Model for stress and volume changes of a thin film on a substrate upon annealing: Application to amorphous Mo/Si multilayers

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Abstract
A model is given for the stress change in a thin film on a thick substrate during annealing. It takes into account temperature changes, volume changes, viscous flow, and anelastic relaxation occurring in the film. The stress change in amorphous Mo/Si multilayer films deposited on Si single-crystal substrates was obtained from in situ wafer-curvature measurements during annealing at temperatures below the glass temperature. The thickness change and the interdiffusion coefficient were obtained from the position and the intensity of the first-order multilayer x-ray reflection. The unconstrained volume change was derived from the measured stress and thickness changes. The free-volume model for amorphous solids has been used to interpret the interdiffusion in and the volume change of the amorphous Mo/Si multilayers. The stress change as it occurred during isothermal annealing was explained by free-volume annihilation, viscous flow, and anelastic relaxation. If anisotropy of the volume change is accepted, the stress change could also be described with free-volume annihilation and viscous flow alone. The product of the experimentally observed viscosity and diffusion coefficient for amorphous Mo/Si multilayers was compared to the value expected from the free-volume-model-based equivalent of the Stokes-Einstein relation.