Glenn E. Beltz

The activation energy for dislocation nucleation at a crack

THE ACTIVATION energy for dislocation nucleation from a stressed crack tip is calculated within the Peierls framework, in which a periodic shear stress vs displacement relation is assumed to hold on a slip plane emanating from the crack tip. Previous results have revealed that the critical G (energy release rate corresponding to the “screened” crack tip stress field) for dislocation nucleation scales with y., (the unstable stacking energy), in an analysis which neglects any coupling between tension and shear along the slip plane. That analysis represents instantaneous nucleation and takes thermal effects into account only via the weak temperature dependence of the elastic constants. In this work, the energy required to thermally activate a stable, incipient dislocation into its unstable “saddle-point” configuration is directly calculated for loads less than that critical value. We do so only with the simplest case, for which the slip plane is a prolongation of the crack plane. A first calculation reported is 2D in nature, and hence reveals an activation energy per unit length. A more realistic scheme for thermal activation involves the emission of a dislocation loop, an inherently 3D phenomenon. Asymptotic calculations of the activation energy for loads close to the critical load are performed in 2D and in 3D. It is found that the 3D activation energy generally corresponds to the 2D activation energy per unit length multiplied by about 5 -10 Burgers vectors (but by as many as 17 very near to the critical loading). Implications for the emission of dislocations in copper. K-iron, and silicon at elevated temperature are discussed. The effects of thermal activation are very significant in lowering the load for emission. Also, the appropriate activation energy to correspond to molecular dynamics simulations of crack tips is discussed. Such simulations, as typically carried out with only a few atomic planes in a periodic repeat direction parallel to the crack tip. are shown to greatly exaggerate the (aheddy large) effects of temperature on dislocation nucleation. WE BUILD on recent advances in the modeling of dislocation nucleation at a crack tip

Modeling of threading dislocation reduction in growing GaN layers

In this work, a model is developed to treat threading dislocation (TD) reduction in (0 0 0 1) wurtzite epitaxial GaN thin films. The model is based on an approach originally proposed for (0 0 1) FCC thin film growth and uses the concepts of mutual TD motion and reactions. We show that the experimentally observed slow TD reduction in GaN can be explained by low TD reaction probabilities due to TD line directions practically normal to the film surface. The behavior of screw dislocations in III-nitride films is considered and is found to strongly impact TD reduction. Dislocation reduction data in hydride vapor phase epitaxy (HVPE) grown GaN are well described by this model. The model provides an explanation for the non-saturating TD density in thick GaN films. r 2001 Elsevier Science B.V. All rights reserved.

Scaling laws for the reduction of threading dislocation densities in homogeneous buffer layers

In the heteroepitaxial growth of films with large misfit with the underlying substrate (linear mismatch strains in excess of 1%–2%) the generation of misfit dislocations and threading dislocations (TDs) is ubiquitous for thicknesses well in excess of the equilibrium critical thickness. Experimental data suggest that the TD density in relaxed homogeneous buffer layers can be divided into three regimes: (i) an entanglement region near the film/substrate interface corresponding to TD densities of ∼1010–1012 cm−2; (ii) a falloff in TD density that is inversely proportional to the film thickness h, applicable to densities in the range ∼107–109 cm−2; and (iii) saturation or weak decay of the TD density with further increase in film thickness. Typical saturation densities are on the order of ∼106–107 cm−2. In this article, we show that the TD reduction may be described in terms of effective lateral motion of TDs with increasing film thickness. An analytic model is developed that successfully predicts both the 1/...

Threading dislocation reduction in strained layers

In this article, we have developed models for threading dislocation (TD) reduction due to the introduction of an intentionally strained layer. Three different types of dislocations have been considered in this model: misfit dislocations (MDs), mobile TDs, and TDs whose glide motion has been blocked by a MD crossing the glide path of the TD (immobile TDs). The models are based on MD formation by the process of lateral TD motion. The strain-induced TD motion leads to possible annihilation reactions of mobile TDs with either other mobile TDs or blocked TDs, or reactions in which a mobile TD is converted to an immobile TD by a blocking reaction with a MD. The evolution of the density of mobile and blocked TDs and the MD density is represented by three coupled nonlinear first order differential equations. When blocking of TDs by MDs is not considered, the equations have an analytical solution that shows that the final TD density should decrease exponentially where the argument of the exponent is proportional t...

Modeling of Threading Dislocation Reduction in Growing GaN Layers

In this work, a model is developed to treat threading dislocation (TD) reduction in (0001) wurtzite epitaxial GaN thin films. The model is based on an approach originally proposed for (001) f.c.c. thin film growth and uses the concepts of mutual TD motion and reactions. We show that the experimentally observed slow TD reduction in GaN can be explained by low TD reaction probabilities due to TD line directions practically normal to the film surface. The behavior of screw dislocations in III-nitride films is considered and is found to strongly impact TD reduction. Dislocation reduction data in hydride vapor phase epitaxy (HVPE) grown GaN is well-described by this model. The model provides an explanation for the non-saturating TD density in thick GaN films.

