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Grazing x-ray reflectometry data processing by fourier transform
JOURNAL OF X-RAY SCIENCE AND TECHNOLOGY 4, 2 0 0 - 2 1 6
(1994)
Grazing X-Ray Reflectometry Data Processing by Fourier Transform F. BRIDOU* AND B. PARDO t *Institut d'Optique, L.A.14 du CNRS, Centre Universitaire, Bat 503, B.P.147, 91403 Orsay Cedex, France; and tlnstitut d'Optique et Universtitd Pierre et Marie Curie, 4 place Jussieu, 75252 Paris Cedex 05, France
Received July 12, 1993; revised December 27, 1993 Grazing x-ray reflectometry allows analysis of thin-layer stacks. The fitting of the reflectivity curve by a trial and error method can determine the parameters of the films. The Fourier analysisof the experimental reflectivitycurve can directlygivea rough determination of the profile index. Such results can be useful in choosing a starting model. With the choice of an appropriate model a fit to the reflecfivitycurve can be undertaken to determine the parameters of the stack. The Fourier analysis method can only be used if the reflectivity data undergo a transformation to produce a periodic curve. Associatedartifacts are studied and discussed. Each Fourier peak is associated with two interfaces. The interface roughness spreads the Fourier peaks, adding the squared roughness values. The sample's absorption of the x-rays does not limit the Fourier analysis. © 1994AcademicPress,Inc. I. INTRODUCTION Grazing x-ray specular reflectivity has, for a long time, been used to characterize thin stratified stacks. (See, for example, (1-5).) In this kind o f study it is assumed that the microcrystals are small and disordered enough that the materials can be t h o u g h t o f as being amorphous. Note that for the grazing incidence (70 m r a d m a x i m u m ) and the wavelengths (>/0.154 n m ) used, the apparent wavelength is between 2.5 and 65 nm. This range is always greater than the lattice spacing. Thus, even if the microcrystals have a plane parallel to the substrate, it can be assumed that there are no Bragg crystalline reflections. In this case, the specular reflectivity o f grazing X-rays depends on the thicknesses, the complex indices o f the layers, and the roughness of the interfaces. N u m e r o u s authors have characterized multilayers by fitting the experimental reflectivity curves. Generally, the fit is obtained by a trial and error m e t h o d which consists o f calculating the theoretical reflectivity and comparing the result with the experimental reflection. In cases where the layers are flat enough, it was shown that it is possible to let a c o m p u t e r do the tedious work o f the trials and comparisons. For a stack which is m a d e up o f at least one layer, the reflectivity curve shows oscillations which reveal the different thicknesses o f the stack. It could be expected that a Fourier transform o f the reflectivity data would give directly the frequencies o f these oscillations and allow us to obtain quickly the thicknesses o f each layer. Even if the Fourier transform gives 0895-3996/94 $6.00 Copyright © 1994 by Academic Press, Inc. All rights of reproduction in any form reserved.
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201
false results it could be expected that these results would be close enough to the real thicknesses to allow them to be used as good starting values for trial and error computing, and thus make this method more efficient. The goal of this study is precisely to show that it is possible to obtain almost directly the real thicknesses of the stack layers by using the Fourier transform. The precautions which must be taken are studied. In order to analyze the effectiveness of this method, we have intentionally worked on calculated reflectivity curves resulting from the fit of experimental curves. So, if the method does not work, poor experimental conditions are not to blame. Furthermore, by using these kinds of synthesized curves, the limitations of the method can be empirically studied by introducing increasingly severe conditions (number of layers, roughness, etc.). II. CLASSICAL METHOD FOR THE CHARACTERIZATION OF STACKS
II. 1. X-Ray Arrangement for the Recording of Experimental Reflectivity Curves (6) The diagram of the goniometer is shown in Fig. 1: The sample is positioned on a fixed stand, and the source unit and the detector are fixed on arms which move around the same axis. The source unit consists of an x-ray source with a copper anode. The Ka radiation is obtained by means of a 220 LiF crystal. A field slit permits us to maintain the Ka 1 radiation, adjust the optical beam numeric aperture, and define the illuminated area of the sample simultaneously. The flat sample is held on the stand by means of a vacuum chuck. The detection unit is made up of a contrast slit, an optical attenuator, and a proportional counter followed by standard electronic devices. The arms are moved with endless screws driven by stepper motors. The maximum angular amplitude is 5 ° . The stepper motors allow an angular resolution of 1/10,000% however, the true angular accuracy is less than 5/1000°. The apparatus is controlled by a personal computer fitted with standard interfaces. The reflectivity curve is obtained by varying the grazing angle while tracking the reflected beam. As the number of detected photons can vary from 1 to 1 million, we use a semilogarithmic scale for graphic representations. The operation is fully automated and controlled by computer except for the sample positioning.
Coun~'eP Affenuafor SIII",/S"f
LIF
I C°ml'uterI
6RX2 FIG. 1. X-ray goniometer sketch.
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BRIDOU
AND
PARDO
An example of an experimental result is given in the dotted line curve of Fig. 2. It shows the reflectivity of a gold layer on a glass substrate.
II.2. Theoretical Reflectivity Computing. The Matrix Method Many methods have been used to compute the theoretical reflectivity of a stratified multilayer. All of them are equivalent. In this paper we use the following so-called matricial formalism (7). One considers the vectors, the components of which are the complex amplitudes of reflected and transmitted plane waves, R and T, respectively.
R z, Rz The matrix M is the product of matrices of two types: (a) diopter crossing type (D.C.), F; (b) propagation type, P. (a) The diopter crossing type matrix involves the components of the wave vectors in the two media. For instance, in the case of T - E polarized electromagnetic plane waves, a D.C. matrix has the following form:
P =2
i + klz
kl±
kl~
kl.
k2±
1+
nm "10"6 I Layer Ithickness ]roughness [ ,ndice [absorption [ 5
"
~
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7
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4 ~
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0-
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6 Grazing angle (seconds)
I
8
I
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I
12x10 3
FIG.2. Experimentalreflectivitycurvefor an Au layer(dottedline),fittedwitha calculatedcurve(continuous line), with the parametersgivenin the table. The layerlabeled0 correspondsto the glass substrate.
