Institute of Fundamental Technological Research

The paper presents the experimental, numerical, and theoretical investigation of the microstructure of nickel aluminide samples manufactured by spark plasma sintering using electron backscatter diffraction and computer assisted software. The aim of the work was to reveal the evolution of the microscopic and macroscopic parameters related to the microstructure of the material and its dependence on the applied sintering parameters—temperature and pressure. The studied porous samples with different relative density were extracted from various planes and then tested by electron backscatter diffraction to evaluate the crystallographic orientation in every spot of the investigated area. On this foundation, the grain structure of the samples was determined and carefully described in terms of the grain size, shape and boundary contact features. Several parameters reflecting the grain morphology were introduced. The application of the electric current resulting in high temperature and the additional external loading leads to the significant changes in the structure of the porous sample, such as the occurrence of lattice reorientation resulting in grain growth, increase in the grain neighbours, or the evolution of grain ellipticity, circularity, grain boundary length, and fraction. Furthermore, the numerical simulation of heat conduction via a finite element framework was performed in order to analyse the connectivity of the structures. The numerical results related to the thermal properties at the micro- and macroscopic scale—local heat fluxes, deviation angles, and effective thermal conductivity—were evaluated and studied in the context of the microstructural porosity. Finally, the effective thermal conductivity of two-dimensional EBSD maps was compared with those obtained from finite element simulations of three-dimensional micro-CT structures. The relationship between the 2D and 3D results was derived by using the analytical Landauer model.

This work presents a comprehensive analysis of heat transfer and thermal conductivity of porous materials manufactured by spark plasma sintering. Intermetallic nickel aluminide (NiAl) has been selected as the representative material. Due to the complexity of the studied material, the following investigation consists of experimental, theoretical and numerical sections. The samples were manufactured in different combinations of process parameters, namely sintering temperature, time and external pressure, and next tested using the laser flash method to determine the effective thermal conductivity. Microstructural characterisation was extensively examined by use of scanning electron microscopy and micro-computed tomography (micro-CT) with a special focus on the structure of cohesive bonds (necks) formed during the sintering process. The experimental results of thermal conductivity were compared with theoretical and numerical ones. Here, a finite element framework based on micro-CT imaging was employed to analyse the macroscopic (effective thermal conductivity, geometrical and thermal tortuosity) and microscopic parameters (magnitude and deviation angle of heat fluxes, local tortuosity). The comparison of different approaches toward effective thermal conductivity evaluation revealed the necessity of consideration of additional thermal resistance related to sintered necks. As micro-CT analysis cannot determine the particle contact boundaries, a special algorithm was implemented to identify the corresponding spots in the volume of finite element samples; these are treated as the resistance phase, marked by lower thermal conductivity. Multiple simulations with varying content of the resistance phase and different values of thermal conductivity of the resistance phase have been performed, to achieve consistency with experimental data. Finally, the Landauer relation has been modified to take into account the thermal resistance of necks and their thermal conductivity, depending on sample densification. Modified theoretical and finite element models have provided updated results covering a wide range of effective thermal conductivities; thus, it was possible to reconstruct experimental results with satisfactory accuracy.

thermal conductivity, porous materials, spark plasma sintering, micro-computed tomography, nickel aluminide, finite element modelling, tortuosity

Factor analysis is a standard statistical technique for reducing data dimensionality, which is widely used in sociology, psychology, and demography. Also, financial and insurance institutions commonly use such a technique for marketing research. In recent years, factor analysis has been used, at the beginning rather diffidently, to analyse selected problems of business management, e.g. to troubleshoot consumer and company communication. There are some literature reports about the successful use of factor analysis in managing a company area. Nevertheless, the literature seems to lack examples with successful use of the method with a clear explanation of its rather difficult application in the field of competitiveness or potential company boost. The modest popularity of such a powerful technique in this particular field seems to be attributed mainly to the complexity of the method and its requirements concerning the data quantity. Besides, the factor analysis technique has great potential and can be used as an efficient tool to reduce the complexity of observed phenomenon or verify the accuracy of theoretical models. Therefore, the purpose of this paper is to present a vast potential of factor analysis (both exploratory and confirmatory) applied to solve various problems in company management, especially related to competitiveness and market success. Two case studies covering the subject of business management are presented to illustrate the benefits of factor analysis application. The exploratory factor analysis is exemplified by the search of factors related to the commercial success of the company, while the confirmatory technique is illustrated by a case study of the intellectual capital of the company and its factors related to competitiveness. The paper also presents the essence of the factor analysis, types of analysis, subsequent procedures, purposes, and its specific features. Finally, the applicability of the factor analysis to solve management issues and possible gain in management are discussed.

