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

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-
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