Multiple scattering effects in strain and composition analysis ... - IEMN

APPLIED PHYSICS LETTERS 94, 013112 2009
Multiple scattering effects in strain and composition analysis
of nanoislands by grazing incidence x rays
M.-I. Richard, 1,2 V. Favre-Nicolin, 1,3 G. Renaud, 1,a T. U. Schülli, 1 C. Priester, 4 Z. Zhong, 5
and T.-H. Metzger 2
1 Institut Nanoscience et Cryogénie/SP2M/NRS, CEA Grenoble, 17 Avenue des Martyrs,
F-38054 Grenoble Cedex, France
2 ID01/ESRF, 6 rue Jules Horowitz, BP220, F-38043 Grenoble Cedex, France
3 Université Joseph Fourier, BP 53, 38041 Grenoble Cedex 9, France
4 Département ISEN, Institut d’Electronique de Microélectronique et de Nanotechnologie, UMR CNRS 8520,
Avenue Poincaré, 59652 Villeneuve d’Ascq Cedex, France
5 Institut für Halbleiter-und Festkrperphysik, Johannes Kepler Universität Linz, Linz 4040, Austria
Received 8 October 2008; accepted 12 December 2008; published online 9 January 2009
Experiments and numerical simulations based on finite element modeling show that the x-ray
intensity scattered by comparatively large nanostructures on a substrate is not simply related to their
strain in experiments using either grazing incidence or exit because of multiple scattering effects.
However, whatever the nanostructure size, the composition profiles are correctly extracted from
grazing incidence multiwavelength anomalous scattering. These effects are illustrated for the
structural analysis of Ge dome-shaped islands grown on Si001. ©2009 American Institute of
Physics. DOI: 10.1063/1.3064157
a Electronic mail: gilles.renaud@cea.fr.
The adjustment of the electronic and optical properties
of nanoislands or nanowires requires a quantitative analysis
of their dimension, strain, and composition, which is a subject
of constant interest. X-ray scattering is widely used for
that sake. 1 Composition analysis can also be performed using
multiwavelength anomalous diffraction MAD. 2 For large
nanoislands with sufficiently large out-of-plane strain profiles,
the isostrain x-ray scattering method 3 can be combined
with anomalous diffraction to determine directly the out-ofplane
strain and composition profiles. However, often the
scattered intensity is assumed to directly reflect the strain
state since it is assumed to be its squared Fourier transform:
the kinematical or Born approximation is applied to extract
the strain state and composition of nano-objects. 4,5 The question
arises to what extent this hypothesis is correct since the
x-ray beam has to be kept grazing with respect to the surface
to avoid background scattering from the substrate, resulting
in multiple scattering effects. In this letter the multiple scattering
effects resulting from the grazing incidence geometry 6
around in-plane Bragg reflections are investigated both experimentally
and numerically, focusing on islands grown using
the Stranski–Krastanow growth mode. 7 They are illustrated
for dome- and pyramid-shaped islands in the widely
studied Ge/Si001 system.
Deposition of 6 ML of Ge at 650 °C on Si001 by
molecular beam epitaxy resulted in fairly monodispersed
124 nm wide and 20 nm high domes, as found by atomic
force microscopy. Grazing incidence x-ray diffraction measurements
wavelength =1.0972 Å as illustrated in the inset
of Fig. 1 were performed at the BM32 and BM02 beamlines
at the European Synchrotron Radiation Facility,
Grenoble. Strain sensitive radial scans in the vicinity of the
Si400 in-plane Bragg reflection were performed as a function
of the Miller index h,0,0 h=2a/sin2/2, a being
the Si lattice parameter and 2 the scattering angle, for different
grazing incident angles i close to the critical angle
for total external reflection, c =0.156°. The scattered intensity
along the exit angle f was integrated from 0° to 1.5°
using a position sensitive detector.
The intensity profile as a function of h Fig. 1 is found
to vary drastically with i . With increasing i , the position of
the scattering peak shifts toward the Si Bragg peak. Above
the critical angle, for i =0.2° and i =0.25°, a second peak
hardly visible because of broadening due to the inhomogeneous
island size and the related inhomogeneous strain
distribution develops at lower h values h3.925 for i
=0.2°, the origin of which is given below. As all the curves
cannot be superimposed by rescaling, the intensity profile
cannot be directly related to the strain state distribution in the
nanoislands by a simple Fourier transform.
