Cracklike surface instabilities in stressed solids

Stress-Induced Acceleration and Ordering in Solid-State Dewetting

We report on the influence of elastic strain on solid-state dewetting. Using continuum modeling, we first study the consequences of elastic stress on the pinching of the film away from the triple line during dewetting. We find that elastic stress in the solid film decreases both the time and the distance at which the film pinches in such a way that the dewetting front is accelerated. In addition, the spatial organization of islands emerging from the dewetting process is affected by strain. As an example, we demonstrate that ordered arrays of quantum dots can be achieved from solid-state dewetting of a square island in the presence of elastic stress.

Dislocation nucleation facilitated by atomic segregation

Surface segregation-the enrichment of one element at the surface, relative to the bulk-is ubiquitous to multi-component materials. Using the example of a Cu-Au solid solution, we demonstrate that compositional variations induced by surface segregation are accompanied by misfit strain and the formation of dislocations in the subsurface region via a surface diffusion and trapping process. The resulting chemically ordered surface regions acts as an effective barrier that inhibits subsequent dislocation annihilation at free surfaces. Using dynamic, atomic-scale resolution electron microscopy observations and theory modelling, we show that the dislocations are highly active, and we delineate the specific atomic-scale mechanisms associated with their nucleation, glide, climb, and annihilation at elevated temperatures. These observations provide mechanistic detail of how dislocations nucleate and migrate at heterointerfaces in dissimilar-material systems.

Modeling elastic anisotropy in strained heteroepitaxy

Using a continuum evolution equation, we model the growth and evolution of quantum dots in the heteroepitaxial Ge on Si(0 0 1) system in a molecular beam epitaxy unit. We formulate our model in terms of evolution due to deposition, and due to surface diffusion which is governed by a free energy. This free energy has contributions from surface energy, curvature, wetting effects and elastic energy due to lattice mismatch between the film and the substrate. In addition to anisotropy due to surface energy which favors facet formation, we also incorporate elastic anisotropy due to an underlying crystal lattice. The complicated elastic problem of the film-substrate system subjected to boundary conditions at the free surface, interface and the bulk substrate is solved by perturbation analysis using a small slope approximation. This permits an analysis of effects at different orders in the slope and sheds new light on the observed behavior. Linear stability analysis shows the early evolution of the instability towards dot formation. The elastic anisotropy causes a change in the alignment of dots in the linear regime, whereas the surface energy anisotropy changes the dot shapes at the nonlinear regime. Numerical simulation of the full nonlinear equations shows the evolution of the surface morphology. In particular, we show, for parameters of the [Formula: see text] [Formula: see text] on Si(0 0 1), the surface energy anisotropy dominates the shapes of the quantum dots, whereas their alignment is influenced by the elastic energy anisotropy. The anisotropy in elasticity causes a further elongation of the islands whose coarsening is interrupted due to [Formula: see text] facets on the surface.

Edge Dislocations Triggered Surface Instability in Tensile Epitaxial Hexagonal Nitride Semiconductor

Understanding the semiconductor surface and its properties including surface stability, atomic morphologies, and even electronic states is of great importance not only for understanding surface growth kinetics but also for evaluating the degree to which they affect the devices' performance. Here, we report studies on the nanoscale fissures related surface instability in AlGaN/GaN heterostructures. Experimental results reveal that edge dislocations are actually the root cause of the surface instability. The nanoscale fissures are initially triggered by the edge dislocations, and the subsequent evolution is associated with tensile lattice-mismatch stress and hydrogen etching. Our findings resolve a long-standing problem on the surface instability in AlGaN/GaN heterostructures and will also lead to new understandings of surface growth kinetics in other hexagonal semiconductor systems.

