Prof. Paweł Dłużewski, Ph.D., Dr. Habil., Eng. 

Doctoral thesis
1985  Zastosowanie teorii poślizgów do opisu procesów deformacji niesprężystych
 422 
Habilitation thesis
1997  Kontynualna teoria dyslokacji jako teoria konstytutywnego modelowania skończonych, sprężystoplastycznych deformacji 
Professor
20091230  Title of professor 
Supervision of doctoral theses
1.  20060629  Jurczak Grzegorz  Anizotropia sprężysta kryształów. Analiza i modelowanie numeryczne metodą elementów skończonych.  595  
2.  20060126  Traczykowski Piotr  Wykorzystanie statyki molekularnej do modelowania procesów deformacji kryształów półprzewodnikowych  590  
3.  20030306  Maciejewski Grzegorz  Zastosowanie metody elementów skończonych do wyznaczania rozkładów naprężeń residualnych w heterostrukturach  568 
Recent publications
1.  Jurczak G., Dłużewski P., Finite element modelling of threading dislocation effect on polar GaN/AlN quantum dot, PHYSICA ELOWDIMENSIONAL SYSTEMS & NANOSTRUCTURES, ISSN: 13869477, DOI: 10.1016/j.physe.2017.08.018, Vol.95, pp.1115, 2018 Abstract: 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 screwtype, common for IIInitride 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 builtin the quantum dot. Local lattice deformation near the dislocation core reduce residual strains in the quantum dot. It is prominent in the case of edgetype 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 bandtoband transition energy for edge and screwtype dislocations are rather small, what suggest low sensitivity to the charge density along dislocation line. Unexpectedly, local strain field around the edgetype dislocation, slightly compensate the negative affect of the electrostatic potential. Keywords:Quantum dot, Threading dislocation, Piezoelectricity, Finite element modelling Affiliations:
 
2.  Domagała J.Z.^{♦}, Morelhão S.L.^{♦}, Sarzyński M.^{♦}, Maździarz M., Dłużewski P., Leszczyński M.^{♦}, Hybrid reciprocal lattice: application to layer stress determination in GaAlN/GaN(0001) systems with patterned substrates, JOURNAL OF APPLIED CRYSTALLOGRAPHY, ISSN: 16005767, DOI: 10.1107/S1600576716004441, Vol.49, No.3, pp.798805, 2016 Abstract: 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 highresolution diffractometer. Keywords:optoelectronics, Group IIInitride semiconductors, epitaxial growth, Xray multiple diffraction, interface defects Affiliations:
 
3.  Jurczak G., Dłużewski P., Finite element modelling of nonlinear piezoelectricity in wurtzite GaN/AlN quantum dots, COMPUTATIONAL MATERIALS SCIENCE, ISSN: 09270256, DOI: 10.1016/j.commatsci.2015.09.024, Vol.111, pp.197202, 2016 Abstract: In this paper we evaluate the effect of higher order elastic and piezoelectric coefficients on elastic and electric fields in IIInitride 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 buildin 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. Keywords:Piezoelectricity, Heterostructure, Nonlinearity, Quantum dot Affiliations:
 
4.  Young T.D., Jurczak G., Lotsari A.^{♦}, Dimitrakopulos G.P.^{♦}, Komninou Ph.^{♦}, Dłużewski P., A study of the piezoelectric properties of semipolar 11(2)overbar2 GaN/AlN quantum dots, PHYSICA STATUS SOLIDI BBASIC SOLID STATE PHYSICS, ISSN: 03701972, DOI: 10.1002/pssb.201552156, Vol.252, No.10, pp.22962303, 2015 Abstract: GaN quantum dots grown in (inline image)’orientated AlN are studied. The inline imagenucleated quantum dots exhibit rectangular or trapezoidbased truncated pyramidal morphology. Another quantum dot type orientated on inline image is reported. Based on highresolution transmission microscopy and crystal symmetry, the geometry of inline imageorientated quantum dots is proposed. A piezoelectric model is used within a finite element method to determine and compare the strainstate and electrostatic potential associated with the quantum dot morphology and an estimation of the bandedge energy is made. We report on some novel properties of the inline imageorientated quantum dot, including mixed strainstates and strainstate bowing. Keywords:III–V semiconductors, AlN, GaN, nanostructures, piezoelectric properties, quantum dots Affiliations:
 
