Institute of Fundamental Technological Research

Cross-sectional transmission electron microscopy studies often reveal a-type dislocations located either below or above the interfaces in
InGaN/GaN structures deposited along the [0001] direction. We show that these dislocations do not emerge during growth but rather are a
consequence of the stress state on lateral surfaces and mechanical processing, making them a post-growth effect. In cathodoluminescence mapping, these defects are visible in the vicinity of the edges of InGaN/GaN structures exposed by cleaving or polishing. Finite element cal-culations show the residual stress distribution in the vicinity of the InGaN/GaN interface at the free edge. The stress distribution is discussed in terms of dislocation formation and propagation. The presence of such defects at free edges of processed devices based on InGaN layers may have a significant negative impact on the device performance.

Luminescence ,Transmission electron microscopy ,Focused ion beam ,Semiconductor materials ,Epitaxy ,Crystal structure ,Crystal lattices ,Crystallographic defects,Mechanical stress,X-ray diffraction

The rocksalt structure of ZnO has a very promising bandgap for optoelectronic applications. Unfortunately, this high-pressure phase is unstable under ambient conditions. This paper presents experimental results for rocksalt-type ZnO/MgO superlattices and theoretical considerations of the critical thickness of MgxZn1−xO layers. The correlations between the layer/spacer thickness ratio, elastic strain, chemical composition, and critical thickness are analyzed. The Matthews and Blakeslee model is revisited to find analytic conditions for the critical layer thickness resulting in phase transition. Our analysis shows that due to the decrease in misfit stresses below some critical limit, the growth of multiple quantum wells composed of rocksalt ZnO layers and MgO spacers is possible only for very large layer/spacer thickness ratios.

We have presented ab initio study, based on density functional theory methods, of full-core edge dislocation impact on basic properties of 4H-SiC semiconductor. To enable calculations in periodic boundary conditions, we have proposed geometry with two dislocations with opposite Burgers vectors. For this geometry, which determines the distance between dislocations, we have estimated the creation energy per unit length of a single-edge dislocation. The radial distribution function has been used to assess the effect of the dislocations on the local crystal structure. The analysis of the electronic structure reveals mid-gap p states induced by broken atomic bonds in the dislocation core. The maps of charge distribution and electrostatic potential have been calculated, and the significant decrease in the electrostatic barriers in the vicinity of the dislocation cores has been quantified. The obtained results have been discussed in the light of previous findings and calculations based mainly on phenomenological models.

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

Epitaxy of semiconductors is a process of tremendous importance in applied science and in the optoelectronics industry. The control of defects introduced during epitaxial growth is a key point in manufacturing devices of high efficiency and durability. In this work, it is demonstrated how useful hybrid reflections are for the study of epitaxial structures with anisotropic strain gradients due to patterned substrates. High accuracy in detecting and distinguishing elastic and plastic relaxations is one of the greatest advantages of measuring this type of reflection, as well as the fact that the method can be exploited in a symmetric reflection geometry on a commercial high-resolution diffractometer.

optoelectronics, Group III-nitride semiconductors, epitaxial growth, X-ray multiple diffraction, interface defects

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

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

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

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.

We present transmission electron microscopy (TEM) and x-ray quantitative studies of the indium distribution in InxGa1−xN/GaN multiple quantum wells (MQWs) with x = 0.1 and 0.18. The quantum wells were grown by low-pressure metalorganic chemical vapour deposition (LP-MOCVD) on a bulk, dislocation-free, mono-crystalline GaN substrate. By using the quantitative TEM methodology the absolute indium concentration was determined from the 0002 lattice fringe images by the strain measurement coupled with finite element (FE) simulations of surface relaxation of the TEM sample. In the x-ray diffraction (XRD) investigation, a new simulation program was applied to monitor the indium content and lateral composition gradients. We found a very high quality of the multiple quantum wells with lateral indium fluctuations no higher than ΔxL = 0.025. The individual wells have very similar indium concentration and widths over the whole multiple quantum well (MQW) stack. We also show that the formation of 'false clusters' is not a limiting factor in indium distribution measurements. We interpreted the 'false clusters' as small In-rich islands formed on a sample surface during electron-beam exposure.

Constitutive equations for interdiffusion and mass transport induced by the stress gradient in crystal lattice is presented in terms of continuum thermodynamics. Only the standard balance laws are analysed for the mass, momentum, moment of momentum, energy and entropy. In consequence the driving forces for interdiffusion of chemical constituents are determined. The forces depend not only on the stress gradient but also on the gradients of chemical components and temperature. The driving forces are used next in constitutive modelling of interdiffusion and mass transport in crystal lattice.

Chemical Strain, Driving Force, Interdiffusion, Kirkendall Effect, Mass Transport

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

Chemical Strain, Dislocation Climbing, Driving Force, Interdiffusion, Kirkendall Effect, Molar Derivative, Molar Motion, Vacancy

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

A finite element algorithm for the determination of both residual stresses and lattice orientation in dislocated crystals is presented. The applied theoretical approach is based on the continuum theory of dislocations. Due to the compatibility condition, an initial plastic distortion field has been used to introduce an elastic incompatibility corresponding to the assumed dislocation density tensor field. The presented method gives the possibility to develop a numerical code suitable for determination of the stress field and lattice orientation. Two numerical examples are given: a low-angle dislocation wall and a subboundary structure. The method can be applied especially to dislocation sets composed of a few groups of monomial dislocations.