factor analysis, confirmatory factor analysis, exploratory factor analysis, management

In this article the consequence of a change of finite strain measure is theoretically considered for nonlinear piezo­electric crystals. Analytical predictions show that second-order piezo­electric coefficients are finite strain measure dependent. Therefore, the use of any finite strain measure in constitutive modelling of piezo­electric materials requires an adequate choice of higher-order piezo­electric coefficients. This allows one to avoid unwanted corrections to the elastic and electric fields in the case of nonlinear modelling of piezo­electric materials, e.g. for piezo­electric hetero­structures such as quantum wells or dots. A general transformation formula for second-order piezo­electric coefficients (elasto­striction) is derived. As an example, specific transformation formulae for two common crystallographic classes, namely 43m and 6mm, are presented. The piezo­electric coefficients for GaN and GaAs crystals, as representative cases of these crystal classes, are recalculated and their dependence on the strain measure is demonstrated. A further implication of that effect is that a complete set of second-order piezo­electric coefficients should contain additional information about the strain measure applied during calculations or measurements.

piezoelectricity, nonlinear piezoelectricity, elastostriction, finite strain measure

In this paper the effect of adjacent threading dislocation at the edge of the GaN/AlN quantum dot is analysed by use of the finite element analysis. Elastic as well electric effects related to dislocation core are taken into account. Two types of threading dislocations: edge- and screw-type, common for III-nitride epitaxial layers, are considered. Also, three different QD geometries are considered to estimate the impact of the threading dislocation on the quantum heterostructure. It is demonstrated that the local elastic and electric fields around dislocation affect local piezoelectric fields built-in the quantum dot. Local lattice deformation near the dislocation core reduce residual strains in the quantum dot. It is prominent in the case of edge-type dislocation. The presence of an electric charge along dislocation line provides significant shift of the total potential towards the negative values. However, estimated difference in band-to-band transition energy for edge- and screw-type dislocations are rather small, what suggest low sensitivity to the charge density along dislocation line. Unexpectedly, local strain field around the edge-type dislocation, slightly compensate the negative affect of the electrostatic potential.

Quantum dot, Threading dislocation, Piezoelectricity, Finite element modelling

In this paper we evaluate the effect of higher order elastic and piezoelectric coefficients on elastic and electric fields in III-nitride wurtzite crystals. To this end, finite element calculations of GaN/AlN QDs of different types are presented and compared. We show that the nonlinear elastic and piezoelectric effects modify the elastic strain field, electrostatic potential, and the build-in electric field in the QDs. These higher order effects lead to corrections of the peak values of the strain field and the electrostatic potential over 8% above the linear model. An even more significant effect, exceeding 13%, is observed for the magnitude of the electric field. Our calculations clearly show that the impact of the nonlinear correction strongly depends on the application, i.e. on the morphology and crystallographic orientation of the quantum dot. It turns out that nonlinear effects play an important role in the semipolar (View the MathML source112¯2) and nonpolar (View the MathML source112¯0) QDs. Because of the theoretical nature of physical parameters describing nonlinear material (obtained by DFT calculations) further studies and experimental verification of the nonlinear effects in nitride structures are necessary.

Piezoelectricity, Heterostructure, Nonlinearity, Quantum dot

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

A proper reconstruction of discrete crystal structure with defects is an important problem in dislocation theory. Currently, procedures for dislocation core reconstruction presented in the literature usually neglect configuration changes. The present paper discusses a new approach, which uses an iterative algorithm to determine an atomistic configuration of the dislocation core. The mathematical background is based on finite deformation theory, in which an iterative algorithm searches for the new atomic configuration corresponding to the actual atomic configuration of the deformed crystal. Its application to the reconstruction of 4H-SiC crystal affected by the system of four threading dislocations is presented as an example. Molecular statics calculations suggest a lower potential energy, as well as dislocation core energy, per-atom energy, and per-atom stresses for the structure reconstructed by use of the iterative algorithm against the classical solution based on the Love's equations.

