Determination of the lattice strain and chemical composition of semiconductor heterostructures by high-resolution x-ray diffraction

In order to determine the strain field of highly mismatched semiconductor heterostructures by high-resolution x-ray diffraction with high accuracy,we derived a new second-order approximation of the incidence parameter considering an arbitrary lattice deformation. Our calculations show that, almost independently from the substrate orientation and the considered material system, for low Miller-index reflections a lattice mismatch greater than 0.004 is the value for which quadratic corrections must be considered. The quadratic approximation increases the range of validity by one order of magnitude, i.e., to a lattice mismatch up to 4%. In addition, the analytical expression which relates the strain components measured by x-ray diffraction to the lattice mismatch is derived for semiconductor epitaxial layers grown on arbitrarily oriented substrate crystals. Using Vegard's rule, our formula allows us to determine the chemical composition of ternary compounds even for low-symmetry substrate orientations. We show that in this case shear strain components have a non-negligible weight in the determination of the chemical composition of ternary compounds. Several examples considering III-V, II-VI, and IV-IV semiconductor material systems are reported and discussed.