The superposition principle and the Fourier integral methodology were used to study the energy of a dislocation array in a strained epitaxial layer which was deposited onto a finite substrate having the same elastic constants. The total energy consisted of the self-energy of the dislocations, the strain energy which arose from the lattice mismatch plus the interaction energy between dislocations and mismatch. The sum of the self-energy and the interaction energy constituted the dislocation formation energy. The criterion of zero formation energy was used to determine the epilayer critical thickness, hc. No dislocations could appear when the epilayer thickness was below hc. When the epilayer thickness was equal to the critical thickness, and the dislocation density was extremely low, the total energy was independent of the dislocation spacing, p. When the critical thickness was less than the substrate thickness, and the epilayer thickness was greater than the critical thickness, the total energy went through a local minimum at a dislocation spacing of p = pmin. Moreover, the corresponding dislocation density was the critical dislocation density. When p was greater than pmin, the total energy decreased upon decreasing the dislocation spacing (increasing the dislocation density). The total-energy curve near to p = pmin changed to a steep valley when the epilayer thickness approached that of the substrate thickness. This corresponded to the experimental observation that a rapid relaxation of misfit strain occurred when the epilayer thickness attained a sufficient thickness. When p was less than pmin, the total energy markedly increased upon decreasing the dislocation spacing. This implied that work-hardening was inevitable; due to dislocation–dislocation core interactions.

Energy of an Array of Dislocations in a Strained Epitaxial Layer Deposited on a Finite Substrate. S.D.Wang: Journal of Applied Physics, 2000, 88[12], 7089-94