Influence of clamp-widening on the quality factor of nanomechanical silicon nitride resonators

Abstract

Nanomechanical resonators based on strained silicon nitride (Si 3 N 4 ) have received a large amount of attention in fields such as sensing and quantum optomechanics due to their exceptionally high quality factors (Qs). Room-temperature Qs approaching 1 billion are now in reach by means of phononic crystals (soft-clamping) and strain engineering. Despite great progress in enhancing Qs, difficulties in fabrication of soft-clamped samples limits their implementation into actual devices. An alternative means of achieving ultra-high Qs was shown using trampoline resonators with engineered clamps, which serves to localize the stress to the center of the resonator, while minimizing stress at the clamping. The effectiveness of this approach has since come into question from recent studies employing string resonators with clamptapering. Here, we investigate this idea using nanomechanical string resonators with engineered clampings similar to those presented for trampolines. Importantly, the effect of orienting the strings diagonally or perpendicularly with respect to the silicon frame is investigated. It is found that increasing the clamp width for diagonal strings slightly increases the Qs of the fundamental out-of-plane mode at small radii, while perpendicular strings only deteriorate with increasing clamp width. Measured Qs agree well with finite element method simulations even for higher-order resonances. The small increase cannot account for previously reported Qs of trampoline resonators. Instead, we propose the effect to be intrinsic and related to surface and radiation losses.

Nanomechanical Dissipation and Strain Engineering

“…Neglecting the elongation energy (and therefore using Equation (11)) and using the corrected modeshape in Equation (12) to account for the sharp curvatures at the clamps we can simplify the dissipation dilution factor of a uniform string resonator to be [ 41,111 ] It is noticeable that a large aspect ratio L / h is advantageous to increase the dissipation dilution factor, with D scaling as L / h in the high stress limit. This is particularly beneficial since the clamping loss also reduces as the device's aspect ratio increases.…”

“…Small changes such as altering the width of the resonator near the clamping points have demonstrated some degree of soft clamping in string and trampoline resonators. [ 90,111 ] For example, the rounding of a string resonator's clamps has shown to increase the quality factor by a factor of two by lessening the curvature of the mode. [ 90,111 ]…”

“…[ 90,111 ] For example, the rounding of a string resonator's clamps has shown to increase the quality factor by a factor of two by lessening the curvature of the mode. [ 90,111 ]…”

Engineering the Dissipation of Crystalline Micromechanical Resonators

“…The radius of the clamping point is engineered to optimize the ratio of elastic energy stored as elongation as opposed to bending which is a source of losses [12,29], enhancing the Q by diluting the dissipation for the fundamental vibration mode. In this approach, two counteracting and competing mechanisms are present; the increased material volume at the widened clamps stores a larger amount of bending energy, while the increased rigidity reduces the overall bending [30,31]. The elongation to bending ratio converges to a maximum value for an optimum radius of curvature R. According to our finite element simulations, the optimal radius of curvature for our crystalline resonators is R = 30 µm, predicting a Q about four times larger compared to rigid clamping (R = 0).…”

Nanomechanical Dissipation and Strain Engineering

“…Neglecting the elongation energy (and therefore using Equation ( 11)) and using the corrected modeshape in Equation (12) to account for the sharp curvatures at the clamps we can simplify the dissipation dilution factor of a uniform string resonator to be [41,111] D ≈ (nπ) 2 12…”