An exactly solvable model for calculating critical misfit and thickness in epitaxial superlattices. II. Layers of unequal elastic constants and thicknesses

Abstract

In this paper we have extended the exact predictions of critical misfit fc and critical thickness hc —based on equilibrium suppositions and a parabolic interfacial potential of shear strength μ—in superlattices of layer pairs A and B introducing the thicknesses ha=ηac and hb=ηbc, and moduli μa, μb, and μ as variables. The extension involved an explicit relation between the homogeneous misfit strains ēa and ēb on the one hand and the thicknesses and moduli on the other in order to ensure lateral force balance. Analytical expressions for the energies ED and Eē per unit area per interface for misfit dislocations and mistfit strains, respectively, have been derived and minimized to obtain the equilibrium value ēm needed to predict fc and hc. The expression for ED is rather complicated. However, a variety of simplifying approximations were feasible, which not only facilitated numerical evaluation but also made it possible to draw various conclusions. The case of equal elastic constants but variable layer thicknesses shows that a maximum value of fc occurred when the superlattice layers have equal thickness and that the layers of equal thickness produced the minimum value of ED. Scaling relations were found that could be used to calculate ED values simply by selecting the appropriate combinations of thicknesses and elastic constants. The effect of changing elastic constants or the values of critical thickness or critical misft could also be represented through scaling relations.