Role of the strained substrate in the x-ray diffraction of free-standing epitaxial nanostructures under grazing incidence conditions

Semiconductor heteronanostructures are of interest for a vast field of applications in optoelectronics. Fast and reliable characterization of their structural properties is of high importance for industry and science. By the example of free-standing In0.6Ga0.4As quantum dots (QDs), we discuss the contributions of the strained epitaxial surface nanostructures and the substrate to the scattered x-ray intensity under grazing incidence conditions. In the frame of the distorted wave Born approximation, we take into account multiple scattering and coherent interactions of the x-ray beams involved in the diffraction process. Employing the finite-element method, we give a detailed analysis of the strain distribution inside the nanostructure and the substrate. Further, analyzing the experimental and simulated x-ray diffraction patterns in x-ray grazing incidence diffraction, we demonstrate the importance of accounting for strain in the substrate and interference effects between the surface and the QDs for strain analysis of the nano-objects. The advantages of our approach are demonstrated by comparison with the standard isostrain approach that is commonly used for strain analysis of the nanostructures. We show that neglecting the substrate contribution to the x-ray scattering causes errors in the standard isostrain approach. Finally, the substrate contribution itself may yield information sufficient for a novel and fast approach for the structural characterization of strained semiconductor heteronanostructures.