dislocation, dislocation core energy, finite deformation, molecular statics

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

Sample thinning for TEM observation introduces large changes with respect to the initial strain state of the bulk sample and particularly relaxation via the free surfaces which leads to HOLZ lines splitting in the CBED pattern. This phenomenon has been simulated owing to extensive calculations either in the kinematical or the dynamical framework of electron diffraction mainly using displacement fields resulting from finite element modelling of the sample relaxation. HOLZ line splitting is well reproduced and numerical fits can be used to compare experimental and calculated curves. This paper proposes new analytical solutions for the kinematical equation of electron diffraction. Simple mathematical functions are used to approximate the deformation profiles. We showed that, under certain conditions, the rocking curve profile can be analytically calculated, thus providing some clue to separate different contributions to the rocking curves against deformation profile. These simplified analytical expressions are used to extract the maximum amplitude displacement within the sample with about 10% accuracy. This accuracy can even be improved to 1% with a short adjustement routine. The influence of the shape of the displacement profile on the rocking curves is demonstrated.

Convergent beam electron diffraction, Kinematical theory of electron diffraction, Analytical expressions, Strain measurement

Problem of locally disordered atomic structure is solved by using a hybrid formulation in which nonlinear elastic finite elements are linked with discrete atomic interaction elements. The continuum approach uses nonlinear hyperelasticity based upon the generalized strain while the atomistic approach employs the Tight-Binding Second-Moment Approximation potential to create new type of elements. The molecular interactions yielding from constitutive models of TB-SMA were turned into interactions between nodes to solve a boundary value problem by means of finite element solver.
In this paper we present a novel way of modelling materials behaviour where both discrete (molecular dynamics) and continuum (nonlinear finite element) methods are used. As an example, the nanoindentation of a copper sample is modelled numerically by applying a hybrid formulation. Here, the central area of the sample subject to a nanoindentation operation is discretised by an atomic net where the remaining area of the sample far from indenters tip is discretised by the use of a nonlinear finite element mesh.

Nanostructure, Nanoindentation, Molecular statics, Finite element modelling

The photoluminescence (PL) from GaN quantum dots (QDs) embedded in AlN has been investigated under hydrostatic pressure. The measured pressure coefficient of emitted light energy [dE / dP] shows a negative value, in contrast with the positive pressure coefficient of the GaN band gap. We also observed that increasing pressure leads to a significant decrease of the light emission intensity and an asymmetric broadening of the PL band. All these effects are related to the pressure-induced increase of the built-in electric field. A comparison is made between experimental results and the proposed theoretical model which describes the pressure behavior of nitride QDs.

III-V semiconductor, Quantum dot, Piezoelectricity, Photoluminescence

We have studied the nonlinear elasticity effects in zinc-blende and wurtzite crystallographic phases of III-N compounds. Particularly, we have determined the pressure dependences of elastic constants in InN, GaN, and AlN by performing ab initio calculations in the framework of plane-wave pseudopotential implementation of the density-functional theory. The calculations have been performed employing two exchange-correlation functionals, one within the local density approximation and the other within the generalized gradient approximation. We have found that C11, C12 in zinc-blende nitrides and C11, C12, C13, C33 in wurtzite nitrides depend significantly on hydrostatic pressure. Much weaker dependence on pressure has been observed for C44 elastic constant in both zinc-blende and wurtzite phases. Further, we have examined the influence of pressure dependence of elastic constants on the pressure coefficient of light emission, dE / dP, in wurtzite InGaN / GaN and GaN / AlGaN quantum wells. We have shown that the pressure dependence of elastic constants leads to a significant reduction of dE / dP in nitride quantum wells. Finally, we have considered the influence of nonlinear elasticity of III-N compounds on the properties of hexagonal nitride quantum dots (QDs). For typical wurtzite GaN / AlN QDs, we have shown that taking into account pressure dependence of elastic constants results in the decrease of volumetric strain in the QD region by about 7%. Simultaneously, the average z component of the piezoelectric polarization in the QDs increases by 0.1 MV/ cm compared to the case when linear elastic theory is used. Both effects, i.e., decrease of volumetric strain as well as increase of piezoelectric field, decrease the band-to-band transition energies in the QDs.