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GRAZING X-RAY REFLECTOMETRY
k denotes the normal component of the wave vector. The subscript number is the medium number. When the interface is rough this matrix must be modified by the so-called Debye-Waller factor, DW. This D W factor simply multiplies the extra diagonal components of the D.C. matrix by DW = e x p ( - 8 • ~r2. ~r2" klj_" k21), where ~ is the root mean squared roughness. In the case of T - M polarization a k component has to be divided by its squared complex index. In the case of x-rays, the indexes are very close to 1 so the correction is not necessary, particularly when the grazing angle is far from Brewster's angle. (b) Propagation matrices are diagonal and have the very simple form
fi=[exp(+i~k±.d)
0
]
e x p ( - i . k ± , d) ' where d is the thickness of the layer. Taking into account that the initial vector in the substrate has a null reflection component, the coefficient of the reflectance intensity is computed as R • R*
T. T* " The reflected intensity is obtained by multiplying I by the incident number of particles.
11.3. Fitting the Experimental Curve The matricial computing enables us to fit the experimental curve of Fig. 2 by means of a trial and error method, as used in a previous paper (8). The result of the fit is summarized in Fig. 2, with its corresponding parameters. III. USE OF FOURIER TRANSFORM IN GRAZING X-RAY REFLECTOMETRY: THEORETICAL APPROACH
III.1. Introduction Fourier transformation has been used in optics in order to synthesize layers with a given transmission and also to make an analysis of layers from their experimental reflectivity curve (9, 10). The wavelength can vary and, just as for optical wavelengths, the absorption and the dispersion can be neglected. Therefore, these methods can be adapted to the case of neutrons and x-rays. Some authors have proposed that Fourier transformation be applied to x-ray specular reflection curves in order to make a rough determination of each layer thickness. One of the main problems is that these reflectivity curves are not periodical and before using the Fourier transform it is necessary to make a preliminary transformation of the reflectivity data.
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BRIDOU AND PARDO
Sakurai and Iida (11) have subtracted an average curve from the logarithm intensity curve. This method is empirical, because it is necessary to define the criteria for the calculation of this average curve and often in the results spurious orders remain. Sivia et aL (12) have used Fourier transform techniques to study neutron reflectivity curves recorded with a grazing incidence neutron reflectometer with a scanned wavelength (time-of-flight technique). For the preliminary transform of the reflecfivity curve, they have multiplied the intensity data by the Q4 factor. This leads to a real improvement in the periodicity of the preliminary transform. But in real cases it seems that, in order to obtain real results, they have made the Fourier transform ofa symmetrized reflectivity curve. So, they found spurious oscillations in the spectrum due to the fact that the constant part of the reflectivity, corresponding to the total reflection, is included in the data. In this paper, the FT is used with grazing x-ray reflectometry. Because the sign of the Fourier transform appears to be of little interest, only the modulus of the Fourier transform is calculated from a curve where the plateau region is eliminated. Furthermore, we take account of the average index of refraction, and the modified reflectivity data have very good periodicity. The roughness has the effect of broadening the peaks in the Fourier transform result. 111.2. Theory
For scattering vectors larger than the critical scattering momentum value Q0, the reflectivity data relate to the derivative of the density profile by (12, 13)
R (Q ) -
(4. ~r)2 Q~
dn ei .O.z. dz2, --~z "
[1]
where the scattering momentum 4 . ~r. sin(0)
Q-
X
[21
is twice the normal component of the wave vector, 0 is the grazing angle, z is the depth, and 2~is the incident wavelength. The formula shows that R is related to the Fourier transform of the derivative with respect to the depth. Then the auto correlation function of the derivative of the density profile can be written as p(z) =
n'(t), n'(t - z). dt
[3]
0O
1 p(z) - ~ (4.~r)
o~
Q4. R(Q). e ioz. dQ.
Starting from these equations, it turns out that, in the case of homogeneous layers, the auto correlation function is simply
GRAZING X-RAY REFLECTOMETRY
p(z) = ~
205
[4]
~ (nk+l - nk)" (nl+l - nl)" 6x~-x,(Z),
k
l
where k and I are the layer numbers, nk are the layer indices, xk is the depth between the kth surface and the origin, and 6h is the Dirac function which corresponds to a translation of length h. If i is the number of interfaces, this p function has i- (i - 1)/2 peaks associated with each pairs of interfaces. The abscissae of these peaks are the distances between the interfaces. There are i other peaks which are superimposed at the origin; these peaks are associated with the self-coupled interfaces. The auto correlation function r is different from the index derivative profile n'(z) = ~ (nk+l - nk)" 6xk(Z).
[5]
k
This last function has i peaks only. The abscissae of the derivative peaks coincide with abscissae of the interface. Equation [4] shows that the auto correlation function has additional peaks, the abscissae of which are the differences between the abscissae of the discontinuities of densities (Fig. 3).
Profile index 1 3
I
Derivative
1
,I2
i
I
1
3
Autocorrelation
I 3-2 I I I-I I 3-I 2-2 3-3
2-I
2il I 1- I 2-2 3-3
3-I
3-2
FIG. 3. Two examples of derivative and autocorrelation spectra. There are i peaks for i interfaces in the derivative spectrum, while, including the zero-order peak, there are 1 + (i(i - 1)/2) peaks in the autoeorrelation spectrum.
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Remarks about the interest of the sign of the auto correlation result. In Eq. [3], the integration is made between - o o and + oo. That implies that the reflectivity curve is symmetrical and that the Fourier transform is real. The sign of the auto correlation function appears in the real part. It can be seen from Fig. 3 that if we have one thick layer of gold on a glass substrate and an additional thin layer in the stack, it is not possible to detect the position of the thin layer in the stack by looking at the sign of the auto correlation result. In both cases, the sign of the auto correlation peaks is the same. Only information on the relative intensity of the peaks can be investigated, in ideal cases. Note that for x-ray reflectivity curves, the plateau, where expression [ 1] is not applicable, is very large and this leads to significant errors in the Fourier reconstruction. This can be seen in Fig. 4a where the theoretical reflectivity curve is obtained from the parameters of Fig. 2, but with no roughness and with a scanning angle between -10000" and 10000" for a symmetrized curve. So, because the sign of the result appears to be not very useful, it can be more interesting to study solely the modulus of the autocorrelation function and to integrate the reflectivity curve over the part beyond the plateau region where Eq. [ 1] is verified. Figure 4b shows an example of the modulus result obtained with the parameters of the preceding reflectivity curve, but with a scanning angle between 2800" and 10000". III.3. Fourier Transform Used in Computing. The Discrete Fourier Transform and Its Limitations A complete study of the properties of the discrete Fourier transform can be found in the referenced papers (14, 15). The main characteristics are recalled here in order to detect and avoid possible errors that can occur when this transformation is used. Using a computer one can apply the discrete Fourier transform (DFT),
F m = Z fq. exp -27r. i q
,
[6]
q=O
where p is the number of sampling points, fq'S are the p values of the direct function at the sampled points, Fm's are the p values of the transformed function, and q and m are the direct and the reciprocal integer indices. The accuracy of the approximation of the continuous Fourier transform by the DFT depends on the choice of some parameters, such as --the sampling frequency, which determines the maximum frequency that can be given by the signal; --the length of the sequence that determines the frequency resolution of the spectrum; --the kind of window that is used to limit the signal. Its shape can lead to spurious components in the DFT that do not exist in the direct function. In the following examples, in order to avoid the well known leakage effect (or Gibbs oscillations), we take care to select minima of the signal to define the limits of the truncation so that the signal is roughly periodic in the selected window.