5.  Cholewiński J., Maździarz M., Jurczak G., Dłużewski P., Dislocation core reconstruction based on finite deformation approach and its application to 4HSiC crystal, INTERNATIONAL JOURNAL FOR MULTISCALE COMPUTATIONAL ENGINEERING, ISSN: 15431649, DOI: 10.1615/IntJMultCompEng.2014010679, Vol.12, No.5, pp.411421, 2014 Abstract: 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 4HSiC 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, peratom energy, and peratom stresses for the structure reconstructed by use of the iterative algorithm against the classical solution based on the Love's equations. Keywords:dislocation, dislocation core energy, finite deformation, molecular statics Affiliations:
 
6.  Dłużewski P., Cholewiński J., Continuum and atomistic modelling of crystal defects, Annual Report  Polish Academy of Sciences, ISSN: 16403754, pp.8082, 2013  
7.  Dimitrakopulos G.P.^{♦}, Kalesaki E.^{♦}, Kioseoglou J.^{♦}, Kehagias T.^{♦}, Lotsari A.^{♦}, Lahourcade L.^{♦}, Monroy E.^{♦}, Häusler I.^{♦}, Kirmse H.^{♦}, Neumann W.^{♦}, Jurczak G., Young T.D., Dłużewski P., Komninou Ph.^{♦}, Karakostas T.^{♦}, Morphology and strain of selfassembled semipolar GaN quantum dots in (1112) AlN, JOURNAL OF APPLIED PHYSICS, ISSN: 00218979, DOI: 10.1063/1.3506686, Vol.108, pp.10430419, 2010 Abstract: GaN quantum dots (QDs) grown in semipolar (1122) AlN by plasmaassisted molecularbeam epitaxy were studied by transmission electron microscopy (TEM) and scanning transmission electron microscopy techniques. The embedded (112)grown QDs exhibited pyramidal or truncatedpyramidal 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 (1122) surface, QDs nucleated at depressions comprising {1011} facets. This was justified by ab initio density functional theory calculations showing that such GaN/AlN facets are of lower energy compared to (1122). Based on quantitative highresolution TEM strain measurements, the threedimensional QD strain state was analyzed using finiteelement simulations. The internal electrostatic field was then estimated, showing small potential drop along the growth direction, and limited localization at most QD interfaces. Keywords:Quantum dots, Transmission electron microscopy, IIIV semiconductors, High resolution transmission electron microscopy, Epitaxy Affiliations:
 
8.  Dłużewski P., Young T.D., Dimitrakopulos G.^{♦}, Komninou Ph.^{♦}, Continuum and atomistic modelling of the mixed straight dislocation, INTERNATIONAL JOURNAL FOR MULTISCALE COMPUTATIONAL ENGINEERING, ISSN: 15431649, DOI: 10.1615/IntJMultCompEng.v8.i3.80, Vol.8, pp.331342, 2010 Abstract: A continuum and atomistic approach to the modeling of dislocations observed by highresolution 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/finiteelement 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. Keywords:dislocations, field theory, atomistic models, finite element method, highresolution transmission electron microscopy Affiliations:
 
9.  Maździarz M., Young T.D., Dłużewski P., Wejrzanowski T.^{♦}, Kurzydłowski K.J.^{♦}, Computer modelling of nanoindentation in the limits of a coupled molecularstatics and elastic scheme, JOURNAL OF COMPUTATIONAL AND THEORETICAL NANOSCIENCE, ISSN: 15461955, DOI: 10.1166/jctn.2010.1469, Vol.7, pp.110, 2010 Abstract: Our numerical approach to modeling elasticplastic deformation comes back to the idea of the timeindependent plasticity developed here at the molecularstatics level. We use a constitutive atomic model based on the secondmoment approximation of the tightbinding 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 timedependency of the nanoindentation problem is reduced to a quasistatic 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 forcedepth response in terms of atomic bondlengths, elasticplastic 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 grainboundaries are responsible for this change in the overall instantaneous stiffness of the material. Keywords:linear elasticity, material science, molecular statics, nanoindentation, quasicontinuum methods Affiliations:
 
10.  Dłużewski P., Maździarz M., Traczykowski P.^{♦}, Jurczak G., Niihara K.^{♦}, Nowak R.^{♦}, Kurzydłowski K.^{♦}, A hybrid atomisticcontinuum finite element modelling of nanoindentation and experimental verification for copper crystal, COMPUTER ASSISTED METHODS IN ENGINEERING AND SCIENCE, ISSN: 22993649, Vol.15, pp.3744, 2008 Abstract: 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 TightBinding SecondMoment Approximation potential to create new type of elements. The molecular interactions yielding from constitutive models of TBSMA were turned into interactions between nodes to solve a boundary value problem by means of finite element solver. Nanostructure, Nanoindentation, Molecular statics, Finite element modelling Affiliations:
 
