Solid-state Dewetting Of Thin Films Research Articles

A symmetrized parametric finite element method for simulating solid-state dewetting problems

We propose an accurate, area/mass-conservative and energy-dissipative parametric finite element method for solving the sharp-interface continuum model of solid-state dewetting with arbitrary anisotropic surface energy. The model is governed by the anisotropic surface diffusion equation with suitable boundary conditions at the contact points, and could be used to describe the kinetic evolution of the film/vapor interface with contact line migration. The sharp-interface continuum model is firstly reformulated into a conservative form by introducing a symmetric positive definite matrix Zk(n) that depends on the Cahn–Hoffman ξ-vector and a stabilizing function k(n). A new symmetrized variational formulation with arbitrary anisotropic surface energy is built and then its area/mass conservation and energy dissipation properties are proved. By using piecewise linear elements in space, we propose the spatial semi-discretized scheme, and using backward Euler in time with an appropriate approximation of the unit normal vector, we further derive the fully discretized parametric finite element approximation. We show, theoretically, that the semi-discretized and fully discretized schemes are both area/mass-conservative and energy-dissipative. Finally, numerical experiments are reported to show the accuracy and efficiency of the numerical method as well as the anisotropic effects on the morphological evolution processes for solid-state dewetting of thin films.

An energy-stable parametric finite element method for anisotropic surface diffusion

We propose an energy-stable parametric finite element method (ES-PFEM) to discretize the motion of a closed curve under surface diffusion with an anisotropic surface energy γ(θ) – anisotropic surface diffusion – in two dimensions, while θ is the angle between the outward unit normal vector and the vertical axis. By introducing a positive definite surface energy (density) matrix G(θ), we present a new and simple variational formulation for the anisotropic surface diffusion and prove that it satisfies area/mass conservation and energy dissipation. The variational problem is discretized in space by the parametric finite element method and area/mass conservation and energy dissipation are established for the semi-discretization. Then the problem is further discretized in time by a (semi-implicit) backward Euler method so that only a linear system is to be solved at each time step for the full-discretization and thus it is efficient. We establish well-posedness of the full-discretization and identify some simple conditions on γ(θ) such that the full-discretization keeps energy dissipation and thus it is unconditionally energy-stable. Finally the ES-PFEM is applied to simulate solid-state dewetting of thin films with anisotropic surface energies, i.e. the motion of an open curve under anisotropic surface diffusion with proper boundary conditions at the two triple points moving along the horizontal substrate. Numerical results are reported to demonstrate the efficiency and accuracy as well as energy dissipation of the proposed ES-PFEM.

An energy-stable parametric finite element method for simulating solid-state dewetting

Abstract We propose an energy-stable parametric finite element method for simulating solid-state dewetting of thin films in two dimensions via a sharp-interface model, which is governed by surface diffusion and contact line (point) migration together with proper boundary conditions. By reformulating the relaxed contact angle condition into a Robin-type boundary condition and then treating it as a natural boundary condition, we obtain a new variational formulation for the problem, in which the interface curve and its contact points are evolved simultaneously. Then the variational problem is discretized in space by using piecewise linear elements. A full discretization is presented by adopting the backward Euler method in time, and the well-posedness and energy dissipation of the full discretization are established. The numerical method is semi-implicit (i.e., a linear system to be solved at each time step and thus efficient), unconditionally energy-stable with respect to the time step and second-order in space measured by a manifold distance between two curves. In addition, it demonstrates equal mesh distribution when the solution reaches its equilibrium, i.e., long-time dynamics. Numerical results are reported to show accuracy and efficiency as well as some good properties of the proposed numerical method.

A Parametric Finite Element Method for Solid-State Dewetting Problems in Three Dimensions

We propose a parametric finite element method (PFEM) for efficiently solving the morphological evolution of solid-state dewetting of thin films on a flat rigid substrate in three dimensions (3D). The interface evolution of the dewetting problem in 3D is described by a sharp-interface model, which includes surface diffusion coupled with contact line migration. A variational formulation of the sharp-interface model is presented, and a PFEM is proposed for spatial discretization. For temporal discretization, at each time step, we first update the position of the contact line according to the relaxed contact angle condition; then, by using the position of the new contact line as the boundary condition, we solve a linear algebra system resulted from the discretization of PFEM to obtain the new interface surface for the next step. The well-posedness of the solution of the PFEM is also established. Extensive numerical results are reported to demonstrate the accuracy and efficiency of the proposed PFEM and to show the complexities of the dewetting morphology evolution observed in solid-state dewetting experiments.

