Stress and dislocations at cross-sectional heterojunctions in a cylindrical nanowire

Abstract

The limit of crystal lattice coherency of a cross-sectional heteroepitaxial junction in a nanowire is calculated in terms of the critical nanowire radius R _c, based on a general calculation of elastic stresses in a long cylindrical rod. R _c is derived from the kinetics of a possible misfit dislocation which can slip in the heterointerface and rest in an energetic minimum, if it occurs at all, regardless of whether it is of zero total energy as assumed in the literature. A close comparison is made with the known models for the critical radius of a dislocation half-loop and the critical thickness h _c of a heteroepitaxial film, where all models are refined by including the energy of the slip step formed or accidentally annihilated. For a symmetrical, abrupt heterojunction, we obtain R _c as a lower and, therefore, quite safe limit, about five times larger than h _c of a comparable thin film. An even larger R _c is found for junctions of finite transition width instead of abrupt transitions. It is estimated that nucleating the dislocations in a perfect nanowire is difficult, in agrement with experimental reports of dislocation-free nanowires with R well above the theoretical R _c.

Acknowledgements

We would like to thank U. Messerschmidt, Max Planck Institute of Microstructure Physics, Halle, for critically reading the manuscript, and W. Seifert, Martin Luther University of Halle, as well as H.-G. Maschke, Fraunhofer Institute for Mechanics of Materials, Halle, for useful hints on the theory of thermal stress.