This work was supported by the Knut and Alice Wallenberg (KAW) Foundation through a Wallenberg Academy Fellowship (W.W.), the KAW project no. 2022.0090, and the Wallenberg Center for Quantum Technology (WACQT, A.C.). H.P. acknowledges funding by the European Union under the project MSCA-PF-2022-OCOMM. MOVPE of AlN on Si was performed at Otto-von-Guericke-University Magdeburg. The mechanical resonators were fabricated in the Myfab Nanofabrication Laboratory at Chalmers. 

Data used for figures in "Membrane phononic crystals for high-Q_m mechanical defect modes in piezoelectric aluminum nitride" by Anastasiia Ciers et.al