Band structure of non-steiochiometric large-sized nanocrystallites
A band structure of large-sized (from 20 to 35nm) non-steichiometric nanocrystallites (NC) of the Si₂₋xCx (1.04 <x< 1.10) has been investigated using different band energy approaches and a modified Car-Parinello molecular dynamics structure optimization of the NC interfaces. The nonsteichio...
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irk-123456789-1189642017-06-02T03:03:22Z Band structure of non-steiochiometric large-sized nanocrystallites Kityk, I.V. A band structure of large-sized (from 20 to 35nm) non-steichiometric nanocrystallites (NC) of the Si₂₋xCx (1.04 <x< 1.10) has been investigated using different band energy approaches and a modified Car-Parinello molecular dynamics structure optimization of the NC interfaces. The nonsteichiometric excess of carbon favors the appearance of a thin prevailingly carbon-contained layer (with thickness of about 1 nm) covering the crystallites. As a consequence, one can observe a substantial structure reconstruction of boundary SiC crystalline layers. The numerical modeling has shown that these NC can be considered as SiC reconstructed crystalline films with thickness of about 2 nm covering the SiC crystallites. The observed data are considered within the different one-electron band structure methods. It was shown that the nano-sized carbon sheet plays a key role in a modified band structure. Independent manifestation of the important role played by the reconstructed confined layers is due to the experimentally discovered excitonic-like resonances. Low-temperature absorption measurements confirm the existence of sharp-like absorption resonances originating from the reconstructed layers. Досліджено зонну структуру широко-розмірних Si₂₋xCx (1.04 <x< 1.10) нанокристалів (НК) з розмірами від 20 до 35 нм, використовуючи різні методи розрахунків зонної структури і модифікований молекулярно-динамічний метод геометричної оптимізації інтерфейсів Кара-Парінелло. Нестехіометричний надлишок вуглецю сприяє появі тонких (біля 1 нм) шарів вуглецю на поверхнях НК. В результаті спостерігається істотна перебудова граничних кристалічних шарів SiC. Числове моделювання показало, що ці НК можуть розглядатись як модифіковані структури кристалів SiC з розмірами біля 2 нм. Отримані результати розглядаються в рамках різних одно-електронних підходів. Незалежним експериментальним проявом цих наноефектів є екситонні ефекти, які проявляються в низькотемпературних спектрах поглинання як гострі абсорбційні резонанси, що походять реконструйованих шарів 2004 Article Band structure of non-steiochiometric large-sized nanocrystallites / I.V. Kityk // Condensed Matter Physics. — 2004. — Т. 7, № 2(38). — С. 401–420. — Бібліогр.: 27 назв. — англ. 1607-324X PACS: 71.20 DOI:10.5488/CMP.7.2.401 http://dspace.nbuv.gov.ua/handle/123456789/118964 en Condensed Matter Physics Інститут фізики конденсованих систем НАН України |
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Digital Library of Periodicals of National Academy of Sciences of Ukraine |
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| language |
English |
| description |
A band structure of large-sized (from 20 to 35nm) non-steichiometric nanocrystallites
(NC) of the Si₂₋xCx (1.04 <x< 1.10) has been investigated
using different band energy approaches and a modified Car-Parinello
molecular dynamics structure optimization of the NC interfaces. The nonsteichiometric
excess of carbon favors the appearance of a thin prevailingly
carbon-contained layer (with thickness of about 1 nm) covering the crystallites.
As a consequence, one can observe a substantial structure reconstruction
of boundary SiC crystalline layers. The numerical modeling has
shown that these NC can be considered as SiC reconstructed crystalline
films with thickness of about 2 nm covering the SiC crystallites. The observed
data are considered within the different one-electron band structure
methods. It was shown that the nano-sized carbon sheet plays a key role in
a modified band structure. Independent manifestation of the important role
played by the reconstructed confined layers is due to the experimentally
discovered excitonic-like resonances. Low-temperature absorption measurements
confirm the existence of sharp-like absorption resonances originating
from the reconstructed layers. |
| format |
Article |
| author |
Kityk, I.V. |
| spellingShingle |
Kityk, I.V. Band structure of non-steiochiometric large-sized nanocrystallites Condensed Matter Physics |
| author_facet |
Kityk, I.V. |
| author_sort |
Kityk, I.V. |
| title |
Band structure of non-steiochiometric large-sized nanocrystallites |
| title_short |
Band structure of non-steiochiometric large-sized nanocrystallites |
| title_full |
Band structure of non-steiochiometric large-sized nanocrystallites |
| title_fullStr |
Band structure of non-steiochiometric large-sized nanocrystallites |
| title_full_unstemmed |
Band structure of non-steiochiometric large-sized nanocrystallites |
| title_sort |
band structure of non-steiochiometric large-sized nanocrystallites |
| publisher |
Інститут фізики конденсованих систем НАН України |
| publishDate |
2004 |
| url |
http://dspace.nbuv.gov.ua/handle/123456789/118964 |
| citation_txt |
Band structure of non-steiochiometric large-sized nanocrystallites / I.V. Kityk // Condensed Matter Physics. — 2004. — Т. 7, № 2(38). — С. 401–420. — Бібліогр.: 27 назв. — англ. |
| series |
Condensed Matter Physics |
| work_keys_str_mv |
AT kitykiv bandstructureofnonsteiochiometriclargesizednanocrystallites |
| first_indexed |
2025-07-08T14:58:55Z |
| last_indexed |
2025-07-08T14:58:55Z |
| _version_ |
1837091245572227072 |
| fulltext |
Condensed Matter Physics, 2004, Vol. 7, No. 2(38), pp. 401–420
Band structure of non-steiochiometric
large-sized nanocrystallites
I.V.Kityk∗
Institute of Materials Science and Metallurgy, Polish Academy of Science.
