scholarly journals Threading Dislocations Piercing the Free Surface of an Anisotropic Hexagonal Crystal: Review of Theoretical Approaches

2018 ◽  
Vol 2018 ◽  
pp. 1-8
Author(s):  
Salem Neily ◽  
Sami Dhouibi ◽  
Roland Bonnet
Keyword(s):  
Free Surface ◽  
Fourier Series ◽  
Dislocation Core ◽  
Surface Region ◽  
Double Fourier Series ◽  
Threading Dislocations ◽  
Core Size ◽  
Hexagonal Crystal ◽  
Integral Formalism ◽  
Theoretical Approaches

Inclined threading dislocations (TDs) piercing the oriented free surface of a crystal are currently observed after growth of oriented thin films on substrates. Up to date the unique way to treat their anisotropic elastic properties nearby the free surface region is to use the integral formalism, which assumes no dislocation core size and needs numerical double integrations. In a first stage of the work, a new and alternative approach to the integral formalism is developed using double Fourier series and the concept of a finite core size, which is often observed in high-resolution transmission electron microscopy. In a second stage, the integral formalism and the Fourier series approaches are applied to the important case of a TD piercing the basal free surface of a hexagonal crystal. For this particular geometry, easy-to-use expressions are derived and compared to a third approach previously known for a plate-like crystal. Finally, the numerical interest and the convergence of these approaches are tested using the basal free surface of the GaN compound, in particular for TDs with Burgers vectors c and (a + c).

Steady-state modelling method for matrix-reactance frequency converter with boost topology

2011 ◽  
Vol 60 (2) ◽  
pp. 137-148
Author(s):  
Igor Korotyeyev ◽  
Beata Zięba
Keyword(s):  
Fourier Series ◽  
Steady State ◽  
Differential Equations ◽  
Numerical Method ◽  
Frequency Converter ◽  
Double Fourier Series ◽  
Galerkin’S Method ◽  
Galerkin's Method ◽  
Modelling Method ◽  
Three Phase

Steady-state modelling method for matrix-reactance frequency converter with boost topologyThis paper presents a method intended for calculation of steady-state processes in AC/AC three-phase converters that are described by nonstationary periodical differential equations. The method is based on the extension of nonstationary differential equations and the use of Galerkin's method. The results of calculations are presented in the form of a double Fourier series. As an example, a three-phase matrix-reactance frequency converter (MRFC) with boost topology is considered and the results of computation are compared with a numerical method.

Double Fourier Series

2018 ◽  
pp. 1
Author(s):  
Amira Ali Een Faied
Keyword(s):  
Fourier Series ◽  
Double Fourier Series

scholarly journals The double Fourier series of a discontinuous function

1924 ◽  
Vol 106 (737) ◽  
pp. 299-314 ◽  
Keyword(s):  
Fourier Series ◽  
Finite Number ◽  
Double Fourier Series ◽  
General Term ◽  
Discontinuous Function ◽  
Lebesgue Integral ◽  
Function Of Two Variables

A function of two variables may be expanded in a double Fourier series, as a function of one variable is expanded in an ordinary Fourier series. Purpose that the function f ( x, y ) possesses a double Lebesgue integral over the square (– π < π ; – π < y < π ). Then the general term of the double Fourier series of this function is given by cos = є mn { a mn cos mx cos ny + b mn sin mx sin ny + c mn cos mx sin ny + d mn sin mx cos ny } There є 00 = ¼, є m0 = ½ ( m > 0), є 0n = ½ ( n > 0), є ms = 1 ( m > 0, n >0). the coefficients are given by the formulæ a mn = 1/ π 2 ∫ π -π ∫ π -π f ( x, y ) cos mx cos ny dx dy , obtained by term-by-term integration, as in an ordinary Fourier series. Ti sum of a finite number of terms of the series may also be found as in the ordinary theory. Thus ∫ ms = Σ m μ = 0 Σ n v = 0 A μ v = 1/π 2 ∫ π -π ∫ π -π f (s, t) sin( m +½) ( s - x ) sin ( n + ½) ( t - y )/2 sin ½ ( s - x ) 2 sin ½ ( t - y ) if f ( s , t ) is defined outside the original square by double periodicity, we have sub S ms = 1/π 2 ∫ π 0 ∫ π 0 f ( x + s , y + t ) + f ( x + s , y - t ) + f ( x - s , y + t ) + f ( x - s , y - t ) sin ( m + ½) s / 2 sin ½ s sin ( n + ½) t / 2 sin ½ t ds dt .

