Page 22 - Mechanical design of microresonators _ modeling and applications

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Design at Resonance of Mechanical Microsystems

Design at Resonance of Mechanical Microsystems 21
Intrinsic losses in the resonator material are important mechanisms
accounting for energy dissipation. In a recent paper, Czaplewski et al. 17
analyzed the loss mechanisms in tetrahedral amorphous carbon (taC)
and concluded that paramount in extrinsic internal dissipation is the
defect motion, which is generated by structural reconfiguration through
atomic motion between equilibrium or metastable states. The nomen-
clature and model of the anelastic or standard solid are utilized to
18
quantify the material losses; see Freudenthal. The model basis is the
incompressible viscoelastic equation (also known as Zener’s model, see
19
Cleland and Roukes ), which describes such a behavior, namely, where
dı dİ
ı + IJ = E(İ + IJ ) (1.58)
1 dt 2 dt

where ı = normal stress
İ = normal strain
IJ = stress-relaxation time
1
IJ = strain retardation time
2
This material model captures both the stress relaxation phenomenon
(which means decrease of the stress in a component when constant
strain is applied) and the strain retardation phenomenon, or creep
(which implies the increase of strain under the application of a constant
19
stress). Cleland and Roukes suggested the following quality factor as
being responsible for intrinsic losses:
1 = ȦIJ E íE
d
Q 2 2 E (1.59)
m 1+ Ȧ IJ
where E d is the dynamic (or unrelaxed) Young’s modulus, which is ex-
perimentally determined by means of rapidly applying the test load,
and ı is an aggregate relaxation time, which is calculated as

IJ = IJ IJ (1.60)
1 2
Other factors accounting for energy dissipation in microresonators
include losses through thin layers (surface losses), losses connected to
the shape of the mechanical components, generated through phonon-
phonon scattering, Nyquist-Johnson noise produced by electronic
circuitry, transduction losses, adsorption-desorption processes from
residual gases, environmental thermal drifts, or material defects such
as those encountered at grain boundaries. By adding the losses that
have been discussed in this section, the overall quality factor is
calculated as

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