Self-organization of quantum dots in epitaxially strained solid films

Why a Large-Scale Mode Can Be Essential for Understanding Intracellular Actin Waves

During the last decade, intracellular actin waves have attracted much attention due to their essential role in various cellular functions, ranging from motility to cytokinesis. Experimental methods have advanced significantly and can capture the dynamics of actin waves over a large range of spatio-temporal scales. However, the corresponding coarse-grained theory mostly avoids the full complexity of this multi-scale phenomenon. In this perspective, we focus on a minimal continuum model of activator-inhibitor type and highlight the qualitative role of mass conservation, which is typically overlooked. Specifically, our interest is to connect between the mathematical mechanisms of pattern formation in the presence of a large-scale mode, due to mass conservation, and distinct behaviors of actin waves.

Orientational competition in quantum dot growth in Si–Ge heteroepitaxy on pit-patterned Si(001) substrates

Growth of quantum dots on patterned substrates shows orientation dependent localization.

Consequences of elastic anisotropy in patterned substrate heteroepitaxy

The role of elastic anisotropy on quantum dot formation and evolution on a pre-patterned substrate is evaluated within the framework of a continuum model. We first extend the formulation for surface evolution to take elastic anisotropy into account. Using a small slope approximation, we derive the evolution equation and show how it can be numerically implemented up to linear and second order for stripe and egg-carton patterned substrates using an accurate and efficient procedure. The semi-infinite nature of the substrate is used to solve the elasticity problem subject to other boundary conditions at the free surface and at the film-substrate interface. The positioning of the quantum dots with respect to the peaks and valleys of the pattern is explained by a competition between the length scale of the pattern and the wavelength of the Asaro-Tiller-Grinfeld instability, which is also affected by the elastic anisotropy. The alignment of dots is affected by a competition between the elastic anisotropy of the film and the pattern orientation. A domain of pattern inversion, wherein the quantum dots form exclusively in the valleys of the patterns is identified as a function of the average film thickness and the elastic anisotropy, and the time-scale for this inversion as function of height is analyzed.

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.

Shape and coarsening dynamics of strained islands

We investigate the formation and the coarsening dynamics of islands in a strained epitaxial semiconductor film. These islands are commonly observed in thin films undergoing a morphological instability due to the presence of the elastocapillary effect. We first describe both analytically and numerically the formation of an equilibrium island using a two-dimensional continuous model. We have found that these equilibrium island-like solutions have a maximum height h_{0} and they sit on top of a flat wetting layer with a thickness h_{w}. We then consider two islands, and we report that they undergo a noninterrupted coarsening that follows a two stage dynamics. The first stage may be depicted by a quasistatic dynamics, where the mass transfers are proportional to the chemical potential difference of the islands. It is associated with a time scale t_{c} that is a function of the distance d between the islands and leads to the shrinkage of the smallest island. Once its height becomes smaller than a minimal equilibrium height h_{0}^{*}, its mass spreads over the entire system. Our results pave the way for a future analysis of coarsening of an assembly of islands.

Stochastic behavior of nanoscale dielectric wall buckling

The random buckling patterns of nanoscale dielectric walls are analyzed using a nonlinear multi-scale stochastic method that combines experimental measurements with simulations. The dielectric walls, approximately 200 nm tall and 20 nm wide, consist of compliant, low dielectric constant (low-k) fins capped with stiff, compressively stressed TiN lines that provide the driving force for buckling. The deflections of the buckled lines exhibit sinusoidal pseudoperiodicity with amplitude fluctuation and phase decorrelation arising from stochastic variations in wall geometry, properties, and stress state at length scales shorter than the characteristic deflection wavelength of about 1000 nm. The buckling patterns are analyzed and modeled at two length scales: a longer scale (up to 5000 nm) that treats randomness as a longer-scale measurable quantity, and a shorter-scale (down to 20 nm) that treats buckling as a deterministic phenomenon. Statistical simulation is used to join the two length scales. Through this approach, the buckling model is validated and material properties and stress states are inferred. In particular, the stress state of TiN lines in three different systems is determined, along with the elastic moduli of low-k fins and the amplitudes of the small-scale random fluctuations in wall properties-all in the as-processed state. The important case of stochastic effects giving rise to buckling in a deterministically sub-critical buckling state is demonstrated. The nonlinear multiscale stochastic analysis provides guidance for design of low-k structures with acceptable buckling behavior and serves as a template for how randomness that is common to nanoscale phenomena might be measured and analyzed in other contexts.

