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SURFACE AND INTERFACE ANALYSIS
Surf. Interface Anal. 28, 240–244 (1999)
Complete Optical Characterization of the
SiO2/Si System by Spectroscopic Ellipsometry,
Spectroscopic Reflectometry and Atomic Force
Microscopy
I. Ohlı́dal,1 * D. Franta,2 E. Pinčı́k3 and M. Ohlı́dal4
1
2
3
4
Department of Solid State Physics, Faculty of Sciences, Masaryk University, Kotlářská 2, 611 37 Brno, Czech Republic
Department of Physical Electronics, Faculty of Sciences, Masaryk University, Kotlářská 2, 611 37 Brno, Czech Republic
Institute of Physics, Slovak Academy of Sciences, Dúbravská cesta 9, 842 28 Bratislava, Slovak Republic
Institute of Physical Engineering, Faculty of Engineering, Technical University, Technická 2, 616 69 Brno, Czech Republic
In this paper results concerning optical analysis of the SiO2 =Si system performed by the combined
ellipsometric and reflectometric method used in multiple-sample modification will be presented. This method
is based on combining both the single-wavelength method and the dispersion method. Three models of the
system mentioned, i.e. the model of the substrate and the layer with the smooth boundaries, the same model
with a transition layer and the model of the substrate and the layer with rough boundaries, will be used to
interpret the experimental data. The spectral dependences of the optical constants of silicon and SiO2 with
the values of the other parameters will be determined. It will be shown that the simplest model with the
smooth boundary is the most convenient with the experimental data. Copyright  1999 John Wiley & Sons,
Ltd.
KEYWORDS: optical constants of Si and SiO2 ; spectroscopic reflectometry; spectroscopic ellipsometry
INTRODUCTION
The optical properties of the SiO2 /Si system have been
studied extensively because of its importance in practice.
However, the results of the optical studies of this system
presented in the literature are somewhat different. Considerable effort was therefore devoted to finding the correct
and precise values of the optical parameters of the SiO2 /Si
system in latter years. The basic problem of optical characterization of the SiO2 /Si system is to find correct and
precise values for the silicon substrate.
In practice, two basic groups of ellipsometric methods
can be utilized for determining the optical parameters
of the Si substrate. The methods of the first group are
based on separate interpretation of the optical quantities of the system measured at individual wavelengths
(single-wavelength methods). The methods of the latter
group are based on interpreting the entire spectral dependences of the optical quantities of the system measured
within a certain spectral region (dispersion methods). The
advantages and disadvantages of the methods belonging
to the two groups were discussed in our earlier paper.1 A
good example of employing the single-wavelength method
* Correspondence to: I. Ohlı́dal, Department of Solid State Physics,
Faculty of Sciences, Masaryk University, Kotlářská 2, 611 37 Brno,
Czech Republic.
E-mail: ohlidal@physics.muni.cz
Contract/grant sponsor: Grant Agency of Czech Republic; Contract/grant number: 202/98/0988.
Contract/grant sponsor: Grant Agency of Ministry of Education of
Czech Republic; Contract/grant number: VS96084.
CCC 0142–2421/99/130240–05 $17.50
Copyright  1999 John Wiley & Sons, Ltd.
for determining the optical constants of a silicon singlecrystal substrate is presented by Aspnes and Theeten.2 The
method used by Herzinger et al.3 is the typical dispersion method for characterizing this substrate but applied
simultaneously to several samples of the system (multiplesample method).
In this paper the results concerning optical characterization of samples of the SiO2 /Si system using a new
combined optical method will be presented.
PREPARATION OF SAMPLES AND
EXPERIMENTAL ARRANGEMENTS
Samples of the SiO2 /Si system studied were prepared
as follows. The surfaces of Si wafers (n-type, (111)orientation) were treated in a standard way. The Si surfaces obtained in this way were treated by a special
procedure based on a combination of thermal oxidation
and etching to remove possible surface layers containing
defects. The SiO2 layers were prepared by thermal oxidation of these Si surfaces at a temperature of 1040 ° C in an
atmosphere of oxygen and HCl (25 : 1). Different values
of the thicknesses of the SiO2 layers were achieved by
different periods of oxidation of the Si surfaces. The samples of the Si surfaces covered with the SiO2 layers were
finally heated in a nitrogen atmosphere at the same temperature (details of this procedure for creating the samples
will be described in a forthcoming paper).
