^{1,a)}and Yuhai Tu

^{1}

### 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 . At a (rescaled) crossover time that increases with , the strain levels off from the universal relaxation curve and saturates to an asymptotic residual strain level, which decreases with . 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.

I. INTRODUCTION

A. Single dislocation in a strained layer

B. Mean-field behavior

II. THE RELAXATION EQUATIONS

III. EVALUATION OF THE FUNCTIONS , , and

IV. COMPARISON OF THE THEORY WITH EXPERIMENT AND SIMULATIONS

V. GENERAL PROPERTIES OF THE SELF-TRAPPING MODEL

A. Decrease of the residual stress with increasing initial thread density

B. Two kinetic regimes of strain relaxation

VI. SUMMARY

### Key Topics

- Mean field theory
- 6.0
- Slip systems
- 6.0
- Stress relaxation
- 6.0
- Number theory
- 5.0
- Interface structure
- 4.0

## Figures

Top: slip systems of the fcc crystal. The triangular faces of the pyramid lie parallel to the allowed glide planes; their edges show the respective directions of the Burgers vectors that can glide on each plane. Only the four Burgers vectors with a component in the [001] direction are active in relieving biaxial strain. Bottom: same-perspective sequential snapshots of a threading segment propagating on a (111) glide plane in a strained layer of height . As the thread propagates, it leaves behind a misfit pair . The analogous free-surface problem is obtained by slicing off the top half of the strained layer.

Top: slip systems of the fcc crystal. The triangular faces of the pyramid lie parallel to the allowed glide planes; their edges show the respective directions of the Burgers vectors that can glide on each plane. Only the four Burgers vectors with a component in the [001] direction are active in relieving biaxial strain. Bottom: same-perspective sequential snapshots of a threading segment propagating on a (111) glide plane in a strained layer of height . As the thread propagates, it leaves behind a misfit pair . The analogous free-surface problem is obtained by slicing off the top half of the strained layer.

The four categories of localized dislocation-dislocation encounters which can occur in the relaxing layer.

The four categories of localized dislocation-dislocation encounters which can occur in the relaxing layer.

Basic thread-trapping functions, obtained by summing over all possible dislocation-dislocation interactions. governs thread-misfit collisions and describes thread-thread collisions. The driving force is the force driving the motion of any particular thread. In the calculation of individual interactions, this is the available applied force . *In situ*, it becomes .

Basic thread-trapping functions, obtained by summing over all possible dislocation-dislocation interactions. governs thread-misfit collisions and describes thread-thread collisions. The driving force is the force driving the motion of any particular thread. In the calculation of individual interactions, this is the available applied force . *In situ*, it becomes .

Distribution of local misfit forces exerted on a test thread by a random misfit array: (a) distribution sampled by an arbitrarily placed thread; (b) distribution filtered so as to approximate sampling by a free thread, where the structure near the cutoff arises from the structure in Fig. 3. As shown by the arrow, the value assumed for here is . The curves show distributions for values of increasing (from left to right) from 0 to , in steps of . Several sets of distributions, more refined and extending to higher values of than the set shown in the figure, were used in the computations.

Distribution of local misfit forces exerted on a test thread by a random misfit array: (a) distribution sampled by an arbitrarily placed thread; (b) distribution filtered so as to approximate sampling by a free thread, where the structure near the cutoff arises from the structure in Fig. 3. As shown by the arrow, the value assumed for here is . The curves show distributions for values of increasing (from left to right) from 0 to , in steps of . Several sets of distributions, more refined and extending to higher values of than the set shown in the figure, were used in the computations.

The functions plotted against . The curves (from left to right) are for of 2, 3, 4, 6, and 10, respectively.

The functions plotted against . The curves (from left to right) are for of 2, 3, 4, 6, and 10, respectively.

Residual strain divided by critical strain observed in helium-implanted and annealed free-surface layers as a function of layer thickness. Solid circles refer to layers grown by ultrahigh vacuum chemical vapor deposition and solid triangles to layers grown by rapid thermal vapor deposition. The × marks show the numerical simulations reported in Ref. 26. The lines show the predictions of the analytical model for different values of (in units of ).

Residual strain divided by critical strain observed in helium-implanted and annealed free-surface layers as a function of layer thickness. Solid circles refer to layers grown by ultrahigh vacuum chemical vapor deposition and solid triangles to layers grown by rapid thermal vapor deposition. The × marks show the numerical simulations reported in Ref. 26. The lines show the predictions of the analytical model for different values of (in units of ).

Model predictions (solid lines) and numerical simulations (dotted lines) for relaxation starting at twice the critical strain. The initial dislocation densities , from top to bottom, are 2, 8, and 32 (in units of ). The error bars denote scatter in the individual simulation runs. The horizontal dashed line indicates the critical strain.

Model predictions (solid lines) and numerical simulations (dotted lines) for relaxation starting at twice the critical strain. The initial dislocation densities , from top to bottom, are 2, 8, and 32 (in units of ). The error bars denote scatter in the individual simulation runs. The horizontal dashed line indicates the critical strain.

Model predictions (solid lines) and numerical simulations (dotted lines) for relaxation starting at four times the critical strain. The initial dislocation densities are the same as in Fig. 7.

Model predictions (solid lines) and numerical simulations (dotted lines) for relaxation starting at four times the critical strain. The initial dislocation densities are the same as in Fig. 7.

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