Obtuse Wedge Research Articles

Identification and Formation Mechanism of a New Interstitial Defect Found in Ion Implanted Silicon

For 40 years, extended defects are found in silicon after ion implantation and annealing. Widely studied for shallow implantation, thermal evolution of these defects follow the same path in the particular case of a Si layer transferred by Smart Cut™ technology.Being extrinsic or intrinsic precipitates, these defects are formed under supersaturated condition and grow by interchanging interstitials, or vacancies, during annealing. All along their growth, the precipitates rearrange them-selves to transform into a more stable structure.A transferred Si layer needs a combination of very high-temperature annealing in neutral and oxidizing atmospheres to thin and smooth the fractured surface. Despite these treatments, irregular topography has been detected on the extremely smooth surface. Studying them in detail, we have found a new type of interstitial-type crystalline defects, extremely stable, that we have named “tent-like” defects in reference to their shape. The presence of such structures was a surprise as it is generally believed that the high-temperature treatment following fracture was sufficient to dissolve all the precipitates initially present in the transferred Si layer.It is the goal of this work to report the characteristics, formation mechanism and origin of these structures. From that, we infer, and then experimentally confirm the process conditions promoting or hampering their formation.For this work, 300 nm-thick Si layers were transferred by using H+ and other species, wafer bonding and thermal initiation of the fracture. Afterwards, the transferred layers were subjected to different annealing steps, oxidizing or not, then annealed at temperatures up to 1200°C under Ar atmosphere.Figure 1a is a typical Scanning Electron Microscopy (SEM) image of one of these structures. Such images were obtained using a low voltage SEM with the ability to image mono-atomic steps separating flat (001) terraces all over the surface of a 300 mm wafer. Step bunching imaged on the right of the defect, as contour lines, reveals some “erosion” around the rectangular shape, at the origin of a shallow hole which was initially detected. In general, these structures are elongated along one of the two in-plane <110> directions (Fig. 1a). Their length increases during oxidizing annealing whereas their width stays almost constant. Since silicon oxidation results in the injection of Si interstitials into the underlying crystal, this growth suggests they are extrinsic defects i.e., contain excess Si atoms.Figure 1b is a HREM image of such a defect, cut in its body part, as shown in Fig. 1a. Two intersecting {111} stacking faults (SF) define a V-shape defect (Fig. 1b). Figure 1c shows the geometric phase distributions around the structure obtained by Dark Field Electron Holography (DFEH) for g= 004, g=111 and g=-1-11. From such images (no phase jump for g=004 and mirror symmetry between g=111 and g=-1-11), the Burgers vector of the dislocation, seen edge-on at the SFs intersection, is found to be 1/6[-1-10], i.e. that of a stair-rod Lomer-Cottrell dislocation. This finding suggests that the tent-like defect probably results from the dissociation of straight dislocation segments, each following a path such as 1/3<-1-11>=1/6<-1-10>+1/6<-1-12>.Based on these experimental data, we propose a scenario describing the formation of such structures. At the beginning an interstitial loop (Frank or perfect) was buried in the matrix. Such loops are known to form in ion implanted silicon annealed at high enough temperature. During oxidation, such a loop can grow up to, or may be cut by, the SiO2/Si interface. It then forms a half-loop. Based on energetic criteria, we show that such a half-loop tends to elongate and becomes segmented. During high-temperature annealing, the dislocation lines dissociate: a 2D defect transforms into a 3D “obtuse wedge elongated defect” bound by four {111} SFs and five segments of stair-rod Lomer-Cottrell dislocations. This dissociation is similar to the one describing the formation of tetrahedral defects from triangular Frank dislocation loops.In summary, we have found a new type of extended defects in Si, of interstitial character, which may appear as a result of the precipitation of silicon interstitials. This defect has not been identified before as the conditions for its formation are quite stringent. Beyond the process conditions under which they have been observed at first, the mechanisms which lead to their formation requires that a highly I-supersaturated region is initially located very close to a free surface (possibly an interface) where a half-dislocation loop can form and then dissociate, a condition rarely found in real cases with the notable exception of layers transferred by the Smart Cut™ process. Figure 1

Rainfall induced groundwater mound in wedge-shaped promontories: The Strack–Chernyshov model revisited

An analytical solution to the Poisson equation governing Strack's discharge potential (squared thickness of a saturated zone in an unconfined aquifer) is obtained in a wedge-shaped domain with given head boundary conditions on the wedge sides (specified water level in an open water body around a porous promontory). The discharge vector components, maximum elevation of the water table in promontory vertical cross-sections, quantity of groundwater seeping through segments of the wedge sides, the volume of fresh groundwater in the mound are found. For acute angles, the solution to the problem is non-unique and specification of the behaviour at infinity is needed. A “basic” solution is distinguished, which minimizes the water table height above a horizontal bedrock. MODFLOW simulations are carried out in a finite triangular island and compare solutions with a constant-head, no-flow and “basic” boundary condition on one side of the triangle. Far from the tip of an infinite-size promontory one has to be cautious with truncation of the simulated flow domains and imposing corresponding boundary conditions. For a right and obtuse wedge angles, there are no positive solutions for the case of constant accretion on the water table. In a particular case of a confined rigid wedge-shaped aquifer and incompressible fluid, from an explicit solution to the Laplace equation for the hydraulic head with arbitrary time-space varying boundary conditions along the promontory rays, essentially 2-D transient Darcian flows within the wedge are computed. They illustrate that surface water waves on the promontory boundaries can generate strong Darcian waves inside the porous wedge. Evaporation from the water table and sea-water intruded interface (rather than a horizontal bed) are straightforward generalizations for the Poissonian Strack potential.

