Fitting X-ray multilayers reflectivities by means of PPM software
D. Spigaa,b*, A. Mironec, C. Ferreroc, V. Cotroneob, G. Pareschib, M. Sanchez del Rioc, D. Vernanib
a
Università di Milano-Bicocca, Dipartimento di Fisica, Piazza della Scienza 3, Milano, Italy
b
INAF/Osservatorio Astronomico di Brera, Via Bianchi 46, 23807 Merate (LC), Italy
c
European Synchrotron Radiation Facility (ESRF), 6 rue Jules Horowitz, Grenoble, France
ABSTRACT
A number of future hard X-ray (10-100 keV) telescopes will implement focusing optics with multilayer coatings. In this
framework, we are developing (at INAF/Brera-Merate Astronomical Observatory) multilayer optics based on the ebeam deposition technique: this approach is suitable to coat very large surfaces at an high deposition rate; in order to
test the performances of the deposited samples, X-ray reflectivity scans at the two "standard" photon energies of 8.05
and 17.4 keV are taken, returning very positive results with high peak reflectivities. However, the exact interpretation of
the reflectivity curves is a complex task since it depends on a large number of parameters: the software PPM (Pythonic
Program for Multilayers) has been recently developed by A. Mirone (ESRF) specifically to the aim of a friendly and
fast determination of the parameters of multilayer structures. In particular, for this paper we present the layer-by-layer
modelization of the characteristics (roughness, density, thickness) of multilayer stacks (Ni/C, Pt/C) by a multiparametric "global" automatic optimization to reach the best fitting performances. In order to physically constrain the
parameters, the data will be compared with the results of TEM measurements performed on the same samples, when
available.
Keywords: X-ray multilayers, X-ray reflectivity scan, automatic fitting procedure
1. INTRODUCTION
An extension of the grazing incidence focusing optics currently operating in the soft X-ray (1-10 keV) imaging
telescopes to the hard X-ray band (10-100 keV) is foreseen in many mission concepts6. The developments of optics with
an enough large effective area in the hard X-ray band may be achieved with an increase of the geometrical area
(XEUS), by using multi-modular telescopes (Con-X) or by adopting a large focal length (Simbol-X, XEUS): a further,
very effective improvement will come from the implementation of graded multilayer-coated mirrors: in fact, a number
of missions will improve their effective area in the hard X-rays band also by enhancing the reflectivity at grazing
incidence angles of 0.4 ÷ 1 deg in photon energy ranges up to 70 - 80 keV. The development of high-reflectance, wideband multilayers is being pursued in the framework of the "High energy astrophysics payload preliminary project"
funded by ASI (the Italian space agency), a research activity that involves the INAF/Brera Astronomical Observatory,
Media-Lario s.r.l., LABEN, Politecnico di Milano, IASF; this research is aimed to produce a complete prototype of an
hard X-ray telescope (mirror shell coated with 150 graded Pt/C bilayers and a CZT detector), but it could also have a
relevant return in spin-off applications, like the nanolithoghapy with Mo/Si, normal incidence multilayers, or for
radiological purposes.
The realization of wide-band, high-reflectance multilayer mirrors requires a careful design and a sharp control of
the layer properties (thickness, density, roughness) in order to achieve the foreseen design. On the other side, the large
surface, uniform coating requirement (e.g. 30 m2 for XEUS1) has suggested the adoption of the e-beam evaporation as
technique for multilayer deposition. It is well known2 that this method is suitable for the uniform coverage of very large
surfaces at a large evaporation rate and for the production in an industrial context (an important requirement if many
mirror shells have to be produced). Moreover, a large coating facility of this type is already installed at Media-Lario and
was already successfully used to deposit the Au single coatings of the soft X-ray mirrors of the Beppo-SAX3, JetX/SWIFT4, Newton-XMM5 missions: the Ni electroforming from mandrel replication technique, used to make the
*
spiga@merate.mi.astro.it; tel. +39-039-9991146, fax +39-039-9991160
mirror substrates of these telescopes, has given very good optical performances and could be easily extended to
replicate hard X-ray multilayer coatings7.
