1 Agentes dermatológicos

The theoretical background of the electrostatic actuation of the parallel plate capacitor is reviewed in section 2.3. There, we found that the actuation of the FPI is achieved through the balancing act of the electrostatic attraction between the opposite charges accumulated on each plate (𝐹_𝑒 = 𝜀𝐴

2 𝑉2

(𝑔0−𝑧)2) and the mechanical, restorative

force exerted by four bar springs in a shape of X-beam (𝐹_𝑘= −𝑘𝑧). Here A is the overlapping area of the capacitor plates, g0 the initial gap distance between two

plates, V the applied bias voltage, z the deflection distance when the bias voltage V is applied, and ε the permittivity for the medium filling the gap. It is a product of 𝜀_𝑟 and 𝜀₀, where 𝜀_𝑟 is dielectric constant or relative electrical permittivity of the medium, and 𝜀0 = 8.85 × 10−14 (F/V) the permittivity of free space. The beam

spring in our design satisfies a ‘fixed-guided’ boundary condition, where far end of the beam is held fixed so that its displacement and the slope cannot be changed by any applied force (‘fixed’). The other end where the beam is attached to the

membrane in the middle at its corner is free to move, but the slope is fixed to match that of the membrane (‘guided’). In section 2.3, we found the spring constant k for the fixed-guided beam spring as 𝑘 =𝐸𝑤𝑡3

𝑙3 , (equation 2. 41), where w, t, and l are the

width, thickness, and length of the beam, respectively. E is the Young’s modulus, a quantity that indicates how stiff the material is. The X-beam structure is analogous to the membrane with four springs attached in parallel. The spring constants add up

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when the springs are attached in parallel. The total spring constant for the entire structure is, therefore, 4𝑘 =4𝐸𝑤𝑡3

𝑙3 .

When the beams spring is forced to deflect more than half of their thickness, the stretching restoring force Fms has to be taken into account, along with the

stretching effect component of the spring constant ks, where 𝐹𝑚𝑠 = 𝑘𝑠𝑧3, and 𝑘𝑠 =

𝜋4𝐸𝑊𝑡3

4𝑙3 [34]. With these equations, the total mechanical, restorative force is modified

to 𝐹𝑚 = 𝑘𝑧 + 𝑘𝑠𝑧3. However, this modification is not applied in the analytic

solution to simplify the derivation of the solution. Moreover, the numerical method presented in the next section uses much more sophisticated model of the X-beam structure to find the actuation solution. Therefore, working with simpler first order approximation in analytic solution is justified.

Substituting the expression for k into Fk and solve the equation Fe = Fk, or 𝐴

2 𝑉2 (𝑔0−𝑧)2 =

4𝐸𝑤𝑡3

𝑙3 𝑧 for the deflection distance z, we find a polynomial in z of degree

three. Instead of using cubic formula1, the equation Fk = Fe is solved to find the

deflection distance as a function of applied bias voltage using the Matlab software. The values for each parameter found in the previous chapter are used in the simulated solution: The area of the plate A = 4 (mm2_{); The permittivity of the gap medium}

𝜀 = 𝜀0 = 8.85 × 10−14 (F/V) for air; The initial gap distance g0 = 25 (µm); The

dimension of the beam spring, l = 2.12 (mm), w = 0.15 (mm), and t = 25 (µm), for

1 _{The solution of 𝑎𝑥}3_{+ 𝑏𝑥}2_{+ 𝑐𝑥 + 𝑑 = 0 is given by 𝑥 = {𝑞 + [𝑞}2_{+ (𝑟 − 𝑝}2₎3_]1_2}1₃₊

{𝑞 − [𝑞2_{+ (𝑟 − 𝑝}2₎3_]1_2}1_{3+ 𝑝, where 𝑝 = −} 𝑏 3𝑎, 𝑞 = 𝑝 3₊(𝑏𝑐−3𝑎𝑑) 6𝑎2 , 𝑟 = 𝑐 3𝑎.

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length, width, and thickness, respectively; The Young’s modulus E = 130 (GPa) for Si in the direction 45 degree to the wafer flat. The bias voltage V is increased from 0 to 130V in an increment of 10V. The graph of the result is given in figure 4.5.

0 20 40 60 80 100 120 0 2 4 6 8 D e fle ct io n D ist a n ce (u m) Bias Voltage (V)

The Deflection vs. Applied Bias Voltage

Fig. 4.5 The graph of Deflection vs. Applied Bias Voltage. The deflection distances are found analytically with various bias voltage values. A cubic relation between the deflection and the bias voltage, as well as onset of the pull-in around 130V, is

demonstrated.

The graphs of the mechanical restorative force Fk and the series of attractive

electrostatic force Fe with several bias voltage values are produced with the Matlab

and given in figure 4.6. The same device parameter values are used to demonstrate the pull-in phenomenon discussed in the chapter 2. At lower values of bias voltage V,

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the graphs of Fk and Fe gives two solutions for the gap distance as two different

intersection points. The larger of these two gap distances obtained at the intersections is discarded as physically impossible solution since it is greater than the pull-in distance. The values of intersections on the graph matches well with the values obtained above by solving an equation Fk = Fe. As the bias voltage V increases, the

graphs of Fk and Fe intersect at a single point. This is an onset of pull-in or snap-in

phenomenon. This critical voltage at which the pull-in sets in is called the pull-in voltage or Vp. With bias voltage larger than Vp, there is no common intersection of

Fk and Fe. Thus, no equilibrium gap distance can be obtained. In section 2.3 we

found that at the onset of pull-in, the function and the first derivative the graphs of Fk

and Fe intersects at a single point. With the air gap, from 𝐹𝑘 = 𝐹𝑒 and

𝑑𝐹𝑘 𝑑𝑧 = 𝑑𝐹𝑒 𝑑𝑧, we have 𝑘𝑧 = 𝜀₀𝐴 2 𝑉2 (𝑔0−𝑧)2 and 𝑘 = 𝜀0𝐴 𝑉2

(𝑔0−𝑧)3 . Canceling k and solving for z, we get = 𝑔0

3 . Therefore, when a parallel plate actuator with the initial gap distance of g0 used,

the maximum actuation distance achievable is 𝑔0

3, and any further increase of bias

voltage results the pull-in and two plates snap together. For the pull-in voltage Vp,

we substitute 𝑧 =𝑔0

3 into any one of equations above and solve it for Vp. The result is

𝑉_𝑃 = √8𝑘𝑔03

27𝜀0𝐴 . This is the maximum bias voltage we need to deflect the top

membrane to the maximum actuation distance of 𝑔0

3 . With the values of parameters

in our design, we have the maximum deflection distance of 8.33 µm at the pull-in voltage, Vp = 129.38 V. The maximum deflection distance and the pull-in voltage we

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found here will be used as a rough guideline in fabrication and characterization process, since it is obtained through first order approximation of the electrostatic actuation system, and we don’t know the exact mechanism of pull-in phenomenon.

a) b)

Fig. 4.6 The graphical Solution of Fk = Fe. a) The bias voltage V lower than Vp, two

solutions are found as two intersections. The larger one is discarded on physical ground. At V = Vp, we have 𝐹_𝑘 = 𝐹_𝑒 and 𝑑𝐹𝑘

𝑑𝑧 =

𝑑𝐹𝑒

𝑑𝑧 . Solving those two equations,

we found z = 8.33 µm at Vp = 129.38 V for maximum deflection at the pull-in. b)

Zoomed-in view of the graph. The solutions at various bias voltage (the smaller x- value of the intersection of the graphs of Fk and Fe) are found to match well with the

values obtained above by analytically solving an equation Fk = Fe.