I. INTRODUCTION

With improved performances, low cost, and compact size, MEMS tuning fork (MEMS-TF) structures are widely used as gyroscopes ^(1-3) and resonators ^(4-6). The MEMS-TF structures consist of two masses and are operated in anti-phase motion. These masses can be either coupled mechanically or electrostatically.

In electrostatically coupled resonators, it is challenging to achieve perfect anti-phase motion since the fabrication errors are inevitable, and it is almost impossible to fabricate two masses with the same frequency ⁽⁷⁾. To overcome this issue, mechanically coupled resonators have been proposed ^(8-10). One of the issues with mechanically coupled structures are the unwanted in-phase motion, which lies at a frequency lower than the required anti-phase frequency. Also, with the fabrication errors, both of the resonant modes can be merged.

MEMS-TF structures are believed to have the advantage of resisting linear accelerations, when operated in an anti-phase mode. However, structure asymmetry and spring unbalancing caused by the fabrication errors make MEMS-TF vulnerable to linear accelerations. The effect of acceleration sensitivity (g-sensitivity), on the performance of MEMS-TF structures has been presented widely ^(7,9,11-14), so it is essential to reduce the effect of linear accelerations (g-sensitivity) on these structures.

Several design approaches have been reported to improve the g-sensitivity of mechanically coupled MEMS tuning fork structures, Singh ^(10,15) studied the effect of linear accelerations, showing that the frequency decoupling ratio between the in-phase and anti-phase modes can improve the acceleration sensitivity. To further improve the resistance to linear accelerations and increase the decoupling ratios, different spring architectures are proposed, such as lever mechanism ^(16,17), z-beam coupling spring ⁽¹⁸⁾, diamond-shaped ⁽¹⁹⁾, circular-shaped ^(8,20), T-shaped ⁽²¹⁾, lozenge-shaped ⁽²²⁾, and polygon-shaped ⁽²³⁾. Most of the proposed spring architectures presented exhibit either large surface area or wide etch cavities. The large size of the spring will increase the device size, making it difficult to use in specific applications such as consumer electronics, whereas the wide etch area will make these springs more prone to fabrication errors.

In this article, a new coupling spring architecture has been proposed to overcome the previous spring designs' issues. The proposed design benefits with the miniaturization and narrow etch cavity, making it more robust to the fabrication errors. The proposed coupling spring always prioritizes the anti-phase motion by keeping the unwanted in-phase motion at a higher frequency, providing robustness to linear accelerations.

II. THE DESIGN CONCEPT OF THE MEMS TUNING FORK RESONATOR

The MEMS tuning fork resonator's conceptual design, which can be modeled as two degrees of freedom system, is shown in Fig. 1. The two identical masses, left mass $(M_{L})~ $and right mass$\left(M_{R}\right)$, are anchored through the left and right spring, $K_{L}$ and $K_{R},$respectively. Also, both the masses are mechanically coupled through coupling spring $K_{c}$ to form a shared resonance. The coupled resonators shown in Fig. 1 can be modeled as a second-order spring-mass system as shown below:

Here, $M,C,$ and $K~$represent the mass, damping, and spring constant of the resonator, and subscripts$R$and$L$represent the right and left sides.$F_{L}$ and $F_{R}$ are the applied forces. Eqs. (1) and (2) can be solved for the in-phase and anti-phase motions by assuming $ M_{L}=M_{R}=M, C_{L}=C_{R}=C$ given by ^(6,19):

In this equation, $x_{an}=x_{L}- x_{R}$ is the anti-phase motion, $x_{in}=x_{L}+x_{R}$ is the in-phase motion, $\omega _{in}=$ $\sqrt{(K_{L}+K_{R})/M}$and$\omega _{an}=\sqrt{(K_{L}+K_{R}+4K_{c})/M}$represents the in-phase and anti-phase resonant frequencies, and $Q_{an}=M\omega _{an}/c$and$Q_{in}=M\omega _{in}/c$are the anti-phase and in-phase quality factors respectively.