11.  Young T.D., Kioseoglou J.^{♦}, Dimitrakopulos G.P.^{♦}, Dłużewski P., Komninou Ph.^{♦}, 3D modelling of misfit networks in the interface region of heterostructures, JOURNAL OF PHYSICS DAPPLIED PHYSICS, ISSN: 00223727, DOI: 10.1088/00223727/40/13/027, Vol.40, pp.40844091, 2007 Abstract: 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. Affiliations:
 
12.  Kret S.^{♦}, Dłużewski P., Szczepańska A.^{♦}, Żak M.^{♦}, Czernecki R.^{♦}, Kryśko M.^{♦}, Leszczyński M.^{♦}, Maciejewski G., Homogenous indium distribution in InGaN/GaN laser active structure grown by LPMOCVD on bulk GaN crystal revealed by transmission electron microscopy and Xray diffraction, NANOTECHNOLOGY, ISSN: 09574484, DOI: 10.1088/09574484/18/46/465707, Vol.18, No.46, pp.4657070, 2007 Abstract: We present transmission electron microscopy (TEM) and xray 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 lowpressure metalorganic chemical vapour deposition (LPMOCVD) on a bulk, dislocationfree, monocrystalline 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 xray 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 Inrich islands formed on a sample surface during electronbeam exposure. Affiliations:
 
13.  Dłużewski P., Nonlinear field theory of stress induced interdiffusion and mass transport, DEFECT AND DIFFUSION FORUM, ISSN: 16629507, DOI: 10.4028/www.scientific.net/DDF.264.63, Vol.264, pp.6370, 2007 Abstract: 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. Keywords:Chemical Strain, Driving Force, Interdiffusion, Kirkendall Effect, Mass Transport Affiliations:
 
14.  Teisseyre H.^{♦}, Suski T.^{♦}, Łepkowski S.P.^{♦}, Perlin P.^{♦}, Jurczak G., Dłużewski P., Daudin B.^{♦}, Grandjean N.^{♦}, Strong electric field and nonuniformity effects in GaN/AlN quantum dots revealed by high pressure studies, APPLIED PHYSICS LETTERS, ISSN: 00036951, DOI: 10.1063/1.2219381, Vol.89, No.5, pp.5190213, 2006 Abstract: 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 pressureinduced increase of the builtin electric field. A comparison is made between experimental results and the proposed theoretical model which describes the pressure behavior of nitride QDs. Keywords:IIIV semiconductor, Quantum dot, Piezoelectricity, Photoluminescence Affiliations:
 
15.  Dłużewski P., Nonlinear field theory of stress induced diffusion in crystalline solids, DEFECT AND DIFFUSION FORUM, ISSN: 16629507, DOI: 10.4028/www.scientific.net/DDF.237240.107, Vol.237240, pp.107114, 2005 Keywords: Chemical Strain, Dislocation Climbing, Driving Force, Interdiffusion, Kirkendall Effect, Molar Derivative, Molar Motion, Vacancy Affiliations:
 
16.  Jurczak G., Maciejewski G., Kret S.^{♦}, Dłużewski P., Ruterana P.^{♦}, Modelling of indium rich clusters in MOCVD InxGa1−xN/GaN multilayers, JOURNAL OF ALLOYS AND COMPOUNDS, ISSN: 09258388, DOI: 10.1016/j.jallcom.2004.05.038, Vol.382, No.12, pp.1016, 2004 Abstract: 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. Keywords:Indium clusters, Vapour deposition, Transmissionelectron microscopy, Elasticity, Finite element method, Lattice distortion, Image simulation Affiliations:
 
17.  Dłużewski P., Maciejewski G., Jurczak G., Kret S.^{♦}, Laval J.Y.^{♦}, Nonlinear FE analysis of residual stresses induced by dislocations in heterostructures, COMPUTATIONAL MATERIALS SCIENCE, ISSN: 09270256, DOI: 10.1016/j.commatsci.2003.10.012, Vol.29, No.3, pp.379395, 2004 Abstract: 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 boundaryvalue 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. Keywords:Microscopy and microanalysis techniques, Nonlinear elasticity, Dislocation structure, Finite element analysis, Residual stresses, Layered structures Affiliations:
 
