Lattice Curvature Research Articles

X-Ray characterisation of indium phosphide substrates

The perfection of vertical gradient freeze (VGF) and liquid encapsulated Czochralski (LEC) grown 〈001〉 sulphur-doped InP crystals has been studied by high resolution X-ray diffraction and synchrotron X-ray topography. Of those examined, all VGF crystals except one were dislocation-free, the exception having a uniform dislocation density of about 200 cm −2 resulting from slip. The double axis diffraction maps and double crystal topographs showed most crystals to have a uniform lattice curvature. The largest radii of curvature were from VGF wafers, while the smallest values were for LEC wafers.

Investigation by synchrotron radiation X-ray topography of lattice tilt formation in partially released InGaAs/GaAs compositionally graded layers

The synchrotron radiation plane wave topography and the Rutherford backscattering technique have been used to investigate the lattice tilts between substrates and layers introduced by the preferential orientation of the Burgers vectors of the misfit dislocations in compositionally graded InGaAs/GaAs heterostructures. A monotonic change of the lattice tilt along the sample surface producing in average a concave curvature of the buffer layer lattice has been found with a nearly complete alignment of the Burgers vector of the misfit dislocations at the sample edges. The topographic observations showed that the buffer layers can follow a nearly continuous curvature or can be sub-divided in large domains with different average lattice tilt. The models recently proposed for the formation of tilt in partially released structures are not able to explain the present observation of a lattice tilt varying coherently along the sample surface.

Domain misorientation in sublimation grown 4H SiC epitaxial layers

High resolution X-ray diffractometry has been applied to study domain misorientation in 4H SiC epitaxial-layers grown by the sublimation epitaxy method. The development of mosaicity of the epitaxial layer has been studied under different growth conditions, i.e. substrate polarity and layer thickness. In the growth experiments on Si-face with a lattice having a concave curvature the mosaicity was increased in the layer. With increasing thickness the structural quality of the layer was improved. Growth of an epilayer on concave C-face resulted in a domain structure comparable to the substrate but the lattice curvature increased considerably. The concave lattice plane curvature is a probable reason for degraded structural quality.

Synchrotron x-ray reticulography: principles andapplications

Synchrotron x-ray reticulography is a versatile new technique for mapping misorientations in single crystals. It is nearly as simple to perform as conventional single-crystal Laue topography, yet it yields quantitative data on misorientations that would demand long sequences of images if the double-crystal technique were applied. In reticulography a fine-scale x-ray absorbing mesh is placed between a Laue-diffracting crystal specimen and the topograph-recording photographic plate. The mesh splits the diffracted beam into an array of individually identifiable microbeams. Direction differences between microbeams, which give the orientation differences between the crystal elements reflecting them, are measured from their relative shifts within the array when mesh-to-plate distance is changed. The angular sensitivity of reticulography depends upon the angular size of the x-ray source. At Station 7.6 at the SRS, Daresbury, 80 m from the tangent point, and with source size FWHM (full width half maximum) = 0.23 mm vertically, the incidence angular range in the vertical plane is only 0.6 arcsec, and misorientations down to this magnitude are measurable. Applications of reticulography to three quite different problems are described, illustrating the method's versatility. The problems are: (1) measuring surface lattice-plane tilts due to an array of dislocations in a large synthetic diamond; (2) determining the sense of the Burgers vector of a giant screw dislocation in SiC; and (3) measuring lattice curvature above an energetic ion implant in a natural diamond.

Lattice curvature of InxGa1−xAs/GaAs [001] graded buffer layers

Ion channeling analysis and x-ray diffraction reciprocal space maps have been performed on InxGa1−xAs buffer layers grown with different composition profiles on well-cut (001) GaAs substrates. On all of the samples analyzed we detect a curvature of the layer lattice, i.e., a tilt of the lattice with respect to the substrate which varies coherently along the sample surface. The layer tilt is directed inward defining a curvature that is concave, large (up to 2.5° cm−1) and that decreases when approaching the substrate. We describe this new phenomenon in terms of a coherent lateral distribution of the orientations of the misfit dislocation Burgers’ vectors.