On the continuum versus atomistic descriptions of dislocation nucleation and cleavage in nickel

A hybrid atomistic-finite-element model is compared with the continuum-based Peierls-Nabarro model for several crack orientations in a nickel crystal. Both methods incorporate the same embedded-atom potential for Ni, in order to make the comparison as valid as possible. The agreement (expressed in terms of a stability diagram showing envelopes in loading space where fracture or dislocation nucleation are likely to occur) is excellent in the case of a crack lying on a (111) plane, with a crack front running along a (211)-type direction, subject to mixed-mode I-II loadings. That orientation involves dislocation nucleation on the prolongation of the crack plane, and hence no ledge is formed upon dislocation nucleation. In other geometries considered (involving a crack on a (100)-type plane), the agreement seems to get poorer with increasing size of the ledge the is created when a dislocation nucleates. In all geometries, the atomistic model shows that incipient dislocation-like features are present before dislocation nucleation takes place, which serves as additional validation of the continuum Peierls-Nabarro model.

An approach to threading dislocation ‘‘reaction kinetics’’

An approach is developed to describe the evolution of threading dislocation (TD) densities in lattice‐mismatched epitaxial films. TD ensembles are treated in close correspondence to chemical species in chemical reaction kinetics. ‘‘Reaction rate’’ equations are derived for changing TD density with increasing film thickness for first‐ and second‐order reactions. Selective area growth is an example of a first‐order reaction. TD annihilation, fusion, and scattering are examples of second‐order reactions. Analytic models are derived for TD behavior in relaxed homogeneous buffer layers, selective area growth, and strained layers.

The effect of crack blunting on the competition between dislocation nucleation and cleavage

Abstract To better understand the ductile versus brittle fracture behavior of crystalline materials, attention should be directed towards physically realistic crack geometries. Currently, continuum models of ductile versus brittle behavior are typically based on the analysis of a pre-existing sharp crack in order to use analytical solutions for the stress fields around the crack tip. This paper examines the effects of crack blunting on the competition between dislocation nucleation and atomic decohesion using continuum methods. We accomplish this by assuming that the crack geometry is elliptical, which has the primary advantage that the stress fields are available in closed form. These stress field solutions are then used to calculate the thresholds for dislocation nucleation and atomic decohesion. A Peierls-type framework is used to obtain the thresholds for dislocation nucleation, in which the region of the slip plane ahead of the crack develops a distribution of slip discontinuity prior to nucleation. This slip distribution increases as the applied load is increased until an instability is reached and the governing integral equation can no longer be solved. These calculations are carried out for various crack tip geometries to ascertain the effects of crack tip blunting. The thresholds for atomic decohesion are calculated using a cohesive zone model, in which the region of the crack front develops a distribution of opening displacement prior to atomic decohesion. Again, loading of the elliptical crack tip eventually results in an instability, which marks the onset of crack advance. These calculations are carried out for various crack tip geometries. The results of these separate calculations are presented as the critical energy release rates versus the crack tip radius of curvature for a given crack length. The two threshold curves are compared simultaneously to determine which failure mode is energetically more likely at various crack tip curvatures. From these comparisons, four possible types of material fracture behavior are identified: intrinsically brittle, quasi-brittle, intrinsically ductile, and quasi-ductile. Finally, real material examples are discussed.

Elastic fields of quantum dots in subsurface layers

In this work, models based on conventional small-strain elasticity theory are developed to evaluate the stress fields in the vicinity of a quantum dot or an ordered array of quantum dots. The models are based on three different approaches for solving the elastic boundary value problem of a misfitting inclusion embedded in a semi-infinite space. The first method treats the quantum dot as a point source of dilatation. In the second approach we approximate the dot as a misfitting oblate spheroid, for which exact analytic solutions are available. Finally, the finite element method is used to study complex, but realistic, quantum dot configurations such as cuboids and truncated pyramids. We evaluate these three levels of approximation by comparing the hydrostatic stress component near a single dot and an ordered array of dots in the presence of a free surface, and find very good agreement except in the immediate vicinity of an individual quantum dot.

Peierls Framework for Dislocation Nucleation from a Crack Tip

Dislocation nucleation from a stressed crack tip is analyzed based on the Peierls concept, in which a periodic relation between shear stress and atomic shear displacement is assumed to hold along a slip plane emanating from a crack tip. This approach allows some small slip displacement to occur near the tip in response to small applied loading and, with increase in loading, the incipient dislocation configuration becomes unstable and leads to a fully formed dislocation which is driven away from the crack. An exact solution for the loading at that nucleation instability was developed using the J-integral for the case when the crack and slip planes coincide (Rice, 1992). Solutions are discussed here for cases when they do not. The results were initially derived for isotropic materials and some generalizations to take into account anisotropic elasticity are noted here. Solutions are also given for emission of dissociated dislocations, especially partial dislocation pairs in fee crystals. The level of applied stress intensity factors required for dislocation nucleation is shown to be proportional to \(\sqrt {{\gamma _{us}}}\) where γus, the unstable stacking energy, is a new solid state parameter identified by the analysis. It is the maximum energy encountered in the block-like sliding along a slip plane, in the Burgers vector direction, of one half of a crystal relative to the other. Approximate estimates of γus are summarized, and the results are used to evaluate brittle versus ductile response in fee and bee metals in terms of the competition between dislocation nucleation and Griffith cleavage at a crack tip. The analysis also reveals features of the near-tip slip distribution corresponding to the saddle point energy configuration for cracks that are loaded below the nucleation threshold, and some implications for thermal activation are summarized. Additionally, the analysis of dislocation nucleation is discussed in connection with the emission from cracks along bimaterial interfaces, in order to understand recent experiments on copper bicrystals and copper/sapphire interfaces, and we discuss the coupled effects of tension and shear stresses along slip planes at a crack tip, leading to shear softening and eased nucleation.