Instabilities of Layers of Deposited Molecules on Chemically Stripe Patterned Substrates: Ridges versus Drops

A mesoscopic continuum model is employed to analyze the transport mechanisms and structure formation during the redistribution stage of deposition experiments where organic molecules are deposited on a solid substrate with periodic stripe-like wettability patterns. Transversally invariant ridges located on the more wettable stripes are identified as very important transient states and their linear stability is analyzed accompanied by direct numerical simulations of the fully nonlinear evolution equation for two-dimensional substrates. It is found that there exist two different instability modes that lead to different nonlinear evolutions that result (i) at large ridge volume in the formation of bulges that spill from the more wettable stripes onto the less wettable bare substrate and (ii) at small ridge volume in the formation of small droplets located on the more wettable stripes. In addition, the influence of different transport mechanisms during redistribution is investigated focusing on the cases of convective transport with no-slip at the substrate, transport via diffusion in the film bulk and via diffusion at the film surface. In particular, it is shown that the transport process does neither influence the linear stability thresholds nor the sequence of morphologies observed in the time simulation, but only the ratio of the time scales of the different process phases.

Localized states in the conserved Swift-Hohenberg equation with cubic nonlinearity

The conserved Swift-Hohenberg equation with cubic nonlinearity provides the simplest microscopic description of the thermodynamic transition from a fluid state to a crystalline state. The resulting phase field crystal model describes a variety of spatially localized structures, in addition to different spatially extended periodic structures. The location of these structures in the temperature versus mean order parameter plane is determined using a combination of numerical continuation in one dimension and direct numerical simulation in two and three dimensions. Localized states are found in the region of thermodynamic coexistence between the homogeneous and structured phases, and may lie outside of the binodal for these states. The results are related to the phenomenon of slanted snaking but take the form of standard homoclinic snaking when the mean order parameter is plotted as a function of the chemical potential, and are expected to carry over to related models with a conserved order parameter.

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.

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.

Creating self-organized submicrometer contact instability patterns in soft elastic bilayers with a topographically patterned stamp

The surface of a thin elastic bilayer becomes spontaneously unstable when it is brought in proximity to another rigid contactor. The instability patterns, which are random and isotropic, exhibit a dominant lateral length scale of instability λ, which linearly scales with the bilayer thickness (h) as: λ = R(F)h. It is known that for an elastic bilayer, R(F) exhibits a nonlinear dependence on the ratios of individual film thicknesses (H) and shear moduli (M) of the two constituent layers, and can have values as low as 0.5 under specific conditions. This is in contrast to a near constant value of R(F) ≈ 3 for a single layer elastic film. (1) These isotropic contact instability patterns in a bilayer can be ordered, aligned and modulated using a topographically patterned stamp. The precise morphology of the aligned structures depends on commensuration between λ and the stamp periodicity (λ(P)), and on the intersurface separation distance. A variety of patterns, like an array of circular holes, double periodic channels, etc., in addition to a positive and a negative replica of the stamp pattern, can be engineered with a simple stamp having 1D grating structure. A lower value of R(F) in a bilayer allows generating patterns with sub 500 nm lateral resolution, which is impossible to create by elastic contact lithography (ECL) of a single layer film due to strong surface tension effects in ultrathin films. Thus, control of elastic instability in a bilayer with a patterned stamp represents a flexible soft lithography tool allowing modulation of length scales, morphology, and order.

Ordering of strained islands during surface growth

We study the morphological evolution of strained islands in growing crystal films by use of a continuum description including wetting, elasticity, and deposition. We report different nonlinear regimes following the elastic instability and tuned by the flux. Increasing the flux, we first find an annealinglike dynamics, then a slower but nonconventional ripening followed by a steady regime, while the island density continuously increases. The islands develop spatial correlations and ordering with a narrow two-peaked distance distribution and ridgelike clusters of islands at high flux.

