Variable capacitor design

From [29], the in-plane gap closing comb structure is used for the variable capacitor, as shown in Fig. 2.11. In Fig. 2.11(a), the dark areas are anchor regions and the light areas are movable regions. The spring of the in-plane gap closing variable capacitor makes it easier to move perpendicular to the finger gaps. Fig.2.11 (b) shows the comb fingers with silicon nitride deposited on the sidewalls for electrical insulation.

Fig. 2.11(a) Top view of the in-plane gap closing variable capacitor, (b) fingers with silicon nitride coating, (c) equivalent capacitance model between fingers

Cdielectric

The symbols used in the following analysis are listed below d : initial gap between comb fingers,

t : thickness of silicon nitride coated on the sidewalls of fingers, Lf : overlap length of comb fingers,

Wf : comb finger width,

h : thickness of comb fingers,

Ng : number of variable capacitor cells,

z : relative displacement between the movable and fixed electrodes, ε0 : permittivity of free space,

εr : relative permittivity of silicon nitride (εr=7)

The thickness of silicon nitride t is 500 Å to avoid the electronic tunneling effect in the charging process. The bumps on the sidewalls of the fingers keep the minimum finger spacing at 0.5 μm. Fig. 2.11 (c) shows the equivalent air gap of the capacitor is _eq

r

d = d - 2t +2t

with dielectric thickness t and relatively permittivity εε r. The total variable capacitance between the comb fingers is [29]

v g o f r

From this equation, the variable capacitance strongly depends on the comb finger structure. A general model of the comb finger structure is shown in Fig. 2.12. The layout includes the movable plate and fingers. A number of free parameters can be adjusted to obtain the optimal output power. The guideline of the optimization of parameters was also presented in [30, 31].

As mentioned above, the output power of the device is produced by the variable capacitor, which is formed by the comb fingers. W0 and L0 are the width and length of

Lf

L0 d

W1 W1

W2

W0

z x

y

Fingers Movable plate

the layout. W1 is the movable plate width on both side in the x-axis direction and W2

is in the y-axis direction. These two parameters determine the mass of the movable plate. W1 should be designed as large as possible because it determines the number of fingers, which is related to the output power. But the initial gap between fingers d also influences the number of fingers for a given chip area constraint. W2 is determined by the overlap length of fingers Lf. In order to have much output power, the comb structure center width should be designed as small as possible to have much number of fingers. But the robustness should be considered during vibration. Without external mass attachment structure design, the comb fingers structure has the center width of 1000 μm to ensure the robust structure and power requirement.

Fig. 2.12 A general model of the movable plate

The output power versus variable capacitor parameters is calculated according to Eq. 2.5 and Eq. 2.6. Device will be fabricated on the SOI wafer. The thickness h is choosing as 200 μm to have large capacitance. The finger width Wf of 10 μm is

restricted by the aspect ratio up of to 20:1 in the deep reactive ion etching process.

The mechanical damping force between fingers for large displacement is [32]

3 3

with b (z) as the equivalent mechanical damping constant. Device was operated in _m low damping environment that can improve the quality factor especially. The squeeze film damping effect can be ignored in the low damping environment. Thus, the mechanical damping constant is independent of the relative displacement z. Thus, the

damping force is proportional to the velocity z . The electrostatic force induced by ^. the charge Q on Cv is [28]

The force is proportional to the negative displacement of the movable plate. The be

can be regarded as the electrostatic spring constant. The electrostatic spring constant is determined by the charge Q on the variable capacitor which varies in the charge-discharge process. The maximum electrostatic spring constant be_max will reduce the total spring constant and make the movement unsteady during the vibration.

Therefore, the be_max should be limited as

2 2 The mass m of the movable plate is related to the density and thickness h of the device. According to Eq. 2.18 and Fig. 2.10, the limitation of the electrostatic spring constant be_max_lim is related to the ratio rd between the mechanical spring k and be_max

which increased with the desired maximum displacement (Eq. 2.14). With the above design parameters, the relationships of the be_max and be_max_lim are the functions of the initial gap d and the finger overlap length Lf, as shown in Fig. 2.13(a). If the be_max

layer is below the be_max_lim layer, it indicated the electrostatic spring softening effect can be ignored for the steady resonance at mentioned above. Fig. 2.13(b) is the two-dimension diagram of Fig. 2.13(a), which shows the available region that indicated be_max layer was below be_max_lim layer. Otherwise it is unavailable region. In Fig. 2.14, we can obtain the maximum power of 0.51 μW (power density of 12 μW /cm³) with the maximum displacement 36.4 μm and finger overlap length 535 μm in the available region. In this analysis, the initial gap between the fingers is 37 μm and the variable capacitance changes from 40 pF to 2080 pF. The output voltage is 5 V and the mass of movable plate is 0.0365g. The spring constant was 21 N/m to fit in with the resonant frequency of 120 Hz. Design parameters of the variable capacitor are listed in Table 2.1.

Fig. 2.13 be_max constraint for (a) three-dimension (b) two-dimension diagram 0 100 200 300 400 500 600

Fig. 2.13 be_max constraint for (a) three-dimension (b) two-dimension diagram (continued)

Fig. 2.14 Output power versus desired maximum displacement and finger overlap length

100 200 300 400 500 600

10

Table 2.1 Variable capacitor design parameters

Variable Description of variables Designed value

h Device thickness 200 μm

Ng Number of variable capacitor cells 788

Wf Finger width 10 μm

Lf Finger overlap length 535 μm

d Finger initial gap 37 μm

Zmax Maximum displacement 36.4 μm

t Silicon nitride sidewall thickness 500 Å Cmax Maximum value of capacitance 2080 pF Cmin Minimum value of capacitance 40 pF

k Mechanical spring const. 21 μN/μm

m Mass of movable plate 36.5mg

RL Driven load resistance 50 MΩ

Cstor Output temporary storage capacitor 5 nF Vout Output voltage (steady state) 5 V

Pout Output power (steady state) 0.51 μW