Discrete models of dislocations in crystal lattices: formulation, analysis and applications

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Real crystal lattices are not perfect. They have defects such as dislocations, vacancies, and cracks that control the mechanical properties of materials, including crystal plasticity, creep, fatigue, ductility, brittleness, hardness and friction. Crystal growth, radiation damage of materials, and their optical and electronic properties are also strongly affected by defects, particularly dislocations. Why is it so important to understand the behavior of defects in crystal lattices? An accurate description of defect dynamics may help to optimize the design and manufacture of important nanoelectronic devices such as those based on self-assembled quantum dots [2, 3] or superlattices [4]. Moreover, assessing how and under which conditions dislocations nucleate may become an essential issue, since they act as scattering centers, degrading charge transport properties in opto-electronic devices. But at the present time, even the homogeneous nucleation of dislocations is not completely understood. While there is a widespread feeling that it is related to some bifurcation occurring once a dislocationfree state becomes unstable, no precise analysis and calculation of this bifurcation has been reported [5, 6]. Think of another example: to build up a superlattice, heteroepitaxial structures of alternate slices of semiconductors having different lattice spacings are grown. But layers with quite different lattice parameters do not fit seamlessly! This typically results in the formation of misfit dislocations at the interfaces that separate different materials. Therefore, it is crucial to compute threshold values for the formation of dislocations in many important experiments: the critical shear stress for homogeneous nucleation of dislocations, the critical thickness of a thin film (and also the critical discrepancy between their lattice constants -the critical misfit-) in heteroepitaxial growth for interfacial misfit dislocations 1 formation, the critical stresses leading to dislocation nucleation from cracks or from nanoindentor tips, and so on. The goal of this thesis is to provide some insight in the aforementioned issues. But, tackling these problems is not simple! Dislocations may affect phenomena such as the strength of materials occurring over many different scales of length and time and the properties at each scale are influenced by the others. At the present time there are different attempts to bridge the gaps between disparate scales by using detailed microscopic calculations such as molecular dynamics in small regions near defect cores and linear elasticity in the far field [7, 8, 9]. In this thesis, we have chosen to model dislocation dynamics at the nanoscale by versions of discrete elasticity that become the proper linear anisotropic elasticity of cubic crystals in the far field and allow motion of dislocations in a natural manner. One important advantage of these models is that they are amenable to analysis using bifurcation theory and numerical continuation methods. We have used these methods to study very simple scalar versions of discrete elasticity models for two-dimensional edge dislocations. Within these limitations, we have analyzed homogeneous nucleation of dislocations in sheared materials, misfit dislocations, nanoindentations and cracks. The simplicity of the models allows us to find a more precise picture of these phenomena that may be useful in the nanoscale. Whether these models can be used as part of multiscale/multiphysics calculations at larger scales, remains as a challenging task for future work. ____________________________________________
La comprensión del comportamiento de los defectos presentes en las redes cristalinas es esencial para el diseño y fabricación de dispositivos nanoelectrónicos porque afectan fuertemente sus propiedades electrónicas, ópticas y magnéticas. Asimismo, defectos como las dislocaciones son esenciales para el proceso de crecimiento de estructuras heteroepitaxiales, para entender la propagación de fisuras o en experimentos de nanoindentación que tratan de aclarar el comienzo de la plasticidad. En la presente tesis doctoral se formulan modelos discretos de dislocaciones en redes cristalinas del sistema cúbico (simple, centrado en las caras o centrado en el cuerpo, con la posibilidad de incluir una base de varios átomos en cada nodo de la red) que recuperan la elasticidad lineal anisotrópa en su límite continuo. En la tesis se analiza la nucleación homogénea de dislocaciones en un cristal bidimensional sujeto a tensiones de cizalladura y se concluye que los estados con dislocaciones aparecen como bifurcaciones subcríticas del estado estacionario sin dislocaciones. Las ramas bifurcadas multiestables se calculan por métodos de continuación numérica y se estudia su selección mediante ramping de la tensión de cizalla. También se calculan valores críticos para la formación de dislocaciones en sistemas heteroepitaxiales, así como en fisuras y experimentos de nanoindentación.
Cristalografía, Física del estado sólido, Redes cristalinas, Semiconductores, Dislocaciones
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