X-ray scattering simulations

Initially, a 3D model of the nanostructure is created, which is based on shape information derived from direct scanning probe techniques. Further, we perform FEM calculations on the model. The FEM considers the full elastic anisotropy and the lattice mismatch governed by the chemical composition. Finally, the displacement field

**u** derived by FEM [

13] may serve as an input for the kinematical scattering simulations (see further below).

As a second input for our nanofocus scattering simulations, we use a beam spot profile which, on the one hand, can be set to any artificial intensity distribution for the desired spot shape. On the other hand, the spot profile can be obtained experimentally on the basis of the actual illumination function, e. g., by X-ray ptychography. Formerly developed for electron microscopical applications [

14,

15], ptychography has been rediscovered for the purpose of scanning X-ray microscopy during the last years [

16-

19]. Ptychography consists of a measurement where a coherent focused beam scans the sample with a step size smaller than the beam diameter, which leads to a spatial overlap of the successively illuminated areas. For each spot position, the scattered intensities are captured by a 2D detector in far-field geometry. From the redundant information about the scanned object within the recorded scattering intensities, due to the spot overlap, the real space image of the object’s structure and the 3D wave field of the focused beam can be calculated.

A picture of an illumination function is shown in Figure , which was gained using ptychography before performing one of the diffraction experiments. To characterize the nanobeam, a ptychogram of a test sample was taken (for details on such a measurement, see [

20]). Figure a depicts the focal plane within the beam caustic, whereas Figure b shows a horizontal line cut through the center of Figure a revealing a beam size of 100 nm full width at half maximum (FWHM). Besides the amplitude values (pixel brightness), Figure a also contains information about the phase within the beam encoded by the hue value (see scale). As focusing optics for hard X-rays, we used refractive nanofocusing lenses made of silicon [

8]. The cross-like vertical and horizontal oscillations VA and HA in Figure a are diffraction phenomena because of the lens aperture and appear slightly axially rotated corresponding to a tilt of the lenses (visible at the dotted line in Figure a) [

20].

In the scattering simulations, we realized the nanofocus spot by introducing a set of weighting factors, 0≤

*w*_{n}≤1, that determine the amplitude of the incident beam for each scatterer

*n*,

*n**N* with 1≤

*n*≤

*N*,

*N* number of scatterers. The factors

*w*_{n}are calculated from the brightness of each RGB pixel within the beam profile image (e. g., Figure ) and, thus, act mathematically as a ‘finite support’ which selects only scatterers within the sample model that should be illuminated by the nanofocused beam. With this approach, beam profiles of different origins can be handled, e.g., experimentally gained profiles (Figure ) or artificially created profiles (for instance, perfect disks or rectangles). The 3D model of the FEM calculations has been used as an input, which yields the positions of each scatterer

**r**_{n} = (

*r*_{x,n},

*r*_{y,n},

*r*_{z,n}) as well as the displacement field

**u**(

**r**_{n}) caused by strain relaxation.

To calculate *w*_{n}for each scatterer within the 3D model, it is necessary to correlate the positions of the scatterers **r**_{n} with the pixel coordinates *p*_{1}, *p*_{2}of the 2D beam profile. Using the geometry and angle definitions illustrated in Figure a, this has been done by a projective transformation formula as follows (in round numbers):

Scaling factor *s* in units of ‘pixels per model length unit’ correlates the length scales of the model and the beam profile; thus, *s* also sets the simulated beam spot size. Constants *c*_{1} and *c*_{2} define the beam center within the beam profile pixel array.

Using Equation 1, the pixel brightness

*b*(

*p*_{1},

*p*_{2}) can be identified with

*b*(

*r*_{n}). Considering a spot brightness minimum

*b*_{min} and maximum

*b*_{max}, we obtain a weighting factor

*w*_{n}=

*b*(

*r*_{n})/(

*b*_{max}−

*b*_{min}) for each scatterer

*n*. In case Equation 1 yields values for

*p*_{1},

*p*_{2} that are beyond the beam profile pixel array,

*w*_{n}is set to zero. The beam phase

*ϕ*_{n}at each scatterer is calculated from the beam profile as well. For that, the pixel hue value

*h*_{n}*h*(

*p*_{1},

*p*_{2}) is multiplied by the number of periods resulting from the distance between scatterer

*n* and a fixed plane orthogonal to the beam path in units of the wave length

*λ*:

If we include the phase

*ϕ*_{n}and weighting factors

*w*_{n}into the coherent sum of the kinematical scattering, e. g. reference [

11], the intensity results in:

The sum runs over all N ‘basic cells’ which the model is consist of (see [

11]). Multiple scattering, refraction, and absorption within the crystal have been neglected. The values for the scattering vector

**q** are calculated from the incidence angle

*α*_{1} and the corresponding azimuth angle

*β* as well as the exit angle for diffusely scattered intensities

with its respective azimuth angle

*γ*^{′}:

wherein the angles

and

*γ*^{′}of diffuse scattering are derived geometrically from the specular beam (at

*α*_{2},

*γ*) and the position of the detector pixels (

*d*_{1},

*d*_{2}) in 3D real space. We applied the nanospot diffraction simulation to SiGe/Si(001) islands, SiGe/Si(001) dot molecules, and InGaAs/GaAs(001) quantum dot molecules using spot sizes of 200, 250, and 100 nm, respectively (see the succeeding sections).