A simplified crystallographic approach of bifurcation for single crystals and polycrystals

Localization of the plastic deformation such as necking, kink and shear bands in polycrystal and single crystals leads to inhomogeneous strain field and lattice rotation field. In order to study at microscopic and macroscopic scales the formation and propagation of such instabilities, single copper crystals and steel polycrystals were deformed in tension within a SEM. The strain field and lattice rotation field were analyzed at the micrometer scale for different steps of the plastic deformation, using simultaneously microextensometry (microgrids) and electronic diffraction (EBSD) within the SEM. Three components of the Green Lagrange tensor, of the strain rate and lattice spin were obtained within the band and the matrix before and during bifurcation. Localization in cfc single crystal and steel (bcc) polycrystals, presented similarities conceming the activation of two shear planes in the necking area and the evolution of the inhomogeneous strain field and lattice rotation. These phenomena can be correlated to the activation of two crystallographic slip systems for the single crystal and two families of parallel slip planes within the grains for the polycrystals. The gradients were analyzed by Cosserat theory: a constitutive law taking into account the lattice curvature (through couple stress tensor) and hardening law dependent on a density of geometrically necessary dislocations within the band. Such analysis previously developed for a two dimensional single crystal, is applied to the polycrystal assumed to a super single crystal.

Finite deformation Cosserat-type modelling of dissipative solids and its application to crystal plasticity

Recently, several proposals were made for the enlargement of the classical field equations in order to solve locally inhomogeneous deformation problems (e.g. at shear banding and damage localization or of composites). In the present paper basic issues of this topic will be treated: i) the implicit dependence within the gradient plasticity theory, ii) the more open micromorphic view-point of the gradient of internal variable approach, iii) a derivation of an elastic-plastic decomposition of the Cosserat strain measures, iv) its application to the description of lattice curvature in crystals. Finally, finite element simulations of strain localization in Cosserat single crystals are presented.

Mesoscale investigation of the deformation field of an aluminum bicrystal

The purpose of this paper is to describe a new experimental approach to the study of deformation heterogeneity in crystalline materials. The approach is similar to that reported by Randle, et al., but extends their work to make more precise connections with the continuum theory of dislocations introduced by Nye, and later developed by Kroner, Bilby, et al. and others. The authors first make a precise connection between the fundamental equation of continuum dislocation theory, linking the curvatures of the lattice and the elastic strain field with the dislocation tensor. A precise connection is made between lattice curvature and orientation measurements obtained from electron backscattering diffraction patterns (EBSP) by the method known as orientation imaging microscopy (OIM).

Influences of crystallographic misorientation of GaAs substrateson misfit stresses and microhardness of InGaP epilayers

InGaP epilayers were grown on the on-axis cut, 2° off, 6° off, and 10° off GaAs substrates by organo-metallic vapor phase epitaxy, and the influences of crystallographic misorientation of the substrate on the structural properties such as lattice mismatch, elastic strain, lattice curvature, misfit stress, X-ray line-width, and microhardness of the epilayers were investigated in this study. The material characterizations were carried out by the TXRD (triple-axis X-ray diffractometer) and the nanoindenter. With an increase of the substrate misorientation angle (S.M.A.), the relative incorporation of Ga atoms on the substrate surface was found to increase. Also, with an increase of S.M.A., the X-ray line-width of the InGaP epilayer was reduced, indicating that the crystal quality of the epilayer could be improved if the misoriented substrates were used. The elastic accommodation of the strain-free lattice misfit was found to be more remarkable in the misoriented samples. It was demonstrated, for the first time that the microhardness of the InGaP epilayer decreased with an increase of S.M.A. This indicated that the variation of microhardness of the epilayer was related to the improvement of the elastic characteristics and atomic displacement induced by the misoriented substrate. The results obtained demonstrated that the elastic characteristics and the crystal quality of the InGaP epilayer were remarkably enhanced and the microhardness of the epilayer decreased when the misoriented substrates were employed.

Selective ion-channeling study of misfit dislocation grids in semiconductor heterostructures: Theory and experiments

