Dislocation-interaction-based model of strained-layer relaxation

Abstract

A strained layer relaxes plastically when dislocations propagate within the layer, leaving behind an array of misfit dislocations at the layer interface. We develop an analytical model of this process, based on the idea that relaxation is frustrated when propagating dislocations are trapped or annihilated by encounters with previously created misfit dislocations or other propagating dislocations. The theory characterizes the evolving density of the misfit array and the density of propagating dislocations in terms of a pair of coupled rate equations. The two trapping functions which appear in these equations are evaluated quantitatively by numerically investigating all possible dislocation-dislocation encounters. Fluctuations in the local stress field driving the individual dislocations are explicitly taken into account when evaluating the trapping functions. Analysis of the rate equations shows that there are two regimes in the strain-relaxation dynamics. Initially, the strain decreases rapidly following a universal dependence on time scaled with the initial dislocation density n0. At a (rescaled) crossover time that increases with n0, the strain levels off from the universal relaxation curve and saturates to an asymptotic residual strain level, which decreases with n0. Microscopically, our model reveals that the initial fast strain-relaxation regime is dominated by collisions between propagating dislocations, while the slow saturation regime is dominated by the trapping of propagating dislocations by the misfits. In the end, the self-trapping of the propagating dislocations by the misfit array they themselves have generated leaves the layer in a frustrated state with residual strain higher than the critical strain. The predictions of the theory are found to be in good agreement with experimental measurements and with large-scale numerical simulations of layer relaxation.