Self-induced parametric amplification arising from nonlinear elastic coupling in a micromechanical resonating disk gyroscope

Parametric amplification, resulting from intentionally varying a parameter in a resonator at twice its resonant frequency, has been successfully employed to increase the sensitivity of many micro- and nano-scale sensors. Here, we introduce the concept of self-induced parametric amplification, which arises naturally from nonlinear elastic coupling between the degenerate vibration modes in a micromechanical disk-resonator, and is not externally applied. The device functions as a gyroscope wherein angular rotation is detected from Coriolis coupling of elastic vibration energy from a driven vibration mode into a second degenerate sensing mode. While nonlinear elasticity in silicon resonators is extremely weak, in this high quality-factor device, ppm-level nonlinear elastic effects result in an order-of-magnitude increase in the observed sensitivity to Coriolis force relative to linear theory. Perfect degeneracy of the primary and secondary vibration modes is achieved through electrostatic frequency tuning, which also enables the phase and frequency of the parametric coupling to be varied, and we show that the resulting phase and frequency dependence of the amplification follow the theory of parametric resonance. We expect that this phenomenon will be useful for both fundamental studies of dynamic systems with low dissipation and for increasing signal-to-noise ratio in practical applications such as gyroscopes.

vapor-phase hydrofluoric acid (HF) etch to remove the sacrificial oxide layer, releasing the resonator structure, after which a second epitaxial encapsulation layer is deposited to seal the vent holes and create the hermetic cavity. Low pressure annealing in a nitrogen environment is then performed to diffuse the residual hydrogen gas out of the cavity. A top aluminum metal layer provides electrical contact, after which the wafer is diced and individual resonator dice are wire-bonded into ceramic packages for testing.

Electrostatic Mode-Matching
The Coriolis coupling between the disk resonator's elliptical vibration modes is maximized when the two modes are degenerate. Imperfections and crystalline anisotropy 2 break the resonator's symmetry, splitting the frequency of the two modes so that they are not perfectly degenerate. The largest component of the frequency split (Δ = δ /2 on the order of 1 kHz) is due to the anisotropic Young's modulus of <100> silicon and was compensated by making small adjustments to the (nominally 45°) angle between the spokes that connect the resonator's concentric rings, as shown in Figure S1 3 . Figure S1: A cartoon showing the layout of the spokes and concentric rings that compose the disk resonator. Small adjustments to the angle between alternating spokes (shown in red) compensate for the anisotropy of Young's Modulus in <100> silicon.
The remaining Δ , on the order of 100 Hz -300 Hz, originates from fabrication defects and must be nulled using electrostatic frequency tuning via the electrodes surrounding the device, which approximate parallel plate capacitors. The energy stored in a parallel plate capacitor operating in a vacuum is given by = 2 0 /2 , where is the applied voltage, 0 is the permittivity of free space, is the electrode area, and = 0 − ( ) is the gap between the electrode plates.
The attractive force exerted by such an electrode is given by for small ( ). Thus, the capacitor acts as a spring with negative stiffness, = − 2 0 / 0 3 . By adjusting the applied voltage, , the stiffness, and therefore the frequency of a given vibration mode, can be tuned. In addition to matching the frequency of two resonant modes, this method can be used to minimize cross-coupling between the two modes that arises from off-diagonal elements of the stiffness matrix. This cross-coupling is the root cause of the quadrature force and occurs because the resonator's primary axis of stiffness is rotated with respect to the drive and sense electrodes. A detailed discussion of electrode configurations for mode-matching of DRGs can be found in 4 and 5 .

Modal Mass and Angular Gain
The modal mass of a given mode shape is calculated by equating the integrated strain energy, = ∫̇2 = 2 ∫ 2 , occurring at a given vibration amplitude, , to the maximum kinetic energy, ̇2/2, and solving for . The resulting expression is where is the resonant frequency of the mode, and is a reference displacement of the mode, chosen here to be the maximum radial displacement. The modal mass for this structure is 3.8 μg.
The angular gain, determined by the mode shape, dictates what percentage of the drive mode's momentum is coupled to the sense mode by an applied Coriolis force. The angular gain is given where and are the two modes under consideration, and ̂ and ̂ are the normalized displacement fields for each mode. The integrals are calculated over the volume of the structure. For a tuning fork gyroscope, which translates linearly, = 1. For a ring gyroscope or DRG operating in the 2θ mode, ≈ 0.8.

Linear Model for the Force Sensitivity of the Sense Axis
Using a linear model for the sense mode, the equation of motion is given by where , and are the modal mass, damping coefficient, and spring constant of the sense mode respectively. For this second-order harmonic oscillator, the quality factor is defined as ≡ √ / and, for a harmonic force input at the resonant frequency , the force-todisplacement sensitivity is given by −1 .

Device Operation and Electrostatic Testing
The block diagram of the disk resonator and associated control electronics is shown in Figure S2.
Following electrostatic mode-matching, the driven mode is excited using a digital phase-locked loop (PLL) that is locked to the driven mode's natural frequency. The PLL amplitude is set to a constant value that is varied to achieve a desired vibration amplitude of the drive mode.
Vibration of the sense mode is detected and subsequently quadrature demodulated using the PLL output as a reference. The in-phase component of the sense mode vibration is used as a measure of rotation rate, and the quadrature component is used to adjust the voltage on a dedicated set of electrodes used to null the quadrature signal. Figure S2: A block diagram of the disk resonator and associated control electronics is shown on the left, while the electrode layout and device design are shown on the right. Following electrostatic mode-matching, the drive mode is excited in a closed loop using a digital PLL. Vibration of the sense mode is demodulated into In-Phase (I) and Quadrature (Q) components, which contain the Coriolis force and quadrature force signals, respectively. The Q channel is used to adjust the voltage on a dedicated set of capacitive electrodes used for quadrature-nulling (shown in green and yellow).
To characterize spontaneous parametric amplification of the sense mode's response to a force with arbitrary phase relative to the drive mode's vibration, tests were conducted wherein a secondary electrostatic force was input to the sense mode, as shown in Figure S2. This testing configuration was used to generate the data shown in Figure 4.

Electrostatic Nonlinearity
The energy stored in an infinitesimal arc of a parallel plate electrode is given by Summing over all electrode locations and voltages, and assuming a very large quadrature displacement of 1% of the gap, yields Δ 2 / = = 0.22 ppm. This value is an order of magnitude smaller than the minimum value for required to explain the observed parametric amplification. Thus, we can safely conclude that the parametric amplification we observe is not due to the electrostatic nonlinearity of the sense electrodes.