Misfit Dislocation Patterning in Thin Films

Abstract

The emergence of non-uniform distributions of misfit dislocations (MDs) in thin films is discussed. A three-element reaction±diffusion model for the kinetics of gliding, climbing and misfit dislocations as proposed by Romanov and Aifantis (R-A model) is used to describe the corresponding pattern. The non-local integral expression for the effective stress field at the film surface, which is the main driving force for MD patterning, is approximated by a gradient expression in the MD density. The corresponding gradient coefficients have an explicit dependence on the film thickness which, thus, defines a characteristic length for the pattern. Analytical solutions of the model are obtained which describe transient spatially uniform dislocation distributions, as well as steady-state spatially periodic dislocation distributions. Linear stability analysis around a uniform steady-state solution demonstrates the formation of MD patches as a result of a dynamical spatial instability. This instability is governed by the competition of the spatial coupling provided by the MD stress field and a diffusion-like term entering the dynamics of the gliding dislocations. A stochastic argument for the corresponding diffusion coefficient, which depends on the average spacing between MDs (thus providing a second characteristic length scale), yields an explanation for not observing MD patterns for film thicknesses below 1 mm.

Strain relaxation models

“…where ρ i , i = g, c, m are the density of the gliding, climbing and misfit dislocations, respectively, A, B, K -corresponding reaction rates [114,115]. The model just described has been applied to the problem of the misfit dislocation patterning [115,116].…”

“…where ρ i , i = g, c, m are the density of the gliding, climbing and misfit dislocations, respectively, A, B, K -corresponding reaction rates [114,115]. The model just described has been applied to the problem of the misfit dislocation patterning [115,116]. The complete system has been used for the linear stability analysis only, while dislocation evolution has been considered for the two limiting cases: the uniform time-dependent solution ρ i = ρ i (t) and the steady-state non-uniform one ρ i = ρ i (z).…”

Investigation of Intrinsic Stress Effects in Cantilever Structures

“…Because of the deposition process the crystal lattice of the thin film and the substrate are forced to line up perfectly at the interface and stress arises. The influence of these misfit stresses is only significant in the initial phase of thin film deposition [7] because of the local lattice adaption at the interface area. The lattice adaption is characterized by the misfit parameter [8] …”

Experimental and Theoretical Studies of Spatio-Temporal Instabilities in Novel Materials

“…[In this connection, it is noted that gradient terms with specific expressions for the corresponding phenomenological coefficients appear in recent constitutive models for random composites (see, for example, Drugan and Willis [61], Buryachenko [62])]. Finally, the work of Romanov and coworkers [63][64][65] on gradient dislocation dynamics for monotonic deformation and thin film problems, as well as an alternative approach of discrete dislocation dynamics simulations by van der Giessen and Needleman [66,67] are worth noting. It may not be an exaggeration to state that the motivation for all aforementioned work on gradient theory with applications to dislocation and deformation patterning may be traced back to the original article of the author [3] (see also [5]).…”