Kinetic Model of Stress Evolution during Coalescence and Growth of Polycrystalline Thin Films

We outline a simple continuum model of the stresses that result from the coalescence and growth of islands during deposition of a polycrystalline thin film. Our model includes a detailed description of attractive forces between neighboring islands, and also accounts for mass transport along surfaces and grain boundaries. The finite element method is used to calculate the island shape changes as well as the stresses and displacements in the film during the growth process. The model reproduces several experimental observations, including the variation of stress with film thickness, the range of observed growth stresses, and the effects of deposition flux and grain boundary diffusivity on stress.

Figure 1

Representative geometry of an array of evolving islands. Because of the symmetry of the system, attention is confined to the region $0<x_1<L$. The color scale is irrelevant here.

Figure 2

Representative set of normalized film force histories [Eq. (1)] for varying normalized growth flux $J$.

Figure 3

Representative behavior of normalized steady state instantaneous stress [Eq. (1)] with normalized growth flux [Eq. (8)] for several values of diffusivity ratio.

Figure 4

Schematic representing variation of chemical potential ($\mu /\Omega$, solid arrows) near the triple junction. Also shown in both figures are the contributions to the chemical potential from the normal stress (${\sigma }_n$, dashed lines) and curvature (shown as $-{\gamma }_s\kappa$, dotted lines). (a) Shows the equilibrium configuration of the triple junction. (b) Corresponds to an identical system in out-of-equilibrium conditions with $J\sim {10}^3$. Both cases have $w_s{D}_s/w_g{D}_g=1$. As shown, $\mu /\Omega$ is the difference between $-{\gamma }_s\kappa$ and ${\sigma }_n$.