GRAZING X-RAY REFLECTOMETRY 0.35
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z(A/ FIG. 4. (a) The symmetrization of a typical x-ray reflectivity curve given by a gold layer leads to a real Fourier transform, but with too many oscillations. (b) Part of the curve shown in Fig. 2, taken outside the critical angle, gives a complex Fourier transform. The corresponding modulus of the Fourier transform is free of oscillations. IV. APPLICATIONS OF THIS TRANSFORMATION In order to estimate the validity o f the results, a study o f synthesized or m o c k curves (perfect theoretical ones) is first undertaken, followed by a study o f theoretical curves with k n o w n defects. Applications to realistic curves are then shown. A real improvement o f the method can be obtained by using an index correction.
IV.1. Introduction Figure 2 shows an experimental reflectivity curve o f a gold layer on float glass. This curve is well fitted (continuous line) with the parameters o f thicknesses, indices, and roughness indicated on the figure.
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It is now possible to make an artificial calculated curve, which looks like the previous experimental one, with known defects such as roughness, background noise, and Poisson's detector noise, as we have done in a previous paper (8). Starting with the possibility of synthesizing a realistic curve, the effect of each kind of defect on the Fourier transform can be studied.
IV.2. Perfect Synthesized Curves In the resulting spectrum shown in Fig. 4b, it can be seen that the two peaks at x = +26.311 n m correspond to the given thickness (d = 24.69 nm); however, the lateral peaks are spread asymmetrically. This is due to the fact that the periodicity of the direct function Q4R(Q) is not exactly constant. Remarks about the indices. It can be seen that usually the Fourier transform only gives the thickness with little accuracy, compared to the direct value thickness found when the curve is fitted by a calculated one. This can be attributed to the fact that the part of the curve related to the total reflectivity is eliminated to avoid artifacts in the transformation, and then information about the order of the peaks, in particular, is lost. Furthermore, there remains asymmetry in the peaks due to the fact that the periodicity of the direct curve is not exactly constant. It would thus be better to make an index correction replacing Eq. [2] with Q = 4~r
Vsin20k -- sin20c X '
[7]
where k is the order of the interference and 0c is the critical grazing angle given at the half maximum reflectivity point of the curve. Figure 5 shows the result of this transform with a reflectivity curve, calculated with the parameters of Fig. 2 but with no roughness. The scanning range is from about 2,800" to 30,000" of arc. In Fig. 5a the X-axis is the "corrected angle" in radians. In Fig. 5b, the peaks are narrow and symmetrical, and the thickness is well defined as 24.61 nm (compared with a given value of 24.69). Influence of the contrast of the indices. In theory, the peak intensity in the spectrum can be linked to the contrast of the indices, but in practice, because of the low thickness resolution due to the scanning range, only an indicative result can be given.
IV.3. Synthesised Curves with Defects Influence of roughness. When the interfaces are rough one can suppose that an interface behaves like a variation of the index with depth. Such a variation is well depicted by an error-type function: IIs (x) =
~+x/se_~. ~2. d~.
[81
Lt--oo
s, which we call the variance, is equal to the interface roughness multiplied by ~/2.7r. The derivative of such a function is a Gauss function:
209
GRAZING X-RAY REFLECTOMETRY
3
2
_..I
I
210
I
410 60 -3 x l 0 Corrected Grazing Angle (rd)
80
30-
25 N
1.1-20
o _<
m,O 1 5 10-
0 -1000
I
I
-500
500
1000
z(A)
FIG. 5. (a) Perfect synthesised reflectivity curve without roughness, with index correction, and over a large scanning angle range between 2800 to 30000" (ideal curve). The x-coordinate is the corrected grazing angle in radians. (b) Modulus of the Fourier transform. The maximum of the satellite peaks is given for E = 246.1 ~, (the real value is 246.9 A).
Gs(x) = ~ e -~(~s)2.
[91
S
This form of Gauss function with ~r explicit is used because it is invariant by the Fourier transform.
e -~-'~2 =
F
~
e 2"~'i'~'x. e -~'x~. dx. oo
[10]
210
BRIDOU AND rARDO
Taking into account the Gaussian profile, it is easy to show that the index profile auto correlation is [11]
P = ~ ~ (nk+l -- nk)" (nl+1 -- nt)" Gsk*G~,*6xk-x r k
l
• denotes the convolution product, and Sk is the variance of the kth interface. The variables of the different functions are not denoted in order to avoid confusing the convolution products. The convolution of two Gauss functions being another Gauss function, G~
[12]
= G,k , G s ,,
it is easy to show that each Dirac peak of the perfect stack becomes a Gaussian peak. The variance of a peak is greater than the variance of each coupled interface. Note that the peak which is at the origin becomes the superposition of n Gaussian peaks, the variances of which are given by Sk,k = ~ " Sk.
[131
The curve of Fig. 6 is obtained with the same parameters as curve 5 but with additional rms roughness: a0 = 0, al = 0.3 nm. It can be seen that, in the spectrum, peaks are broadened. The broadening increases with roughness and corresponds to the convolution of the peak obtained without roughness with the Gaussian function of the derivative of the indices introduced by the roughness.
1614-
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,,?,
8-
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20-
-1000
I
-500
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500
1000
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FIG. 6. (a) The same synthesized reflectivity curve as in Fig. 5 but for roughness: a0 = 0, ~1 = 3 A gives satellite peaks for E = 264.09 ~ in the Fourier transform with index correction. The peaks are not shifted, but they are broadening.
211
G R A Z I N G X-RAY REFLECTOMETRY
2.5-
2.0-
N v LL1.5-
o,< co 0
•,~ 1 . 0 -
0.5-
0.0-1000
I
I 500
-500
I 1000
z(A) FIG. 7. With the realistic data of the curve given in Fig. 2, with index correction, the peaks are not shifted (at E = 242.58 h), taking into account the resolution of the spectrum (20 A).
It was also verified that, according to the above equations, permuting the roughnesses a0 and o-1 gives the same peak intensities, but it was seen that the zero-order intensity is not the same. A more complete study of the effect of roughness shall be made later.
IV.4. Realistic Mocked Curves Roughness. Let us study the theoretical reflectivity curve given by the fit parameters of the Au layer shown in Fig. 2. Figure 7 shows the result obtained with the roughness fit parameters, with index correction limited to a realistic scanning range 2800"-8000", and with the number of sampled points limited to 28. It can be seen that, with realistic parameters, the result is good. We find E = 24.258 nm with a resolution of 2 nm. Note that the roughness influence is visible in the Fourier transform spectrum only if the broadening is greater than the thickness resolution, that is, the inverse of the scanning range. Influence ofPoisson's noise. Realistic calculated Poisson's noise corresponding to the detection noise was added. There was no difference between the spectrum with or without Poisson's noise, so it is not shown. In fact, the frequency of this noise is too high to be seen. Influence of background noise. It is absolutely necessary to subtract the background noise (thermal or cosmic spurious noise) before data processing. Experimentally, the background noise is measured after each recording, so it is possible to take it into account.