18.  Maciejewski G., Dłużewski P., Nonlinear finite element calculations of residual stresses in dislocated crystals, COMPUTATIONAL MATERIALS SCIENCE, ISSN: 09270256, DOI: 10.1016/j.commatsci.2004.01.027, Vol.30, No.12, pp.4449, 2004 Abstract: 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 lowangle dislocation wall and a subboundary structure. The method can be applied especially to dislocation sets composed of a few groups of monomial dislocations. Affiliations:
 
19.  Dłuzewski P., Jurczak G., Antúnez H.J., Logarithmic strain measure in finite element modelling of anisotropic hyperelastic materials, Computer Assisted Mechanics and Engineering Sciences, ISSN: 1232308X, Vol.10, No.1, pp.6979, 2003 Abstract: 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. Keywords:Anisotropic material, Constitutive behaviour, Elastic material, Finite element method, Logarithmic strain measure Affiliations:
 
20.  Dłużewski P., Rodzik P., Elastic eigenstates in finite element modelling of large anisotropic elasticity, COMPUTER METHODS IN APPLIED MECHANICS AND ENGINEERING, ISSN: 00457825, DOI: 10.1016/S00457825(97)002958, Vol.160, No.34, pp.325335, 1998 Abstract: FE algorithm for large anisotropic hyperelastic 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. Affiliations:
 
21.  Dłużewski P., On geometry and continuum thermodynamics of movement of structural defects, MECHANICS OF MATERIALS, ISSN: 01676636, DOI: 10.1016/01676636(95)000208, Vol.22, No.1, pp.2341, 1996 Abstract: 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 wellknown 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 wellknown, e.g. PeachKoehler 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. Oriented continuum, Dislocation density tensor, Driving forces, Curvature tensors Affiliations:
 
22.  Dłużewski P., Kontynualna teoria dyslokacji jako teoria konstytuwnego modelowania skończonych sprężystoplastycznych deformacji (Praca habilitacyjna), Prace IPPT  IFTR Reports, ISSN: 22993657, No.13, pp.1112, 1996  
23.  Dłużewski P., Finite deformations of polar elastic media, INTERNATIONAL JOURNAL OF SOLIDS AND STRUCTURES, ISSN: 00207683, DOI: 10.1016/00207683(93)90087N, Vol.30, No.16, pp.22772285, 1993 Abstract: 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.
 
24.  Dłużewski P., Crystal orientation spaces and remarks on the modelling of polycrystal anisotropy, JOURNAL OF THE MECHANICS AND PHYSICS OF SOLIDS, ISSN: 00225096, DOI: 10.1016/00225096(91)90045P, Vol.39, No.5, pp.651661, 1991 Abstract: recently, in many microstructural models of polycrystals the micromacro 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. Affiliations:
 
25.  Dłużewski P., Zastosowanie teorii poślizgów do opisu procesów deformacji niesprężystych (Praca doktorska), Prace IPPT  IFTR Reports, ISSN: 22993657, No.37, pp.191, 1985 
List of chapters in recent monographs
1. 399  Jurczak G., Maździarz M., Dłużewski P., Nanomechanics: Selected problems, rozdział: Atomisticcontinuum modelling of coupled fields and defects in semiconductor crystals, Wydawnictwa Politechniki Krakowskiej , pp.7798, 2014 
2. 3  Dłużewski P., Young T.D., Dimitakopulos G.^{♦}, Kioseoglou J.^{♦}, Komninou Ph.^{♦}, Computer Methods in Mechanics, Advanced Structured Materials, rozdział: Nonlinear finite element and atomistic modelling of dislocations in heterostructures, Springer Verlag, Kuczma M., Wilmański K. (Eds.), 1/III, pp.239253, 2010 
Editor of monographs
1. 425  Dłużewski P., Jurczak G., Young T.D., Book of abstracts of 3rd International Conference on Material Modelling incorporating 13th > European Mechanics of Materials Conference, IPPT PAN, pp., 2013 
Conference papers
1.  Jurczak G., Young T.D., Dłużewski P., A quantum dot nucleated on the edge of a threading dislocation: elastic and electric field effects, PHYSICA STATUS SOLIDI C: CURRENT TOPICS IN SOLID STATE PHYSICS, ISSN: 16101642, DOI: 10.1002/pssc.201200551, Vol.10, pp.97100, 2013 Abstract: 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 electroelastic line defect originating in the matrix material. Two common types of dislocation are considered: an edgetype and a screwtype. quantum dot, threading dislocation, piezoelectricity, band edge structure Affiliations:
 
