  We present a highly reflective, sub-wavelength-thick membrane resonator
featuring high mechanical quality factor and discuss its applicability for
cavity optomechanics. The $88.5~\text{nm}$ thin stoichiometric silicon-nitride
membrane, designed and fabricated to combine 2D-photonic and phononic crystal
patterns, reaches reflectivities up to $99.89~\%$ and a mechanical quality
factor of $2.9 \times 10^7$ at room temperature. We construct a
Fabry-Perot-type optical cavity, with the membrane forming one terminating
mirror. The optical beam shape in cavity transmission shows a stark deviation
from a simple Gaussian mode-shape, consistent with theoretical predictions. We
demonstrate optomechanical sideband cooling to mK-mode temperatures, starting
from room temperature. At higher intracavity powers we observe an
optomechanically induced optical bistability. The demonstrated device has
potential to reach high cooperativities at low light levels desirable for e.g.
optomechanical sensing and squeezing applications or fundamental studies in
cavity quantum optomechanics, and meets the requirements for cooling to the
quantum ground state of mechanical motion from room temperature.
