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The phase-field crystal (PFC) model describes crystal lattices at diffusive timescales. Its amplitude expansion (APFC) can be applied to the investigation of relatively large systems under some approximations. However, crystal symmetries accessible within the APFC model are limited to basic ones, namely triangular and square in two dimensions, and body-centered cubic and face-centered cubic in three dimensions. In this work, we propose a general, amplitudes-based description of virtually any lattice symmetry. To fully exploit the advantages of this model, featuring slowly varying quantities in bulk and localized significant variations at dislocations and interfaces, we consider formulations suitable for real-space numerical methods supporting adaptive spatial discretization. We explore approaches originally proposed for the PFC model which allow for symmetries beyond basic ones through extended parametrizations. Moreover, we tackle the modeling of non-Bravais lattices by introducing an amplitude expansion for lattices with a basis and further generalizations. We study and discuss the stability of selected, prototypical lattice symmetries. As pivotal examples, we show that the proposed approach allows for a coarse-grained description of the kagome lattice, exotic square arrangements, and the diamond lattice, as bulk crystals and, importantly, hosting dislocations., Comment: 14 pages, 7 figures

We present a phase-field crystal model for solidification that accounts for thermal transport and a temperature-dependent lattice parameter. Elasticity effects are characterized through the continuous elastic field computed from the microscopic density field. We showcase the model capabilities via selected numerical investigations which focus on the prototypical growth of two-dimensional crystals from the melt, resulting in faceted shapes and dendrites. This work sets the grounds for a comprehensive mesoscale model of solidification including thermal expansion.

An intrinsic feature of nearly all internal interfaces in crystalline systems (homo- and hetero-phase) is the presence of disconnections (topological line defects constrained to the interface that have both step and dislocation character). Disconnections play a major role in determining interface thermodynamics and kinetics. We demonstrate that elastic interactions between disconnections lead to a thermodynamic, first-order, finite-temperature, faceting-defaceting transition, in agreement with experiments. These elastic interactions strongly modify equilibrium interface morphologies (compared with those solely determined by anisotropic surface energy) as well as the kinetics and morphologies of migrating interfaces. We demonstrate these phenomena through numerical simulations based upon a general, continuum disconnection-based model for interface thermodynamics and kinetics applied to embedded particles/grains, steady-state interface migration geometries, and nominally flat interfaces.

We derive the amplitude expansion for a phase-field-crystal (APFC) model that captures the basic physics of magneto-structural interactions. The symmetry breaking due to magnetization is demonstrated, and the characterization of the magnetic anisotropy for a BCC crystal is provided. This model enables a convenient coarse-grained description of crystalline structures, in particular when considering the features of the APFC model combined with numerical methods featuring inhomogeneous spatial resolution. This is shown by addressing the shrinkage of a spherical grain within a matrix, chosen as a prototypical system to demonstrate the influence of different magnetizations. These simulations serve as a proof of concept for the modeling of manipulation of dislocation networks and microstructures in ferromagnetic materials within the APFC model., 17 pages, 9 figures; extended and accepted (Modelling and Simulation in Materials Science and Engineering)

We present an experimental and theoretical analysis of the formation of nanovoids within Si microcrystals epitaxially grown on Si patterned substrates. The growth conditions leading to the nucleation of nanovoids have been highlighted, and the roles played by the deposition rate, substrate temperature, and substrate pattern geometry are identified. By combining various scanning and transmission electron microscopy techniques, it has been possible to link the appearance pits of a few hundred nanometer width at the microcrystal surface with the formation of nanovoids within the crystal volume. A phase-field model, including surface diffusion and the flux of incoming material with shadowing effects, reproduces the qualitative features of the nanovoid formation thereby opening new perspectives for the bottom-up fabrication of 3D semiconductors microstructures., Ordered 3D arrays of nanovoids can be formed during the epitaxial growth of Si microcrystals on patterned substrates. Phase-field simulations provide a clear link between nanovoid formation and the ratio between diffusion coefficient and deposition rate. With the appropriate combination of pattern geometry, deposition rate and deposition temperature control over nanovoids formation could be achieved also with other semiconductors.

