Elastic and plastic anisotropy during growth from the melt of single semiconductor crystals

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Abstract
The effect on dislocation density of elastic anisotropy (relation of growth direction to elastic stiffness tensor of cubic symmetry) and of plastic anisotropy (relation of growth direction to crystallographic slip systems of {111} ≪110≫ type) is investigated for the growth from the melt of shaped, single-crystal semiconductors of III–V compounds (typically GaAs, InP) by the Czochralski method. The thermal stresses are determined numerically by using three-dimensional finite element techniques for growth along the ≪100≫, ≪111≫ or ≪110≫ directions. The equivalent shear stress is calculated and it is assumed that the dislocation density is proportional to the sum over all slip systems of the excess of the resolved thermal shear stress over the crystal's yield stress in shear. It is shown that proper account of both elastic and plastic anisotropy is necessary in order to correlate numerical estimates of dislocation density to experimentally determined patterns. In particular, it is shown that the effect of elastic anisotropy is to produce significantly higher dislocation densities, especially for the case of ≪111≫ and ≪110≫ growth. For moderate levels of elastic anisotropy, the dislocation density levels during ≪111≫ and ≪110≫ growth become significantly larger than density, during ≪100≫ growth, at specific locations within the crystal. The correlation between dislocation density and equivalent shear stress is presented, and the effect of plastic anisotropy is thus discussed. It is shown that the assumption of elastic isotropy leads to dislocation density levels that considerably underestimate the levels but, for a given growth direction, are qualitatively similar to the dislocation density patterns corresponding to the three-dimensional numerical solutions in which elastic anisotropy is accounted in full.