Page 65 - Principles and Applications of NanoMEMS Physics

52 Chapter 2

The devices are essentially parallel-plate capacitors, of nominal plate
separation g , in which the top plate (beam) is free to move in response to
0
an electrostatic force developed between it and the rigid bottom plate, as a
result of a voltage applied between the two.

2.2.2.1.1 Large-signal Actuation—Switch

For typical dimensions employed in MEMS [48], e.g., beam gaps, lengths,
widths, and thicknesses of about µ , 100− 250 µ s of µ m , and
2
m
m , 10
'
1− 10 µ m , respectively, the displacement behavior of the beams, which
manifests itself as continuous gap reduction versus applied voltage, is
dictated by the equilibrium F + F = 0 established between the
Coulomb Spring 2
quadratic electrostatic force , F = 1 ε 0 AV , and the linear spring
Coulomb 2
2 (g + ) z
0
force, F = − k z , (Hooke’s law) which attempts to bring the beam
Spring Beam
back to its undeflected position. This dynamic equilibrium, and its
accompanying smooth displacement, is maintained up to about one-third of
the beam-to-substrate distance, at which point it is lost and the beam
collapses onto the bottom plate, abruptly reducing the gap to zero. The
voltage demarcating these two regimes is called pull-in voltage and is given
by [49],

8k g 3
V Pull− = Beam 0
27 ε , (34)
in
A
0
where k is the spring constant of the beam, and A is the electrode area.
Beam

2.2.2.1.2 Small-signal Actuation—Resonator

For application as resonators [54], an AC voltage, together with a so-
called DC polarization voltage, introduced to enhance the current elicited by
the variable beam capacitance, are applied. Since the resonators are intended
for application as stable frequency standards, with frequency given by [18],

f = . 1 03κ E h , (35)
r ,nom 2
ρ L
r