Depth profiling of Si nanocrystals in Si-implanted SiO2 films by spectroscopic ellipsometry

Article abstract of DOI:10.1063/1.1528286

An approach to determine depth profiles of silicon nanocrystals in silica films was developed. In the spectral fittings, the dielectric function of silicon nanocrystal was calculated based on two different models for the band-gap expansion due to the nano

Full text of DOI:10.1063/1.1528286

Depth profiling of Si nanocrystals in Si-implanted SiO 2 films by spectroscopic
ellipsometry
T. P. Chen, Y. Liu, M. S. Tse, P. F. Ho, Gui Dong, and S. Fung
Citation: Applied Physics Letters 81, 4724 (2002); doi: 10.1063/1.1528286
View online: http://dx.doi.org/10.1063/1.1528286
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APPLIED PHYSICS LETTERS
VOLUME 81, NUMBER 25
16 DECEMBER 2002
Depth profiling of Si nanocrystals in Si-implanted SiO films
2
by spectroscopic ellipsometry
T. P. Chen,a) Y. Liu, M. S. Tse, and P. F. Ho
School of Electrical and Electronic Engineering, Nanyang Technological University, Singapore 639798
Gui Dong
Institute of Microelectronics, Singapore Science Park II, Singapore 117685
S. Fung
Department of Physics, The University of Hong Kong, Pokfulam Road, Hong Kong
͑
Received 30 April 2002; accepted 16 October 2002͒
In this letter, we report an approach to depth profiling of Si nanocrystals embedded in SiO film
2
based on spectroscopic ellipsometry. The SiO film is divided into many sublayers with equal
2
thickness, and each sublayer is characterized by its nanocrystal concentration. In the spectral fittings,
the effective dielectric function of each sublayer is obtained from an effective medium
approximation by using the dielectric function of Si nanocrystal that is calculated with either the
bond contraction or the phenomenological models for the band gap expansion of nanocrystals. The
fittings yield the nanocrystal depth profiles and the nanocrystal sizes as well. The depth profiles from
the two models are similar, and they are in good agreement with secondary ion mass spectroscopy
analysis. © 2002 American Institute of Physics. ͓DOI: 10.1063/1.1528286͔
SiO films containing Si nanocrystals have recently at-
m sublayers with equal thickness dϭTox /m, where Tox is the
total thickness of the film, that is, sublayers 1, 2, ... m from
the surface to the SiO2 /Si interface (mϭ22 in this study͒,
and the nanocrystal concentration is considered constant
within each sublayer. Each sublayer has its own complex
refractive index, N ϭn ϩjk (iϭ1,2,... m) where n and k
2
tracted much attention because of their light-emitting ability
1
that can be used for Si-based optoelectronic applications. In
addition, they have also regained interest as possible candi-
dates for the application of single electron memory devices
or other single-electron devices.2 One of the promising
techniques being used to elaborate nanocrystals is the im-
–5
i
i
i
i
i
are the refractive index and extinction coefficient for the ith
sublayer, respectively. Note that N is also a function of
plantation of Si ions into SiO films that are thermally grown
i
2
6
–8
wavelength ͑␭͒. Therefore, one can use a (mϩ2)-phase
model, that is, air/sublayer 1/.../sublayer m/Si substrate, for
the elliposmetry analysis. Each phase is characterized by its
on Si substrates.
The SiO has proven to be a robust ma-
2
trix that provides good chemical and electrical passivation of
the nanocrystals. In addition, the fabrication is fully compat-
ible with the mainstream complementary metal–oxide–
semiconductor processes, allowing the integration of the op-
toelectronic devices into the Si circuits. Therefore, this
technique is very attractive. For applications it is essential to
have detailed information on the depth distributions of the
complex refractive index, N (iϭ0,1,... m,mϩ1). Note that
i
N ϭ1 for air and Nmϩ1ϭNSi ͑i.e., the Si complex refractive
0
index͒ for the Si substrate. For a fixed incident angle ␾ and
0
a given d (␾ ϭ75° and dϭ25 nm in this study͒, the ellip-
0
sometric angles ␺ and ⌬ are functions of the parameters N ,
1
N ,...N , N and ␭.1
0,11
2
m
Si
nanocrystals in the SiO films. In this letter, we report a
2
For each sublayer, its effective complex dielectric func-
cheap and nondestructive approach, based on spectroscopic
elliposmetry, to determine the depth profiles of nanocrystals
2
tion ␧ (ϭN ,iϭ1,2,... m) can be calculated with the fol-
i
i
1
0,12
lowing effective medium approximation
in Si-implanted SiO films.