IV.5. Other Mock Examples Mock three-layer curve. As above, a realistic curve, obtained after fitting a real one with known parameters (Fig. 8), is used with the corresponding stack model.
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BRIDOU AND PARDO
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3
2.35,
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0.16£ 1,7!
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t
3:>. 2o~
.t
-1 20100
4 0 0I 0
60100
Grazing
angle
8 0 0I 0
1 0 0I 0 0
(seconds)
FIG. 8. The dotted line is the experimental reflectivity curve (stack made at SIV, San Salvo, Italy) and the continuous line is the calculated reflectivity curve with parameters given in the table.
For this realistic curve, taken in the bandwidth 1500"- 10000", the information seems to be a little degraded, but in fact the peak positions are close to the real ones in the limits of the resolution (= 1.82 nm in the spectrum) (Fig. 9). As there are four interfaces, following the above-mentioned general rule, six peaks are expected in the spectrum, the position of each of them corresponding to the distance
2o01
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ox 100-
18.22
\
273.3"
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50-i000
I -500
I 0
I 500
I ~000
z(A)
FIG. 9. Calculated spectrum from the realistic theoretical parameters of the previous curve. Five peaks appear. The result is satisfactory, taking into account the resolution of the spectrum (18 ~,).
213
GRAZING X-RAY REFLECTOMETRY 4-
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10-y,
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FIG. 10. (a) Calculated reflectivity curve for a multilayer without roughness, (b) Q4 transformation, (c) Fourier transform, and (d) enlarged part of the Fourier transform at the beginning of the spectrum.
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FIG. 11. (a) Realistic calculated reflectivity curve given by a stack of 50 layers with alternative roughness of 2 and 3 A, (b) Q4 transformation, (c) Fourier transform, (d) enlarged part of the Fourier transform at the beginning of the spectrum.
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BRIDOU AND PARDO
between the interfaces taken two by two. They are, in this case, (1) E3, (2) E l , (3) E2, (4) E2 + E3, (5) E1 + E2, and (6) E1 + E2 + E3. Due to the roughness, the widening of peak 1, related to the top interface, means that this peak is unresolvable from the central one.
Multilayers. Example 1. The reflectivity of a twenty-layer periodic stack with a 1.4 n m thick Mo layer and a 3.2 n m thick layer of Si is calculated in the range of 1000" to 30000". For ~ = 0.15405 nm, the corresponding Q range is 5 to 12 n m -1 (Fig. 10). The roughness of all the layers is taken as zero. Note that the Q4 transform exhibits the Bragg peaks, while the Kissig fringes are minimized. The Fourier transform curve cancels for a value of about 46 rim, which is equal to the total thickness of the stack. It can be seen in an expanded result curve that the thickness of the period is clearly exhibited, and also that the thicknesses of the two components in a period can be evaluated despite insufficient resolution by averaging the peak positions at the beginning of the curve. Example 2. The same indices are taken in a 50-layer periodic stack with roughness taken as 0.2 nm on odd interfaces, and as 0.3 nm on even interfaces (Fig. 11). Despite the lightening of the reflectivity as a result of the roughness, it is still possible to evaluate the thickness ratio in the period. The index correction does not improve the result.
Au/FG
249.2
o <_ 3 ¸
'O
-1000
I -500
I 0
I 500
z(A) FIG. 12. Result of the Fourier transform of the real curve of Fig. 2.
I 1000
GRAZING X-RAY REFLECTOMETRY
215
IV. APPLICATION TO THE STUDY OF REAL CURVES
Some of the reflectivity curves studied here are realistic ones, because they are calculated from parameters of fitted curves. So it can be assumed that the real results are close to the given ones. But, in the case where an index correction is required, it is desirable to make a scan with a step proportional to the "corrected grazing angle" instead of the grazing angle. If this is not possible, an interpolation can be tried. Obtained by such a method, Fig. 12 is the Fourier transform of the real curve described in Fig. 2. It can be seen that the result is very close to the expected one, and this shows that the method works. A further study will be made with only actual curves. V. CONCLUSION
One of the main difficulties in fitting a real curve by a direct optimization method, like a trial-and-error one, is in starting with a good stack model because, most of the time, even if the materials deposited are known, the deposition introduces some perturbation or there is a degradation during the air exposure of the multilayer. The real model is therefore often more complicated than the expected one. Despite the constraints of its use, the Fourier transform can be helpful in providing an expectation for a stack model, even when the number of layers is not a priori clearly determined. The number of peaks in the spectrum can be related to the number of interfaces even if some peaks disappear or overlap. For example, in Fig. 9, the first peak spreads in the zero order. When the correction of an index is undertaken for stacks with few layers, a good evaluation of thicknesses can be made, despite the roughness. The spreading of peaks is dependent on the roughness of the corresponding interface so the stack model can be refined. The thickness accuracy is dependent on the resolution of the spectrum, that is, on the bandpass recording of the direct curve. To determine the thicknesses with good resolution, the initial curve has to be fitted by, for example, a trial-and-error method, but with a stack model better known with the help of Fourier transformation. ACKNOWLEDGMENTS The authors thank F. Sacchetfi from SIV (San Salvo, Italy) for the growth of the three-layer stacks mentioned in this paper. This work was supported by the European BCR Commission.
REFERENCES P. CROCE, L. NEVOT, AND B. PARDO, Nouv. Rev. Opt. Appl. 3(1), 37 (1972). L. NEVOT, B. PARDO, AND C. CORNO, Analusis 16(7), 381 (1988). L. NEVOT, B. PARDO, AND J. CORNO, Rev. Phys. AppL 23, 1675 (1988). E. SPILLER, Rev. Phys. AppL 23, 1687 (1988). Ph. HOUDV, "La pulvtrisation diode RF, l'ellipsometrie cinttique in situ, la rtflectomttrie d'X rasants, un ensemble performant pour la rtalisation et l'analyse d'empilements de couches/t l'tchelle atomique," Th~se d'habilitation fi diriger des recherches, Universit6 Paris VII, 1989. 6. J. CORNO AND B. PARDO, A new x-ray reflectometer, in "Proceedings, SPIE conference, San Diego, 1988," Vol. 984, pp. 119-123.
1. 2. 3. 4. 5.