The amplitude expansion for a magnetic phase-field-crystal (magnetic APFC) model enables a convenient coarse-grained description of crystalline structures under the influence of magnetic fields. Considering higher-order magnetic coupling terms, we demonstrate the possibility of tuning the magnetic anisotropy in these models. This allows for reproducing the easy and hard direction of magnetization. Such a result can be achieved without increasing the computational cost, enabling simulations of the manipulation of dislocation networks and microstructures in ferromagnetic materials. As a demonstration, we report on the simulation of the shrinkage of a spherical grain with the magnetic anisotropy of Fe., Comment: 13 pages, 11 figures

Comprehensive investigations of crystalline systems often require methods bridging atomistic and continuum scales. In this context, coarse-grained mesoscale approaches are of particular interest as they allow the examination of large systems and time scales while retaining some microscopic details. The so-called Phase-Field Crystal (PFC) model conveniently describes crystals at diffusive time scales through a continuous periodic field which varies on atomic scales and is related to the atomic number density. To go beyond the restrictive atomic length scales of the PFC model, a complex amplitude formulation was first developed by Goldenfeld et al. [Phys. Rev. E 72, 020601 (2005)]. While focusing on length scales larger than the lattice parameter, this approach can describe crystalline defects, interfaces, and lattice deformations. It has been used to examine many phenomena including liquid/solid fronts, grain boundary energies, and strained films. This topical review focuses on this amplitude expansion of the PFC model and its developments. An overview of the derivation, connection to the continuum limit, representative applications, and extensions is presented. A few practical aspects, such as suitable numerical methods and examples, are illustrated as well. Finally, the capabilities and bounds of the model, current challenges, and future perspectives are addressed., Comment: 57 pages, 15 figures

We introduce a dislocation density tensor and derive its kinematic evolution law from a phase field description of crystal deformations in three dimensions. The phase field crystal (PFC) model is used to define the lattice distortion, including topological singularities, and the associated configurational stresses. We derive an exact expression for the velocity of dislocation line determined by the phase field evolution, and show that dislocation motion in the PFC is driven by a Peach-Koehler force. As is well known from earlier PFC model studies, the configurational stress is not divergence free for a general field configuration. Therefore, we also present a method (PFCMEq) to constrain the diffusive dynamics to mechanical equilibrium by adding an independent and integrable distortion so that the total resulting stress is divergence free. In the PFCMEq model, the far-field stress agrees very well with the predictions from continuum elasticity, while the near-field stress around the dislocation core is regularized by the smooth nature of the phase-field. We apply this framework to study the rate of shrinkage of an dislocation loop seeded in its glide plane.

We optimize a numerical time-stabilization routine for the phase-field crystal (PFC) models of solidification. By numerical experiments, we showcase that our approach can improve the accuracy of underlying time integration schemes by a few orders of magnitude. We investigate different time integration schemes. Moreover, as a prototypical example for applications, we extend our numerical approach to a PFC model of solidification with an explicit temperature coupling., Comment: 5 pages, 3 figures

In many processes for crystalline materials such as precipitation, heteroepitaxy, alloying, and phase transformation, lattice expansion or compression of embedded domains occurs. This can significantly alter the mechanical response of the material. Typically, these phenomena are studied macroscopically, thus neglecting the underlying microscopic structure. Here we present the prototypical case of an elastic inclusion described by a mesoscale model, namely a coarse-grained phase-field crystal model. A spatially-dependent parameter is introduced into the free energy functional to control the local spacing of the lattice structure, effectively prescribing an eigenstrain. The stress field obtained for an elastic inclusion in a 2D triangular lattice is shown to match well with the analytic solution of the Eshelby inclusion problem., 5 pages; 3 figures

Fabrication and scaling of disordered hyperuniform materials remain hampered by the difficulties in controlling the spontaneous phenomena leading to this novel kind of exotic arrangement of objects. Here, we demonstrate a hybrid top-down/bottom-up approach based on sol-gel dip-coating and nanoimprint lithography for the faithful reproduction of disordered hyperuniform metasurfaces in metal oxides. Nano- to microstructures made of silica and titania can be directly printed over several cm

International audience; Fabrication and scaling of disordered hyperuniform materials remain hampered by the difficulties in controlling the spontaneous phenomena leading to this novel kind of exotic arrangement of objects. Here, we demonstrate a hybrid top-down/ bottom-up approach based on sol−gel dip-coating and nanoimprint lithography for the faithful reproduction of disordered hyperuniform metasurfaces in metal oxides. Nano-to microstructures made of silica and titania can be directly printed over several cm 2 on glass and on silicon substrates. First, we describe the polymer mold fabrication starting from a hard master obtained via spontaneous solid-state dewetting of SiGe and Ge thin layers on SiO 2. Then, we assess the effective disordered hyperuniform character of master and replica and the role of the thickness of the sol− gel layer on the metal oxide replicas and on the presence of a residual layer underneath. Finally, as a potential application, we show the antireflective character of titania structures on silicon. Our results are relevant for the realistic implementation over large scales of disordered hyperuniform nano-and microarchitectures made of metal oxides, thus opening their exploitation in the framework of wet chemical assembly.