Feedback control of subcritical Turing instability with zero mode

A global feedback control of a system that exhibits a subcritical monotonic instability at a nonzero wave number (short-wave or Turing instability) in the presence of a zero mode is investigated using a Ginzburg-Landau equation coupled to an equation for the zero mode. This system is studied analytically and numerically. It is shown that feedback control, based on measuring the maximum of the pattern amplitude over the domain, can stabilize the system and lead to the formation of localized unipulse stationary states or traveling solitary waves. It is found that the unipulse traveling structures result from an instability of the stationary unipulse structures when one of the parameters characterizing the coupling between the periodic pattern and the zero mode exceeds a critical value that is determined by the zero mode damping coefficient.

Nanostructures with long-range order in monolayer self-assembly

A key ingredient for continued expansion of nanotechnologies is the ability to create perfectly ordered arrays on a small scale with both site and size control. Self-assembly-i.e., the spontaneous formation of nanostructures-is a highly promising alternative to traditional fabrication methods. However, efforts to obtain perfect long-range order via self-assembly have been frustrated in practice as ensuing patterns contain defects. We use an idea based on the fundamental physics of pattern formation to introduce a strategy to consistently obtain perfect patterns.

Simulation of self-assembled nanopatterns in strained 2D alloys on the face centered cubic (111) surface

We investigate the formation of nanostructures in 2D strained alloys on face centered cubic (111) surfaces by means of equilibrium Monte Carlo simulations. In the framework of an off-lattice model, we consider one monolayer of two bulk-immiscible adsorbates A and B with negative and positive misfit relative to the substrate, respectively. Simulations show that the adsorbates partly self-organize into island or stripe-like patterns. We show how these structures depend on the relative misfits, interaction, and concentration of components. The morphology is quite different for phase separation and intermixing regimes.

Renormalization of stochastic lattice models: epitaxial surfaces

We present the application of a method [C. A. Haselwandter and D. D. Vvedensky, Phys. Rev. E 76, 041115 (2007)] for deriving stochastic partial differential equations from atomistic processes to the morphological evolution of epitaxial surfaces driven by the deposition of new material. Although formally identical to the one-dimensional (1D) systems considered previously, our methodology presents substantial additional technical issues when applied to two-dimensional (2D) surfaces. Once these are addressed, subsequent coarse-graining is accomplished as before by calculating renormalization-group (RG) trajectories from initial conditions determined by the regularized atomistic models. Our applications are to the Edwards-Wilkinson (EW) model [S. F. Edwards and D. R. Wilkinson, Proc. R. Soc. London, Ser. A 381, 17 (1982)], the Wolf-Villain (WV) model [D. E. Wolf and J. Villain, Europhys. Lett. 13, 389 (1990)], and a model with concurrent random deposition and surface diffusion. With our rules for the EW model no appreciable crossover is obtained for either 1D or 2D substrates. For the 1D WV model, discussed previously, our analysis reproduces the crossover sequence known from kinetic Monte Carlo (KMC) simulations, but for the 2D WV model, we find a transition from smooth to unstable growth under repeated coarse-graining. Concurrent surface diffusion does not change this behavior, but can lead to extended transient regimes with kinetic roughening. This provides an explanation of recent experiments on Ge(001) with the intriguing conclusion that the same relaxation mechanism responsible for ordered structures during the early stages of growth also produces an instability at longer times that leads to epitaxial breakdown. The RG trajectories calculated for concurrent random deposition and surface diffusion reproduce the crossover sequences observed with KMC simulations for all values of the model parameters, and asymptotically always approach the fixed point corresponding to the equation proposed by Villain [J. Phys. I 1, 19 (1991)] and by Lai and Das Sarma [Phys. Rev. Lett. 66, 2899 (1991)]. We conclude with a discussion of the application of our methodology to other growth settings.

Dewetting of a solid monolayer

We report on the dewetting of a monolayer on a solid substrate, where mass transport occurs via surface diffusion. For a wide range of parameters, a labyrinthine pattern of bilayer islands is formed. An irreversible regime and a thermodynamic regime are identified. In both regimes, the velocity of a dewetting front, the wavelength of the bilayer island pattern, and the rate of nucleation of dewetted zones are obtained. We also point out the existence of a scaling behavior, which is analyzed by means of a geometrical model.

