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We investigate the performance of microwave-frequency phononic crystalresonators fabricated on thin-film lithium niobate for integration withsuperconducting quantum circuits. For different design geometries atmillikelvin temperatures, we achieve mechanical internal quality factors $Q_i$above $10^5 - 10^6$ at high microwave drive power, corresponding to$5\times10^6$ phonons inside the resonator. By sweeping the defect size ofresonators with identical mirror cell designs, we are able to indirectlyobserve signatures of the complete phononic bandgap via the resonators'internal quality factors. Examination of quality factors' temperaturedependence shows how superconducting and two-level system (TLS) loss channelsimpact device performance. Finally, we observe an anomalous low-temperaturefrequency shift consistent with resonant TLS decay and find that materialchoice can help to mitigate these losses.

Loss channels affecting lithium niobate phononic crystal resonators at cryogenic temperature