2
SiO films 550 nm thick were grown on p-type Si ͑100͒
␧iϪ␧SiO2
␧ncϪSiϪ␧SiO2
2
substrates by wet oxidation of Si at 1000 °C. The SiO films
ϭ␯i
,
͑1͒
2
␧iϩ2␧SiO2
␧ncϪSiϩ2␧SiO2
were implanted with a dose of 1ϫ1017 atoms/cm of Si at
2
ϩ
50 keV. Afterwards, the samples were annealed at 1000 °C
where ␯ (iϭ1,2,... m) is the volume fraction ͑it will be
i
for duration of 1 h. Upon such an annealing, one would
converted to atom percentage later͒ of the nanocrystals in the
ith sublayer, ␧SiO2 is the dielectric function of host material
8
,9
expect the formation of Si nanocrystals in the SiO film.
2
As revealed by the spectroscopic elliposmetry analysis and
secondary ion mass spectroscopy ͑SIMS͒ measurements dis-
cussed later, the nanocrystals distribute from the surface to a
depth of about 250 nm.
SiO , and ␧
is the dielectric function of Si nanocrystal.
ncϪSi
2
Using Eq. ͑1͒, the depth distribution of the complex refrac-
tive index is converted to the depth distribution of the vol-
ume fraction (␯ ). Thus, ␺ and ⌬ can be symbolically written
i
As the concentration of the nanocrystals in the SiO film
as
2
varies with the depth, the optical properties of the film will
vary with the depth also. To reflect the variation of the nano-
crystal concentration with the depth, the film is divided into
⌿
ϭf ͑␯ ,␯ ,...␯ ,N
,NSiO2,N ,␭͒
͑2͒
͑3͒
1
1
2
m
ncϪSi
Si
and
a͒Electronic mail: echentp@ntu.edu.sg
⌬ϭf ͑␯ ,␯ ,...␯ ,N
2
1
2
m
ncϪSi
,NSiO2,N ,␭͒.
Si
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003-6951/2002/81(25)/4724/3/$19.00 4724 © 2002 American Institute of Physics
On: Mon, 22 Dec 2014 13:43:37
0

Appl. Phys. Lett., Vol. 81, No. 25, 16 December 2002
Chen et al.
4725
The functions f1 and f2 cannot be expressed as analytical
formulas, but ␺ and ⌬ can be calculated numerically with the
functions. As the complex refractive indexes of SiO and Si
2
͑
i.e., NSiO2 and N ) are well known, if the complex refrac-
Si
tive index of Si nanocrystal is also known, the depth profile
of the nanocrystal concentration can be found by searching
for one set of (␯ ,␯ ,...␯ ) such that the following error
1
2
m
function:
␭
m
␭
c
2
␭
m
␭
2
Fϭ
͓͑⌿ Ϫ⌿ ͒ ϩ͑⌬ Ϫ⌬ ͒ ͔
͑4͒
͚
c
␭
is a minimum for the whole measured spectral range, where
␭
␭
⌿
and ⌬ are the respective measured ⌿ and ⌬ values at ␭,
m
m
␭
␭
and ⌿ and ⌬ are the calculated ⌿ and ⌬ values with Eqs.
c
c
͑2͒ and ͑3͒, respectively, at the same wavelength. Actually,
this is a spectral fitting over a wide range of wavelengths. In
this study, this spectral fitting was carried out in the wave-
length range of 400 to 1200 nm. This fitting procedure has
been proven correct and effective by its successful applica-
tion to the well-known system of pure SiO film on Si sub-
2
strate.
