Institute of Fundamental Technological Research

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.

Atomic physics, Finite element method, Numerical analysis, Quantum confinement

In 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.

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.

Piezoelectric fields, Quantum wells, Polarization, Zinc oxide films, High pressure

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.

Hartree–Fock method, Confined atom model, Galerkin finite element method, Modified Slater-type orbitals

GaN quantum dots grown in (inline image)’orientated AlN are studied. The inline image-nucleated quantum dots exhibit rectangular- or trapezoid-based truncated pyramidal morphology. Another quantum dot type orientated on inline image is reported. Based on high-resolution transmission microscopy and crystal symmetry, the geometry of inline image-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 inline image-orientated quantum dot, including mixed strain-states and strain-state bowing.

III–V semiconductors, AlN, GaN, nanostructures, piezoelectric properties, quantum dots

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.

Beryllium, Gallium Nitride (GaN), Light Emitting Diode (LED), Optical Converters

In this communication computational methods that facilitate finite element analysis of density functional computations are developed. They are: (i) hh–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.

Density functional theory, Finite element discretization, Grid refinement, Large-scale eigenvalue problem, Message-passing parallelization

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.

Piezoelectricity, Continuum mechanics, Finite element method, Quantum dots

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.

molecular statics, molecular dynamics, nanoindentation, copper

GaN quantum dots (QDs) grown in semipolar (11-22) AlN by plasma-assisted molecular-beam epitaxy were studied by transmission electron microscopy (TEM) and scanning transmission electron microscopy techniques. The embedded (11-2)-grown QDs exhibited pyramidal or truncated-pyramidal morphology consistent with the symmetry of the nucleating plane, and were delimited by nonpolar and semipolar nanofacets. It was also found that, in addition to the (11-22) surface, QDs nucleated at depressions comprising {10-11} facets. This was justified by ab initio density functional theory calculations showing that such GaN/AlN facets are of lower energy compared to (11-22). Based on quantitative high-resolution TEM strain measurements, the three-dimensional QD strain state was analyzed using finite-element simulations. The internal electrostatic field was then estimated, showing small potential drop along the growth direction, and limited localization at most QD interfaces.

Quantum dots, Transmission electron microscopy, III-V semiconductors, High resolution transmission electron microscopy, Epitaxy

A continuum and atomistic approach to the modeling of dislocations observed by high-resolution transmission electron microscopy (HRTEM) is discussed in terms of the continuum theory of dislocations. The atomistic models are obtained by means of the use of a mathematical formula for discrete dislocations. A new analytical solution for a continuously distributed dislocation core is presented. This solution is employed in the finite element modeling of residual stresses induced by the net of dislocations visible on an HRTEM image of GaN structure. This paper terminates with some comments on the atomistic/finite-element modeling of dislocation fields. Because of some confusion concerning notations used in the literature, the mathematical foundations of the continuum theory of dislocations are revisited.

dislocations, field theory, atomistic models, finite element method, high-resolution transmission electron microscopy

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.

linear elasticity, material science, molecular statics, nanoindentation, quasicontinuum methods

A Schrödinger 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.

adaptive finite element analysis, Harmonic oscillator problems, Schrödinger equation

We present a methodology for the stress–strain analysis of a film/substrate interface by combining crystallographic and continuum modelling. Starting from measurements of lattice parameters available from experimental observations, the heterostructure is recast initially in the form of a crystallographic model and finally as a continuum elastic model. The derived method is capable of handling dense arrays of misfit dislocations as well as large areas of the interface between two crystal structures. As an application we consider the misfit dislocation network in the GaN/Al2O3 interface region through determination of strain relaxation and associated residual stresses. Our calculated results are referred back to and found to be in good agreement with the experimental observations of misfit dislocation arrays obtained from high resolution transmission electron microscopy.

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.

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-to-band transition energy.

quantum dot, threading dislocation, piezoelectricity, band edge structure

It is known that heterojunctions give rise to changes in the elastic and piezoelectric properties of a given semiconductor nanostructure. For practical applications it is decisive to understand the influence of these strain and piezoelectric fields on the electronic response of these systems, both as distinct and as coupled fields. In the present study, the confinement properties of the electron and hole wavefunctions are of interest. Therefore, a model based on a traditional envelope 8-band k · p Hamiltonian is implemented and used to explore a GaN/AlN quantum dot/matrix system.

It was observed experimentally by Rouviere et al. that GaN/AlN Quantum Dots (QDs) nucleate at the edge of threading dislocations (Appl. Phys. Lett. 75, 2632 (1999) [1]). The preferred nucleation of QDs in this way is generally assumed to be due to the influence of the stress/strain field around the dislocation core, which in turn, gives the chemical and geometric conditions for nucleation of the QDs. We solve the finite element problem for QDs situated at the edge of threading dislocations where different lattice parameters, piezoelectric and spontaneous polarisation coefficients are assumed for the QD and its matrix. By solving the elastic and electric equilibrium problems we obtain both the residual stress and electric fields. The computational scheme employed here was obtained by linking two previous finite element algorithms described inreferences (P. Dłu ̇zewski et al., Comput. Mater. Sci. 29, 379 (2004) [2]) and (G. Jurczak et al., phys. stat. sol. (c) 2, 972 (2005) and S.P. Łepkowski et al., Phys. Rev. B 73, 245201 (2005) [3, 4], respectively). This approach allows us to get a deeper physical insight into the mechanics and electrical properties of QDs and ultimately determine the efficiency of light emission from these objects.

Nanostructure, III-V semiconductor, Piezoelectricity, Threading dislocation

Semiconductor, Quantum dot, Piezoelectricity, Band-edge structure

III-V semiconductors, piezoelectricity, high resolution transmission electron microscopy, band edge structure

quantum dot, threading dislocation, piezoelectricity, band edge structure

Piezoelektryczność, Mechanika Ośrodków Ciągłych, Metoda Elementów Skończonych, Nanostruktura

molecular statics, molecular dynamics, nanoindentation, copper