2.  Maździarz M., Nalepka K.T., Dłużewski P., Cholewiński J., Reconstruction of dislocations in interface layer CuAl2O3, MMM2010, 5th International Conference Multiscale Materials Modeling, 20101004/1008, Freiburg (GE), pp.482485, 2010 Abstract: 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. Affiliations:
 
3.  Dłużewski P., Young T.D., Jurczak G., Majewski J.A.^{♦}, Nonlinear piezoelectric properties of GaN quantum dots nucleated at the edge of threading dislocations, IWN 2006, International Workshop on Nitride Semiconductors, 20061022/027, Kyoto (JP), DOI: 10.1002/pssc.200674866, Vol.4, pp.23992402, 2007 Abstract: 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. Keywords:Nanostructure, IIIV semiconductor, Piezoelectricity, Threading dislocation Affiliations:
 
4.  Dłużewski P., Belkadi A., Chen J.^{♦}, Ruterana P.^{♦}, Nouet G.^{♦}, FE simulation of InGaN QD formation at the edge of threading dislocation in GaN, IWN 2006, International Workshop on Nitride Semiconductors, 20061022/027, Kyoto (JP), DOI: 10.1002/pssc.200674870, Vol.7, pp.24032406, 2007 Abstract: The stress induced diffusion process of InGa 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. Affiliations:
 
5.  Postek E.W., Hardy S.J.^{♦}, Dłużewski P., Stress analysis of a quantum dot region using an elastic anisotropic model, Plasticity 2006, 12th International Symposium on Plasticity and Its Current Applications, 20060717/0722, Halifax (CA), pp.304306, 2006  
6.  Jurczak G., Łepkowski S.P.^{♦}, Dłużewski P., Suski T.^{♦}, Modeling of elastic, piezoelectric and optical properties of vertically correlated GaN/AlN quantum dots, EMRS 2004, Symposium on Science and Technology of Nitrides and Related Materials/Wide Band Gap IIVI Semiconductors, EMRS 2004 FALL MEETING SYMPOSIA C AND F, 20040806/0810, Warszawa (PL), DOI: 10.1002/pssc.200460604, Vol.2, No.3, pp.972975, 2005 Abstract: We theoretically investigate elastic, piezoelectric and optical properties of wurtzite GaN/AlN quantum dots, having hexagonal pyramidshape, 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, bandtoband transition energies in the vertically correlated quantum dots show rather weak dependence on the width of AlN barriers. Keywords:IIIV semiconductor, quantum dot, piezoelectricity, elastic strain, electrostatic potential Affiliations:
 
7.  Ruterana P.^{♦}, Singh P.^{♦}, Kret S.^{♦}, Cho H.K.^{♦}, Lee H.J.^{♦}, Suh E.K.^{♦}, Jurczak G., Maciejewski G., Dłużewski P., Size and shape of In rich clusters and InGaN QWs at the nanometer scale, IWN 2004, International Workshop on Nitrides Semiconductors, 20040619/0623, Pittsburgh (US), DOI: 10.1002/pssc.200461463, Vol.2, No.7, pp.23812384, 2005 Abstract: 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. Affiliations:

Conference abstracts
1.  Jurczak G., Maździarz M., Dłużewski P., Finite element modelling of threading dislocation effect on GaN/AlN quantum dot, ICMM5, 5th International Conference on Material Modeling, 20170614/0616, Rome (IT), pp.11, 2017 Keywords: Quantum dot, Threading dislocation, Piezoelectricity, Finite element modelling Affiliations:
 
2.  Dłużewski P., Maździarz M., Tauzowski P., Constitutive and FE modeling of Residual Stresses and Kirkendall Effect in Semiconductor Structures, 4th International Conference on Material Modeling, 20150527/0529, Berkeley (US), pp.7980, 2015 Abstract: 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 chemomechanical 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). Keywords:Constitutive modelling, Finite Element Method Affiliations:
 
3.  Dłużewski P., Tauzowski P., The FE modelling of residual stresses and void formation observed during the growth of semiconductor layers, PCMCMM 2015, 3rd Polish Congress of Mechanics and 21st Computer Methods in Mechanics, 20150908/0911, Gdańsk (PL), pp.553554, 2015 Abstract: 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 chemomechanical 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. Keywords:residual stresses, Kirkendall effect, diffusion, mass transport, semiconductor layers, crystal growth Affiliations:
 
