Design and optimization of novel RF Nano-Electro-Mechanical (NEM) resonators such as Resonant Body Transistors (RBT) require modeling across multiple domains, including mechanical (distributed stress and elastic wave models), electrical (semiconductor devices and RF small signal models), and thermal. These domains are all cross-coupled in nonlinear ways and require lengthy finite element multi-physics analyses to solve. Due to the complexity of these structures embedded in the CMOS stack and sensed using active FETs, the day-long time scale of each finite element simulation prevents quick, intuitive parameterization of device design. A reduced model parameterized across all three domains is therefore necessary both for rapid prototyping and for device optimization.

In this work, we are developing an algorithm to automatically generate compact models for NEM resonators. Our compact models are suitable for AC, DC and RF operation of the device and allow the circuit designers to run circuit-level time-domain simulations using any commercial circuit simulator ^{[1] [2]} . The compact models are “parameterized,” so that the circuit designer will be able to instantiate instantaneously models within the circuit simulator for different values of the key device parameters.  Key resonator parameters included in the compact parameterized model are resonant frequency, quality factor, signal strength, isolation, presence of spurious modes, and operating temperature. Values for the model coefficients are calibrated using measurements from NEMS resonator devices. A critically important feature of our models is to guarantee that when circuit designers change arbitrarily values for the device parameters, the compact models will always preserve the physical properties of the original device and will never cause numerical instabilities and convergence issues when connected to other device models and circuits within the circuit simulator ^[3] . Figure 1 shows the layout of a Si-based NEMS-CMOS resonator. Numerical results show a great promise for our technique. We have achieved high quality fit to the measured data, as Figure 2 shows, which offered modeling challenges including the presence of noise and spurious resonant peaks.

1. Z. Mahmood and L. Daniel, “Circuit synthesizable guaranteed passive modeling for multiport structures,” in Proc. of Behavioral Modeling and Simulation Conference (BMAS), Sept. 2010.
2. Z. Mahmood, R. Suaya and L. Daniel, “An efficient framework for passive compact dynamical modeling of multiport linear systems,” in Proc. of Design, Automation and Test in Europe, (DATE), Mar. 2012.
3. Z. Mahmood and L. Daniel, “Guaranteed passive parameterized modeling of multiport passive circuit blocks,” in Proc. of TECHCON, Sept. 2011.

This work presents the first hybrid RF MEMS-CMOS resonators demonstrated in silicon at the transistor level of IBM’s 32-nm SOI CMOS process, without the need for any post-processing or packaging. The unreleased, Si bulk acoustic resonators are driven capacitively and sensed using a field effect transistor (FET). MEMS-CMOS Si resonators with acoustic Bragg reflectors (ABRs) are demonstrated at 11.1 GHz with Q~18 and a footprint of 5µm × 3µm.

The majority of electromechanical devices require a release step to freely suspend moving structures, which necessitate costly complex encapsulation methods and back-end-of-line (BEOL) processing of large-scale devices ^[1] . Development of unreleased Si-based MEMS resonators in CMOS allows integration into front-end-of-line (FEOL) processing with no post-processing or packaging. We have previously demonstrated the Resonant Body Transistor (RBT), which employs active FET sensing of acoustic vibrations ^{[2] [3]} , which amplifies the mechanical signal before parasitics. Realization of the RBT in CMOS technology leverages high f_T, high-performance transistors, enabling RF-MEMS resonators at frequencies orders of magnitude higher than possible with passive devices.

The hybrid MEMS-CMOS RBT presented in this work is a Si bulk-acoustic resonator with electrostatic drive formed using the gate dielectric and a body-contacted nFET sense transducer (see Figure 1). Acoustic vibrations in the unreleased resonator are confined using 7 pairs of 1D ABRs surrounding the device, which are patterned using shallow trench isolation (STI). The DC characteristics of the sense transistor are similar to standard body-contacted nFETs of the 32-nm SOI process and show no direct effect of the capacitor drive voltage on the FET behavior. The frequency response of an 11.1-GHz resonator is shown in Figure 2 for multiple bias conditions, verifying the mechanical nature of the resonance.

This first demonstration of an unreleased hybrid MEMS-CMOS resonator paves the way for monolithically integrated RF MEMS frequency sources and signal processors.

1. H. Xie, L. Erdmann, X. Zhu, K. Gabriel, and G. Fedder, “Post-CMOS processing for high-aspect-ratio integrated silicon microstructures,” Journal of Microelectromechanical Systems, vol. 11, no. 2, pp. 93-101, 2002.
2. D. Weinstein and S. A. Bhave, “The resonant body transistor,” Nano Letters, vol. 10, no. 4, pp. 1234-37, 2010.
3. D. Weinstein and S. A. Bhave, “Acoustic resonance in an independent-gate FinFET,” in Solid State Sensor, Actuator and Microsystems Workshop(Hilton Head), 2010, pp. 459-462.