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7. B. PARDO,T. MEGADEMINI,AND J. M. ANDRE,Rev. Phys. Appl. 23, 1579 (1988). 8. F. BRIDOU AND B. PARDO, J. Opt. (Paris) 21(4), 183 (1990). 9. J. A. DOBROWOLSKI AND D. LOWE, AppL Opt. 17(19), 3039 (1978).
10. 11. 12. 13.
P. L. SWARTAND B. M. LACQUET, J. AppL Phys. 70(2), 1069 (1991). K. SAKURAIAND A. IIDA, Adv. X-RayAnal. 35, 813 (1992). D. S. SIVIA, W. A. HALMILTON, G. S. SMITH,T. P. RIEKER,AND R. PYNN,dr. AppL Phys. 70(2), (1991).
See, for example, G. P. FELCHER,R. FELICI,R. T. KAMPWlRTH,AND K. E. GRAY, J. Appl. Phys. 57, 3789 (1985). 14. A. PREUMONT,"Vibrations al6atoires et analyse spectrale," Presses polytechniques et universitaires rornandes, 1970. 15. K. R. CASTELEMAN,"Digital image processing," Prentice-Hall, Englewood Cliffs, NJ, 1979.
(1994)
Grazing X-Ray Reflectometry Data Processing by Fourier Transform F. BRIDOU* AND B. PARDO t *Institut d'Optique, L.A.14 du CNRS, Centre Universitaire, Bat 503, B.P.147, 91403 Orsay Cedex, France; and tlnstitut d'Optique et Universtitd Pierre et Marie Curie, 4 place Jussieu, 75252 Paris Cedex 05, France
Received July 12, 1993; revised December 27, 1993 Grazing x-ray reflectometry allows analysis of thin-layer stacks. The fitting of the reflectivity curve by a trial and error method can determine the parameters of the films. The Fourier analysisof the experimental reflectivitycurve can directlygivea rough determination of the profile index. Such results can be useful in choosing a starting model. With the choice of an appropriate model a fit to the reflecfivitycurve can be undertaken to determine the parameters of the stack. The Fourier analysis method can only be used if the reflectivity data undergo a transformation to produce a periodic curve. Associatedartifacts are studied and discussed. Each Fourier peak is associated with two interfaces. The interface roughness spreads the Fourier peaks, adding the squared roughness values. The sample's absorption of the x-rays does not limit the Fourier analysis. © 1994AcademicPress,Inc. I. INTRODUCTION Grazing x-ray specular reflectivity has, for a long time, been used to characterize thin stratified stacks. (See, for example, (1-5).) In this kind o f study it is assumed that the microcrystals are small and disordered enough that the materials can be t h o u g h t o f as being amorphous. Note that for the grazing incidence (70 m r a d m a x i m u m ) and the wavelengths (>/0.154 n m ) used, the apparent wavelength is between 2.5 and 65 nm. This range is always greater than the lattice spacing. Thus, even if the microcrystals have a plane parallel to the substrate, it can be assumed that there are no Bragg crystalline reflections. In this case, the specular reflectivity o f grazing X-rays depends on the thicknesses, the complex indices o f the layers, and the roughness of the interfaces. N u m e r o u s authors have characterized multilayers by fitting the experimental reflectivity curves. Generally, the fit is obtained by a trial and error m e t h o d which consists o f calculating the theoretical reflectivity and comparing the result with the experimental reflection. In cases where the layers are flat enough, it was shown that it is possible to let a c o m p u t e r do the tedious work o f the trials and comparisons. For a stack which is m a d e up o f at least one layer, the reflectivity curve shows oscillations which reveal the different thicknesses o f the stack. It could be expected that a Fourier transform o f the reflectivity data would give directly the frequencies o f these oscillations and allow us to obtain quickly the thicknesses o f each layer. Even if the Fourier transform gives 0895-3996/94 $6.00 Copyright © 1994 by Academic Press, Inc. All rights of reproduction in any form reserved.
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201
false results it could be expected that these results would be close enough to the real thicknesses to allow them to be used as good starting values for trial and error computing, and thus make this method more efficient. The goal of this study is precisely to show that it is possible to obtain almost directly the real thicknesses of the stack layers by using the Fourier transform. The precautions which must be taken are studied. In order to analyze the effectiveness of this method, we have intentionally worked on calculated reflectivity curves resulting from the fit of experimental curves. So, if the method does not work, poor experimental conditions are not to blame. Furthermore, by using these kinds of synthesized curves, the limitations of the method can be empirically studied by introducing increasingly severe conditions (number of layers, roughness, etc.). II. CLASSICAL METHOD FOR THE CHARACTERIZATION OF STACKS
II. 1. X-Ray Arrangement for the Recording of Experimental Reflectivity Curves (6) The diagram of the goniometer is shown in Fig. 1: The sample is positioned on a fixed stand, and the source unit and the detector are fixed on arms which move around the same axis. The source unit consists of an x-ray source with a copper anode. The Ka radiation is obtained by means of a 220 LiF crystal. A field slit permits us to maintain the Ka 1 radiation, adjust the optical beam numeric aperture, and define the illuminated area of the sample simultaneously. The flat sample is held on the stand by means of a vacuum chuck. The detection unit is made up of a contrast slit, an optical attenuator, and a proportional counter followed by standard electronic devices. The arms are moved with endless screws driven by stepper motors. The maximum angular amplitude is 5 ° . The stepper motors allow an angular resolution of 1/10,000% however, the true angular accuracy is less than 5/1000°. The apparatus is controlled by a personal computer fitted with standard interfaces. The reflectivity curve is obtained by varying the grazing angle while tracking the reflected beam. As the number of detected photons can vary from 1 to 1 million, we use a semilogarithmic scale for graphic representations. The operation is fully automated and controlled by computer except for the sample positioning.
Coun~'eP Affenuafor SIII",/S"f
LIF
I C°ml'uterI
6RX2 FIG. 1. X-ray goniometer sketch.
202
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PARDO
An example of an experimental result is given in the dotted line curve of Fig. 2. It shows the reflectivity of a gold layer on a glass substrate.
II.2. Theoretical Reflectivity Computing. The Matrix Method Many methods have been used to compute the theoretical reflectivity of a stratified multilayer. All of them are equivalent. In this paper we use the following so-called matricial formalism (7). One considers the vectors, the components of which are the complex amplitudes of reflected and transmitted plane waves, R and T, respectively.
R z, Rz The matrix M is the product of matrices of two types: (a) diopter crossing type (D.C.), F; (b) propagation type, P. (a) The diopter crossing type matrix involves the components of the wave vectors in the two media. For instance, in the case of T - E polarized electromagnetic plane waves, a D.C. matrix has the following form:
P =2
i + klz
kl±
kl~
kl.
k2±
1+
nm "10"6 I Layer Ithickness ]roughness [ ,ndice [absorption [ 5
"
~
I
°l
o
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7
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4 ~
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2
]
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I
2
I
4
I
6 Grazing angle (seconds)
I
8
I
10
I
12x10 3
FIG.2. Experimentalreflectivitycurvefor an Au layer(dottedline),fittedwitha calculatedcurve(continuous line), with the parametersgivenin the table. The layerlabeled0 correspondsto the glass substrate.
