Depth profile determination of Si nanocrystals embedded in SiO2 films by spectroscopic
ellipsometry
Y. Liu*1, T. P. Chen1, M.S.Tse1, P.H.Ho1, T.B. Chong1, D. Gui2, J. H. Hsieh3
*1
School of Electrical and Electronic Engineering,
Nanyang Technological University, Singapore 639798
2
Institute of Microelectronics, Singapore 117685
3
School of Mechanical and Production Engineering
Nanyang Technological University, Singapore, 639798
Abstract
report an approach to determination of depth
profiles of both optical constants and excess Si nc
fraction in SiO2 films. In this approach, the depth
profiles are quantified in an inexpensive and nondestructive way that is based on spectroscopic
elliposmetry (SE). The results of the depth profile
of excess Si fraction are in good agreement with
secondary ion mass spectroscopy (SIMS)
analysis, indicating that the approach is reliable.
100
No Implantation
Dose=2x1016/cm2
Dose=6x1016/cm2
Dose=1x102/cm2
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60
degree)
In this paper, we report an approach to
determine the depth profiles of Si nanocrytals
fraction distributed in SiO2 films based on
spectroscopic ellipsometry (SE). In the SE
analysis, a Si implanted SiO2 film is divided into
m sub-layers with equal thickness (a better depth
resolution for a larger m), and an effective
medium approximation (EMA) is used to convert
the depth profile of the complex refractive index
to the depth profile of the excess Si fraction. With
this approach, the depth profiles of both excess Si
nanocrystal (nc) fraction and optical constants are
determined quantitatively in a inexpensive and
non-destructive way. The depth profiles of excess
Si nc fraction obtained are in good agreement
with secondary ion mass spectroscopy (SIMS),
indicating that the approach is reliable.
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Introduction
0
*
Corresponding author. Email: p150531616@ntu.edu.sg
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Wavelength (nm)
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No implantation
Dose=2x1016/cm2
Dose=6x1016/cm2
Dose=1x1017/cm2
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(degree)
SiO2 films containing Si nanocrystals
(nc-Si) have recently attracted much attention
because of their light-emitting ability that can be
used for Si-based optoelectronic applications [1].
In addition, they have also regained the interest as
a possible candidate for the application of single
electron memory devices or other single electron
devices [2-5]. One of the promising techniques
being used to elaborate nc-Si is the implantation
of Si ions into SiO2 films that are thermally
grown on Si substrates [6-8]. The SiO2 has
proven to be a robust matrix that provides good
chemical and electrical passivation of the
nanocrystals. In addition, the fabrication is fully
compatible with the mainstream CMOS
processes, and this allows the integration of the
optoelectronic devices into the Si circuits.
Therefore, this technique is very attractive. For
applications such as light-emitting structures,
single-electron memories, optical storage devices
or wave guides, it is essential to have detailed
information on the exact depth distribution of the
excess Si concentration and of optical properties
of the Si-doped SiO2 films. In this paper, we
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Wavelength (nm)
Fig1. Measured  and  as a function of wavelength.
Experiment
550nm-thick SiO2 films were grown on ptype Si (100) substrates by wet oxidation of Si at
1000 oC. The SiO2 films were implanted with
doses from 21016 to 11017 atoms/cm2 of Si+ at
50 keV. Thermal annealing at 1000 oC for 1 hour
led to the crystallization of the implanted Si ions
in nano-size. As revealed by the SE analysis and
SIMS measurements discussed below, the excess
Si atoms distribute from the surface to a depth of
about 250 nm. SE measurements before and after
Si implantation were carried out in the wavelength
range of 400 to 1200 nm. A continuous significant
change in the ellipsometric angles  and  with
the implant dose was observed as shown in Fig.1.
As discussed below, the depth profiles of the
excess Si nc fraction in the SiO2 films were
determined from these SE measurements.
SE analysis and results
As the concentration of excess Si atoms in
the SiOx films varies with the depth, namely, the x
is a function of the depth; the optical properties of
the films will also vary with the depth. To model
the SiOx films, the films are divided into m sublayers with equal thickness d = Tox / m where Tox
is the total thickness of the films, i.e., layers 1, 2,
...m from the surface to the SiO2/Si interface. The
x (2) is considered constant within each sublayer, and the corresponding x for the m layers are
x1, x2, ...xm, respectively. Actually, the depth
profile of x can be translated into the depth profile
of excess Si fraction in the films. Each sub-layer
has its own complex refractive index. For
example, for the ith sub-layer, its complex
refractive index is Ni = ni + jki (i=1, 2, ...m) where
ni and ki are the refractive index and extinction
coefficient for the ith sub-layer, respectively.
Note that Ni is also a function of wavelength.
For SE analysis, an appropriate optical
model is required.
Based on the above
discussions, we can use a (m + 2)-phase model,
i.e., air/sub-layer 1/.../sub-layer m/Si substrate to
describe the system of Si implanted SiO2 film on
Si substrate. Each phase is characterized by its
complex refractive index, namely, Ni (i=0, 1, ...m,
m+1). Note that N0 = 1 for air and Nm+1 = NSi for
the Si substrate. For the total system, the ratio 
of the complex reflection coefficients for the p
and s polarizations is given by [10]
  Rp / Rs  tan( ) exp( j ) ,
(1)
where  and  are the ellipsometric angles . The
complex reflection coefficient R ( =p, s) for the
p and s polarizations is given by
R =( r0,1 + R1,2  X1)/ (1+ r0,1 R1,2X1) ,
(2)
where X1=exp(-j4 N1d cos 1 /  )
,
R1,2=(r1,2+R2,3X2)/(1+r1,2R2,3X2) ,
X2=exp (-j4 N 2d cos 2 /  )
,
...