Optimal streaks in a Falkner–Skan boundary layer

This paper deals with the optimal streaky perturbations (which maximize the perturbed energy growth) in a wedge flow boundary layer, which involves a favorable streamwise pressure gradient. These three-dimensional perturbations are governed by a system of linearized boundary layer equations around the Falkner–Skan base flow. Based on an asymptotic analysis of this system near the free-stream and the leading edge singularity, we show that for acute wedge semi-angle, all solutions converge after a streamwise transient to a single streamwise-growing solution of the linearized equations, whose initial condition near the leading edge is given by an eigenvalue problem first formulated in this context by Tumin [Phys. Fluids 13, 1521 (2001)]. Such a solution may be regarded as a streamwise evolving most unstable streaky mode, in analogy with the usual eigenmodes in strictly parallel flows, and shows an approximate self-similarity, which was partially known and is completed in this paper. An important consequence of this result is that the optimization procedure based on the adjoint equations heretofore used to define optimal streaks is not necessary. Instead, a simple low-dimensional optimization process is proposed and used to obtain optimal streaks. Comparison with previous results by Tumin and Ashpis [AIAA J. 41, 2297 (2003)] shows an excellent agreement. The unstable streaky mode exhibits transient growth if the wedge semi-angle is smaller than a critical value that is slightly larger than π/6, and decays otherwise. Thus, the cases of right and obtuse wedge semi-angles exhibit less practical interest, but they show a qualitatively different behavior, which is briefly described in the supplementary material.

Model of wear of a stiff-plastic material under sliding contact

We consider the interaction of an obtuse wedge (a rigid shaft micro-roughness model) with a stiff-plastic semi-space (a plain bearing bushing), with the formation of a plastic deformation wave. Multiple passing of the wedge through the wave causes accumulation of damage to the bushing material as a result of the loss of plasticity and subsequent separation of the wear particle. The problem is solved in the plane statement using the sliding-line technique. The calculation formulas are presented, as well as a flow-chart of the algorithm of the stressed-deformed-state calculation in the plastic deformation wave and the surface layer damage; an example is given of the plain bearing bushing. Resulting from numerical modeling, dependences are obtained to find the limit number of cycles before destruction of the bearing surface layer.

Exact analytic solutions in plane obtuse wedge's problems based on rigid perfect-plasticity analysis

Exact analytic solutions are provided for the stress and velocity states in statically determinate rigid perfectly-plastic bodies under plane-strain conditions. The analysis concerns straight boundaries with obtuse wedge's geometry under arbitrarily distributed lateral normal and shear loadings. The constructed solutions are expressed either explicitly in the form of specific polynomials (in case of linearly distributed lateral loadings), or implicitly including arbitrary functions (in case of arbitrarily distributed lateral and shear loadings). Plastic regions, fan zones, lines of discontinuity and stress states are analytically determined for obtuse wedges under the action of the prescribed kinds of loading.

Closed-form solutions of the nonlinear partial differential equations governing plane rigid perfect plasticity problems by ad hoc assumptions

Closed-form solutions, including arbitrary functions, of the system of nonlinear partial differential equations in plane rigid perfect plasticity are extracted. This system is reduced to a strongly nonlinear ordinary differential equation which brings, to the solution of the Abel equation of the second kind, exact analytic solutions examples of which were recently constructed. The stress and strain-velocity fields are given and an example for an obtuse wedge under unilateral, uniformly distributed load, under plane-stress conditions, is described indicating that the present method can treat problems yet unsolved. The stress field is described analytically and results for the limiting load are analogous to the classical Prandtl formula.

On sliding an obtuse wedge along the boundary of a perfectly plastic half-space

On sliding an obtuse wedge along the boundary of a perfectly plastic half-space

Electric-dipole radiation over a wedge with imperfectly conductive faces: a first-order physical-optics solution

We solve a three-dimensional (3-D) electromagnetic diffraction problem involving an obtuse wedge with penetrable planar faces and an electric dipole which is parallel to the edge of the wedge. The analytical formulation is based on Stratton-Chu (1941) integrals of the electromagnetic field, which is excited by the dipole source on infinitely extending planes that coincide with the faces of the wedge. Fictitious charges are introduced along the edge to account for the discontinuity of the electromagnetic field on the faces across the edge. We evaluate asymptotically the integral expressions for the electric-field intensity far from the edge to obtain uniformly valid formulas. Our first-order physical-optics solution incorporates single reflection from both faces, the lateral wave, the edge-diffracted space wave, the edge-diffracted lateral wave, and transition terms which ensure that the electromagnetic field is finite and continuous at the single-reflection and lateral-wave boundaries. The numerical results establish the validity of this solution through a reciprocity check and comparisons with other analytical solutions.