Planar dechanneling by networks of misfit dislocations was measured in a series of ${\mathrm{In}}_{x}{\mathrm{Ga}}_{1\ensuremath{-}x}\mathrm{A}\mathrm{s}/\mathrm{G}\mathrm{a}\mathrm{A}\mathrm{s}$ samples (001) grown by molecular-beam epitaxy. At the beginning of the strain-relaxation process the dechanneling probability exhibits different values for nominally equivalent (110) planes. At larger strain relaxation, the dechanneling probability saturates at a value around $$$\frac{1}{2}$ as the beam-energy increases. In order to explain these results a new model for planar dechanneling by dislocations is proposed. This model is based on the harmonic approximation of the continuum potential but anharmonicity effects are taken into account. The perturbation to the harmonic oscillations caused by the lattice plane curvature around a dislocation is written in terms of a distortion function that depends only on the geometrical configuration of the channeling direction and of the dislocation line. This function is explicitly calculated for geometrical configurations relevant to the present samples allowing us then to solve the equation of motion. The results show that the dechanneling probability saturates at a level sensibly lower than 100% due to the quasiplanar distribution of dislocations. Without any adjustable parameter, the comparison between computed and measured dechanneling probabilities supplies dislocation density values in excellent agreement with those measured by transmission electron microscopy and in good agreement with results deduced from previous strain-relaxation data.

Orientation imaging microscopy: Emerging and future applications

Orientation imaging microscopy (OIM), which images the microstructure using the electron backscattered diffraction probe, has proven to be a powerful tool for the study of polycrystalline materials. OIM's chief strength lies in its ability to couple the morphological aspects of microstructure, with lattice orientation. With spatial and angular resolution of at least 100 nm and 1°, OIM is ideally suited to investigations at the mesoscale. In this paper three examples are described which illustrate emerging and future applications of OIM to important problems in materials science at the mesoscale. The first example is the use of higher-order point statistics to obtain improved bounds on structure-insensitive properties. It is emphasized that the approach may be used, not only to enhance material properties, but also to control their variability. The second example is an emerging analysis to map the relationships between the character (type) of grain boundaries and their intrinsic properties. The main idea is illustrated for mappings linking character and excess free energy. The third example focuses on the ability of OIM to measure lattice curvature, and to thereby obtain information about the geometrically necessary dislocation content of the microstructure. This example is illustrated by analysis to extract the dislocation density tensor.

Curved crystal lattice in resolidified submicron Al lines

The microstructure of Al lines resolidified in an amorphous groove pattern, is investigated by backscatter Kikuchi diffraction, x-ray diffraction, and transmission electron microscopy (TEM). If a temperature gradient is present during cooling, the Al lines do not have the common (111) fiber texture. Instead, the (111) crystal planes are parallel to the vertical sidewalls of the grooves and the crystal lattice shows a remarkable, constant curvature over the entire line length of 1 cm. TEM revealed parallel dislocations in the bulk of the line and small crystallites at the interfaces. It is suggested that the (111) sidewall texture is a consequence of the sidewall area being larger than the bottom area. The observed lattice curvature is explained by a nonisotropic movement of dislocations during cooling.

A Cosserat theory for elastoviscoplastic single crystals at finite deformation

IN THIS WORK, displacement and lattice rotation are regarded as independent degrees of freedom. They are connected only on the constitutive level and by the balance equations. The description of plastic deformation is based on the slip theory. Elastic lattice curvature and torsion are associated with couple-stresses. The continuum theory of dislocations has been revisited to derive the kinematics of plastic lattice torsion-curvature. Explicit constitutive equations and hardening rules are proposed to close the theory in the case of elastoviscoplasticity. The thermodynamical formulation of the model involves internal variables which are similar to the densities of statistically stored dislocations and the densities of geometrically necessary dislocations. Accordingly, the proposed Cosserat theory can be regarded, on the one hand, as the classical crystal plasticity theory complemented by lattice curvature and torsion variables and, on the other hand, as the continuum theory of dislocations closed by the missing hardening variables and constitutive equations within the appropriate micropolar framework. A generalization of Mandel's elastoviscoplastic decomposition of strain is used especially for the torsion-curvature measure at finite deformation.

On The Measurement Of Residual Stress In Thin Films

ABSTRACTThis work reports a comparative study on three curvature based techniques for stress measurement in thin films, including double crystal diffraction topography (DCDT), profilometry, and laser scanning technique (LST). These techniques measure different physical quantities. In the former case, the curvature of crystal lattice planes is assessed while in the latter two cases, the curvature of physical surface of a sample is measured. The current experiments use these three techniques to determine the quantities of interest for a variety of films deposited on Si (100) wafers. In general, profilometry and LST produce similar results on surface curvature. For specimens where the residual stresses produces large curvatures of both types (lattice curvature and surface curvature), the results by DCDT and LST agree fairly well. When small to moderate curvatures are present, the two methods deviate to varying degrees on absolute curvatures. Nevertheless, DCDT and LST generally yield similar results on differential curvatures, i.e., the stress induced curvature differentials. When proper consideration is taken for the inherent limits of each technique, both DCDT and LST can be used as valid procedures for stress measurement in thin film-substrate systems.

