Radiation and Internal Loss Engineering of High-Stress Silicon Nitride Nanobeams

High-stress Si₃N₄ nanoresonators have become an attractive choice for electro- and optomechanical devices. Membrane resonators can achieve quality factor (Q)-frequency (f) products exceeding 10¹³ Hz, enabling (in principle) quantum coherent operation at room temperature. String-like beam resonators possess smaller Q × f products; however, on account of their significantly lower mass and mode density, they remain a canonical choice for precision force, mass, and charge sensing, and have recently enabled Heisenberg-limited position measurements at cryogenic temperatures. Here we explore two techniques to enhance the Q of a nanomechanical beam. The techniques relate to two main loss mechanisms: internal loss, which dominates for high aspect ratios and f ≲ 100 MHz, and radiation loss, which dominates for low aspect ratios and f ≳ 100 MHz. First, we show that by embedding a nanobeam in a 1D phononic crystal (PnC), it is possible to localize its flexural motion and shield it against radiation loss. Using this method, we realize f > 100 MHz modes with Q ≈ 10⁴, consistent with internal loss and contrasting sharply with unshielded beams of similar dimensions. We then study the Q × f product of high-order modes of millimeter-long nanobeams. Taking advantage of the mode-shape dependence of stress-induced "loss dilution", we realize a f ≈ 4 MHz mode with Q × f ≈ 9 × 10¹² Hz. Our results complement recent work on PnC-based "soft-clamping" of nanomembranes, in which mode localization is used to enhance loss dilution. Combining these strategies should enable ultra-low-mass nanobeam oscillators that operate deep in the quantum coherent regime at room temperature.