R (m-1),m=(r(m-1),m+rm,(m+1)Xm)/(1+r(m-1),mrm,(m+1)Xm),
Xm=exp (-j4 N m d cos m /  ).
In the above equations,  is the freespace wavelength of light; ri,(i+1) (i =O, 1, ...m) is
the Fresnel complex-amplitude reflection
coefficient for - (=p, s) polarized light at the
interface between the ith phase and the (i+1)th
phase[11]; and the angles i and the complex
refractive index Ni (i=0, 1, ...m, m+1) are related
by Snell’s law. In the present study, 0 is fixed at
75o.
As N0 (=1) and Nm+1 (= Si complex
refractive index) are known, for a given d, from
the above equations,  and  can be symbolically
written as = f1(N1, N2, …Nm, ) and = f2(N1, N2,
…Nm, ). The functions f1 and f2 cannot be
expressed as analytical formulae, but the  and 
can be calculated numerically. To determine the
depth profile of the optical constants, one can
search for one set of the parameters (N1, N2, ...Nm)
which are dependent of wavelength , by
comparing the calculated (, ) with the
experimental (, ). To obtain the definite values
of the complex refractive indices of the m sublayers at various wavelengths, an appropriate
constraint that is independent of wavelength, is
required. .
Fig2. Comparison of theoretical computation (solid
lines) with experimental (* and o) data for dose=0.
i   h
  h
 vi a
 i  2 h
 a  2 h
(3)
where i is the volume fraction (it will be
converted to atom percentage later) of the excess
Si atoms in the ith sub-layer, h is the dielectric
function of host material SiO2, and a is the
dielectric function of the Si inclusion. With (3),
the depth profile of the complex refractive index
is converted to the depth profile of the volume
fraction (i). Note that the i is independent of the
wavelength. Therefore, there are only m (the
number of the sub-layers) unknown parameters,
i.e., 1, 2, ...m in the spectral fitting of
experimental (, ), which greatly reduces the
complexity of the fitting. Obviously, a larger m
means a better depth resolution but a longer
computation time
more complicated compared with that of pure
SiO2 film on Si substrate. However, the fitting
procedure is also able to produce a very good
fitting over the whole spectral range. One typical
example is shown in Fig.3. As can be seen in this
figure, all the complicated spectral features of
both  and  can be fitted excellently. Such a
fitting can yield the depth profile of the volume
fraction of the excess Si atoms in the SiO2 films.
The volume fraction can be simply converted to
the atom percentage that can be compared with
the SIMS analysis. One example of the depth
profile of the excess Si atom percentage obtained
is shown in Fig.4. As shown in this figure, the
depth profile obtained agrees well with the SIMS
result.
This indicates that our approach is
reliable. On the other hand, as revealed by the
TRIM simulation, the depth profile follows a
Gaussian distribution approximately.
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SE calculation
Normalized SIMS
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Excess Si fraction (at %)
The constraint can be established by using
an effective medium approximation (EMA) to
relate the depth profile of the complex refractive
index to the depth profile of the excess Si fraction
that is independent of wavelength. According to
the EMA [10,12], the effective complex dielectric
function of each sub-layer, i (=Ni2, i=1, 2, ...m),
can be approximately calculated using the
following expression
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 measurement
 measurement
 calculation
 calculation
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Fig.4. Comparison of the depth profiles of excess Si
nanocrystal fraction obtained with the SE and SIMS
analysis. The Si implant dose is 11017 atoms/cm2.
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Degrees
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Depth (nm)
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Conclusion
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 (nm)
Fig.3. Typical example of spectral fitting of  and 
with the (m+2)-phase model and the constraint that
uses the EMA to convert the depth profile of the
complex refractive index to the depth profile of the Si
nanocrystal fraction. The Si implant dose is 11017
atoms/cm2.
The above models and the fitting
procedure are proven correct and effective by
applying them to a system of pure SiO2 film on Si
substrate. The fitting yields i =0 (i=1, 2, ...m)
and the calculated (, ) fit the experimental (,
) perfectly over the whole spectral range (400 1200nm), as illustrated in Fig2. For Si implanted
SiO2 films on Si substrate, the situation is much
In conclusion, we have developed an
approach to determination of depth profiles of the
excess Si nc fraction distributed in SiO2 films
based on the SE. In the SE analysis, a Si
implanted SiO2 film is divided into m sub-layers
with equal thickness (a better depth profile
resolution for a larger m), and an optical model of
(m+2) phases is used. In the spectral fitting of
experimental  and , an effective medium
approximation (EMA) is used to convert the depth
profile of the complex refractive index to the
depth profile of the excess Si fraction, which
greatly reduces the complexity of the fitting.
With this approach, the depth profiles of Si nc
fraction is determined quantitatively in an
inexpensive and non-destructive way.
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