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GRAZING X-RAY REFLECTOMETRY
k denotes the normal component of the wave vector. The subscript number is the medium number. When the interface is rough this matrix must be modified by the so-called Debye-Waller factor, DW. This D W factor simply multiplies the extra diagonal components of the D.C. matrix by DW = e x p ( - 8 • ~r2. ~r2" klj_" k21), where ~ is the root mean squared roughness. In the case of T - M polarization a k component has to be divided by its squared complex index. In the case of x-rays, the indexes are very close to 1 so the correction is not necessary, particularly when the grazing angle is far from Brewster's angle. (b) Propagation matrices are diagonal and have the very simple form
fi=[exp(+i~k±.d)
0
]
e x p ( - i . k ± , d) ' where d is the thickness of the layer. Taking into account that the initial vector in the substrate has a null reflection component, the coefficient of the reflectance intensity is computed as R • R*
T. T* " The reflected intensity is obtained by multiplying I by the incident number of particles.
11.3. Fitting the Experimental Curve The matricial computing enables us to fit the experimental curve of Fig. 2 by means of a trial and error method, as used in a previous paper (8). The result of the fit is summarized in Fig. 2, with its corresponding parameters. III. USE OF FOURIER TRANSFORM IN GRAZING X-RAY REFLECTOMETRY: THEORETICAL APPROACH
III.1. Introduction Fourier transformation has been used in optics in order to synthesize layers with a given transmission and also to make an analysis of layers from their experimental reflectivity curve (9, 10). The wavelength can vary and, just as for optical wavelengths, the absorption and the dispersion can be neglected. Therefore, these methods can be adapted to the case of neutrons and x-rays. Some authors have proposed that Fourier transformation be applied to x-ray specular reflection curves in order to make a rough determination of each layer thickness. One of the main problems is that these reflectivity curves are not periodical and before using the Fourier transform it is necessary to make a preliminary transformation of the reflectivity data.
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Sakurai and Iida (11) have subtracted an average curve from the logarithm intensity curve. This method is empirical, because it is necessary to define the criteria for the calculation of this average curve and often in the results spurious orders remain. Sivia et aL (12) have used Fourier transform techniques to study neutron reflectivity curves recorded with a grazing incidence neutron reflectometer with a scanned wavelength (time-of-flight technique). For the preliminary transform of the reflecfivity curve, they have multiplied the intensity data by the Q4 factor. This leads to a real improvement in the periodicity of the preliminary transform. But in real cases it seems that, in order to obtain real results, they have made the Fourier transform ofa symmetrized reflectivity curve. So, they found spurious oscillations in the spectrum due to the fact that the constant part of the reflectivity, corresponding to the total reflection, is included in the data. In this paper, the FT is used with grazing x-ray reflectometry. Because the sign of the Fourier transform appears to be of little interest, only the modulus of the Fourier transform is calculated from a curve where the plateau region is eliminated. Furthermore, we take account of the average index of refraction, and the modified reflectivity data have very good periodicity. The roughness has the effect of broadening the peaks in the Fourier transform result. 111.2. Theory
For scattering vectors larger than the critical scattering momentum value Q0, the reflectivity data relate to the derivative of the density profile by (12, 13)
R (Q ) -
(4. ~r)2 Q~
dn ei .O.z. dz2, --~z "
[1]
where the scattering momentum 4 . ~r. sin(0)
Q-
X
[21
is twice the normal component of the wave vector, 0 is the grazing angle, z is the depth, and 2~is the incident wavelength. The formula shows that R is related to the Fourier transform of the derivative with respect to the depth. Then the auto correlation function of the derivative of the density profile can be written as p(z) =
n'(t), n'(t - z). dt
[3]
0O
1 p(z) - ~ (4.~r)
o~
Q4. R(Q). e ioz. dQ.
Starting from these equations, it turns out that, in the case of homogeneous layers, the auto correlation function is simply
GRAZING X-RAY REFLECTOMETRY
p(z) = ~
205
[4]
~ (nk+l - nk)" (nl+l - nl)" 6x~-x,(Z),
k
l
where k and I are the layer numbers, nk are the layer indices, xk is the depth between the kth surface and the origin, and 6h is the Dirac function which corresponds to a translation of length h. If i is the number of interfaces, this p function has i- (i - 1)/2 peaks associated with each pairs of interfaces. The abscissae of these peaks are the distances between the interfaces. There are i other peaks which are superimposed at the origin; these peaks are associated with the self-coupled interfaces. The auto correlation function r is different from the index derivative profile n'(z) = ~ (nk+l - nk)" 6xk(Z).
[5]
k
This last function has i peaks only. The abscissae of the derivative peaks coincide with abscissae of the interface. Equation [4] shows that the auto correlation function has additional peaks, the abscissae of which are the differences between the abscissae of the discontinuities of densities (Fig. 3).
Profile index 1 3
I
Derivative
1
,I2
i
I
1
3
Autocorrelation
I 3-2 I I I-I I 3-I 2-2 3-3
2-I
2il I 1- I 2-2 3-3
3-I
3-2
FIG. 3. Two examples of derivative and autocorrelation spectra. There are i peaks for i interfaces in the derivative spectrum, while, including the zero-order peak, there are 1 + (i(i - 1)/2) peaks in the autoeorrelation spectrum.
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Remarks about the interest of the sign of the auto correlation result. In Eq. [3], the integration is made between - o o and + oo. That implies that the reflectivity curve is symmetrical and that the Fourier transform is real. The sign of the auto correlation function appears in the real part. It can be seen from Fig. 3 that if we have one thick layer of gold on a glass substrate and an additional thin layer in the stack, it is not possible to detect the position of the thin layer in the stack by looking at the sign of the auto correlation result. In both cases, the sign of the auto correlation peaks is the same. Only information on the relative intensity of the peaks can be investigated, in ideal cases. Note that for x-ray reflectivity curves, the plateau, where expression [ 1] is not applicable, is very large and this leads to significant errors in the Fourier reconstruction. This can be seen in Fig. 4a where the theoretical reflectivity curve is obtained from the parameters of Fig. 2, but with no roughness and with a scanning angle between -10000" and 10000" for a symmetrized curve. So, because the sign of the result appears to be not very useful, it can be more interesting to study solely the modulus of the autocorrelation function and to integrate the reflectivity curve over the part beyond the plateau region where Eq. [ 1] is verified. Figure 4b shows an example of the modulus result obtained with the parameters of the preceding reflectivity curve, but with a scanning angle between 2800" and 10000". III.3. Fourier Transform Used in Computing. The Discrete Fourier Transform and Its Limitations A complete study of the properties of the discrete Fourier transform can be found in the referenced papers (14, 15). The main characteristics are recalled here in order to detect and avoid possible errors that can occur when this transformation is used. Using a computer one can apply the discrete Fourier transform (DFT),
F m = Z fq. exp -27r. i q
,
[6]
q=O
where p is the number of sampling points, fq'S are the p values of the direct function at the sampled points, Fm's are the p values of the transformed function, and q and m are the direct and the reciprocal integer indices. The accuracy of the approximation of the continuous Fourier transform by the DFT depends on the choice of some parameters, such as --the sampling frequency, which determines the maximum frequency that can be given by the signal; --the length of the sequence that determines the frequency resolution of the spectrum; --the kind of window that is used to limit the signal. Its shape can lead to spurious components in the DFT that do not exist in the direct function. In the following examples, in order to avoid the well known leakage effect (or Gibbs oscillations), we take care to select minima of the signal to define the limits of the truncation so that the signal is roughly periodic in the selected window.
