**Motivation:**

**Motivation:**

In this section, we demonstrate the effect of nonlinear coupling of between different bulk modes of silicon MEMS resonators (e.g. Lame mode resonator and Length extensional resonator). We observed experimentally that the coupling behavior has a strong dependence on the resonant mode order, the mode shape of the coupled modes, as well as the doping type, concentration, and crystal orientation, leading to a variety of complex phenomena which may need to be taken notice of when devices are used under dual-mode operations.

Nonlinear mode coupling arises from the strain induced within a resonator when one mode is excited to large vibration amplitude and the large strain affects the other modes that are simultaneously driven. To explore the effect of mode coupling, two resonant modes (within one device) are excited simultaneously, of which one can be denoted as the ‘driven mode’ (DR) and the other as ‘detection mode’ (DET). When the DR mode is excited to large amplitude, a frequency shift of the DET mode, which is proportional to the square of the DR mode’s displacement amplitude, can be observed.

Part of this work is published in [1].

Nonlinear mode coupling arises from the strain induced within a resonator when one mode is excited to large vibration amplitude and the large strain affects the other modes that are simultaneously driven. To explore the effect of mode coupling, two resonant modes (within one device) are excited simultaneously, of which one can be denoted as the ‘driven mode’ (DR) and the other as ‘detection mode’ (DET). When the DR mode is excited to large amplitude, a frequency shift of the DET mode, which is proportional to the square of the DR mode’s displacement amplitude, can be observed.

Part of this work is published in [1].

Finite Element Analysis of Resonator Mode Shape within One Device

Finite Element Analysis of Resonator Mode Shape within One Device

Finite Element Analysis of Resonator Mode Shape within One Device

**Modeling of Nonlinear Coupling**

**Modeling of Nonlinear Coupling**

Experiment setup:

Experiment setup:

Experiment setup:

The setup used for characterizing these resonators is shown in Fig. 3. Dual phase-lock loops, which are independent of each other on the Zurich HF2LI-PLL, are used to simultaneously excite and monitor the responses of both the DR and DET modes separately.
A DC voltage is applied on the resonator body to bias the device. To combine the two PLL output signals, the AC output signals from the two PLLs are summed using a 0° combiner (also a splitter), and the combined output signal is then used to excite the resonator. The resonator output, after amplification through a transimpedance amplifier, is divided using a splitter and fed back to the two PLLs individually, and internal filters within the PLL are used to obtain the corresponding frequency signal. |

**Measurement result:**

**Measurement result:**

**- Effect of Relative Mode Order**

**- Effect of Relative Mode Order**

**- Effect of Doping and Crystal Orientation**

**- Effect of Doping and Crystal Orientation**

**Reference:**

**Reference:**

[1]. Yushi Yang, Eldwin Ng, Pavel Polunin, Yunhan Chen, Scott Strachan, Vu Hong, Chae Hyuck Ahn, Ori Shoshani, Steven Shaw, Mark Dykman, and Thomas Kenny, "Experimental investigation on mode coupling of bulk mode silicon MEMS resonators," 2015 28th IEEE International Conference on Micro Electro Mechanical Systems (MEMS), pp. 1008-1011, Jan 2015.