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Current-Induced Stabilization of Surface Morphology in Stressed Solids

“…1 for the functional form f 1 Acos 2 m for a facecentered cubic metal such as Al or Cu; the dimensionless parameters A, m (an integer parameter), and determine the strength of the anisotropy, symmetry due to surface crystallographic orientation, and misorientation of a fast surface diffusion direction with respect to the applied electric-field direction, respectively. This anisotropy function has given results in excellent qualitative agreement with experiments of void morphological evolution in metallic thin films under EM conditions [10,15]. The results of Fig.…”

“…For example, it has been demonstrated that a planar surface of a stressed elastic solid can evolve rapidly into a cusped surface, with smooth tops and deep cracklike grooves by surface diffusion [6], in agreement with experimental observations [7]. Moreover, dynamical simulations of electromigration (EM)-driven void morphological evolution in stressed metallic thin films demonstrated the formation of cracklike features emanating from void surfaces [8][9][10], including propagating faceted slits [9,10], due to an ATG instability. Recent theoretical work also addressed problems of EM-induced surface morphological evolution and stability of unstressed conductors and predicted the formation of various interesting surface features, including surface wave patterns [11,12], and complex oscillatory response of single-layer islands on crystalline substrates [13].…”

“…(3) expressed in the local surface coordinates, according to the interface tracking method of Refs. [10,15]. In this notation,n ŝ 0 and u n is the local surface displacement normal to the surface.…”

“…The mathematical formulation of the boundary-value problems for and u has been presented in detail in Ref. [10] and can be easily adapted to the present computational domain. At a given set of parameters ( , A, m, , k), hx;t is monitored forhx; 0 ~0 sinkx with~0k 1.…”

“…(2), 0 is the chemical potential for a flat unstressed surface, is the isotropic [10] surface free energy per unit area, M is the elastic modulus, is the stress component tangential to the surface, and is the local surface curvature: parameterizing the surface morphology with the height function y h x; t gives…”