215 EE Homework 3
You can work on the layout problems in your design groups and just turn in one layout per group. Be sure to list the names of all the group members. Please do the non-layout problems individually. 1) Draw the cross section A-A of the following layout in the MUMPS process. The red material is Poly 1, the gray material is Poly 2 and the cross-hatched area is Anchor 2. What would be the function of this device once it has been released in a sacrificial HF etch?
This would function as a hinge once it has been released.
EE 215 Homework 3
2) 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 2331 kg/m3 and that the acceleration due to gravity is 9.8 m/sec2. Fracture = 1 GPa, = 2331 kg/m3 = F/A = mg/A = ((/4)d2Lg)/ (/4)d2 = Lg L = Fracture/g = (1x109 N/m2)/(2331 kg/m3)(9.8 m/sec2) = 4.38x104 meters 3) A silicon cantilever of length 500 m, width 50 m, and thickness 2 m is subjected to a uniform distributed transverse load P. Find the tip deflection, and calculate an effective spring constant for this beam. For a load that produces a tip deflection of 2 m, calculate the maximum stress at the support. Assume Youngs modulus = 160 GPa. From page 186 of Kovacs:
(1) y(x) = (Px2/24EI)(6L2 4Lx + x2) (2) P = F/L Substituting for P in (1) and finding the tip deflection at L: (3) y(L) = ((F/L)L2/24EI))(6L2 4L2 + L2) = 3FL3/24EI = FL3/8EI This is the tip deflection. To find the spring constant solve for F: (4) F = (8EI/L3)y(L) This is now in the form of Hooks Law; (5) F = -k
EE 215 Homework 3
Where k is the spring constant and is the spring deflection. From (4) & (5) we get:
(6) k = -(8EI/L )
3
(7) max = PL2t/4I = (F/L)(L2t/4I) = FLt/4I For a load that produces a 2 m tip deflection, from (3): FL3/8EI = y(L) = 2 m So that: (8) F = (2 m)(8EI)/L3 Substituting (8) into (7) and setting t = 2 m and L = 500 m: max = [(2 m)(8EI)Lt/L3]/4I = (2 m)(8E)(2 m)/[(4)(500 m)2]= 8E/(500)2 Subsituting E = 160 GPa: max = 8(160 GPa)/(500)2 = 5.12 MPa 4) A parallel plate electrostatic actuator with a variable capacitance C1 is in series with a fixed feedback capacitance C2, as shown in the figure below.
Spring constant K
C1
g0
Vs C2
a) The total capacitance of the system at zero-voltage would be given by the sum of the two capacitors in series: Ctotal = C1C2/(C1 + C2)
EE 215 Homework 3
With R = C1(V=0)/C2 and C1(V=0)= 0A/g0: Ctotal = 0A/(1 + R)g0 b) The voltage across the variable capacitor is determined by the voltage divider formed by the two capacitors C1 & C2: V = VS/(1 + C1/C2) = VS/(1 + R(g0/g)) c) The electrostatic force as a function of the supply voltage is give by: Felectrostatic = 0AV2/2g2 = (0A/2)(Vs2/g2(1+R(g0/g))2 = (0AVs2/2)/(g+Rg0)2 d) To determine when pull-in will occur, we can set the electrostatic force equal to the mechanical restoring force of the spring, and solve for the voltage. Pull-in will occur when dV/dg = 0 as we discussed in class. Felectrostatic = (0AVs2/2)/(g+Rg0)2 = Fmechanical = k(g0g) Solving for Vs: Vs = A[((g0g)(g+Rg0)2]1/2 where A = [2k/0A]1/2 dVs/dg = (A/2)[((g0g)(g+Rg0)2]-1/2[-(g+Rg0)2 + 2(g+Rg0)( g0g)] Setting dVs/dg = 0 gives: [-(g+Rg0)2 + 2(g+Rg0)( g0g)] = 0 2(g+Rg0)( g0g) = (g+Rg0)2 2(g0g) = g+Rg0 3g = 2g0-Rg0 g = (g0/3)(2-R) We see that the pull-in displacement depends on the value of the feedback capacitance. For R>2 there are not solutions (g<0) so all operating conditions are stable For R<2 there is still pull-in, but the stable operating range can be extended beyond g0/3.
