Generation of misfit dislocations in a core-shell nanowire near the edge of prismatic core

Abstract

The misfit strain relaxation via generation of partial and perfect misfit dislocations and their dipoles at the interface in core-shell nanowires with faceted cores is considered. The core has the shape of a long parallelepiped of a square cross-section and is placed symmetrically with respect to the cylindrical shell surface. The energy change caused by dislocation generation in such nanowires is calculated. Critical conditions for the onset of such dislocations are calculated and analyzed. The energy barriers for the dislocation generation are defined with respect to different values of the nanowire parameters. The preference of dislocation configuration and generation sites in the nanowire are determined.

Introduction

For a long time, the problem of misfit strains and mechanisms of their relaxation through generation of various misfit defects in advanced functional composite materials and solid heterostructures has remained in the focus of many researchers in over the world. In the second half of the 20th century, the efforts were mainly concentrated on studying this problem in planar heterostructures, from multilayered structural composites (for example, lamellar metallic-intermetallic eutectics) to thin-film metallic and semiconductor heterostructures (see, for example, the reviews [1], [2], [3], [4], [5], [6], [7], [8], [9], [10], [11], [12] and book [13]).

From the beginning of the 1990-es, the attention has been gradually shifted to the same problem in such novel low-dimensional nanocomposite structures as nanoislands [14], [15], [16], [17], [18], [19], [20], [21], [22], [23], [24], embedded quantum dots [20,[25], [26], [27], [28], [29], [30], [31], [32], [33]] and wires [13,20,26,[34], [35], [36], [37], [38], [39], [40], [41], [42]], core-shell nanoparticles [43], [44], [45], [46], [47], [48], [49], [50], [51], [52], [53], [54], [55], [56], [57], axially- [58], [59], [60], [61], [62], [63], [64] and radially-inhomogeneous (core-shell) [20,52,63,[65], [66], [67], [68], [69], [70], [71], [72], [73], [74], [75], [76], [77], [78], [79], [80], [81], [82], [83], [84], [85], [86], [87], [88], [89], [90], [91], [92], [93], [94], [95], [96]] nanowires (NWs). A large part of this recent work is theoretical studies [18,20,[24], [25], [26], [27], [28], [29], [30],[34], [35], [36], [37], [38], [39], [40], [41], [42], [43],46,[49], [50], [51], [52], [53],[57], [58], [59],63,65,66,[68], [69], [70],72,73,75,78,[86], [87], [88], [89],92,93,96] done mainly within various continuum approaches and aimed at the calculation of critical conditions for the onset of misfit strain relaxation by nucleation of the first misfit defects (commonly, perfect and partial dislocations, rarely, wedge disclinations and edge dislocation walls [66]). As is the case with planar heterostructures [1], [2], [3], [4], [5], [6], [7], [8], [9], [10], [11], [12], [13], the authors have commonly used the energetic approach, considering either (in most of the cases) the energy change due to the formation of the final configuration of individual misfit defects – straight dislocations [35,58,59,65,69,87,88], dislocation dipoles [34,[36], [37], [38], [39], [40], [41],73], closed [[26], [27], [28], [29], [30],37,46,49,50,53,57,[68], [69], [70],72,75,78,86,89] or disclosed [37] dislocation loops, straight disclinations [66], disclination dipoles [66], and dislocation walls [66] – or, relatively rarely, the energy barriers for nucleation and evolution of the defect configuration, as for individual straight dislocations [18], their dipoles [42] and dislocation loops and semi-loops [[51], [52], [53],96]. Most of these works (although not all) have been reviewed by Ovid'ko and Sheinerman [20], Glas [63], Kavanagh [97], and Gutkin et al. [98], so we only notice here that the models of the first group obviously prevail in their number over the models of the second one. This imbalance looks rather strange due to the evident fact that the initial stage of the misfit defect nucleation is of primary importance. Whilst the first-group models do not describe in detail neither the mechanism of misfit defect generation, nor the energy barriers which must be overcome by the system to pass in a new partly relaxed state, the second-group models naturally include both these factors. Moreover, the height of the energy barrier may look more reliable criterion for the onset of misfit strain relaxation than such critical parameters, calculated within the first-group models, as, for example, critical misfit value or critical sizes of a composite nanostructure. Another argument in favor of the second-group models is that in experimental observations of misfit dislocations (MDs) and stacking faults in core-shell nanoparticles [44,45,47,48], the authors show asymmetric images of MD cross sections, which certifies that the MD lines are not closed around cores but rather form an intermediate disclosed configuration when the dislocation lines end at the free surface of the shell.

