Continuum Relaxation of Interacting Steps on Crystal Surfaces in $2+1$ Dimensions

Abstract

Abstract. We study the relaxation of crystal surfaces in 2+1 dimensions via the motion of interacting atomic steps. The goal is the rigorous derivation of the continuum limit. The starting point is a discrete scheme from the Burton, Cabrera, and Frank model, which accounts for diffusion of point defects ("adatoms") on terraces and attachment-detachment of atoms at step edges. It is shown that the macroscopic adatom current involves a tensor mobility, which describes different fluxes in directions parallel and transverse to step edges, although the physics of each terrace is assumed isotropic. When the steps are everywhere parallel (straight or circular) the tensor character of the mobility is unimportant; in the general (2+1)-dimensional setting, however, it is crucial. Our methods consist of (i) the solution of the diffusion equation for adatoms via the separation of local space variables into "fast" and "slow," and (ii) the treatment of the step chemical potential for a wide class of step energies and repulsive interactions. Previous works using similar methods were mainly restricted to (1+1)-dimensional or axisymmetric geometries and entirely missed the tensor character of the mobility. The continuum limit of the step flow yields a fourth-order partial differential equation for the surface height profile.

Continuum model for the long-range elastic interaction on stepped epitaxial surfaces in2+1dimensions

“…The step line energy and the elastic energy due to force dipole interaction are relatively well known in the continuum framework in the literature. 1,4, [31][32][33][34][35]37,[39][40][41][42]46 A stepped surface changes its morphology by motion, nucleation, and annihilation of steps, instead of changing continuously. Based on the Burton-Cabrera-Frank ͑BCF͒ theory, 17 the evolution equation is given by 30,[32][33][34][35]37,[39][40][41][42]46 …”

“…The anisotropic mobility tensor D on a stepped surface has been obtained by Margetis and Kohn. 42 In this paper, we focus on the long-range elastic effect and assume that the mobility D is isotropic which corresponds to the case of no barriers for the adatom attachment to the steps. 42 Note that our continuum model is derived from a discrete model based on the simplifications of elastic isotropy and same elastic constants in the film and the substrate.…”

“…34 The anisotropic mobility due to step edge barriers in 2 + 1 dimensions was obtained by Margetis and Kohn. 42 The kinetic instabilities and step edge barriers are neglected in this paper and will be considered in the future work.…”

“…1,4,23,36 The force dipole effect between curved steps on unstrained epitaxial surfaces are relatively well modeled in the frameworks of both the discrete step dynamics 37,38 and the continuum model. 32,37,[39][40][41][42] In the framework of the continuum theory, it is well known that the elastic effect on such a stepped surface of unstrained film is quite different from that on an epitaxial surface above roughening transition temperature due to the discrete structures on the stepped surface.…”

Singular Neumann problems and large-time behavior of solutions of noncoercive Hamilton-Jacobi equations

“…(This type of derivation is rigorously justified by W. E and N. K. Yip [13] and studies on relation between models via step-motion and via macroscopic partial differential equations are still an active research field in various settings; see e.g. [20].) In a macroscopic model the dimensionless growth speed V in the direction normal to a crystal surface Γ t is generally expressed using the surface supersaturation σ(≥ 0) and the dimensionless kinetic coefficient M (p) depending on the modulus p of a local slope (gradient) of the crystal surface as in [8] and [11] of the form…”