EE 215 Homework 3
e) If we would like to eliminate pull-in completely, what capacitance value (measured in terms of the capacitance value of the parallel plate actuator) should we use? As shown above, we can eliminate the pull-in instability completely for R=2 where R = C1(V=0)/C2 = 2, so we can eliminate the pull-in instability by setting C2 = C1(V=0)/2. 5) (a) Using either the closed form M-TEST solutions 1 or the finite difference iterative method solver implemented in MATLAB 2 find the pull-in voltage the for following structures fabricated in Poly1 of the PolyMUMPS process:
Assume the following materials properties:
EE 215 Homework 3
Youngs Modulus = 160 GPa, Poissons ratio = 0.2, Transverse stress gradient = 0
Film Nitride Poly0 Oxide1 Poly1 Oxide2 Poly2 Metal T 6,171 5,020 20,452 20,108 7,581 15,015 5,188 SD 195 84 1267 307 446 200 SR 28.45 11.98 30.48 0.05 R 1.43E-03 2.41E-03 4.58E-03 2.59E-06 S 68 T 24 C 7C 15 C 21 T T-Thickness (A); SD-Standard Deviation (A); SR-Sheet Resistance; (ohm/sq); R-Resistivity (ohm-cm); "S-Stress (MPa)"
Assume the following dimensions: Cantilever beam (CB): Length 300 m 400 m 500 m Same as above Radius = 300, 400, 500 m Width = 100 m Width = 100 m Width = 100 m
Fixed-fixed beam (FB): Circular Diaphragm (CD):
Approximate pull-in voltages from MatLab: 300 m 400 m 500 m Fixed beam (FB): 300 m 400 m 500 m Circular Diaphragm (CD): 300 m 400 m 500 m Cantilever beam (CB): 6.3 V 3.5 V 2.3 V 45.1V 27.2V 18.7V 24.6V 15.9V 11.7V
6) Layout a comb drive resonator in the PolyMUMPS process using L-Edit in MEMS Pro as shown in the figure below. The length of the folded springs will be 150 m, the width of the beams will be 2 m wide and they will be separated by 18 m. The comb drive fingers will be 40 m long and 3 m wide, with a 3 m gap between the fingers. The fixed and released fingers should have an undeflected overlap of 20 m. You should not need to use poly 2 in your layout. Be sure to include metal on the top of the bond pads. Save the final layout file as Group Name_Resonator.tdb
EE 215 Homework 3
3 m lines and spaces
Resonator
2 m 18 m 150 m
40 m
20 m
7) Calculate the resonant frequency for the comb drive resonator in problem 6 above. a. Break the folded spring down into parts. First, consider the two beams, each of length L, shown in the figure below. Each of these beams is a fixed-guided beam, and they are connected in series so that the spring constant is given by 1/ktotal=1/k1+1/k2:
Two fixed-guided beams of length L connected in series Keffective=1/k1+1/k2 where k1=k2=12EI/L3 so that keffective=6EI/L3
EE 215 Homework 3
Next, consider the two springs shown below that are connected in parallel. The effective spring constant for springs connected in parallel is given by ktotal=k1+k2:
Two springs connected in parallel Keffective=k1+k2 where k1=k2=6EI/L3 so that keffective=12EI/L3
Finally, consider the two springs shown below that are connected in parallel:
Two springs connected in parallel Keffective=k1+k2 where k1=k2=12EI/L3 so that keffective=24EI/L3
So that the total effective spring constant for the two folded springs that are connected in parallel is given by:
k sys = 24 EI
L
3
= 2 Eh W
( L)
3
where
I = 1 hW 3 12
()
EE 215 Homework 3
The spring constant for the system is then given by:
Spring Constant Young's Modulus (E) Beam Thickness (h) Beam Width (W) Beam Length (L) k [N/m] 1.60E+11 2.00E-06 2.00E-06 1.50E-04 1.52E+00
b. Use the effective spring constant ksys found above to estimate the resonant frequency of the comb drive resonator using the following formula:
1 fr = 2
k sys M p + 0.3714M
1/ 2
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. The effective mass can be found from summing each of the individual components of the resonator:
EE 215 Homework 3
Plate Element Yellow Blue Cyan Green Fingers Total (um)^3 Total (m)^3 Density (kg/m^3) Mass Plate (kg) Beams Mass Beams (kg) Effective Mass (kg)
Length (um) Width (um) Thickness (um) Volume (um)^3 78 15 2 2340 170 12 2 4080 90 17 2 3060 54 20 2 2160 40 3 2 240
Total (um)^3 4680 8160 6120 2160 7200 28320 2.832E-14 2330 6.60E-11
150
2
2
600
4800 1.12E-11 7.01E-11
The resonant frequency is then given by:
Resonant Frequency k Effective mass f [Hz] 1.52E+00 7.01E-11 23,407
1
Peter M. Osterberg and Stephen 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 http://www-mtl.mit.edu/researchgroups/MEMCAD/mtest.html