By using the superposition principle, and Fourier-integral methods, a study was made of the energy of a dislocation array in a strained epitaxial layer that was deposited onto a finite substrate having the same elastic constants. The total energy included the self-energy of the dislocations, the strain energy which arose from the lattice mismatch and the interaction energy between dislocations and mismatch. The sum of the self-energy and the interaction energy constituted the dislocation formation energy. A zero formation energy was used as a criterion for determining the critical epilayer thickness. No dislocation could appear when the epilayer thickness was below this critical value. When the epilayer thickness equalled the critical thickness, and the dislocation density was extremely low, the total energy was independent of the dislocation spacing. If the critical thickness was less than the substrate thickness, and the epilayer thickness was greater than the critical value, the total energy was a local minimum at a certain dislocation spacing. The corresponding dislocation density was the critical dislocation density. When the spacing was greater than the minimum-energy value, the total energy decreased by decreasing the dislocation spacing (increasing the dislocation density). The total-energy curve near to the minimum-energy value changed to a steep valley when the epilayer thickness approached the substrate thickness. This corresponded to the experimental observation that a rapid relaxation of misfit strain occurred when the epilayer thickness increased to a sufficient value. If the dislocation spacing was below the minimum-energy value, the total energy increased markedly with decreasing dislocation spacing. This implied that work-hardening was inevitable, due to dislocation-dislocation core interaction.

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