Defect Structure of MEV Si Implantation in GaAs

1988 ◽  
Vol 126 ◽  
Author(s):  
S.-Tong Lee ◽  
G. Braunstein ◽  
Samuel Chen
Keyword(s):  
Free Surface ◽  
Defect Structure ◽  
Room Temperature ◽  
Surface Region ◽  
Peak Position ◽  
Lattice Disorder ◽  
Depth Profiles ◽  
Trim Calculation ◽  
Good Agreement

ABSTRACTThe defect and atomic profiles for MeV implantation of Si in GaAs were investigated using He++ channeling, TEM, and SIMS. Doses of 1–10 × 1015Si/cm2 at 1–3 MeV were used. MeV implantation at room temperature rendered only a small amount of lattice disorder in GaAs. Upon annealing at 400°C for 1 h or 800°C for 30 a, we observed a ‘defect-free’ surface region (- 1 μ for 3 MeV implant). Below this region, extensive secondary defects were formed in a band which was 0.7 μ wide and centered at 2 μ for 3 MeV implant. These defects were mostly dislocations lying in the [111] plane. SIMS depth profiles of Si implants showed the Si peak to be very close to the peak position of the defects. The experimental profiles of Si were compared to the TRIM calculation; generally good agreement existed among the peak positions.

Damping of floor vibrations by constrained viscoelastic layers

1977 ◽  
Vol 4 (4) ◽  
pp. 405-411
Author(s):  
A. Farah ◽  
I. M. Ibrahim ◽  
R. Green
Keyword(s):  
Fourier Series ◽  
Loss Modulus ◽  
Optimization Technique ◽  
Double Fourier Series ◽  
Viscoelastic Layer ◽  
Sandwich Beams ◽  
Algorithm Optimization ◽  
Floor Systems ◽  
Floor Vibrations ◽  
Viscoelastic Layers

Formulations are presented for enhancing the serviceability of one-way floor systems subjected to dynamic loading through the use of constrained viscoelastic layers. The constrained layers are combined with the floors and the resulting systems are analyzed as sandwich structures using a double Fourier series approach. Results indicate that the damping of the resulting sandwich beams is governed by factors related to the elastic and geometric properties of the constrained layer and the constraining system (i.e. cover plates and beams) and the loss modulus of the viscoelastic material, and is highly influenced by the location of the viscoelastic layer in the sandwich beams. Optimum designs of the sandwich beams are obtained using the box algorithm optimization technique.

scholarly journals Annihilation Radii for Dislocations Intercepting a Free Surface with Application to Heteroepitaxial Thin Film Growth

1998 ◽  
Vol 535 ◽  
Author(s):  
M. Chang ◽  
S.K. Mathis ◽  
G.E. Beltz ◽  
C.M. Landis
Keyword(s):  
Free Surface ◽  
Film Thickness ◽  
Film Growth ◽  
Interaction Force ◽  
Critical Issue ◽  
Dislocation Motion ◽  
Threading Dislocations ◽  
Fusion Reactions ◽  
Gaas Films ◽  
Preliminary Experimental Data

AbstractOne critical issue in heteroepitaxial, lattice mismatched growth is the inevitable appearance of threading dislocations which reside in the relaxing film and degrade its semiconducting properties. It has been shown in previous work that threading dislocations interact with each other through a series of annihilation and fusion reactions to decrease their density as the film thickness increases and follow a 1/h decay, where h is the film thickness. A characteristic reaction radius is associated with these interactions. In previous simulations, the reaction radius was taken to be a constant value estimated using a simple approximation based on infinite, parallel dislocation lines. Here, a continuum-based elasticity approach is taken to more accurately quantify the reaction radius by comparing the Peach-Koehler force of one dislocation acting on another at a free surface with the lattice resistance to dislocation motion. The presence of the free surface gives rise to a moderate reduction of the interaction force. Results are compared with preliminary experimental data for GaAs films grown on InP.

Interaction of surface waves with turbulence: direct numerical simulations of turbulent open-channel flow

1995 ◽  
Vol 286 ◽  
pp. 1-23 ◽  
Author(s):  
Vadim Borue ◽  
Steven A. Orszag ◽  
Ilya Staroselsky
Keyword(s):  
Free Surface ◽  
Numerical Simulations ◽  
Channel Flow ◽  
Gravity Waves ◽  
Open Channel ◽  
Open Channel Flow ◽  
Surface Region ◽  
Direct Numerical Simulations ◽  
Turbulence Production ◽  
Wind Sea

We report direct numerical simulations of incompressible unsteady open-channel flow. Two mechanisms of turbulence production are considered: shear at the bottom and externally imposed stress at the free surface. We concentrate upon the effects of mutual interaction of small-amplitude gravity waves with in-depth turbulence and statistical properties of the near-free-surface region. Extensions of our approach can be used to study turbulent mixing in the upper ocean and wind–sea interaction, and to provide diagnostics of bulk turbulence.

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