TABLE I. Parameters obtained from the fit for wet and dry oxidised gate
oxides.
below. For the analysis we incorporated correction of geo-
metrical factors ͑foot-print correction͒ and substracted the
offspecular diffuse background to obtain true specular reflec-
tivity intensities. Figure 1 shows the corrected x-ray reflec-
tivity data. We observe a sudden drop in reflectivity above
the critical angle c , indicating that both samples have large
roughness amplitudes at the poly-Si electrode surfaces. How-
ever, we can see the difference between the two curves at the
large angle of incidence (). For the wet oxidized sample
oscillations in the reflectivity curve are clearly observable,
while in the case of a dry oxidized sample the oscillations are
not observed. The period of the oscillation corresponds to
102 Å which is the nominal thickness of the buried oxide
layers. The oscillations in the reflectivity are due to a large
contrast in the average electron density ͑AED͒ between the
layers ͑i.e., a large difference in AED values of the layers͒
and smooth interfaces.6,7 If either of these conditions are not
satisfied then the oscillations in the reflectivity are not ob-
served as illustrated for the dry oxidized sample in Fig. 1.
The oscillation in reflectivity curve is observed only at large
angles due to the large thickness of poly-Si layer on top of
the oxide layer.
Thickness
AED
Roughness Absorption
Sample
dry
Layer
͑Å͒
͑ÅϪ3
͒
͑Å͒
͑ϫ10Ϫ6
͒
poly-Si
SiO2
Si
poly-Si
SiO2
Si
SiO2
Si
SiO2
Si
2000Ϯ20
0.69
0.65
0.70
0.69
0.63
0.70
0.667
0.706
•••
27
8
2
34
2
1.61
0.87
1.61
1.61
0.87
1.61
0.84
1.59
0.84
1.59
98
•••
1980Ϯ20
102
•••
wet
2
SiO2/Si
͑model A͒
SiO2/Si
77
•••
77
•••
3.37
2.09
2.38
2.00
a
a
͑model B͒
0.706
aParameters of a bare oxide layer obtained using Parratt’s recursion relation.
the inset. We found that increased interface roughness for a
layer even with a larger AED contrast reduces the oscilla-
tions in the reflectivity curve very rapidly. The reflectivity
curve ͑similar to that of the dry oxidized sample͒ could be
achieved as shown in curve ͑b͒ of the inset assuming a
smooth interface ͑ϳ2 Å͒ and a rough interface ͑ϳ8 Å͒ for
a buried SiO2 layer with an AED contrast of 0.06 ÅϪ3. Thus
we conclude that smooth interfaces of poly-Si/SiO2/Si sub-
strate for the wet oxidized sample gives rise to oscillations in
the reflectivity curve at high angles while absence of oscil-
lations in the reflectivity for dry oxidized sample is due to
the large interface roughness at one of the SiO2 interfaces.
For the structure of gate oxides it is very difficult to
To analyze our x-ray reflectivity data quantitatively we
have used a simple semi-kinematical approximation. The ex-
plicit formula for the reflectivity amplitude ͑XϭErefl./Einc.
)
can be expressed as
N
lϪ1
l
Xϭ
r eϪi
,
ϭϪ
l
q d ,
j j
͑1͒
͚
͚
l
lϭ1
jϭ1
determine which interface of the dry oxide layer has a larger
2
roughness amplitude. This is evident from
͉
X
͉
of Eq. ͑1͒
where N(ϭ3) is the number of layers, rl is the Fresnel co-
efficient of the lth interface which determines the amplitude
of oscillation and depends on the contrast of AED, dj is the
thickness of the jth layer, and qj is the normal component of
the incident wave vector of the jth layer containing both real
and imaginary components of the layer’s refraction index.
Hence, the above expression includes the effects of absorp-
tion and refraction. Roughness at each interface was intro-
duced using a Debye–Waller like term such that rl is multi-
plied by a factor of exp(ϪqlqlϪ12), where is the interface
which contains a symmetric oscillatory term with respect to
the exchange of roughness amplitudes. In order to determine
which interface is rough, we prepared a dry oxide thin film
on a Si substrate without a cap layer to avoid symmetric
interfaces; the interfaces are now those of air/SiO2 and
SiO2/Si-substrate. The x-ray reflectivity data of this sample
are shown in Fig. 2. We have used Parratt’s formalism9 to
analyze the x-ray reflectivity data. First we considered a
single layer of SiO2 with no variation in AED in the layer
͑model a͒; the result of the fit is presented by a solid line. We
tried to improve the fit by including variations in AED in the
oxide layer10,11 ͑model b͒; and got consistent values for
l
l
roughness amplitude.8 We did not include multiple scattering
effects since the reflected intensities are very weak ͑less than
10Ϫ4 of the incident beam intensity͒.
The parameters obtained from fitting the above expres-
sion to the specular x-ray reflectivity data are given in Table
I. The top surface roughness of poly-Si is estimated to be
about 30 Å for both samples. We observe that one of the
interface roughness of SiO2 in the dry oxidized sample is
larger than that of the wet oxidized sample. We also observe
smaller contrast in AED between poly-Si and SiO2 of the dry
oxidized sample than that of the wet oxidized sample. To
pinpoint whether the large amplitude of roughness or the
small AED contrast is responsible for the absence of oscilla-
tions in the reflectivity of the dry oxidized sample, we have
simulated the reflectivity curves by varying the contrast in
AED and the interface roughness. We show the simulated
result in the inset of Fig. 1. We find that if both interface
roughness of poly-Si/SiO2 and SiO2/sub-Si is 2 Å, then even
with a very small AED contrast of 0.01 ÅϪ3 we still observe
oscillation in the reflectivity curve as shown in curve ͑a͒ of
FIG. 2. X-ray reflectivity of a thermally grown SiO2 film by dry oxidation
process. Solid lines represent the fit using Parratt’s recursion relation for
model a ͑a͒, and model b ͑b͒. Inset: Electron density profiles for model a and
model b. Sizes of errors are equivalent to those of symbols.
434 Appl. Phys. Lett., Vol. 72, No. 4, 26 January 1998 Banerjee et al.
130.88.90.110 On: Sat, 20 Dec 2014 10:36:54
Banerjee
