Utilization of the sum rule for construction of advanced dispersion model
of crystalline silicon containing interstitial oxygen
Daniel Franta a,b,
⁎, David Nečas a,b
, Lenka Zajíčková a,b
, Ivan Ohlídal a
a
Department of Physical Electronic, Faculty of Science, Masaryk University, Kotlářská 2, 61137 Brno, Czech Republic
b
Plasma Technologies, CEITEC — Central European Institute of Technology, Masaryk University Kamenice 5, 62500 Brno, Czech Republic
a b s t r a c ta r t i c l e i n f o
Available online xxxx
Keywords:
Dispersion model
Sum rule
Transition strength
Crystalline silicon
The distribution of the total transition strength, i.e. the right hand side of the integral form of Thomas–Reiche–
Kuhn sum rule, into individual absorption processes is described for crystalline silicon containing interstitial oxygen.
Utilization of the sum rule allows the construction of a dispersion model covering all elementary excitations
from phonon absorption to core electron excitations. The dependence of transition strength of individual
electronic and phonon contributions on temperature and oxygen content is described.
© 2014 Elsevier B.V. All rights reserved.
1. Introduction
The classical theory of dispersion provides three general conditions
for the linear dielectric response that physically consistent models
must satisfy [1–4]: Kramers–Kronig relation, time reversal symmetry
and sum rule. While the first two conditions are well known and widely
used in construction of dispersion models, the sum rule is rarely utilized.
Classical f-sum rule is usually expressed for the imaginary part of dielectric
function εi as follows
Z ∞
0
εi ωð Þω dω ¼
π
2
ω
2
p; ð1Þ
where ω is light frequency and ωp is a constant called the plasma frequency.
In the frame of classical physics plasma frequency is proportional
to density of electrons N e
ω
2
p ¼
e2
N e
ϵ0me
; ð2Þ
where e, ϵ0 and me are physical constants, i.e. electron charge, vacuum
permittivity and electron mass. It is clear that Eq. (1) together with
Eq. (2) form a bridge between dielectric response and structural parameters
of the system such as atomic or mass density. One reason why the
sum rule is seldom used for construction of dispersion models is that it
is a global condition which can be applied correctly only to the entire
dielectric response covering all elementary excitations of the system.
However, thanks to the progress in instrumentation we are nowadays
able to measure dielectric response in broad spectral range from far
infrared (FIR) to vacuum ultraviolet (VUV) using commercial table top
instruments. Moreover, using synchrotron facilities it is possible to extend
the spectral range to the X-ray region and cover the full range of
electronic excitations in solids. Therefore, modeling of the complete
dielectric response in the entire spectral region has become important.
Silicon wafers are widely used as substrates both in the microelectronic
industry and fundamental research of thin films. Although the
ideal substrates would be pure silicon single crystals, in practice silicon
wafers produced using Czochralski process [5] are mostly utilized. In
general, these wafers differ from ideal Si crystals by the presence of interstitial
oxygen (Oi) and concentration of dopants [6]. Moreover, the
concentrations of interstitial oxygen and dopants differ between wafers
and even within one wafer. Therefore, it is not possible to use single tabulated
optical constants for all Si pieces, especially in the IR region, and
models describing the dielectric function of real silicon wafers must be
developed. It is also important that such models should contain the minimum
numbers of parameters necessary to express the variations
between individual silicon wafers. This means that models should be
parametrized by concentration of interstitial oxygen, oxygen precipitates
[7], substitutional carbon [8] and dopants such as boron, phosphorus
or arsenic. In addition, since the optical measurements can be
carried out at different temperatures, the models should also include
temperature as a parameter.
In this work it will be shown how the sum rule can be utilized for
construction of dispersion model of c-Si wafers applicable in the entire
spectral region from FIR to VUV. This model will be temperature dependent,
nevertheless, it will only contain the influence of interstitial oxygen
which is perhaps the most important effect since interstitial
Thin Solid Films xxx (2014) xxx–xxx
⁎ Corresponding author at: Department of Physical Electronic, Faculty of Science,
Masaryk University, Kotlářská 2, 61137 Brno, Czech Republic. Tel.: +420 54949 3836;
fax: +420 541211214.
E-mail address: franta@physics.muni.cz (D. Franta).
TSF-33311; No of Pages 6
http://dx.doi.org/10.1016/j.tsf.2014.03.059
0040-6090/© 2014 Elsevier B.V. All rights reserved.
Contents lists available at ScienceDirect
Thin Solid Films
journal homepage: www.elsevier.com/locate/tsf
Please cite this article as: D. Franta, et al., Utilization of the sum rule for construction of advanced dispersion model of crystalline silicon containing
interstitial oxygen, Thin Solid Films (2014), http://dx.doi.org/10.1016/j.tsf.2014.03.059
oxygen is present in all Czochralski wafers [9]. The presented results
summarize ellipsometric and spectrophotometric characterizations of
a set of double side polished wafers and slabs produced using float
zone and Czochralski processes. Data sets from several instruments
were combined to cover a wide spectral range from 8.7 meV (70 cm−1
)
to 8.7 eV and temperature range from 300 to 600 K. Experimental details
are out of scope of this paper and will be published elsewhere.
2. Theoretical background
Thomas–Reiche–Kuhn (TRK) sum rule for system of Ne electrons
can be written as [10]
Xf ≠i
f
2
me
〈 fj j^pxe i 〉j j
2
Ef −Ei
¼
Xf ≠i
f
f if ¼ Ne; ð3Þ
where me and ^pxe denote electron mass and total momentum operator
of electrons, respectively. The symbols |i 〉 and |f 〉 represent a
complete set of many-body eigenstates of the system with eigenvalues
of energy Ei and Ef. The quantity fif is called oscillator
strength. It was shown that the discrete TRK sum rule (3) can be rewritten
in an integral form [11]
Z ∞
0
Fe Eð Þ dE ¼
Ne
V
¼ N e ; ð4Þ
where V is volume of the system and the transition strength function
Fe is defined as
Fe Eð Þ ¼
1
V
Xf≠i
f
f if δ Ef −Ei−E
 
þ δ Ei−Ef −E
 h i
: ð5Þ
The same TRK sum rule can be also written for nuclei
Z ∞
0
Fn Eð Þ dE ¼ N n; ð6Þ
where n distinguishes the type of nuclei (n = Si or O for silicon containing
interstitial oxygen) and N n are the corresponding nuclei densities.
Within dipole approximation the three quantum-mechanical transition
strength functions Fe, FSi and FO can be linearly combined to form
a new quantity F, also called (optical) transition strength function,
directly related to the dielectric function:
F Eð Þ ¼ M
–1
Fe Eð Þ þ
X
n
Z2
nme
mn
Fn Eð Þ
" #
≈ εi Eð ÞE; ð7Þ
where M
–1
= (eh)2
/(8πϵ0me) is a combination of fundamental physical
constants and the symbols Zn and mn denote the proton number and
mass of nucleus n, respectively. The transition strength function F
satisfies the following sum rule which is a linear combination of the
TRK sum rules (4) and (6):
Z ∞
0
F Eð Þ dE ¼ M
–1
N e þ
X
n
Z2
nme
mn
N n
!
¼ N: ð8Þ
The quantity N on the right hand side is called total transition
strength of the system. Thus, the transition strength function is the
spectral distribution of the transition strength. Utilization of the
sum rule for construction of dispersion models consists in the distribution
of total transition strength among individual contributions of
elementary excitations [11–13].
The symbol ≈ in formula (7) is used to emphasize that the quantum
mechanical quantity F(E) and macroscopic quantity εi representing dielectric
response are connected by the dipole approximation. Within
classical physics this distinction disappears and the description using
transition strength function is equivalent to other representations of linear
dielectric response:
F Eð Þ ¼ Eεi Eð Þ ¼
ℏ
ϵ0
σr Eð Þ ; ð9Þ
where σr is the real part of conductivity. The linear dielectric response is
then given as the sum of damped harmonic oscillators (DHOs):
F Eð Þ ¼
4
π
X
j
NjBjE2
E2
c;j−E2
 2
þ 4B2
j E2
;
X
j
Nj ¼ N; ð10Þ
where Nj, Ec,j and Bj are the transition strength, central energy and
broadening of DHO, respectively. The quantity Nj has unit eV2
and is
also often called oscillator strength even though it differs from unitless
quantity fif occurring in Eq. (3). In formula (10) the terms satisfying inequality
Ec,j N Bj can be rewritten by the Lorentz functions:
F j Eð Þ ¼
NjBjE
πEj
1
Ej−E
 2
þ B2
j
−
1
Ej þ E
 2
þ B2
j
0
B
@
1
C
A; ð11Þ
where Ej is the energy of transition related to the central energy of DHO
by relation Ec,j
2
= Ej
2
+ Bj
2
.
Note that classical systems with parameters Ec,j b Bj (overdamped
oscillators) or Ec,j = 0 (Drude formula) are impossible to describe
using Eq. (11) as Lorentzian broadened discrete spectrum. Formula (11)
is equivalent to quantum mechanical description represented by Eq. (5)
for limit Bj → 0. On the other hand, quantum mechanical systems exhibit
behavior that cannot be described using a finite number of DHOs, for
instance bandgap. Moreover, Gaussian broadening is usually more appropriate
for quantum systems [14].
The approach based on the transition strength function takes advantage
of a clear connection between dielectric response and microscopic
quantities as oscillator strengths or momentum matrix elements. If Nj is
the transition strength of an individual transition i → j corresponding to
absorbed energy Ej = Ef − Ei then it holds:
Nj ¼
M
–1
V
f if ¼
ehð Þ2
8πϵ0meV
f if ¼
ehð Þ2
4πϵ0m2
e VEj
j 〈 f j^pxeji 〉 j
2
: ð12Þ
The constant M represents a link between optical quantities,
i.e. transition strength, and density of the system. For example, if Nex
is the transition strength of an excitonic peak then the relation
Nex ¼
M–1
V
X
f
f if ¼ M
–1
N ex ð13Þ
gives the ‘volume density of excitons’N ex (the summation is performed
over f states representing all the excitonic excitations). However, it is
not clear how is such quantity related to the density of real particles
(electrons, nuclei). Since all elementary excitations in solids involve
quasiparticles the only formula with a clear interpretation is Eq. (8)
for total transition strength N. For a more detailed discussion see [11].
3. Absorption processes in c-Si wafers
The absorption in c-Si wafers is caused by the following effects corresponding
to the individual contributions to the transition strength
function:
dt Direct interband transitions from valence to conduction
band.
idt Indirect interband transitions from valence to conduction
band.
2 D. Franta et al. / Thin Solid Films xxx (2014) xxx–xxx
Please cite this article as: D. Franta, et al., Utilization of the sum rule for construction of advanced dispersion model of crystalline silicon containing
interstitial oxygen, Thin Solid Films (2014), http://dx.doi.org/10.1016/j.tsf.2014.03.059
fc Indirect intraband transitions in valence and conduction
bands, i.e. contribution of free carriers.
vx Excitations of valence electrons to higher energy states
above the conduction band.
K, L Excitations of electrons from the core levels to the electron
states above the Fermi energy.
1 ph One-phonon absorption, i.e. excitation of vibrational states
of Si nuclei. It is forbidden by selection rules for ideal c-Si,
however, it is present in real wafers due to the presence of
imperfections and foreign atoms.
mph Multi-phonon absorption, in our model considered up to
four phonons, i.e. mph = 2 ph + 3 ph + 4 ph.
A2u One-phonon absorption corresponding to antisymmetric
stretching vibrational mode of interstitial oxygen nuclei.
A2u + A1g Two-phonon absorption corresponding to combination of
A2u and A1g vibrational modes of interstitial oxygen nuclei.
General ideas that can be used for construction of dispersion models
related to individual absorption processes are given in [11]. Application
of these ideas to amorphous hydrogenated silicon (a-Si:H) is shown in
[12,13]. Detailed description of individual dispersion models for c-Si is
out of this paper because it deals with dependences of integral transition
strength of individual processes on temperature and concentration
of interstitial oxygen.
4. Distribution of the transition strength
The total transition strength N is the sum of contributions of electrons
and nuclei of Si and O, i.e.
N ¼ Ne þ NSi þ NO: ð14Þ
All the individual contributions depend on temperature T and atomic
fraction of interstitial oxygen f Oi
as follows
Ne T; f Oi
 
¼ ZSi 1−fOi
 
þ ZO f Oi
h i
Na Tð Þ; ð15Þ
NSi T; f Oi
 
¼
Z2
Sime
mSi
1−f Oi
 
Na Tð Þ; ð16Þ
NO T; f Oi
 
¼
Z2
Sime
mO
f Oi
Na Tð Þ: ð17Þ
Symbols ZSi, ZO, mSi and mO denote proton number of Si, proton number
of O, mass of Si nucleus and mass of O nucleus, respectively. The
density parameter Na depends on temperature as follows
Na Tð Þ ¼ Na 300 Kð Þ
1 þ e 300 Kð Þ
1 þ e Tð Þ
!3
ð18Þ
where e(T) is the linear thermal expansion coefficient [15] and
Na(300 K) = 108.2 eV2
[11]. The thermal dependence of Na is illustrated
in Fig. 1 for the temperature range 300–1000 K. In principle, Na depends
also on the concentrations of impurities, but this dependence
can be disregarded [8] for common concentrations.
From the point of view of absorption processes the total transition
strength N can be split into electron excitation Nee and phonon Nph
parts:
N ¼ Nee þ Nph: ð19Þ
Even though the electron excitation transition strength Nee differs
from the electronic part Ne of the sum introduced in Eq. (14) their values
are very close. This is because N ¼ NeU , where U is close to unity,
specifically for silicon U ¼ 1:0002735, while for Nph it holds Nph b NSi +
NO [11]. Hence both differ from N only slightly and thus
N ≈ Nee ≈ Ne ≈ ZSiNa; ð20Þ
where ZSi = 14 is the approximate mean number of electrons per atom in
silicon wafers. It is evident that temperature dependence of Nee is then
given by the temperature dependence of Na in Eq. (18).
4.1. Transitions above the bandgap
The dielectric response above the bandgap, i.e. K, L, vx and vc
(interband valence-to-conduction transitions vc = dt + idt) absorption
processes, is insensitive to presence of interstitial oxygen
in concentrations typical for Czochralski silicon. Therefore, the
dependence of all electronic transitions on parameter f Oi
can be
disregarded.
Concerning temperature, the transition strengths Nt of individual
contributions t = K, L, vx, vc depend on temperature only through Na.
The spectral distribution of transition strength of electronic excitations
is of course temperature dependent as shown in Fig. 2 for 300 and
600 K. Therefore, it is useful to introduce the effective number of
electrons nt via the following relation:
Nt Tð Þ ¼ ntNa Tð Þ: ð21Þ
107.2
107.4
107.6
107.8
108.0
108.2
300 400 500 600 700 800 900 1000
densityparam.,Na(eV2
)
temperature, T (K)
Fig. 1. Temperature dependence of density parameter Na for c-Si calculated using Eq. (18).
10-6
10-4
10-2
100
102
100
101
102
103
104
transitionstrengthfunction,F(eV)
photon energy, E (eV)
Eg
E0
E1
E2
Eh
EL
EK
K
L
vx
dt
idt
c-Si 300 K
c-Si 600 K
Fig. 2. Spectral distribution of transition strength of c-Si in the region of electronic excitations
above the bandgap for two selected temperatures. Important energies are marked
with arrows: Eg — minimum energy of interband transitions (bandgap); E1 — energy of
1st direct transition structure; E2 — energy of 2nd direct transition structure; Eh — maximum
energy of interband transitions; EL — minimum energy of core level L-electron
excitations; and EK — minimum energy of core level K-electron excitations.
3D. Franta et al. / Thin Solid Films xxx (2014) xxx–xxx
Please cite this article as: D. Franta, et al., Utilization of the sum rule for construction of advanced dispersion model of crystalline silicon containing
interstitial oxygen, Thin Solid Films (2014), http://dx.doi.org/10.1016/j.tsf.2014.03.059
The effective number of electrons nt defined in this way is then temperature
independent. It is also possible to introduce the effective number
of valence electrons nve as
nve ¼ nvc þ nvx: ð22Þ
The effective numbers nve, nL and nK are not equal to the numbers
of electrons in the corresponding shells, i.e. 4, 8 and 2, respectively.
However, their sum must correspond to the number of electrons in
Si:
nve þ nL þ nL≈ZSi ¼ 14 : ð23Þ
It was shown that values nve = 4.12, nL = 8.33 and nK = 1.55 calculated
by Shiles for aluminum [16] can be used for a-Si:H if the measurements
do not extend to the X-ray region [12,13]. These values should be
suitable also for crystalline silicon.
It follows from the foregoing that the effective number of electrons
calculated up to a certain energy E
neff Eð Þ ¼
1
Na Tð Þ
Z E
0
F E
0À Á
dE
0
ð24Þ
is a useful quantity since it can be interpreted as the sum per one atom.
Its limit for E → ∞ is
neff ∞ð Þ ¼ ZSi 1−fOi
 
þ ZO f Oi
h i
U≈14 : ð25Þ
The course of neff(E) is plotted in Fig. 3 for the same two temperatures
as in Fig. 2. It is practically identical for both the temperatures
except for the region of interband transitions which is shown in the
inset. It is interesting that neff(E) starts to increase significantly
above the onset of direct transitions E0, and not from the band gap
Eg.
Similar to Eq. (21) we can introduce the effective numbers of electrons
also for t = dt, idt
Nt Tð Þ ¼ nt Tð ÞNa Tð Þ; ð26Þ
but they depend on temperature. The dependence of nidt(T) on temperature
is almost linear and it is given by the following formula [17,18,11]:
nidt Tð Þ∝1 þ 2f
BE
Ep; T
 
¼ 1 þ
2
exp Ep=kBT
 
−1
; ð27Þ
where fBE
denotes Bose–Einstein statistics, kB is the Boltzmann constant
and Ep is the effective mean energy of phonons participating
in the indirect interband transitions. The dependence of ndt(T) follows
from
nidt Tð Þ þ ndt Tð Þ ¼ nvc: ð28Þ
The values of effective numbers of electrons are tabulated for selected
temperatures in Table 1.
4.2. Transitions below the bandgap
Transitions below the bandgap include phonon and free carriers absorption
processes. The transition strength in spectral region where
these processes are important is plotted for selected temperatures
from 300 to 600 K in Fig. 4. Note that the curves correspond to pure cSi
hence no one-phonon absorption is present. The phonon contribution
to the sum is then dominated by two-phonon absorption processes.
Therefore, temperature dependence of Nmph is practically linear around
and above room temperature because phonon occupation numbers
depend on temperature linearly in the considered range. This linear
dependence can be seen in Fig. 5 showing Nmph of pure c-Si calculated
from the temperature dependence of all multi-phonon processes [19,
0
1
2
3
3 4 5
E0
E1
E2
0
2
4
6
8
10
12
14
10-2
10-1
100
101
102
103
104
105
106
effectivenumberofelectrons,neff
photon energy, E (eV)
Eg
E0 nve = 4.12
nL = 8.33
nK = 1.55
c-Si 300 K
c-Si 600 K
Fig. 3. The evolution of effective number of electrons neff with photon energy calculated
using Eq. (24) for two selected temperatures. Dash-and-dot lines extrapolate ve and L contributions
above EL and EK energies. The inset shows the region of onset of direct interband
electronic transitions in detail.
Table 1
Effective numbers of electrons corresponding to the individual absorption processes for
selected temperatures. Note that nve = nvx + nvc, nvc = nidt + ndt and nmph =
n2ph + n3ph + n4ph. The last column describes the approximate temperature
dependence.
300 K 400 K 500 K 600 K
Above Eg
nK 1.55 Const.
nL 8.33 Const.
nve 4.12 Const.
nvx 0.24 Const.
nvc 3.88 Const.
nidt 0.50 0.61 0.72 0.84 Lin.
ndt 3.38 3.27 3.16 3.04 Lin.
below Eg
nfc 3.23 × 10−12
1.01 × 10−9
3.42 × 10−8
3.76 × 10−7
Exp.
nmph 1.16 × 10−7
1.48 × 10−7
1.84 × 10−7
2.22 × 10−7
Lin.
n2ph 1.00 × 10−7
1.26 × 10−7
1.52 × 10−7
1.79 × 10−7
Lin.
n3ph 1.51 × 10−8
2.08 × 10−8
2.84 × 10−8
3.78 × 10−8
Quad.
n4ph 1.09 × 10−9
1.91 × 10−9
3.17 × 10−9
5.01 × 10−9
Cubic
10-8
10-7
10-6
10-5
10-4
10-3
10-2
10-2
10-1
100
transitionstrengthfunction,F(eV)
photon energy, E (eV)
2ph
3ph
4ph
c-Si 300 K
c-Si 400 K
c-Si 500 K
c-Si 600 K
Fig. 4. Spectral distribution of transition strength of pure c-Si in the region of phonon and
free carrier absorption below the bandgap for selected temperatures.
4 D. Franta et al. / Thin Solid Films xxx (2014) xxx–xxx
Please cite this article as: D. Franta, et al., Utilization of the sum rule for construction of advanced dispersion model of crystalline silicon containing
interstitial oxygen, Thin Solid Films (2014), http://dx.doi.org/10.1016/j.tsf.2014.03.059
11]. The transition strength of free carriers is also plotted in this figure.
The free carrier part is given by the well-known expression [20]:
Nfc∝T
3
2
exp −
Eg
2kBT
 
: ð29Þ
The values of effective numbers of electrons corresponding to free
carriers and phonons are tabulated for selected temperatures in
Table 1. In this table multi-phonon absorption is split to individual contributions
2 ph, 3 ph and 4 ph. It can be seen that transition strengths of
3 ph and 4 ph contributions are 10 and 100 times smaller than 2 ph. It is
clear that contribution of free carriers and phonons to the sum is insignificant
even though it is certainly not negligible from the point of view
of dielectric response.
In contrast to electronic excitations, the dielectric response corresponding
to phonon absorption processes depends considerably
on concentration of interstitial oxygen. In Fig. 6 the transition
strength functions at room temperature is plotted for pure c-Si and
typical Czochralski silicon with 20 ppm of interstitial oxygen (c-Si:
O20). The contribution to transition strength function caused by interstitial
oxygen is proportional to its atomic fraction f Oi
.
The most prominent structure corresponds to antisymmetric
stretching vibrational mode of Oi at 1107 cm−1
labeled as A2u. The transition
strength of this vibrational mode was assumed as follows:
NA2u
¼ αA2u
NO; ð30Þ
where NO is given by formula (17) and the relative transition strength
αA2u
¼ 0:1914 corresponds to the effective charge of 3.5e determined
by ab initio calculations [21]. Note that the value of αA2u
calculated
from the relation between Oi concentration and α determined by another
independent method such as ASTM F1188 [22]. The obtained value of
αA2u
was approximately 5% larger than the value given above.
Other Oi vibrational modes are not IR active. However, a twophonon
A2u + A1g absorption peak can be observed at 1721 cm−1
[23]. In contrast to one-phonon A2u vibrational mode, the strength of
this two-phonon vibrational mode is temperature dependent:
NA2uþA1g
¼ αA2uþA1g
Tð Þ NO; ð31Þ
where relative transition strength is given as follows:
αA2uþA1g
Tð Þ ¼ αA2uþA1g
300 Kð Þ
f A2uþA1g
Tð Þ
f A2uþA1g
300 Kð Þ
; ð32Þ
f A2uþA1g
Tð Þ ¼ f
BE
EA2u
; T
 
þ f
BE
EA1g
; T
 
þ 1: ð33Þ
The relative transition strength at room temperature is αA2uþA1g
300 Kð Þ ¼ 0:0025.
In addition to oxygen's own vibrational modes the presence of Oi enhances
the absorption band in the region of one-phonon Si modes
(0–520 cm−1
) due to locally broken symmetry of the Si crystal. Two
sharp structures corresponding to the transversal acoustic (TA) and
transversal optical (TO) modes are marked in Fig. 6. Their integral transition
strength can be written as follows
N1ph ¼ α1ph
2f Oi
1−f Oi
NSi; ð34Þ
where the relative transition strength α1ph = 0.0172 is related to the
two Si atoms closest to the interstitial oxygen atom. It is temperature independent
because the absorption processes are one-phonon.
5. Conclusion
Parametrization of the transition strength function is a suitable
starting point for the construction of dispersion models because the
sum rule for transition strength function can be derived without any approximation
in frame of non-relativistic quantum mechanics. It was
shown how to distribute the total transition strength, i.e. the right
hand side of sum rule, into individual absorption processes occurring
in crystalline silicon containing interstitial oxygen. Individual dispersion
formulas were not discussed but the dependences of integral transition
strength of individual processes on temperature and concentration of
interstitial oxygen were presented. Since some contributions depend
on temperature explicitly while other only through thermal expansion
the latter were described using temperature independent effective numbers
of electrons or relative transition strengths. Interstitial oxygen influences
the dielectric response only in the region of the phonon absorption
where one-phonon absorption structures, which are not present in pure
c-Si, appear. The dispersion model constructed on the basis of distribution
of the total transition strength can be then parametrized using
only temperature and atomic fraction of interstitial oxygen. The specific
transition strength functions describing individual absorption processes
will be published in forthcoming papers together with analyses of
experimental ellipsometric and spectrophotometric data.
Acknowledgments
This work was supported by projects “R&D center for low-cost plasma
and nanotechnology surface modifications” (CZ.1.05/2.1.00/
03.0086) and “CEITEC — Central European Institute of Technology”
(CZ.1.05/1.1.00/02.0068) from the European Regional Development
Fund and project TA02010784 of Technological Agency of Czech
Republic.
0 × 10-5
1 × 10-5
2 × 10-5
3 × 10-5
4 × 10-5
5 × 10-5
6 × 10-5
7 × 10-5
300 350 400 450 500 550 600
transitionstrength(eV2
)
temperature, T (K)
Nph
Nfc
Nph + Nfc
Fig. 5. Temperature dependence of transition strengths of absorption processes below the
bandgap for pure c-Si.
10-7
10-6
10-5
10-4
10-3
0 500 1000 1500 2000
transitionstrengthfunction,F(eV)
wavenumber, ν (cm-1
)
A2u
A2u+A1g
TO
TA
1ph
2ph
3ph
4ph
c-Si
c-Si:O20
Fig. 6. Comparison of transition strength function of pure c-Si (full line) and c-Si with
20 ppm of interstitial oxygen (dotted line) at room temperature.
5D. Franta et al. / Thin Solid Films xxx (2014) xxx–xxx
Please cite this article as: D. Franta, et al., Utilization of the sum rule for construction of advanced dispersion model of crystalline silicon containing
interstitial oxygen, Thin Solid Films (2014), http://dx.doi.org/10.1016/j.tsf.2014.03.059
References
[1] F. Wooten, Optical Properties of Solids, Academic Press, New York, 1972.
[2] M. Altarelli, D.L. Dexter, H.M. Nussenzveig, D.Y. Smith, Phys. Rev. B 6 (1972) 4502.
[3] D.Y. Smith, in: E.D. Palik (Ed.), Handbook of Optical Constants of Solids, vol. 1, Academic
Press, 1985, p. 35.
[4] V. Lucarini, K.-E. Peiponen, J.J. Saarinen, E.M. Vartiainen, Kramers–Kronig Relations
in Optical Materials Research, Springer, Berlin, 2005.
[5] J. Czochralski, Z. Phys. Chem. 92 (1918) 219.
[6] R.C. Newman, Rep. Prog. Phys. 45 (1982) 1163.
[7] S.M. Hu, J. Appl. Phys. 51 (1980) 5945.
[8] J.A. Baker, T.N. Tucker, N.E. Moyer, R.C. Buschert, J. Appl. Phys. 39 (1968) 4365.
[9] R.C. Newman, J. Phys. Condens. Matter 12 (2000) R335.
[10] H.A. Bethe, E.E. Salpeter, Quantum Mechanics of One- and Two-Electron Atoms,
Springer-Verlag, Berlin, 1957.
[11] D. Franta, D. Nečas, L. Zajíčková, Thin Solid Films 534 (2013) 432.
[12] D. Franta, D. Nečas, L. Zajíčková, I. Ohlídal, J. Stuchlík, D. Chvostová, Thin Solid Films
539 (2013) 233.
[13] D. Franta, D. Nečas, L. Zajíčková, I. Ohlídal, J. Stuchlík, Thin Solid Films 541 (2013) 12.
[14] C.C. Kim, J.W. Garland, H. Abad, P.M. Raccah, Phys. Rev. B 45 (1992) 11749.
[15] H. Watanabe, N. Yamada, M. Okaji, Int. J. Thermophys. 25 (2004) 221.
[16] E. Shiles, T. Sasaki, M. Inokuti, D.Y. Smith, Phys. Rev. B 22 (1980) 1612.
[17] S. Flügge, H. Marshall, Rechenmethoden der Quantentheorie, 2nd edition Springer,
Berlin, 1952.
[18] P.Y. Yu, M. Cardona, Fundamentals of Semiconductors, Springer, Berlin, 2001.
[19] F.A. Johnson, Proc. Phys. Soc. 73 (1959) 265.
[20] C. Kittel, Introduction to Solid State Physics, 5th edition Wiley, New York, 1976.
[21] R. Jones, A. Umerski, S. Öberg, Phys. Rev. B 45 (1992) 11321.
[22] ASTM, Standard Test Method for Interstitial Atomic Oxygen Content of Silicon by
Infrared Absorption, F11882000.
[23] B. Pajot, in: R. Hull (Ed.), Properties of Crystalline Silicon, The Institution of
Engineering and Technology, 1998, p. 492.
6 D. Franta et al. / Thin Solid Films xxx (2014) xxx–xxx
Please cite this article as: D. Franta, et al., Utilization of the sum rule for construction of advanced dispersion model of crystalline silicon containing
interstitial oxygen, Thin Solid Films (2014), http://dx.doi.org/10.1016/j.tsf.2014.03.059