Thermodlastic analysis of periodic thin lines deposited on a substrate

摘要

Thermoelastic stresses and curvatures arising from patterned thin lines on initially flat isotropic substrates are analyzed. A connection is made between substrates with patterned lines and laminated anisotropy composites containing transverse matrix cracks. Using this analogy along with anisotropy plate theories, approximate analytical expressions are derived for volume-averaged stresses as well as curvatures along and normal to the lines, for any thickness, width and spacing of the lines. The predictions of the analysis are shown to compare favorably with finite element simulations of stresses and curvatures for Si substrates with Al, Cu or SiO_2 lines. The predictions also match prior experimental measurements of curvatures along and normal to patterned SiO_2 lines on Is wafers, and further capture the general experimental trends reported previously for curvature evolutions in Si wafers with Al lines. The model presented here thus provides a very convenient and simple analytical tool for extracting stresses in thin lines on substrates from a knowledge of experimentally determined film stress, thereby circumventing the needs for detailed computations for a wide range of unpassivated line geometries of interest in microelectronic applications.