Geometric Tuning of Stress in Predisplaced Silicon Nitride Resonators

Abstract

We introduce a novel method to geometrically tune the tension in prestrained resonators by making Si 3 N 4 strings with a designed predisplacement. This enables us, for example, to study their dissipation mechanisms, which are strongly dependent on the stress. After release of the resonators from the substrate, their static displacement is extracted using scanning electron microscopy. The results match finite-element simulations, which allows a quantitative determination of the resulting stress. The in-and out-of-plane eigenmodes are sensed using on-chip Mach−Zehnder interferometers, and the resonance frequencies and quality factors are extracted. The geometrically controlled stress enables tuning not only of the frequencies but also of the damping rate. We develop a model that quantitatively captures the stress dependence of the dissipation in the same SiN film. We show that the predisplacement shape provides additional flexibility, including control over the frequency ratio and the quality factor for a targeted frequency.

Relaxation and dynamics of high-stress pre-displaced string resonators

“…So far, the discussion of relaxation of high-stress beams has focused on straight strings. Figure 1(b) shows a schematic of our pre-displaced strings [21]. Here, the beam is made with a center line that is not straight, but has a x-dependent displacement in the y-direction u 0 (x).…”

“…A resonator with Σ y 1 behaves as a string, whereas one with Σ y 1 acts as a tensionless beam. Interestingly, when the "stringness" [21] Σ y 1, i.e. a resonator where the tension dominates over the bending rigidity, u 0 drops out of Eq.…”

“…( 14). Thus, after relaxing, the pre-displaced beams simply behave as strings under tension and the only effect of the pre-displacement u 0 will be the geometric tuning of the tension T [21]. Irrespective of the value of Σ y , the u 0 term in Eq.…”

“…In the latter case, it is important to also include the ac part of the tension [34]. Although the mode shapes χ n (x) can be solved analytically, finding the tension and eigenvalues ω n typically has to be done numerically [21,34].…”

“…For all these experiments, it is imperative to engineer the geometry of the resonator [9,[13][14][15][16][17][18] including the stress [19,20], to reach the best performance. Our recent work [21] experimentally demonstrates the geometrical tuning of the stress of resonators made out of pre-strained SiN films. This not only provides a new approach for a systematic study of dissipation dilution [10,22] but also raises interesting questions, like their potential to buckle and the influence of the geometry on the eigenmodes.…”