Growth of Highly Strained CeO2 Ultrathin Films

Large biaxial strain is a promising route to tune the functionalities of oxide thin films. However, large strain is often not fully realized due to the formation of misfit dislocations at the film/substrate interface. In this work, we examine the growth of strained ceria (CeO₂) thin films on (001)-oriented single crystal yttria-stabilized zirconia (YSZ) via pulsed-laser deposition. By varying the film thickness systematically between 1 and 430 nm, we demonstrate that ultrathin ceria films are coherently strained to the YSZ substrate for thicknesses up to 2.7 nm, despite the large lattice mismatch (∼5%). The coherency is confirmed by both X-ray diffraction and high-resolution transmission electron microscopy. This thickness is several times greater than the predicted equilibrium critical thickness. Partial strain relaxation is achieved by forming semirelaxed surface islands rather than by directly nucleating dislocations. In situ reflective high-energy electron diffraction during growth confirms the transition from 2-D (layer-by-layer) to 3-D (island) at a film thickness of ∼1 nm, which is further supported by atomic force microscopy. We propose that dislocations likely nucleate near the surface islands and glide to the film/substrate interface, as evidenced by the presence of 60° dislocations. An improved understanding of growing oxide thin films with a large misfit lays the foundation to systematically explore the impact of strain and dislocations on properties such as ionic transport and redox chemistry.

In situ stress observation in oxide films and how tensile stress influences oxygen ion conduction

Many properties of materials can be changed by varying the interatomic distances in the crystal lattice by applying stress. Ideal model systems for investigations are heteroepitaxial thin films where lattice distortions can be induced by the crystallographic mismatch with the substrate. Here we describe an in situ simultaneous diagnostic of growth mode and stress during pulsed laser deposition of oxide thin films. The stress state and evolution up to the relaxation onset are monitored during the growth of oxygen ion conducting Ce0.85Sm0.15O2-δ thin films via optical wafer curvature measurements. Increasing tensile stress lowers the activation energy for charge transport and a thorough characterization of stress and morphology allows quantifying this effect using samples with the conductive properties of single crystals. The combined in situ application of optical deflectometry and electron diffraction provides an invaluable tool for strain engineering in Materials Science to fabricate novel devices with intriguing functionalities.

Morphological Evolution in Heteroepitaxial Thin Film Structures at the Nanoscale

The aim of this study is to resolve the phenomenon of formation of mesoscopic structures on the surface of heteroepitaxial thin film system due to surface diffusion by considering the effects of both surface and interface stresses. Elastic stress field caused by curved surface is solved by using the constitutive equations of linear elasticity for the bulk and surface phases. Based on the method of superposition, a boundary perturbation technique, Goursat-Kolosov complex potentials and Muskhelishvili representations, the boundary value problem is reduced to the successive solution of a system of singular and hypersingular integral equations for any order of approximation. This solution and thermodynamic approach allows us to derive a governing equation which gives the amplitude changing of a surface roughness with time.

Interrupted self-organization of SiGe pyramids

We investigate the morphological evolution of SiGe quantum dots deposited on Si(100) during long-time annealing. At low strain, the dots' self-organization begins by an instability and interrupts when (105) pyramids form. This evolution and the resulting island density are quantified by molecular-beam epitaxy. A kinetic model accounting for elasticity, wetting, and anisotropy is shown to reproduce well the experimental findings with appropriate wetting parameters. In this nucleationless regime, a mean-field kinetic analysis explains the existence of nearly stationary states by the vanishing of the coarsening driving force. The island size distribution follows in both experiments and theory the scaling law associated with a single characteristic length scale.

Self-assembled pit arrays as templates for the integration of Au nanocrystals in oxide surfaces

We report on the fabrication of long-range ordered arrays of Au nanocrystals (sub-50 nm range) on top of manganite (La(2/3)Sr(1/3)MnO(3)) thin films achieving area densities around 2 × 10(10) gold nanocrystals per cm(2), well above the densities achievable by using conventional nanofabrication techniques. The gold-manganite interface exhibits excellent conduction properties. Long-range order is achieved by a guided self-assembling process of Au nanocrystals on self-organized pit-arrays acting as a template for the nucleation of gold nanocrystals. Self-organization of pits on the manganite film surface promoted by the underlying stepped SrTiO(3) substrate is achieved by a fine tuning of the growth kinetic pathway, taking advantage of the unusual misfit strain relaxation behaviour of manganite films.