The development of three-dimensional architectures in semiconductor technology is paving the way to new device concepts for various applications, from quantum computing to single photon avalanche detectors. In most cases, such structures are achievable only under far-from-equilibrium growth conditions. Controlling the shape and morphology of the growing structures, to meet the strict requirements for an application, is far more complex than in close-to-equilibrium cases. The development of predictive simulation tools can be essential to guide the experiments. A versatile phase-field model for kinetic crystal growth is presented and applied to the prototypical case of Ge/Si vertical microcrystals grown on deeply patterned Si substrates. These structures, under development for innovative optoelectronic applications, are characterized by a complex three-dimensional set of facets essentially driven by facet competition. First, the parameters describing the kinetics on the surface of Si and Ge are fitted on a small set of experimental results. To this goal, Si vertical microcrystals have been grown, while for Ge the fitting parameters have been obtained from data from the literature. Once calibrated, the predictive capabilities of the model are demonstrated and exploited for investigating new pattern geometries and crystal morphologies, offering a guideline for the design of new 3D heterostructures. The reported methodology is intended to be a general approach for investigating faceted growth under far-from-equilibrium conditions.

We extend the doubly degenerate Cahn-Hilliard (DDCH) models for isotropic surface diffusion, which yield more accurate approximations than classical degenerate Cahn-Hilliard (DCH) models, to the anisotropic case. We consider both weak and strong anisotropies and demonstrate the capabilities of the approach for these cases numerically. The proposed model provides a variational and energy dissipative approach for anisotropic surface diffusion, enabling large scale simulations with material-specific parameters., 15 pages; 6 figures

A long-standing goal of materials science is to understand, predict and control the evolution of microstructures in crystalline materials. Most microstructure evolution is controlled by interface motion; hence, the establishment of rigorous interface equations of motion is a universal goal of materials science. We present a new model for the motion of arbitrarily curved interfaces that respects the underlying crystallography of the two phases/domains meeting at the interface and is consistent with microscopic mechanisms of interface motion; i.e., disconnection migration (line defects in the interface with step and dislocation character). We derive the equation of motion for interface migration under the influence of a wide range of driving forces. In Part II of this paper [Salvalaglio, Han and Srolovitz, 2021], we implement the interface model and the equation of motion proposed in this paper in a diffuse interface simulation approach for complex morphology and microstructure evolution., Comment: 15 pages, 12 figures, supplementary material (5 pages, 3 figures) enclosed

The motion of interfaces is an essential feature of microstructure evolution in crystalline materials. While atomic-scale descriptions provide mechanistic clarity, continuum descriptions are important for understanding microstructural evolution and upon which microscopic features it depends. We develop a microstructure evolution simulation approach that is linked to the underlying microscopic mechanisms of interface migration. We extend the continuum approach describing the disconnection-mediated motion of interfaces introduced in Part I [Han, Srolovitz and Salvalaglio, 2021] to a diffuse interface, phase-field model suitable for large-scale microstructure evolution. A broad range of numerical simulations showcases the capability of the method and the influence of microscopic interface migration mechanisms on microstructure evolution. These include, in particular, the effects of stress and its coupling to interface migration which arises from disconnections, showing how this leads to important differences from classical microstructure evolution represented by mean curvature flow., Comment: 14 pages, 8 figures, supplementary material (2 pages, 1 figure) enclosed

Large-scale, defect-free, micro- and nano-circuits with controlled inter-connections represent the nexus between electronic and photonic components. However, their fabrication over large scales often requires demanding procedures that are hardly scalable. Here we synthesize arrays of parallel ultra-long (up to 0.75 mm), monocrystalline, silicon-based nano-wires and complex, connected circuits exploiting low-resolution etching and annealing of thin silicon films on insulator. Phase field simulations reveal that crystal faceting and stabilization of the wires against breaking is due to surface energy anisotropy. Wires splitting, inter-connections and direction are independently managed by engineering the dewetting fronts and exploiting the spontaneous formation of kinks. Finally, we fabricate field-effect transistors with state-ofthe- art trans-conductance and electron mobility. Beyond the first experimental evidence of controlled dewetting of patches featuring a record aspect ratio of 1/60000 and selfassembled mm long nano-wires, our method constitutes a distinct and promising approach for the deterministic implementation of atomically-smooth, mono-crystalline electronic and photonic circuits.

