  We study the mechanical quality factors of bilayer aluminum/silicon-nitride
membranes. By coating ultrahigh-Q Si3N4 membranes with a more lossy metal, we
can precisely measure the effect of material loss on Q's of tensioned resonator
modes over a large range of frequencies. We develop a theoretical model that
interprets our results and predicts the damping can be reduced significantly by
patterning the metal film. Using such patterning, we fabricate Al-Si3N4
membranes with ultrahigh Q at room temperature. Our work elucidates the role of
material loss in the Q of membrane resonators and informs the design of hybrid
mechanical oscillators for optical-electrical-mechanical quantum interfaces.