FIG. 1. Color online Experimental radial scans around the Si400 Bragg
peak for Ge dome-shaped islands on Si001 for various values of i . Note
the variation in the Ge peak position from h=3.95 to 3.965 as the incident
angle increases. The inset displays the experimental setup. The incident and
exit wave vectors k i and k f are kept grazing angles i and f with respect
to the surface. The scattering angle 2 is the in-plane angle between them.
The one dimensional detector slit setup allows summing the scattering contributions
along f .
0003-6951/2009/941/013112/3/$23.00
94, 013112-1
© 2009 American Institute of Physics

013112-2 Richard et al. Appl. Phys. Lett. 94, 013112 2009
FIG. 2. Color online a Simulated radial scan intensities around the
Si400 reflection for Ge islands on Si001 whose strain has been calculated
by FDM: a 124 nm wide and 20 nm high dome. The intensity is integrated
between f 0° and f 1.5°. The total DWBA intensity is reported for
different incident angles and compared to the BA first path intensity. The
substrate scattering has been omitted. b Evolution of t i ,z 2 inaGe
dome on Si001 as a function of height z and i / c for =1.097 Å. As the
intensity scattered by each z-layer above the surface is proportional to
t i ,z 2 , an x-ray scattering experiment will predominantly reflect the
strain state of a given height depending on i rather than the full island
strain state distribution. c Same as a but for a 70 nm wide and 7 nm high
pyramid.
To understand this effect, four paths according to the
distorted wave Born approximation DWBA have to be
taken into account, 8,9
F DWBA =
f j e ik f −k i ·r j + r i f j e ik f +k i ·r j
jQDs
jQDs
+ r f f j e i−k f −k i ·r j
jQDs
+ r i r f f j e i−k f +k i ·r j,
jQDs
where f j is the atomic scattering factor and r j is the position
of an atom j in the island; k i and k f are the incoming and exit
wave vectors and r i,f are the reflectivities of the incident
and scattered waves. The first term in Eq. 1, which we call
first path, corresponds to the kinematical scattering of the
incident wave by the islands or Born approximation BA.
The three other terms imply single or double reflections by
the substrate. 8
To interpret these observations, the strain field in a coherent
Ge/Si001 dome-shaped island with the experimental
width and height was calculated with the finite difference
method FDM, 10 assuming continuum theory of elasticity.
For simplicity, a pure Ge dome was modeled. The scattered
intensity see Fig. 2a was then calculated using Eq. 1
around the in-plane Si400 reflection for different values
of i around c and summed up between f 0° and f
1.5° to account for the experimental conditions. The island
form factor given by the BA shows two broad peaks that
arise from two prevalent strain states in the dome-shaped
island: the first peak close to the Si Bragg peak corresponds
to the basis and also to the main part of the island, which is
surrounded by the 113 and 15323 facets, while the second
peak results from the strain state of the top part of the
island, terminated by 105 facets. For i =0.62 c , only one
1
maximum is present in the simulated radial scan. It is positioned
at h=3.944, i.e., displaced compared to the real
maxima h=3.925 and 3.954 of the strain state given by the
first path. Above the critical angle, the positions of the intensity
maxima are close to the real positions of the maxima of
the strain state given by the first path.
To discuss these effects, the islands can be decomposed
into layers of different heights z, each having its own lattice
parameter az. With this hypothesis, a radial position h corresponds
to the scattering by a given layer at z. We also
suppose case of Ge islands on Si that the higher z, the
larger the lattice parameter and hence the smaller h. Within
the DWBA and as k z i,f = 2/ sin i,f , for small incident
and exit angles, the amplitude scattered at a given h by the
portion of the island at height z can be written as 11
F z DWBA i , f = F z BA h, i , f t i ,zt f ,z,
where F z BA is the island form factor at height z and t,z
=1+re −4iz/ is a generalized optical function. t i ,z
varies as a function of i and z, as illustrated in Fig. 2b.
