This contribution presents the numerical analysis of Hartree-Fock’s method of computing electron atomic statefunctions using Galerkin’s finite element method. The underlying theory and computational implementation are presented in some detail for the first time and highly accurate energies are presented for free neutral atoms, ions, and for the spatially confined He atom. The method of using local basis sets is shown to be competitive with global basis sets of the Slater and modified Slater types in terms of accuracy and use.

We used high hydrostatic pressure to perform photoluminescence measurements on polar ZnO/ZnMgO quantum well structures. Our structure oriented along the c-direction (polar direction) was grown by plasma-assisted molecular beam epitaxy on a-plane sapphire. Due to the intrinsic electric field, which exists in polar wurtzite structure at ambient pressure, we observed a red shift of the emission related to the quantum-confined Stark effect. In the high hydrostatic pressure experiment, we observed a strong decrease of the quantum well pressure coefficients with increased thickness of the quantum wells. Generally, a narrower quantum well gave a higher pressure coefficient, closer to the band-gap pressure coefficient of bulk material 20 meV/GPa for ZnO, while for wider quantum wells it is much lower. We observed a pressure coefficient of 19.4 meV/GPa for a 1.5 nm quantum well, while for an 8 nm quantum well the pressure coefficient was equal to 8.9 meV/GPa only. This is explained by taking into account the pressure-induced increase of the strain in our structure. The strain was calculated taking in to account that in-plane strain is not equal (due to fact that we used a-plane sapphire as a substrate) and the potential distribution in the structure was calculated self-consistently. The pressure induced increase of the built-in electric field is the same for all thicknesses of quantum wells, but becomes more pronounced for thicker quantum wells due to the quantum confined Stark effect lowering the pressure coefficients.

SummaryIn this paper, details of an implementation of a numerical code for computing the Kohn–Sham equations are presented and discussed. A fully self‐consistent method of solving the quantum many‐body problem within the context of density functional theory using a real‐space method based on finite element discretisation of realspace is considered. Various numerical issues are explored such as (i) initial mesh motion aimed at co‐aligning ions and vertices; (ii) a priori and a posteriori optimization of the mesh based on Kelly's error estimate; (iii) the influence of the quadrature rule and variation of the polynomial degree of interpolation in the finite element discretisation on the resulting total energy. Additionally, (iv) explicit, implicit and Gaussian approaches to treat the ionic potential are compared. A quadrupole expansion is employed to provide boundary conditions for the Poisson problem. To exemplify the soundness of our method, accurate computations are performed for hydrogen, helium, lithium, carbon, oxygen, neon, the hydrogen molecule ion and the carbon‐monoxide molecule. Our methods, algorithms and implementation are shown to be stable with respect to convergence of the total energy in a parallel computational environment. Copyright © 2015 John Wiley & Sons, Ltd.

Two different basis set methods are used to calculate atomic energy within Hartree–Fock theory. The first is a local basis set approach using high-order real-space finite elements and the second is a global basis set approach using modified Slater-type orbitals. These two approaches are applied to the confined helium atom and are compared by calculating one- and two-electron contributions to the total energy. As a measure of the quality of the electron density, the cusp condition is analyzed.

GaN quantum dots grown in (11 (2) over bar2)'orientated AlN are studied. The (11 (2) over bar2)-nucleated quantum dots exhibit rectangular- or trapezoid-based truncated pyramidal morphology. Another quantum dot type orientated on (10 (1) over bar1) is reported. Based on high-resolution transmission microscopy and crystal symmetry, the geometry of (10 (1) over bar1)-orientated quantum dots is proposed. A piezoelectric model is used within a finite element method to determine and compare the strain-state and electrostatic potential associated with the quantum dot morphology and an estimation of the band-edge energy is made. We report on some novel properties of the (10 (1) over bar1)-orientated quantum dot, including mixed strain-states and strain-state bowing.

In this communication, the use of gallium nitride doped with beryllium as an efficient converter for white light emitting diode is proposed. Until now beryllium in this material was mostly studied as a potential p-type dopant. Unfortunately, the realization of p-type conductivity in such a way seems impossible. However, due to a very intensive yellow emission, bulk crystals doped with beryllium can be used as light converters. In this communication, it is demonstrated that realisation of such diode is possible and realisation of a colour rendering index is close to that necessary for white light emission.

