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Le projet de recherche Modélisation multiéchelle de la rhéologie des minéraux du manteau (dont l’acronyme anglais est RheoMan) a été lauréat de l’ERC (European research Council) suite à l’appel d’offre Advanced Grant de 2011. Il visait à proposer une nouvelle approche de la rhéologie du manteau basée sur une approche de physique des minéraux qui intègre les avancées les plus récentes de la modélisation numérique des matériaux. Ce ebook rédigé en anglais décrit le projet, son contexte et les principaux résultats obtenus. Partant de l’échelle des défauts cristallins à l’échelle atomique, le projet RheoMan a permis d’obtenir les premières lois rhéologiques purement numériques qui sont applicables aux déformations naturelles sans extrapolation. Au cours de ce travail, de nouveaux mécanismes sont apparus : les disclinaisons aux joints de grains ou la montée pure qui éclairent la rhéologie du manteau terrestre (et des intérieurs planétaires) sous un nouveau jour. The research project Multiscale modelling of the rheology of mantle minerals (acronym RheoMan) was awarded by the European Research Council in response to its 2011 Advanced Grant call. Its aim was to provide a new approach of the rheology of the mantle from the mineral physics point of view which would take advantage of the most recent developments of computational materials science. The RheoMan project ended in May 2018. This ebook describes the project and the main results obtained. Starting from the atomic scale and the structure of crystal defects, the RheoMan project provides the first numerical rheological laws applicable without extrapolation to natural conditions. In the course of this project, new mechanisms emerged: disclinations at grain boundaries and pure climb creep, providing new paradigms for the rheology of the Earth’s mantle and planetary interiors.

The hetero-epitaxial strain relaxation in nano-scale systems plays a fundamental role in shaping their properties. Here, the elastic and plastic relaxation of self-assembled SiGe islands grown by surface-thermal-diffusion from a local Ge solid source on Si(100) are studied by atomic force and transmission electron microscopies, enabling the simultaneous investigation of the strain relaxation in different dynamical regimes. Islands grown by this technique remain dislocation-free and preserve a structural coherence with the substrate for a base width as large as 350 nm. The results indicate that a delay of the plastic relaxation is promoted by an enhanced Si-Ge intermixing, induced by the surface-thermal-diffusion, which takes place already in the SiGe overlayer before the formation of a critical nucleus. The local entropy of mixing dominates, leading the system toward a thermodynamic equilibrium, where non-dislocated, shallow islands with a low residual stress are energetically stable. These findings elucidate the role of the interface dynamics in modulating the lattice distortion at the nano-scale, and highlight the potential use of our growth strategy to create composition and strain-controlled nano-structures for new-generation devices.

Thanks to their unique morphology, nanowires have enabled integration of materials in a way that was not possible before with thin film technology. In turn, this opens new avenues for applications in the areas of energy harvesting, electronics, and optoelectronics. This is particularly true for axial heterostructures, while core-shell systems are limited by the appearance of strain-induced dislocations. Even more challenging is the detection and understanding of these defects. We combine geometrical phase analysis with finite element strain simulations to quantify and determine the origin of the lattice distortion in core-shell nanowire structures. Such combination provides a powerful insight in the origin and characteristics of edge dislocations in such systems and quantifies their impact with the strain field map. We apply the method to heterostructures presenting single and mixed crystalline phase. Mixing crystalline phases along a nanowire turns out to be beneficial for reducing strain in mismatched core-shell structures.