Large-scale, defect-free, micro- and nano-circuits with controlled inter-connections represent the nexus between electronic and photonic components. However, their fabrication over large scales often requires demanding procedures that are hardly scalable. Here we synthesize arrays of parallel ultra-long (up to 0.75 mm), monocrystalline, silicon-based nano-wires and complex, connected circuits exploiting low-resolution etching and annealing of thin silicon films on insulator. Phase field simulations reveal that crystal faceting and stabilization of the wires against breaking is due to surface energy anisotropy. Wires splitting, inter-connections and direction are independently managed by engineering the dewetting fronts and exploiting the spontaneous formation of kinks. Finally, we fabricate field-effect transistors with state-of-the-art trans-conductance and electron mobility. Beyond the first experimental evidence of controlled dewetting of patches featuring a record aspect ratio of \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\sim$$\end{document}~1/60000 and self-assembled \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\sim$$\end{document}~mm long nano-wires, our method constitutes a distinct and promising approach for the deterministic implementation of atomically-smooth, mono-crystalline electronic and photonic circuits., Fabricating defect-free micro- and nano-circuits over large scales with controlled interconnections remains a challenge. Here, Bollani et al. show a dewetting strategy for engineering arrays of parallel Si-based nanowires up to 0.75 mm and complex interconnected circuits of monocrystalline Si on a chip.

We address the solid state dewetting of ultra-thin and ultra-large patches of monocrystalline silicon on insulator. We show that the underlying instability of the thin Si film under annealing can be perfectly controlled to form monocrystalline, complex nanoarchitectures extending over several microns. These complex patterns are obtained guiding the dewetting fronts by etching ad-hoc patches prior to annealing. They can be reproduced over hundreds of repetitions extending over hundreds of microns. We discuss the effect of annealing temperature and patch size on the stability of the final result of dewetting showing that for simple patches (e.g. simple squares) the final outcome is stable and well reproducible at 720 degrees C and for similar to 1 mu m square size. Finally, we demonstrate that introducing additional features within squared patches (e.g. a hole within a square) stabilises the dewetting dynamic providing perfectly reproducible complex nanoarchitectures of 5 pm size. (C) 2018 Elsevier B.V. All rights reserved.

In this work we present an innovative approach to realise coherent, highly-mismatched 3-dimensional heterostructures on substrates patterned at the micrometre-scale. The approach is based on the out-of-equilibrium deposition of SiGe alloys graded at an exceptionally shallow grading rate (GR) of 1.5% µm −1 by low energy plasma enhanced chemical vapour deposition (LEPECVD). Fully coherent SiGe/Si crystals up to 6 µm in width were achieved as confirmed by defect etching and transmission electron microscopy (TEM) analyses. The experimental results are supported by calculations of the energy for dislocation formation which indicate that elastic relaxation is energetically favoured over plastic relaxation in the narrower heterostructures. X-ray diffraction measurements show that the SiGe crystals are strain-free irrespective of the stress relieving mechanism which changes from elastic to plastic by increasing their width. The impact of dislocations on the SiGe crystal quality is analysed by comparing the width of X-ray diffraction peaks as a function of the heterostructure size.

Lateral ordering of heteroepitaxial islands can be conveniently achieved by suitable pit-patterning of the substrate prior to deposition. Controlling shape, orientation, and size of the pits is not trivial as, being metastable, they can significantly evolve during deposition/annealing. In this paper, we exploit a continuum model to explore the typical metastable pit morphologies that can be expected on Si(001), depending on the initial depth/shape. Evolution is predicted using a surface-diffusion model, formulated in a phase-field framework, and tackling surface-energy anisotropy. Results are shown to nicely reproduce typical metastable shapes reported in the literature. Moreover, long time scale evolutions of pit profiles with different depths are found to follow a similar kinetic pathway. The model is also exploited to treat the case of heteroepitaxial growth involving two materials characterized by different facets in their equilibrium Wulff’s shape. This can lead to significant changes in morphologies, such as a rotation of the pit during deposition as evidenced in Ge/Si experiments.