In this communication computational methods that facilitate finite element analysis of density functional computations are developed. They are: (i) h-adaptive grid refinement techniques that reduce the total number of degrees of freedom in the real space grid while improving on the approximate resolution of the wanted solution; and (ii) subspace recycling of the approximate solution in self-consistent cycles with the aim of improving the performance of the generalized eigenproblem solver. These techniques are shown to give a convincing speed-up in the computation process by alleviating the overhead normally associated with computing systems with many degrees-of-freedom. (C) 2012 Elsevier B.V. All rights reserved.

This paper provides an overview of the new features of the finite element library deal.II version 8.1.

No 241 187

The applicability of a grid-based method for determining the wavefunctions of atomic systems is explicitly demonstrated by calculations based on direct wavefunction theory. Integration over the real space domain is approximated by a sum over numerically determined values in Gauss quadrature points lying inside finite elements. As examples of application, calculations have been performed for the electronic configurations of hydrogen- and helium-like atoms. The results of calculations for the total energy, orbital values and functions, and some derived quantities are given.

No 252 315 Fig.1. Example of numerical foam structure used in numerical simulations of the compression tests with 90 % porosity. Third International Conference on Material Modelling

This paper describes results of a finite element analysis of the elastic and electric field distribution in a semipolar and a nonpolar isolated quantum dot based on previously obtained measurements from transmission electron microscopy. The two quantum dot orientations are each investigated and compared in terms of the resultant piezoelectric fields and their redistribution due to growth orientation and quantum dot geometry/surface effects. Alongside that, a standard polar quantum dot is investigated as a reference-state system. It is found that the geometry of quantum dots grown in alternative orientations affect the elastic strain and, along with orientation dependent spontaneous polarisation, modify the electrostatic potential and the built-in electric fields. A theoretical verification of a reduction in the quantum confined Stark effect by determining the band edge splitting energies for electron and hole states is given. (C) 2012 Elsevier B. V. All rights reserved.

III-nitride based layers generally contain many thr eading dislocations that can be characterized by continuous displacement fields ari ing from a crystallographic half-plane inclusion [1]. It is observed by experiment that th e atomic half-plane terminates at the edge of the quantum dot and does not propagate into the dot and the influence of the strain field around the dislocation core is assumed to provide p referential geometric conditions for the nucleation of the QD at this site [2]. The spontane ous electrostatic polarization found in wurtzite crystals tends to orientate the electrosta tic potential with a dipole parallel to the crystal c axis [3]. This dominates the electronic and optica l properties of isolated quantum dots that is further modified by the built-in piezo lectric field [4]. Nevertheless, the presence of long-range elastic and electric fields appearing in the proximity of a negatively charged dislocation line [5,6] lead to the expectation that additionally these fields will affect the optoelectronic properties of the QD. The quantifica tion of these effects remain an open question. To investigate these phenomena, a model o f a hexagonal-based GaN/AlN QD nucleated at the edge of a TD is studied by solving a fully coupled elastic-electric problem in an adaptive finite element basis. Results are prese nt d in terms of the quantitative and qualitative differences in these fields due to the inclusion of a charged-core dislocation.

In this work the affect of a threading dislocation localised on the edge of GaN/AlN quantum dot is analysed. A standard piezoelectric continuum model is extended to allow the embodiment of threading dislocations that are modelled as a continuous electro-elastic line defect originating in the matrix material. Two common types of dislocation are considered: an edge-type and a screw-type. It is demonstrated that the presence of a TD provides local region of tensile strain as a preferential condition for GaN QD growth by reduction of the GaN/AlN lattice mismatch. It is found that dislocation induced potential causes a measurable in-plane shift of the electron/hole localisation and an asymmetric decrease in the band-toband transition energy. (C) 2012 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim

III-Nitride structures grown along nonpolar and semipolar crystallographic directions are attracting attention as promising materials for next-generation nitride-based optoelectronic devices. Where for the nonpolar case we can expect a lack of a built-in electric field, for semipolar direction the electric field exists but is effectively reduced by the crystallographic reorientation. Different thickness of GaN/Al0.5Ga0.5N Quantum Dot (QD) structures were grown by ammonia molecular beam epitaxy along the (11-22) direction. As a substrate we used 2 μm thick (11-22) GaN layer grown by metal organic vapour phase epitaxy on m-plane sapphire substrate. During QD formation, when a small amount of GaN is deposited, the geometrical morphology is dominated by the presence of isolated islands but with increased GaN fraction. The nanostructures are elongated along the (1-100) direction and formed structures resembling quantum wires rather than dots [1]. Analysis of the observed photoluminescence spectrum is facilitated by theoretical means within a framework where by a coupled electro-elastic model is solved, followed by an 8band k.p Schrödinger equation [2]. This amounts to calculating the electronic structure quantum mechanically and under conditions that reproduce those of the experimental analysis, at finite temperatures of up to 300K. To gain insight into the dimensionality of the electron-hole confinement (dots/wires), numerical experiments are performed on a series of multi-dimensional systems and their opto-electronic behaviour referred back to the experimentally obtained data. These structures are found to exhibit strong photoluminescence excitations despite the embodiment of a large density of basal stacking faults in the samples. The emission energy is shifted towards higher energies due to reduction in the built-in electric field (see figure) and, due to this, a reduction of the quantum confined Stark effect is observed. These effects are explained by examining a self-consistently determined electron density function and exciton energy levels, that are significantly perturbed by the morphology and the growth direction of the sample.

THIS PAPER EXAMINES how prerelaxation effects the development of the mechanics of a nanoindentation simulation. In particular, the force-depth relation, indentation stress-strain curves, hardness and elastic modulus, are investigated through molecular statics simulations of a nanoindentation process, starting from initial relaxation by: (i) molecular dynamics and (ii) molecular statics. It; is found that initial relaxation conditions change the quantitative response of the system, but not the qualitative response of the system. This has a significant; impact on the computational time and quality of the residual mechanical behaviour of the system. Additionally, the method of determining of the elastic modulus is examined for the spherical and planar indenter; and the numerical results are compared. An overview of the relationship between the grain size and hardness of polycrystalline copper is examined and conclusions are drawn.

To qualitatively determine the behaviour of micro-macro properties of a quantum dot grown in a non-polar direction, we propose a simple semi-classical model based on well established ideas. We take into account the following empirical phenomena: (i) The displacement and induced strain at heterojunctions; (ii) The electrostatic potential arising from piezoelectric and spontaneous polarisation; and (iii) The localisation of excitons (particle-hole pairs) arising from quantum confinement. After some algebraic manipulation used to cast the formalism into an arbitrarily rotated frame, a numerical model is developed for the case of a semi-polar wurtzite GaN quantum dot buried in a wurtzite AlN matrix. This scheme is found to provide a satisfying qualitative description of an isolated semi-polar quantum dot in a way that is accessible to further physical interpretation and quantification.

Our numerical approach to modeling elastic-plastic deformation comes back to the idea of the time-independent plasticity developed here at the molecular-statics level. We use a constitutive atomic model based on the second-moment approximation of the tight-binding potential coupled to a linear theory of elasticity solved simultaneously within the finite element method. Our model is applied to the nanoindentation problem for copper in which the indenter is represented by the equations of a sphere. For convenience the time-dependency of the nanoindentation problem is reduced to a quasi-static adiabatic scheme. A recurring theme in this paper is to determine the response of the proposed model for two differing systems: mono and polycrystalline copper. This paper discusses the force-depth response in terms of atomic bond-lengths, elastic-plastic deformations, and the instantaneous stiffness of the material. We report on an increased instantaneous stiffness of polycrystalline copper compared to that of its monocrystalline counterpart. From both a distinct and a comparative analysis of both systems, based on the relaxed positions of the atoms in the structure during the simulation, we deduce that plastic deformations at grain-boundaries are responsible for this change in the overall instantaneous stiffness of the material.

A Schrodinger eigenvalue problem is solved for the 2D quantum simple harmonic oscillator using a finite element discretization of real space within which elements are adaptively spatially refined. We compare two competing methods of adaptively discretizing the real-space grid on which computations are performed without modifying the standard polynomial basis-set traditionally used in finite element interpolations; namely, (i) an application of the Kelly error estimator, and (ii) a refinement based on the local potential level. When the performance of these methods are compared to standard uniform global refinement, we find that they significantly improve the total time spent in the eigensolver.