Subgrain growth and misorientation of the α matrix in an ( α + γ) microduplex stainless steel

The change of the ( α + γ) microduplex structure during annealing in an Fe-26%Cr-7%Ni alloy has been investigated with emphasis on the change in the misorientation of the α matrix subgrain boundaries with subgrain growth in a large range of subgrain size. A long time holding up to 360 ks at 1273 K caused significant structure coarsening, and the growth of the α matrix subgrains and γ particles obeyed the third power law. On the other hand, the average misorientation of the α subgrain boundaries changed insignificantly, even though the α subgrain size markedly increased from 2.6 to 15.6 μm. This insignificance in the misorientation resulted from the fact that the variation of the local lattice rotation in the α matrix is not monotonic but periodic, and no long range lattice curvature exists.

Subgrain growth in heavily deformed aluminium—experimental investigation and modelling treatment

A comprehensive study of the annealing behaviour of two variants of commercial purity aluminium alloys has been carried out. The primary objectives of this work have been to improve the understanding of the recovery processes in itself as well as to study the effect of recovery on the annealing behaviour of heavily deformed aluminium. The experimental part has focused, firstly on describing the as deformed structure with emphasis on sub-boundary misorientations and long range lattice curvature, secondly on following subgrain growth and characterization of the softening reaction. The modelling part treats subgrain growth as a reaction controlled by sub-boundary migration. The significance of sub-boundary flexibility in this context is emphasized, an aspect which requires lateral drift of ledges as an important element in the boundary migration mechanism. A subgrain growth model based on thermally activated migration of ledges has been developed and applied to the experimental results.

Lattice curvature induced substrate X-ray line broadening in InGaP/GaAs heterostructures

The interfacial coherency of InGaP/GaAs heterostructure wafers grown by the metal-organic chemical vapor deposition technique is examined and its effects on the X-ray line broadening of the GaAs substrate are investigated. Lattice mismatches are measured using both (400) symmetric and [511] asymmetric reflections. The strain-free chemical lattice misfit and the elastic strain are also calculated. The X-ray full width at half maximum (FWHM) of the substrate is measured, and its dependence on the degree of lattice mismatch is found. In order to understand this observation, measurements of lattice curvature are carried out using the automatic Bragg angle control method. It is found that the radius of curvature varies with the elastic strain and the substrate FWHM. Using the results obtained, the misfit stress is also determined. Various contributions to the substrate line broadening are separated out. Calculated line widths are shown to be in good agreement with measured values. It is thus concluded that the primary factor affecting the X-ray line width of the substrate is misfit stress induced lattice curvature. The results presented demonstrate that interfacial coherency is of major importance in influencing the structural properties of InGaP/GaAs heterostructures through elastic strain and lattice curvature.

Evaluation of substructure characteristics from the X-ray divergent beam back reflection patterns

en

On the effect of persistent slip band (PSB) parameters on fatigue life: Cao, W.-D. and Conrad, H. Fatigue Fract. Eng. Mater. Struct. June 1992 15, (6), 573–583

We study the mechanisms of slip transfer at a grain boundary, in titanium, using Differential Aperture X-ray Laue Micro-diffraction (DAXM). This 3D characterisation tool enables measurement of the full (9-component) Nye lattice curvature tensor and calculation of the density of geometrically necessary dislocations (GNDs). We observe dislocation pile-ups at a grain boundary, as the neighbour grain prohibits easy passage for dislocation transmission. This incompatibility results in local micro-plasticity within the slipping grain, near to where the slip planes intersect the grain boundary, and we observe bands of GNDs lying near the grain boundary. We observe that the distribution of GNDs can be significantly influenced by the formation of grain boundary ledges that serve as secondary dislocation sources. This observation highlights the non-continuum nature of polycrystal deformation and helps us understand the higher order complexity of grain boundary characteristics.

EBSP, Progress in Technique and Applications

To measure grain orientation in SEM with high accuracy a new EBSP calibration software is developed. Grain orientation determination is simply based on marking three zone axes in the live EBSP. The EBSP-technique has been applied to study misorientation between subgrains in high purity and commercially pure aluminium. The lattice curvatures, which develop during deformation, increase with the amount of deformation. During annealing the subgrains grow, resulting in an increased average neighbouring misorientation due to misorientation accumulation.