We developed a lattice dynamical theory of an atomically thin compressional piezoelectric resonator, based on a two-dimensional (2D) ionic crystal with $D_{3h}$ point group symmetry and finite lateral size. Starting from an anharmonic vibrational Hamiltonian, we derive in-plane acoustic and optical displacement propagators using linear response theory, incorporating symmetry-based selection rules. These propagators satisfy Dyson equations with polarization operators dependent on frequency, temperature, and crystal size. The dynamic piezoelectric susceptibilities governing the direct and converse piezoelectric effects are shown to be equal. Resonant behavior of in-plane longitudinal and transverse acoustic waves is analyzed in both classical and quantum regimes. In the quantum limit near zero temperature, resonance broadenings scale inversely with crystal size. Below a crossover temperature, the quantum zero-point fluctuations become dominant and put an upper limit on the quality factor which is size independent. In the classical regime, the broadenings scale with temperature and the quality factor decreases inversely with crystal size. Scaling relations for temperature, size, and quality factor are established. Anharmonic scattering is shown to be dominated by out-of-plane flexural modes. The theory is showcased numerically on 2D hexagonal boron nitride (h-BN) and $\mathrm{2H}\text{−}{\mathrm{MoS}}_2$. The results suggest that h-BN, despite a smaller static piezoelectric coefficient, is a more robust candidate for resonator applications due to its higher in-plane rigidity. The framework applies broadly to 2D piezoelectric crystals of $D3h$ symmetry, including transition-metal dichalcogenides and dioxides.

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• Electric polarization
• Piezoelectricity
• Nano-oscillators
• Nanomechanical devices