Page 224 - An Introduction to Microelectromechanical Systems Engineering

Microelectromechanical Resonators 203

total spring constant for the system of spring beams is k =2 k . Because the
total beam
springs do not move as much as the main shuttle mass, only a fraction of their mass
is added to that of the shuttle mass in determining the resonant frequency.
A representative comb-drive resonator made of polycrystalline silicon using
standard surface-micromachining techniques has beams with a thickness 2 µm,
widths of 2 µm, and lengths of 185 µm, resulting in a system spring constant of 0.65
N/m. With an effective motional mass equal to 5.7 × 10 −11 kg, the structure reso-
nates at 17 kHz [17]. Keeping the same beam thickness and width but reducing the
length to 33 µm gives a structure that resonates at 300 kHz [18]. The Q can be over
50,000 in vacuum but rapidly decreases to below 50 at atmospheric pressure due to
viscous damping in air [19]. Thus, vacuum packaging is necessary to commercialize
these high-Q devices.
To attain a higher resonant frequency, the total spring constant must be
increased or the motional mass must be decreased. The former is done by increasing
the beam width and decreasing its length; the latter is difficult to do while retaining
a rigid shuttle with the same number of comb fingers. Using electron-beam lithogra-
phy to write submicron linewidths, single-crystal silicon beams with lengths of 10
µm and widths of 0.2 µm reached a resonant frequency of 14 MHz [20]. Alterna-
tively, while using the same resonator dimensions, the resonant frequency can be
increased by using a material with a larger ratio of Young’s modulus, E, to density,
ρ, than silicon. Metals known in engineering for their high stiffness-to-mass ratio,
such as aluminum and titanium, have a ratio E/ρ that is actually lower than for sili-
con. Two materials with higher E/ρ ratios are silicon carbide and polycrystalline
diamond; the latter is a research topic for high-frequency resonators.

Beam Resonators
To build a micromachined structure with higher resonant frequency than that read-
ily achievable with a comb drive, the mass must be further reduced. Beam resona-
tors have been studied extensively at the University of Michigan, Ann Arbor
[21–23], for this purpose, and Discera, Inc., of Ann Arbor, Michigan, is commer-
cializing them for reference frequency oscillators to replace quartz crystals in cellu-
lar phones. The advantages include a much smaller size, the ability to build several
different frequency references on a single chip, higher resonant frequencies, more
linear frequency variation with temperature over a wide range, and the ability to
integrate circuitry, either on the same chip or on a circuit chip bonded to the MEM
chip, all at a lower cost than the traditional technology.
The simplest beam resonator is rigidly clamped on both ends and driven by an
underlying electrode (see Figure 7.10). A dc voltage applied between the beam and
the drive electrode causes the center of the beam to deflect downward; removal
allows it to travel back upward. An ac drive signal ν causes the beam to flex up and
a
down. As with the comb-drive actuator, when a large dc bias V is superimposed to
D
the ac drive signal, the beam oscillates at the same frequency as the drive signal. At
resonance, the deflection amplitude is at its greatest. An example polysilicon beam
is 41 µm long, 8 µm wide, 1.9 µm thick, with a gap of 130 nm [21]. Applying volt-
ages V = 10V and ν = 3 mV, the measured resonant frequency is 8.5 MHz, the
D a
quality factor Q is 8,000 at a pressure of 9 Pa, and the deflection amplitude is a mere
4.9 nm at the center of the beam. At atmospheric pressure, Q drops to less than