Frequency fluctuations in nanomechanical silicon nitride string resonators

Open Access

Frequency fluctuations in nanomechanical silicon nitride string resonators

Pedram Sadeghi, Alper Demir, Luis Guillermo Villanueva, Hendrik Kähler, and Silvan Schmid

Phys. Rev. B 102, 214106 – Published 7 December 2020

Abstract

High quality factor ( $Q$ ) nanomechanical resonators have received a lot of attention for sensor applications with unprecedented sensitivity. Despite the large interest, few investigations into the frequency stability of high- $Q$ resonators have been reported. Such resonators are characterized by a linewidth significantly smaller than typically employed measurement bandwidths, which is the opposite regime to what is normally considered for sensors. Here, the frequency stability of high- $Q$ silicon nitride string resonators is investigated both in open-loop and closed-loop configurations. The stability is here characterized using the Allan deviation. For open-loop tracking, it is found that the Allan deviation gets separated into two regimes, one limited by the thermomechanical noise of the resonator and the other by the detection noise of the optical transduction system. The point of transition between the two regimes is the resonator response time, which can be shown to have a linear dependence on $Q$ . Laser power fluctuations from the optical readout are found to present a fundamental limit to the frequency stability. Finally, for closed-loop measurements, the response time is shown to no longer be intrinsically limited but instead given by the bandwidth of the closed-loop tracking system. Computed Allan deviations based on theory are given as well and found to agree well with the measurements. These results are of importance for the understanding of fundamental limitations of high- $Q$ resonators and their application as high performance sensors.

Received 25 September 2020
Revised 9 November 2020
Accepted 12 November 2020

DOI:https://doi.org/10.1103/PhysRevB.102.214106

Published by the American Physical Society under the terms of the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to the author(s) and the published article's title, journal citation, and DOI.

Physics Subject Headings (PhySH)

Mechanical & acoustical properties

Thin films

Optical interferometry

Condensed Matter, Materials & Applied Physics

Authors & Affiliations

Pedram Sadeghi ¹, Alper Demir ², Luis Guillermo Villanueva³, Hendrik Kähler ¹, and Silvan Schmid ^1,*

¹Institute of Sensor and Actuator Systems, TU Wien, 1040 Vienna, Austria
²Department of Electrical Engineering, Koç University, Istanbul 34450, Turkey
³Advanced NEMS Laboratory, École Polytechnique Fédérale de Lausanne (EPFL), CH-1015 Lausanne, Switzerland

^*silvan.schmid@tuwien.ac.at

Article Text

Click to Expand

References

Click to Expand

Issue

Vol. 102, Iss. 21 — 1 December 2020

Reuse & Permissions

Author publication services for translation and copyediting assistance advertisement

Images

Figure 1
(a) Schematic of the measurement setup. Mechanical vibrations of the string resonator are measured using a commercial laser-Doppler vibrometer. The sample is placed inside a vacuum chamber equipped with a needle valve for pressure control and resonances are actuated using a piezo. (b) Exemplary plot of the square root of the power spectral density ${PSD}^{1 / 2}$ versus frequency around the fundamental resonance, highlighting the two sources of noise: thermomechanical noise $N_{th}$ and detection noise $N_{d}$ . The resonance linewidth $Γ$ and the resolution bandwidth BW are shown as well. (c) Allan deviation curve showing the limits presented by both noise sources along with the resonator response time $τ_{r} = 1 / (π Γ) = Q / (π f_{0})$ . For longer integration times ( $τ > τ_{r}$ ), the Allan deviation is limited by thermomechanical noise (green dotted line) and thermal drift, while detection noise (red dashed line) limits the stability at shorter integration times ( $τ < τ_{r}$ ).
Reuse & Permissions
Figure 2
Effect of varying detection noise on the Allan deviation for open-loop tracking. Experimental Allan deviations were recorded by varying the vibrometer laser power $P$ between 1.5 $μ W$ to 270 $μ W$ , while setting the piezo driving voltage $U = 5 μ V$ (a), $U = 50 μ V$ (b), and $U = 500 μ V$ (c). Corresponding theoretical Allan deviations are shown in (d)–(f). Calculations using equation (4) are shown as well with $N_{l} = N_{th}$ (green dotted lines) and $N_{l} = N_{d}$ (red dashed lines). Black vertical dashed lines highlight $τ = τ_{r}$ . In (c), an Allan deviation curve corresponding to laser power fluctuations for $P = 270 μ W$ is given as well (blue dash-dotted curve), which is low-pass filtered during post-processing at a frequency corresponding to the resonator thermal time constant $τ_{th}$ (blue vertical dash-dotted line).
Reuse & Permissions
Figure 3
Effect of varying vibrational amplitude on the Allan deviation for open-loop tracking. (a) Measured Allan deviations for varying piezo driving voltages from $U = 5 μ V$ to $U = 1.5 mV$ . (b) Theory-based Allan deviations for the data presented in (a). Red dashed lines are detection noise limits calculated using equation (4). Black vertical dashed lines highlight $τ = τ_{r}$ .
Reuse & Permissions
Figure 4
$Q$ dependence of the frequency stability in open-loop configuration. Measured (left column) and theory based (right column) Allan deviations for resonance linewidths of 0.67 Hz (a), 3.59 Hz (b), 36.3 Hz (c), 363.6 Hz (d), and 1869 Hz (e). Green dotted lines and red dashed lines are calculations using equation (4) for thermomechanical and detection noise limits, respectively. Black vertical dashed lines highlight $τ = τ_{r}$ .
Reuse & Permissions
Figure 5
Influence of varying phase-locked loop target bandwidth on the frequency stability. (a) Measured and (b) theory-based Allan deviations for target bandwidths of (from bottom to top): 0.25 Hz, 0.5 Hz, 1 Hz, 2.5 Hz, 5 Hz, 10 Hz, 25 Hz, 50 Hz, and 719.6 Hz. Blue dash-dotted curve is an open-loop measurement. Calculations based on equation (4) for thermomechanical (green dotted lines) and detection (red dashed lines) noise limits are given as well.
Reuse & Permissions
Figure 6
Transient response of the phase. (a) Extraction of response times from an exponential fit to the phase change. (b) Dependence of the resonator time constant $τ_{r}$ on the mechanical quality factor $Q$ as extracted from the exponential fits and calculations using $τ_{r} = Q / (π f_{0})$ .
Reuse & Permissions

Physical Review B

covering condensed matter and materials physics