Critical Thickness for Three-Dimensional Epitaxial Island Growth

1988 ◽  
Vol 130 ◽  
Author(s):  
K. Jagannadham ◽  
J. Narayan

AbstractThe generation of misfit dislocation loops in three-dimensional epitaxial islands grown on thick substrates is analyzed. The coherent strain in the island is described by virtual interfacial dislocation loops situated in the interface. The traction free surface boundary conditions are satisfied by the surface dislocation loops situated on the surface of the island. A misfit dislocation loop is formed and the changes in the energy of the configuration used to determine if the total energy is lowered. The numerical analysis is carried out forhemispherical islands of GaAs grown on (100) silicon with a misfit dislocation of Burgers vector 3.84 Å. It has been found energetically favorable to nucleate a misfit dislocation loop at a distance of 3 å from the interface when the radius of the hemispherical island is equal to or greater than 40 å. In addition, a misfit dislocation loop could be nucleated at a larger distance from the interface when the size of the island is larger.

1999 ◽  
Vol 396 ◽  
pp. 37-71 ◽  
Author(s):  
LEONID BREVDO ◽  
PATRICE LAURE ◽  
FREDERIC DIAS ◽  
THOMAS J. BRIDGES

The film flow down an inclined plane has several features that make it an interesting prototype for studying transition in a shear flow: the basic parallel state is an exact explicit solution of the Navier–Stokes equations; the experimentally observed transition of this flow shows many properties in common with boundary-layer transition; and it has a free surface, leading to more than one class of modes. In this paper, unstable wavepackets – associated with the full Navier–Stokes equations with viscous free-surface boundary conditions – are analysed by using the formalism of absolute and convective instabilities based on the exact Briggs collision criterion for multiple k-roots of D(k, ω) = 0; where k is a wavenumber, ω is a frequency and D(k, ω) is the dispersion relation function.The main results of this paper are threefold. First, we work with the full Navier–Stokes equations with viscous free-surface boundary conditions, rather than a model partial differential equation, and, guided by experiments, explore a large region of the parameter space to see if absolute instability – as predicted by some model equations – is possible. Secondly, our numerical results find only convective instability, in complete agreement with experiments. Thirdly, we find a curious saddle-point bifurcation which affects dramatically the interpretation of the convective instability. This is the first finding of this type of bifurcation in a fluids problem and it may have implications for the analysis of wavepackets in other flows, in particular for three-dimensional instabilities. The numerical results of the wavepacket analysis compare well with the available experimental data, confirming the importance of convective instability for this problem.The numerical results on the position of a dominant saddle point obtained by using the exact collision criterion are also compared to the results based on a steepest-descent method coupled with a continuation procedure for tracking convective instability that until now was considered as reliable. While for two-dimensional instabilities a numerical implementation of the collision criterion is readily available, the only existing numerical procedure for studying three-dimensional wavepackets is based on the tracking technique. For the present flow, the comparison shows a failure of the tracking treatment to recover a subinterval of the interval of unstable ray velocities V whose length constitutes 29% of the length of the entire unstable interval of V. The failure occurs due to a bifurcation of the saddle point, where V is a bifurcation parameter. We argue that this bifurcation of unstable ray velocities should be observable in experiments because of the abrupt increase by a factor of about 5.3 of the wavelength across the wavepacket associated with the appearance of the bifurcating branch. Further implications for experiments including the effect on spatial amplification rate are also discussed.


