Elastic relaxation of coherent epitaxial deposits
Résumé
The epitaxial contact between a three-dimensional (3D) deposited crystal A and its lattice mismatched substrate B may be coherent and remain coherent during the elastic relaxation of the 3D crystal. In this case, during the relaxation, the 3D crystal drags the atoms of the contact area and produces a strain field in the substrate. In this paper we calculate the equilibrium stresses and strains in the substrate as well as in the deposit by using a method initiated by Hu [J. Appl. Phys. 50 (1979) 4661] where continuous forces that the deposit exerts on its substrate are calculated in a self-consistent way. Approximated analytical solutions for equilibrium stresses and strains are given and discussed for a thin infinite deposited ribbon. These results are extended with some approximation to thick ribbons. For nanostructures, size effects have to be taken into account. In this case we show that the two facing crystals no longer share the natural misfit mo but share a so-called active misfit m involving the surface stresses of A. This surface stress may enhance, compensate or decrease the epitaxial misfit mo. Furthermore, since surface stresses are adsorption sensitive, a surface adsorption may force the equilibrium strains and stresses to change in a reversible fashion. For a periodic set of ribbons, the induced substrate strains may overlap and a back stress effect increases the strain in the ribbons when the ribbons grow laterally. Owing to these overlapping fields, the ribbons communicate. Just before coalescence, the substrate strain in between the ribbons increases dramatically and the ribbons become pseudomorphous with the substrate. Our theoretical results for ribbons are compared with experimental ones obtained on islands of Ge0.ssSiols/Si(001), InAs/GaAs and then on observation of in situ growing two-dimensional islands of ln0.sGao.sAs/GaAs. In all cases an acceptable agreement is obtained between our ribbon calculations (without any adjustable parameter) and experimental results on islands. Finally, we calculate the minimal elastic energy stored by a deposited-isolated ribbon. By extension we give that for one isolated island in a simple approximation. A consequence is that the relaxation by striction of the substrate may be the driving force for the Stranski Krastanov growth mode. @ 1997 Elsevier Science B.V.