Facet Edges Research Articles

Model for molecular-beam-epitaxy growth over nonplanar surfaces

A model for molecular-beam-epitaxy growth is proposed to describe the morphology formed on a nonplanar, profiled surface that may consist of multifaceted crystal structures. Anisotropy of growth rate is introduced as a main factor for explaining the observed growth morphology. We show that on the nonplanar surface, the migration of adatoms modifies the growth rate at each point of the growing surface and thus affects the formation of the morphology within the range of migration length. The model can successfully describe the growth behavior including the development of sharp facet edges, since it is formulated in terms of a difference (rather than differential) equation which does not require the continuity of surface slope. The morphology simulated on the basis of this model agrees to experimental results of GaAs growth.

Transmutation of the vicinal surface exponent due to gravity.

Near the edge of a facet, vicinal surfaces curve away from the plane with exponent (3/2. With gravity normal to the facet the exponent may change to 3. Generally, there will be a crossover from 3 to (3/2 as the facet edge is approached. Comparison with experiments is made.

Finite-size effects cannot explain experimental equilibrium crystal shape

Experiments on equilibrium metallic and helium crystals show an 'anomalous' mean-field behaviour of crystal shape near facet edges, in contradiction with existing microscopic theories. The possible finite-size, or 1/R, region of the observed behaviour is analysed. It is shown that these 1/R effects are negligible, even for very small crystals ( approximately 1 mu m). The author conclude that, neither finite-size effects nor classical long-range step-step interaction can account for the observed experimental behaviour.

Universal Behavior of the Equilibrium Crystal Shape near the Facet Edge. I. A Generalized Terrace-Step-Kink Model

A generalized version of the terrace-step-kink (TSK) model is introduced to study in detail the shape of a crystal near the facet edge. The model is analyzed by the transfer-matrix method combined with the lattice fermion approach. As a result, behavior of the curvature near the facet edge is found to have unexpected new features: finite Gaussian curvature jump with universal amplitude and vanishing transverse curvature with universal exponent.

Roughening, Faceting and Equilibrium Shape of Two-Dimensional Anisotropic Interface. I. Thermodynamics of Interface Fluctuations and Geometry of Equilibrium Crystal Shape

Thermodynamics and geometry of two-dimensional anisotropic interfaces are discussed, which serves as a foundation for detailed quantitative studies of the `shape transitions' ( e.g. faceting transition, facet edge problem, etc.) in the equilibrium crystal shape (ECS). A fluctuation theory based on the Andreev's formalism for the ECS is utilized to derive a generalized relationship between the interface stiffness tensor and the curvature tensor of a surface describing the ECS. As an application, new Monte-Carlo method to calculate the curvature of the ECS is proposed.

Morphologic edge detection

Edge operators based on gray-scale morphologic operations are introduced. These operators can be efficiently implemented in near real time machine vision systems which have special hardware support for gray-scale morphologic operations. The simplest morphologic edge detectors are the dilation residue and erosion residue operators. The underlying motivation for these and some of their combinations are discussed and justified. Finally, the blur-minimum morphologic edge operator is defined. Its inherent noise sensitivity is less than the dilation or the erosion residue operators. Some experimental results are provided to show the validity of these morphologic operators. When compared with the enhancement/thresholding edge detectors and the cubic facet second derivative zero-crossing edge operator, the results show that all the edge operators have similar performance when the noise is small. However, as the noise increases, the second derivative zero-crossing edge operator and the blur-minimum morphologic edge operator have much better performance than the rest of the operators. The advantage of the blur-minimum edge operator is that it is less computationally complex than the facet edge operator.

Statistical Mechanical Theory of the Facet Edge of a Crystal

The facet edge of the crystal is investigated theoretically on the basis of the Terrace-Step-Kink (TSK) model. The statistical mechanics of this model is formulated in terms of the lattice Fermion approach. The analysis leads always to the Pokrovsky-Talapov-Gruber-Mullins transition. At the facet edge, the facet connects with the neighboring curved surface without discontinuity of the slope. In such model the lattice underlying the surface is assumed to be rigid without any deformation caused by emergence of the steps. We extend the TSK model to a more general model including relaxation of the lattice bonds caused by the steps. The lattice deformation increases the elastic energy but can decrease the step creation energy. We show that the extended model leads always to a sharp edge; there is a discontinuity of the slope at the facet edge.