The spectral dependences of the ellipsometric parameters  (azimuth) and  (phase change) of the
samples were measured using a Jobin–Yvon ellipsometer
Received 30 November 1998
Accepted 8 March 1999
OPTICAL CHARACTERIZATION OF SiO2 /Si
(UVISEL) in the spectral range 240–830 nm at angles
of incidence of 55–75° . The spectral dependences of
the reflectance of these samples were measured by a
spectrophotometer (Varian Cary 5E) in the same spectral
region at an angle of incidence of 10° .
Surface roughness of the SiO2 layers was measured
by means of an atomic force microscope (Accurex,
TopoMetrix Inc.).
241
E qj denotes the vectors whose components are
where A
formed by the complex amplitudes ACqj and Aqj of the
electric fields belonging to the right-going and left-going
waves inside the jth medium, respectively5 (the incident
wave on the system is the right-going wave). It holds
that j D 0, 1, 2, where the indices 0, 1 and 2 denote the
ambient, layer and substrate, respectively. Then
E qj D
A
DATA PROCESSING
The ellipsometric parameters  and  and the reflectance
R of isotropic thin-film systems are defined as
tan ei D
rOp
,
rOs
RD
jOrp j2 C jOrs j2
2
O ii D tan i eii ,
O ij D
Ri
Rj
Ci
Rj
Ri
E denotes the vector whose components are idenwhere X
tical to the parameters sought. The index k represents the
serial number of experimental value of the complex quanexp
tity O k , wk are the weights of the experimental values
and k and/or 0k is the wavelength and/or the angle of
incidence of light falling onto the upper boundary of the
system SiO2 /Si belonging to the corresponding experimental value (the indices i and j are functions of the index
E k , 0k / is calculated using Eqns (1)
k). The function O ij .X,
and (2). The values of the parameters sought correspondE are considered
ing to the minimum of the function S.X/
to be the true values of these parameters characterizing
the samples studied. In the procedure of searching the
minimum of S, the Marquardt–Levenberg algorithm was
used.4 This LSM is employed for treating the experimental
data in a combined way (see Results section).
Mq21
Mq11
.6/
Substrate covered by a single film with smooth
boundaries
Q q of this system is expressed as
The transfer matrix M
Q q D BQ q1 TQ 1 BQ q2
M
.7/
1 1
BQ qj D
Otqj rOqj
rOqj
1
.8/
The symbol rOqj and/or Otqj represents the reflection and/or
transmission Fresnel coefficient of the jth boundary for
the wave incident on this boundary from the left side. The
amplitudes of the waves inside the film corresponding to
the first and second boundaries are connected by means
of the elements of the phase matrix TQ 1 . i.e.
TQ 1 D
eik0 nO 1 d
0
0
e
.9/
ik0 nO 1 d
where the symbols d and/or n1 represent the thickness
and/or refractive index of the film and k0 D 2/.
Substrate covered by a transition layer and a single
film
For this system one can express the transfer matrix in the
form
Q q1 1 XQ q W
Q q2
Q q D BQ q1 TQ 1 W
.10/
M
Q q1 1 XQ q W
Q q2 represents the transition layer
where the matrix W
between the substrate and the single film. The matrices
Q qk are expressed as
W
MODELS OF THE SiO2 /Si SYSTEM
Let us consider an isotropic thin-film system placed on
a semi-infinite substrate. The amplitudes of the waves
propagating within both the ambient and the substrate then
fulfil the following matrix equation5
Copyright  1999 John Wiley & Sons, Ltd.
