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Dielectric membranes with exceptional mechanical and optical propertiespresent one of the most promising platforms in quantum opto-mechanics. Theperformance of stressed silicon nitride nanomembranes as mechanical resonatorsnotoriously depends on how their frame is clamped to the sample mount, which inpractice usually necessitates delicate, and difficult-to-reproduce mountingsolutions. Here, we demonstrate that a phononic bandgap shield integrated inthe membrane's silicon frame eliminates this dependence, by suppressingdissipation through phonon tunneling. We dry-etch the membrane's frame so thatit assumes the form of a $\mathrm{cm}$-sized bridge featuring a 1-dimensionalperiodic pattern, whose phononic density of states is tailored to exhibit one,or several, full band gaps around the membrane's high-$Q$ modes in theMHz-range. We quantify the effectiveness of this phononic bandgap shield byoptical interferometry measuring both the suppressed transmission ofvibrations, as well as the influence of frame clamping conditions on themembrane modes. We find suppressions up to $40~\mathrm{dB}$ and, for threedifferent realized phononic structures, consistently observe significantsuppression of the dependence of the membrane's modes on sample clamping - ifthe mode's frequency lies in the bandgap. As a result, we achieve membrane modequality factors of $5\times 10^{6}$ with samples that are tightly bolted to the$8~\mathrm{K}$-cold finger of a cryostat. $Q\times f$-products of $6\times10^{12}~\mathrm{Hz}$ at $300~\mathrm{K}$ and $14\times 10^{12}~\mathrm{Hz}$ at$8~\mathrm{K}$ are observed, satisfying one of the main requirements foroptical cooling of mechanical vibrations to their quantum ground-state.

Demonstration of suppressed phonon tunneling losses in phononic bandgap shielded membrane resonators for high-Q optomechanics