SiGe heterostructures have gained a lot of interests in view of developing devices integrated into the main-stream Silicon technology and also from a scientific point of view as a prototypical system to understand the properties of more complex systems, such as III-V semiconductors. Si-Ge epitaxial structures, as well as other mismatched heteroepitaxial materials, have a high potential to improve the state-of-the-art of Si devices, thanks to the fact that the strain modifies the band structures of this material class, opening new possibility of band-gap engineering. Since the nineties, the development of devices having strained-SiGe layers as the active part occurred, in particular the heterojunction bipolar transistors, further developed to what is presently the fourth-generation of SiGe technology. Also the introduction of strained Si layers by using relaxed SiGe virtual substrates, is very important, for example, for the complementary metal–oxide–semiconductor (CMOS) technology. In order to effectively exploit SiGe or strained-Si layers in any application, it is fundamental to growth high quality single crystalline materials, reducing as much as possible the defect density in the active volume and the surface corrugation, and to obtain the desired strain state in the epitaxial layers. However the possibility of using such heterostructures for any application, is hindered by the nucleation of dislocation, which is often an unavoidable strain-relief mechanisms. Dislocation formation affects both the final material quality and the relaxation degree of mismatched layers. These defects are often charged and act as non-radiative recombination centers and it is generally accepted that they are detrimental for opto-electronic devices based on Si-Ge semiconductors. In the past years, a lot of effort has been devoted reduce the defect density or to segregate dislocations in non-active regions. However, dislocation engineering, intended as the precise control of dislocation position, has always been a goal out of reach, because of the nucleation of such defects at unpredictable sites at the surface or at other heterogeneities. It is clear that predicting the extent of the plastic relaxation process and governing dislocation nucleation and positioning would be of the utmost importance. Self-assembled nanoislands and nanowires, represent other novel heterostructures that can be exploited to obtain defect-free configurations with the desired strain state. Even in this case, very high stresses arise from the epitaxial integration of lattice mismatched materials and dislocation formation remains a competitive strain relief mechanism. Hence it is of fundamental importance to determine the coherency limits of such nanostructures and to elucidate the main strain relief mechanisms in the attempt to predict the final dislocation microstructure and strain state in heteroepitaxial systems. The main goal of this work, is the understanding of the fundamental mechanisms of dislocation nucleation and propagation in Si-Ge nanostructures (i.e. films, nanoislands and nanowires) through dislocation modelling. Even if dislocation formation and motion relies on a sequence of discrete atomic displacements, such defects induce in a crystal a smooth deformation field in the entire structure. The elastic theory of dislocations provides a good description of such stress field and of the elastic energy, as produced by dislocations in bulk materials or in finite size solids with simple geometries. In order to assess the stresses and the energetics of plastically relaxed multifaceted structures, characterized by an high surface to volume ratio and typical length scale in the order of tens or hundreds of nanometers, linear elasticity theory numerically solved by finite element methods is the most suitable tool, since in this approximation the dislocation-surface interaction can be correctly taken into account. Moreover, the motion of dislocations in nanostructures can be handled by using three-dimensional dislocation dynamics simulations. This simulation technique, originally developed to study plasticity in bulk materials, has been demonstrated to give accurate results also for nanometric systems, and is the tool of choice to study the motion and interactions of a large density of dislocation in thin films or three-dimensional nanocrystals. Important properties determined by the atomistic nature of dislocations moving in a discrete lattice, can be included, both in the finite element calculations and in dislocation dynamics simulations, by adopting simple rules that take into account such atomistic features. The first topic addressed in this work, is the investigation of plastic relaxation in SiGe epitaxial films aimed at governing dislocation nucleation and positioning. In particular, we show with the help of finite element calculations and dislocation dynamics simulations that a turning point to direct dislocation formation and propagation in predefined regions, is the introduction of preferential nucleation sites through substrate nanopatterning. Theoretical predictions indicating effective dislocation trapping along the features of trench- or pit-patterned substrates are discussed and compared with tailored experiments of SiGe deposition on nanopatterned substrate. The second issue investigated here concerns self-assembled SiGe nanoislands. In these epitaxial nanostructures an intriguing mechanism of dislocation ordering is observed. In this work we reproduced such behavior by using a simple analytical model based on energetics considerations. Furthermore, the plastic relaxation onset for dislocation formation has been determined in epitaxial islands grown on pit-patterned substrates and nucleated in pits. The key factors influencing dislocation formation in such structures have been identified, opening new possibility to grow large defect-free islands on nanopattered substrates. Finally, dislocation formation in core-shell nanowires has been considered. Elastic and plastic strain relaxaion has been investigated in such structures and a mechanism for dislocation nucleation and propagation in core-shell nanowires is presented. This allowed us to predict dislocation configurations that are more efficient in the strain relief process and the expected misfit dislocation pattern at the core-shell interface.

The growth of heteroepitaxially strained semiconductors at the nanoscale enables tailoring of material properties for enhanced device performance. For core/shell nanowires (NWs), theoretical predictions of the coherency limits and the implications they carry remain uncertain without proper identification of the mechanisms by which strains relax. We present here for the Ge/Si core/shell NW system the first experimental measurement of critical shell thickness for strain relaxation in a semiconductor NW heterostructure and the identification of the relaxation mechanisms. Axial and tangential strain relief is initiated by the formation of periodic a/2 ⟨110⟩ perfect dislocations via nucleation and glide on {111} slip-planes. Glide of dislocation segments is directly confirmed by real-time in situ transmission electron microscope observations and by dislocation dynamics simulations. Further shell growth leads to roughening and grain formation which provides additional strain relief. As a consequence of core/shell strain sharing in NWs, a 16 nm radius Ge NW with a 3 nm Si shell is shown to accommodate 3% coherent strain at equilibrium, a factor of 3 increase over the 1 nm equilibrium critical thickness for planar Si/Ge heteroepitaxial growth.