.5/
The matrix BQ qj (refraction matrix) is defined as
k
q D p, s
rOq D
.2/
where the indices i and j express the numbers of the
samples .i, j D 1, 2 . . . 6/. For finding the values of the
parameters characterizing the samples, the least-squares
method (LSM) can be used. One can then construct the
following merit function
X
E k , 0k / O kexp j2 wk
E D
jO ij .X,
.3/
S.X/
E q0 D M
E q2 ,
Q qA
A
Q q represents the transfer matrix of the sysThe symbol M
tem for q-polarization. The reflection Fresnel coefficient
of the whole system is then given as5
.1/
where rOp and / or rOs denotes the reflection Fresnel coefficient of the systems for p- and/or s-polarization. In our
studies we treated the spectral dependences of the ellipsometric parameters and relative reflectances measured for
the six samples and therefore we introduced the complex
quantities O ij defined by
ACqj
Aqj
.4/
Q sk D
W
1
ik0 nO k cos O k
and
Q pk D
W
nO k
ik0 cos O k
1
ik0 nO k cos O k
nO k
ik0 cos O k
.11/
.12/
Surf. Interface Anal. 28, 240–244 (1999)
242
I. OHLÍDAL ET AL.
If the thickness of the transition layer dT is much
smaller than the wavelength, then matrices XQ q exhibit the
forms
1
dT
.13/
XQ s D
dT k02 .n20 sin2 0 P/ 1
1
dT P
.14/
XQ p D
2
2
2
dT k0 .Qn0 sin 0 1/
1
where
1
PD
dT
Z
dT
2
T
nO .z/dz,
0
1
QD
dT
Z
0
dT
1
dz
nO 2T .z/
.15/
by two quantities characterizing the roughness, i.e. the
rms value of the heights of the irregularities and the
autocorrelation length of these irregularities T.
MODELS OF DISPERSION OF MEDIA OF THE
SiO2 /Si SYSTEM
The ambient is always formed by air and so its refractive
index n0 is equal to unity. It is assumed that dispersion of
the refractive indices of all of the thermal SiO2 layers
studied is the same. Moreover, it is assumed that the
formula expressing this dispersion is given as
We assume that the refractive index of the transition layer
nO T .z/ exhibits the profile expressed by the equation
nO 2T .z/ D n21 C
z 2
.nO
dT 2
n21 /
.16/
After integrating in Eqn (15) one obtains the following
expressions for P and Q
PD
nO 22 C n21
,
2
QD
ln nO 22
nO 22
ln n21
n21
.17/
Note that the matrix formalism presented is equivalent
to the Drude approximate approach for inhomogeneous
layers.5,6
Substrate covered by a single film with rough
boundaries
If the boundaries of the system are not ideally smooth
then Fresnel coefficients of these boundaries differ from
those describing the smooth boundaries. In the case when
the root-mean-square (rms) values of the heights of the
irregularities of the boundaries j are much smaller than
the wavelength of the incident wave, the Rayleigh–Rice
theory (RRT) can be employed for expressing the Fresnel
coefficients.7 Moreover, if the autocorrelation length of
roughness of the boundaries is larger than the thickness
of the film, then the matrix formalism presented for the
substrate covered by a single film with smooth boundaries
cannot be used. One must then employ the formalism
derived in our earlier paper.8 Within this formalism the
Q q exhibits the general form
transfer matrix M
Qq D 1
M
OtqR
1
rOqR
OtqR OtqL
rOqL
rOqR rOqL
.18/
In the foregoing equations the indices ‘R’ and ‘L’ mark
the Fresnel coefficients of the whole system for the wave
incident on this system from the left-side and right-side,
respectively. From Eqn (6) it is evident that the reflection
Fresnel coefficients of the system rOq that are important for
our purposes are identical with the coefficients rOqR . Note
that the expressions for the Fresnel coefficients considered
above must be calculated using a numerical procedure.8
In our studies it is assumed that the film with the rough
boundaries is represented by the model of the identical
thin films whose boundaries are described by the Gaussian
statistics.8 This model of the rough film is parameterized
Surf. Interface Anal. 28, 240–244 (1999)
n1 D A C
B
C
C 4
2
.19/
where A, B and C are material parameters. As for the
silicon substrate, no dispersion formulae are assumed
for the refractive index n or the extinction coefficient k
.nO 2 D n ik/.
