This document discusses microelectromechanical systems (MEMS) devices such as cantilever beams that can be used for sensing applications. It describes the governing equations of motion for cantilever beam oscillation and methods for analyzing beam frequency and damping. Experimental data on beam resonant frequency is shown for various beam lengths. Applications discussed include logic gates built using electrostatically actuated beams. Pull-in voltage is measured and hysteresis due to stiction is observed. In summary, the document analyzes the dynamics and operation of MEMS cantilever beams and their potential use in sensors and basic logic circuits.

1. Subha Chakraborty
Department of Electronics and Electrical Communication Engineering
(E&ECE)
June 25, 2015

2. Low thickness materials are deposited on top of
June 25, 2015

3.  Low and high frequency (RF) switches:
Resistive and Capacitive
 Tunneling accelerometers
 Chemical and Bio sensors
June 25, 2015
 Logic operation

4. June 25, 2015

5. June 25, 2015
y0
L
b
h
insulating base
anchor
conducting cantilever beam
W
actuation pad
V
 Governing equation of motion
Aρ
txF
t
y
t
y
ζω
x
y
Aρ
EI e ),(
2 2
2
04
4
=
∂
∂
+
∂
∂
+
∂
∂
External force, for example
electrostatic actuation
Spring force
Damping force
Acceleration

6. June 25, 2015
Aρ
txF
t
y
t
y
ζω
x
y
Aρ
EI e ),(
2 2
2
04
4
=
∂
∂
+
∂
∂
+
∂
∂
 For frequency analysis reduce it to homogenous form
02 2
2
04
4
=
∂
∂
+
∂
∂
+
∂
∂
t
y
t
y
ζω
x
y
Aρ
EI
Use the method of separation of variables
)()(),( tTxXtxy =
Apply boundary conditions
0;00;0),0(
,
3
3
,
2
2
,0
=
∂
∂
=
∂
∂
=
∂
∂
=
tLtLt x
y
x
y
x
y
ty
Eigen Value Equation
01coshcos
4/1
2
4/1
2
=+
































L
EI
ωAρ
L
EI
ωAρ nn

7. June 25, 2015
Eigen Value Equation
01coshcos
4/1
2
4/1
2
=+
































L
EI
ωAρ
L
EI
ωAρ nn
ρ
ω
E
L
hkn
n 













= 2
2
1
4
12

8. June 25, 2015
ρ
ω
E
L
hkn
n 













= 2
2
1
4
12

9. June 25, 2015
ρ
ω
E
L
hkn
n 













= 2
2
1
4
12
Length
of beam
in
microm
eter
Theoretical
undamped
natural
frequency
Theoretical
damping
ratio for
squeezed
film
damping
Theoretical
resonant
frequency
under forced
oscillation
Measured
resonant
frequency in
LDV
100 275.4 KHZ 0.177 266.63 KHZ 265.3 KHZ
130 163.1 KHZ 0.203 156.23 KHZ 163.1 KHZ
160 107.5 KHZ 0.223 102.01 KHZ 93.75 KHZ
190 76.4 KHZ 0.244 71.71 KHZ 78.75 KHZ
220 56.9 KHZ 0.258 52.98 KHZ 45.0 KHZ
250 44.1 KHZ 0.281 40.47 KHZ 38.75 KHZ
280 35.2 KHZ 0.297 31.94 KHZ 32.5 KHZ
310 28.6 KHZ 0.313 25.65 KHZ 27.25 KHZ
340 23.8 KHZ 0.328 21.08 KHZ 21.79 KHZ
370 20.1 KHZ 0.342 17.59 KHZ 19.15 KHZ
400 17.2 KHZ 0.356 14.86 KHZ 16.33 KHZ
410 16.5 KHZ 0.359 14.22 KHZ 15.89 KHZ
420 15.7 KHZ 0.364 13.46 KHZ 14.65 KHZ
430 14.9 KHZ 0.369 12.71 KHZ 14.11 KHZ
440 14.3 KHZ 0.372 12.16 KHZ 13.65 KHZ
450 13.7 KHZ 0.379 11.57 KHZ 13.26 KHZ
 Quality factor
)1(2
1
2
ζζ
Q
−
=
ω
ω
Δ
0
=

10. June 25, 2015
 Quality factor
)1(2
1
2
ζζ
Q
−
=
ω
ω
Δ
0
=
Different damping mechanisms
affect the response of the cantilever
• Thermo-elastic Damping
(QTED ~ 105
– 107
)
• Attachment loss
(Qattch ~ 103
- 105
)
• Squeezed film Damping
(Qsqueeze) ~ 1 – 102
Most dominant source of
damping is squeezed film
damping
squeezeTEDsqueezeattch QQQQQ
11111
≅++=

11. June 25, 2015
Aρ
txF
t
y
t
y
ζω
x
y
Aρ
EI e ),(
2 2
2
04
4
=
∂
∂
+
∂
∂
+
∂
∂
 Euler Bernoulli equation under applied force Fe
For electrostatic actuation
2
2
0
0
)(
)(2
),( tV
yy
bε
txF se
−
=
Under steady state
2
0
2
0
4
4
)(2 yy
bVε
dx
yd
EI
−
=
Integrating this fourth order
differential equation twice using the
boundary conditions
')'(
)]'([2
1
2
0
2
0
2
2
dxxx
xyy
bVε
dx
yd
EI
L
x
∫ −
−
=
;0;0)0(
0
=
∂
∂
=
x
y
y
Boundary conditions
0;0 3
3
2
2
=
∂
∂
=
∂
∂
LL
x
y
x
y

12. June 25, 2015
Integrating this fourth order
differential equation twice using the
boundary conditions
')'(
)]'([2
1
2
0
2
0
2
2
dxxx
xyy
bVε
dx
yd
EI
L
x
∫ −
−
=
When the beam end deflection exceeds nearly one third of the initial gap
between the beam and the actuation electrode, the equilibrium between
electrostatic and spring force becomes unstable and the beam collapses on the
bottom electrode. This phenomenon is called PULL-IN.
 Once the beam pulls in contact stiction forces, namely Van der Waal force and
Casimir force come into action and the beam stays stuck to the bottom
electrode . This phenomenon is called STICTION.
 Due to stiction, the beam cannot release from the bottom electrode come
back to its initial position at the same voltage at which it the beam pulled in.
Therefore HYSTERISIS takes place.

13. June 25, 2015
The stiction forces strongly depend on
the roughness of the contact surface
between the beam and the bottom
electrode. As the roughness increases,
the stiction effects reduce, and hence
chances that the beam may release are
more.
Pull-in Pull-out characteristics
have been experimentally observed
for a fabricated cantilever of length
400 μm, width 10 μm, thickness 2 μm.
It shows pull-in voltage 5.4 V
(design value 4.85 V) and pull-out
voltage 0.8 V.

14. June 25, 2015
Vd
0
Vin Vout
0
0
0
Von
Von
0
Von
 When input is ‘0’, ‘p_beam’ is turned on
while ‘n_beam’ remains off, pulling output
to high level.
 When input is ‘1’, ‘n_beam’ is turned
on while ‘p_beam’ turns off, pooling
output to low level.
The working principle is similar to CMOS inverter
circuit.

15. June 25, 2015

16. June 25, 2015
Static transfer characteristics of
the inverter
Dynamic test results

17. For more details: subha.chakraborty@epfl.ch