Capillarity-driven Surface Diffusion Research Articles

Electrode–electrolyte interface stability in solid state electrolyte systems: influence of coating thickness under varying residual stresses

We introduce a model of electrode–electrolyte interfacial growth which focuses on the effect of thin coating layers on the interfacial stability in prestressed systems. We take into account transport resulting from deposition from the electrolyte, from capillarity driven surface diffusion, and from changes of the chemical potential due to the elastic energy associated with the interface profile. As model system, we use metallic lithium as electrode, LLZO as electrolyte and Al2O3 as a thin film interlayer, which is a highly relevant interfacial system in state of the art all-solid-electrolyte batteries. We consider the stability of the electrode-coating-electrolyte interface depending on the thickness of the thin film interlayer and the magnitude of the elastic prestresses. Our central approach is a linear stability analysis based on the mass conservation at the planar interface, employing approximations which are appropriate for solid state electrolytes (SSEs) like LLZ, a thin Li metal electrode and a thin coating layer with a thickness in the range of nanometres.

Capillarity-Driven Welding of Semiconductor Nanowires for Crystalline and Electrically Ohmic Junctions.

Semiconductor nanowires (NWs) have been demonstrated as a potential platform for a wide-range of technologies, yet a method to interconnect functionally encoded NWs has remained a challenge. Here, we report a simple capillarity-driven and self-limited welding process that forms mechanically robust and Ohmic inter-NW connections. The process occurs at the point-of-contact between two NWs at temperatures 400-600 °C below the bulk melting point of the semiconductor. It can be explained by capillarity-driven surface diffusion, inducing a localized geometrical rearrangement that reduces spatial curvature. The resulting weld comprises two fused NWs separated by a single, Ohmic grain boundary. We expect the welding mechanism to be generic for all types of NWs and to enable the development of complex interconnected networks for neuromorphic computation, battery and solar cell electrodes, and bioelectronic scaffolds.

Linear stability of circular micro- and nanowires with facets

Micro- and nanowires are commonly used in biological sciences, micro- and nanoelectronics, and optoelectronics, and their morphological stability needs to be understood and controlled. We study the linear stability of equilibrium circular wires with length to diameter ratio of 1, 2, 3.5, 6, and 11, assuming that the wire surface can deform by capillarity-driven surface diffusion. The facetted equilibrium wire shape is modeled by the Dirac delta function and is perturbed by an infinitesimal axisymmetric disturbance, leading to an eigenvalue problem for the growth rate, which is solved by a finite-difference method. Numerical accuracy is checked by grid refinement. All converged eigenvalues are negative, indicating that the wires are linearly stable. The first six eigenvalues are listed for all the wires which show that, for the same eigenmode, the eigenvalue decreases in magnitude as the wire length increases. The eigenfunctions for the longest wire studied are plotted and reveal how a non-equilibrium wire finally approaches the equilibrium state. The linear-stability formulation is then extended to an infinitely-long circular wire. The wire is stable for all wavelengths if its surface coincides with a facet plane. Hence, Rayleigh’s instability is completely suppressed in faceted circular wires.

Capillarity-driven migration of a thin Ge wedge in contact with a bicrystalline Au film

We have investigated the retraction of a single-crystalline Ge wedge in epitaxial contact with a bicrystalline Au film using in situ electron microscopy. The rate of retraction was close to that predicted for capillarity-driven surface diffusion, following kinetics proportional to t n , with n = 0.22–0.35, but crystal anisotropy caused migration to be significantly faster along 〈1 0 0〉 directions than along 〈1 1 0〉. The bicrystalline Au substrate was not inert, but underwent abnormal grain growth in the area swept by the receding Ge wedge. Cross-sections made from plan-view transmission electron microscopy samples revealed that this was related to ridge formation during the retraction process. In situ observations of the process in an inclined orientation showed direct evidence of substrate grain boundaries being dragged by the receding Ge wedge. The results can be understood in the framework of capillarity models for isotropic solid-state wedges and reactive wetting in high-temperature liquid–solid experiments.