Capillary-driven mass transport in solids is typically understood in terms of surface-diffusion limited kinetics, leading to conventional solid-state dewetting of thin films. However, another mass transport mechanism, so-called surface-attachment and detachment limited kinetics, is possible. It can shrink a solid film, preserving its original topology without breaking it in isolated islands, and leads to faster dynamics for smaller film curvature in contrast with the opposite behavior observed for surface-diffusion limited kinetics. In this work, we present a rimless dewetting regime for Si, which is ascribed to effective attachment-limited kinetics mediated by the coexistence of crystalline and amorphous Si phases. Phase-field numerical simulations quantitatively reproduce the experimental observations, assessing the main mass transport mechanism at play. The process can be exploited to obtain in a deterministic fashion monocrystalline islands (with 95% probability) pinned at ≈500 nm from a hole milled within closed patches.

The phase-field crystal model in its amplitude equation approximation is shown to provide an accurate description of the deformation field in defected crystalline structures, as well as of dislocation motion. We analyze in detail the elastic distortion and stress regularization at a dislocation core and show how the Burgers vector density can be directly computed from the topological singularities of the phase-field amplitudes. Distortions arising from these amplitudes are then supplemented with non-singular displacements to enforce mechanical equilibrium. This allows for the consistent separation of plastic and elastic time scales in this framework. A finite element method is introduced to solve the combined amplitude and elasticity equations, which is applied to a few prototypical configurations in two spatial dimensions for a crystal of triangular lattice symmetry: i) the stress field induced by an edge dislocation with an analysis of how the amplitude equation regularizes stresses near the dislocation core, ii) the motion of a dislocation dipole as a result of its internal interaction, and iii) the shrinkage of a rotated grain. We also compare our results with those given by other extensions of classical elasticity theory, such as strain-gradient elasticity and methods based on the smoothing of Burgers vector densities near defect cores., 16 pages, 7 figures

Capillary-driven mass transport in solids is typically understood in terms of surface-diffusion limited kinetics, leading to conventional solid-state dewetting of thin films. However, another mass transport mechanism, so-called surface-attachment/detachment limited kinetics, is possible. It can shrink a solid film, preserving its original topology without breaking it in isolated islands, and leads to faster dynamics for smaller film curvature in contrast with the opposite behavior observed for surface-diffusion limited kinetics. In this work, we present a rimless dewetting regime for Si, which is ascribed to effective attachment-limited kinetics mediated by the coexistence of crystalline and amorphous Si phases. Phase-field numerical simulations quantitatively reproduce the experimental observations, assessing the main mass transport mechanism at play. The process can be exploited to obtain in a deterministic fashion monocrystalline islands (with $95%$ probability) pinned at $\ensuremath{\approx}500$ nm from a hole milled within closed patches.

Semiconductor heteroepitaxy involves a wealth of qualitatively different, competing phenomena. Examples include three-dimensional island formation, injection of dislocations, mixing between film and substrate atoms. Their relative importance depends on the specific growth conditions, giving rise to a very complex scenario. The need for an optimal control over heteroepitaxial films and/or nanostructures is widespread: semiconductor epitaxy by molecular beam epitaxy or chemical vapour deposition is nowadays exploited also in industrial environments. Simulation models can be precious in limiting the parameter space to be sampled while aiming at films/nanostructures with the desired properties. In order to be appealing (and useful) to an applied audience, such models must yield predictions directly comparable with experimental data. This implies matching typical time scales and sizes, while offering a satisfactory description of the main physical driving forces. It is the aim of the present review to show that continuum models of semiconductor heteroepitaxy evolved significantly, providing a promising tool (even a working tool, in some cases) to comply with the above requirements. Several examples, spanning from the nanometre to the micron scale, are illustrated. Current limitations and future research directions are also discussed.