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z(A/ FIG. 4. (a) The symmetrization of a typical x-ray reflectivity curve given by a gold layer leads to a real Fourier transform, but with too many oscillations. (b) Part of the curve shown in Fig. 2, taken outside the critical angle, gives a complex Fourier transform. The corresponding modulus of the Fourier transform is free of oscillations. IV. APPLICATIONS OF THIS TRANSFORMATION In order to estimate the validity o f the results, a study o f synthesized or m o c k curves (perfect theoretical ones) is first undertaken, followed by a study o f theoretical curves with k n o w n defects. Applications to realistic curves are then shown. A real improvement o f the method can be obtained by using an index correction.
IV.1. Introduction Figure 2 shows an experimental reflectivity curve o f a gold layer on float glass. This curve is well fitted (continuous line) with the parameters o f thicknesses, indices, and roughness indicated on the figure.
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It is now possible to make an artificial calculated curve, which looks like the previous experimental one, with known defects such as roughness, background noise, and Poisson's detector noise, as we have done in a previous paper (8). Starting with the possibility of synthesizing a realistic curve, the effect of each kind of defect on the Fourier transform can be studied.
IV.2. Perfect Synthesized Curves In the resulting spectrum shown in Fig. 4b, it can be seen that the two peaks at x = +26.311 n m correspond to the given thickness (d = 24.69 nm); however, the lateral peaks are spread asymmetrically. This is due to the fact that the periodicity of the direct function Q4R(Q) is not exactly constant. Remarks about the indices. It can be seen that usually the Fourier transform only gives the thickness with little accuracy, compared to the direct value thickness found when the curve is fitted by a calculated one. This can be attributed to the fact that the part of the curve related to the total reflectivity is eliminated to avoid artifacts in the transformation, and then information about the order of the peaks, in particular, is lost. Furthermore, there remains asymmetry in the peaks due to the fact that the periodicity of the direct curve is not exactly constant. It would thus be better to make an index correction replacing Eq. [2] with Q = 4~r
Vsin20k -- sin20c X '
[7]
where k is the order of the interference and 0c is the critical grazing angle given at the half maximum reflectivity point of the curve. Figure 5 shows the result of this transform with a reflectivity curve, calculated with the parameters of Fig. 2 but with no roughness. The scanning range is from about 2,800" to 30,000" of arc. In Fig. 5a the X-axis is the "corrected angle" in radians. In Fig. 5b, the peaks are narrow and symmetrical, and the thickness is well defined as 24.61 nm (compared with a given value of 24.69). Influence of the contrast of the indices. In theory, the peak intensity in the spectrum can be linked to the contrast of the indices, but in practice, because of the low thickness resolution due to the scanning range, only an indicative result can be given.
IV.3. Synthesised Curves with Defects Influence of roughness. When the interfaces are rough one can suppose that an interface behaves like a variation of the index with depth. Such a variation is well depicted by an error-type function: IIs (x) =
~+x/se_~. ~2. d~.
[81
Lt--oo
s, which we call the variance, is equal to the interface roughness multiplied by ~/2.7r. The derivative of such a function is a Gauss function:
209
GRAZING X-RAY REFLECTOMETRY
3
2
_..I
I
210
I
410 60 -3 x l 0 Corrected Grazing Angle (rd)
80
30-
25 N
1.1-20
o _<
m,O 1 5 10-
0 -1000
I
I
-500
500
1000
z(A)
FIG. 5. (a) Perfect synthesised reflectivity curve without roughness, with index correction, and over a large scanning angle range between 2800 to 30000" (ideal curve). The x-coordinate is the corrected grazing angle in radians. (b) Modulus of the Fourier transform. The maximum of the satellite peaks is given for E = 246.1 ~, (the real value is 246.9 A).
Gs(x) = ~ e -~(~s)2.
[91
S
This form of Gauss function with ~r explicit is used because it is invariant by the Fourier transform.
e -~-'~2 =
F
~
e 2"~'i'~'x. e -~'x~. dx. oo
[10]
210
BRIDOU AND rARDO
Taking into account the Gaussian profile, it is easy to show that the index profile auto correlation is [11]
P = ~ ~ (nk+l -- nk)" (nl+1 -- nt)" Gsk*G~,*6xk-x r k
l
• denotes the convolution product, and Sk is the variance of the kth interface. The variables of the different functions are not denoted in order to avoid confusing the convolution products. The convolution of two Gauss functions being another Gauss function, G~
[12]
= G,k , G s ,,
it is easy to show that each Dirac peak of the perfect stack becomes a Gaussian peak. The variance of a peak is greater than the variance of each coupled interface. Note that the peak which is at the origin becomes the superposition of n Gaussian peaks, the variances of which are given by Sk,k = ~ " Sk.
[131
The curve of Fig. 6 is obtained with the same parameters as curve 5 but with additional rms roughness: a0 = 0, al = 0.3 nm. It can be seen that, in the spectrum, peaks are broadened. The broadening increases with roughness and corresponds to the convolution of the peak obtained without roughness with the Gaussian function of the derivative of the indices introduced by the roughness.
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12-N10-O ,<
,,?,
8-
O
"~
64-
20-
-1000
I
-500
I
I
500
1000
z(A)
FIG. 6. (a) The same synthesized reflectivity curve as in Fig. 5 but for roughness: a0 = 0, ~1 = 3 A gives satellite peaks for E = 264.09 ~ in the Fourier transform with index correction. The peaks are not shifted, but they are broadening.
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G R A Z I N G X-RAY REFLECTOMETRY
2.5-
2.0-
N v LL1.5-
o,< co 0
•,~ 1 . 0 -
0.5-
0.0-1000
I
I 500
-500
I 1000
z(A) FIG. 7. With the realistic data of the curve given in Fig. 2, with index correction, the peaks are not shifted (at E = 242.58 h), taking into account the resolution of the spectrum (20 A).