III-V semiconductor, ab-initio calculation, nonlinear elasticity, third order elastic coefficient

The elastic stiffness tensors for wurtzite GaN and AlN show a significant hydrostatic pressure dependence, which is the evidence of nonlinear elasticity of these compounds. We have examined how pressure dependence of elastic constants for wurtzite nitrides influences elastic and piezoelectric properties of GaN/AlN planar superlattices and quantum dots. Particularly, we show that built-in hydrostatic pressure, present in both quantum wells of the GaN/AlN superlattices and GaN/AlN quantum dots, increases significantly by 0.3–0.7 GPa when nonlinear elasticity is used. Consequently, the compressive volumetric strain in quantum wells and quantum dots decreases in comparison to the case of the linear elastic theory. However, the z-component of the built-in electric field in the quantum wells and quantum dots increases considerably when nonlinear elasticity is taken into account. Both effects, i.e., a decrease in the compressive volumetric strain as well as an increase in the built-in electric field, decrease the band-to-band transition energies in the quantum wells and quantum dots.

III-V semiconductor, nonlinear elasticity, third order elasic coefficient

Chemical composition in a ternary alloy is examined using a quantitative high resolution transmission electron microscopy, finite element modelling of the thin foil relaxation phenomena and microscopy image simulation. The measurement of local lattice distortion on transmission electron microscopy images is a powerful tool for chemical composition determination. However, for the correct interpretation of the results, one needs to take into account the inhomogeneous relaxation of the sample and the strain averaging across the sample. The 3D finite element modelling of such phenomena have been performed as a function of chemical composition and geometry of an indium rich cluster in a MOCVD InxGa1−xN/GaN quantum well. Lattice distortion field measured on: experimental transmission electron microscopy image and simulated one, obtained on the basis of finite element simulation, are compared. This procedure allows an accurate determination of chemical composition in such heterostructures.

Indium clusters, Vapour deposition, Transmissionelectron microscopy, Elasticity, Finite element method, Lattice distortion, Image simulation

In this paper the field theory of dislocations is used in the finite element analysis of residual stresses in epitaxial layers. By digital processing of the HRTEM image of a GaAs/ZnTe/CdTe system the tensor maps of dislocation distribution are extracted. Such obtained maps are used as the input data to the finite element code. The mathematical foundations of this code are based on the compatibility equations for lattice distortions. The surface tension induced by misfit dislocations is considered here in terms of a 3D boundary-value problem for stress equilibrium in the interfacial zone. The numerical results show how strongly the surface tension depends on the nonlinear elastic behaviour of the crystal structure.

Microscopy and microanalysis techniques, Nonlinear elasticity, Dislocation structure, Finite element analysis, Residual stresses, Layered structures

The cover picture of this issue depicts indium composition fluctuations in InGaN/GaN multi quantum wells. The coded color strain distribution (left) was derived from finite element method calculations of the strain relaxation process and high‐resolution transmission electron microscopy (HRTEM) image simulations, superimposed on the HRTEM image of the quantum wells. The possible corresponding shape and εxx strain profiles in the indium rich clusters (right) hint at a concentration close to pure InN in their core. The paper by Pierre Ruterana et al. [1] was presented at the 5th International Symposium on Blue Laser and Light Emitting Diodes (ISBLLED‐2004), held in Gyeongju, Korea, 15–19 March 2004.

HRTEM, quantum well, composition fluctuation, strain distribution

A new finite element to analyze problems of anisotropic hyperelasticity is presented. The constitutive equations are derived by means of the energy method, which leads to the stress measure conjugate to the logarithmic strain. Equilibrium equation are integrated in the current configuration. Multiplicative instead of additive - decomposition of the time derivative of a strain tensor function is applied as a crucial step that makes possible the formulation for anisotropic hyperelastic materials. Unlike previous known anisotropic large deformation models, the one here presented assures the energy conservation while using the anisotropic elastic constants and the logarithmic strain measure. It is underlined that for the first time a model including all these features is presented. Some numerical examples are shown to illustrate the results obtained with this model and to compare them with other known anisotropic models.

Anisotropic material, Constitutive behaviour, Elastic material, Finite element method, Logarithmic strain measure

A nonlinear finite element approach presented here is based on the constitutive equations for anisotropic hyperelatic materials. By digital image processing the elastic incompatibilities (lattice mismatch) are extracted from the HRTEM image of GaN epilayer. Such obtained tensorial field of dislocation distribution is used next as the input data to the FE code. This approach is developed to study the stress distribution associated with lattice defects in highly mismatched heterostructures applied as buffer layers for the optically active structures.