We address a three-dimensional, coarse-grained description of dislocation networks at grain boundaries between rotated crystals. The so-called amplitude expansion of the phase-field crystal model is exploited with the aid of finite element method calculations. This approach allows for the description of microscopic features, such as dislocations, while simultaneously being able to describe length scales that are orders of magnitude larger than the lattice spacing. Moreover, it allows for the direct description of extended defects by means of a scalar order parameter. The versatility of this framework is shown by considering both fcc and bcc lattice symmetries and different rotation axes. First, the specific case of planar, twist grain boundaries is illustrated. The details of the method are reported and the consistency of the results with literature is discussed. Then, the dislocation networks forming at the interface between a spherical, rotated crystal embedded in an unrotated crystalline structure, are shown. Although explicitly accounting for dislocations which lead to an anisotropic shrinkage of the rotated grain, the extension of the spherical grain boundary is found to decrease linearly over time in agreement with the classical theory of grain growth and recent atomistic investigations. It is shown that the results obtained for a system with bcc symmetry agree very well with existing results, validating the methodology. Furthermore, fully original results are shown for fcc lattice symmetry, revealing the generality of the reported observations., 11 pages, 9 figures

Nanoscale membranes have emerged as a new class of vertical nanostructures that enable the integration of horizontal networks of III-V nanowires on a chip. To generalize this method to the whole family of III-Vs, progress in the understanding of the membrane formation by selective area epitaxy in oxide slits is needed, in particular for different slit orientations. Here, it is demonstrated that the shape is primarily driven by the growth kinetics rather than determined by surface energy minimization as commonly occurs for faceted nanostructures. To this end, a phase-field model simulating the shape evolution during growth is devised, in agreement with the experimental findings for any slit orientations, even when the vertical membranes turn into multifaceted fins. This makes it possible to reverseengineer the facet-dependent incorporation times, which were so far unknown, even for common low-index facets. The compelling reproduction of the experimental morphologies demonstrates the reliability of the growth model and offers a general method to determine microscopic kinetic parameters governing out-of-equilibrium three-dimensional growth.

Si-based nanoarchitectures are formed with unprecedented precision and reproducibility via templated dewetting of thin SOI., Dewetting is a ubiquitous phenomenon in nature; many different thin films of organic and inorganic substances (such as liquids, polymers, metals, and semiconductors) share this shape instability driven by surface tension and mass transport. Via templated solid-state dewetting, we frame complex nanoarchitectures of monocrystalline silicon on insulator with unprecedented precision and reproducibility over large scales. Phase-field simulations reveal the dominant role of surface diffusion as a driving force for dewetting and provide a predictive tool to further engineer this hybrid top-down/bottom-up self-assembly method. Our results demonstrate that patches of thin monocrystalline films of metals and semiconductors share the same dewetting dynamics. We also prove the potential of our method by fabricating nanotransfer molding of metal oxide xerogels on silicon and glass substrates. This method allows the novel possibility of transferring these Si-based patterns on different materials, which do not usually undergo dewetting, offering great potential also for microfluidic or sensing applications.

We simulate the morphological evolution of Ge microcrystals, grown out-of-equilibrium on deeply patterned Si substrates, as resulting from surface diffusion driven by the tendency toward the minimization of the surface energy. In particular, we report three-dimensional phase-field simulations accounting for the realistic surface energy anisotropy of Ge/Si crystals. In Salvalaglio et al. (2015) [10] it has been shown both by experiments and simulations that annealing of closely spaced crystals leads to a coalescence process with the formation of a suspended film. However, this was explained only by considering an isotropic surface energy. Here, we extend such a study by showing first the morphological changes of faceted isolated crystals. Then, the evolution of dense arrays is considered, describing their coalescence along with the evolution of facets. Combined with the previous results without anisotropy in the surface energy, this work allows us to confirm and assess the key features of the coalescence process.

The structure, interface abruptness and strain relaxation in InAs/GaAs nanowires grown by molecular beam epitaxy in the Ga self-catalysed mode on (111) Si have been investigated by transmission electron microscopy. The nanowires had the zincblende phase. The InAs/GaAs interface was atomically and chemically sharp with a width around 1.5 nm, i.e. significantly smaller than previously reported values. This was achieved by the consumption of the Ga droplet and formation of a flat top facet of the GaAs followed by the growth of InAs by splitting the depositions of In and As. Both elastic and plastic strain relaxation took place simultaneously. Experimental TEM results about strain relaxation very well agree with linear elasticity theory calculations by the finite element methods. (C) 2016 Elsevier B.V. All rights reserved.