Nonlinear dynamics of island coarsening and stabilization during strained film heteroepitaxy

Nonlinear evolution of three-dimensional strained islands or quantum dots in heteroepitaxial thin films is studied via a continuum elasticity model and both perturbation analysis of the system and numerical simulations of the corresponding nonlinear dynamic equation governing the film morphological profile. Three regimes of island array evolution are identified and examined, including a film instability regime at early stage, a nonlinear coarsening regime at intermediate times, and the crossover to a saturated asymptotic state, with detailed behavior depending on film-substrate misfit strains but not qualitatively on finite system sizes. The phenomenon of island array stabilization, which corresponds to the formation of steady but nonordered arrays of strained quantum dots, occurs at later time for smaller misfit strain. It is found to be controlled by the strength of film-substrate wetting interaction which would constrain the valley-to-peak mass transport and hence the growth of island height, and also determined by the effect of elastic interaction between surface islands and the high-order strain energy of individual islands at late evolution stage. The results are compared to previous experimental and theoretical studies on quantum dot coarsening and stabilization.

Shape evolution of a core-shell spherical particle under hydrostatic pressure

The morphological evolution by surface diffusion of a core-shell spherical particle has been investigated theoretically under hydrostatic pressure when the shear modulii of the core and shell are different. A linear stability analysis has demonstrated that depending on the pressure, shear modulii, and radii of both phases, the free surface of the composite particle may be unstable with respect to a shape perturbation. A stability diagram finally emphasizes that the roughness development is favored in the case of a hard shell with a soft core.

SURFACE MORPHOLOGY EVOLUTION OF STRAINED InAs/GaAs(331)A FILMS

Surface morphology evolution of strained InAs/GaAs (331) A films was systematically investigated in this paper. Under As -rich conditions, InAs elongated islands aligned along [Formula: see text] are formed at a substrate temperature of 510°C. We explained it as a result of the anisotropic diffusion of adatoms. Under In -rich conditions, striking change has occurred with respect to the surface morphology of the InAs layers. Instead of anisotropic InAs elongated islands, unique island-pit pairs randomly distributed on the whole surface were observed. Using cooperative nucleation mechanisms proposed by Jesson et al. [Phys. Rev. Lett.77, 1330 (1996)], we interpret the resulting surface morphology evolution.

Brittle fracture in viscoelastic materials as a pattern-formation process

A continuum model of crack propagation in brittle viscoelastic materials is presented and discussed. Thereby, the phenomenon of fracture is understood as an elastically induced nonequilibrium interfacial pattern formation process. In this spirit, a full description of a propagating crack provides the determination of the entire time dependent shape of the crack surface, which is assumed to be extended over a finite and self-consistently selected length scale. The mechanism of crack propagation, that is, the motion of the crack surface, is then determined through linear nonequilibrium transport equations. Here we consider two different mechanisms, a first-order phase transformation and surface diffusion. We give scaling arguments showing that steady-state solutions with a self-consistently selected propagation velocity and crack shape can exist provided that elastodynamic or viscoelastic effects are taken into account, whereas static elasticity alone is not sufficient. In this respect, inertial effects as well as viscous damping are identified to be sufficient crack tip selection mechanisms. Exploring the arising description of brittle fracture numerically, we study steady-state crack propagation in the viscoelastic and inertia limit as well as in an intermediate regime, where both effects are important. The arising free boundary problems are solved by phase field methods and a sharp interface approach using a multipole expansion technique. Different types of loading, mode I, mode III fracture, as well as mixtures of them, are discussed.

A Numerical Study of Stress Controlled Surface Diffusion During Epitaxial Film Growth

A two-dimensional numerical simulation is performed to model the morphological evolution of a strained film growing heteroepitaxially on a substrate under simultaneous action of vapor deposition and surface diffusion. To facilitate numerical implementation, a continuum boundary layer model is proposed to account for the influence of film/substrate interface on the film growth pattern. Discussions are focused on the Stranski-Krastanow growth mode, although our model is capable of explaining Frank-van der Merwe and Volmer-Weber growth modes as well. Both first-order perturbation and numerical results are developed to demonstrate that the film surface tends to remain flat during the initial stage of growth and that surface roughening occurs once the film thickness exceeds a critical value, in consistency with experimentally observed patterns of S-K growth. Numerical results further show that, depending on the deposition rate, the surface evolution could lead to a steady state morphology, unstable cusp formation, or growing islands with flattened valleys.