RESULTS
First the model formed by the substrate covered by the
single film with smooth boundaries was used to interpret
the experimental data.
In the first step we used the single-wavelength method
for determining the values of all the optical parameters,
i.e. the values of the refractive index n and the extinction
coefficient k of the Si substrates and A, d1 , d2 , . . . , d6 of
the SiO2 layers (B and C were fixed). In the second step
we employed the dispersion method for determining the
values of the parameters A, B, C, d1 , d2 , . . . , d6 . In the
second step the values of both the refractive index and
the extinction coefficient of the Si substrate were fixed
for all the wavelengths in the values determined in the
first step. These two steps were repeated in successive
iterations, i.e. every iteration was formed by the two steps
described above. In the first step of the second iteration the
parameters d6 , B and C were fixed in the values obtained
within the second step of the first iteration. In the first step
of the third iteration the parameters d5 , d6 , B and C were
fixed in the values found in the second step of the second
iteration, etc. This means that from the eighth iteration the
optical constants of the Si substrates n and k were only
determined in the first step. After performing a sufficient
number of iterations, this numerical procedure converges
to the values sought for the thicknesses d1 , d2 , . . . , d6
and the spectral dependences of the optical constants of
the system.
In this way we obtained the following results: A D
1.45698 š 0.00008, B D .2689 š 21/ nm2 , C D .5.99 š
0.12/ ð 107 nm4 , d1 D .27.635 š 0.007/ nm, d2 D
.44.654 š 0.005/ nm, d3 D .67.344 š 0.005/ nm, d4 D
.86.057 š 0.005/ nm, d5 D .101.138 š 0.005/ nm and
d6 D .13.853 š 0.006/ nm. The values of the spectral
dependence of the refractive index of SiO2 calculated
by means of the values of A, B and C presented above
are greater than the values of bulk amorphous SiO2 9 (the
difference between both sets of data is ¾0.007 over the
entire spectral region of interest).
Copyright  1999 John Wiley & Sons, Ltd.
OPTICAL CHARACTERIZATION OF SiO2 /Si
The spectral dependences of the optical constants of the
silicon single-crystal substrate determined by the procedure described are not presented completely in this paper.
(They will be published elsewhere.) Here we shall perform a comparison of our values of the silicon optical
constants with those found by other researchers (Aspnes
and Studna10 and Herzinger et al.3 ) at selected wavelengths (see Tables 1 and 2). Furthermore, in Fig. 1 a
comparison of the values of the extinction coefficient
determined by our procedure with those published by
the other researchers is performed in the spectral region
450–830 nm. From Table 1 it can be seen that there are
certain differences between the positions and magnitudes
of the peaks of the imaginary part of the dielectric function ε2 .ε2 D 2nk/ corresponding to the critical points E1
and E2 . This same statement is also true for the values of
the optical constants corresponding to the popular wavelengths in the visible region (see Table 2). Note that in
the region of weak absorption of silicon our values of
the extinction coefficient are greater than those presented
by the researchers mentioned (see Table 2 and Fig. 1).
However, our values of k agree with the values of this
quantity published by Hulthèn11 determined using transmission measurements (see Fig. 1).
If the model containing the transition layer is considered, the number of parameters sought is greater (the
Table 1. The position and magnitudes of the "2 peaks of Si
corresponding to critical points E1 and E2
Data
Aspnes & Studna
Herzinger et al.3
Our data
10
E1 (eV)
ε2 .E1 /
E2 (eV)
ε2 .E2 /
3.429
3.418
3.415
36.44
36.62
37.70
4.249
4.239
4.233
46.81
47.64
46.91
Table 2. Values of the refractive index n and extinction
coefficient k of Si for two selected walelengths
D 546.1 nm
n
k
Aspnes & Studna10
Herzinger et al.3
Our data
4.098
4.090
4.105
0.044
0.026
0.055
D 632.8 nm
n
k
3.882
3.874
3.879
0.019
0.015
0.030
243
thickness of the transition layer dT is added). When treating the experimental data by our numerical procedure we
fixed the value of dT to 1 nm in the first nine iterations
(the fixed value of dT was chosen on the basis of results
achieved for the transition layer by other researchers2,3).