One of the major difficulties in employing phase field crystal (PFC) modeling and the associated amplitude (APFC) formulation is the ability to tune model parameters to match experimental quantities. In this work we address the problem of tuning the defect core and interface energies in the APFC formulation. We show that the addition of a single term to the free energy functional can be used to increase the solid-liquid interface and defect energies in a well-controlled fashion, without any major change to other features. The influence of the newly added term is explored in two-dimensional triangular and honeycomb structures as well as bcc and fcc lattices in three dimensions. In addition, a finite element method (FEM) is developed for the model that incorporates a mesh refinement scheme. The combination of the FEM and mesh refinement to simulate amplitude expansion with a new energy term provides a method of controlling microscopic features such as defect and interface energies while simultaneously delivering a coarse-grained examination of the system., Comment: 14 pages, 9 figures

Shadowing effects during the growth of nano- and microstructures are crucial for the realization of several technological applications. They are given by the shielding of the incoming material flux provided by the growing structures themselves. Their features have been deeply investigated by theoretical approaches, revealing important information to support experimental activities. However, comprehensive investigations able to follow every stage of the growth processes as a whole, particularly useful to design and understand targeted experiments, are still challenging. In this work, we study the thin-film growth dynamics by means of a diffuse interface approach accounting for both deposition with shadowing effects and surface diffusion driven by the minimization of the surface energy. In particular, we introduce the coupling between a phase-field model and the detailed calculation of the incoming material flux at the surface deposited from vacuum or vapor phase in the ballistic regime. This allows us to finely reproduce the realistic morphological evolution during the growth on nonflat substrates, also accounting for different flux distributions. A general assessment of the method, focusing on two-dimensional profiles, is provided thanks to the comparison with a sharp-interface approach for the evolution of the early stages. Then, the long-time-scale dynamics is shown in two and three dimensions, providing a general overview of the features observed during deposition on corrugated surfaces involving flattening, increasing of surface roughness with the growth of columnar structures, and voids formation.

We present a detailed experimental and theoretical analysis of the epitaxial stress relaxation process in micro-structured compositionally graded alloys. We focus on the pivotal SiGe/Si(001) system employing patterned Si substrates at the micrometre-size scale to address the distribution of threading and misfit dislocations within the heterostructures. SiGe alloys with linearly increasing Ge content were deposited by low energy plasma enhanced chemical vapour deposition resulting in isolated, tens of micrometre tall 3D crystals. We demonstrate that complete elastic relaxation is achieved by appropriate choice of the Ge compositional grading rate and Si pillar width. We investigate the nature and distribution of dislocations along the [001] growth direction in SiGe crystals by transmission electron microscopy, chemical defect etching and etch pit counting. We show that for 3 μm wide Si pillars and a Ge grading rate of 1.5% μm−1, only misfit dislocations are present while their fraction is reduced for higher Ge grading rates and larger structures due to dislocation interactions. The experimental results are interpreted with the help of theoretical calculations based on linear elasticity theory describing the competition between purely elastic and plastic stress relaxation with increasing crystal width and Ge compositional grading rate.

We demonstrate that the elastic stress relaxation mechanism in micrometre-sized, highly mismatched heterostructures may be enhanced by employing patterned substrates in the form of necked pillars, resulting in a significant reduction of the dislocation density. Compositionally graded Si1−xGex crystals were grown by low energy plasma enhanced chemical vapour deposition, resulting in tens of micrometres tall, three-dimensional heterostructures. The patterned Si(001) substrates consist of micrometre-sized Si pillars either with the vertical {110} or isotropically under-etched sidewalls resulting in narrow necks. The structural properties of these heterostructures were investigated by defect etching and transmission electron microscopy. We show that the dislocation density, and hence the competition between elastic and plastic stress relaxation, is highly influenced by the shape of the substrate necks and their proximity to the mismatched epitaxial material. The SiGe dislocation density increases monotonically with the crystal width but is significantly reduced by the substrate under-etching. The drop in dislocation density is interpreted as a direct effect of the enhanced compliance of the under-etched Si pillars, as confirmed by the three-dimensional finite element method simulations of the elastic energy distribution.

