Modeling and Observation of Nonlinear Damping in Dissipation-Diluted Nanomechanical Resonators

Dissipation dilution enables extremely low linear loss in stressed, high aspect ratio nanomechanical resonators, such as strings or membranes. Here, we report on the observation and theoretical modeling of nonlinear dissipation in such structures. We introduce an analytical model based on von Kármán theory, which can be numerically evaluated using finite-element models for arbitrary geometries. We use this approach to predict nonlinear loss and (Duffing) frequency shift in ultracoherent phononic membrane resonators. A set of systematic measurements with silicon nitride membranes shows good agreement with the model for low-order soft-clamped modes. Our analysis also reveals quantitative connections between these nonlinearities and dissipation dilution. This is of interest for future device design and can provide important insight when diagnosing the performance of dissipation dilution in an experimental setting.

Figure 1

(a) Optical interferometer for displacement measurement [acousto-optic modulator (AOM), piezoelectric actuator (PZT)]. The detection is realized with a heterodyne receiver. (b) Soft-clamped membrane pattern. (c) Nonlinear amplitude decay for 19-nm-thick membrane and corresponding fit. Linear exponential decay (gray) is extrapolated to highlight deviation arising from nonlinear damping. (d) Duffing frequency shift as a function of displacement amplitude and corresponding fit. The abscissa is the fit result from (c).

Figure 2

Nonlinear parameters. (a)–(d) Measured nonlinear parameters as a function of membrane thickness $h$. Blue (green) points are the Duffing (nonlinear damping) parameters ${\omega }_i^{\text{sD}}$ (${\gamma }_i^{\mathrm{nl}}$), estimated from the median of each statistical ensemble. Dashed lines represent simulated values for the Duffing (blue) and nonlinear damping (green) parameters.

Figure 3

Nonlinear loss angles. (a)–(d) Measured quality factors against ${\theta }_{\mathrm{nl}}^{-1}$. The gray line is the expected ${\theta }_{\mathrm{lin}}^{-1}$ [35] and gray area is the uncertainty in that value. The black line is the limit $Q_{\mathrm{meas}}\le D_Q{\theta }_{\mathrm{lin}}^{-1}$ with a dissipation dilution factor $D_Q$ obtained from simulations. The hatched area is inaccessible under the most obvious sources of excess dissipation. (e)–(h) Measured ${\theta }_{\mathrm{nl}}^{-1}$ as a function of the membrane thickness. The stars represent the median of the points shown in (a)–(d). The gray line is the ${\theta }_{\mathrm{lin}}^{-1}$ as a function of thickness, expressed in Eq. (12), whereas the gray area reflects the uncertainty in the parameters.

Figure 4

Nonlinear loss angles at different temperatures. (a) Measured ${\theta }_{\text{nl}}^{-1}$ as a function of the cryostat temperature $T_{\mathrm{mxc}}$. The gray line is a polynomial fit, roughly showing the behavior. (b) Measured quality factors versus ${\theta }_{\text{nl}}^{-1}$, taken at room temperature (orange) and cryogenic temperatures (blue). The gray line is the expectation value of ${\theta }_{\mathrm{lin}}^{-1}$ at room temperature, and the gray area reflects uncertainty [35]. The black line is the simulated quality factor, from the dissipation dilution factor $D_{Q_1}$. Error bars are the mean absolute deviation among three repetitions.