Variational method applied to determine the lattice parameter profiles in semiconductor heterostructures

We have obtained the lattice parameter profiles for Ge/Si (1 1 n) and InAs/GaAs (1 1 n) heterostructures using the elasticity continuum theory in the linear approximation. In our model we assume that a small fraction of the substrate participates in the heterostructures relaxation in the non-rigid approximation. Minimization of the free energy by the Euler-Lagrange method allows us to predict the evolution of the lattice parameter with the film coverage. A sigmoidal-like law for the lattice parameter profile is observed in the rigid and non-rigid approximations. This behaviour is qualitatively similar to that obtained for lower Miller indices in Ge/Si and InAs/GaAs heterostructures. As the aspect ratio changes, we observed a significant dependence of the lattice parameter slope basically for higher aspect ratio values. Therefore, the difference in the depletion parameter does not sensitively affect the lattice parameter changes for higher aspect ratios in some range of investigated materials. So the aspect ratio is seen to play a key role in relaxation mechanisms and can explain different shapes observed in the formation of heterostructures. From these results, we have deduced that the aspect ratio and subsequently the misfit strain and substrate orientations are the key parameters in relaxation processes, since they define changes in the lattice parameter with the height of deposited film.