6.777J/2.732J Design and Fabrication of Microelectromechanical Devices Spring Term Massachusetts Institute of Technology

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1 6.777J/.7J Design and Fabrication of Microelectromecanical Devices Spring Term 007 Massacusetts Institute of Tecnology PROBLEM SET 4 SOLUTIONS (5 pts) Issued /7/07 Due /6/07 Problem 6.5 (5 pts): Te in-plane interdigitated electrostatic (or comb drive) transducer a). Current drive case Te gap g between te finger tip and te electrode can be expressed in terms of te displacement z as follow, g g 0 z Te effective capacitor lengt is ence, l g l g 0 + z. Te capacitance of te upper part of comb is (by neglecting te electric fields around te finger tip), ε (l g) t 0 C upper Since te one-finger model is equivalent to two capacitors in parallel (upper and lower parts), te total capacitance is multiplied by two, ε 0 ( l g) t C Te energy stored in te system is Substitute C into te equation above, we ave Te force becomes To find te gap, we ave Te voltage can be expressed as, Q W vdq Q dq Q C C Q 0 0 Q W 4 ε 0 ( l g) t W Q F g Q 4 ε 0 tl ( g) F kz k(g g 0 ) F Q g g 0 g 0 k 4ε 0 tk ( l g) W Q v Q ε g 0 t( l g) Cite as: Carol Livermore and Joel Voldman, course materials for 6.777J Design and Fabrication of Microelectromecanical Devices, Spring 007. MIT OpenCourseWare (ttp://ocw.mit.edu/),

2 b). Assumptions made in calculating te force in a). a. Te upper and lower gaps of te comb fingers are exactly te same as fabricated b. Fringing effects are neglected. c. Te energy stored between te finger tip and te electrode (gap g) is neglected. d. Te electrical permittivity of air is assumed to be te one of free space. e. Tere is no space carge in te space between te plates. f. Te actuation of te comb drive is quasi-static. g. Carge distributed uniformly on te plates c). Voltage drive case We can use co-energy to find force and carge. Te co-energy in te capacitor is V V W * (, ) Qdv Cvdv Cv ε 0 (l g) tv v g Te force And te carge To find te gap, we again ave 0 0 ε tv 0 W * F ε 0 tv F W * Q v g g V Q ε 0 (l g)tv F kz k(g g 0 ) k( g g 0 ) g g ε 0 tv 0 k d). Te net force for te voltage drive case: F net ε 0 tv + K (g 0 g) F net k < 0 g Te effective spring constant is a negative constant, wic means te increase in gap will cause te decrease in force, terefore, te system is always stable and tere is no spring softening/ardening. e). Te net force for te current drive case: F net Q 4 F g ε 0 t(l g) + k(g 0 g) net Q k eff ε0 k < 0 t( l g) Te effective spring constant is also negative, owever, it s not a constant. Since g always decreases from g 0 to 0 wen actuation starts, decreases as te comb drive is actuated, creating te spring softening effect. However, k eff since k eff is always negative regardless of te value of g, te system is always stable, and ence, no pull-in will occur. Tis conclusion is based on te assumption tat te electric fields around te fingertips are negligible. Also, it is assumed tat te upper gap and lower gap of te comb fingers are te same, wile in reality, tey migt vary due to nonuniform etcing or oter fabrication effects. Pull-in, ence, can occur due to tese secondary effects. Cite as: Carol Livermore and Joel Voldman, course materi als for 6.777J Design and Fabrication of Microelectromecanical Devices, Spring 007. MIT OpenCourseWare (ttp://ocw.mit.edu/),