The deposition of flat, multilayer Pt/C and Ni/C coated samples has given since now very interesting and
encouraging results8. However, a typical e-beam problem is the non-steady evaporation rate, especially with Carbon:
this can in turn cause an instability of the layer thickness, and a consequent deviation of the thickness trend from the
designed one. This problem is being resolved with the achievement of steady evaporation conditions, as well as the
interfacial roughness growth (which will be lowered by using an ion-assistance device), but a diagnostic technique to
analyze in detail the layered structure is a fundamental need. The TEM sections are a viable solution and they can be
used for selected samples: as a direct performance test, we always measure X-ray reflectivity (XRR) scans at the two
standard energies 8.05 keV and 17.4 keV with a Bede-1 diffractometer installed at OAB. This kind of measurement
takes the advantage of being sensitive to some physical layer properties (density, high-frequency microroughness) in
addition to the properties (thickness, interdiffusion, crystallization state) that can be studied with a TEM.
The interpretation of the XRR is nevertheless difficult as it depends on a large number of parameters, namely all
the thickness, density, roughness…values in the stack. To interpret the multilayer structure, a software like IMD9 is very
useful as it is able to provide the average properties of the layers; an alternative software is PPM (Pythonic Program for
Multilayers) developed by one of us (A. Mirone - ESRF) aimed to perform a multiparametric optimization of the
reflectivity scan at one or more energies at the same time. The reflectivity fitting can work with an arbitrary number of
free parameters searching the combination of values that represents the best possible fit of the scan. Once found a set of
values returning a modeled reflectivity well superposed to the experimental one, we can be confident that the found
solution represents a good approximation of the multilayer structure. PPM has recently become a subject of interest in
the COST cooperation promoted by the European Science Foundation.
PPM has been tested on reflectivity scans taken at the OAB on some multilayer samples (with almost constant dspacing), deposited at the Media-Lario coating facility. In the sect. 2 we will describe these Ni/C and Pt/C multilayer
samples reflectivity curves: a short description of PPM will be provided in the sect. 3, and some early results coming
from the use of this tool will be provided in the sect. 4, with a short discussion in the sect. 5.
2. E-BEAM DEPOSITED MULTILAYER MIRRORS
We have deposited by simple e-beam evaporation several Ni/C and Pt/C multilayers on flat, non-rotating Si wafer
substrates. An example of a Ni/C multilayer mirror reflectivity curves at 8.05 and 17.4 keV is shown in the fig.1.
Fig.1: XRR results at 8.05 (left) and 17.4 keV (right) of a Ni/C sample with 19 bilayers: the first Bragg peak reflectivity is very good
(95% @ 8.05 keV): an uncontrolled period variation in the stack of about 2 nm is evident from the 2nd peak dispersion. The
roughness inferred from the fit is very good (3 Å, assuming the natural carbon density) starting from the Si wafer roughness value.
The reflectivity scans return results in a good agreement: the reflectivity at the first Bragg peak is very high (95%
@ 8.05 keV) and the IMD fit shows that there is a period variation in the stack of about 2 nm along the 19 bilayers,
probably caused by the drift of the Carbon layers. The average period (13 nm) is much larger than the programmed one
(9 nm). An interesting result is the Carbon density value (1.6 g/cm3), much lower than the natural one (2.3 g/cm3). The
Nickel has instead deposited at densities (8.7 g/cm3) near to the bulk value (8.9 g/cm3) The roughness is also good (4 Å
with the actual C density, 3 Å if we assume the bulk value) as we started from a Si wafer with a similar roughness
value. The high contrast density Ni/C, together with the low roughness, is probably responsible for the high reflectivity.
The following activity was aimed to stabilize and calibrate the deposition conditions and the Carbon thickness: an
example of the results achieved with the Pt/C multilayers is shown in fig.2.
Fig.2: Reflectivity scan of a Pt/C multilayer (20 bilayers) deposited on a Si wafer: the period drift has been better controlled as the
peaks are narrow and clearly defined. The average period is 9 nm with Γ∼0.33, the inferred roughness from the fit is 3 Å, very similar
to that of the substrate.