To apply the fitting procedure to SiO films containing
2
Si nanocrystals, we must know the dielectric function of Si
nanocrystal. It is questionable to use the dielectric function
of bulk crystalline Si for the nanocrystal, as it is expected
that with size reduction of a nanometric system, its dielectric
function will change. Nevertheless, as a trial, we used the
dielectric function of bulk crystalline Si in the fitting, and the
best fitting is shown in Fig. 1͑a͒ As can be seen in Fig. 1͑a͒,
although the agreement between the measurement and the
calculation is not too good, all the complicated spectral fea-
tures of the experimental ⌿ and ⌬ are basically fitted by the
calculation. This indicates that the difference of dielectric
functions between bulk crystalline Si and Si nanocrystal is
not huge, implying that the sizes of the nanocrystals are not
too small ͑say, not less than 3 nm͒ in this study.
FIG. 1. Best spectral fittings of ⌿ and ⌬ with 161 wavelengths: ͑a͒ using the
dielectric function of bulk crystalline Si; ͑b͒ using the dielectric function of
Si nanocrystal calculated with the bond contraction model for ⌬E ; and ͑c͒
The dielectric function of Si nanocrystal can be calcu-
g
lated by using the simple formulas proposed in13 which are
using the dielectric function of Si nanocrystal calculated with the phenom-
enological model for ⌬E . The values of F defined in Eq. ͑4͒ are 25679,
g
duplicated in the following for easy reference. As a result of
11331, and 8477 for the fittings shown in ͑a͒, ͑b͒, and ͑c͒, respectively.
the band-gap expansion (⌬E ) due to the reduction of nano-
g
crystal size, the change ͑⌬␹͒ of the dielectric susceptibility is
given by
n͑2nϩ5͒/3
C
d
m
⌬
E ϭ ͩ ͪ
,
͑7͒
g
n
d
d
0
0
⌬
␹/␹ ϭϪ2͑⌬E /Eg0͒,
͑5͒
0
g
where d ͑in nm͒ is the mean size of nanocrystals and d is
the size for which the maximum occurs in the log-normal
distribution, nϭ1.22, Cϭ3.9, and d /d ϭ0.7. The second
model is the bond contraction model.
0
m
and the change (⌬␧ ) of the imaginary part of the complex
m
dielectric function is given as
1
4
m
0
⌬
␧m /␧m0ϭϪ2͓E /͑EϪEg0͔͒͑⌬E /Eg0͒,
͑6͒
13,15
g0
g
From this model, if
the surface bond contraction is significant for only the two
outermost atomic layers of a spherical dot and the dot size
where ␹ , ␧ , and Eg0 are the corresponding physical pa-
0
m0
rameters of bulk crystalline Si, and E is photon energy. It
should be pointed out that the formulas are just a first-order
approximation valid for a small ⌬E /E . To calculate the
͑i.e., the diameter D͒ is much larger than the atomic diameter
d, the band-gap expansion is given as
g
g0
dielectric function of Si nanocrystal, the band-gap change
E /E should be known.
⌬
Eg
Ϫm
Ϫm
2
⌬
ϭ␥Јc ͑c Ϫ1͒ϩ␥Јc ͑c Ϫ1͒,
͑8͒
g
g0
1
1
1
2
2
Eg0
For comparison, here two different models are used to
2
calculate the band-gap change. The first model is a phenom-
enological model, which predicts the trend for quantum con-
finement, that is, the inverse dependence of the expanded
band gap on the nanocrystallite size, and in which a log-
where ␥Јϭ(3/k)͓1Ϫ(iϪ0.5)/k͔ (iϭ1,2), kϭD/(2d), m
ϭ1, c ϭ0.88, and c ϭ0.94. For a given nanocrystal size
͑i.e., the mean size d in the first model or the diameter D in
the second model͒, ⌬E /E can be calculated by using
these two models, and thus the dielectric function of the Si
nanocrystal can be obtained with Eqs. ͑5͒ and ͑6͒. Therefore,
i
1
3
1
2
0
g
g0
1
4
normal distribution of nanocrystallite sizes is considered.