From Eqs. (3) and (4), it is clear that the in-phase and anti-phase resonant modes depend on the polarity of the applied force and the coupling spring. The design goal for the coupling spring, as discussed above, is to have an anti-phase resonant frequency lower than the spurious in-phase resonant frequency, occupying a small surface area and narrow etch cavity. This can be achieved by the proposed anti-phase coupling spring architecture shown in the Fig. 2.

The proposed anti-phase spring architecture is comprised of outer, inner, and torsional arms. The outer arms are connected to the masses, $M_{L}$ and $M_{R}$. Meanwhile, one side of the inner arms is anchored, while the other side is connected to the outer arm through the torsional arm in a rotationally symmetric manner, constituting $180^{\circ}$ in relation to the center of the coupling spring. The deformation of the designed spring during the lateral load and axial load is shown in the Fig. 2(a) and (c) respectively. During the lateral load the outer arms move in the opposite direction, whereas the inner arms move in the same direction, in such a way that the torsional arm rotates along the lateral axis. The rotation of the torsional arm lessens the resistance for a low frequency anti-phase resonant mode. However, during the axial load, both the outer arms bend in the same direction, whereas the inner arms bend in the opposite direction. This opposite motion of the inner arm resist the motion of the torsional arm, resulting in a high frequency in-phase resonant mode.

The proposed spring architecture prioritizes the anti-phase motion, providing resistance to linear acceleration and robustness to fabrication imperfections with the narrow cavity holes. Resonant modes of the designed MEMS-TF resonator utilizing the anti-phase coupling spring architecture are shown in the Fig. 3.

III. FEM SIMULATIONS

The designed MEMS-TF resonator was investigated through FEM simulation using COMSOL Multiphysics. The resonator was simulated as single-crystal silicon having a Young modulus of 179GPa and poison ratio of 0.29. A triangular mesh with a minimum element size of 0.1$\mathrm{\mu}m$ was used, consisting of 120 $\textit{K}$ triangular elements. Frequency domain study was used to investigate (i) the anti-phase operation of the proposed coupling spring and (ii) the effect of linear acceleration on the resonant modes of the MEMS-TF resonator.

IV. EXPERIMENTAL RESULTS

The equivalent electrical model and test method of the fabricated MEMS-TF resonator is shown in Fig. 7(a), while the optical image of the fabricated device using the designed coupling spring is shown in Fig. 7(b). The resonator was fabricated using three wafer stacked processes (i.e., device wafer, via wafer, and cap wafer), with a device thickness of $30\mu m$ and vacuum-sealed at $100\textit{mtorr}$.

The resonator was actuated by electrostatic force with an $AC~ $ voltage of$v_{ac}=100mV_{pk}$ and a $DC$ polarization voltage of $V_{DC}=5V$. The drive circuitry generates the differential signal; i.e., $V_{DP}=V_{DC}+v_{ac}$ and $V_{DN}=V_{DC}- v_{ac}$. The sense current was converted to the sense voltage using a charge amplifier, and the frequency response function (FRF) was computed through the Zurich instrument (MFLI).

V. CONCLUSION

This study has successfully demonstrated a new coupling spring design for MEMS tuning fork structures. The new coupling spring design provides perfect anti-phase motion at a lower resonant mode while keeping the surplus in-phase resonant mode high. The designed coupling spring has low etched cavities, making it more robust to fabrication errors and small surface area, resulting in a wide range of device applications. The FEM simulations were carried out to study the effect of linear accelerations and un-balanced forces. The in-phase mode is more susceptible to the linear accelerations, showing that the individual masses have an acceleration sensitivity of $89nm/G$, whereas the differential output corresponds to $76.4nm/G$. This shows that the anti-phase mode is 69.45 times more resilient to the linear acceleration. Furthermore, experimentally, the designed coupling spring architecture suppresses the in-phase mode by 97 %.

Author Contributions: Conceptualization, Faisal Iqbal, Seungoh Han and Byeungleul Lee; Formal analysis, Faisal Iqbal; Software, Faisal Iqbal and Hussamud Din; Experiment, Faisal Iqbal; Supervision, Byeungleul Lee; Writing—original draft, Faisal Iqbal; Writing—review and editing, Faisal Iqbal, Hussamud Din, and Byeungleul Lee.