For quantum computing to become fault tolerant, the underlying quantum bits must be effectively isolated from the noisy environment. It is well known that including an electromagnetic bandgap around the qubit operating frequency improves coherence for superconducting circuits. However, investigations of bandgaps to other environmental coupling mechanisms remain largely unexplored. Here we present a method to enhance the coherence of superconducting circuits by introducing a phononic bandgap around the device operating frequency. The phononic bandgaps block resonant decay of defect states within the gapped frequency range, removing the electromagnetic coupling to phonons at the gap frequencies. We construct a multi-scale model that derives the decrease in the density of states due to the bandgap and the resulting increase in defect state $T_1$ times. We demonstrate that emission rates from in-plane defect states can be suppressed by up to two orders of magnitude. We combine these simulations with theory for resonators operated in the continuous-wave regime and show that improvements in quality factors are expected by up to the enhancement in defect $T_1$ times. Furthermore, we use full master equation simulation to demonstrate the suppression of qubit energy relaxation even when interacting with 200 defects states. We conclude with an exploration of device implementation including tradeoffs between fabrication complexity and qubit performance.
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