Resolution Limits of Resonant Sensors

Resonant sensors hold great promise in measuring small masses, to enable future mass spectrometers, and small forces in applications like atomic and magnetic force microscopy. During the last decades, scaling down the size of resonators has led to huge enhancements in sensing resolution, but has also raised the question of what the ultimate limit is. Current knowledge suggests that this limit is reached when a resonator oscillates at the maximum amplitude for which its response is predominantly linear. We present experimental evidence that it is possible to obtain better resolutions by oscillation amplitudes beyond the onset of nonlinearities. An analytical model is developed that explains the observations and unravels the relation between ultimate sensing resolution and speed. In the high-speed limit, we find that the ultimate resolution of a resonator is improved when decreasing its damping. This conclusion contrasts with previous works, which proposed that lowering the damping does not affect or even harms the ultimate sensing resolution.

Figure 1

Log-log plots of the minimum Allan deviation that can be obtained by a thermomechanically limited Duffing resonator when optimizing the actuation power, as a function of the integration time $\tau$. Each curve within the same plot corresponds to a different value of the quality factor. (a) Limit found by Roy et al. [8] by assuming that the minimum Allan deviation is reached for actuation at the critical amplitude. (b) Limit found by Olcum et al. [9] for high quality factor resonators, and by Demir and Hanay [7] for any quality factor but assuming a closed-loop operation. Both works use the assumption that the minimum Allan deviation is reached at or close to the critical amplitude. (c) Limit for Duffing resonators under closed-loop operation, by using the optimum actuation amplitude found in this work, that depends on the integration time. For short integration times, the minimum Allan deviation is independent of the integration time. For long integration times, the minimum Allan deviation shows a ${\tau }^{-1/2}$ dependence and matches the result in (b). The transition between both regimes occurs at ${\tau }_c=\sqrt{3}Q/(\pi f_0$).

Figure 2

(a) Top-view optical picture of the ${\mathrm{Si}}_3{\mathrm{N}}_4$ membrane used in the experiments. (b) Cross-section representation of the chip containing the membrane in the experimental setup. (c) Measured noise spectrum in the absence of actuation. The peak corresponds to the thermomechanical noise of the membrane’s fundamental resonance. (d) The solid lines represent experimental upward frequency sweeps around the fundamental resonance for different actuation levels. For low levels, the membrane behaves as a linear resonator. For 4.4 nN and above, a stiffening effect is observed that produces bifurcation for 11 nN and above. The dotted lines represent fittings to the Duffing model for each actuation level, using the values in Table 1. The backbone curve described by Eq. (1) is plotted for reference. The critical amplitude calculated from $\gamma$ and $Q$ [13], $a_{\mathrm{crit}}=0.59\text{μ}\mathrm{m}$, is also indicated. (e) Dimensionless Allan deviation for the same actuation levels as in (d), when the membrane is driven by a PLL controller at the displacement maxima. The solid lines represent measurements, where white force noise with PSD $S_0=2.5\times {10}^{-21}{\mathrm{N}}^2{\mathrm{Hz}}^{-1}$ is added to the actuation to ensure that the response is dominated by white force noise. This force noise resembles the thermomechanical noise the resonator would experience at temperatures higher than room temperature. The dotted colored lines represent the theoretical values for the membrane assuming a linear response under the experimental conditions. The dotted black line indicates the previously known limit for the Allan deviation, defined by actuation at the critical amplitude. The horizontal dashed line indicates the lower boundary for the Allan deviation found by our model at integration times below ${\tau }_c$.

Figure 3

(a) Laplace-domain block diagram of the linearized small-signal model of the resonator and the resonance-tracking control loop. The closed-loop system matrix $J(s)$ governs the conversion of amplitude and phase perturbations at the input (force) into amplitude and phase perturbations at the output (displacement). We highlight blue the input and output magnitudes used in the analysis. Amplitude and phase random perturbations due to thermomechanical noise, with PSDs $S_{a|\mathrm{thm}}$ and $S_{\varphi |\mathrm{thm}}$, respectively, translate into phase perturbations at the output with PSD $S_{\varphi x}$, from which the Allan deviation (${\sigma }_y$) is calculated. (b) Log-log plot of the Allan deviation predicted by the model in (a) for a Duffing resonator at its thermomechanical limit oscillating at $\varphi =-\pi /2$. Each curve represents a different amplitude of oscillation. A characteristic minimum ${\sigma }_{y|\min }$ is reached at an integration time ${\tau }_c/\chi$ that depends on the amplitude of oscillation. For $\tau \ll {\tau }_c/\chi$, the Allan deviation is governed by the linear response and the Duffing term can be neglected ($\gamma =0$).

Figure 4

(a) Factor $\chi$ accounting for the excess Allan deviation at $\tau \gg {\tau }_c$ due to the amplitude-phase noise conversion, with the phase shift $\varphi$ in the horizontal axis. Each curve, obtained from Eq. (23), represents a different value of the displacement amplitude normalized to the critical value ($\hat{a}$), indicated by labels. (b) Factor $\chi$ for a resonator operated at $\varphi =-\pi /2$, with $\hat{a}$ in the horizontal axis. For low values of $\hat{a}$, a Duffing resonator behaves as a linear resonator, with $\chi \approx 1$. (c) Force needed to set a given value of $\hat{a}$ with $\varphi ={\varphi }^{\ast }$, divided by the force needed to set the same value of $\hat{a}$ with $\varphi =-\pi /2$. This curve is obtained from Eq. (10).

Figure 5

Simulated Allan deviation for a Duffing resonator embedded in a direct feedback oscillator. The resonator parameters are $f_0=1\mathrm{Hz}$, $k_1=1{\text{N m}}^{-1}$, $\gamma =1{\mathrm{m}}^{-2}$. The temperature is assumed to be 300 K. (a),(b) Each curve represents the Allan deviation as a function of the integration time for a different displacement amplitude in the range from ${10}^{-4}$ to ${10}^{-1}\mathrm{m}$ for $\varphi =-\pi /2$ and a quality factor of (a) ${10}^3$ and (b) ${10}^4$. The minimum Allan deviation found by the model is indicated for each case by black dashed lines. (c),(d) Color maps based on the same simulations. The Allan deviation is represented as a function of the displacement amplitude and integration time for a quality factor of (c) ${10}^3$ and (d) ${10}^4$. The white solid lines (smoothed) mark the displacement amplitude that minimizes the Allan deviation for each integration time simulated. The black dashed lines indicate the optimal displacement predicted by the model.