The behaviour of steps on a crystal facet under small supersaturation

The geometry of individual steps on a crystal facet is fixed by the anisotropic step energy β(ϕ). On the basis of effective stresses on a step element, we briefly discuss the step shape and its matching angles with the facet edge or with a wall. When the step originates from a single screw dislocation, the facet may be metastable if the dislocation lies off centre. When the number of screw dislocations is large, we show that for a large enough supersaturation F the trapped steps behave as if they were free (through successive rearrangements). For long nearly aligned steps, this threshold F is much lower than the usual Frank-Read threshold for neighbouring ± dislocations. We illustrate this « collective behaviour » of Frank-Read sources on a few examples.

Morphologic Edge Detection

Edge operators based on grayscale morphologic operations are introduced. These operators can be efficiently implemented in near real time machine vision systems which have special hardware support for grayscale morphologic operations. The simplest morphologic edge detectors are the dilation residue and erosion residue operators. The underlying motivation for these is discussed. Finally, the blur minimum morphologic edge operator is defined. Its inherent noise sensitivity is less than the dilation or the erosion residue operators. Some experimental results are provided to show the validity of the blur minimum morphologic operator. When compared with the cubic facet second derivative zero-crossing edge operator, the results show that they have similar performance. The advantage of the blur-minimum edge operator is that it is less computationally complex than the facet edge operator.

The structure of the adenovirus capsid: II. The packing symmetry of hexon and its implications for viral architecture

The orientation and location of the 240 hexons comprising the outer protein shell of adenovirus have been determined. Electron micrographs of the capsid and its fragments were inspected for the features of hexon known from the X-ray crystallographic model as described in the accompanying paper. A capsid model is proposed with each facet comprising a small p3 net of 12 hexons, arranged as a triangular sextet with three outer hexon pairs. The sextet is centrally placed about the icosahedral threefold axis, with its edges parallel to those of the facet. The outer pairs project over the facet edges on one side of the icosahedral twofold axes at each edge. The model capsid is defined by the underlying icosahedron, of edge 445 Å, upon which hexons are arranged. The hexons are thus bounded by icosahedra with insphere radii of 336 Å and 452 Å. A quartet of hexons forms the asymmetric unit of an icosahedral hexon shell, which can be closed by the addition of pentons at the 12 vertices. Considering the hexon trimer as a complex structure unit, its interactions in the four topologically distinct environments are very similar, with conservation of at least two-thirds of the inter-hexon bonding. The crystal-like construction explains the flat facets and sharp edges characteristic of adenovirus. Larger “adenovirus-like” capsids of any size could be formed using only one additional topologically different environment. The construction of adenovirus illustrates how an impenetrable protein shell can be formed, with highly conserved intermolecular bonding, by using the geometry of an oligomeric structure unit and symmetry additional to that of the icosahedral point group. This contrasts with the manner suggested by Caspar & Klug (1962), in which the polypeptide is the structure unit, and for which the number of possible bonding configurations required of a structure unit tends to infinity as the continuously curved capsid increases in size. The known structures of polyoma and the plant viruses with triangulation number equal to 3 are evaluated in terms of hexamer-pentamer packing, and evidence is presented for the existence of larger subunits than the polypeptide in both cases. It is suggested that spontaneous assembly can occur only when exact icosahedral symmetry relates structure units or sub-assemblies, which would themselves have been formed by self-limiting closed interactions. Without such symmetry, the presence of scaffolding proteins or nucleic acid is necessary to limit aggregation.

Simple model for crystal shapes: Step-step interactions and facet edges

The terrace-step-kink model for equilibrium crystal shapes is considered. Noninteracting steps are known to correspond to a free-fermion model which leads to a continuous transition from facets to curved surfaces. We study both short- and long-ranged interactions between steps within meanfield theory. For nearest-neighbor step interactions, the model can be solved exactly, and details are given. The possibility of a slope discontinuity between facets and curved surfaces is explored within the interacting terrace-step-kink model (which ignores voids and overhangs); this possibility is realized only for sufficiently long-ranged attractive interactions. As physical examples of such interactions we consider both elastic and dipolar interactions between steps. These have the same range, and may be comparable in magnitude. It is argued that elastic interactions (which are repulsive) do not change the free-fermion predictions for the exponent governing the facet-curved surface edge while dipolar interactions, if attractive, can lead to slope discontinuities and associated tricritical phenomena.