Equilibrium Shapes of Small Strained Islands

We calculate the equilibrium morphology of a strained layer, for the case where it wets the substrate (Stranski-Krastonow growth). Assuming isotropic surface energy and equal elastic constants in the film and substrate, we are able to calculate two-dimensional equilibrium shapes as a function of the island size and spacing. We present asymptotic results for the equilibrium shape of a thin island where the island height is much smaller than the island width. We also present numerical results of the full equations to describe the island shape when the islands are widely separated. From these solutions we are able to determine the chemical potential of the island as a function of island volume and the strain energy density along the surface of the island for small to medium-sized islands.

Surface Stress, Morphological Development, and Dislocation Nucleation During Strained-Layer Epitaxy

Utilizing Marker layer experiments and Z-contrast imaging, we have observed the formation of surface cusps during SixGe1−x alloy growth. The formation of cusps can be understood in terms of stress-driven surface diffusion, and we consider the large stress build-up at the cusp tip as a potential source for the nucleation of misfit dislocations.

Understanding the epitaxial growth of SexTey@Te core-shell nanorods and the generation of periodic defects

This study demonstrates solution-processed epitaxial growth of Te on Se(x)Te(y) nanorods and the generation of periodic defects in the core. We investigated Se(x)Te(y)@Te core-shell nanorods with a diameter of 40-50 nm and a length of 600-700 nm. In spite of a large lattice mismatch between the Se(x)Te(y) core and the Te shell, the soft character of the core and the shell at a high reaction temperature allowed epitaxial growth of Te on the Se(x)Te(y) nanorods. During the cooling process to room temperature (below the glass transition temperatures), the lattice mismatch between the core and the shell led to homogeneous stress along the epitaxial interface so that periodic defects were generated in the core.

Finite-time singularities in surface-diffusion instabilities are cured by plasticity

A free material surface which supports surface diffusion becomes unstable when put under external nonhydrostatic stress. Since the chemical potential on a stressed surface is larger inside an indentation, small shape fluctuations develop because material preferentially diffuses out of indentations. When the bulk of the material is purely elastic one expects this instability to run into a finite-time cusp singularity. It is shown here that this singularity is cured by plastic effects in the material, turning the singular solution to a regular crack.

Comparison of phase-field models for surface diffusion

The description of surface-diffusion controlled dynamics via the phase-field method is less trivial than it appears at first sight. A seemingly straightforward approach from the literature is shown to fail to produce the correct asymptotics, albeit in a subtle manner. Two models are constructed that approximate known sharp-interface equations without adding undesired constraints. Numerical simulations of the standard and a more sophisticated model from the literature as well as of our two models are performed to assess the relative merits of each approach. The results suggest superior performance of the models in at least some situations.

Current-induced stabilization of surface morphology in stressed solids

We examine the surface morphological evolution of a conducting crystalline solid under the simultaneous action of an electric field and mechanical stress based on a fully nonlinear model and combining linear stability theory with self-consistent dynamical simulations. We demonstrate that electric current, through surface electromigration, can stabilize the surface morphology of the stressed solid against cracklike surface instabilities. The results also have more general implications for the morphological response of solid surfaces under the simultaneous action of multiple external forces.

Phase field modeling of fracture and stress-induced phase transitions

We present a continuum theory to describe elastically induced phase transitions between coherent solid phases. In the limit of vanishing elastic constants in one of the phases, the model can be used to describe fracture on the basis of the late stage of the Asaro-Tiller-Grinfeld instability. Starting from a sharp interface formulation we derive the elastic equations and the dissipative interface kinetics. We develop a phase field model to simulate these processes numerically; in the sharp interface limit, it reproduces the desired equations of motion and boundary conditions. We perform large scale simulations of fracture processes to eliminate finite-size effects and compare the results to a recently developed sharp interface method. Details of the numerical simulations are explained, and the generalization to multiphase simulations is presented.

Phase field modeling of fast crack propagation

We present a continuum theory which predicts the steady state propagation of cracks. The theory overcomes the usual problem of a finite time cusp singularity of the Grinfeld instability by the inclusion of elastodynamic effects which restore selection of the steady state tip radius and velocity. We developed a phase-field model for elastically induced phase transitions; in the limit of small or vanishing elastic coefficients in the new phase, fracture can be studied. The simulations confirm analytical predictions for fast crack propagation.

