Intrinsic dissipation in nanomechanical resonators due to phonon tunneling

State of the art nanomechanical resonators present quality factors $Q\sim {10}^3–{10}^5$, which are much lower than those that can be naively extrapolated from the behavior of micromechanical resonators. We analyze the dissipation mechanism that arises in nanomechanical beam structures due to the tunneling of mesoscopic phonons between the beam and its supports (known as clamping losses). We derive the environmental force spectral density that determines the quantum Brownian motion of a given resonance. Our treatment is valid for low frequencies and provides the leading contribution in the aspect ratio. This yields fundamental limits for the $Q$ values, which are described by simple scaling laws and are relevant for state of the art experimental structures. In this context, for resonant frequencies in the $0.1–1\mathrm{GHz}$ range, while this dissipation mechanism can limit flexural resonators, it is found to be negligible for torsional ones. In the case of structureless three-dimensional supports, the corresponding environmental spectral densities are Ohmic for flexural resonators and super-Ohmic for torsional ones, while for two-dimensional slab supports, they yield $1∕f$ noise. Furthermore, analogous results are established for the case of suspended semiconducting single-walled carbon nanotubes. Finally, we provide a general expression for the spectral density that allows us to extend our treatment to other geometries and illustrate its use by applying it to a microtoroid. Our analysis is relevant for applications in high precision measurements and for the prospects of probing quantum effects in a macroscopic mechanical degree of freedom.

Figure 1

(Color online) Schematic diagram of suspended mechanical beam structures presenting abrupt junctions with the supports. Throughout the paper, the $z$ axis corresponds to the beam’s axis, while the $x$ and $y$ axes are the principal axes of inertia of the beam’s cross section at the left junction (the origin is set at the center of mass of the latter).

Figure 2

(Color online) Schematic diagram of a microtoroid structure with largest diameter $D$ and membrane thickness $h$ (the origin is set at the center of the contact area $S$ with the pedestal).

Figure 3

Dimensionless displacements ${\tilde{u}}_c$ (dashed line), ${\tilde{u}}_{v∕h}$ (solid line), and dimensionless angle ${\tilde{u}}_t^{\left(S\right)}$ (dotted line) as a function of the ratio $\alpha \equiv {(c_t∕c_l)}^2$. For typical materials of interest, ${\sigma }_s\approx 1∕3$, which yields ${\tilde{u}}_c(1∕4)=0.13$, ${\tilde{u}}_t^{\left(S\right)}(1∕4)=0.019$, and ${\tilde{u}}_{v∕h}(1∕4)=0.12$.