From the tenth iteration the value of dT also was searched.
By using this procedure we found that the thickness dT
converges to zero and the other optical parameters converge to the values determined by means of the foregoing
model. This fact thus implies that between the silicon substrates and thermal oxide layers there are relatively sharp
boundaries.
When treating the experimental data using the rough
model, we applied our numerical procedure in a similar
way to that employed in the case of the model with the
transition layer. We found that the roughness parameters and T converge to the values corresponding to the smooth
boundaries. One can thus state that roughness taking place
in the boundaries cannot be observed within the optical
methods. This conclusion is supported by our atomic force
microscopy measurements performed for the samples of
SiO2 /Si studied. From our atomic force microscopy measurements the following typical values of and T were
determined: D 0.2 nm and T D 80 nm. From a numerical analysis it is clear that roughness characterized with
such the values of and T cannot affect the ellipsometric
and reflectometric data in a substantial way.
CONCLUSION
In this paper the results concerning optical analysis of
the SiO2 /Si system by the combined ellipsometric and
reflectometric method used in multiple-sample modification are presented. This method is based on combining
both the single-wavelength method and the dispersion
method. From the results one can imply that the best physical model for the experimental data is identical with the
model formed by a silicon single-crystal substrate covered
by as SiO2 layer with smooth and sharp boundaries. This
conclusion is in contrast to the results of other researchers,
who found that a transition layer existed between the Si
substrate and the SiO2 layer on the basis of interpretation of ellipsometric data.2,3 This fact may be caused by
the different technological conditions when preparing the
samples of SiO2 /Si studied by the different researchers.
Within our analysis the spectral dependences of the optical constants of Si and SiO2 are found, together with
the values of the thicknesses of the SiO2 layers. Furthermore, it is shown that there are certain differences
between our values of the optical constants mentioned
and those presented by other researchers. The complete
spectral dependences of the optical constants of silicon
will be presented elsewhere.
Acknowledgements
Figure 1. Spectral dependence of the extinction coefficient k of
Si determined by different researchers in the region of weak
absorption of this material.
Copyright  1999 John Wiley & Sons, Ltd.
We wish to thank Dr C. M. Herzinger for providing us with his data
of the optical constants of silicon and Dr K. Navrátil for measuring the
reflectance data. Moreover, we wish to thank the researchers of SCB of
the Masaryk University for their help with the numerical computations.
This work was supported by the Grant Agency of the Czech Republic
and the Grant Agency of the Ministry of Education of the Czech
Republic under contract nos 202/98/0988 and VS96084, respectively.
Surf. Interface Anal. 28, 240 244 (1999)
244
I. OHLÍDAL ET AL.
REFERENCES
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2. D. E. Aspnes and J. B. Theeten, J. Electrochem. Soc. 127,
1359 (1980).
3. C. M. Herzinger, B. Johs, W. A. McGahan, J. A. Woollam and
W. Paulson, J. Appl. Phys. 83, 3323 (1998).
4. W. D. Marquardt, J. Soc. Ind. Appl. Math. 11, 431 (1963).
5. Z. Knittl, Optics of Thin Films. Wiley, Chichester (1976).
Surf. Interface Anal. 28, 240–244 (1999)
P. Drude, Wied. Ann. 43, 136 (1891).
S. O. Rice, Commun. Pure Appl. Math. 4, 351 (1951).
D. Franta and I. Ohlı́dal, J. Mod. Optics 45, 903 (1998).
D. F. Edwards, in Handbook of Optical Constants of Solids,
ed. by E. D. Palik, pp. 547 569. Academic Press, New York
(1985).
10. D. E. Aspnes and A. A. Studna, Phys. Rev. B 27, 985 (1983).
11. R. Hulthèn, Phys. Scr. 12, 342 (1975).
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Copyright  1999 John Wiley & Sons, Ltd.
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