The reflectivity scan at 8.05 keV shows in this case clearly defined and narrow peaks (at the first Bragg peak it
reflects 83% of the incident radiation): the period in this case has been kept more constant (the estimated drift is 5 Å
along 20 bilayers) and it very close (9.3 nm) to the programmed one (9 nm). The roughness is very near to that of the Si
wafer substrate (3 Å) and the C has deposited at 1.6 g/cm3, as in the previous case. Pt has the density of 21.1 g/cm3, its
natural value. These results are very encouraging as they indicate a good quality of the film in view of the production of
graded multilayers.
3. PPM: PYTHONIC PROGRAM FOR MULTILAYERS
A layer-by-layer analysis of the stack may be performed with PPM. This software is open-source, downloadable
from the URL ftp://www.esrf.fr/pub/scisoft/ESRF_sw/linux_i386_00, it is specifically conceived for a fast determination
of the stack parameters by fitting the reflectivity scans even in the details: this allows us to take into account the
reflectance features caused by the deepest layers. A scheme of the PPM working principle is shown in the figure 3. The
program takes as input the reflectivity curves and a file in XML format that defines the structure scheme. The number
of necessary parameters may vary according to the supposed complexity of the stack. For example we can take all the
layer thicknesses as free variables, or assume a gradual drift of the layers. Other variables define the layer density and
their roughness. The user must also enter the lower and the upper limits and the initial guessed value for every defined
variable.
Starting from the initial values, PPM computes the reflectivity curve Rc for every available experimental scan Rm
and computes the Figure Of Merit (FOM),
FOM =
Σ
(log
Rm(i) - log Rc(i))2
__________________
i
|log Rm(i)|
where i is the index of the angular sampled point. The minimizing algorithm consists in the variation of the stack
parameters since the FOM reaches its minimum - a situation corresponding to the best possible fit, that is, the best
approximation of the defined structure to the real multilayer structure.
Fig.3: A scheme of the working principle of PPM: the reflectivity curves are initially calculated from the parameters values defined
by the user for the chosen scheme which approximates the stack structure. The computed reflectivity (at one or more photon
energies) is then compared to the experimental one(s). The Figure Of Merit (FOM) to be minimized is the χ2 of the logarithmic plots:
the minimization algorithm (downhill - simulated annealing) varies the stack parameters within the fixed limits by the user. At every
cycle the FOM is computed and the parameter set with the minimum FOM value is selected.
The choice of minimizing the χ2 of the logarithmic reflectivity permits to give more weight in the fit to the
reflectance details (which in a linear plot would be likely neglected) that carry important information about the
reflection in the deepest layers.
The adopted algorithm that varies the parameters to search the global minimum in the parameter space is a
derivation of the well-known downhill simplex1,11 (or amoeba) algorithm. A set of N+1 points in the N-dimensional
parameter space follows a series of moves that converge to the nearest minimum: however, the downhill simplex alone
would lead very likely to a local minimum, that could be very far from the global one. For this reason an annealing
function12,13,14, which is able to take us out from the local minima, is also implemented.
The simulated annealing function associates to every point of the parameter space a microscopic state of N
particles: in this comparison, the convergence to a local minimum is compared to a thermalization process, leading to a
state of definite temperature T. The system of particles is assumed to obey the Boltzmann distribution, where the FOM
(depending on the actual particle state) plays the role of the energy. Using the downhill simplex alone we would
exclude completely the moves that increase the FOM, trapping so the system in a local minimum: if the system
stabilizes itself according the Boltzmann distribution at a temperature large enough, there is instead a finite likelihood
that the system goes out from the local minimum. Usually the initial temperature is set at a high value in order to allow
a large number of transitions increasing the FOM, and when the system reaches the equilibrium T is slowly decreased.
When the temperature is decreased, the probability of increasing the FOM during a transition is lower and lower.
At this point the system tends to converge towards a more global minimum till a minimum temperature value is
reached, where the calculation stops. The probability of finding the global minimum is so increased as a much more
large fraction of the parameter space is explored, at a cost of an increase of the computing time (with respect to the
downhill simplex case).
The annealing procedure has also some critical points, like the choice of the initial temperature and the cooling rate
(usually an exponential decrease is adopted). A very slow cooling rate gives a more degree of confidence that the global
minimum has been found, but it is more expensive in term of the computing time. These parameters, as well as the
multilayer structure and the initial values for the parameters, have also to be carefully chosen in every single case.
The capability of searching the minimum FOM within the allowed values of the stack parameters makes PPM a
very powerful tool to fit X-ray reflectivity curves.