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

FE algorithm for large anisotropic hyper-elastic deformations is presented. This algorithm is based on the concept of the local reference configuration. The spectral decomposition of the elastic modulus tensor into eigentensors of elastic deformations has been utilized in the constitutive equations. Some numerical results are also presented.

A thermodynamic theory of the movement of point, line and surface defects is considered at finite deformations. This theory is based on the balance laws for crystal defects. The defects balance laws together with the well-known balance laws for the mass, momentum, moment of momentum, energy and entropy have been utilized to find the driving forces acting on crystal defects. Some of the derived formulae are well-known, e.g. Peach-Koehler formula, nevertheless, many new relations are obtained, e.g. for osmotic forces and for the energy flux due to the movement of crystal defects. The driving force acting on a grain boundary is found as a thermodynamic force needed to balance the jump in energy density across the moving discontinuity surface.

Using the relations derived for driving forces the problem of the constitutive modelling of the crystal defect movement is considered. The elastic behaviour of materials with structural defects is determined by a constitutive equation imposed on the free energy density. This equation takes into account the elastic strain, crystal defect densities and temperature. The crystal plasticity is described by vector constitutive equations stated between the defects velocities and the respective driving forces.

Oriented continuum, Dislocation density tensor, Driving forces, Curvature tensors

In many papers on oriented continua some orthonormal angular coordinates were proposed. With respect to the curvature of the orientation space, it is obvious that such coordinates could not be applied in practice. Therefore, instead of these coordinates a tensor field of rotations had to be used to define the wryness tensor.

In this paper the curvilinear coordinates in orientation space are considered. The Euler angles are an example of such coordinates. The inertia conservation law is replaced here by a constitutive relation. From the physical point of view this relation is more general and seems to be better justified than the mentioned law. Within the framework of the polar continuum theory a micromorphic structure is discussed. Some remarks on the principle of a material frame-indifference are also presented.

The dislocation density tensor is considered as a measure of angular inelastic deformation of crystal lattice. The modelling of plastic deformation of dislocated crystal is done by means of the vector constitutive relation imposed on the force-velocity dependence for dislocations. The force exerted on a dislocation is treated here as a direct reason of the dislocation movement. This force is induced by the elastic deformation of crystal lattice and by the osmotic stress due to the unbalanced concentration of vacancies.

Activation Energy of Dislocation Motion, Constitutive Equations of Dislocation Movement, Continuum Theory of Dislocations, Curvature Tensor, Dislocation Balance Law, Dislocation Density Tensor, Force on Dislocation, Internal Stress, Osmotic Pressure, Peach-Koehler Formula, Residual Stress, Thermodynamic Restrictions, Vacancy Concentration

recently, in many microstructural models of polycrystals the micro-macro transition is based on the averaging of crystal behaviours over all the crystal orientations. In this paper the elements of the theory of crystal orientation spaces are presented from the viewpoint of Riemannian geometry. The Euler angles are used as coordinate systems on these spaces. The angular metric tensor is introduced. It is shown that each turn of a crystal describes a geodesic line in the crystal orientation space. A simple formula to determine the finite distance between two different crystal orientations is given. It appears that the orientation region frequently used to describe the orientation distribution of f.c.c. crystals does not take into account all orientations of the crystals. Therefore a new, more suitable region has been proposed. A continuity equation for the crystal orientation distribution function (CODF) is obtained as a conservation law in this geometry. It is shown that the frequently used form of this equation is not correct.

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

Using three different methods namely, CDT (Continuous Dislocation Theory), molecular TB - SMA (Tight Binding Second Moment Approximation) type many - body potential, and MEM (Molecular Effective Medium) theory, we are looking for the best possible reconstruction of dislocations in Cu - Al 2 O 3 heterostructure.

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

The stress induced diffusion process of In-Ga segregation in InxGa1–xN layer deposited on GaN is simulated step by step by using a 3D nonlinear FE method. From the thermodynamical point of view this process is governed by the driving force induced by the gradient of residual stresses operating in an anisotropic nonlinear elastic structure. The source of stresses we consider is the set of threading dislocations examined in the plane view HRTEM investigation of GaN layer deposited on sapphire.

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.

