EE215 Spring 2012 Homework 2

1) Layout a comb drive resonator in the PolyMUMPS process using L-Edit in MEMS Pro as

shown in the figure below. Use the poly 1 layer (h = 2 µm) for fabrication of the comb drives and folded springs. You can construct the resonator using elements from the MEMS Pro cell library. The length of the folded springs will be L = 150 µm, the width of the beams will be W = 2 µm wide and they will be separated by 18 µm. The comb drive fingers will be 40 µm long and 3 µm wide, with a 3 µm gap between the fingers. The fixed and released fingers should have an un-deflected overlap of 20 µm. You should not need to use poly 2 in your layout except for the bond pads. Be sure to use bond pads from the MEMS Pro cell library! Save the final layout file as Your-Name_Resonator.tdb and mail it to the instructor. Make a solid model of your resonator. Experiment with the scaling in the z-direction to make the solid model easier to visualize.

Resonator

150 µm

40 µm20 µm

2 µm

3 µm lines and spaces

18 µm

Figure 2.20. Comb drive resonator.

2) Calculate the resonant frequency for the comb drive resonator in problem 1 above.

a. First find the effective spring constant ksys for the 2 folded springs where each folded

spring is made up of four beams of length L (i.e. L = 150 µm in the layout in problem 1). Assume the trusses joining the folded spring segments are rigid, that Young’s modulus for polysilicon is E = 160 GPa and that the density of polysilicon is ρ = 2330 kg/m3. Hint: A folded spring can be broken down into fixed-guided springs that are connected in parallel and series. Springs that are connected in parallel have an effective spring constant that is the sum of the individual spring constants ktotal=k1+k2, like capacitors that are connected in parallel where Ctotal=C1+C2.

EE215 Spring 2012 Homework 2

Springs that are connected in series have an effective spring constant given by 1/ktotal=1/k1+1/k2, like capacitors that are connected in series where 1/Ctotal=1/C1+1/C2. The spring constant for a single fixed-guided beam is given by:

( ) ( ) 33

3 12112 hWIwhereL

WEhL

EIk guidedfixed ===−

b. Use the effective spring constant ksys found above to estimate the resonant frequency

of the comb drive resonator using the following formula:

2/1

3714.02

1

+=

MM

kf

p

sysr π

Where MP and M are the masses of the shuttle plate and of the supporting beams, respectively. This approximate expression is found using the Rayleigh Ritz energy method.

3) Assume that silicon will fracture when the axial stress reaches ≈ 1 GPa. Find the

maximum length of a vertical silicon rod which, under the action of its own gravitational load, will not exceed this fracture stress. Assume the density of silicon is ρ = 2331 kg/m3 and that the acceleration due to gravity is g = 9.8 m/sec2. Are you surprised?

4) Model a released mirror for a MEMS Fabry-Perot interferometer in the SOI-MUMPS

process as a stiff plate (2mm x 2mm) fabricated in the device layer of the 10 µm thick SOI wafer suspended by four diagonal fixed-guided cantilever beams (h = 10µm thick, W = 100 µm wide, L long) at each corner of the mirror as shown in the figure below. Find an expression for the deflection δ of the mirror as a function of the support arm length L assuming a force F is acting vertically on the mirror. What is the longest length L for the support arms that you can fit into the allowed 8mm x 8mm die site including the central 2mm x 2mm mirror? You can assume Young’s modulus for silicon is E = 169 GPa parallel to the wafer flat for a (100) wafer, 130 GPa at 45 degrees to the wafer flat and that Poisson’s ratio is ν = 0.28.

5) Calculate the voltage that will be required to deflect the mirror for the Fabry-Perot interferometer you designed in problem 4 using an electrostatic actuator. You can assume that the area of the parallel plate capacitor is the same as the 2 mm x 2mm mirror surface.

EE215 Spring 2012 Homework 2

Assume the initial gap from a second bonded wafer with a counter-electrode is 2.4 µm and that you need to be able to deflect the mirror surface until the final gap is 2/3 of the initial value, or 1.6 µm.

6) Two M-Test structuresi are shown in the figure below. These test structures are useful for characterizing the mechanical properties of thin films such as Young’s modulus, Poisson’s ratio and residual stress. They are also very common elements in MEMS design. In this problem we will determine the “pull-in” voltage of the cantilever and fixed-fixed beams analytically.

Figure 3.9. M-Test structures. Cantilever Beam (top), Fixed-Fixed Beam (bottom).

a. Layout each of these structures in L-Edit. Define the released structure (cantilever

beam or fixed-fixed beam, both 500 µm long and 100 µm wide) in Poly1. Instead of the dielectric spacer, as shown in the figure above, use a polysilicon anchor to nitride. Make the ground plane in Poly0. You can start the ground plane after the nitride anchor. Use Oxide1 to define a 2 µm gap between the released structure and the Poly0 ground plane.

b. Make a solid model of each of these structures using MEMS Pro. Draw

representative cross-sections through your structures.

c. Calculate the pull-in voltage for each of these structures analytically for the following dimensions:

Cantilever beam (CB): Length Width

300 µm 100 µm 400 µm 100 µm 500 µm 100 µm

EE215 Spring 2012 Homework 2

Fixed-fixed beam (FB): Same as above

Assume the following materials properties:

• Young’s Modulus E = 160 GPa • Poisson’s ratio ν = 0.2 • Transverse stress gradient = 0 • Oxide1 thickness = 2 µm • Poly1 thickness = 2 µm

You may also assume that each of these structures can be approximated as an ideal parallel plate actuator, and that there is a uniform electrostatic load on each of them. You can also ignore the length of the beam taken up by the anchors in your calculations, and any fringing effects of the fields. The displacements would then be given by:

Cantilever beam:

ρ

l

ρ

ll

( )222

4624

)( xxEI

xxy +−= ll

ρ

Fixed-fixed beam:

ρρ

( )22

24)( x

EI

xxy −= l

ρ

Where the moment of inertia I is given by:

12

3wtI =

Figure 3.10. Cantilever beam under uniform load ρ = F/L [N/m] (top), Fixed-fixed beam (bottom).

EE215 Spring 2012 Homework 2

To calculate the pull-in voltages, first determine an effective spring constant km for each of the structures. Note that the maximum displacement for the cantilever beam occurs at x = L and that the maximum displacement of the fixed-fixed beam occurs at x = L/2. Then calculate their initial capacitance C0. The pull-in voltage can then be found from equation 4.18 in Liu2:

0

0

5.13

2

C

kxV m

p =

7) Choose an article to review on optical MEMS for Thursday 5/3. It may be helpful for

your design project to review some of the articles on MEMS deformable mirrors from Tom Bifano’s group.3

i P.M. Osterberg and S.D. Senturia, M-Test: A Test Chip for MEMS Material Property Measurement Using Electrostatically Actuated Test Structures, J. Microelectromechanical Systems, Vol. 5, No. 2, pp. 107-118 (1997). 2 Chang Liu, Foundations of MEMS, Pearson Prentice Hall, p. 113 (2006). 3 http://people.bu.edu/bifano/Publications.html