Chapter 4: Strain and composition determination in semiconducting heterostructures by high-resolution X-ray diffraction

In this chapter, the basics of the methods for measuring the strain and the composition in two- and zero-dimensional structures by means of high-resolution X-ray diffraction techniques and laboratory X-ray sources are presented, with the aim of introducing these techniques to the reader. The main physical properties for defining the strain state of a heterostructure are given and the basics of the elasticity theory for cubic and hexagonal crystals are also introduced. The X-ray diffraction method for determining the composition in semiconducting alloys is explained, allowing to conclude that the lattice parameter measurement method is one of the most accurate way to determine the composition, provided that the composition versus the lattice parameter dependence is known. The comparison between composition values obtained from X-ray diffraction method and that determined by other analytical techniques has allowed to measure a deviation from the linear Vegard law in several semiconducting alloys. The experimental determination of the deviation of the Vegard law in the InGaAs alloy is reported. The methods of asymmetric diffraction geometries and reciprocal lattice maps to characterize lattice-matched and lattice-mismatched heterostructures are briefly introduced. Some of the most cited theories describing the strain release in semiconductor heterostructures are introduced, even if a semi-empirical approach has to be used to fit the experimental data. By using a method similar to that proposed by Tersoff, the theory is extended to composition graded heterostructures, which are of great interest for obtaining virtual substrates or strain engineered heterostructures. The theoretical models are then compared with experimental results for GaAlSb/GaSb single heterostructures and InGaAs/GaAs composition graded heterostructures. Finally, the kinematical theory of X-ray scattering from quantum dot (QD)-based heterostructures is briefly introduced and discussed. The independent determination of strain and composition profiles in QD heterostructures is a very complex task as laboratory X-ray sources do not allow to change the X-ray wavelength. The characterization of heterostructures containing QDs is based on the comparison between simulations of X-ray scattering based on finite element calculations of the heterostructure containing the dots and reciprocal lattice maps obtained in different symmetrical and asymmetrical geometries. A satisfactory agreement between experimental and simulated reciprocal lattice maps of samples containing several stacks of InAs/GaAs QD is reported.