Ab-initio, atomistic modeling, tensor fields, dislocation fields algebra, lattice distortions, visualization methods

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

Zinc oxide has wurtzite structure (wz-ZnO) at ambient conditions. Due to the promising bandgap (4.0-7.8eV) we consider the misfit stress for the growth of rock salt rs-Zn$_x$Mg$_{1-x}$O layers on rock salt MgO. At the ambient conditions, a solid solution of ZnO in MgO is stable only up to 13%. Nevertheless, due to the misfit stress the range of chemical composition of thermodynamically stable layers can be extended. We consider a mechanism of the dislocation network formation at the interface rs-Zn$_x$Mg$_{1-x}$O/MgO. Based on the dislocation theory, many different analytic formulas for critical layer thickness have been derived, cf. Hu (1991), Brown (2002). The formulas concern the critical thickness of the layers which retain thermodynamically stable at atmospheric pressure. On the other hand, for thin layers which lose the stability earlier, before the stress relaxation, we can expect a lower critical thickness. We present a derivation of an analytic formula for the critical thickness of rs-Zn$_x$Mg$_{1-x}$O layers which lose the stability due to the rocksalt-wurtzite phase transition, cf. Lu et al. (2016). In the new formula the dependency of the onset elastic energy $E(sigma, x)$ of the rs$ ightarrow$wz phase transition is taken into account. In the general case this energy depends on the misfit stress and chemical composition.

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

The interdiffusion of chemical components coupled with vacancy movement can cause void formation and/or spinodal decomposition in crystal growth. In the case of SiC growth on Si, the higher mobility of Si atoms compared to C results in the migration of SiC/Si interface and formation of voids in the substrate in some thermodynamic conditions. In the case of In -rich InGaN layers deposited on GaN a precipitation of metallic indium bordering with voids is observed. In the current approach we consider interdiffusion, lattice distortion and chemical maps extracted from HRTEM images of SiC/Si and InGaN/GaN. Dislocations and void surface are treated as local regions of nucleation and annihilation of the vacancies transporting the mass in FE mesh. In result, the interface and FE mesh are convected with the crystal lattice drift. In the constitutive modeling applied [1] the lattice strain and the atom fraction of chemical component are used as two independent thermodynamic variables. Due to climbing of misfit dislocations the plastic distortion tensor field is taken into account in the form of additional nodal variables. This tensor field is spanned on corner nodes of Lagrangian finite elements (FE) which gives the possibility for reconstruction of the atomistic model of dislocation network interpenetrating the considered FE mesh [2,3]. The chemo-mechanical coupling is based on the use of Vegard's law formulated in terms of Biot strain. Due to the logarithmic strain applied in hyperelastic modeling, some transformation rule is considered for Vegard's law. This rule allowed us to eliminate artificial residual stresses yielding from incompatibe fields of the atom fraction and plastic distortions spanned on nodes by means of shape functions [2]. In the case of single finite elements, the mentioned approach allowed us to reduce spurious stresses in integration points from the level 100 MPa to 10^ -5 MPa, while at the same time holding the stress components yielding from Vegard's law at the level of 1 GPa (relaxed by plastic distortions).

Constitutive modelling, Finite Element Method

In the current approach we consider interdiffusion, lattice distortions and chemical maps corresponding the growth of SiC/Si and In- GaN/GaN layers. Dislocations and free surfaces are treated as local regions for nucleation and annihilation of the vacancies transporting the mass between finite elements (FEs). In result, the interface and FE mesh are convected with the crystal lattice drift. In the constitutive modelling applied [1] the lattice distortion and the Si and vacancy molar fractions are used as independent nodal variables. Due to the climbing down of misfit dislocations the plastic distortion tensor field is taken into account in the form of additional nodal variables. This tensor field is spanned on corner nodes of the second order Lagrangian finite elements [2]. The chemo-mechanical coupling is based on the use of Vegard’s law formulated alternatively in terms of Biot or Hencky strains. Due to the logarithmic strain applied in hyperelastic modelling, some transformation rule is considered for Vegard’s law. This rule allowed us to eliminate artificial residual stresses yielding from incompatibility of the fields of atom fraction and plastic distortions spanned by means of the same shape functions on the corner nodes.

residual stresses, Kirkendall effect, diffusion, mass transport, semiconductor layers, crystal growth

Contrary to higher order elastic constants for momentum stresses the second (classical) and third-order elastic coefficients (TOEC) for symmetric elasticity are measured and tabulated successfully with good accuracy for tens of years. In the classical experimental measurements of TOEC, the correct recalculation of instantaneous stiffness changes onto TOEC has an important role. A similar problem arises in the constitutive and finite element (FE) modelling. Namely, because of a very strong dependency of TOEC on the strain measure choice, the constitutive and FE modelling of elastic materials is considered here in terms of different finite strain measures. To this aim, the known analytical formulae for calculation of two first derivatives of the isotropic tensor function of tensor variable are verified by means of the finite difference method. In result, the revised formulae are used for calculation of the tangent stiffness matrix. This paper closes with some remarks on the use of TOEC in finite element modelling.

nonlinear elasticity, third-order elastic coefficients, logarithmic strain

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

A method for reconstruction of atomistic models of dislocations and stacking faults in the interfacial region of heterostructures is presented. Its mathematical foundations come back to the algebra of the finite deformation fields related to introducing of discrete dislocations into an initially coherent interface. From the practical point of view the method concerns generation of interfacial regions with misfit/treading partial dislocations and stacking faults being formed in the interfacial region between crystal structures of different crystallographic type.

atomistic models, dislocations, stacking faults, lattice distortion