4. SOME EARLY RESULTS
Reflectivity @ 8.05 keV
The fitting capabilities of PPM were tested on the multilayers shown in the fig 1 and 2. The Ni/C multilayer
presented in fig.1 was schematized as a succession of 6 blocks formed by 3 bilayers each: in each block the bilayers
have a linearly drifting thickness to allow short-term thickness fluctuations. The Ni and the C density, also parameters
to be fitted, are assumed to be constant along the stack, whereas the Ni and C roughness are independent in every block.
The modeled structure has been fitted simultaneously to both reflectance scans at the photon energies of 8.05 and 17.4
keV.
The excellent agreement of the fitting model with the experimental curve is apparent in fig.4 and 5 (8.05 and 17.4
keV respectively) . The reflectance features are fitted in the all the details, either in the linear scale or in the logarithmic
scale. The main deviations are oscillations of the modeled reflectivity around the experimental curve in correspondence
to the near-zero reflectance angular regions: they can be eliminated by taking into account the finite angular resolution
of the measurement: in fact, if we set the convolution parameter in the source file to the actual measurement angular
resolution, the agreement is complete also in those points.
Ni/C, measured data
Ni/C, PPM fit
0.8
0.6
0.4
0.2
0.0
1000
2000
3000
4000
5000
Grazing incidence angle (arcsec)
6000
0
Reflectivity @ 8.05 keV
10
10
10
10
10
Ni/C, measured data
Ni/C, PPM fit
-1
-2
-3
-4
1000
2000
3000
4000
5000
Grazing incidence angle (arcsec)
6000
Fig. 4: the experimental reflectance scan @ 8.05 keV of the Ni/C multilayer as in fig. 1 compared to the fitting model found by PPM.
The fit was run simultaneously on both scans at 8.05 and 17.4 keV (fig. 5). Linear plot (above) and logarithmic plot (below). The
model agreement with the experimental curve is very good and detailed.
Ni/C, measured data
Ni/C, PPM fit
Reflectivity @ 17.4 keV
0.8
0.6
0.4
0.2
0.0
Reflectivity @ 17.4 keV
400
600
800
1000
1200
1400
1600
Grazing incidence angle (arcsec)
1800
2000
Ni/C, measured data
Ni/C, PPM fit
0.1
0.01
0.001
400
600
800
1000
1200
1400
1600
Grazing incidence angle (arcsec)
1800
2000
Fig. 5: the experimental reflectance scan @ 17.4 keV of the Ni/C multilayer as in fig. 1 compared to the fitting model found by PPM.
The fit was run simultaneously on both scans at 8.05 (fig.6) and 17.4 keV. Linear plot (above) and logarithmic plot (below). The
model agreement with the experimental curve is very good even in details.
The main result of this two-energy fit may be summarized as follows: the obtained density of Ni is 8.7 g/cm3 and
that of C is 1.7 g/cm3, in agreement with the found values with IMD. The average roughness is 3.5 Å for both Pt and C.
The layer thickness trend shows (see fig. 6) a slow decrease of the Ni layers and a very strong increase of the C layers:
the main responsible for the thickness drift is C, as previously supposed. The overall period variation over all the stack
is 2 nm.
The thickness trend has been compared to the data derived from the TEM section of the same multilayer (see
fig.6). The structure obtained by PPM is in disagreement with the TEM: this is probably due to the choice of the
modeled structure of the stack as a succession of blocks. A better structure should be defined letting all the layers
independent in their variations (within quite stringent limits in order to avoid the calculation divergence). This has been
done with the next example, a Pt/C multilayer.
Layer thickness (angstrom)
100
90
80
Nickel layers
Carbon layers
70
60
50
40
0
2
4
6
8
10
Bilayer number
12
14
16
Fig. 6: (above) The trend of the Ni and Carbon layer thickness as resulting from the PPM fit. The multilayer was modeled with 6
blocks formed by 3 bilayers each, with in addition a linear drift in every block (as a consequence, the C layers appear grouped). The
drift is mainly due to the Carbon layers as in the Pt/C case, but also the Ni contribution is not negligible. The period drift (2 nm) is in
agreement with the IMD results.