Sharp-Interface Model for Simulating Solid-State Dewetting in Three Dimensions

The problem of simulating solid-state dewetting of thin films in three dimensions (3D) by using a sharp-interface approach is considered in this paper. Based on the thermodynamic variation, a speed method is used for calculating the first variation to the total surface energy functional. The speed method shares more advantages than the traditional use of parameterized curves (or surfaces), e.g., it is more intrinsic and its variational structure (related with Cahn-Hoffman $\boldsymbol{\xi}$-vector) is clearer and more direct. By making use of the first variation, necessary conditions for the equilibrium shape of the solid-state dewetting problem is given, and a kinetic sharp-interface model which includes the surface energy anisotropy is also proposed. This sharp-interface model describes the interface evolution in 3D which occurs through surface diffusion and contact line migration. By solving the proposed model, we perform lots of numerical simulations to investigate the evolution of patterned films, e.g., the evolution of a short cuboid and pinch-off of a long cuboid. Numerical simulations in 3D demonstrate the accuracy and efficacy of the sharp-interface approach to capture many of the complexities observed in solid-state dewetting experiments.

Deterministic 3D self-assembly of Si through a rim-less and topology-preserving dewetting regime

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 novel 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.

Deterministic three-dimensional self-assembly of Si through a rimless and topology-preserving dewetting regime

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.

An Efficient and Unconditionally Energy Stable Scheme for Simulating Solid-State Dewetting of Thin Films with Isotropic Surface Energy<sup>†</sup>

An Efficient and Unconditionally Energy Stable Scheme for Simulating Solid-State Dewetting of Thin Films with Isotropic Surface Energy<sup>†</sup>

A sharp-interface model and its numerical approximation for solid-state dewetting with axisymmetric geometry

Based on the thermodynamic variation, we rigorously derive a sharp-interface model for solid-state dewetting of thin films on a flat substrate under the assumption that the morphology of thin film is axisymmetric. The governing equations for the sharp-interface model belong to fourth-order geometric evolution partial differential equations, and we rigorously prove that the total volume of the thin film is conserved and the total free energy is dissipative during the evolution. Furthermore, we propose a parametric finite element method for solving the above sharp-interface model. Extensive numerical results demonstrate the high performance of the numerical method and capture many significant morphological evolution features for solid-state dewetting of thin films with axisymmetric geometry.

Sharp-interface approach for simulating solid-state dewetting in two dimensions: A Cahn–Hoffman [formula omitted]-vector formulation

By using a Cahn–Hoffman ξ-vector formulation, we propose a sharp-interface approach for solving solid-state dewetting problems in two dimensions. First, based on the thermodynamic variation and smooth vector-field perturbation method, we rigorously derive a sharp-interface model with weakly anisotropic surface energies, and this model describes the interface evolution which occurs through surface diffusion flow and contact line migration. Second, a parametric finite element method in terms of the ξ-vector formulation is proposed for numerically solving the sharp-interface model. By performing numerical simulations, we examine several specific evolution processes for solid-state dewetting of thin films, e.g., the evolution of small islands, pinch-off of large islands and power-law retraction dynamics of semi-infinite step films, and these simulation results are consistent with experimental observations. Finally, we also include the strong surface energy anisotropy into the sharp-interface model and design its corresponding numerical scheme via the ξ-vector formulation.

Solid-state dewetting on curved substrates

Based on the thermodynamic variation to the free energy functional, we propose a sharp-interface model for simulating solid-state dewetting of thin films on rigid curved substrates in two dimensions. This model describes the interface evolution which occurs through surface diffusion-controlled mass transport and contact point migration along the curved substrate. Furthermore, the surface energy anisotropy is easily included into the model, and the contact point migration is explicitly described by the relaxed contact angle boundary condition. We implement the mathematical model by a semi-implicit parametric finite element method to study several interesting phenomena, such as "small" particle migration on curved substrates and templated solid-state dewetting on a pre-patterned substrate. Based on ample numerical simulations, we demonstrate that, the migration velocity of a "small" solid particle is proportional to the substrate curvature gradient $\hat{\kappa}'$ and inversely proportional to the square root of the area of the particle $\sqrt{A}$, and it decreases when the isotropic Young angle $\theta_i$ increases. In addition, we also observe four periodic categories of dewetting on a pre-patterned sinusoidal substrate. Our approach can provide a convenient and powerful tool to exploring how to produce well-organized nanoparticles by making use of template-assisted solid-state dewetting.

