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Subha Chakraborty
Department of Electronics and Electrical Communication Engineering (E&ECE)
April 18, 2023April 18, 2023
Low thickness materials are deposited on top of
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Low and high frequency (RF) switches: Resistive and Capacitive
Tunneling accelerometers
Chemical and Bio sensors
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Logic operation
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y0
Lbh
insulating base
anchorconducting cantilever beam
W
actuation pad
V Governing equation of motion
Aρ
txF
t
y
t
yζω
x
y
Aρ
EI e ),(2
2
2
04
4
External force, for example electrostatic actuation
Spring force
Damping force
Acceleration
April 18, 2023April 18, 2023
Aρ
txF
t
y
t
yζω
x
y
Aρ
EI e ),(2
2
2
04
4
For frequency analysis reduce it to homogenous form
022
2
04
4
t
y
t
yζω
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
yty
Eigen Value Equation
01coshcos
4/124/12
L
EI
ωAρL
EI
ωAρ nn
April 18, 2023April 18, 2023
Eigen Value Equation
01coshcos
4/124/12
L
EI
ωAρL
EI
ωAρ nn
E
L
hknn
2
2
14
12
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E
L
hknn
2
2
14
12
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E
L
hknn
2
2
14
12
Length of beam in micrometer
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
12ζζ
Q
ω
ω
Δ0
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Quality factor
)1(2
12ζζ
Q
ω
ω
Δ0
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
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Aρ
txF
t
y
t
yζω
x
y
Aρ
EI e ),(2
2
2
04
4
Euler Bernoulli equation under applied force Fe
For electrostatic actuation2
20
0 )()(2
),( tVyy
bεtxF se
Under steady state
20
20
4
4
)(2 yy
bVε
dx
ydEI
Integrating this fourth order differential equation twice using the boundary conditions
')'()]'([2
12
0
20
2
2
dxxxxyy
bVε
dx
ydEI
L
x
;0;0)0(0
x
yy
Boundary conditions
0;03
3
2
2
LLx
y
x
y
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Integrating this fourth order differential equation twice using the boundary conditions
')'()]'([2
12
0
20
2
2
dxxxxyy
bVε
dx
ydEI
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.
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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.
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Vd
0
Vin Vout
0
0
0
Vo
n
Von
0
Vo
n
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.
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Static transfer characteristics of the inverter
Dynamic test results
For more details: [email protected]