(below) The TEM section of the same multilayer. The Ni layers are the dark bands, the C layers are the clear bands. The layer
deposition has a very good quality as the layers are completely amorphous and homogeneous.
The Pt/C in fig.2 has been fitted assuming a structure with all the layers thicknesses (both Pt and C) as independent
parameters. Also the density and the roughness of both Pt and C, assumed to be constant along the stack, are considered
as fitting parameters. The multilayer is so fitted with an overall set of 44 free parameters: in the figure 7 we show a
comparison of the achieved fit and the experimental scan. The agreement between model and experiment (only at 8.05
keV) is apparent in both the linear and logarithmic plot.
The result of this fit procedure has returned values in qualitative agreement with the IMD fit. In particular, the Pt
density turns out to be 20.6 g/cm3 and the C density 1.55 g/cm3. Moreover, the model can be fitted assuming an
interfacial density for both C and Pt of 3.5 Å, very near to the substrate value. In fig. 8 the thickness trend of all the
layers is shown: the Pt layers are almost constant and have the desired value, whereas the Carbon is apparently
responsible for the most of the period variation in the stack. Moreover, except for three very exceeding values, the layer
drift range is quite close to the obtained value with IMD. Unfortunately, for this sample a TEM section is not yet
available and we cannot verify the results.
Pt/C, measured data
Pt/C, PPM fit
Reflectivity @ 8.05 keV
0.8
0.6
0.4
0.2
0.0
Reflectivity @ 8.05 keV
2
10
10
10
10
10
4
6
8
Grazing incidence angle (arcsec)
3
10
12x10
Pt/C, measured data
Pt/C, PPM fit
-1
-2
-3
-4
-5
2
4
6
8
Grazing incidence angle (arcsec)
3
10
12x10
Layer thickness (angstrom)
Fig. 7: the experimental reflectance scan @ 8.05 keV of the Pt/C multilayer as in fig. 2 compared to the fitting model found by PPM.
Linear plot (above) and logarithmic plot (below). The layer-by-layer analysis has found a model in very good agreement with the
experiment. Despite the large number of parameters, the calculation has converged to a very good solution.
70
60
Errore.
50
Platinum layers
Carbon layers
40
30
0
2
4
6
8
10
12
Bilayer number
14
16
18
Fig.8: the thickness trend of the Pt and C layers, as inferred by PPM. The Pt layers have a more constant thickness (3 nm, as
programmed), whereas almost all the period instability was caused by the C layers. Three values are very exceeding, probably due to
the too large limits allowed for the C layers.
5. CONCLUSIONS AND FINAL REMARKS
PPM has turned out to be a very powerful fitting tool. PPM models very well the reflectance scans, even at two
energies at the same time. Like all software, it has also some limitations: for instance it seems to manage with difficulty
the Mo/Si multilayers fit (fig. 9), probably because of the well-known interdiffusion layer MoxSiy with very uncertain
composition. Even a 4-layer model produces a not-so performing fitting: it is probable that the composition of this layer
is so uncertain that the number of freedom degrees becomes too large. More stringent limits about the interlayer
composition are necessary in order to return a performing fit.
Like in all the fitting software, the risk of the local minimisation is always present, especially when the number of
free parameters is large: PPM seems to have optimal capabilities to reduce this risk: this occurrence may be reduced by
slowing the cooling rate of the annealing algorithm. This would in turn increase even more the computing time, but
PPM (which has been developed in a Linux ambient) can be also easily run in a parallel-computing network: this
configuration permits to reduce the time necessary to the elaboration.
In the next developments, a number of multilayers will be produced and the thickness data returned by PPM will
be compared to the TEM measurements: in order to take PPM in agreement with the TEM results, it will be necessary
to give more stringent limits to the input parameters and to vary the assumed model of the stack, deriving it from the
TEM results. We still need a number of cases to be studied, but, by means of the TEM sections, a calibration of PPM in
order to provide a complete realistic analysis of the stack structure can be done.
Once calibrated, PPM will be an important tool in the development and use of a multilayer diagnostic. The
systematic use of PPM will be very useful in the development of e-beam evaporated multilayers and the production of
hard X-rays mirrors for the next hard X-ray telescopes.