Author(s):  
Xinshu Zhang ◽  
Robert F. Beck

Three-dimensional, time-domain, wave-body interactions are studied in this paper for cases with and without forward speed. In the present approach, an exact body boundary condition and linearized free surface boundary conditions are used. By distributing desingularized sources above the calm water surface and using constant-strength panels on the exact submerged body surface, the boundary integral equations are solved numerically at each time step. Once the fluid velocities on the free surface are computed, the free surface elevation and potential are updated by integrating the free surface boundary conditions. After each time step, the body surface and free surface are regrided due to the instantaneous changing submerged body geometry. The desingularized method applied on the free surface produces non-singular kernels in the integral equations by moving the fundamental singularities a small distance outside of the fluid domain. Constant strength panels are used for bodies with any arbitrary shape. Extensive results are presented to validate the efficiency of the present method. These results include the added mass and damping computations for a hemisphere. The calm water wave resistance for a submerged spheroid and a Wigley hull are also presented. All the computations with forward speed are started from rest and proceed until a steady state is reached. Finally, the time-domain forced motion results for a modified Wigley hull with forward speed are shown and compared with the experiments for both linear computations and body-exact computations.


Author(s):  
Mahmoud Alidadi ◽  
Sander Calisal

The effects of two base-potentials on the accuracy of a slender-body method are studied in this paper. In the formulation for this method which is developed for the slender ships, the velocity potential is decomposed into a base-potential and a perturbation potential. Then using an order of magnitude analysis, the three-dimensional flow problem is simplified into a series of two-dimensional problems for the perturbation potential. These two-dimensional problems are solved with the linearized free surface boundary conditions, using a mixed Eulerian-Lagrangian method. Finally for the two base-potentials, the numerical wave elevation along a Wigleyull are compared with the experimental results.


1978 ◽  
Vol 5 (4) ◽  
pp. 479-488 ◽  
Author(s):  
William James ◽  
Basem Eid

This paper discusses the formulation of surface boundary conditions for a three-dimensional transport model for shallow lakes, specifically for Hamilton Harbour. The same hydrodynamic equations that describe the circulation of the ocean and the Great Lakes were used in this study. However, the boundary conditions (bed topography, shoreline configuration, and surface and bottom shear stress fields) have bigger effects on circulation in shallow enclosed lakes.In this study the flow is assumed to be incompressible and in hydrostatic equilibrium. A layered system is used in which the lake is considered to consist of a number of unequal layers in the vertical. The hydrodynamic equations are integrated vertically over each layer, and both vertical and horizontal eddy viscosities are introduced.The over-water wind stress is determined using the logarithmic wind velocity distribution and Von Karman's integral equation for turbulent flow over a rough movable surface of variable roughness, in conjunction with equations for wind–wave generation. Thus the wind drag coefficient is determined as a function of wind and wave characteristics, and is time- and space-dependent.


1989 ◽  
Vol 160 ◽  
Author(s):  
Eric P. Kvam ◽  
D.M. Maher ◽  
C.J. Humphreys

AbstractWe have observed that the nature of misfit dislocations introduced near the critical thickness in GexSi1-x alloys on (001)Si changes markedly in the region 0.4 ≤ x ≤ 0.5. At or below the lower end of this compositional range, the observed microstructure is comprised almost entirely of 60° type dislocations, while at the high end, the dislocation structure is almost entirely Lomer edge type. Concurrent with this change, the dislocation density at the top of the epilayer varies by a factor of about 60X. Similarly, several other observables (e.g. dislocation length and spacing) also change appreciably.Part of the reason for the morphological variation seems to be a change in the source for dislocation introduction, in conjunction with a change in glide behaviour of dislocations as a function of film thickness. Evidence will be presented that indicates strain, as well as thickness, has a critical value for some dislocation introduction mechanisms, and that these together determine the resulting microstructure.Furthermore, it appears unlikely that the edge-type Lomer dislocations which appear at about x = 0.5 are either introduced directly, by climb, or grown in, as in the three-dimensional island growth and coalescence which occurs when x approaches unity. Instead, a two-step mechanism involving glissile dislocations is proposed and discussed.


