Analysis of the strained-layer critical thickness concept based on a Peierls—Nabarro model of a threading dislocation

Abstract

The Peierls–Nabarro theory of crystal dislocations is applied to estimate the critical thickness of a strained layer bonded to a substrate for a given mismatch strain. Strained epitaxial layers are common in microelectronic structures. The electronic properties of a film depend in most cases on the misfit strain (and hence the misfit stress). Unfortunately, an excessive strain may lead to the nucleation and propagation of defects, such as misfit dislocations. One such defect that has received wide attention in recent years is the so-called threading dislocation, which runs from the free surface of the strained layer to the interface, where it continues as an interfacial misfit dislocation. The critical thickness is achieved when the net driving force necessary to extend the misfit dislocation becomes positive. Previous analyses were based on the continuum theory of elastic dislocations, and hence depended on the artificial core cut-off parameter r₀. The Peierls–Nabarro theory makes use of an interplanar shear law based on atomistic interactions to lead to a more realistic description of the stresses and displacements in the vicinity of a dislocation core, thus eliminating the need for the core cut-off parameter. The dependence of the critical layer thickness on the mismatch strain is found to be similar to that predicted by the continuum elastic dislocation theory, provided that effective core cut-off parameters, ranging from 1/10b (for materials with a relatively high unstable stacking energy) to 1/2b (for materials with a relatively low unstable stacking energy) are used.