Misfit relaxation in heteroepitaxy: A theoretical method for inferring individual dislocation positions from tilting maps

Dislocations are defects unavoidably created during deposition of lattice-mismatched films. Controlling their distribution and the density of threading arms reaching the film surface is fundamental for applications, as dislocations can seriously hinder the desired performances of various devices. A detailed, quantitative understanding of their nucleation, subsequent gliding and final distribution is still lacking. One of the main problems stems in the difficulties encountered when attempting to experimentally provide a complete characterization of the defects. Dislocations are characterized by misfit segments (with full or partial edge character), suitably connected to threading arms. The distribution of threading arms at the free surface can be easily analyzed by various techniques. Inner misfit segments, on the other hand, can be imaged only by time-consuming and destructive techniques such as TEM analysis, usually leading to poor statistics. Recent developments in fast-scanning X-rays microscopy have recently opened up the possibility to record detailed tilting-angle maps [1,2], as caused by the dislocation distribution in the film and in the substrate. Here we present a theoretical procedure allowing one to extract from such maps, and from the knowledge of the residual strain R in the film, the position of individual dislocations. Based on the exact (within linear elasticity theory) expression of the tilting angle produced in the crystal by a dislocation, and by fixing the density of misfit segments based on the R value, we generate distributions of dislocations and compare the predicted and measured tilt map. Based on local deviations between simulated and experimental maps, we further change the dislocation distribution until a good match is obtained. To make the procedure efficient we directly exploit the physics of the system: instead of sampling all possible dislocation positions in the simulation cell, we use dislocation dynamics simulations to evolve the system from a starting guess to a local equilibrium position. The procedure is here applied to Ge0.07 Si0.93 films grown on Si(001) by Chemical Vapor Deposition. A thicker (1500nm) and thinner sample (600nm) were grown and the corresponding tilting maps were obtained as in [2]. Our methodology was then applied, leading to interesting results: in both cases the model predicted the occurrence of several identical dislocations piling-up on the same glide plane, and penetrating the Si substrate, a clear indication of the presence of multiplication processes [3]. Interestingly, the number of dislocations not produced by multiplication turned out to be almost constant in both samples, in spite of the different residual strain (as expected, the thicker film is more relaxed). What changes when the film gets thicker, instead, is the number (growing with thickness) of dislocations in isolated pile ups. Based on these results, an interpretative model of relaxation in the samples is built, distinguishing between a first phase, where ?standard? dislocation s form and glide to the film/substrate interface, and a second where nucleation of additional defects mainly takes place by multiplication, likely produced by crossing and self-blocking between dislocations. The model predictions are further checked by selected TEM images, confirming both the presence of misfit-segments pile ups and of dislocations penetrating the Si substrate. [1] V. Mondiali et al., APL 105, 242103 (2014); [2] M.I. Richard et al., ACS Appl. Mater. & Interfaces 7, 26696 (2015); [3] F.K. LeGoues et al, Phys. Rev. Lett. 66, 2903 (1991)