Islands in the Stream: Electromigration-Driven Shape Evolution with Crystal Anisotropy

Abstract

We consider the shape evolution of two-dimensional islands on a crystal surface in the regime where mass transport is exclusively along the island edge. A directed mass current due to surface electromigration causes the island to migrate in the direction of the force. Stationary shapes in the presence of an anisotropic edge mobility can be computed analytically when the capillary effects of the line tension of the island edge are neglected, and conditions for the existence of non-singular stationary shapes can be formulated. In particular, we analyse the dependence of the direction of island migration on the relative orientation of the electric field to the crystal anisotropy, and we show that no stationary shapes exist when the number of symmetry axes is odd. The full problem including line tension is solved by time-dependent numerical integration of the sharp-interface model. In addition to stationary shapes and shape instability leading to island breakup, we also find a regime where the shape displays periodic oscillations. (2000). Primary 53C44; Secondary 82C24. Mathematics Subject Classification

Complex Shape Evolution of Electromigration-Driven Single-Layer Islands

“…For small islands (R 0 ≤ 2) the evolution converges to a sta- tionary shape which moves in the direction of the field. For large S the shapes develop facets [22], similar to what has been observed for void electromigration [5]. Increasing the island radius the direction of migration starts to deviate from the field direction, and we enter the regime of oblique stationary (os) motion (Fig.2).…”

“…Beyond the instability a bifurcation to two branches of non-circular stationary solutions occurs [20]. Numerical integration of the time-dependent problem [22] shows that only one of the branches, corresponding to islands elongated in the field direction, is realized. At large radii island breakup occurs, mediated by the outgrowth of a finger of the kind found in [19].…”

“…The simplest solutions in the anisotropic case are stationary islands moving at constant speed, which satisfy the equation v n = V sin(θ + φ); here the angle φ accounts for the fact that the island does not necessarily move in the field direction. A complete analysis of stationary island shapes has been achieved in the limiting case of zero stiffness [22]. For an even number n of symmetry axes smooth stationary shapes are found for small S, while for larger anisotropy the shapes develop self-intersections; no stationary shapes exist when n is odd.…”

Nonlinear Dynamics of Surface Steps

“…Throughout this section the electromigration force is assumed to be constant in magnitude and direction. This implies that Beyond the linear instability of the circular solution one finds a family of stationary shapes which are elongated in the direction of the force and become increasingly sensitive to breakup with increasing size [15,16]. The effect of crystalline anisotropy in the mobility σ was explored, mostly numerically, in [15,17].…”

“…This implies that Beyond the linear instability of the circular solution one finds a family of stationary shapes which are elongated in the direction of the force and become increasingly sensitive to breakup with increasing size [15,16]. The effect of crystalline anisotropy in the mobility σ was explored, mostly numerically, in [15,17]. Using the expression [18] σ(θ) = σ 0 [1 + S cos 2 (nθ)], (1.5) where 2n denotes the number of symmetry axes, a surprisingly rich phase diagram of migration modes was obtained in the plane spanned by the anisotropy strength S and the dimensionless island radius R 0 = R/l E for the case of sixfold anisotropy (n = 3), see Fig.1.2.…”