Application of Onsager's variational principle to the dynamics of a solid toroidal island on a substrate

In this paper, we consider the capillarity-driven evolution of a solid toroidal island on a flat rigid substrate, where mass transport is controlled by surface diffusion. This problem is representative of the geometrical complexity associated with the solid-state dewetting of thin films on substrates. We apply Onsager's variational principle to develop a general approach for describing surface diffusion-controlled problems. Based on this approach, we derive a simple, reduced-order model and obtain an analytical expression for the rate of island shrinking and validate this prediction by numerical simulations based on a full, sharp-interface model. We find that the rate of island shrinking is proportional to the material constants B and the surface energy density γ0, and is inversely proportional to the island volume V0. This approach represents a general tool for modeling interface diffusion-controlled morphology evolution.

Stable Equilibria of Anisotropic Particles on Substrates: A Generalized Winterbottom Construction

We present,a new approach for predicting stable equilibrium shapes of two-dimensional crystalline islands on flat substrates, as commonly occur through solid-state dewetting of thin films. The new theory is a generalization of the widely used Winterbottom construction (i.e., an extension of the Wulff construction for particles on substrates). This approach is equally applicable to cases where the crystal surface energy is isotropic, weakly anisotropic, strongly anisotropic, and “cusped”. We demonstrate that, unlike in the classical Winterbottom approach, multiple equilibrium island shapes may be possible when the surface energy is strongly anisotropic. We analyze these shapes through perturbation analysis, by calculating the first and second variations of the total free energy functional with respect to contact locations and island shape. Based on this analysis, we find the necessary conditions for the equilibria to be stable to two-dimensional perturbations and exploit this through a generalization of th...

A parametric finite element method for solid-state dewetting problems with anisotropic surface energies

We propose an efficient and accurate parametric finite element method (PFEM) for solving sharp-interface continuum models for solid-state dewetting of thin films with anisotropic surface energies. The governing equations of the sharp-interface models belong to a new type of high-order (4th- or 6th-order) geometric evolution partial differential equations about open curve/surface interface tracking problems which include anisotropic surface diffusion flow and contact line migration. Compared to the traditional methods (e.g., marker-particle methods), the proposed PFEM not only has very good accuracy, but also poses very mild restrictions on the numerical stability, and thus it has significant advantages for solving this type of open curve evolution problems with applications in the simulation of solid-state dewetting. Extensive numerical results are reported to demonstrate the accuracy and high efficiency of the proposed PFEM.

Power-law scaling regimes for solid-state dewetting of thin films

The capillary force drives the edges of solid thin films to retract. The distance a film edge has retracted over time is usually fitted to a power law. However, experiments and numerical simulations suggest that edge retraction does not follow a power-law. In this work, a simple geometric model for edge retraction is presented that reproduces the retraction distance versus time scalings of simulations for both isotropic and highly-anisotropic films, and is consistent with experiments. The earliest time at which a power-law fit becomes a reasonable approximation is calculated as a function of substrate–film contact angle.

Self-assembled silver nanoparticles for plasmon-enhanced solar cell back reflectors: correlation between structural and optical properties

The spectra of localized surface plasmon resonances (LSPRs) in self-assembled silver nanoparticles (NPs), prepared by solid-state dewetting of thin films, are discussed in terms of their structural properties. We summarize the dependences of size and shape of NPs on the fabrication conditions with a proposed structural-phase diagram. It was found that the surface coverage distribution and the mean surface coverage (SC) size were the most appropriate statistical parameters to describe the correlation between the morphology and the optical properties of the nanostructures. The results are interpreted with theoretical predictions based on Mie theory. The broadband scattering efficiency of LSPRs in the nanostructures is discussed towards application as plasmon-enhanced back reflectors in thin-film solar cells.

Solid-State Dewetting of Thin Films

Solid films are usually metastable or unstable in the as-deposited state, and they will dewet or agglomerate to form islands when heated to sufficiently high temperatures. This process is driven by surface energy minimization and can occur via surface diffusion well below a film's melting temperature, especially when the film is very thin. Dewetting during processing of films for use in micro- and nanosystems is often undesirable, and means of avoiding dewetting are important in this context. However, dewetting can also be useful in making arrays of nanoscale particles for electronic and photonic devices and for catalyzing growth of nanotubes and nanowires. Templating of dewetting using patterned surface topography or prepatterning of films can be used to create ordered arrays of particles and complex patterns of partially dewetted structures. Studies of dewetting can also provide fundamental new insight into the effects of surface energy anisotropy and facets on shape evolution.

Solid-state dewetting for ordered arrays of crystallographically oriented metal particles

We demonstrate that topographically modified substrates can be used as templates to control solid-state dewetting of thin films by creating a periodic variation in the initial curvature of the film. The dewetting of gold films on oxidized silicon surfaces patterned with arrays of inverted pyramid shaped pits is investigated. For specific ranges of relative film thickness and topographic dimension, dewetting results in arrays of submicrometer-scale particles with uniform periodic spacing and nearly uniform size and crystallographic orientation.