  We present techniques to model and design membrane phononic crystals with
low-mass defects, optimized for force sensing. Further, we identify the
importance of the phononic crystal mass contrast as it pertains to the size of
acoustic bandgaps and to the dissipation properties of defect modes. In
particular, we quantify the tradeoff between high mass contrast phononic
crystals with their associated robust acoustic isolation, and a reduction of
soft clamping of the defect mode. We fabricate a set of phononic crystals with
a variety of defect geometries out of high stress stoichiometric silicon
nitride membranes, and measured at both room temperature and 4 K in order to
characterize the dissipative pathways across a variety of geometries. Analysis
of these devices highlights a number of design principles integral to the
implementation of low-mass, low-dissipation mechanical modes into
optomechanical systems.