Among the models of the second group [18,42,[51], [52], [53],96], only two models [18,42] describe the generation of MDs by glide along the interface, namely, between the nanoisland and the substrate [18], and the nanolayer and the NW of rectangular cross section, embedded to it [42]. The other models [[51], [52], [53],96] dealt with rectangular prismatic dislocation loops or semi-loops which were supposed to nucleate by coagulation of point defects, either vacancies or interstitials. Meanwhile, for today there are some evident experimental observations of misfit strain relaxation by the glide of partial or perfect MDs along flat areas of interface in Au-Pd [44,47] and Au-FePt [45] core-shell nanoparticles. Glide of MDs along the interface between CoSi₂ nanoislands and (111) Si substrates from the edges of the islands to their centers was observed and simulated by Liu et al. [17].

In core-shell NWs, to the best of our knowledge, there are still no experimental evidence of MD glide along the interface, although Nguyen et al. [90] have recently shown the images of many MDs lying in the {111} facets of the interface in Si-Ge core-shell NWs. It seems that these MDs could be generated by glide in the interface from the edges to the centers of the facets. In this case, the edges of the facets (the corners of polygonal cross sections of the Si cores) can play the role of concentrators for misfit shear stresses in such core-shell NWs [92]. Moreover, the misfit shear stress concentration also occurs in the areas of the shell surface, which are close to the corners of the interface contour [92] that makes these surface areas favorable for generation of partial or perfect dislocations. Since the majority of cores in core-shell NWs have polygonal (triangle [79,99,100], square [80,81], pentagonal [101,102], hexagonal [76,77,79,80,82,84,85,94,100], octagonal [90], etc.) or close-to-polygonal cross sections, this mechanism of MD generation should be rather common when the interface facets are the planes of easy dislocation glide for the core and/or shell materials.

To the best of our knowledge, no theoretical models to describe the mechanisms of MD generation by glide along flat facets of interface in core-shell NWs have been suggested until now. Moreover, there is a very limited number of analytical solutions for stress/strain fields of faceted misfitting inclusions in cylindrical NWs [92,93,103], which could be used in elaboration of the theoretical models. All these solutions were found for long inclusions (cores) in the form of a long parallelepiped with square [92,103] or rectangular [93] cross section, either centered at the cylinder axis [92,103] or being in an eccentric position [93]. The inclusion eigenstrain was supposed to be 1D [93], 2D (the plane strain problem) [103] and 3D [92] homogeneous dilatation. The solutions were found by the complex potentials method. In the 1D-dilatation case [93], the stress tensor was given in a closed analytical form. In the 2D-dilatation case [103], the solution was represented by the complex potentials and illustrated by stress maps in Cartesian coordinates. However, the authors [103] did not demonstrate the fulfillment of the boundary conditions of this problem. In the 3D-dilatation case [92], the misfit stresses were found in a concise and transparent form of trigonometric series which is very useful for theoretical modeling of misfit relaxation mechanisms in similar NWs.

In the present paper, we apply the 3D-dilatation solution [92] to development of theoretical models describing the generation of partial and perfect MDs by glide in vicinity of edges of a prismatic core of square cross section centered at the axis of a core-shell NW. Within these models, we consider the nucleation of partial and perfect edge dislocations either at the free surface of the NW or at the interface edge and their further glide in the slip planes coming along the interface, in either shell-and-core or shell materials. At the interface edge, the dislocations nucleate by dipoles when one dislocation glides to the center of the interface, while the other one to the free surface of the shell. For comparison, we also consider a perfect dislocation which nucleates at the shell free surface and climbs to the interface. The energy barriers, which are characteristic for every case of MD generation, are compared with each other, in which case the lowest barrier indicates on the easiest (and therefore the most probable) way of MD generation.