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Characterization of thin films and multilayers by specular X-ray reflectivity
W. Plotz, K. Lischka
To cite this version:
W. Plotz, K. Lischka. Characterization of thin films and multilayers by specular X-ray reflectivity.
Journal de Physique III, EDP Sciences, 1994, 4 (9), pp.1503-1511. �10.1051/jp3:1994303�. �jpa- 00249200�
Classification Fhj'si<.s Ahstra<.ts
78.70C
Characterization of thin films and multilayers by specular X-ray reflectivity
W. M. Plotz and K. Lischka
Forschungsinstitut fir Optoelektronik, Johannes-Kepler-Universitfit Linz, Altenbergerstra8e 69, A-4040 Linz-Auhof, Austria
(Recei;ed J9 November 1993, revised 9 Maich 1994, accepted 26 April 1994)
Abstract. The application of glancing incidence X-ray reflectivity measurements for the
investigation of submicrometer thick layers and muJtiJayer stacks which are produced in modern technology processes is discussed. We describe different setups for X-ray reflectivity measure-
ments and computer methods which allow the extraction of various layer parameters from the
experimental data.
Introduction.
The structural characterization of submicrometer thick layers and layer stacks is an important issue in modern industrial production processes, e.g. in the semiconductor industry and in the glass coating industry. Transmission Electron Microscopy (TEM), Rutherford Backscattering (RBS), Atomic Force Microscopy (AFM) and Glancing Incidence X-Ray Reflectivity (GIXR)
can provide valuable structural information. However GIXR offers somi advantages
over
other methods for the use in a production line because (a) GIXR reveals the layer thickness and
roughness with high accuracy (b) is non destructive and (c) is able to scan macroscopic sample
areas within reasonable short times [I].
In this paper we shall demonstrate that GIXR is an effective method to obtain layer
thicknesses and roughnesses when careful modeling of the experimental data is performed.
Theory.
For X-rays striking an ideally flat surface at low incidence angles @, the reflectivity R~ is given by the Fresnel forrnula
RF ~ ~~ ~ ~~ ~~~ ~ ~
(l)
sin + (n cos )
1504 JOURNAL DE PHYSIQUE III N° 9
In equation (I) n is the refractive index of the material which is slightly less than unity for X- rays
n = I I# (2)
where and # depend on the composition and the density of the layer material and on the
wavelength of the X-rays. When using Cu K~ radiation typical values for are in the range from 5 x 10~~ to 50
x 10~ ~, the values for # are typically one order of magnitude smaller.
The reflectivity R of a rough surface or interface can be calculated by multiplying the Fresnel
reflectivity R~ with a Debye-Waller Iike attenuation factor [2, 3]
R
= R~ exp («~Q~ Q[ ) (3)
where « denotes the root-mean-square roughness, Q~ and Q[ are the components of the wave vector transfer normal to the interface above and below the interface, respectively.
The reflectivity of a layer stack is calculated using the formalism described in reference [4], replacing each Fresnel reflectivity R~ by the corresponding reflectivity R.
From the measured X-ray reflectivity the layer thickness and roughness as well as the layer
material density can be obtained. However, it is not possible to extract all these parameters in a
direct way from the experimental curve. Therefore it is necessary to use a fitting procedure
which starts with a description of the sample based on available information. The fit is
improved by looking for a set of parameters which yield an optimum match of the calculated and measured reflectivity. A number of algorithms for a systematic search for the optimum fit have been described in the literature [5-8]1 We have used a simplex [6] and the Levenberg- Marquadt [5 algorithm. These algorithms essentially minimize the area between the calculated and measured reflectivity curves.
Experimental set-ups.
Figure I shows three experimental set-ups that have been used for our X-ray reflectivity
measurements. All system components are available from various vendors. The set-up shown
in figure la is h Philips MRD, a system designed for X-ray diffraction measurements. When used with a Bartels monochromator, it offers a monochromatic (AAIA
= 10~ ~) primary beam with a low divergence (= 10" ). However, these beam conditions are in general not required for
X-ray reflectivity measurements and limit the dynamical range to less than six orders of
magnitude. A system especially designed for glancing incidence X-ray analysis (GIXA) is
shown in figure16. The use of a multilayer mirror as monochromator yields much higher
intensities and still provides a sufficiently monochromatic and parallel primary beam. The system is equipped with a detector for analyzing X-ray fluorescence emitted from the sample surface, allowing chemical analysis and depth profiling. The set-up depicted in figure lc
provides an even higher intensity of the primary beam by using a graphite monochromator.
This equipment offers a dynamical range of eight orders of magnitude when a standard 2 kW fixed anode tube is used.
Results and discussion.