0
Reflectivity @ 8.05 keV
10
10
10
10
10
10
10
Mo/Si, measured data
Mo/Si, PPM fit
-1
-2
-3
-4
-5
-6
2
4
6
Grazing incidence angle (arcsec)
8
10x10
3
Fig. 9: A Mo/Si multilayer XRR at 8.05 keV, as fitted by PPM. In this case the fit is not so performing because of the uncertainty in
the composition of the interdiffusion layer, that has doubled the number of parameters to be fitted. Some additional constraints about
the interdiffusion layer would be necessary.
ACKNOWLEDGMENTS
This research is funded by ASI (the Italian space agency). O. Citterio, L. Maraschi (INAF / Osservatorio Astronomico
di Brera), G. Nocerino and A. Valenzuela (Media-Lario s.r.l., Bosisio Parini, Italy) are acknowledged to support this
work. D. Spiga acknowledges the European Science Foundation - COST office, for the grant that made possible the
training in PPM at the ESRF, and he thanks especially T. Krist (Hahn-Meitner Institut, Germany) and M. Idir
(Synchrotron Soleil, France) for the support. V. Cotroneo is grateful to Fondazione Cariplo for the grant
2003.1995/10.5940. The valuable collaboration of G. Valsecchi, G. Grisoni, M. Cassanelli (Media-Lario s.r.l.) is
acknowledged. We thank C. Ferrari (IMEM-CNR, Parma, Italy) for performing the TEM analysis.
REFERENCES
1.
2.
3.
4.
5.
6.
7.
8.
9.
10.
11.
12.
13.
14.
G. Pareschi, V.Cotroneo, “Soft (0.1 -10 keV) and hard (>10 keV) X-ray multilayer mirrors for the XEUS
astronomical mission”, SPIE Proc. 5168, 53 (2004)
E. Spiller, “Soft X-ray optics”, SPIE Press, 1994
O. Citterio, G. Bonelli, G. Conti, E. Mattaini, E. Santambrogio, B. Sacco, E. Lanzara, H. Brauninger, and W.
Burkert, “Optics for the X-ray imaging concentrators aboard the X-ray astronomy satellite SAX”, Appl. Opt., 27,
1470 (1988).
D. N. Burrows, J. E. Hill, J. A., Nousek, A. A. Wells, A. D. Short, R. Willingale, O. Citterio, G. Chincarini, and
G.Tagliaferri, “Swift X-Ray Telescope”, SPIE Proc., 4140, 64 (2000).
P. Gondoin et al., “Calibration of the first XMM flight mirror module I – image quality”, SPIE Proc. 3444, 278
(1998)
G. Pareschi et al. “The HEXIT (High Energy X-ray Imaging Telescope) balloon-borne mission”, Proc. of the 16th
ESA Symposium on European Rocket and Balloon Programmes and Related Research", St. Gallen (CH) 2-5 June
2003
G.Pareschi et al., "Nickel replicated multilayer optics for soft and hard X-ray telescopes", SPIE Proc. 4012, 284
(2000)
D. Spiga et al., “Development of multilayer coatings (Ni/C – Pt/C) for hard X-ray telescopes by e-beam
evaporation with ion assistance”, SPIE Proc. 5488, 134, in press
D. W. Windt, "IMD: software for modeling the optical properties of multilayer films", Computer in Physics, 12,
360 (1998)
O. Citterio, M. Ghigo, F. Mazzoleni, G. Pareschi, L. Peverini, “Development of soft and hard X-ray optics for
astronomy”, SPIE Proc. 4138, 43 (2000)
V. Cotroneo, G. Pareschi, “Global optimization of X-ray astronomical multilayer mirrors by an iterated simplex
procedure” SPIE Proc. 5536, 05 (present issue)
S. Kirkpatrick, C. D. Gelatt, M. P. Vecchi, “Optimization by simulated annealing”, Science 220, 4598 (1983)
V. Cerny, “A thermodynamical approach to the travelling salesman problem: an efficient simulation algorithm”,
Report, Comenius University; Bratislava, Slovakia (1982)
E. Ziegler, C. Ferrero, F. Lamy, C. Chapron, Ch. Morawe, "Characterisation of nanometer layered structures using
multiple wavelength X-ray reflectometry and simulated annealing data analysis", CPDS 2002, Advances in X-ray
Analysis, 45