Misfit dislocations and critical thickness of heteroepitaxy

An expression is derived for the critical thickness of heteroepitaxial films by taking the energy minimization approach. It is exactly identical to the critical thickness obtained by Matthew and Blakeslee [J. Cryst. Growth 27, 118 (1974)] using a force balance approach, after allowing for a discrepancy judged to be an unmentioned approximation in the latter. The equivalence of the two formulations is explained conceptually. The reason is pointed out for the very different critical thickness obtained by People and Bean [Appl. Phys. Lett. 47, 322 (1985); 49, 229 (1986)] using the energy minimization approach. It is further shown that misfit dislocations do not appear en masse catastrophically when the critical thickness is reached. Rather, their density increases gradually with the epitaxial thickness, approaching only asymptotically a value required for a complete relief of film stress as the thickness tends to infinity. Any observed sudden mass appearance of misfit dislocations must be attributed to a substantial supercritical thickness actually existing, prior to nucleation of dislocations.