Surface instability of confined elastic bilayers: Theory and simulations

The surface of a soft elastic film becomes unstable and forms a self-organized undulating pattern because of adhesive interactions when it comes in contact proximity with a rigid surface. For a single film, the pattern length scale lambda , which is governed by the minimization of the elastic stored energy, gives lambda approximately 3h , where h is the film thickness. Based on a linear stability analysis and simulations of adhesion and debonding, we consider the contact instability of an elastic bilayer, which provides greater flexibility in the morphological control of interfacial instability. Unlike the case of a single film, the morphology of the contact instability patterns, debonding distance, and debonding force in a bilayer can be controlled in a nonlinear way by varying the thicknesses and shear moduli of the films. Interestingly, the pattern wavelength in a bilayer can be greatly increased or decreased compared to a single film when the adhesive contact is formed by the stiffer or the softer of the two films, respectively. In particular, lambda as small as 0.5h can be obtained. This indicates a new strategy for pattern miniaturization in elastic contact lithography.

Dynamics of wrinkle growth and coarsening in stressed thin films

A stressed thin film on a soft substrate can develop complex wrinkle patterns. The onset of wrinkling and initial growth is well described by a linear perturbation analysis, and the equilibrium wrinkles can be analyzed using an energy approach. In between, the wrinkle pattern undergoes a coarsening process with a peculiar dynamics. By using a proper scaling and two-dimensional numerical simulations, this paper develops a quantitative understanding of the wrinkling dynamics from initial growth through coarsening till equilibrium. It is found that, during the initial growth, a stress-dependent wavelength is selected and the wrinkle amplitude grows exponentially over time. During coarsening, both the wrinkle wavelength and amplitude increases, following a power-law scaling under uniaxial compression. More complicated dynamics is predicted under equibiaxial stresses, which starts with a faster coarsening rate before asymptotically approaching the same scaling under uniaxial stresses. At equilibrium, a parallel stripe pattern is obtained under uniaxial stresses and a chaotic labyrinth pattern under equibiaxial stresses. Under stresses of the same magnitude, the two patterns have the same average wavelength, but different amplitudes. It is noted that the dynamics of wrinkling, while analogous to other phase ordering phenomena, is distinct and rich under the effects of stress and substrate elasticity.

Langevin equations for fluctuating surfaces

Exact Langevin equations are derived for the height fluctuations of surfaces driven by the deposition of material from a molecular beam. We consider two types of model: deposition models, where growth proceeds by the deposition and instantaneous local relaxation of particles, with no subsequent movement, and models with concurrent random deposition and surface diffusion. Starting from a Chapman-Kolmogorov equation the deposition, relaxation, and hopping rules of these models are first expressed as transition rates within a master equation for the joint height probability density function. The Kramers-Moyal-van Kampen expansion of the master equation in terms of an appropriate "largeness" parameter yields, according to a limit theorem due to Kurtz [Stoch. Proc. Appl. 6, 223 (1978)], a Fokker-Planck equation that embodies the statistical properties of the original lattice model. The statistical equivalence of this Fokker-Planck equation, solved in terms of the associated Langevin equation, and solutions of the Chapman-Kolmogorov equation, as determined by kinetic Monte Carlo (KMC) simulations of the lattice transition rules, is demonstrated by comparing the surface roughness and the lateral height correlations obtained from the two formulations for the Edwards-Wilkinson [Proc. R. Soc. London Ser. A 381, 17 (1982)] and Wolf-Villain [Europhys. Lett. 13, 389 (1990)] deposition models, and for a model with random deposition and surface diffusion. In each case, as the largeness parameter is increased, the Langevin equation converges to the surface roughness and lateral height correlations produced by KMC simulations for all times, including the crossover between different scaling regimes. We conclude by examining some of the wider implications of these results, including applications to heteroepitaxial systems and the passage to the continuum limit.

Rotating hexagonal pattern in a dielectric barrier discharge system

Here, we report on the experimental observation of a rotating hexagonal pattern in a continuous dissipative medium. The system under investigation is a planar dielectric barrier gas-discharge cell. The pattern consists of a set of current filaments occupying the whole discharge area and rotating as a rigid body. The symmetry of the rotating hexagons is lower than the symmetry of the stationary hexagonal pattern. We study the dynamics of the pattern, especially peculiarities of its rotational velocity. The temperature of the gas is found to be an important quantity influencing the rotating hexagons.