Ad hoc exact solutions for the stress and velocity fields in rigid perfectly plastic materials under plane-stress conditions

Making use of convenient and original ad hoc assumptions we construct closed-form solutions for the problem of stress and velocity states in statically determinate rigid, perfectly plastic materials under plane-stress conditions. The obtained solutions are expressed implicitly including arbitrary functions. For the stresses they are extracted by means of the two equilibrium linear PDEs and the appropriate strongly non-linear von Mises–Hencky condition; for the velocities we use the Saint Venant–von Mises theory of plasticity PDEs. Finally, an application concerning the analytical derivation of the stress state, the lines of discontinuity and the limiting load for an obtuse wedge under lateral normal pressure on one side is developed. The advantage of the proposed analytical solution methodology compared to the technique of characteristics is the general applicability delivering from the a priori construction of the slip-lines, as well as the demanded numerical solutions of the corresponding equations of characteristics.

The Plane Asymmetric Problem of Impact of a Solid Obtuse Wedge on the Surface of a Compressible Fluid

The plane problem of vertical impact of a solid obtuse wedge, with sides having different inclinations to the surface of a compressible fluid, onto this surface, is analyzed. The solution of the mixed boundary-value problem is reduced to solving an infinite set of linear integral Volterra equations of the second kind for the coefficients of the expansion of the hydrodynamic pressure into a Fourier series by employing methods of integral Laplace transformations in time, separation of variables an expansion into cosine and sine Fourier series. A numerical example of the time dependence of the fluid force, moment of reaction, angle of symmetry and Mach numbers of the sides of submerging wedges of different mass at different deadrise angles is presented.

Diffraction of a supercompressed detonation wave regularly reflected from the wall of an obtuse wedge

The problem of the diffraction of a supercompressed detonation wave representing the regular reflection of a plane shock wave, by a corner of a wall turned by a small angle ε is solved. The properties of the medium, the intensity and the angle of incidence of the shock wave are assumed to be such that the front does not cause a detonation, while the reflected front initiates it and causes it to disappear. An exact solution is constructed differing from the solutions for the shock wave /1, 2/ only in more complicated expressions for the parameters appearing in it and expressed in terms of the parameters of the regular unperturbed reflection /3/.

An Elongated Porous Wedge under the Action of Pressure

AbstractThe present paper performs an example towards the theory of the slip line field for fluid‐saturated porous solids. The limit load, the stress distribution in the plastic region and the slip line field are determined for an elongated obtuse wedge under the action of one‐sided pressure. The non‐porous solid wedge arises from the present calculations as a limiting case.

The diffraction accompanying the regular reflexion of a plane obliquely impinging shock wave from the walls of an obtuse wedge

The diffraction problem of a plane shock wave at the apex of an obtuse wedge treated by Lighthill (1950) is extended by assuming that the shock wave strikes the walls of the obtuse wedge at some finite oblique angle of incidence (not exceeding the critical angle). Transformations similar to that performed in the above-mentioned paper lead to a non-symmetrical boundary-value problem for an analytic function of a complex variable having a non-homogeneity in the form of a delta-function. It was found possible to extend, for the case considered, the method developed by Lighthill and construct the solution in almost as simple a form as given in the above-mentioned paper. The case of three-dimensional stationary flow is considered when the line of reflexion makes a finite angle with the edge of the wedge.

The elastic behaviour of a sharp obtuse wedge impressed on a plane

The conditions have been examined experimentally under which a sharp but sufficiently obtuse wedge of elastic material may be forced on to a plane of similar elastic material with no consequent permanent deformation. Applications to the design of knife-edges for precision balances are dicussed. The torque developed when such an elastic system is tilted under load is considered theoretically and experimentally. Applications in the design of fluid seals and the generation of high pressures are discussed.

On the cutting diamond

The author, having never met with any satisfactory explanation of the property which the diamond possesses of cutting glass, has endeavoured, by experiment, to determine the conditions necessary for this effect, and the mode in which it is produced. The diamonds chosen for this purpose are naturally crystallized, with curved surfaces, so that the edges are also curvilinear. In older to cut glass, a diamond of this form requires to be so placed that the surface of the glass is a tangent to a curvilinear edge, and equally inclined laterally to the two adjacent surfaces of the diamond. Under these circumstances the parts of the glass to which the diamond is applied are forced asunder, as by an obtuse wedge, to a most minute distance, without being removed; so that a superficial and continuous crack is made from one end of the intended cut to the other. After this, any small force applied to one extremity is sufficient to extend this crack through the whole substance, and successively across the whole breadth of the glass. For since the strain at each instant in the progress of the crack is confined nearly to a mathematical point at the bottom of the fissure, the effort necessary for carrying it through is proportionally small. The author found by trial that the cut caused by the mere passage of the diamond need not penetrate so much as 1/200th of an inch.