic wave functions (BWF) in the form
[10, 11]:
ml
rn YerN µ∗ α−−
µµ =ς 1 , (2)
where Nµ is constant; µα is the Slater’s parameter which
is defined by an algorithm [11]; mlY are spherical
harmonic functions, and n* is the effective quantum
number. The LCAO ansatz leads to the secular problem.
The matrix elements of the Hamiltonian vH µ and the
overlap matrix elements vSµ are defined as
νµµν ξξ≡ HH ˆ , νµµν ξξ≡S . (3)
In the offered method of PDFT, the Hamiltonian of
the system looks as
)(ˆˆ
eff rVTH
r
+= , (4)
where T
)
is the operator of kinetic energy, and ( )rV r
eff
is the effective KS potential. It is approximated as a
simple superposition of the potentials of neutral atoms
V j
0 :
( ) ( )∑=
j
jj rVrV
rr 0
eff , (5)
where jj Rrr
rr
−≡ . Such an approximation for the
potential is consistent with the following approximations
in the matrix elements of the Hamiltonian:
{ }
⎪⎩
⎪
⎨
⎧ ∈ν∩µξ++ξ νµ
µν
0
,, kjVVT
H kj . (6)
Thus, only two-center terms in the Hamiltonian
matrix are considered, but all two-center terms
H µν , S µν are calculated exactly.
The approximation formulated above leads to the
same structure of the secular equations as in the (non-
orthogonal) tight-binding (TB) or Iterative Extended
Huckel (IEHT) [11] schemes, but it has the important
advantage that all matrix elements are calculated, and
none of them is handled as an empirical parameter.
P. Blaudeck and co-workers [18] showed that the quality
of results, especially those concerning the binding
energies, can be improved distinctly by using basis
functions and potentials from slightly “compressed
atoms,” rather than from free atoms. This “compression”
is achieved by the introduction of an artificial additional
repulsive potential of the form ( )κ0rr in the atomic
calculations. The authors of works [16, 17] originally
developed such a potential for the optimization of
LCAO-LDA band structure calculations. The main
effect of this term is a damping of the long-range part of
basis functions. Practically, it optimizes the basis
functions concerning the approximations, mainly by
neglecting the 3-center terms.
The total energy ( )[ ]rE
r
ρ can be written in a form
using the KS eigenvalues iε as
[ ]
.
2
1
2
1
)]([
3
ext
3
eff
3
NXCXC
occ
i
i
ErVdE
rVdrVdrE
+ρ−+
+ρ−ρ−ε=ρ
∫
∫∫∑r
(7)
The external potential Vext is the electron-nucleus
potential, EXC is the exchange-correlation energy, VXC is
the corresponding potential, and EN stands for the
nuclear repulsion energy. The representation of the
density and the potential as a superposition of atomic-
like contributions, i.e., ∑ρ=ρ
i
i , ∑=
i
iVV , allows us
to write the total energy as
Semiconductor Physics, Quantum Electronics & Optoelectronics, 2007. V. 10, N 4. P. 81-86.
© 2007, V. Lashkaryov Institute of Semiconductor Physics, National Academy of Sciences of Ukraine
84
,
2
1~
2
1
2
1
2
1)]([
33
3
3
∑∑∫∑∑∫
∑∑∫
∑∑∫∑
′≠ ′
′
′
′
′
′
′
′
+ρ+
+
ρ
−
−ρ−ε=ρ
j j jj
jj
j j
jXCj
j j j
jj
j j
jj
occ
i
i
R
ZZ
rdVrd
r
Z
rd
rVdrE r
(8)
where jjR ′ is the distance between nuclei:
jjjj RRR ′′ −=
rr
, jZ is the charge of the j-th atom, ρj is
the electronic density localized on the j-th atom, and εi is
the one-electron energy.
For large internuclear distances, the electron-
nucleus energy compensates strongly the nuclear
repulsion energy, and the two-center terms with the
potential vanish also: ∫ =ρ ′ 03
jjrVd , jj ′≠ , due to the
screening of the potential. Assuming additionally that
∫ =ρ ′ 0~3
jXCVrd and writing the total energy )]([ rE r
ρ
of a single atom in the corresponding form as (8), one
can evaluate the binding energy approximately by the
KS eigenvalues εB and the KS energies
jnε of the atomic
orbital nj:
εB ≡ ∑∑∑ ∑ ε−ε≈−
j n
n
j
occ
i
ij
j
j
EE . (9)
The binding energy calculated in this way and the
binding energy from the full SCF-LDA calculation, by
using algorithm GAMESS [16] (∆E ≡ εB – EB
GAMESS) for
the Si-AC, are in good agreement, as it was illustrated in
[8, 9]. The differences between these energies increase
with decrease in the internuclear distance (r). However,
this increase is rather smooth, and the difference
practically vanishes already at distances about 3/2 r0.
Within the PDFT approximation, the forces acting
on the atoms (Fm) can be calculated easily using the
LCAO coefficients (Ciµ) and the derivatives of the
Hamilton and overlap matrices ⎟
⎠
⎞
⎜
⎝
⎛
∂
∂
∂
∂ µµ
j
v
j
v
x
S
x
H
, ,
respectively:
.∑
∑∑∑
′≠
′
µµ
µ
µ
∂
∆
+
+
⎥
⎥
⎦
⎤
⎢
⎢
⎣
⎡
∂
∂
+
∂
∂
−=
jj j
jj
j
v
i
j
v
vi
occ
i v
im
x
E
x
S
E
x
H
CCF
(10)
Having the forces, MD simulations can be easily
performed (see, e.g. [19]). In order to obtain the relaxed
atomic geometries for the models described below, we
have performed extensive energy-minimization calcu-
lations. The PDFT Si-AC database includes the total
energies of clusters, energy fragmentation, the overlap
integral matrix, charges on atoms, etc.
We have tested a new PDFT approach to
calculations of the total energy for a complicated atomic
geometry, which has included the electron-electron
interaction in a self-consistent manner.
We start with a bulk-like Si-AC size of 4 atoms and
examine the cluster-size dependence of electronic states
for Si up to 60 atoms. The AC-size dependence of the
calculated energy levels, local densities of states, and
charge distributions are examined for Si-AC, it is found
that a 10-atom cluster, which had three possible space
configurations was good enough for the AC-core to be
identified and to study the optical spectra [10]. To study
the regularity of the behaviour of big ACs (including
ACs with n = 18-41), it is very important to define the
properties of a Si-AC with 10 atoms. The PDFT method
(with different basis sets: STO-3G and HF/3-21G) was
utilized for the analysis of the geometry and electronic
structures of Si-ACs with the size range 1 < n ≤ 60.
In addition, the surface elementary Si-AC models
are investigated. The PDFT method is applied to the
isolated 5-, 6-, and 7- atomic rings reconstructed as a
fragment of the Si(111)2×1 surfaces of materials
crystallizing in the diamond structure. In Fig. 1, an
example of the simulation of a 5-atom silicon ring is
presented. We researched the possibility to derive a
planar Si-cyclic 5-AC structure. The energy transition is
present in Fig. 1. The displacements of the 1st, 4th, and
5th atoms are equal to ∆Rkj = 0.11 Å; for the 2nd and 3rd
atoms − ∆Rkj = 0.18 Å. We note that, as the surface Si-
atom is moved inward, the hybridization of the local
orbitals on the surface atom is changed to keep them
orthogonal. Therefore, we are forced to study the dehyb-
ridization of Si-atomic sp3-orbitals from the equivalent
form in terms of the relative positions of Si-atoms.
Fig. 1. The illustration of the possible space orientation of a 5-
membered cyclic Si-AC and the transition energy E (a), and
the direction of a displacement of Si-atom for obtaining the
planar AC (b).
Semiconductor Physics, Quantum Electronics & Optoelectronics, 2007. V. 10, N 4. P. 81-86.
© 2007, V. Lashkaryov Institute of Semiconductor Physics, National Academy of Sciences of Ukraine
85
The detailed algorithm to carry out the estimations
of a hybridization degree is described in [1]. Clearly, this
is consistent with the intuitive consideration that, upon
the relaxation (to the planar position), the dangling-
orbital becomes more pz-like and the back-bonds become
more s-like tending toward the trigonal sp2 form. The
dangling-orbitals on the raised atoms become s-like,
while those on the lowered atoms become more pz-like.
The calculated lengths of Si-Si bonds (rkj) and angles
between chemical bonds (α, β, γ) for the most probable
space configurations are given in Table 1.
Altogether, the theoretical Si-AC structure seems to
be somewhat less strained than the experimental ones.
The bond lengths obtained from the total energy
minimization vary in the interval 2.34−2.215 Å. The
bond angle variation around the ideal tetrahedron value
of 109.5° increases. These values indicate a slightly
higher strain in the same sequence.
The π-bonded chain structure with the 7- and 5-
atomic Si-clusters includes two critical bond angles.
They are remarkably smaller than the ideal tetrahedron
bond angle of 109.5° due to the adaptation of the 7-
atomic ring and the 5-atomic ring. From the chemical
point of view, such bond angles, < 90°, cannot occur
since the underlying structures are not stable.
The calculated geometry of the clusters is in good
agreement with results from the corresponding DFT ab
initio calculations, provided using the GAMESS
algorithm [15]. The nearest neighbour distances in the Si
lattice are obtained correctly. The energetic positions
and the equilibrium distances of high-pressure modifi-
cations of silicon are described rather well.
In the next step, the Si-NS was constructed. For the
interaction of Si-polyhedral clusters (Si-PCs), the
pairwise additive approximation (PAA) [9] was used.
The calculation analysis of Si-PCs, which have spherical
symmetry, was performed. Formally, the total energy of
an ensemble of N Si-ACs can be written as:
( ) ( )
( ) ....,,,,,,...
...,,,
+∑ Ω+
+Ω∑+∑ Ω=
<<<
<<<
N
lkji
ijkllkjljkilikijijkl
ijkjkijijk
N
kji
N
ji
ijijijN
rrrrrrE
rrErEE
(11)
Table 1. Bond lengths rkj (in Å) and bond angles α, β, γ (in
degrees) for 5-membered Si-ACs with different symmetries
(C2, Cs, D5h). The basis set is STO-3G (or HF/3-21G).