Figure 2 shows the measured X-ray reflectivity of our sample i# I which is a ZrB~ layer on a
silicon substrate (dots) and a computer simulation (dashed line). The reflectivity was measured
on a system as shown in figure la. For the simulation we used the density of bulk ZrB~ and a layer thickness which was estimated from the deposition rate and deposition time. The value of the interface roughness must be estimated. The roughness of the substrate may be determined
X-Ray Source a)
/ 4,Crystal
-/ Monochromator i~~
Sample ~~~~~~°~
Multilayer Detector For Fluorescent b)
Monochromator Radiation
/ i
~ ~
Graphite c)
Monochromator
@_/ I ~
/ i
Fig. I. Different experimental set-ups for glancing incidence X-ray reflectivity measurements al
high resolution diffractometer using a four crystal Bartels monochromator and a receiving slit
(Philips MRD), b) a combined mea~urement of reflectivity and X-ray fluorescence (Philips GIXA), cl a dedicated X-ray reflectometer using a graphite monochromator.
10~
~~
10~
Ti~
~ /
10~ ~,,~ /~,
~ '' ,',
I
~
, l',
g lo ' ii
- '~
10~
~o
0.00 0.05 o-lo o-15
Q
~
lA~]
Fig. 2. Glancing incidence X-ray reflectivity of sample # I (dois), a 500 h ZrB~ layer
on a silicon substrate, a simulation based on the grower's estimate (dashed curve), and the optimum fit simulation (full curve). The parameters of the simulations are listed in table1.
1506 JOURNAL DE PHYSIQUE III N° 9
Table I. Parameters for the simulation of the X-ray reflectii>ity from sample i# I as shown in figure 2. Parameters obtained by an algorithm for fit optimization are indicated by bold
letters.
Model Matedal Thickness Roughness 6 fi p
Initial Guess 10.0 0.78
Si « 7.59
Best fit 498.9 17.4 14.05 0.M 4.97
« ii.o
before the deposition of the layer. In order to demonstrate that for the evaluation of X-ray reflectivity data the knowledge of the substrate roughness is not required, its initial value has been chosen extremely small. These parameters which are given in table I (initial guess) were
used with a computer program which uses algorithms for fit optimization. It allows to find a set of parameters for an optimum match of the simulated and the measured reflectivity. The result of such a fitting procedure using the Levenberg-Marquadt algorithm is shown in figure 2 (full line). The resulting parameter set for the « best fit » is given in table I. As an important result
we find that the density obtained from X-ray reflectivity is significantly smaller than the bulk density.
Figure 3 depicts the Zr X-ray fluorescence from sample i# I (dots) measured as a function of the angle of incidence of the exciting X-rays. Experimental data were obtained with the
Philips GIXA set-up shown in figure 16. The full curve is calculated [9- Ill using the set of parameters obtained by the reflectivity measurement (Tab. I, best fit).
For samples which consist of two or more individual layers the experimental reflectivity
curve can become quite structured due to the interference of the radiation reflected from
io
8
~ 6
.
£ *
. .-
6i ~
fi 4
-
0
0.00 0.05 o-lo Ol 5
Q~ [A'~j
Fig. 3. The Zr X-ray fluorescence of sample # I (dots) and a simulation (full curve) based on the parameters given in table I, best fit.
10~
i~s
_10~
I
3l10~
a~ ,
6i ~~2 '$
j
G 10~
; 'J,-
~~o _'t
10"~
0.00 0.05 0.10 0.15 0.20
Q~ IA-i
Fig. 4. Glancing incidence X-ray reflectivity measurement of sample # 2 (dots), a loo A ZrBN layer
on a 300 h ZrB~ layer on a silicon substrate and a simulation (full curve) based on the grower's estimate
as described in table II, initial guess.
different interfaces. As an example the reflectivity from a ZrBN/ZrB~ double layer on a silicon substrate (sample ~# 2) is shown in figure 4. The dots are experimental data, the full curve is
calculated using estimated values for the layer thicknesses and the density of the bulk materials, respectively.
The parameters used in the simulation are given in table II (initial guess). Since the difference between experiment and simulation is quite large, the use of an algorithm for
optimization will not yield a proper description of the measurement. In this and similar cases a
Table II. -Parameter"s for the siniulation of sample i# 2 using nvo layers. Parameters ohtained hi, an algorithm for fit optimization are indicated by bold letter"s.
Model Material Thickness Roughness 6 fi p
Initial guess, 100.0 15.0 18.85 0.83
see fig. 4 300.0 10.0 17.21 0.78 6.09
Si « 5.0 7.59 0.18
FTbased ZrBN 175.0 15.0
guess, 500.0
see 6 « 5.0 7.59 0.18 2.33
Best fit using 12,1 10.97 0.48 3.83
mo layers, 0.69 5.37
co
1508 JOURNAL DE PHYSIQUE III N° 9
Fourier transform (FT) of the reflectivity curve [12] reveals a reasonable good estimate of the
layer thicknesses. The FT of the reflectivity curve from sample i# 2 is shown in figure 5. It
yields values for the thickness of the individual layers (175 and 500 1, the peak at 675 1 is the
sum peak, which is always present in the FT of a multilayer system) which are much larger
than that estimated from the growth conditions. Using the thicknesses obtained from the FT and a reduced density for the top layer we obtain a better match of the calculated and the measured reflectivity as can be seen in figure 6 (dotted line). The parameters describing this simulation are given in table II (FT based simulation). The result of an optimization procedure
loo
« Ln
« ~i
80 ~
; -
m-
60
i~
I ~°
(
«
20 o
~
0
0 500 1000
Q~ lA~l
Fig. 5. Result of a Fourier transform of the measurement shown in figure 5. The peaks at 175 h and
675 1 indicate positions of interfaces below the surface, yielding layer thicknesses of 175 h and 500 h.