Ul. Reymonta, 25, Cracow, Poland
Received December 8, 2004, in final form April 26, 2004
A band structure of large-sized (from 20 to 35nm) non-steichiometric nano-
crystallites (NC) of the Si2−xCx (1.04 < x < 1.10) has been investigat-
ed using different band energy approaches and a modified Car-Parinello
molecular dynamics structure optimization of the NC interfaces. The non-
steichiometric excess of carbon favors the appearance of a thin prevailingly
carbon-contained layer (with thickness of about 1 nm) covering the crystal-
lites. As a consequence, one can observe a substantial structure recon-
struction of boundary SiC crystalline layers. The numerical modeling has
shown that these NC can be considered as SiC reconstructed crystalline
films with thickness of about 2 nm covering the SiC crystallites. The ob-
served data are considered within the different one-electron band structure
methods. It was shown that the nano-sized carbon sheet plays a key role in
a modified band structure. Independent manifestation of the important role
played by the reconstructed confined layers is due to the experimental-
ly discovered excitonic-like resonances. Low-temperature absorption mea-
surements confirm the existence of sharp-like absorption resonances orig-
inating from the reconstructed layers.
Key words: pseudopotential, band structure, nanocrystallites
PACS: 71.20
1. Introduction
Recently, there appeared a new direction in the nano-technology consisting in
the possibility of using the nanocrystallites as materials for optics and electronics
due to the size-dependent electronic and optical properties of these materials [1–8].
The so-called large-size nanocrystallites (NC) (larger than 10 nm) are of particular
interest. They demonstrate properties typical of the bulk crystalline materials as
well as nano-sized effects originating from SiC reconstructed NC layers similar to
quantized nano-dot structures. One can expect that the latter will be directly de-
pendent on relative sizes of particular crystallites and the corresponding interface
∗E-mail: i.kityk@wsp.czest.pl; kityk@torus.uck.pk.edu.pl
c© I.V.Kityk 401
I.V.Kityk
environment. Moreover, one can expect the occurrence of excitonic interactions. Co-
existence of bulk-like long-range ordered and quantum size-confined layers opens a
rare possibility to operate by electron energy dispersion within the same crystallites
contrary to the thin microcrystalline films deposited on crystalline surfaces [8].
Among many NC materials, the SiC crystallites were chosen for the modelling,
since the technology of their production to obtain desired sizes was well developed.
An energy gap of the SiC can vary within wide spectral ranges (from 2 to 4.5 eV),
depending on a polytype structure. Furthermore, SiC is substantially more stable
towards external mechanical and thermal treatments compared to other semicon-
ductors.
The main goal of the present work is to model and simulate the effect of the
NC sizes and carbon excess on the optical absorption of the large sized Si2−xCx.
Contributions of the reconstructed near-the-interface states as well as the excitonic
resonances to the observed absorption spectra have been separately evaluated.
The nano-sized confined effects are usually studied for the NC with sizes below 8-
10 nm, where clear effects of quantization are observed. In such cases, k-space bulk-
like dispersion disappears and discrete excitonic-like levels appear in the energy
gap. However, the quantum-confined interface possessing the bulk-like as well as
the dot-like quantized excitonic properties can be of interest for the experimental
optical phenomena. These features are caused by the reconstructed SiC surfaces
with the thickness up to 2 nm, separating the bulk-like crystallites and surrounding
amorphous-like background.
Depending on a relative content of the quantum confined states in the total
NC, one can evaluate relative contributions of the reconstructed SiC near the grain
boundary to the band energy (BE) dispersion as well as to the optical absorption
(directly connected with the imaginary part of dielectric susceptibility).
In section 2, several technological details of the Si2−xCx large-sized NC synthesis
are presented. Monitoring the Si2−xCx NC structural and morphological features is
achieved by the use of the TEM, NMR, Raman and IR spectroscopic methods. A
calculation technique for the molecular dynamics interface structure optimization is
described in section 3. Results of the band structure (BS) calculations performed
by different calculation techniques and comparison with the experimental optical
data are presented in section 4. Particular attention is devoted to simulation of
optical absorption induced by the BS of NC reconstructed levels as well as to the
corresponding excitonic effects.
2. Sample features
The SiC large-sized nanopowders were synthesized by a CO2 laser pyrolysis of
silane and acetylene gaseous mixtures following a technical device described in [2].
402
Band structure of non-steiochiometric large-sized nanocrystallites
2.1. Specimen synthesis, structure and morphology characterizations
The reactant fluxes monitor the C/Si ratios or silicon-rich network in the outer-
most Si2−xCx nanoparticle surfaces. The present work bears on two SiC batches
with atomic carbon excess of about 4%(x = 1.04) and 10%(x = 1.10). The former
was synthesized by a flow rate of about 602 cm3/mn for silane and 314 cm3/mn for
acetylene.
The Si2−xCx (with x = 1.10) was sunthesized by a flow rate of about 120 cm3/mn
and 66 cm3/mn, for silane and acetylene, respectively. Chemical analyses of the as-
formed powders shows only a low oxygen contamination (< 3% weight) through
the formation of silicate. Annealing at 1200◦C was performed under argon in a high
temperature graphite furnace at 600◦C/min heating rate and a dwell time of about
1 hour. Such a treatment favors homogenization and densification of the powders
without altering the average size of the nanoparticles (26 nm for x = 1.04) and
(50 nm for x = 1.10). All the forthcoming experimental investigations concern the
nanopowders annealed at 1200◦C.
In TEM investigations for x = 1.10 batch, the NC exhibit rounded shapes with
sizes between 10nm and 100 nm and mono-domain-like single crystals with stacking
faults. The external particle surface is covered with a thin (≈1 nm) layer of carbon.
The major (SiC) phase, more or less coating the grain surface, is identified as a
polyaromatic carbon.
amorph.
a+b
a
a
12 0 -12 -24 -37
Figure 1. NMR spectrum of the Si0.96C1.04. The decomposition in several lines
reflects coexisting structures (6H-SiC, 3C-SiC and amorphous SiC).
NMR investigations of short range were performed for x = 1.04 specimens. The
Si spectrum (figure 1) forms the polytypes involved in the sample. The triplet lines
located at −25,−20 and −16 ppm with 1:1:2 relative ratios are associated with the
6H-SiC structure. The single −16 ppm line is connected to 3C-SiC. The content of
the 6H-SiC is of about 26%, whereas the 3C-SiC has a relative concentration of 59%.
403
I.V.Kityk
An amorphous structure is also involved with a relative contribution of about 12%.
For x = 1.04 batch, the powder is formed by agglomerated SiC grains with round
shapes and a size ranging from 3 to 40 nm. A layer of about ≈ 0.5 nm thickness covers
the crystals and consists of amorphous silicon oxide or possibly silicon oxycarbon.
500 750 1000 1250 1500 1750 500 750 1000 1250 1500 1750
SiC177-1200 C SiC212-1200 C
Frequency(cm )-1Frequency(cm )-1
Figure 2. Raman spectra of the Si2-xCx with different x: (a) x = 1.10; (b)
x = 0.96.
2.2. Raman and IR investigations
The Raman measurements were performed using an Ar+ ion laser (λ = 488 nm).
The spectral width of about 4 cm−1 was a compromise between good resolution,
sufficiently scattered light and low power absorbed by the sample (3 mW) in order
to avoid the specimen overheating. The Raman spectra of the Si0.90C1.10 batches
are presented in figure 2. These spectra demonstrate the presence of graphite with
sp2 and sp3 bonding. The corresponding spectral bands located between 1347 and
1595 cm−1originate from the in-plane E2g mode (1595 cm−1) and in-plane zone-
boundary A1g mode (1347 cm−1). However, finite crystallite sizes or disorder in
the Graphtec arrangement can lead to an active Raman of the A1g mode. Anyway,
the Raman investigations are understandable through a rich carbon composition in
external particle surfaces with high scattering efficiencies capable of screening the
bulk particle mainly composed of SiC structures. No Raman signature from SiC was
clearly detected by the Raman spectra. This means a large absorption coefficient of
the surface composition.
The IR transmission spectrum of the Si0.96C1.04 sample incorporated in KBr
platelet is shown in figure 3. The main features consist in a characteristic SiC IR
transmission band centered at 830 cm−1 and a weak band located at 1150 cm−1.
The latter is accounted for by the Si-O stretching vibrations that occur in the SiO2.
Similar features are observed in the IR spectrum of the Si0.90C1.10 sample where a
complete coverage of the particle by thin carbon layer is realized. This permits to
assertain the transparency of the Si2−xCx in the IR range in contrast to the behavior
in the visible range as developed above for Raman investigations.
404
Band structure of non-steiochiometric large-sized nanocrystallites
Frequency(cm )-1
IR
T
ra
n
sm
is
si
o
n
(a
.u
.)
Figure 3. IR absorption spectra for the Si1.04C0.96 after oxidation.
3. Molecular dynamics NC layer’s structure optimization
Silicon carbide occurs in several closely inter-related structural modifications that
hardly differ one from another.
3.1. Simulations of the carbon sheets
All the structural modifications considered have tetrahedral bonds and their local
arrangement is the same up to the second neighbors. All these structural types,
expressed in terms of hexagonal units, have almost the same value of the lattice
constant a = 5.825 a.u., but a value of c depends on the polytype kind. Their
structural difference may be understood by considering the stacking sequence of the
hexagonal-like bi-layers (formed with pairs of C and Si layers). The atoms in each
bi-layer have the following positions: α (0,0,c1); β (1/3, 2/3, c2) and γ (2/3,1/3,c3).
At the same time, a vertical sequence of the bi-layers for the cubic polytypes are:
αβγαβγu These bi-layers have cubic-like as well as hexagonal-like site-symmetries.
Due to the dominant role played by the hexagonal-like structural component, the
effective hexagonal-like (wurtzite) lattice and corresponding Brillouine zone (BZ)
will be considered here to be the basic ones during the BE calculations. Ratios of
different structural fragments (α-SiC and β-SiC ) were assumed when determining
the starting structural factors used for the BE calculations and were determined
from the NMR data (see figure 1).
405
I.V.Kityk
3.2. Covering carbon sheet
The structure and coordination bonding of carbon sheet covering the SiC nanopar-
ticles were optimized using molecular dynamics (MD) calculations. Reconstructions
with the carbon sp-bonded chains are energetically favored compared with the sp3-
like C bonds. The energy difference between two successive reconstruction geome-
try differs only by a few meV/dimmer. Moreover, the formation energy of sp3-like
disturbances on a sp-bonded interface rapidly decreases with the increasing defect
concentration. These lead to sp→ sp3 transition on the crystallite surfaces.
Recent experiments [9] have independently confirmed that stable sp3-like bonded
carbon lines can be formed on the C-terminated SiC〈001〉 surface, by inducing an
irreversible sp→ sp3 phase transition in the substrate. These sp3 bonds are possible
precursors for the diamond growth on SiC surfaces.
Self-consistent calculations [7,8] using the DFT approach within the local-density
approximation (LDA) have shown that the most probable reconstruction of the C-
terminated surface of cubic SiC〈001〉 is a carbon possessing sp3-like carbon chains.
Independently, experimental investigations [10] have shown that hydrocarbon evap-
oration on Si-terminated substrates yields sp-bonded carbon atoms, usually referred
to as bridge reconstruction. The stability of sp3- and sp-bonded reconstruction,
as inferred from first principles calculations using DFT and gradient corrected ex-
change energy functional was determined by the so-called generalized-gradient ap-
proximations (GGA) [11]. Possible coexistence of the sp2 → sp3 transition observed
experimentally was analyzed and has shown little probability.
3.3. Molecular dynamics simulations of reconstructed SiC layers
The geometry optimization of the reconstructed Si2−xCx surfaces was started
from the interface between the NC and carbon sheet background (calculation details
are similar to the described by Albercht et.al., [12] (see figure 4).
Figure 4. Principal schema of the grain obtained from the TEM data: I – crys-
tallite phases; II – interface region; III – amorphous-like region.
The TEM and other structural data indicate that the investigated NC are formed
by proper crystallites with a given α/β SiC ratio. The TEM data show that near-
the-interface layers have thickness changing from 2 to 4 nm. The EPR, X-ray and
406
Band structure of non-steiochiometric large-sized nanocrystallites
other structural investigations [2] have shown that the near-the-interface levels play
a key role in the electronic properties of the SiC NC. It is also seen from the above
mentioned figures that one chosen segment of the interface (squared in figure 4)
is responsible for electronic properties of the whole composite. Sizes of the large-
sized NC will define only a partial contribution of the interface regions to the total
electronic behavior.
The optimization was started from four neighboring layers: two layers from
the crystallite side (indicated by I) and two layers in the direction of the carbon
amorphous-like sheet background (indicated by II and III). 50–60 atoms from both
sides of the interface have been taken into account. The MD procedure was carried
out until the total energy minimum value for a given cluster was the same as for
the whole cluster (re-normalized by one ion). At the second step, the next proper
layers were considered and the procedure was repeated for the new total energy per
ion. The iteration process was repeated until the relative ion displacement for two
successive layers was less than 0.2 Å. The latter was found to be a limit in a precision
when determining the atom position within the framework of the adopted model.
More details concerning structural optimizations of the interfaces are given in the
subsequent sections.
Afterwards, the structure of the surrounding (near-the-interface SiC) sheets has
been additionally optimized using the Car-Parrinello ab initio molecular dynamics
(CPMD) [13] within a DFT pseudopotential description. The functional used in the
CPMD was of the BLYP type [14,15]. The atom core positions and the PW expan-
sion coefficients were treated as simultaneous dynamic variables. Simulations in this
approach were performed using super-cells containing 128 atoms with a PW cutoff
of 37 Ry. Periodic boundary conditions and Γ point sampling of the Brillouin zone
were used. The calculated total energies using Γ point were found to be accurate
to better than 0.05 eV per atom. Martins-Trouiller pseudopotentials ([16]) in the
Kleinmann-Bylander form were used to represent the carbon ion cores. This pseu-
dopotential was checked for accuracy against the well-known molecules (benzene,
methane, acetylene) possessing different carbon configurations.
At a PW cutoff of 40 Ry, the computed bond lengths fell within 2% of the
observed experimental values. The so-called “liquid quench” method was employed
to produce the final network at each density C configuration. The relative stability of
the reconstructed interfaces was studied in order to address the controversy between
the experiment and the existing DFT-LDA calculations. Previous evaluations have
shown that the LDA approach underestimates the SiC bulk lattice constant by
1.69% and the pseudopotential approximation overestimates it by 0.33%. The more
preferable geometry corresponds to the lattice constant of about 4.312 Å.
3.4. Optimization results of the reconstructed NC layer structure
The performed calculations have shown that the near-the-interface SiC region
(II in the figure 4) may be considered to be the reconstructed SiC quantum confined
surfaces deposited on the SiC. The evaluated structural data are presented in ta-
ble 1. Since they are dependent on the distance from SiC layer-crystalline interface,
407
I.V.Kityk
Table 1. Changes of effective structural parameters of the SiC reconstructed
surfaces while moving from the amorphous-like level (region II–I) (carbon layer-
SiC crystallite) towards the deeper crystalline layers (III).
Distance deff from the
carbon – SiC interface
in the direction of the
crystallite states [nm]
a [nm], SiC-3C
(cubic structural
component)
a [nm], SiC-6H
(hexagonal struc-
tural component)
c/p (hexago-
nal structural
component)
0.2 4.231(3) 2.889(4) 2.515(2)
0.4 4.256(4) 2.916(3) 2.515(1)
0.6 4.286(8) 2.937(1) 2.515(6)
0.8 4.309(1) 2.953(5) 2.514(4)
1.0 4.321(5) 2.974(4) 2.514(2)
1.2 4.326(7) 2.989(2) 2.514(3)
1.4 4.331(3) 3.036(4) 2.514(9)
1.6 4.346(1) 3.068(1) 2.513(3)
1.8 4.352(3) 3.071(3) 2.513(1)
2.0 4.359(4) 3.072(1) 2.513(0)
appropriate results are presented versus the distance d.
One can see that deff shows a considerable increase of lattice parameters while
moving from the carbon sheet towards deeper SiC layers. For the deff larger than
2 nm, the reconstruction of the structure disappears and the bulk-like crystallite
structure begins to dominate. It is striking that for the hexagonal structural compo-
nent the c/p parameters are more stable comparing with the a ones. The presented
data show that the thickness of interface region is equal to about 2 nm which is
an additional confirmation of the good agreement between the experimental and
theoretical data.
Simulated by the above mentioned method, differential local charge density dis-
tributions versus the deff distance are presented in figure 5. One can clearly notice a
substantial charge density redistribution determining the electronic structure of the
reconstructed surfaces. Glans et al., in 2000 independently demonstrated that the
SiC epilayers deposited on the SiC-3C surfaces show a clear interface reconstruction.
Since the sizes of the SiC crystallites are relatively large (of about 20 nm), a model
similar to the SiC epilayers deposited on the SiC crystallites can be proposed.
4. Band energy structure calculations
4.1. Choice criterion for calculation technique
Based on the structural data for the NC layered interfaces, one can expect the
appearance of the BS with a substantially modified k-dispersion. Due to the de-
creasing effective lattice constants, the existence of a flatter band energy dispersion
408
Band structure of non-steiochiometric large-sized nanocrystallites
(a) deff=0.6 nm (b) deff=1.1 nm
(c) deff=1.5 nm
Figure 5. Differential charge density of the interface depending on the distance
d from the interfaces.
and an increasing effective energy gap value can be expected.
There exist several works devoted to the BS calculations of the perfect bulk SiC
crystals ([17,18]). A choice of the BS calculation technique depends on the kind of
properties we would like to simulate. Because we are interested in simulations of the
optical features, a main criterion consists in maximal coincidence of the experimental
optical parameters with the simulated ones. Therefore, we apply several band energy
approaches, such as a self-consistent full linear augmented plane wave approach
within the local density approximation (FLAPW-LDA) ([19]), a modified norm-
conserving pseudopotential (NCPP) ([20]), a semi-empirical pseudopotential method
(SEPM) ([21]). Among the BS calculation techniques, these methods seem to be
more appropriate today in order to interpret the optical data. We will thus perform
a comparison of the obtained theoretical data with the experimental ones.
4.2.1. Norm-conserving pseudopotential method
The total energy of crystalline system is expressed within a local density func-
tional approximation (LDFA) with respect to the charge density ρ(r). The nuclear-
electron functional potential Vn−e[ρ(r)] is chosen to be in the form described by
409
I.V.Kityk
BACHELET, 1982. During the calculations there were included screened potentials
Ve−e[ρ(r)] and Ve−c[ρ(r)], corresponding to Coulomb contribution and exchange-
correlation energy, respectively.
A nonlinear extrapolation procedure was carried out in order to calculate the
fitting coefficients of the corresponding pseudo-wave-functions as well as of corre-
sponding derivatives, in a form convenient for analytic evaluations of secular equa-
tion matrix elements, in particular :
ψ(l, r, β) =
∑
n
aβ
nr
n exp
[−a(l,β)
n rn
]
, (1)
where β denotes the atom type; l is a corresponding angular momentum; n deter-
mines a level of precision of the nonlinear fitting procedure and can vary from 1
to 5. The aβ and α
(l,β)
n coefficients are the fitting coefficients evaluated during the
nonlinear fitting procedure with a precision of the fit less than 0.03 eV.
The atomic pseudopotential was chosen in non-local (l-dependent) form:
V (l,β)
ps =
3∑
i=1
Aβ
i + r2Aβ
i+3 exp
[
−a(l,β)
l r2
]
, (2)
where Ai,Ai+3, α
(l,β)
1 are the fitting coefficients. The calculated atomic potentials
were used as starting ones in forming the solid-state form-factors.
When we go from the atoms to a long-range plane wave basis set, we do the
Fourier transformation from the real r-space up to the reciprocal k-space. The ob-
tained Fourier transformers served as basic for the secular equation. The secular
equation was solved appropriately to a pseudopotential expressed by equation 2:∥∥∥∥∥
[
h2 (k +Gn)2
2m
− E(k)
]
δn,n′ +
∑
β
Vβ (G′
n −Gn)Sβ (G′
n −Gn)
∥∥∥∥∥ = 0, (3)
where E(k) is an eigenenergy for the k-point in the Brillouin zone, G′
n, Gn are
wavevectors of the interacting basis plane waves. A structural form-factors for βth
type atoms can be expressed as follows:
Sβ (G′
n −Gn) =
g(β)
ΩNa
∑
exp [−l (G′
n −Gn) τβ]
. (4)
The positions of the βth atoms are determined from the above described molecular
dynamics procedure.
Here g(β) is a weighting factor determining a partial contribution to the to-
tal potential of a given structural component (for example α/β-SiC ratio) or of
the reconstructed layer presented above. A similar approach for different structural
fragments has been successfully applied for binary solid alloys, glasses and organic
materials ([23]). One can expect that this approach would be also appropriate for
different disordered and partially ordered solids including the investigated SixC1−x
410
Band structure of non-steiochiometric large-sized nanocrystallites
NC. Hexagonal-like BZ was chosen as a basic one because the wurtzite-like structure
is more important (see section 2).
Therefore, the g(β)weighting factors (determining the atom positions of the recon-
structed surfaces) were introduced into appropriate structural factors corresponding
to the contributions of different crystal subsystems. As a consequence, the recon-
structed interface contribution is directly taken into account within the mentioned
approach.
Electron screening effects were taken into account during the calculations of
the total potential (7) using a parameterized Perdew-Zunger ([24]) approach. A
special Chadhi-Cohen point method was applied in calculating the spatial electron
charge density distribution. A diagonalization procedure was carried out at special
weighting points of the BZ for each structural type.
Acceleration of the iteration convergence was achieved by transferring 75% of
the (m−1)th iteration result to the mth iteration. The following condition was taken
as a criterion for self-consistency:∣∣∣∣(ρm − ρm−1)
ρm
∣∣∣∣ < ε. (5)
During the calculations, a level of calculation error (ε) was smaller than 0.10%.
To verify the dependence of BE calculation results on the choice of the α-SiC/β-SiC
ratio, additional calculations were done for the perfect α-SiC and β-SiC single crys-
tals. The level of agreement with previously performed calculations of the electron
density of states using LMTO methods was equal to about 0.3 eV. However, the
main drawback of all one-electron LDF-LDSA calculations consists in underestima-
tion of the band gap values. For this reason, a self-energy correction renormalization
was carried out.
A basic BZ wurtzite-like hexagonal structure was chosen because it corresponds
to the major structural component of the investigated NC. While varying struc-
tural factors obtained from the structural data and the MD geometry optimization,
appropriate changes were introduced in equation (6).
In order to take into account the core-like state contributions, the norm-conserving
PP wavefunctions were additionally modified through their orthogonalization to the
LCAO wavefunctions. As a result, the total energy deviation from the energy cutoff
and the Perdew-Alder screening parameter was stabilized within 0.22 eV.
4.2.2. Local-density-derived semi-empirical pseudopotentials (SEPM)
As a second method in calculating the semiconductor quantum structures, we
have applied a modified semi-empirical pseudopotential (SEPM) approach intro-
duced in [21]. This approach is proposed explicitly for the quantum dots, wires,
wells and films with typical linear dimensions of 2–10 nm. A description of the elec-
tronic properties of such cluster structures using first-principles methods pertinent
to bulk solids ([25]) is currently prohibitive. The SEPM approach is much faster
than the self-consistent first-principles methods used in the case of the bulk solids.
411
I.V.Kityk
Indeed, the Schrodinger equation is solved and an efficient diagonalization method
providing energy levels in a fixed “energy window” is available and appropriate.
Unlike effective-mass-based approaches ([17]), this method uses explicit and varia-
tionally flexible basis functions, permitting a direct comparison of the wave functions
with local density approximation LDA studies when available. However, contrary to
the LDA approach, the current method provides only electronic structure informa-
tion, transition probabilities, wave functions but not ground-state properties such
as equilibrium geometries, which have to be assumed at the outset.
The general mathematical formalism is similar to that described in section 4.2.1.
In a scalar local-density approximation (SLDA), a scalar-relativistic atomic pseu-
dopotential is obtained using the Troullier–Martins ([16]) procedure in order to
perform a core-like correction. A plane-wave basis with a kinetic energy cutoff of
36.5 Ry was used. The non-local part of the ionic pseudopotential V was obtained
numerically. It was kept unchanged while moving from the LDA to the SLDA. The
“small box” implementation was used to handle the nonlocal part of pseudopoten-
tial. This approach takes an advantage of the short-range nature of the nonlocal
part and uses the plane waves (PW) basis set with large momentum to expand it in
the Fourier space.
It is well known that the LDA band gaps are usually underestimated. Therefore,
it is adjusted to reproduce the experimentally observed BE structures. The changes
in the SLDA potentials are small comparing with the LDA results. The performed
calculations have shown that sufficient sizes of the secular matrix varied from 170
to 240.
Since the final pseudopotential was rather smooth, a rapidly converged PW ex-
pansion was possible. In fact, using the efficient diagonalization method and the
full semi-empirical pseudopotential, the electronic structure for 90 atoms of the SiC
supercluster was calculated.
4.2.3. FLAPW-LDA method
Among different BE calculation methods, the augmented plane wave (APW)
methods were effectively developed. The well-known WIEN97 30 version is the full
linear augmented plane wave local density approximation (FLAPW-LDA) and, in
accordance with ([19]), the exchange-correlation Hedin-Lundquist ([26]) parameter-
ization was performed. The SiC PW basis set was taken for the energy cutoffs of
about 15 eV and potential representation of about 52 eV. The expansion by angular-
dependent spherical harmonics was done for l < 10. Approximately 16 states have
been taken into account including 6 empty ones with a criterion of the self-consistent
eigenenergy value stabilization of about 0.016 eV. Summation within the BZ was
performed at 8 special points of the BZ. All the core states (1sC, 1sSi, 2sSi, 2pSi)
were recalculated during all iteration steps. A Thomas-Fermi screening potential was
applied at the starting step and afterwards it was replaced by the Engel screened
functions ([26]). Additional corrections were done using the generalized gradient
approximation (GGA) to prevent energy gap underestimation.
The plane wave basis set was limited by wave vectors smaller than 9/Rm ( where
412
Band structure of non-steiochiometric large-sized nanocrystallites
Figure 6. Fragment of the BE structure of the SiC calculated with inclusion of the
near-the-interface inclusion projected into the hexagonal-like structure calculated
by the modified NCPP method. The points indicate the reconstructed BE points.
Rm is a muffin-tin sphere radius). The 4H structure requires 1340 plane waves and
3C-structure needs 312 ones. The radius of the Si sphere was varied within the range
RSi = (1.08−1.16)RC. The interacting spheres were chosen as non-overlapping (just
touching) ones. The 1s C and 1,2s Si wavefunctions were taken as core states. All
remaining orbitals were considered to be valence states orbitals.
For numerical calculations of the total and partial density of states (DOS), nu-
merical simulations were conducted by means of a tetrahedron method using 96 k-
points for the β-polytype and 122 k-points for the α-polytype with energy resolution
of 0.12 eV. The considered atoms are relatively light and the spin-orbit interaction
was not necessary to be included in the considerations. Accounting for the scalar-
relativistic part leads to the compulsory inclusion of corresponding perturbations.
In classifying the BZ, the notions presented in [9] were used.
Typical fragments of the BE dispersions calculated for the hexagonal-like 6H
wurtzite structure modified by near-the-interface structure are shown in figure 6
and figure 7. The presentation is limited only by the NCPP as well as by the SEPM
scheme because the FLAPW data give the results much further from the experi-
mental data compared with the mentioned ones. One can observe the appearance
of substantial deviations from the perfect BE structure. It is worth emphasizing
that within the SEPM scheme a considerable deviation is observed not only for
the valence states, but for the conduction band states as well. This fact confirms
the structural reconstruction for the near-the-interface state and the drawbacks of
the semi-empirical method in describing the excited states. For comparison, a corre-
sponding behavior achieved for the cubic fragment using the FLAPW-LDA approach
413
I.V.Kityk
(a)
(b)
Figure 7. (a) Fragment of the BE structure of the SiC calculated by the SEPM
approach with inclusion of the near-the-interface inclusion projected into the
hexagonal-like structure. The points indicate the reconstructed BE points. (b)
Fragment of the reconstructed BE structure modified by the reconstructed surface
calculated within the FLAPW-LDA approach. The projection is done to the
cubic-like structure.
414
Band structure of non-steiochiometric large-sized nanocrystallites
0
5
10
15
4 5 6 7 8 9 10
E[eV]
�2
Figure 8. Spectral dependence of the e2(E) measured by ellipsometry and eval-
uated using different approaches: solid line – FLAPW method, dotted line –
modified NCPP, dashed line – SEPM. Experimental points are taken from [27]
is presented in figure 8. One can notice that the deviation is very large in this case.
The latter may reflect a substantial presence of the non-cubic structural phase.
As an important criterion of the applicability of a given method, the structural
(see table 2) and the BE parameters (see table 3) can be compared. It is obvious
that the maximal agreement between the experimental energy gap and the theo-
retically calculated optical data and electronic structure is given in the case of the
modified NCPP approach. The SEPM and FLAPW methods lead to results being
substantially different from the experimental data.
4.3. Optical absorption
In order to simulate the optical properties, calculations of the imaginary part of
dielectric susceptibility optical tensors were done using an expression:
ε2(E) =
2πNe2
h3
∑
v.b.k
∫
H|〈ΨΨv.b.(k, r)|∇rΨΨc.b.(k, r)|〉
× d3rδ(Ev.b(k).−Ec.b(k).− E), (6)
where ΨΨv.b.(k,r) and ΨΨc.b.( k,r) corresponds to the valence and conduction band
pseudowavefunction, respectively. All the summations should be carried out over the
whole BZ.
The general expression was introduced with accounting for the matrix dipole
moments. The matrix dipole moments were evaluated using commutation of a space
and single-particle Hamiltonian. An additionally performed numerical verification
has shown that the nonlocal effects change the average oscillator strength up to
415
I.V.Kityk
Table 2. Lattice constants for the 4H- and 3C-polytypes calculated using different
methods
4H-SiC 3C-SiC
a[A]
NCPP modified as in the present work 3.071 4.359
FLAPW-LDA 3.067 4.341
SEPM 3.053 4.336
c/p [A]
NCPP modified as in the present work 2.511
FLAPW-LDA 2.503
SEPM 2.489
Experimental [27] a = 3.073; c/p = 2.513 a = 4.360
Table 3. Principal band energy parameters of the SiC crystals.
3C-SiC 4H-SiC
VB [eV] (MNCPP-8.24; FLAPW-
LDA-8.07; SEPM-7.97)
(MNCPP-8.06; FLAPW-
LDA-7.93; SEPM-7.89)
CB[eV] (MNCPP-6.68; FLAPW-
LDA-6.10; SEPM-6.06)
(MNCPP-6.15; FLAPW-
LDA-6.18; SEPM-6.67)
Eg [eV] (MNCPP-3.024; FLAPW-
LDA-2.894; SEPM-2.74)
(MNCPP-3.63; FLAPW-
LDA-3.37; SEPM-;3.24)
Experimental 8.28-VB; 6.45-CB;
Eg = 3.02 eV
8.04-VB; 6.23-CB;
Eg = 3.5 eV
12%. A plasmon peak in the energy loss spectra is shifted towards higher energies
up to 1.5–2 eV. As a consequence, all the calculations were done with the inclusion
of the nonlocal contributions. The real parts of the dielectric tensors were evaluated
using Kramers-Kronig dispersion relations. The reflectivity and energy loss functions
were obtained from a Fresnel’s formula.
A crucial point in the calculations of the ε2(E) was the k-space integration. A
linear tetrahedron method for the 286 (324,436) k points in the 1/24 irreducible
part of the BZ for 2H(3C,4H) SiC was applied. In the present paper during the
band summation, 24 (56,82) conduction bands were included in order to achieve
convergent results for the 2H(3C,4H) structures. This convergence is a crucial point
in calculating the energy loss-electron spectra, since their main peaks occur at higher
energies compared with the experimental ones [27]. To account for the excitation
aspect not included in the DFT-LDA electronic structure, the theoretical curves
were shifted to higher energies by a constant value of 0.66 eV. The important point
is that the mean value of the scissor operator was smaller than the quasiparticle
shifts calculated for various transitions. It nearly approaches half of the quasiparticle
416
Band structure of non-steiochiometric large-sized nanocrystallites
shift calculated for the indirect fundamental gaps. This fact seems to be a general
one considering optical gaps in contrast to the case where these gaps are derived
from photoemission and inverse photoemission. We believe that the reason for the
reduction is a consequence of the electron-hole interaction (excitonic effects) omitted
in the theoretical description but present in the experiment.
The calculated BE gaps were of the direct type (with the top of the valence
band and the bottom of the conduction band situated at the same point of the BZ).
Figure 8 presents spectral dependences of the imaginary part of dielectric suscepti-
bility ε2(E) calculated by the three mentioned methods. The ε2(E) calculated within
the orthogonalization NCPP method gives a better agreement with the experiment
compared with the other two methods. This fact together with the better structural
(see table 2) and energy gap (see table 3) parameters indicates the choice of the
NCPP method in performing the calculations. The modified NCPP method is thus
more preferable in simulating the optical and electronic properties of the SiC.
To describe the excitonic-like near-the-interface confined SiC states, a microscop-
ic expression for the nonlocal susceptibility was derived both in linear and nonlinear
regimes, including a three-dimensional description of electronic quantum states and
their electron-hole Coulomb interactions. The knowledge of the nonlocal suscepti-
bility allows us to calculate the excitonic matrix dipole moments determining the
optical absorption. Explicit calculations of the local linear response of the excitons
confined in SiC reconstructed surfaces were reported. Significant interference effects
in the interacting SiC reconstructed layers, electron-hole wave functions (particu-
larly in the coupled nanostructures) turn out to be proved. The response of the
SiC quantized dot-like structure is based on a microscopic description of electronic
quantum states and their Coulomb interactions.
Absorption spectra simulations in the linear-response regime for the SiC struc-
tures with actual geometry and composition were calculated. The new features in
excitonic spectra arise, particularly in the coupled nanostructures, owing to the in-
terference effects in the interacting electron-hole wave functions. One can conclude
that the Coulomb effects on the local spectra should be taken into account in order
to correctly assign the experimental features.
The microscopic expression of the nonlocal susceptibility, including the Coulomb
interactions between electrons and holes, is described within the space dispersion
regime. It was revealed that the energy exciton levels are practically insensitive to
the α/β ratio.
Let us note that there is no sign of spatially indirect transitions connecting an
electron and a hole localized in different SiC sublayers. More substantial was the
inclusion of the inter-electron Coulomb correlation. For the larger carbon sheet, the
two main peaks, arising from direct transitions located in either SiC layers, are
red-shifted by the exciton binding energy towards the high-energy continuum.
Table 4 compares the contribution of the reconstructed SiC surface to the ad-
ditional absorption for 550 nm wavelength that dominates the absorption behavior
compared with the crystalline ones. However, the main contribution to the light
absorption is caused by the excitons originating from the near-the-interface states
417
I.V.Kityk
Table 4. The origin of absorption caused by different mechanisms.
Structural fragment Absorption value, cm−1
SiC crystalline contribution 50–70
Reconstructed BE state 200–250
Graphite-like layer 820–850
Excitonic state 103 − 104 cm−1
possessing a giant absorption compared with the traditional BE structural recon-
struction (see table 2 and table 3).
Figure 9. Absorption excitonic spectra of the Si2−xCx at room (n) and helium
liquid temperature (�).
In order to verify the prediction about the appearance of the reconstructed layers,
additional low-temperature measurements of absorption for the Si2−xCx crystallites
were done. Figure 9 presents two fragments of the spectra for two different tempera-
tures together with the theoretically simulated excitonic-resonance absorption. One
can notice that the exitonic absorption plays a major role in the absorption. This
fact directly confirms the physical model considering the reconstructed surfaces as
thin layers covering the crystallites.
The contribution of excitons leads to a substantial rearrangement of the remain-
ing spectra and particularly to a substantial re-building of all the optical spectra.
The data presented demonstrate that the dependence of the ε2(E) spectra versus the
film thickness is vanishingly weak. Therefore, one can conclude that the main role in
the features of excitonic-orginated ε2(E) spectra will be played by the reconstructed
SiC surfaces and the thickness of the carbon sheet. This is in contradiction to the
418
Band structure of non-steiochiometric large-sized nanocrystallites
expectation from a reconstruction of the charge density distribution and reflects a
relative stability of the interlayer interactions within a few surface layers.
5. Conclusions
A new model of describing the structural and BE properties of the large sized
Si2−xCx NC is presented. The subsystem may be considered as reconstructed SiC
crystalline layers (with thickness of about 2–4 nm) covering the SiC crystallites with
the sizes 30–50 nm. The reconstructed SiC layers possess a flatter BE k-dispersion.
Among the calculation methods, the best agreement with the experimental data
is given by the norm-conserving non-local pseudopotential method together with
the modified Car-Parinello MD simulations of interfaces. The varying reconstruct-
ed sheet thickness stimulates the appearance of disordered SiC crystallites NC
crystalline state. Low-temperature experimental measurements confirm the domi-
nant role of the excitonic resonances originated from the reconstructed layers. The
excitonic-like states give a substantially higher contribution to the visible absorption
compared with the reconstructed BE sub-band contributions. The reconstructed lay-
ers may be considered to be a new type of high performance electronic materials
possessing quantum-confinement effects manifested in the excitonic spectra.
Acknowledgements
The author is grateful to CNRS – Region Rennes Delegation.
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Зонна структура нестехіометричних
широко-розмірних нанокристалів
І.В.Кітик
Інститут Матеріалів Науки та Металургії, Польська Академія Наук.
Вул. Реймонта, 25, Краків, Польща
Отримано 8 грудня 2003 р., в остаточному вигляді – 26 квітня
2004 р.
Досліджено зонну структуру широко-розмірних Si2−xCx (1.04 < x <
1.10) нанокристалів (НК) з розмірами від 20 до 35 нм, використо-
вуючи різні методи розрахунків зонної структури і модифікований
молекулярно-динамічний метод геометричної оптимізації інтерфей-
сів Кара-Парінелло. Нестехіометричний надлишок вуглецю сприяє
появі тонких (біля 1 нм) шарів вуглецю на поверхнях НК. В результа-
ті спостерігається істотна перебудова граничних кристалічних шарів
SiC. Числове моделювання показало,що ці НК можуть розглядатись
як модифіковані структури кристалів SiC з розмірами біля 2 нм. От-
римані результати розглядаються в рамках різних одно-електронних
підходів. Незалежним експериментальним проявом цих наноефек-
тів є екситонні ефекти, які проявляються в низькотемпературних
спектрах поглинання як гострі абсорбційні резонанси, що походять
реконструйованих шарів.
Ключові слова: псевдопотенціал, зонна струтура, нанокристаліти
PACS: 71.20
420
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