MEASUREMENT OF THE NONLINEAR ELASTICITY OF DOPED BULK-MODE MEMS RESONATORS

In this paper, we present measurements of the nonlinear amplitudefrequency (A-f) coefficients for three types of bulk-mode silicon MEMS resonators with different doping type and concentration. Our experimental results clearly show a strong dependence of the nonlinearity character on the orientation, doping level, and vibrational mode shape. These results indicate that the doping can have a significant impact on the character and strength of the elastic nonlinearities in MEMS devices. INTRODUCTION High performance MEMS oscillators have gained much attention as an alternative for quartz since they offer potential benefits such as volume and cost reduction, low power consumption, and IC compatibility [1, 2]. The use of MEMS oscillators in wireless communication requires device performance characteristics such as high stability, high quality factor (Q), and large power handing capability. These will provide a larger signalto-noise ratio and reduced phase noise [3]. However, because the power handling capability of MEMS oscillators is inversely related to the resonator size, oftentimes the device has to be driven into nonlinear regime to obtain suitable signal-to-noise ratio and performance [4]. Another aspect that has gained wide attention is frequencytemperature compensation. Recent research has demonstrated the use of heavily doped por n-type silicon for passive frequencytemperature compensation [5, 6], thus making it attractive for this study. Adding dopants to the silicon lattice will not only introduce free charge carriers, but will also cause a strain in the lattice due to a size dissimilarity between the silicon and the dopant atoms. These effects shift the semiconductor energy bands and result in a change in the material elastic constants, since any change in the total electronic energy content of the crystal will also contributes to the total elastic energy [7]. Nonlinearities in lightly doped silicon resonators have been studied [8], and the effect has also been noted in heavily doped silicon [9]. This examines the nonlinear effects for a variety of doping types/levels and devices, showing that substantial changes in the strength and character of the nonlinearity should be expected in doped silicon resonators One common way of characterizing the nonlinear behavior of a resonator is to analyze the amplitude-frequency effect (A-f effect). For nonlinear analysis, correction terms are added to the equation of motion (EOM) of the resonator, which yields: 2 2 (1) where k1 and k2 are the 2nd and 3rd order nonlinear forcing terms respectively. The linear natural frequency is defined as / . Analytically it has been shown that the peak frequency is a function of the peak amplitude in the presence of the nonlinear terms [10]: ′ (2) where X0 is the resonator peak amplitude, ′ is the shifted frequency at large displacement, and is the A-f coefficient, which is calculated as: (3) In this paper, we present the modeling and experimental observation of how doping affects the devices’ nonlinear behavior. DESIGN AND FABRICATION Three types of bulk-mode resonators are designed and fabricated (See Table.1 for resonator dimensions): length extensional mode resonator (LE), Lamé mode resonator (Lamé), and dual breathe mode ring resonator (Ring). Nonlinearities in capacitive resonators are usually classified into electrostatic and mechanical (geometric and material). Compared with flexural-mode resonators, bulk-mode resonators are usually dominated by material nonlinearity effects, thus making them suitable candidates for studying the effect of doping [8]. Figure 1: Mode shape of the three types of bulk-mode resonators Four different types of doping levels are investigated, with the measured device layer resistivity listed as follows: Phosphorus (NP) 1. 78mΩ-cm, Arsenic (NA) 3. 1mΩ-cm, Antimony (NS) 17. 1mΩcm, and Boron (PB) 15. 8mΩ-cm. These doping densities are representative of the characteristics of commercially-available silicon-on-insulator (SOI) wafers The devices are fabricated using a wafer-level encapsulation process (epi-seal). The process starts off with (100) SOI wafer with a 40μm single-crystal device layer doped as described above, and a 2μm buried oxide layer. First, deep reactive ion etching is used to etch the device structural outline. A 2μm sacrificial oxide layer is then deposited and electrical contacts to the device layer are etched. Afterwards, the first epitaxial polysilicon cap layer is deposited. Vent holes are then etched and the resonant structures are released using vapor phase HF. Immediately after the release, the second polysilicon cap layer, which acts as the final encapsulation layer is deposited. Residual hydrogen is then diffused out to create a low pressure cavity (< 1Pa). Finally, electrical contacts are etched and aluminum is deposited and patterned. This process provides a clean and low pressure environment, which allows the devices to achieve long-term stability [11]. Figure 2: Final cross section view of the epi-seal process 9781940470016/HH2014/$25©2014TRF 285 Solid-State Sensors, Actuators and Microsystems Workshop Hilton Head Island, South Carolina, June 8-12, 2014 RESONATOR MODELING In this section, we present lumped mass models following the method given in [8] to analyze the nonlinear dynamics of these continuum systems. To begin, the resonators are transformed into single degree-of-freedom systems with the lumped EOM given as: