World Journal of Nano Science and Engineering, 2012, 2, 1-5 Published Online March 2012 (
New Numerical Method to Calculate the True Optical
Absorption of Hydrogen ated Nanocrystalline
Silicon Thin Films
Fatiha Besahraoui, Larbi Chahed, Yahia Bouizem, Jamal Dine Sib
Laboratory of Physics of Thin Films and Materials for Electronics, University of Oran Es-Senia, Oran, Algeria
Email:, {larbi.chahed, yahia.bouizem, jamaldine.sib}
Received November 14, 2011; revised December 18, 2011; accepted January 12, 2012
The enhanced optical absorption measured by Constant Photocurrent Method (CPM) of hydrogenated nanocrystalline
silicon thin films is due mainly to bulk and/or surface light scattering effects. A new numerical method is presented to
calculate both true optical absorption and scattering coefficient from CPM absorption spectra of nanotextured nano-
crystalline silicon films. Bulk and surface light scattering contributions can be unified through the correlation obtained
between the scattering coefficient and surface roughness obtained using our method.
Keywords: Solution Hydrogenated Nanocrystalline Silicon; Constant Photocurrent Method; Optical Absorption; Bulk
Light Scattering; Surface Roughness; Film Thickness
1. Introduction
Materials with nanometer crystallites constitute an im-
portant class with some of their properties distinctly dif-
ferent from either amorphous or large grain materials or
single crystals. Especially, hydrogenated nanocrystalline
and polymorphous silicon thin films. Hydrogenated nano-
crystalline silicon (nano-Si:H) is an heterogeneous mate-
rial. It consists of an ordered nanocrystallites of spherical
form with size varies from 2 to 100 nm embedded in
amorphous matrix [1]. These semiconducting nanomate-
rials are very promising to the photovoltaic applications
[2]. The important key for the success of nano-Si:H films
as a PV absorbent materials is their enhanced absorption
compared to the monocrystalline silicon (c Si), mainly in
infrared region [3]. The main reason of this optical be-
havior is due to their particular structure which gives
place to the bulk and/or surface light scattering phenom-
ena [3-5]. The Constant Photocurrent Method measure-
ments of these heterogeneous mediums give us an “ap-
parent” optical absorption coefficient at en-
ergy E. is affected by light scattering effects
different from true
app E
app E
measured for homogeneous medi-
ums, i.e., amorphous or monocrystalline silicon [3]. In
this paper, we suggest a new numerical method to calcu-
late the true optical absorption. A numerical resolution of
Equation (1) is given at energy E = 1.1 eV. The contribu-
tion of bulk light scattering, from the spectral depend-
ence of apparent optical absorption coefficient
app E
in thin films of nano-Si:H is measured by Constant
Photocurrent Method (CPM). The details of CPM method
and its experimental configuration are described else-
where [6]. This method is developed in order to under-
stand the subgap part of the true optical absorption spec-
tra related to the defect states within the energy gap. It’s
introduced with the help of a recent theory [3] which
describes the different contributions of light scattering in
terms of photocurrent
deduced from CPM meas-
urements. This procedure has a crucial importance to
determine true
in the case of weak bulk light scattering,
without need to compare a several CPM measurements
(for different interelectrodes spacing) [3]. We will pre-
sent the details of our procedure and discuss its validity
by comparing a calculated spectrum with the
experimental one obtained by CPM for the same sample
of nano-Si:H. Finally, we will apply this method on a
series of nano-Si:H which have a different surface rough-
ness in order to determine the correlation between this
material parameter and the scattering coefficient
true E
2. Numerical Method
In the following, let us consider a nano-Si:H thin film
with a typical thickness df configured according to the
CPM setup. If the standard CPM evaluation method will
be applied to the measured spectrum, we will obtain an
“apparent” optical absorption coefficient app
the following equation is valid [3]:
or w
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 
1exp 1exp
cos1 exp
cos exp1
cos exp1
app fff
true sc
 
 
is the total optical absorption in the film
given by:
sc true
 (2)
bulk is the number of the scattering events between
the electrodes (to calculate this parameter see reference
[3]) and θ is the critical angle for total reflections. θ de-
fines the escape cone and can be calculated from the total
internal reflection condition
, where n is the
refractive index of the outer medium. The function
“cosθ” presents the propability that the incident photon is
scattered outside the escape cone. The parameters,
and θ are known experimentally. We have considered the
experimental values measured by the authors of the ref-
erence [3], in order to compare our method with their
experimental one. We have taken cosθ = 0.5 calculated
for 2 µm thick nano-Si:H with an spectrum measured by
CPM method for 8 × 2 mm interelectrode spacing (Fig-
ure 3). The Equation (1) (with the variable
) cannot
be solved analytically and an adequate numerical method
will be used.
First, we give some estimated values for the true opti-
cal absorption coefficient true
, and we inject them in
Equation (1) in order to get a solution. We must take into
account that
apptrue .We note that for each
value, it corresponds a value at the
considered energy. After that, we choose the limited
true E
app E
values (minimum and maximum values) in order to
get an ,min,maxsc sc
interval. For each interval of
values, it corresponds an interval of
,min ,max
. Second, in order to calculate values
with a good precision at the considered energy, we select
the true
sc sc
values in which the interval ,min ,maxscsc
will be much reduced. It means that
values which
favour the following condition:
true true
true true
Finally, we calculate the mean value ,true mean
of the
selected interval and we replace it in Equation (1) in or-
der to determine the corresponding solution
. Figure
1 shows the numerical resolution of Equation (1) ac-
cording to this process. After that, we suppose that the
spectral dependence of the calculated scattering coeffi-
cient has the following form in the energy range which
favours the previous inequality:
sc EE
where β and γ are a constants.
Using the values of
sc E
at energy E estimated in
the first step and by a simple convolution, we can deter-
mine the constants β and γ (Figure 2).
In Figure 2, one can observe that the convoluted spec-
trum of scattering coefficient corresponds to the Rayleigh
type of scattering, i.e., a dependence of E4. Consequently,
sc E
can be calculated in all the considered energy
Figure 1. Numerical resolution of Equation (1) given at en-
ergy E = 1.1 eV.
Figure 2. Convoluted
spectrum in all considereted
energy rang e.
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range. Then, we inject values in Equation (1)
but this time with variable
sc E
in order to calculate the
spectral dependence of . However, in reference
[3], the authors have their specific experimental proce-
dure to extract true and spectra of nano-
Si:H samples. Indeed, they estimate the scattering coeffi-
cient from a comparison of several CPM measurements
(for different interelectrodes spacing) [3,7] and they sup-
posed a Rayleigh light scattering. Using the first equation
and by an iterative procedure, they obtained the true op-
tical absorption spectrum . Figure 3 shows a
perfect agreement of the calculated and measured true
optical absorption coefficient spectra.
true E
true E
3. Experiment
Nanocrystalline silicon layers were deposited by RF
magnetron sputtering of a silicon target, under different
pressure (2, 3 and 4 Pa) with different substrate tempera-
ture (100˚C, 150˚C, 200˚C). This process enables to de-
posit the silicon thin films on all types of substrates. With
respect to the deposition conditions, the layers result in a
rough (textured). The measured typical root mean square
surface roughness of the nanotextured silicon
samples varies between 4 and 12 nm. The CPM results of
nano-Si:H series deposited at 2 Pa are shown in Figure
We note that rms
of this series is about 6 nm for
about 2.5 µm thick films. In order to exclude the influ-
ence of surface light scattering on the calculations, we
have polished numerically the surfaces of the samples.
Figure 3. Comparison of true optical absorption spectra of
thin nano-Si:H film deposited by Very High Frequency
Glow Discharge (VHFGD) method. The results plotted by
black bolls correspond to the values calculated by our nu-
merical method and that plotted by empties bolls for αtrue
evaluated by the authors using their experimental proce-
dure [3].
Figure 4. Fitted apparent optical absorption spectra of
nano-Si: H series deposited by RF magnetron sputtering
method under different substrate temperature.
values are calculated after a numerical polishing of the film
surface (
rms = 0 nm).
The polishing procedure has been done using the surface
light scattering theory presented in reference [3]. Indeed,
in the calculations, we have supposed that the textured Si
has a smooth surface (rms
= 0 nm) and we have calcu-
true E
values which are in reality the apparent
optical absorption coefficient
app E
values influ-
enced only by bulk light scattering effects (Figure 4).
The evaluated
app E
values will be after that, in-
jected in our program in order to calculate
true E
sc E
4. Results
We have applied our numerical method on the silicon
series with their CPM data given in Figure 4. Figure 5
shows the calculated spectral dependence of the true op-
tical absorption
true E
and the scattering coefficient
sc E
of the sample deposited at 200˚C.
The calculated spectra of the samples of this series are
presented together in Figure 6.
5. Discussions
Results presented in Figure 5 and Figure 6 demonstrate
that the samples deposited at 2 Pa exhibit an important
bulk light scattering contribution especially in low ab-
sorption range (0.9 - 1.6 eV spectral region) compared
with the samples deposited at 3 and 4 Pa. These samples
have a considerable surface roughness, which leads to a
remarkably surface light scattering. Consequentially, the
contribution of the scattered light at the rough surface
will be dominant. Furthermore, the scattering coefficient
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Figure 5. Scattering and true optical absorption spectra
calculated from CPM data of nano-Silicon thin film depos-
ited under pressure of 2 Pa and substrate temperature of
Figure 6. True optical absorption spectra calculated by the
numerical method for a nano-Si:H samples deposite d by RF
magnetron sputtering method under different substrate
evaluated for each sample deposited at 2 Pa has the
Rayleigh scattering form . This result is very
convenient with the nature of our sample. Indeed, we
study here nanotextured thin films, i.e., samples which
have a random rough surface with rms roughness smaller
than the wave length of the incident light. Furthermore,
they possess the property to have small heterogeneities
included in their volume similarly to the case of amor-
phous silicon. We must note that there are some nano-
Si:H rough samples which demonstrate an important
surface scattering and by a standard mechanical polishing,
the surface scattering disappears and the true optical ab-
sorption is directly measured by CPM. But there are
sc E
some other ones which exhibit a bulk scattering contribu-
tion, which cannot remove by polishing. We calculate
their true and scattering coefficients spectra using the
proposed numerical method. We can unify the bulk and
surface scattering in terms of the scattering coefficient
given by the following equation:
n n and a are respectively, the refraction in-
dexes of nano-Si:H film and the ambient. λ is the incident
wave length.
6. Conclusion
In this paper, we have developed a new numerical me-
thod in order to calculate the true optical absorption
spectra and the contribution of light scattering in CPM
measurements of hydrogenated nanocrystalline silicon
thin films. With the help of surface and bulk light scat-
tering theories presented by Poruba and al, we have con-
cluded that the contribution of bulk light scattering in
CPM spectra of nanocrystalline silicon samples depos-
ited at low pressure which have a low surface roughness
is due mainly to the included heterogeneities (nanocrys-
tallites, voids,...) similarly to the case of amorphous sili-
7. Acknowledgements
We gratefully acknowledge Prof. Kacem Zellama from
Picardie Jules Vernes University in France for providing
the nano-Si:H samples. Thanks to all who contributed to
this work.
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