The stresses due to arrays of inclusions and dislocations of infinite length in an anisotropic half space - Application to strained semiconductor structures

Exact closed-form solutions are derived for the stresses due to periodic arrays of dislocations and arbitrarily polygonal inclusions of infinite length in a generally anisotropic half space. These solutions allow the analysis of the stability of a range of strained semiconductor structures. Critical thickness calculations are performed for unburied strained layers and for buried strained layers and quantum wire arrays in the GeSi/Si system. For strained layers (buried and unburied) these are in agreement with experiment. The results for quantum wires suggest that, once buried, they may be able to support, without loss of coherency, up to seven times the lattice mismatch that may be accommodated by a strained layer of comparable thickness. Comparison of these results with those obtained in the isotropic approximation show that for the GeSi system the effect of anisotropy is to increase the predicted critical thickness by more than 30%. Dislocation formation above the critical thickness is studied for unburied strained layers. A stability criterion is presented, based on modelling the dislocation distribution as periodic and considering the driving force on a threading segment gliding through the periodic distribution. A closed-form solution for this driving force is presented, taking the effects of anisotropy fully into account. The configuration is defined to be stable when there is no path of positive driving force for the threading segment through the periodic distribution. The stability criterion yields results for the equilibrium dislocation density that are in reasonable agreement with experiment. In contrast with approaches that incorporate only the mean stress due to the background dislocation distribution, the present approach predicts that for a given layer thickness the equilibrium residual strain should depend upon the initial strain due to the lattice mismatch.