4.  Dłużewski P., Maździarz M., Tauzowski P., Thirdorder elastic coefficients and logarithmic strain in finite element modelling of anisotropic elasticity, SolMech 2014, 39th Solid Mechanics Conference, 20140901/0905, Zakopane (PL), pp.3536, 2014 Abstract: Contrary to higher order elastic constants for momentum stresses the second (classical) and thirdorder 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. Keywords:nonlinear elasticity, thirdorder elastic coefficients, logarithmic strain Affiliations:
 
5.  Jurczak G., Young T.D., Dłużewski P., Dimitrakopulos G.P.^{♦}, Elastic and electric properties of a semipolar (1122) GaN/AlN quantum dots, ICMM3, 3rd International Conference on Material Modelling incorporating 13th European Mechanics of Materials Conference, 20130908/0911, Warszawa (PL), pp.243244, 2013 Keywords: Semiconductor, Quantum dot, Piezoelectricity, Bandedge structure Affiliations:
 
6.  Jurczak G., Maździarz M., Dłużewski P., Effect of threading dislocation on elastic and electric properties of semipolar GaN/AlN quantum dot, ICMM3, 3rd International Conference on Material Modelling incorporating 13th European Mechanics of Materials Conference, 20130908/0911, Warszawa (PL), pp.186187, 2013 Keywords: Semiconductor, Piezoelectricity, Dislocation Affiliations:
 
7.  Young T.D., Teisseyre H.^{♦}, Dłużewski P., Kahouli A.^{♦}, Kriouche N.^{♦}, Brault J.^{♦}, Damilano B.^{♦}, Leroux M.^{♦}, Courville A.^{♦}, de Mierry P.^{♦}, GaN/AlGaN (1122) semipolar nanostructures: Observations by theory and experiment, 41st International School and Conference on the Physics of Semiconductors, 20120608/0615, KrynicaZdrój (PL), 2012  
8.  Kret S.^{♦}, Bilska M.^{♦}, Ivaldi F.^{♦}, Leszczyński M.^{♦}, Czernecki R.^{♦}, Dłużewski P., Jurczak G., Young T.D., Determination of the nanoscale structural properties of the InAlN based devices by advanced TEM methods, EMRS 2012 FALL MEETING, 20120917/0921, Warszawa (PL), 2012 Keywords: IIIV semiconductors, piezoelectricity, high resolution transmission electron microscopy, band edge structure Affiliations:
 
9.  Jurczak G., Young T.D., Dłużewski P., Elastic and Electric Field Effects of a Quantum Dot Nucleated on the Edge of a Threading Dislocation, EDS 2012, International Conference on Extended Defects in Semiconductors, 20120624/0629, Thessaloniki (GR), pp.103, 2012 Keywords: quantum dot, threading dislocation, piezoelectricity, band edge structure Affiliations:
 
10.  Young T.D., Jurczak G., Dłużewski P., Wpływ defektów struktury krystalicznej na elektromechaniczne własności nanostruktur, III National Conference of Nano and Micromechanics, 20120704/0706, Warszawa (PL), pp.125127, 2012 Keywords: Piezoelektryczność, Mechanika Ośrodków Ciągłych, Metoda Elementów Skończonych, Nanostruktura Affiliations:
 
11.  Young T.D., Teisseyre H.^{♦}, Brault J.^{♦}, Kahouli A.^{♦}, Vennegues P.^{♦}, Leroux M.^{♦}, Courville A.^{♦}, de Mierry P.^{♦}, Damilano B.^{♦}, Dłużewski P., Optoelectronic properties of a GaN quantum dot grown on a Al_{0.5}Ga_{0.5}N (1122)orientated surface, IWN 2012, International Workshop on Nitride Semiconductors, 20121014/1019, Sapporo (JP), pp.12, 2012  
12.  Dłużewski P., Cholewiński J., Maździarz M., Tauzowski P., Nalepka K.T., Atomistic/continuum reconstruction of misfit dislocations and stacking faults in Cu/sapphire interfacial region, CMM 2011, 19th International Conference on Computer Methods in Mechanics, 20110509/0512, Warszawa (PL), pp.25712, 2011 Abstract: 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. Keywords:atomistic models, dislocations, stacking faults, lattice distortion Affiliations:
