Grain-boundary electromigration in thin films. I. Low-temperature theory

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Abstract
A Macroscopic mathematical theory is presented accounting for grain‐boundary diffusion and electromigration in the presence of a simultaneous flux of atoms into the surrounding bulk lattice. The model employs a semi‐infinite bicrystal geomettry with a constant source at the origin, and both integral and numerical solutions to the subsequent non‐steady‐state transport equations are given. ONly the case where V2/4Db‐DL/δ2≳0, which is ttrue at low temperatures, is considered in detail. (V, Db, and aδ are the grain‐boundary ion drift velocity, diffusivity, and width, respectively, while DL is the lattice diffusivity.). It is predicted that the time dependence of the displacement of a front of fixed concentration is linear only at short times and parabolic at longer times or with increasing amounts of atoms leakage into the bulk. In addition, the concentration profiles are predicted to have a plateau which becomes less apparent at higher temperatures. A comparison between the present theory and a previous treatment based on an extension of the Fisch analysis will be made. Application to recent results in thin films will be discussed.