Coarsening kinetics of a spinodally decomposed vicinal Si(111) surface

Regularized anisotropic motion-by-curvature in phase-field theory: Interface phase separation of crystal surfaces

The kinetic equation for anisotropic motion-by-curvature is ill posed when the surface energy is strongly anisotropic. In this case, corners or edges are present on the Wulff shape, which span a range of missing orientations. In the sharp-interface problem the surface energy is augmented with a curvature-dependent term that rounds the corners and regularizes the dynamic equations. This introduces a new length scale in the problem, the corner size. In phase-field theory, a diffuse description of the interface is adopted. In this context, an approximation of the Willmore energy can be added to the phase-field energy so as to regularize the model. In this paper, we discuss the convergence of the Allen-Cahn version of the regularized phase-field model toward the sharp-interface theory for strongly anisotropic motion-by-curvature in three dimensions. Corners at equilibrium are also compared to theory for different corner sizes. Then we investigate the dynamics of the faceting instability, when initially unstable surfaces decompose into stable facets. For crystal surfaces with trigonal symmetry, we find the following scaling law L∼t^{1/3}, for the growth in time t of a characteristic morphological length scale L, and coarsening is found to proceed by either edge contraction or cube removal, as in the sharp-interface problem. Finally, we study nucleation of crystal surfaces in a two-phase system, as for a terrace-and-step surface. We find that, as compared with saddle-point nucleation, ridge crossing is dynamically favored. However, the induced nucleation mechanism, when a facet induces at its wake formation of additional facets, is not evidenced with a type-A dynamics.

A regularized phase-field model for faceting in a kinetically controlled crystal growth

At equilibrium, the shape of a strongly anisotropic crystal exhibits corners when for some orientations the surface stiffness is negative. In the sharp-interface problem, the surface free energy is traditionally augmented with a curvature-dependent term in order to round the corners and regularize the dynamic equations that describe the motion of such interfaces. In this paper, we adopt a diffuse interface description and present a phase-field model for strongly anisotropic crystals that is regularized using an approximation of the Willmore energy. The Allen-Cahn equation is employed to model kinetically controlled crystal growth. Using the method of matched asymptotic expansions, it is shown that the model converges to the sharp-interface theory proposed by Herring. Then, the stress tensor is used to derive the force acting on the diffuse interface and to examine the properties of a corner at equilibrium. Finally, the coarsening dynamics of the faceting instability during growth is investigated. Phase-field simulations reveal the existence of a parabolic regime, with the mean facet length evolving in t , with t the time, as predicted by the sharp-interface theory. A specific coarsening mechanism is observed: a hill disappears as the two neighbouring valleys merge.

Enhancing resolution in coherent x-ray diffraction imaging

Achieving a resolution near 1 nm is a critical issue in coherent x-ray diffraction imaging (CDI) for applications in materials and biology. Albeit with various advantages of CDI based on synchrotrons and newly developed x-ray free electron lasers, its applications would be limited without improving resolution well below 10 nm. Here, we review the issues and efforts in improving CDI resolution including various methods for resolution determination. Enhancing diffraction signal at large diffraction angles, with the aid of interference between neighboring strong scatterers or templates, is reviewed and discussed in terms of increasing signal-to-noise ratio. In addition, we discuss errors in image reconstruction algorithms-caused by the discreteness of the Fourier transformations involved-which degrade the spatial resolution, and suggest ways to correct them. We expect this review to be useful for applications of CDI in imaging weakly scattering soft matters using coherent x-ray sources including x-ray free electron lasers.

Emergent parabolic scaling of nano-faceting crystal growth

Nano-faceted crystals answer the call for self-assembled, physico-chemically tailored materials, with those arising from a kinetically mediated response to free-energy disequilibria (thermokinetics) holding the greatest promise. The dynamics of slightly undercooled crystal–melt interfaces possessing strongly anisotropic and curvature-dependent surface energy and evolving under attachment–detachment limited kinetics offer a model system for the study ofthermokineticeffects. The fundamental non-equilibrium feature of this dynamics is explicated through our discovery of one-dimensional convex and concave translating fronts (solitons) whose constant asymptotic angles provably deviate from the thermodynamically expectedWulffangles in direct proportion to the degree of undercooling. Thesethermokineticsolitons induce a novel emergent facet dynamics, which is exactly characterized via an original geometric matched-asymptotic analysis. We thereby discover an emergent parabolic symmetry of its coarsening facet ensembles, which naturally implies the universal scaling lawL∼t1/2for the growth in timetof the characteristic lengthL.

Nonlinear wavelength selection in surface faceting under electromigration

We report on the control of the faceting of crystal surfaces by means of surface electromigration. When electromigration reinforces the faceting instability, we find perpetual coarsening with a wavelength increasing as t(1/2). For strongly stabilizing electromigration, the surface is stable. For weakly stabilizing electromigration, a cellular pattern is obtained, with a nonlinearly selected wavelength. The selection mechanism is not caused by an instability of steady states, as suggested by previous works in the literature. Instead, the dynamics is found to exhibit coarsening before reaching a continuous family of stable nonequilibrium steady states.

One-dimensional to three-dimensional ripple-to-dome transition for SiGe on vicinal Si (1 1 10)

SiGe heteroepitaxy on vicinal Si (1 1 10) is studied as a model system for one-dimensional (1D) to three-dimensional growth mode transitions. By in situ scanning tunneling microscopy it is shown that the 1D-3D transition proceeds smoothly from perfectly facetted 1D nanoripples to coarsened superripples, tadpoles, asymmetric domes, and barns without involving coalescence or agglomeration. By extension of the studies to a wide range of SiGe compositions, a 1D-3D growth phase diagram is obtained. Total energy calculations reveal that the observed critical transition volumes are fully consistent with thermodynamic driven strain relaxation.