Online proceedings - EDA Publishing Association

11-13
May 2011, Aix-en-Provence, France
⎛ ⎞
⎜ −=
M
out PP 2P
⎛ 2P
⎞
characteristic frequencies ω M and ω long .
in 1 ⎟ ;
⎝ P
⎜ −=
M
IL Log 11 ⎟
The 0
approximated value for the actuation voltage for a
in ⎠
⎝ Pin
⎠ (7)
central actuation is obtained by using the definition of the
spring constant for the entire structure. The spring constant
The approach for calculating the absorbed power by
k is a measure of the potential energy of the bridge
longitudinal modes is the same given in Eq. (5), thus leading
accumulated as a consequence of its mechanical response to
to:
the electrical force due to the applied voltage V. An
approximated definition of it for central actuation can be
given by [8],[10]:
1
2
Plong
= ωlongCVRF
(8)
2
3
k k k =+=
K 32 Ewr
i.e. the capacitance will be affected by both longitudinal
and transversal modes, and, by using the same formalism
introduced in the previous equations, the full power
transferred to the mechanical system by the RF signal
passing through the line will be:
2
( ) CV
' 1
PM
M ωω+= long RF
(9)
2
The above equation is the measure of the total power
transferred to the beam because of the RF signal to both
longitudinal fundamental mode and transversal mode.
Higher order longitudinal modes will absorb a power
fraction scaled by the order of the excited mode, with lower
amount of power for the highest modes.
It is worth noting that in the spectrum reconstruction the
above contributions have to be separated, leading to
different values of the peak power. In particular, we should
have the following distribution, which accounts also for the
excitation of the two satellites:
I
I
I
out
out
out
( ω )
RF
( )
RF
long
( ωω ) ω CZ
RF
M
in
long
that the frequency of resonance and the capacitance
associated to the beam can be calibrated to fully absorb the
RF signal. Such a result is particularly interesting for
resonating structures based on double clamped
configurations, because the maximum absorption
corresponds to a resonance condition. Actually, such a
device is a notch filter and it could be used as a feedback
element in a one port oscillator. Another conclusion coming
out from Eq. (10) is that the power released to the
mechanical structure does not depend on the frequency of
the carrier, but just on the geometry of the beam and its
+ K
2
where:
K
1
r =
=
t
L
E
σ
[ ( − ) wr ]
1
( )
18 νσ
1
; K
2
⎛ L ⎞⎛
L ⎞
⎜ 22
−− ⎟⎜
⎟
⎝ L ⎠⎝
L ⎠
1
=
Lc
2 −
L
2
cc
(11)
(12)
L is the bridge total length, L c is the switch length in the
RF contact region (width of the central conductor of the
CPW), w is the bridge width, t is the Au thickness of the
bridge. The other parameters are the Young modulus E, the
residual stress σ and the Poisson coefficient ν. As well
established, the Young modulus is an intrinsic property of
the material, and specifically it is a measure of its stiffness.
Let’ use, as an example, the following structure for a RF
MEMS switch in coplanar waveguide (CPW) configuration:
L=600 μm as the bridge total length, L
2( M
PP
long
)
c =300 μm as the
+
1−=
Z021
(
M
+− ωω
long
) C
switch length
=
in the RF contact region (width of the central
P
conductor of the CPW), w=100 μm as the bridge width,
in
w S =100 μm for the switch width (transversal dimension of
PM
ωωω
M
==±
0 MCZ
(10) the switch, parallel with respect to the CPW direction),
Pin
d=thickness of the dielectric material=0.2 μm, with
dielectric constant ε=3.94 (SiO
P
2 ), t=1.5 μm for the gold
long
==± bridge, ρ=19320 kg/m 3 for the gold density, E=Young
0 long
P
modulus=80×10 9 Pa, ν=0.42 for the metal Poisson
coefficient and σ=18 MPa as the residual stress of the metal
From the first of Eq. (10) it is worth noting that the (measured on specific micromechanical test structures). A
intensity of the central peak could vanish under the uniform distribution of holes with 5 µm radius and distant
condition 21 Z 0 ( ω ω ) C =+−
0 . This is an evidence 10 µm each other has been also considered, leading to
effective values in terms of the beam area and spring
constant.
A recent experimental approach was also adopted for
evaluating the contribution of the spring constant and for
modeling it on the base of nano-indentation techniques[9].
All the quantities previously introduced have to be redefined
because of the presence of holes in the released
beam. The holes need to be used for an easier removal of
the sacrificial layer under the beam, and for mitigating the
stiffness of the gold metal bridge, i.e. for better controlling
the applied voltage necessary for collapsing it, to have not
values too high because of the residual stress.
265