A method for growing suspended Ge films on micron-sized Si pillars in Si(001) is discussed. In [C.V. Falub et al., Science 335 (2012) 1330] vertically aligned three-dimensional Ge crystals, separated by a few tens of nanometers, were obtained by depositing several micrometers of Ge using Low-Energy Plasma-Enhanced Chemical Vapor Deposition. Here a different regime of high growth temperature is exploited in order to induce the merging of the crystals into a connected structure eventually forming a continuous, two-dimensional film. The mechanisms leading to such a behavior are discussed with the aid of an effective model of crystal growth. Both the effects of deposition and curvature-driven surface diffusion are considered to reproduce the main features of coalescence. The key enabling role of high temperature is identified with the activation of the diffusion process on a time scale competitive with the deposition rate. We demonstrate the versatility of the deposition process, which allows to switch between the formation of individual crystals and a continuous suspended film simply by tuning the growth temperature.

In this work, we demonstrate the growth of Ge crystals and suspended continuous layers on Si(001) substrates deeply patterned in high aspect-ratio pillars. The material deposition was carried out in a commercial reduced-pressure chemical vapor deposition reactor, thus extending the "vertical-heteroepitaxy" technique developed by using the peculiar low-energy plasma-enhanced chemical vapor deposition reactor, to widely available epitaxial tools. The growth process was thoroughly analyzed, from the formation of small initial seeds to the final coalescence into a continuous suspended layer, by means of scanning and transmission electron microscopy, X-ray diffraction, and μ-Raman spectroscopy. The preoxidation of the Si pillar sidewalls and the addition of hydrochloric gas in the reactants proved to be key to achieve highly selective Ge growth on the pillars top only, which, in turn, is needed to promote the formation of a continuous Ge layer. Thanks to continuum growth models, we were able to single out the different roles played by thermodynamics and kinetics in the deposition dynamics. We believe that our findings will open the way to the low-cost realization of tens of micrometers thick heteroepitaxial layer (e.g., Ge, SiC, and GaAs) on Si having high crystal quality.

Monolithic integration of III-V compounds into high density Si integrated circuits is a key technological challenge for the next generation of optoelectronic devices. In this work, we report on the metal organic vapor phase epitaxy growth of strain-free GaAs crystals on Si substrates patterned down to the micron scale. The differences in thermal expansion coefficient and lattice parameter are adapted by a 2-μm-thick intermediate Ge layer grown by low-energy plasma enhanced chemical vapor deposition. The GaAs crystals evolve during growth towards a pyramidal shape, with lateral facets composed of {111} planes and an apex formed by {137} and (001) surfaces. The influence of the anisotropic GaAs growth kinetics on the final morphology is highlighted by means of scanning and transmission electron microscopy measurements. The effect of the Si pattern geometry, substrate orientation, and crystal aspect ratio on the GaAs structural properties was investigated by means of high resolution X-ray diffraction. The thermal strain relaxation process of GaAs crystals with different aspect ratio is discussed within the framework of linear elasticity theory by Finite Element Method simulations based on realistic geometries extracted from cross-sectional scanning electron microscopy images.

Dislocation networks are one of the most principle sources deteriorating the performances of devices based on lattice-mismatched heteroepitaxial systems. We demonstrate here a technique enabling fully coherent germanium (Ge) islands selectively grown on nanotip-patterned Si(001) substrates. The silicon (Si)-tip-patterned substrate, fabricated by complementary metal oxide semiconductor compatible nanotechnology, features ∼50-nm-wide Si areas emerging from a SiO2 matrix and arranged in an ordered lattice. Molecular beam epitaxy growths result in Ge nanoislands with high selectivity and having homogeneous shape and size. The ∼850 °C growth temperature required for ensuring selective growth has been shown to lead to the formation of Ge islands of high crystalline quality without extensive Si intermixing (with 91 atom % Ge). Nanotip-patterned wafers result in geometric, kinetic-diffusion-barrier intermixing hindrance, confining the major intermixing to the pedestal region of Ge islands, where kinetic diffusion barriers are, however, high. Theoretical calculations suggest that the thin Si/Ge layer at the interface plays, nevertheless, a significant role in realizing our fully coherent Ge nanoislands free from extended defects especially dislocations. Single-layer graphene/Ge/Si-tip Schottky junctions were fabricated, and thanks to the absence of extended defects in Ge islands, they demonstrate high-performance photodetection characteristics with responsivity of ∼45 mA W(-1) and an Ion/Ioff ratio of ∼10(3).