Si-AC rkj, Å
α β γ
k j rkj
1 3 2.234
3 5 2.271
C2
5 4 2.269
110.0º
(110.2º)
103.5º
(103.3º)
103.6º
(103.6º)
1 3 2.234
2 4 2.271
Cs
5 4 2.269
108.5º
(109.0º)
108.0º
(108.0º)
101.0º
(101.0º)
1 3 2.215
3 5 2.215
D5h
5 4 2.215
108.0º
(108.0º)
108.0º
(108.0º)
108.0º
(108.0º)
Fig. 2. Atomic cluster as a block of the nanodimensional
structure.
where the first term is the sum of all two-body (Eij)
interactions (each as a function of the separation rij and
the relative orientation Ωij of the two “balls”). The three-
body term Eijkl provides the difference between the
actual energy for a trio of Si-PCs in a given orientation
and the sum of the three pair potential terms; similarly,
Eijkl is the corresponding correction to give the correct
energy for a quartet of “balls”; etc. For the system which
consist of Si-PCs, we used the pairwise additive
approximation, by taking only the first term in the
above-presented expansion. The NS consists of Si-PCs
with average diameter d (as shown in Fig. 2). For the
choice of a potential (Eij, Eijkl), the stick-and-ball model
may be recommended.
In conclusion, we present the results of PDFT-
calculations for nanometer-size particles of silicon. The
geometric characteristics of Si60, having symmetry Ih, are
listed in Table 2. “Si-ball” is constructed from twelve 5-
atom Si-rings (5-Si-AC) and twenty 6-Si-ACs. Using
PDFT, we have estimated the diameter d of such a Si-
PC.
As seen from Table 2, the Si-Si interatomic distances
in 6- and 5- AC rings are reduced. These distances are
designated as rr66 and rr65 , respectively. The calculated
bond lengths and the diameter of Si-AC are given together
with errors. These data are in a quite good agreement with
Table 2. Geometric characteristics of Ih − Si60 Si-ACs.
°A,66r
r
°A,65r
r
°∆ A,r °A,d
Method and
basis set
2.189 2.226 0.037 123 ±
0.4
MIEHT-α [11]
2.062 ±
0.04
2.152 ±
0.04
0.085 ±
0.01
11.4 ±
0.4
STO-3G PDFT
2.213 ±
0.04
2.301 ±
0.04
0.088 ±
0.01
11.6 ±
0.4
HF/3-21G
PDFT
Semiconductor Physics, Quantum Electronics & Optoelectronics, 2007. V. 10, N 4. P. 81-86.
© 2007, V. Lashkaryov Institute of Semiconductor Physics, National Academy of Sciences of Ukraine
86
the other results (see [8, 9]). The Si-Si bond length was
generally overestimated by about 2 %, and the diameter of
Si-PC was calculated to within 3.5 %.
We have analyzed other polyhedral ACs of silicon
as well. Among them, we separate the following ones:
1) Si50 (D5h) −∆E = 0.0868 eV; 2) Si30 (C2v) − ∆E =
0.252 eV; 3) Si24 (D3) − ∆E = 0.404 eV. Here, ∆E
denotes the difference between the total energies of PCs
Si60 and Si50 (Si30, Si24). The 60-atomic Si-structure has
appeared more stable than 50, 30, 24 Si-ACs. The given
fact testifies that the probability of the synthesis for such
stable structures as AC Ih − Si60 is extremely high.
Therefore, polyhedral silicon structures took part in the
creation of the Si-NS and were identified experi-
mentally [1, 2, 4].
Summary. We have tested the modified DFT
scheme in calculations of the total energy for a
complicated atomic geometry which included some
useful simplicity. Due to both the neglect of all three-
center integrals and the use of a short-range repulsive
interaction potential in calculations of the energies, the
method is computationally extremely fast. It gives
reliable results for the geometries, binding energies, and
vibration frequencies of different ACs. The method is
applied to the building of the Si-PC from the Si-surface
“blocks” reconstructed as planar. The bond lengths and
the angles are obtained with errors less than 5 %.
Three important reconstruction steps have been
considered: 1) the isolated Si-cyclic ACs (with 5-, 6- and
7-atoms), 2) the Si-PC which consists of Si-cyclic ACs,
and 3) the cluster model of Si-NS. We have found a
reasonable agreement of the different structures for the i-
fold Si-AC rings on the elementary cell surface.
However, there are also the characteristic discrepancies
with respect to the order of magnitude for several effects
and quantities: the buckling amplitude and the bond
length in a chain, relaxation in the first and second
atomic layers, bond angles. But these problems will be
considered in the following publications.
The PDFT scheme may be viewed as a “hybrid”
between the pure ab initio method − based on DFT − and
the use of purely empirical potentials. It has the
advantage over the latter in overcoming the
"transferability" problem, and it requires much less
computational efforts than full ab initio methods. Even
in comparison with the traditional TB schemes which are
computationally as fast as our method, the transferability
is much better, since few parameters are used, and their
determination is straightforward.
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