Grain-boundary grooving by surface diffusion with asymmetric and strongly anisotropic surface energies

Grain-boundary migration controls grain growth, which is important in material processing and synthesis. When a vertical grain boundary ends at a horizontal free surface, a groove forms at the tip to reduce the combined grain-boundary and surface energies. The groove affects the migration of the grain boundary, and its effect must be understood. This work studies grain-boundary grooving by capillarity-driven surface diffusion with asymmetric and strongly anisotropic surface energies. The surface energies are described by the delta-function facet model that holds for temperatures above the roughening temperature of the bicrystal. Since the asymmetric anisotropy does not introduce a length scale, the asymmetric groove still grows with time t as t1∕4. Thus, the nonlinear partial differential equation that governs grooving is reduced by a self-similar transformation to an ordinary differential equation, which is then solved numerically by shooting methods. We vary systematically the crystallographic orientations of the bicrystal and the normalized grain-boundary energy. We find that the self-similar groove profile is faceted and asymmetric for most bicrystals. However, a few asymmetric bicrystals can also yield smooth and symmetric grooves. Some bicrystals may even form ridges instead of grooves. We also find that the asymmetric surface energies tilt the grain-boundary tip sideways, which induces the grain boundary to migrate. We show that anisotropic groove profiles measured in SrTiO3 and Ni can be well fitted by our model.

Fingering instability of a retracting solid film edge

A thin gold film under annealing on a silica substrate can develop “fingers” at the perimeter of the film. The perimeter retracts to leave behind longer fingers, which eventually pinch off to reduce the surface energy of the system. New fingers then form at the film edge and the process continues until the entire film disintegrates. To maintain the structure integrity of annealed thin films, this fingering instability must be understood. The retraction of a straight film edge via capillarity-driven surface diffusion has been analyzed in two dimensions by Wong et al.[Acta Mater. 48, 1719 (2000)]. They found that a retracting film is thickened at the edge followed by a valley before the film thickness becomes uniform. We study the three-dimensional linear stability of this two-dimensional film profile and find one unstable mode of perturbation. The growth rate of the perturbation is determined as a function of the wavelength of the perturbation and the speed of the receding edge. The results show that a straight film edge becomes wavy when perturbed. The wavelength λm of the fastest growing perturbation agrees with the distance between adjacent fingers observed in a gold-film experiment. Fingers can also form during annealing at the retracting edges of cracks in sapphire, and our predicted λm compares well with the measured finger spacing.

Grain-boundary grooving by surface diffusion with strong surface energy anisotropy

A vertical grain boundary intercepting a horizontal free surface forms a groove to reduce the combined surface energy of the system. The groove grows with time and is commonly used for measuring surface diffusion coefficients. This work studies grooving by capillarity-driven surface diffusion with strong surface energy anisotropy and finds that faceted grooves still grow with time t as t 1/4. However, an anisotropic groove can be smooth if the groove surface does not cross a facet orientation. The groove has the same shape as the corresponding isotropic groove, but the growth rate is reduced by a factor that depends on the degree of anisotropy. This reduction induces an error in the surface diffusion coefficient if the isotropic model is applied to a smooth, but anisotropic groove. We show how to correct for this error.

Periodic mass shedding of a retracting solid film step

A semi-infinite, uniform film on a substrate tends to contract from the edge to reduce the surface energy of the system. This work studies the two-dimensional retraction of such a film step, assuming that the film evolves by capillarity-driven surface diffusion. It is found that the retracting film edge forms a thickened ridge followed by a valley. The valley sinks with time and eventually touches the substrate. The ridge then detaches from the film. The new film edge retracts to form another ridge accompanied again by a valley, and the mass shedding cycle is repeated. This periodic mass shedding is simulated numerically for contact angle α between 30 and 180°. For smaller α, a small-slope late-time solution is found that agrees with the numerical solution for α=30°. Thus, the complete range of α is covered. The long-time retraction speed and the distance traveled per cycle agree quantitatively with experiments.

