Determination of bulk mismatch values in transmission electron microscopy cross-sections of heterostructures by convergent-beam electron diffraction

The relaxation which occurs along the thinning direction of transmission electron microscopy (TEM) cross-sections of heterostructures is still poorly known. This has so far prevented the accurate determination of the corresponding bulk mismatch values from convergent-beam electron diffraction (CBED) patterns. In this paper it is demonstrated that, by using elasticity theory, it is possible to deduce a simple relationship for (001) heterostructures which relates the lattice mismatches along the different crystallographic directions of the TEM specimen, as deduced from a single CBED pattern, to the bulk value. Both [001] and [110] orientations of the TEM cross-sections are considered. However, to obtain accurate results: the validity of the kinematical approach used to deduce mismatches from high-order Laue zone line patterns must be critically checked; it depends on the crystallographic projection, on the beam voltage and, in the case of Si1-xGex heterostructures: on the Ge concentration. It is found that the best results are obtained at 100 kV, working in the [013] projection, and for Ge concentrations up to 20 at.%. The method has been applied to both uniform and graded Si1-xGex/Si heterostructures, as well as to a B+-implanted, epitaxially regrown Si wafer. Good agreement is found between the CBED results and the bulk values obtained from double-crystal X-ray diffractometry.