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From this model, ⌬E is given by
g
On: Mon, 22 Dec 2014 13:43:37

4726
Appl. Phys. Lett., Vol. 81, No. 25, 16 December 2002
Chen et al.
FIG. 2. Depth profiles of Si nanocrystals obtained from the spectral fittings
by using the dielectric function of bulk crystalline Si or the dielectric func-
tion of Si nanocrystal calculated with either the bond contraction or phe-
nomenological models for ⌬Eg .
FIG. 3. Comparison of the result of SIMS with the depth profile obtained
from the spectral fitting by using the dielectric function of Si nanocrystal
calculated with the bond contraction model for ⌬Eg .
mine depth profiles of Si nanocrystals in SiO films based on
spectroscopic ellipsometry. In the ellipsometry analysis, the
2
the spectral fitting discussed previously is now carried out by
searching for the nanocrystal size and one set of
SiO film with a depth distribution of Si nanocrystals is di-
2
(
␯ ,␯ ,...␯ ) such that the error function ͑F͒ is a minimum.
1 2 m
vided into m sublayers with equal thickness ͑a better depth
resolution for a larger m͒, and each sublayer is characterized
by its nanocrystal concentration. As the dielectric function of
Figures 1͑b͒ and 1͑c͒ show the best fittings with the band
contraction model ͑the second model͒ and the phenomeno-
logical model ͑the first model͒ for ⌬E , respectively. As can
g
SiO is well known, the effective dielectric function of each
2
be seen clearly in Fig. 1, these fittings, which have consid-
ered the nanosize effect on the band gap, are superior to the
fitting with the dielectric function of bulk crystalline Si, in-
dicating that the nanosize effect plays a role in the dielectric
function. The depth profiles of Si nanocrystal concentration
obtained from the fittings based on the two models are
shown in Fig. 2, and they are also compared in this figure
with the depth profile obtained from the fitting using the
dielectric function of bulk crystalline Si. As can be seen in
Fig. 2, the three depth profiles appear to be very similar ͑only
some small difference exist in the peak regions of the pro-
files͒. The reason for this is that, as indicated by our calcu-
lations based on the two models, the difference of dielectric
functions between the bulk crystalline Si and the Si nano-
crystal is not large, as the size of the Si nanocrystals is not
very small. The fitting based on the bond contraction model
yields a nanocrystal size of about 5 nm, which agrees with
the result reported in Ref. 8 for similar nanocrystal formation
conditions, while the fitting based on the phenomenological
model yields a size of about 9 nm. For such nanocrystal
sizes, the bond contraction model ͑and the phenomenological
model͒ gives a decrease of 9% ͑and 16%͒ in the real part and
a decrease of 5% ͑and 9%͒ in the imaginary part of the
dielectric function at the wavelength of 400 nm. Therefore,
although the fittings can be improved by taking into account
the small reductions in the dielectric function due to the size
effect ͑as shown in Fig. 1͒, the reductions do not have a very
significant impact on the depth profiles. The depth profiles
agree with that from the SIMS measurement. Figure 3 shows
the comparison of the depth profile obtained from the spec-
tral fitting based on the bond contraction model with that
from the secondary ion mass spectroscopy measurement. The
good agreement indicates that the above approach for depth
profiling is reliable.
sublayer can be calculated with the effective medium ap-
proximation if the dielectric function of Si nanocrystal is also
given. In the spectral fittings, the dielectric function of Si
nanocrystal is calculated based on two different models ͑i.e.,
the bond contraction model and the phenomenological
model͒ for the band-gap expansion due to the nanocrystal
size reduction. The fittings yield the nanocrystal depth profile
and the nanocrystal size as well. The depth profiles from the
two models are very similar, and they are in good agreement
with the SIMS measurement.
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In conclusion, we have developed an approach to deter-
͑1999͒.
On: Mon, 22 Dec 2014 13:43:37