Nano-beam clamping revisited

Within recent years, the field of nano-mechanics has diversified in a variety applications, ranging from quantum information processing to biological molecules recognition. Among diversity devices produced these days, simplest (but versatile) element remains doubly-clamped beam: it can store very large tensile stresses (producing high resonance frequencies $f_0$ and quality factors $Q$), is interfaceable with electric setups (by means conductive layers), be easily clean rooms (with scalable designs including multiplexing). Besides, its mechanical properties are describe. Resonance $Q$s being modeled, as specific achievement ultra-high resonances based on ``soft clamping'' ``phonon shields''. Here, we demonstrate that fabrication undercut clamping regions basic nano-beams produces ``natural soft clamping'', given for free. We present analytic theory enables fit experimental data, which used $\{ Q , f_0 \}$ design: beyond Finite Element Modeling validation, presented expressions provide profound understanding phenomenon, both enhancement downwards frequency shift.