Ordering of misfit dislocation segments in concentric polygons at the base of SiGe epitaxial islands on Si(001) has been recently indicated by in situ Transmission Electron Microscope observation. In this paper we confirm the very regular spacing by Atomic Force Microscope and He-ion Microscope measurements of the footprint carved in the Si substrate by the plastic events. We explain the intriguing ordering, as obtained with no gliding rearrangements, by cyclic occurrence of the thermodynamic critical conditions for plastic events. Quantitative predictions by a fully analytical model, which includes the dependence on island shape and composition in the generation of misfit dislocations, matched very well experimental measurements.

We demonstrate the possibility of growing SiGe islands on patterned Si(001) substrates with pits having a continuous variation of the sidewall inclination angle \ensuremath{\alpha} from \ensuremath{\alpha} \ensuremath{\sim} 4\ifmmode^\circ\else\textdegree\fi{} to \ensuremath{\alpha} \ensuremath{\sim} 54\ifmmode^\circ\else\textdegree\fi{}. Experiments show that the pit-inclination angle critically affects island positioning. While for shallow pits (pit-sidewall inclination angle \ensuremath{\alpha} \ensuremath{\sim}30\ifmmode^\circ\else\textdegree\fi{}) islands are observed solely within the pits, at higher angles islands grow outside the pits. In particular a progressive (complete at \ensuremath{\alpha} \ensuremath{\sim} 54\ifmmode^\circ\else\textdegree\fi{}) decoration of the pit rim by several islands is observed. We use elasticity theory to compare strain relaxation of a single island inside and outside the pit, as a function of \ensuremath{\alpha}. The theoretical results show that there exists an elastic driving force for island positioning inside shallow enough pits, which reaches its maximum at \ensuremath{\alpha} \ensuremath{\sim} 20\ifmmode^\circ\else\textdegree\fi{} and changes sign for \ensuremath{\alpha} g 40\ifmmode^\circ\else\textdegree\fi{}. At the same time, the calculations indicate a progressive relaxation of the wetting layer (WL) outside the pit with increasing \ensuremath{\alpha}. The consistency between numerical results and experimental observations gives a clear indication on the important role played by the elastic relaxation in this system.

We demonstrate dislocation engineering without oxide masks. By using finite element simulations we show how nanopatterning of Si substrates with {111} trenches provides anisotropic elastic relaxation in a SiGe film, generates preferential nucleation sites for dislocation loops, and allows for dislocation trapping, leaving wide areas free of threading dislocations. These predictions are confirmed by atomic force and transmission electron microscopy performed on overcritical Si0.7 Ge0.3 films. These were grown by molecular beam epitaxy on a Si(001) substrate patterned with periodic arrays of selectively etched {111}-terminated trenches. © 2011 American Institute of Physics.

We show that suitable pit-patterning of a Si(001) substrate can strongly influence the nucleation and the propagation of dislocations during epitaxial deposition of Si-rich Si1-xGex alloys, preferentially gettering misfit segments along pit rows. In particular, for a 250 nm layer deposited by molecular beam epitaxy at x(Ge) - 15%, extended film regions appear free of dislocations, by atomic force microscopy, as confirmed by transmission electron microscopy sampling. This result is quite general, as explained by dislocation dynamics simulations, which reveal the key role of the inhomogeneous distribution in stress produced by the pit-patterning.

A simple, but still three-dimensional, model describing the morphological stability of realistic SiGe islands on Si(001) is presented. The experimental evolution toward steeper islands with volume can be predicted for any average composition. Despite the use of elastic theory for stress relaxation under the assumption of a uniform SiGe distribution, and of a common mean surface energy of the faceted islands, the model seems to capture the essence of the energetic balance determining the morphological evolution with volume, with no fitting parameters. This is suggested by close comparison with molecular beam epitaxy data at three different temperatures (i.e. compositions). The good agreement also allows for interpreting the minor scattering of experimental data with temperature and provides a reliable tool for extracting the average Ge content from standard atomic force microscopy analysis.

The onset of plastic relaxation in SiGe islands grown on pit-patterned Si(001) substrates is investigated using elasticity theory solved by finite element methods. Larger critical island volumes with respect to the unpatterned case are predicted. A justification based on the qualitatively different stressors acting on the substrate in the presence of pits is provided. Experimental results in terms of critical SiGe-island volumes as a function of the Ge content are nicely reproduced by the model.