Snmek 1

1
Powder x-ray diffraction from multilayer in a
grazing-incidence non-coplanar geometry
L. Horák and V. Holý, Department of Electronic
Structures, Charles University, Prague
X-ray powder diffraction in a non-coplanar
grazing-incidence geometry makes it possible to
determine lateral grain sizes in polycrystaline
multilayers. Due to the low incidence angle close
to the critical angle the refracted beam is
nearly parallel to surface and diffracts on
perpendicular planes. Thats the reason, why we
can obtain lateral grains parameters.
It should be mentioned that it is not only low
angle diffraction. There are comparable
amplitudes of reflected and transmited waves in
the multilayer, so a standing x-ray wave is
generated by the interference of the primary and
specularly reflected beams 1. The grains of a
polycrystalline layer are irradiated by the
standing wave so that the intensity of the
diffracted beam is modulated by the standing wave
pattern. Similarly, the radiation diffracted by
the polycrystalline grains reflects specularly
from the interfaces in the multilayer and another
standing wave pattern results, which affects the
diffracted intensity as well.
Whole these processes in sample we describe using
distorted-wave Born approximation, the goal is to
solve the wave equation. We consider scattering
potential divided to two potentials, undisturbed
system of interfaces without crystal lattice, we
are able to solve exactly using dynamic theory.
This potential leads to wave field in sample 2.
The second potential is given by crystal lattice
and it is a disturbance of the first potential.
Assuming that grains are much smaller than
extinction length and they diffract
kinematically, we solve disturbance using
kinematical theory and wave field given by
undisturbed solution 1.
Changing the incidence angle, we move the
positions of the antinodes of the standing-wave
pattern so here is an expectation of obtaining
vertical profile of the parameters of the
polycrystalline layer. Total intensity are mostly
given by the contribution of grains near
antinodes.
We use this concept for the investigation of
polycrystalline single layers and periodic
multilayers. For both sample types, we have
measured the distribution of the diffracted
intensity in the angular (ai,2Q) plane the
measurements have been carried out using a
conventional laboratory diffractometer allowing
for a non-coplanar scattering geometry equipped
with a polycapillary optics and a secondary flat
monochromator, intensity was integrated over af
in a range given by detectors window. We have
measured these samples using a synchrotron
radiation with position sensitive detector so the
intensity has been measured in af axis too.
Multilayer
Monolayer
SiO2 / 9x ( Si / W ) / quartz s 0.8 0.3
nm rSiO2/9x(Si/W)/quartz, rel 81 / 9x (77
100 / 50 68) / 100 T SiO2/9x(Si/W) 2nm /
9x (44nm 47nm / 11.8nm 13 nm)
Cr/Quartz TCr 54nm, rCr, rel 87, sCr 1.9
nm rsklo, rel 70, ssklo 0.9 nm
The reflectivity curve shows maximas given by
thickness of the periodical silicon-tungsten
bilayer. Total thickness of multilayer causes
high frequency modulation. Blue curve is fit of
red measured reflectivity.
Parameters of the layers such as thickness,
roughness and relative density we have obtained
from x-ray reflectivity.
Red curve shows measured reflectivity with blue
fit. One can see modulation with frequency given
by thickness of chromium layer
Distribution of intensity in directions ai 2Q,
comparision of measured and simulated map
Parameters of multilayer obtained from
reflectivity we have used to calculate wavefield
in sample. It has made possible to simulate a
distribution of intensity. We can see, that
simulated and measured map have both the same
shape and number of maximas.
The measured intensity distributions we have
compared with simulations based on the
distorted-wave Born approximation and kinematical
scattering theory 1,2 to determine lateral
grain sizes.
Distribution of intensity in directions ai 2Q,
comparision of measured and simulated map
Parameters of chromium layer from reflectivity
shows that surface is quite rough. With inclusion
od this roughness we simulated distribution od
intensity. As one can see on reflectivity curve,
it has no maximas caused by periodicity because
it is one layer, so in non-coplanar GID measured
map we can see only mudulation from total
thickness. There is one maximum corresponding to
critical angle of chromium. Measured and
simulated map have the same shape, there is
discrepancy in ai position. It is most probably
caused by inaccurate adjustment of sample
measured with coplanar laboratory diffractometer
in non-coplanar geometry.
Measured bragg peak
Computation of standing wave in sample
af 2Q scan - simulation
These scans shows dependence of intensity on
angle exit (incidence) angle with constant 2Q.
Simulations and measurements have the same number
and positions of maximas, but there is a
difference in shape of curves. The main maximum
corresponds to incident angle equal with exit
angle.
2Q - af - measurement
Measured bragg peak
Computation of standing wave in sample
We can see that width od Bragg peak is the same
for various incident angle. There is no strong
modulation of standing wave in monolayer as in
multilayer, all the grains has almost the same
contribution to diffracted intensity. Lateral
grain size is (9.62)nm. From coplanar powder
diffraction with constant angle 1 we have
determined using Williamson-Hall plot size of
grains in perpendicular direction, it is
(6.21.5)nm.
The standing wave in sample changes its frequency
for different incident angle and also moves its
antinodes. The lateral grain sizes profile can be
detrmined by fitting of measured bragg peak with
various incident angle. In this case lateral
grain size is (102)nm and it doesnt change with
depth.
Coplanar powder difraction
From coplanar powder diffraction with constant
angle of incidence we have measured grain sizes
in a direction perpendicular to the surface.
Coplanar powder difraction
We have measured coplanar powder diffraction with
constant incident angle 2 to determine grain
size. Perpendicular size to surface of tungsten
grains is (4.61)nm from Williamson-Hall
plot. Lateral size is greater than perpendicular,
moreover perpendicular size agrees with tungsten
layers thickness. This means that tungsten
grains grow through layer and they are
cylindrical.
Comparison of simulated and measured map shows
agreement in shape and positions of main maximas,
which confirms assumpted model of diffracting
standing wave (interference of transmited and
reflected waves in sample). This method allows to
measure lateral grain size and its depth profile
in multilayers, it is suitable for structures
with thin layers, where conventional coplanar
powder diffraction gives measurable intensity
only for small angle of incidence. Noncoplanar
GID has sensitivity to multilayers parameters
grater than method of x-ray reflectivity.
Reference 1 P. F. Fewster, N. L. Andrew, V.
Holý, and K. Barmak, Phys. Rev. B 72, 174105
(2005). 2 U. Pietsch, V. Holý, and T.
Baumbach, High Resolution X-Ray Scattering - From
Thin Films to Lateral Nanostructures (Springer,
2004).