Electrode configuration and electrical dissipation of mechanical energy in quartz crystal resonators

Mechanical resonators made with monolithic piezoelectric quartz crystals are promising for studying new physical phenomena. High mechanical quality factors ($Q$) exhibited by the mm-sized quartz resonators make them ideal for studying weak couplings or long timescales in the quantum regime. However, energy losses through mechanical supports pose a serious limiting factor for obtaining high quality factors. Here we investigate how the $Q$ of quartz resonators at deep cryogenic temperatures can be limited by several types of losses related to anchoring. We first introduce means to reduce the mechanical losses by more than an order of magnitude in a no-clamping scheme, obtaining $Q$-factors of $10^8$ of the lowest shear mode. We can exclude a wide coverage of aluminum metallization on the disk or bond wires as sources of dissipation. However, we find a dramatic reduction of the $Q$-factor accompanying an electrode configuration that involves strong focusing of the vibrations in the disk center. We propose a circuit model that accounts for the reduced mechanical $Q$-factor in terms of electrical losses. In particular, we show how the limiting factor for losses can be small ohmic dissipation in a grounding connection, which can be interpreted as electrical anchor losses of the mechanical device.