Island, pit, and groove formation in strained heteroepitaxy

We study the morphological evolution of strained heteroepitaxial films using a kinetic Monte Carlo method in three dimensions. The elastic part of the problem uses a Green's function method. Isolated islands are observed under deposition conditions for deposition rates slow compared with intrinsic surface roughening rates. They are hemispherical and truncated conical for high and low temperature cases, respectively. Annealing of films at high temperature leads to the formation of closely packed islands as in instability theory. At low temperature, pits form via a multistep layer-by-layer nucleation mechanism in contrast to the conventional single-step nucleation process. They subsequently develop into grooves, which are energetically more favorable.

Consecutive rotation of crystallographic orientation in lateral growth

A consecutive rotation of crystallographic orientation has been observed in lateral crystallization of NH4Cl on a glass substrate, which induces a periodic distribution of faceted and roughened regions on the surface of a crystallite aggregate. Experimental observation indicates that this phenomenon derives from the asymmetric surface energies at the growth front, which deform the nascent nucleus and tilt the crystallographic orientation in the nucleation-mediated layered growth. We suggest that this effect is significant for a class of lateral growth where nucleation plays a dominate role.

Modeling elastic and plastic deformations in nonequilibrium processing using phase field crystals

A continuum field theory approach is presented for modeling elastic and plastic deformation, free surfaces, and multiple crystal orientations in nonequilibrium processing phenomena. Many basic properties of the model are calculated analytically, and numerical simulations are presented for a number of important applications including, epitaxial growth, material hardness, grain growth, reconstructive phase transitions, and crack propagation.

Self-organization of quantum dots in epitaxially strained solid films

A nonlinear evolution equation for surface-diffusion-driven Asaro-Tiller-Grinfeld instability of an epitaxially strained thin solid film on a solid substrate is derived in the case where the film wets the substrate. It is found that the presence of a weak wetting interaction between the film and the substrate can substantially retard the instability and modify its spectrum in such a way that the formation of spatially regular arrays of islands or pits on the film surface becomes possible. It is shown that the self-organization dynamics is significantly affected by the presence of the Goldstone mode associated with the conservation of mass.

Pattern selection in biaxially stressed solids

We analyze the morphological instability of the surface of a solid which is subject to a biaxial stress. The stability calculation reveals a new favored pattern: a diamond morphology. This occurs if the stress is tensile in one direction and compressive in the orthogonal one and the ratio exceeds a certain value. A nonlinear analysis shows that the bifurcation is subcritical and hints to a nontrivial competition between tilted stripes and diamonds. This study opens a new line of inquiries in the field of stress-induced pattern selection.

Fast crack propagation by surface diffusion

We present a continuum theory which describes the fast growth of a crack by surface diffusion. This mechanism overcomes the usual cusp singularity by a self-consistent selection of the crack tip radius. It predicts the saturation of the steady state crack velocity appreciably below the Rayleigh speed and tip blunting. Furthermore, it includes the possibility of a tip splitting instability for high applied tensions.

Competing roughening mechanisms in strained heteroepitaxy: a fast kinetic Monte Carlo study

We study the morphological evolution of strained heteroepitaxial films using kinetic Monte Carlo simulations in two dimensions. A novel Green's function approach, analogous to boundary integral methods, is used to calculate elastic energies efficiently. We observe island formation at low lattice misfit and high temperature that is consistent with the Asaro-Tiller-Grinfeld instability theory. At high misfit and low temperature, islands or pits form according to the nucleation theory of Tersoff and LeGoues.

Grinfeld instability on crack surfaces

The surface of a propagating crack is shown to be morphologically unstable because of the nonhydrostatic stresses near the surface (Asaro-Tiller-Grinfeld instability). We find numerically that the energy of a wavy crack becomes smaller than the energy of a straight crack if the crack length exceeds a critical length L(c)=5.18 L(G) (L(G) is the Griffith length). We analyze the dynamic evolution of this instability, governed by surface diffusion or condensation and evaporation. It turns out that the curvature of the crack surface becomes divergent near the crack tips. This implies that the widely used condition of the disappearance of K(II), the stress intensity factor of the sliding mode, is replaced by the more general requirement of matching chemical potentials of the crack surfaces at the tips. The results are generalized to situations of different external loading.