2010 ◽  
Vol 54 (02) ◽  
pp. 79-94 ◽  
Author(s):  
Xinshu Zhang ◽  
Piotr Bandyk ◽  
Robert F. Beck

Large-amplitude, time-domain, wave-body interactions are studied in this paper for problems with forward speed. Both two-dimensional strip theory and three-dimensional computation methods are shown and compared by a number of numerical simulations. In the present approach, an exact body boundary condition and linearized free surface boundary conditions are used. By distributing desingularized sources above the calm water surface and using constant-strength flat panels on the exact body surface, the boundary integral equations are solved numerically at each time step. The strip theory method implements Radial Basis Functions to approximate the longitudinal derivatives of the velocity potential on the body. Once the fluid velocities on the free surface are computed, the free surface elevation and potential are updated by integrating the free surface boundary conditions. After each time step, the body surface and free surface are regrided due to the instantaneous changing wetted body geometry. Extensive results are presented to validate the efficiency of the present methods. These results include the added mass and damping computations for a Wigley III hull and an S-175 hull with forward speed using both two-dimensional and three-dimensional approaches. Exciting forces acting on a Wigley III hull due to regular head seas are obtained and compared using both the fully three-dimensional method and the two-dimensional strip theory. All the computational results are compared with experiments or other numerical solutions.


2020 ◽  
Author(s):  
xinlin zhang

<p>This study presents a new method to calculate displacement and potential changes caused by an earthquake in a three-dimensional viscoelastic earth model. It is the first time to compute co- and post-seismic deformation in a spherical earth with lateral heterogeneities. Such a method is useful to investigate the 3-dimensional viscoelastic structure of the earth by interpreting precise satellite gravity and GPS data. Firstly, we concern with Maxwell’s constitutive equation, the linearized equation of momentum conservation and Poisson’s equation, and obtain the solution in the Laplace domain in a spherical symmetric viscoelastic earth model. Furthermore, we employ the perturbed method to deal with the effect of lateral heterogeneities and obtain the relation between the solutions of the spherical symmetric earth model, the three-dimension earth model with lateral inhomogeneity and the auxiliary solutions. Then, using the given surface boundary conditions to determine the auxiliary solutions, we obtain the perturbed solutions of lateral increment in the Laplace domain. Finally, taking the inverse Laplace transforms of solutions in a spherical symmetric viscoelastic earth model and perturbed solutions with respect to lateral hetergeneities, we obtain the solutions of deformation in a three-dimensional viscoelastic earth model. </p>


