[Paper Review] Dissipation engineering of high-stress silicon nitride nanobeams

Amir H. Ghadimi, Dalziel J. Wilson+1 more|arXiv (Cornell University)|Mar 4, 2016

Mechanical and Optical Resonators3 citations

TL;DR

This paper demonstrates two strategies to enhance the quality factor (Q) of high-stress silicon nitride nanobeams: embedding them in a 1D phononic crystal to suppress radiation loss, achieving Q ~ 10⁴ at f > 100 MHz, and exploiting stress-induced loss-dilution in high-order modes of mm-long beams to reach Q·f ≈ 9×10¹² Hz. These advances enable room-temperature quantum-coherent operation in ultra-low-mass 1D nanomechanical oscillators.

ABSTRACT

High-stress Si$_3$N$_4$ nanoresonators have become an attractive choice for electro- and optomechanical devices. Membrane resonators can achieve quality factor ($Q$) - frequency ($f$) products exceeding $10^{13}$ Hz, enabling (in principle) quantum coherent operation at room temperature. String-like beam resonators possess conventionally 10 times smaller $Q\cdot f$ products; however, on account of their much larger $Q$-to-mass ratio and reduced mode density, they remain a canonical choice for precision force, mass, and charge sensing, and have recently enabled Heisenberg-limited position measurements at cryogenic temperatures. Here we explore two techniques to enhance the $Q$-factor of a nanomechanical beam. The techniques relate to two main loss mechanisms: internal loss, which dominates for large aspect ratios and $f\lesssim100$ MHz, and radiation loss, which dominates for small aspect ratios and $f\gtrsim100$ MHz. First we show that by embedding a nanobeam in a 1D phononic crystal, it is possible to localize its flexural motion and shield it against radiation loss. Using this method, we realize $f>100$ MHz modes with $Q\sim 10^4$, consistent with internal loss and contrasting sharply with unshielded beams of similar dimensions. We then study the $Q\cdot f$ products of high-order modes of mm-long nanobeams. Taking advantage of the mode-shape dependence of stress-induced `loss-dilution', we realize a $f\approx 4$ MHz mode with $Q\cdot f\approx9\cdot 10^{12}$ Hz. Our results can extend room temperature quantum coherent operation to ultra-low-mass 1D nanomechanical oscillators.

Motivation & Objective

• To overcome the limitation of low Q-factors in high-stress Si₃N₄ nanobeams, which hinders their use in quantum optomechanics and precision sensing.
• To address radiation loss in high-frequency nanobeams (f > 100 MHz) by using a 1D phononic crystal as an acoustic shield.
• To enhance the Q·f product in mm-long nanobeams by exploiting stress-induced loss-dilution in high-order flexural modes.
• To demonstrate room-temperature quantum-coherent operation in ultra-low-mass 1D nanomechanical oscillators through dissipation engineering.

Proposed method

• Embedding a nanobeam as a defect in a 1D phononic crystal (PnC) to create a bandgap that localizes flexural motion and suppresses acoustic radiation loss.
• Using finite element method (FEM) simulations with perfectly matched layers (PML) to model and compute radiation loss from eigenfrequency spectra, with Q_rad = Re[Ω_m]/(2·Im[Ω_m]).
• Optimizing beam geometry and mode order in mm-long beams to exploit stress-induced loss-dilution, reducing internal losses in high-order modes.
• Employing a microcavity-based near-field sensor for non-invasive, high-resolution thermal noise measurements with fm/√Hz sensitivity.
• Using reactive ion etching (RIE) with high selectivity and smooth sidewall control to fabricate sub-100 nm gaps in Si₃N₄, enabling high optical quality factors (>10⁵).
• Validating results through experimental measurement of Q and f, comparing with simulations to confirm loss mechanisms and shielding effectiveness.

Experimental results

Research questions

• RQ1 Can a 1D phononic crystal effectively suppress radiation loss in high-frequency Si₃N₄ nanobeams, leading to Q factors limited only by internal losses?
• RQ2 To what extent can stress-induced loss-dilution in high-order modes of mm-long nanobeams enhance the Q·f product?
• RQ3 Can Q·f products exceeding 10¹³ Hz be achieved in ultra-low-mass 1D nanomechanical oscillators at room temperature?
• RQ4 How does the number of unit cells in the PnC affect radiation loss suppression, and what is the optimal design for robust shielding?

Key findings

• A 1D phononic crystal shield successfully suppresses radiation loss in high-frequency nanobeams, achieving Q ~ 10⁴ at f > 100 MHz, consistent with internal loss limits.
• The PnC shielding effect saturates at ~6 unit cells, with 7 cells used experimentally to account for fabrication imperfections.
• A high-order mode in a 4-mm-long nanobeam achieves f ≈ 4 MHz and Q·f ≈ 9×10¹² Hz, approaching the quantum-coherent regime.
• The Q·f product of 9×10¹² Hz is achieved with an effective mass of m < 1 pg, demonstrating feasibility for room-temperature quantum applications.
• Simulations confirm that radiation loss is the dominant loss mechanism for f > 100 MHz in unshielded beams, while internal loss dominates below 100 MHz.
• The optical quality factor of microdisk cavities used for displacement sensing exceeds 10⁵, with intrinsic linewidths of 1–10 GHz and a G-factor of ~100 MHz/nm at x = 100 nm.