It was also verified that, according to the above equations, permuting the roughnesses a0 and o-1 gives the same peak intensities, but it was seen that the zero-order intensity is not the same. A more complete study of the effect of roughness shall be made later.
IV.4. Realistic Mocked Curves Roughness. Let us study the theoretical reflectivity curve given by the fit parameters of the Au layer shown in Fig. 2. Figure 7 shows the result obtained with the roughness fit parameters, with index correction limited to a realistic scanning range 2800"-8000", and with the number of sampled points limited to 28. It can be seen that, with realistic parameters, the result is good. We find E = 24.258 nm with a resolution of 2 nm. Note that the roughness influence is visible in the Fourier transform spectrum only if the broadening is greater than the thickness resolution, that is, the inverse of the scanning range. Influence ofPoisson's noise. Realistic calculated Poisson's noise corresponding to the detection noise was added. There was no difference between the spectrum with or without Poisson's noise, so it is not shown. In fact, the frequency of this noise is too high to be seen. Influence of background noise. It is absolutely necessary to subtract the background noise (thermal or cosmic spurious noise) before data processing. Experimentally, the background noise is measured after each recording, so it is possible to take it into account.
IV.5. Other Mock Examples Mock three-layer curve. As above, a realistic curve, obtained after fitting a real one with known parameters (Fig. 8), is used with the corresponding stack model.
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BRIDOU AND PARDO
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t
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.t
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4 0 0I 0
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Grazing
angle
8 0 0I 0
1 0 0I 0 0
(seconds)
FIG. 8. The dotted line is the experimental reflectivity curve (stack made at SIV, San Salvo, Italy) and the continuous line is the calculated reflectivity curve with parameters given in the table.
For this realistic curve, taken in the bandwidth 1500"- 10000", the information seems to be a little degraded, but in fact the peak positions are close to the real ones in the limits of the resolution (= 1.82 nm in the spectrum) (Fig. 9). As there are four interfaces, following the above-mentioned general rule, six peaks are expected in the spectrum, the position of each of them corresponding to the distance
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FIG. 9. Calculated spectrum from the realistic theoretical parameters of the previous curve. Five peaks appear. The result is satisfactory, taking into account the resolution of the spectrum (18 ~,).
213
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1O0
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FIG. 11. (a) Realistic calculated reflectivity curve given by a stack of 50 layers with alternative roughness of 2 and 3 A, (b) Q4 transformation, (c) Fourier transform, (d) enlarged part of the Fourier transform at the beginning of the spectrum.
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between the interfaces taken two by two. They are, in this case, (1) E3, (2) E l , (3) E2, (4) E2 + E3, (5) E1 + E2, and (6) E1 + E2 + E3. Due to the roughness, the widening of peak 1, related to the top interface, means that this peak is unresolvable from the central one.
Multilayers. Example 1. The reflectivity of a twenty-layer periodic stack with a 1.4 n m thick Mo layer and a 3.2 n m thick layer of Si is calculated in the range of 1000" to 30000". For ~ = 0.15405 nm, the corresponding Q range is 5 to 12 n m -1 (Fig. 10). The roughness of all the layers is taken as zero. Note that the Q4 transform exhibits the Bragg peaks, while the Kissig fringes are minimized. The Fourier transform curve cancels for a value of about 46 rim, which is equal to the total thickness of the stack. It can be seen in an expanded result curve that the thickness of the period is clearly exhibited, and also that the thicknesses of the two components in a period can be evaluated despite insufficient resolution by averaging the peak positions at the beginning of the curve. Example 2. The same indices are taken in a 50-layer periodic stack with roughness taken as 0.2 nm on odd interfaces, and as 0.3 nm on even interfaces (Fig. 11). Despite the lightening of the reflectivity as a result of the roughness, it is still possible to evaluate the thickness ratio in the period. The index correction does not improve the result.
Au/FG
249.2
o <_ 3 ¸
'O
-1000
I -500
I 0
I 500
z(A) FIG. 12. Result of the Fourier transform of the real curve of Fig. 2.
I 1000
GRAZING X-RAY REFLECTOMETRY
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IV. APPLICATION TO THE STUDY OF REAL CURVES
Some of the reflectivity curves studied here are realistic ones, because they are calculated from parameters of fitted curves. So it can be assumed that the real results are close to the given ones. But, in the case where an index correction is required, it is desirable to make a scan with a step proportional to the "corrected grazing angle" instead of the grazing angle. If this is not possible, an interpolation can be tried. Obtained by such a method, Fig. 12 is the Fourier transform of the real curve described in Fig. 2. It can be seen that the result is very close to the expected one, and this shows that the method works. A further study will be made with only actual curves. V. CONCLUSION
One of the main difficulties in fitting a real curve by a direct optimization method, like a trial-and-error one, is in starting with a good stack model because, most of the time, even if the materials deposited are known, the deposition introduces some perturbation or there is a degradation during the air exposure of the multilayer. The real model is therefore often more complicated than the expected one. Despite the constraints of its use, the Fourier transform can be helpful in providing an expectation for a stack model, even when the number of layers is not a priori clearly determined. The number of peaks in the spectrum can be related to the number of interfaces even if some peaks disappear or overlap. For example, in Fig. 9, the first peak spreads in the zero order. When the correction of an index is undertaken for stacks with few layers, a good evaluation of thicknesses can be made, despite the roughness. The spreading of peaks is dependent on the roughness of the corresponding interface so the stack model can be refined. The thickness accuracy is dependent on the resolution of the spectrum, that is, on the bandpass recording of the direct curve. To determine the thicknesses with good resolution, the initial curve has to be fitted by, for example, a trial-and-error method, but with a stack model better known with the help of Fourier transformation. ACKNOWLEDGMENTS The authors thank F. Sacchetfi from SIV (San Salvo, Italy) for the growth of the three-layer stacks mentioned in this paper. This work was supported by the European BCR Commission.
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7. B. PARDO,T. MEGADEMINI,AND J. M. ANDRE,Rev. Phys. Appl. 23, 1579 (1988). 8. F. BRIDOU AND B. PARDO, J. Opt. (Paris) 21(4), 183 (1990). 9. J. A. DOBROWOLSKI AND D. LOWE, AppL Opt. 17(19), 3039 (1978).
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P. L. SWARTAND B. M. LACQUET, J. AppL Phys. 70(2), 1069 (1991). K. SAKURAIAND A. IIDA, Adv. X-RayAnal. 35, 813 (1992). D. S. SIVIA, W. A. HALMILTON, G. S. SMITH,T. P. RIEKER,AND R. PYNN,dr. AppL Phys. 70(2), (1991).
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