10~
10~
_10~
m cL
° 10~ ~
a~
10~
~ .l',
~ 10~ ."'
,~>
l0~
'
10"~
0.00 0.05 0.10 o-15 0.20
Q~jA"~j
Fig. 6. Glancing incidence X-ray reflectivity of sample # 2 (dots), an improved simulation (dotted curve) based on the result of the Fourier transform shown in figure 5, a simulation (dashed curve) of the
optimum match two-layer system as described, and a simulation (full curve) of the optimum match system using interface layers. The parameters of these simulations are listed in tables lI and III.
using these parameters is shown in figure 6 (dashed line), the corresponding parameter set is
given in table II (best fit).
The fit of the experimental to the calculated reflectivity curve from sample i# 2 can be improved further by the introduction of « interface layers ». This is demonstrated in figure 6 (full line) where an optimized five layer model has been used for the calculation of the
reflectivity. We find an excellent match of the simulation using parameters as given in table III. It is important to note that the layer thickness of the interface layers is comparable to the root mean square roughness of their interfaces. Since the reflectivity is influenced only by the density of the layer material these interface layers can be interpreted as a region of varying density which may be due to (a) chemical intermixing of adjacent layers and/or (b) two chemical distinct layer with an extremely non Gaussian interface profile. By using X-ray reflectivity measurements these two cases cannot be distinguished. Therefore it seems
reasonable to plot the results of X-ray reflectivity investigations in the form of a &profile [2, 13], which is essentially a density profile of the sample. Other techniques which are beyond the scope of this paper may be used for furthe~ investigation of the nature of these interfacial
layers [14].
Table III. Parameters for the simulation ofsample i# 2 using a model with interface layers.
Parameters obtained by an optimization algorithm are indicated by bold letters.
Model Material Thickness Roughness 6 fi p
Interface 38.9 9.9 8.59 0.01 3.01
Best fit using ZrBN 4.31
interface Interface 15,1 23.9 12. II 0. 14 4. 22
layers, Layer
see fig. 6
487.3 11.7 14.58 0.29 5.07
Interface 26.7 20.8 5.31 0.02 .74
Si oo 8.0 7.59 0.18 2.33
In figure 7 the &-depth profiles of sample i# 2 are plotted which correspond to the sets of parameters given in tables II and III. Comparing the initial guess (circles) with the FT-based guess (filled circles) and the best fit of a 2 layer system (squares) shows the fact that the initial guess underestimated the layer thicknesses. It can also be seen that the density of the layer
material is smaller than that of the bulk material which has been used for the initial guess. It is obvious that the main difference between the best fit of a 2 layer system (squares) and best fit
using interface layers (filled squares) are the different density profiles at the interfaces and the surface.
1510 JOURNAL DE PHYSIQUE III N° 9
20
15
_
© '
~ iQ
~l '
t$~
5 ,
o
-200 0 200 400 600 800
z [Al
Fig. 7. The b-depth profiles of the sample descriptions as listed in tables II and III. The parameter given in table II are marked as filled circles (initial guess), circles (FT based guess), and squares (best fit). The parameter set given in table III is marked by filled squares.
Conclusions.
We have demonstrated the application of glancing incidence X-ray reflectivity measurements as an effective and non destructive tool for the characterization of submicrometer thick layers
and layer stacks. We have shown how layer parameters may be obtained by computer
simulation of the reflectivity data. The influence of the starting parameters on this procedure is discussed. For layer stacks of two or more individual layers a Fourier transform of the measured reflectivity curve can help to improve the fitting procedure considerably. The use of
an algorithm for the optimization of the computer fit is stressed. Since all our measurements were performed using standard laboratory equipment our method may be used for the quality
control of nanometer layers produced in modern technology processes.
Acknowledgment.
This work is supported by the « Fonds zur F6rderung der wissenschaftlichen Forschung in
Osterreich »~ the EC project « NAMIX », and Philips Analytical X-ray BV.
References
Ii For a recent review on X-ray reflectivity see : MiceJJi P. F., X-ray reflectivity from Heteroepitaxial Layers, in Semiconductor Interfaces, Microstructures and Devices : Properties and Appli- cations~ Z. C. Feng Ed. (Adam IOP Publishing Ltd., Bristol, 1992).
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[4] Parrat L. G., Surface studies of solids by total reflection of X-rays, Fhys. Ret<. 9s (1954) 359.
[5] Marquadt D. W.~ An algorithm for least-squares estimation of nonlinear parameters, J. Soc. lnd.
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[9] de Boer D. K. G.~ Glancing-incidence X-ray fluorescence of layered materials, Fhys. Rei~. B 44 (199J) 498.
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