Energy of an array of dislocations in a strained epitaxial layer deposited on a finite substrate

Abstract

By adopting the superposition principle and Fourier integral methodology, this work studies the energy of a dislocation array in a strained epitaxial layer deposited on a finite substrate with the same elastic constants. The total energy comprises the self-energy of the dislocations, the strain energy arising from the lattice mismatch, and the interaction energy between the dislocations and the mismatch. The sum of the self-energy and the interaction energy constitute the dislocation formation energy. Zero formation energy is used as the criterion to determine the epilayer critical thickness hc. No dislocation can appear when the epilayer thickness is below hc. When the epilayer thickness equals the critical thickness and the dislocation density is extremely low, the total energy is independent of the dislocation spacing p. If the critical thickness is less than the substrate thickness and the epilayer thickness is greater than the critical thickness, the total energy has a local minimum at dislocation spacing p=pmin; in addition, the corresponding dislocation density is the critical dislocation density. When p>pmin, the total energy decreases by decreasing the dislocation spacing, i.e., increasing the dislocation density. The total energy curve near p=pmin changes to a steep valley when the epilayer’s thickness approaches that of the substrate thickness. This corresponds to the experimental observation that a fast relaxation of misfit strain occurs when the epilayer thickness grows to a sufficient thickness. If p<pmin, the total energy markedly increases by decreasing the dislocation spacing. This phenomenon implies that work hardening is inevitable due to the dislocation–dislocation core interaction.