Geometrical Mode-Matching in a (100) Single-Crystalline Silicon, Smooth-Quatrefoil Disk Resonator

We present, for the first time ever, a disk resonator made from (100) single-crystalline silicon (SCS) in which the two conjugate vibration modes are frequency-matched. This mode matching is achieved by shaping the disk as a smooth-quatrefoil, where the uneven distribution of inertia balances the inherent anisotropic elasticity of SCS. The average diameter of the disk is <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$400 \ \mu \mathrm{m}$</tex> and it is supported by concentric stems that allow free in-plane vibrations, yielding a quality factor of <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$Q=9.7\times 10^{5}$</tex>. The measured average frequency of the two conjugate modes is 9.737 MHz, and the mismatch is 105 Hz (a relative mismatch of 11 ppm). For comparison, in a circular disk resonator made from (100) SCS, the mismatch between the two conjugate modes is ~24% (a relative mismatch of 248,647 ppm).