Page 19 - Mechanical design of microresonators _ modeling and applications

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Design at Resonance of Mechanical Microsystems

18 Chapter One

0.8
Q / Q s,u 0.00001
0.4
1 ×10 -6 y 0

z 0 1 × 10 -6
0.00001
Figure 1.17 Quality factor contribution by the Stokes flow underneath the movable plate.

as mechanical noise mechanisms, are particularly important when the
microdevice operates in vacuum, and therefore the energy dissipation
through viscous damping is minimized. A global estimator of the me-
chanical noise in mechanical microoscillators is the total noise equiva-
12
lent acceleration (TNEA) (see Yazdi, Ayazi, and Najafi ), defined as
k Tc k TȦ r
b
b
TNEA = 2 =2 (1.54)
m Qm
where k b is the Boltzmann constant and T is the absolute temperature.
One source of internal energy losses in microresonators is the
thermoelastic energy dissipation (TED) mechanism. More details on
this temperature-related loss mechanism are given in the works of
14
13
13
Roszhart and Lifshitz and Roukes. Roszhart mentions that this
mechanism is marked for beam resonators as thick as 10 ȝm, whereas
15
Yasumara et al. report experimental data showing that TED is
significant down to beam thicknesses of 2.3 ȝm. For a bent beam, such
as the one sketched in Fig. 1.18, the lower fibers are in tension and are
cooler, whereas the upper fibers, which are in compression, are warmer
than the undeformed beam. As a consequence of this temperature
difference, a temperature gradient is set over the beam thickness which
generates energy flow in the opposite direction, as shown in Fig. 1.18,
and this mechanism generates irreversible energy losses in the
mechanical microresonator.
The TED depends on material properties such as the coefficient of
thermal expansion Į, specific heat c , thermal conductivity ț, specific
p
mass (density) ȡ, elastic modulus (Young’s modulus for bending) E, as
well as on the temperature T and geometry. The quality factor which is
related to TED can be quantified into the following form (after
13
Roszhart ):

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