Dislocations, Anisotropic Hyperelasticity, Residual Stresses

Elastic model of continuum material is often used to simulate the relaxation of crystalline heterostructures. There are many reports on the successful application of the theory of elasticity to nano-sized crystalline heterostructures, even if the continuum condition for them is hardly fulfilled. On the other hand, progress in epitaxial growth allows for the preparation of stable ultra-thin layers with thickness of few monolayers. For such ultra-thin layers, results provided by continuum model and molecular statics/dynamics calculations become diverging. The key problem seems to be located at the modelling of the interface between layers, which is problematic in the continuum approach. By applying a step-wise substitutive compositional interfacial function, it is possible to obtain good agreement with molecular dynamics calculations, even for a single monolayer heterostructure. We propose another approach that uses composition as an extra parameter during finite element calculations, along with classical nodal displacements. Such an approach creates a chemo-elastic coupling that allows to interpolate the composition much like in the case of atomistic calculations.

ultra-thin layers, elastic relaxation, molecular statics, finite elemenet modelling

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

Convergent-beam electron diffraction, elastic strain relaxation, finite element modelling

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

We investigate Quantum Confined Stark Effect (QCSE) in vertically correlated wurtzite GaN/AlN QDs, having hexagonal pyramid-shape. We show that the QCSE in these structures depends not only on the vertical dimensions, i.e., the height of the QDs and the thickness of the barriers, but also on their base diameter. We show that for typical wurtzite GaN/AlN QDs, having the base diameter of 19.5nm, drop of the electrostatic potential in the QD region slightly increases with increasing the width of barriers. Consequently, the band-to-band transition energies in the QDs decrease. Qualitatively similar but a factor of two stronger dependences is obtained for superlattices of QWs having the same vertical dimensions. The increase of the base diameter of the dots results in stronger dependences of both the electrostatic potential and the band-to-band transition energy on the thickness on barriers.

III-V semiconductor, Quantum dot, Piezoelectricity, Quantum confined Stark effect

We theoretically investigate elastic, piezoelectric and optical properties of wurtzite GaN/AlN quantum dots, having hexagonal pyramid-shape, stacked in a multilayer. We show that the strain existing in quantum dots and barriers depends significantly on the distance between the dots i.e. on the width of AlN barriers. Drop of the electrostatic potential in the quantum dot region slightly increases with increasing of the barrier width. This increase is however much smaller for QDs than for superlattice of quantum wells. Consequently, band-to-band transition energies in the vertically correlated quantum dots show rather weak dependence on the width of AlN barriers.

III-V semiconductor, quantum dot, piezoelectricity, elastic strain, electrostatic potential

Following the need to accurately understand the In composition fluctuations and their role on the optical properties of the GaN based heterostructures, an investigation of MOCVD InGaN/GaN quantum wells is carried out. To this end, quantitative High Resolution Transmission Electron Microscopy (HRTEM) is coupled with image simulation and Finite Element Method (FEM) for the thin foil relaxation modelling. The results show that the indium content can reach x = 1 in the clusters inside the core. In these MOCVD QWs, we attempt to connect the Quantum dot density, composition, and shape to the growth conditions, in order to help the engineering process of highly efficient devices.

The effect of adjacent threading dislocation at the edge of polar GaN/AlN quantum dot was widely discussed in the literature, see e.g. [1]. Anyway, development of growth techniques for the III-nitrides is moving towards semipolar or even nonpolar orientations, where more efficient radiative recombination is expected due to less significant quantum confinement Stark effect and elimination of spontaneous polarisation. New growth orientations entails entirely new geometry of quantum structures, what calls into question already done analyzes carried out for polar setup. First off all, there are only a few experimental reports showing the real geometry of semipolar and nonpolar quantum dots which differs significantly from well known truncated hexagonal pyramid shape, see e.g. [2]. Secondly, there is no clear information about the geometric relation of the dislocation line and the quantum dot as it was clearly presented in the polar case. However, such relation definitely exists as it is well documented that its dislocation density is much higher compared to crystals grown in the polar regime. Finally, the possible effect of charged dislocation line may additionally alter the optoelectronic properties of the quantum dot [3].

In this work, finite element method is used to determine how the threading dislocation affects semipolar and nonpolar quantum dots and alternates its build-in elastic and electric fields, so in this way modify band-to-band transition energy for the recombining pair of carriers. Threading dislocation, modeled by use of classical continuum dislocation theory via polynomial approximation for distortion field, generates axisymmetric elastic and electric fields. Coupled fields around dislocation line affect neighbouring quantum dot with its build-in fields related to lattice mismatch between GaN dot and AlN matrix and in a limited extent to spontaneous polarisation. Additionally, electric charge localised along the dislocation line is taken into account, and generates extra negative potential field affecting close surroundings of the threading dislocation. Two common types of threading dislocations for III-nitride epitaxial layers are considered: perfect edge- and perfect screw-type dislocation.

It is demonstrated that local elastic and electric fields around threading dislocation together with the presence of an electric charge along dislocation line affect local piezoelectric field build-in the quantum dot, creates geometrical shift of the carrier localization regions, and reduce band-to-band transition energy.

REFERENCES
[1] Rouviere, P.J.L., et al., Appl. Phys. Lett. 75, 1999, 2632.
[2] Dimitrakopulos, G.P., et al., J. Appl. Phys. 108(10), 2010, 104304.
[3] Jurczak, G., Dłużewski, P., Phys. E: Low-Dimens. Syst. Nanostructures, 95, 2018, pp. 11-15.

Quantum dot, Threading dislocation, Piezoelectricity, Finite element modeling

Very rapid technological development in the electronic branch of the industry observed during last decades, together with the progressive miniaturisation of electronic devices induce increasing interest in the subject of piezoelectric semiconducting heterostructures. In some cases, the linearity of the piezoelectric effect under extreme strain and electric field conditions is challenged for these heterostructures. There are many experimental reports in the literature dealing with nonlinear piezoelectricity as well as theoretical calculations which predict the nonlinear behaviour of such crystalline heterostructures. If, as stated above, the nonlinearity appears under extreme load conditions, therefore from the point of view of mechanics a finite deformation approach shoud be applied to properly describe the kinematics of the deformed crystal. Thus, problem of the choice of a proper strain measure appears as far as many different finite strain measures can be used to describe deformation of the body. Furthermore, higher order piezoelectric coefficients which are derivatives of the heterostructure energy (deformation in the vicinity of the natural state of the body) over the strain depends on the choice of the strain measure [1,2]. Theoretical prediction shows that second-order piezoelectric coefficients are finite strain measure dependent. Therefore, the use of any finite strain measure in constitutive modelling of nonlinear piezoelectric materials requires an adequate choice of higher-order piezoelectric coefficients. Otherwise, erroneous elastic and electric fields may appear in the case of modelling of nonlinear piezoelectric phenomena, e.g. for quantum heterostructures such as wells or dots. A further implication of that effect is that a complete set of second-order piezoelectric coefficients should contain additional information about the strain measure applied during calculations or measurements. General transformation formula for second-order piezoelectric coefficients (elastostriction) is derived as well as individual transformation formulae for common crystallographic classes (e.g. cubic, hexagonal).

piezoelectricity, second order piezoelectric coefficients

Theory of elasticity, a continuum model of a macroscopic material is commonly used to model a relaxation of a crystalline heterostructures. There are many reports on the successful application of theory of elasticity to nanometer crystalline heterostructures, even if the continuum condition for these structures is hardly fulfilled. On the other hand progress in epitaxial growth techniques allows to prepare the stable ultra thin layers with the thickness about a single monolayer. For such extremely thin layers the theory of elasticity seems to fail in describing the relaxation process. The results provided by theory of elasticity and experimental measurements or molecular statics/dynamics become diverging. The key problem in that case seems to be located at the interface between layers and related to composition change, which is problematic in classic, elastic approach. By applying a "substitutive" composition of the interface layers which is just an interpolation, it is possible to obtain a good agreement with molecular statics, even for 1 monolayer heterostructure. Instead of classic approach to the composition within the theory of elasticity, we propose another approach which takes into account the composition as an extra degree of freedom along with classical displacement. Such approach creates a chemo-elastic coupling with composition interpolated by use of the Vegard's law. This allows to take into account a composition changes at the interface and avoid mesh refining necessary at the classic approach.

theory of elasticity, semiconductor, monolayer, relaxation

Quantum dot, Threading dislocation, Piezoelectricity, Finite element modelling

Semiconductor, Quantum dot, Piezoelectricity, Band-edge structure

Semiconductor, Piezoelectricity, Dislocation

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