Phononic crystals have numerous potential applications including use as filters and oscillators in communications systems and as acoustic isolators for resonant sensors such as gyroscopes. These applications are based on the ability of phononic crystals to exhibit elastic band gaps, frequency bands where the propagation of acoustic waves is forbidden. Here, we focus on solid-solid phononic crystals (solid inclusions in a solid matrix), since they typically exhibit wider band gaps than those observed with air-solid phononic crystals (air inclusions in a solid matrix). We present a micromachined solid-solid phononic crystal operating at 1.4 GHz center frequency with an ultrawide 800 MHz band gap.
High-Q ͑quality factor͒ resonators are a versatile class of components for radio frequency micro-electromechanical systems. Phononic crystals provide a promising method of producing these resonators. In this article, we present a theoretical study of the Q factor of a cavity resonator in a two-dimensional phononic crystal comprised of tungsten rods in a silicon matrix. One can optimize the Q of a phononic crystal resonator by varying the number of inclusions or the cavity harmonic number. We conclude that using higher harmonics marginally increases Q while increasing crystal length via additional inclusions causes Q to increase by orders of magnitude. Incorporating loss into the model shows that the silicon material limit on Q is achievable using a two-dimensional phononic crystal design with a reasonable length. With five layers of inclusions on either side of the cavity, the material limit on Q is achieved, regardless of the harmonic number.
Solid–solid phononic crystals exhibit wider band gaps than those observed with air–solid phononic crystals. For micromachined phononic crystal devices it is advantageous to release the phononic crystal to avoid propagation losses. In a solid–solid phononic crystal operating in the low megahertz range, due to the large lattice constant, it is necessary to place release holes in the center of the inclusions to release devices from the substrate while minimizing the effect the release hole has on the band gap. In this report, we investigate the effect of release holes on phononic band gaps and highlight the need for careful design. It was determined that release holes of radius rair holes/rinclusion=0.26 can reliably release a phononic crystal membrane composed of W inclusions in SiO2 without significantly compromising the phononic band gap.
Plane wave expansion analyses that use the inverse rule to obtain the Fourier coefficients of the elastic tensor instead of the more conventional Laurent's rule, exhibit faster convergence rates for solid-solid phononic crystals. In this work, the band structure convergence of calculations using the inverse rule is investigated and applied to the case of high acoustic impedance contrast solid-solid phononic crystals, previously known for convergence difficulties. Results are contrasted to those obtained with the conventional plane wave expansion method. The inverse rule is found to converge at a much rate for all ranges of impedance contrast, and the ratio between the computational times needed to obtain a convergent band structure for a high-contrast solid-solid phononic crystal with the conventional plane wave expansion method using 1369 reciprocal lattice vectors is as large as 6800:1. This ratio decreases for material sets with lower impedance contrast; however, the inverse rule is still faster for a given error threshold for even the lowest impedance contrast phononic crystals reported in the literature. This convergence enhancement is a major factor in reconsidering the plane wave expansion method as an important tool in obtaining propagating elastic modes in phononic crystals.
Solid-solid phononic crystals (solid inclusions in a solid matrix) exhibit wider bandgaps than those observed with air-solid phononic crystals (air inclusions in a solid matrix). In a solid-solid phononic crystal operating in the low MHz range, it is essential to place release holes in the center of the inclusions to release devices from the substrate. It is necessary to release, and therefore suspend the phononic crystal to avoid propagation losses through the substrate. In this report we investigate the effect of release holes on phononic bandgaps and highlight the need for careful design to avoid compromising the phononic bandgap. Studying release issues for solid-solid phononic crystals is essential for the successful fabrication of such devices.
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