In natural diamonds of optical classification type la , nitrogen is the major identified impurity and is distributed mainly in point defects known as A defects (probably a pair of nitrogen atoms substituting for a pair of adjacent carbon atoms) and B defects (probably four substituted nitrogen atoms tetrahedrally surrounding a carbon vacancy), and also in the electron-microscopically visible platelet precipitates on {100}. This paper is concerned with other electron-microscopically detectable defects, discovered by R. F. Stephenson (Ph.D. thesis, University of Reading (1977)), that lie in {100} planes in circumstances strongly suggesting that they result from the decomposition of platelets. High-resolution electron microscopy shows these defects to be {111}-faceted cavities. They behave as pure phase-contrast objects whose interior density does not exceed about one-third that of the diamond matrix: we call them ‘voidites’. The experimental background to voidite observation is reviewed, including electron-microscopic measurements on normal {100} platelets and models of their structure, and the optical, X-ray diffraction and cathodoluminescence evidence for unusually large platelets whose presence, together with a relative richness in B defects, indicates an environment in which voidites are likely to be discovered. Almost all observed voidites are confined to sheets strictly parallel to {100}. Some voidite sheets occur in ‘ partial platelets’, where they replace part of the original area of normal platelet. Other voidite sheets occur within dislocation loops whose size and shape are similar to those of the peripheries of normal platelets in the specimen. Voidites occur in a wide range of sizes. The largest equiaxed voidites observed measure about 10 nm between opposite {111} facets, and the smallest resolved about 0.5 nm. Many voidites are elongated in one of the <110> directions in the plane of the voidite sheet: the most highly elongated voidites seen approach 100 nm in length, with diameters of a few nanometres. Variations in size, shape and number density of voidites, together with many other characteristics relevant to the microscopic processes of voidite formation, are discussed in detailed descriptions of about 40 voidite sheets occurring in partial platelets and dislocation loops in two diamond specimens. One specimen was free from both grown-in dislocations and dislocations associated with plastic deformation. It contained zones of highly elongated platelets and it appeared that transformation of a platelet into a voidite sheet surrounded by a dislocation loop was triggered by the mutual very close approach of platelets. The second voidite-containing specimen had suffered plastic deformation at some stage in its history, but did not exhibit direct evidence that glide dislocations had triggered the transformation. The Burgers vectors of 24 dislocation loops enclosing voidite sheets in the second specimen were determined. Twelve were of normal ½<110> type having a component ½a 0 normal to the voidite sheet, and twelve were non-primitive, the Burgers vector being a 0 normal to the voidite sheet (a 0 is the diamond face-centred cubic (fcc) unit cell edge). The volumes of over 2000 individual voidites, representing all or major parts of 12 voidite sheets, have been measured. Values found for the ratio ∑ V /Aa 0 (where ∑ V is the aggregate voidite volume in a sheet area A ) averaged about unity for 9 sheets of generally similar, voidite-rich appearance. Other sheets are poorer in voidites of measurable dimensions: the ratios for two such sheets averaged 0.25. In the concluding analysis, a reaction involving A and B point defects is proposed for the production of platelets. Other reactions including voidites (but no dislocations) are suggested in which both platelet production and elimination might occur. For the dominating reaction, when a platelet is replaced by a voidite sheet surrounded by an interstitial dislocation loop, models are developed for the cases when the Burgers vector component perpendicular to the loop is either a 0 or ½ a 0 , with the assumption that the platelet nitrogen is dispersed partly into B defects and partly into the voidites. The predicted values of ∑ V /Aa 0 come out as about unity and as 0.25 (or lower) for the larger and smaller Burgers vectors, respectively.


2001 ◽  
Vol 686 ◽  
Author(s):  
S.H. Christiansen ◽  
P.M. Mooney ◽  
J.O. Chu ◽  
A. Grill

AbstractStrain relaxation in He+-implanted and annealed Si(001)/Si1−xGex heterostructures was investigated using transmission electron microscopy techniques and x-ray diffraction. Depending on the implant conditions, bubbles and/or platelets form below the Si/Si1−xGex interface upon annealing and act as nucleation sources for dislocation loops. The dislocation loops extend to the interface and form a misfit dislocation network there, resulting in relaxation of 30-80% of the strain in layers as thin as 100-300 nm. When bubbles form close to the interface, dislocations nucleate by a climb loop mechanism. When smaller bubbles form deeper in the Si substrate an irregular three-dimensional dislocation network forms below the interface resulting in an irregular misfit dislocation network at the interface. When platelets form deeper in the Si substrate, prismatic punching of dislocation loops is observed and dislocation reactions of misfit dislocations at the interface result in Lomer dislocation formation.


Author(s):  
C.D. Cann ◽  
M. Griffiths ◽  
R.C. Styles

In hexagonal metals, such as Zr and Ti, both vacancy and interstitial loops may form (or develop) under neutron irradiation. These loops generally have a perfect Burgers vector, b, although faulted vacancy loops have been observed in Zr. Recently, Griffiths et al. have reported the observation of fringe contrast associated with dislocation loops in Ti. These loops had a b in the direction, but the presence of the fringe contrast was evidence that it differed from a perfect vector. An example of this fringe contrast is shown in Fig. 1. Griffiths et al. suggested two possibilities for the origin of this fringe contrast. The first was that this contrast resulted from impurities in the vicinity of these loops and the second was that the loops were faulted with . However, for both of these possibilities it was difficult to explain that the fringes were observed only under conditions of inside contrast and that fringed loops were observed in only certain grains.


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