Theoretical analysis of electromigration-induced failure of metallic thin films due to transgranular void propagation

Failure of metallic thin films driven by electromigration is among the most challenging materials reliability problems in microelectronics toward ultra-large-scale integration. One of the most serious failure mechanisms in thin films with bamboo grain structure is the propagation of transgranular voids, which may lead to open-circuit failure. In this article, a comprehensive theoretical analysis is presented of the complex nonlinear dynamics of transgranular voids in metallic thin films as determined by capillarity-driven surface diffusion coupled with drift induced by electromigration. Our analysis is based on self-consistent dynamical simulations of void morphological evolution and it is aided by the conclusions of an approximate linear stability theory. Our simulations emphasize that the strong dependence of surface diffusivity on void surface orientation, the strength of the applied electric field, and the void size play important roles in the dynamics of the voids. The simulations predict void faceting, formation of wedge-shaped voids due to facet selection, propagation of slit-like features emanating from void surfaces, open-circuit failure due to slit propagation, as well as appearance and disappearance of soliton-like features on void surfaces prior to failure. These predictions are in very good agreement with recent experimental observations during accelerated electromigration testing of unpassivated metallic films. The simulation results are used to establish conditions for the formation of various void morphological features and discuss their serious implications for interconnect reliability.

Universal pinch off of rods by capillarity-driven surface diffusion

Interfacial energy is a central factor in setting the morphology of phases and in determining the stability of equilibrium morphologies. here the authors examine the morphological evolution of a rod via capillary-driven surface diffusion as it both approaches and departs the topological singularity of pinch off. During the final stages of pinching the neck radius approaches zero, and self-similar solutions are sought. The authors have derived local similarity solutions for the axisymmetric pinch off of rods when the morphological evolution is by capillarity-driven surface diffusion. These local solutions describe the approach to and departure from the topological singularity where a rod pinches into two separate bodies. During pinching, the self-similar surface profile far away from the neck approaches two opposing cones with a unique half-cone angle of 46.04{degree}. It is thus likely that all rods must pinch off with this cone angle. This assertion is supported by several numerical simulations. After pinch off, the smoothening of the cone tip is again self-similar. The results obtained here for rods also apply to the pinch off of cylindrical pore channels.

Capillarity driven motion of solid film wedges

A solid film freshly deposited on a substrate may form a non-equilibrium contact angle with the substrate, and will evolve. This morphological evolution near the contact line is investigated by studying the motion of a solid wedge on a substrate. The contact angle of the wedge changes at time t = 0 from the wedge angle α to the equilibrium contact angle β, and its effects spread into the wedge via capillarity-driven surface diffusion. The film profiles at different times are found to be self-similar, with the length scale increasing at t 1 4 . The self-similar film profile is determined numerically by a shooting method for α and β between 0 and 180°. In general, we find that the film remains a wedge when α = β. For α < β, the film retracts, whereas for α > β, the film extends. For α = 90°, the results describe the growth of grain-boundary grooves for arbitrary dihedral angles. For β = 90°, the solution also applies to a free-standing wedge, and the thin-wedge profiles agree qualitatively with those observed in transmission electron microscope specimens.

Capillary instabilities of a catenoidal hole in a solid film

Holes formed in a solid film during annealing can grow and lead to the formation of many isolated islands. The morphological evolution of holes is thus of primary importance in the production of planar films. This work studies the linear instability of a stationary axisymmetric hole in a film with zero surface mean curvature. At the hole, the film forms a contact angle α with the substrate at a circular contact line of radius a0. The film is bounded by an outer wall at a distance a0L from the center. An infinitesimal disturbance in the form of a normal mode is applied and its stability analyzed for 0⩽α⩽180° and 1⩽L&amp;lt;∞. Capillarity-driven surface diffusion is taken to dominate the mass transport. As L→1, the film is a ring that is unstable to periodic disturbances along the ring. For an unbounded film with L→∞, only axisymmetric disturbances can grow, and the growth rates become independent of L or the boundary conditions at the outer wall. This instability persists even when the film is “flat” in the limit α→0, in contrast to the stability results of a uniform film without a hole. The growth rates agree qualitatively with those observed in experiments.