Continuum models of heteroepitaxial growth on patterned substrates

The continuous advancing of semiconductor technologies in micro-electronics, optics, energy production, ..., constantly demands for an improvement in the choice of suitable materials. Heteroepitaxial systems represents a class of materials that has been largely exploited to achieve such enhancements thanks to the possibility to combine properties of the different material with a large degree of customization and tuning. Many different heteroepitaxial systems are possible, from small nanostruc- tures (e.g. quantum dots, islands), to microscopic crystals or films. Generally such structures form spontaneously during the growth in consequence of a self- assembly process. If the growth is operated on flat undifferentiated substrate the self-assembly process can be controlled only on a rather limited extent, by properly tuning the growth conditions responsible for the material rearrangement during the growth. The possibility to achieve a major control on the formation of the different structures along the surface is highly desirable. Substrate patterning techniques strongly evolved in the last decade allowing for a better control of the system evolution through the introduction of preferential sites where the growing material can accumulate and evolve in a more controllable way. With this respect, the understanding of the mechanisms underlying the specific growth modes observed experimentally is crucial to proficiently define the optimal growth conditions for the desired behavior. Growth models for heteroepitaxial systems are then widely used on very different scales. Due to the large variability of the systems, different models need to be in general considered for the several typologies of materials and growth methods. In this Thesis we essentially focus our attention on two different physical systems: the Stranski-Krastanow growth of three-dimensional islands on pit- patterned substrates, and the growth of 3D crystals on substrate patterned with pillar structures, on the micrometer size scale. In order to define the appropriate way of modeling, the characteristic size and time scale of the systems should be taken into account. The typical periodicity of patterns used for island growth in Stranski-Krastanow pit is larger than 100 nm and the grown islands can in general become very large, including several milions of atoms. Deposition of few atomic layers are usually considered but the growth rate is typically slow so that the growth process lasts for minutes. The growth on pillar patterns is performed on even larger size scale, with typical dimensions of μm; also, several μm of material are deposited so that the growth can even require hours. Evidently, an atomistic description is not feasible so that a coarsened scale must be considered. Continuum models are hence the best choice for the description of these kind of systems, as they allow to efficiently describe the overall profile evolution for long time scales, including most of the physics resulting from the underlying atomic events in an average way. The problem of Stranski-Krastanow island growth can be mainly defined on the basis of thermodynamic arguments as it is typically obtained by slow Molecular Beam Epitaxy at temperature high enough to guarantee an effective diffusion dynamics of the atoms along the surface. Such conditions can be considered close enough to equilibrium so that the main driving force for the system evolution is the free energy minimization, although, as largely discussed in this work, thermodynamics applies only to the surface region. Thermodynamic- driven profile evolution has been object of large interest since the seminal work by W.W. Mullins, more than fifty years ago. Several models accounting for the effects of surface energy and profile faceting have been developed. Strain effects have also been successfully included, offering a deeper insight on the islanding mechanisms in connection with the raise of morphological instability of a flat film. Several studies have been devoted also to the characterization of the role of substrate patterning. However, most of these models were based on a single component picture loosing a crucial element that proved to play an essential role in the evolution of the growth process: intermixing between the deposited material and the substrate one. Some attempts to account for this effect, accounting for the evolution of the composition profile have been proposed. In 2003, J. Tersoff developed a successful description of the coupled evolution of profile and composition, based on the restriction of intermixing within a small layer around the surface, that proved to be particularly well suited to capture the main physics of Stranski-Krastanow heteroepitaxial growth on flat substrate. In this Thesis we discuss of an extension of the original Tersoff model to the characterization of the growth on pit-patterned substrate, developed in direct collaboration with J. Tersoff himself. Ge/Si system is considered. In particular, a realistic description of the surface dynamics for the adatoms of both components, accounting for their different mobilities and its dependence on the local environment, is introduced into the model derivation. Simulations are then implemented for an initial pit-patterned geometry of the substrate and the evolution is investigated in comparison with the experimental phenomenology. An effect of anomalous smoothing of the substrate profile induced in the initial stages of the growth and recently observed in experiments is analyzed in details and explained on the basis of simulation results. Islands formation is also inspected and their morphological evolution is considered. Tersoff approach was proved to be effective to capture the main physical as- pects related to the intermixing dynamics, crucial for the understanding of many aspects of heteroepitaxy. Some technical limitations however reduce the possibilities to extend it to complex system geometries, in particular three-dimensional systems. In order to allow for a future study of more general problems, during this Thesis work an original phase-field model for heteroepitaxial growth has been developed, in collaboration with A. Voigt and his group. Phase-field technique was found to be particularly well suited for the analysis of surface processes, offering an efficient way to represent the profile evolution of a free surface upon surface diffusion, eventually including also strain effects. Multiple component systems are also considered in phase-field models for alloy solidification. An exhaustive model for the heteroepitaxial processes, including both surface dynamics and intermixing effects, is not yet available in literature. The purpose of this part of the Thesis then consists in the definition of a suitable model accounting for such peculiar features of Stranski-Krastanow heteroepitaxial systems. A proof of concepts is provided by example simulations, testing the effects of the various contribution defining the problem. The modeling of the growth processes leading to the self-assembly of crystals on pillar patterned substrates is much different as the growth in this case is obtained in conditions of high growth rates and relatively low temperature, driving the system well far from equilibrium, in conditions such that the thermodynamic driving forces are frustrated by kinetic effects. In this case, a different approach is needed, properly accounting for the different kinetics at the surface. In particular, a modeling for such conditions can be defined by exploiting the local mechanisms of atomic exchange between the grow flux and the advancing crystal front by means of suitable rate equations. The dynamics on the different facets of crystals is then characterized in terms of an orientation dependent growth rate, mainly determined by the impinging flux. In this Thesis a model including both intrinsic different growth properties and local deposition fluxes is defined and used to interpret the growth of micrometric 3D crystals on a suitable pattern of pillars, obtained in experiments for both Ge/Si and GaAs/Si pillars. Simulation results allow to inspect the key features of the observed growth modality, showing the crucial role of frustrated diffusion and mutual flux shielding between neighboring crystals. A close comparison with the experiments is reported.