Figure 3. Limiting values for the f × Q product of BAW resonators with phonon-electron interaction.

(a) The f × Q limits for the various intrinsic loss mechanisms in PS GaN. For low to moderately doped PS GaN, the phonon-electron mechanism is the significant limiting factor, lower than the anharmonic phonon-scattering (Akhieser and Landau-Rumer losses) and thermoelastic damping (TED) loss. (b) For a representative frequency, (here, 1.5 GHz), the piezoelectrically coupled model for phonon-electron interaction is significant only for a range of values of moderate doping. Beyond this range, other interactions dominate. The f × Q limit due to phonon-electron interaction is highly dependent on the effective electron concentration. The two important mechanisms of phonon-electron interaction are (c) piezoelectric coupling and (d) deformation-potential (DP) coupling. At low to moderate doping, the piezoelectric interaction dominates, and the two loss mechanisms achieve parity only at a concentration of ~1 × 10¹⁸ cm⁻³ for GaN. Above this value, the deformation-potential coupling will dominate the phonon-electron loss. Deformation-potential coupling is more significant for weakly piezoelectric or non-piezoelectric semiconductors.