Hence, the intensity distribution in Fig. 2a results from the
product of the island form factor by the generalized optical
function. Because z is directly correlated with h, the optical
function varies along the radial direction. If the value of i is
below or at the critical angle of the substrate, the optical
function can be very strongly enhanced for certain heights z
and thus for the corresponding radial positions h. Low incident
angles maximize the generalized optical function for
high z values and thus maximize the scattered intensity at
lower h values. For i =0.62 c , this maximum corresponds
to z20 nm see Fig. 2b, i.e., the intensity scattered by
the island’s top, where it is the most relaxed, is strongly
enhanced. Above the critical angle, the optical function does
not display strong minima and maxima; the multiple scattering
effects can be neglected, and the kinematical approximation
is recovered. The condition for these DWBA effects to
be significant is 11
= rzH2/ahd /dz 1,
where H is the total height of the nanocrystal, z is the
parallel strain of an isostrain area at height z, and rz is its
radius. Hence, this phenomenon is observed only for large
enough islands, with large enough vertical strain gradient.
For the simulated Ge dome, 10. For smaller islands with
smaller strain gradient, these multiple scattering effects are
not observed. This was checked in the case of a small Ge
pyramid on Si001 see Fig. 2c for which the intensities
simulated for different i values can be superimposed. Indeed,
in that case, the DWBA analysis for islands on a substrate
is identical to that of islands buried below the surface:
the total scattered amplitude see Eq. 2 is proportional to
the form factor amplitude, F BA k f −k i , according to
F DWBA i , f = F BA k f − k i t i t f ,
where t i,f is the usual amplitude transmission function as
a function of incident exit angle.
Having determined for which conditions the DWBA has
to be used instead of the BA, the question arises whether the
standard MAD analysis using the BA is still valid for large
islands under grazing incidence. Equation 2 shows that the
intensity scattered by a stripe at height z is simply the BA
intensity multiplied by the incident and exit generalized op-
2
3
4

013112-3 Richard et al. Appl. Phys. Lett. 94, 013112 2009
Ge content
1
α
0.8
i
=0.12°
α i
=0.2°
0.6
0.4
0.2
0
3.9 3.92 3.94 3.96 3.98
h (r.l.u.)
FIG. 3. Color online Ge content as a function of h as extracted using
grazing incidence MAD with 12 different energies around the Ge K-edge.
The Ge content extracted for two different incident angles remains consistent
in spite of the strong Ih dependence with i .
tical functions, which are energy independent far from c in
the case of a buffer containing nonanomalous atoms. Therefore
during a MAD experiment the extracted structure factors
are F Ge and F Si multiplied by these functions. Since the Ge
composition is extracted by a ratio of these factors according
to x Ge = f Si F Ge /f 0 0
Ge F Si + f Si F Ge , where f Ge and f Si are the
Ge Thomson atomic form factor and the Si atomic form factor,
respectively, the generalized optical functions disappear
from the result, and the deduced composition is not affected
by multiple scattering effects. This result was experimentally
checked by comparing the results of grazing-incidence MAD
analysis of the dome sample for different incident angles.
Figure 3 shows the Ge composition calculated from the Ge
and Si structure factors F Si and F Ge , which were extracted
for two incident angles without any structural hypothesis,
and refined through a least-squares minimization using
a dedicated program. 12 This program evaluates the uncertainties
of F Ge and F Si and hence the error bars on the Ge
composition. The partial structure factors are affected by the
generalized optical function as is the total scattering amplitude.
However, only slight variations in the Ge composition
are observed, with overlapping error bars. Hence, the correct
Ge content is extracted, whatever the conditions of incidence.
A BA treatment of anomalous diffraction can thus be
performed to extract the composition of isostrain regions of
the islands. The subcritical regime i c yields smaller
scattering from the substrate and is thus preferred.
To conclude, for large islands, multiple scattering effects
strongly distort the intensity distribution in reciprocal space
and need a complete DWBA analysis to extract the lattice
parameter distribution. By contrast, the in-plane strain in
small islands can be investigated using a BA treatment.
Hence, scattering from small and large islands can be distinguished
simply by varying the incident angle around the
critical angle c , checking if the intensities can be superimposed
to each other by rescaling. This may be most useful for
bimodal island growth e.g., pyramids and domes in the
Ge/Si case to discriminate the origin of the measured intensity.
Most importantly, the standard MAD analysis can be
applied to all islands as long as the x-ray energy is far from
an absorption edge of the substrate and the substrate does not
contain any elements violating this condition.
Z.Z. acknowledges support from the FWF, Vienna. We
acknowledge Hubert Renevier for fruitful discussions.
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