High-quality submicron niobium tunnel junctions with reactive-ion-beam oxidation

A new reactive-ion-beam oxidation technique has been applied to the fabrication of rugged, high-quality niobium-lead alloy Josephson tunnel junctions. Control of critical current density over a wide range is possible, and critical current densities exceeding 106 A/cm2 have been obtained. In addition, a process-compatible edge geometry has been developed which allows a junction to be formed on the faceted edge of a niobium base electrode, yielding devices with areas of 10−9 cm2 using 1-μm photolithography.

The cracking of Czochralski-grown crystals

Crystals of brittle materials are easily cracked during manufacture and processing. This paper examines the thermal strains existing in crystals during and after pulling and the strains which can be built-in by the facet effect. It is shown that during the growth of a crystal with a radius R, there is a maximum acceptable axial gradient roughly proportional to R −1.5 and that to prevent cracking after growth there is a maximum rate of cooling proportional to R −2. Results obtained with bismuth silicon oxide for which all the relevant parameters (breaking strain, expansion coefficient, thermal diffusivity and cooling constant) have been measured are in accord with these relations at least for 4 < R<23 mm. The analysis seems valid for other materials (e.g. lithium niobate) with small breaking strains and negligible plastic ranges. For bismuth silicon oxide, (110) and (100) facets have lattice constants larger than the bulk of the crystal by about 4 and 6 parts in 10 5 respectively. The edges of these facets are thus regions which are strained by amounts corresponding to 20 or 30% of the breaking strain. Under normal growth conditions, this strain is not particularly significant, but it does affect post-growth handling and during rapid growth, interface instabilities develop first at the facet edges.

Precipitation in thin foils of Al-4 wt.% Cu alloy—II. Growth kinetics of θ precipitates

The mechanisms and kinetics of precipitates grown at the surfaces of thin foils have been studied by hot-stage electron microscopy. Emphasis has been placed on three typical morphologies: circular plates, rectangular plates and needles. It has thus been found that rounded parts of the edges of these precipitates, where the interfacial structure is essentially disordered, migrate in a diffusion controlled manner. Faceted edges of the plates migrate by the ledge mechanism. These results are consistent with general morphological ideas developed for bulk-grown precipitates. In thin foils, however, the effective diffusivity is much larger than that of the bulk, because the surface provides a short circuit diffusion path. Quantitative support for this conclusion has been obtained from detailed analyzes of the kinetics.

On a Fossil Bird and a Fossil Cetacean from New Zealand

Some time ago, my friend Mr. Walter Mantell submitted to my examination two fossil bones from tertiary deposits at Kakaunui and Parimoa in New Zealand. Of these, the one is the right tarso-metatarsal bone of a Bird belonging to the Penguin family, the other the humerus of a Cetacean of small size. Fossil Bird. —The former bone (of which a front view is represented in fig. 1, and a back view in fig. 2) measures two inches and a half in extreme length, and rather more than an inch and a quarter across its proximal end. The precise width at the distal end cannot be given, as the innermot part of this extremity ofthe bone has been broken away; what remains measures one and one-sixteenth of an inch. The proximal end of the bone presents two articular facets,—the one internal, an oval, shallow concavity, looking upwards and a little inwards, the other, external, quadrilateral, slightly convex from before backwards, slightly concave from side to side, and inclined more obliquely upwards and outwards. The two facets are separated by a stout median ridge, which rises into a conical tuberosity anteriorly, but dies away posteriorly into a shallow triangular pit. The posterior edges of both facets are rather more raised than the anterior ones; and marked transverse depressions separate both from the upper extremities of the four strong calcaneal ridges which project from the upper part of the posterior face of the bone (fig. 2). Of these, the innermost is the strongest and longest; and a deep groove divides it from the two middle ones,