Stress Distribution and Critical Thicknesses of Thin Epitaxial Films
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STRESS DISTRIBUTION AND CRITICAL THICKNESSES OF THIN EPITAXIAL FILMS S. SHARAN, K. JAGANNADHAM AND J. NARAYAN Dept. of Materials Science and Engineering North Carolina State University Raleigh, N. C. 27695 ABSTRACT We have calculated stress distribution and critical thickness for the growth of strained epilayers having various values of mismatch between the epilayer and the substrate. The present analysis has been carried out assuming that the nucleation of a misift dislocation is controlled by the activation energy. Further, a misfit dislocation is nucleated when the areal strain energy density of the coherent film exceeds the activation energy associated with the misfit dislocation S9nfiguration. The latter term has been deternmined using the discrete dislocation method in conjunction with a surface dislocation analysis. The surface dislocation configuration is obtained by minimizing the total energy associated with the two-phase medium. It is verified that the free surface boundary conditions and the continuity of stresses acrosss the interface are satisfied by the surface dislocation array. These energy calculations are expected to be more accurate than those performed by previous workers. INTRODUCTION The possibility of growing high quality epitaxial layers of different materials on lattice mismatched semiconductor substrates is a topic of considerable interest. The range of useful devices available within a given substrate is considerably enhanced by this method. During epitaxial growth if the misfit between a substrate and the growing epilayer is small, the first atomic layers which are deposited will be strained to match the substrate lattice and a coherent interface is formed. However, as the thickness of the growing epilayer increases, the homogeneous strain energy of the epilayer becomes large enough to make the nucleation of misfit dislocations energetically favorable. The thickness of the epilayer at which misfit dislocations will be nucleated to relieve the homogeneous strain energy corresponds to the critical thickness for the system. The existence of the critical thickness was first considered by Van der Merwel, 2 , later by numerous authors icluding Matthews et. al. 3 -5 and People and Bean 6,7 . Van der Merwe 2 initially determined the critical thickness of a lattice mismatched epilayer by equating the areal strain energy density in the film with the interfacial energy per unit length between the film and the substrate arising as a result of the generation of interface misfit
Mat. Res. Soc. Symp. PRoc. Vol. 91. ý 1987 Materials Research Sociely
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dislocations. The interfacial energy was calculated using the Peierls - Nabarro potenial to represent the interatomic forces at the interface. However, the value of the interfacial energy obtained becomes significantly inaccurate when the misfit dislocation spacing is greater than twice the thickness of the epilayer. Specifically, the expression for the interfacial energy in the analysis has a term which Van der Merwe calls" the shear modulus of the
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