Phase-field modeling of stress-induced instabilities

A phase-field approach describing the dynamics of a strained solid in contact with its melt is developed. Using a formulation that is independent of the state of reference chosen for the displacement field, we write down the elastic energy in an unambiguous fashion, thus obtaining an entire class of models. According to the choice of reference state, the particular model emerging from this class will become equivalent to one of the two independently constructed models on which brief accounts have been given recently [J. Müller and M. Grant, Phys. Rev. Lett. 82, 1736 (1999); K. Kassner and C. Misbah, Europhys. Lett. 46, 217 (1999)]. We show that our phase-field approach recovers the sharp-interface limit corresponding to the continuum model equations describing the Asaro-Tiller-Grinfeld instability. Moreover, we use our model to derive hitherto unknown sharp-interface equations for a situation including a field of body forces. The numerical utility of the phase-field approach is demonstrated by reproducing some known results and by comparison with a sharp-interface simulation. We then proceed to investigate the dynamics of extended systems within the phase-field model which contains an inherent lower length cutoff, thus avoiding cusp singularities. It is found that a periodic array of grooves generically evolves into a superstructure which arises from a series of imperfect period doublings. For wave numbers close to the fastest-growing mode of the linear instability, the first period doubling can be obtained analytically. Both the dynamics of an initially periodic array and a random initial structure can be described as a coarsening process with winning grooves temporarily accelerating whereas losing ones decelerate and even reverse their direction of motion. In the absence of gravity, the end state of a laterally finite system is a single groove growing at constant velocity, as long as no secondary instabilities arise (that we have not been able to see with our code). With gravity, several grooves are possible, all of which are bound to stop eventually. A laterally infinite system approaches a scaling state in the absence of gravity and probably with gravity, too.

Equilibrium Shapes of Solid Particles on Elastically Mismatched Substrates

Small particles or islands bonded to a substrate can be profoundly influenced by both interfacial and elastic driving forces that tend to have opposing influences on the apparent wetting behavior. The superposition of these two driving forces can therefore lead to a rich set of particle properties, most notably their equilibrium shapes. Here we present a variational analysis leading directly to an Euler-Lagrange equation that can be solved to yield the equilibrium shapes of partially wetting particles as a function of their size, interface energy densities, and elastic interaction with a rigid substrate. The solutions are used to gain insight into the variables that most significantly influence the equilibrium morphology, and to derive the approximate driving force for surface area reduction by coarsening among a dispersion of unequally sized particles. The relatively simple analytical model can also form a foundation upon which more realistic numerical simulations may be built and compared. Copyright 1999 Academic Press.

Mass-conserved morphological evolution of hypocycloid cavities: a model of diffusive crack initiation with no associated energy barrier

An analytic proof is developed for an energy variation theorem which states that in a homogeneous isotropic linear elastic solid under equibiaxial loading, a cusp-shaped hypocycloid cavity satisfying the Griffith conditionG≽ 2γcan be formed through mass-conserved shape evolution with no associated energy barrier. In this process, the sum of the strain energy and the surface energy continuously decreases as the hypocycloid changes from a perfectly circular hole to an array of cusp cracks. The hypocycloid evolution provides a unified model of diffusive crack formation from Griffith cracks to cycloid surfaces; the latter has been used recently to model stress driven surface evolution in heteroepitaxial thin films. A number of interesting observations are made regarding features that are common to the entire hypocycloid family. It is shown in Appendix A that the assumption of a finite energy release rate at diffusive crack initiation leads to the conclusion that the only kind of geometric singularities which may be formed by stress controlled surface diffusion is a mode I cusp crack exactly satisfying the Griffith equalityG= 2γ. If thermal nucleation of dislocations has not occured when such a cusp crack is formed, it is only logical to assume cleavage decohesion at a tension cusp and diffusive trapping of dislocations at a compression cusp.

Crack-Like Sources of Dislocation Nucleation and Multiplication in Thin Films

With the combination of the height sensitivity of atomic force microscopy and the strain sensitivity of transmission electron microscopy, it is shown that near singular stress concentrations can develop naturally in strained epitaxial films. These crack-like instabilities are identified as the sources of dislocation nucleation and multiplication in films of high misfit. This link between morphological instability and dislocation nucleation provides a method for studying the basic micromechanisms that determine the strength and mechanical properties of materials.