3 Problem 6.9 ( pts): Design te spring Since we ave springs in parallel (as sown in te figure), te effective spring constant of eac is k eq k parallel / 0.5 kn/m Eac spring acts as a beam wit a fixed support at one end (te ancor) and subjected at te oter end to a zero slope boundary condition (because of its connection to te mass), despite its ability to translate. Tis BC imposes a moment reaction at te oter end. Hence te resultant deflection is te superposition of tabulated deflections: tat of a cantilever beam subjected to a point force at its end and tat of a cantilever beam subjected to a moment at its end. Te moment is unknown yet, and will be found by substituting te zero-slope BC into te resultant deflection profile. F For a cantilever (lengt L, widt a, tickness t) wit point force F at te end: wˆ( x ) ( x + x L ) Eta For a cantilever (lengt L, widt a, tickness t) wit a moment M at te end: wˆ ( x ) 6M x Eta F 6M Te resulting deflection is tus: wx ˆ ( ) ŵ ( x ) + w ˆ ( x ) ( x + x L ) + x Eta Eta Using te boundary condition, dw ˆ( x ) F M ( x + 6xL ) + x 0 dx Eta Eta x L x L we can solve for M and find: FL M We ten plug back into our deflection equation to find: Evaluating at xl provides: wx ˆ ( ) F ( x +.5 xl ) Eta FL wl ˆ( ) Eta Springs Mass We can ten solve for te equivalent spring constant. F Eta k eq wl ˆ( ) L Subsituting E 50 GPa, a 0 µm, and t 00 µm, we get: L 0.7 µm. We can fit suc a long spring into a minimal wafer area by folding it, as sown: Cite as: Carol Livermore and Joel Voldman, course materi als for 6.777J Design and Fabrication of Microelectromecanical Devices, Spring 007. MIT OpenCourseWare (ttp://ocw.mit.edu/),

4 Problem 6.0 (7 pts): Design a simple switc (a) Using t, l, w, and g in [m] to avoid confusion, and all oter parameters in standard SI units: 8 kg V PI 0V () 7 εa cap (assuming electrical force applied as point force at tip) Ewt k () 4l (assuming ideal cantilever support) A cap l o w () (assuming capacitor area does not vary muc as switc is deflected) l R 0Ω (4) σ wt (assuming canges in lengt and widt of cantilever between te closed and open switc modes are negligible) l 5w (5) w 0t (6) Cost constraint: lw $ / m + (t 0 6 ) u(t 0 6 m) 0 6 $ / m $ (7) (u is te unit step function, since we start paying for te tickness once it exceeds µm, according to te problem statement). (b) Substituting E 50 GPa, σ 0 5 S/m, ρ00 kg/m, l 0 0 µm, and ε ε x 0 - F/m (i.e. assuming te gap is vacuum), we get: gt m l (A) (from (), (), and ()) l 0 6 m wt (B) l 5 w (C) Combining (B) and (C) yields te following relation: l 5 t 0 w 6 5 t 0 m m t 6 m 6 Terefore, from pysical constraints alone t min 5 0 m 5µm Because we are forced to use a beam tickness greater tan µm, in te cost function te term tat involves te step function is in te active state. We terefore find ourselves in a linear regime were any increases in te beam area and/or te beam tickness beyond prescribed pysical minima will add to te cost of te part. From (A), l gt / Te l corresponding wit te t min of 5 µm and te minimum gap (0.5 µm) is given by: l min /( ) m 70µm. 4 Cite as: Carol Livermore and Joel Voldman, course materials for 6.777J Design and Fabrication of Microelectromecanical Devices, Spring 007. MIT OpenCourseWare (ttp://ocw.mit.edu/), Massacusetts Institute of Tecnology. Downloaded on [DD Mont YYYY].

5 Te w follows from (B): l m w t m 54 0 m w ( ) 0 Te smallest w we can ave and still satisfy all of te pysical constraints associated wit te problem is terefore: w min m 54µm 6 We also know from (6) tat te condition w 0t must be satisfied. Since t min 5 0 m 5µm and w min m 54 µm, tis last condition is met, and we ave terefore found our design values. Tus g 0.5µm, t 5µm, w 54µm, l 70µm. Substituting into equations ()-(4) and (7), we get: V c 9.98 V, R 0Ω, and te minimum cost is: (5 ) 8.9$ (c) Process: (Note tat te actual dimensions of te device are sligtly larger tan tose of te cantilever. Hence te cost will be sligtly iger tan tat calculated above.. Start wit a silicon wafer, perform RCA clean wit HF dip.. LPCVD 0. µm of silicon nitride to act as insulator between te electrode and switc along te substrate pat.. LPCVD 0. µm of polysilicon. 4. Perform potolitograpy using positive potoresist (not sown) and Mask to define te electrode. 5. Dry-etc te polysilicon using reactive-ion etcing. Ten as resist and perform RCA clean (witout HF dip). 6. PECVD 0.5 µm of sacrificial oxide 7. Perform potolitograpy using positive potoresist (not sown) and Mask to pattern te sacrificial layer. 8. Wet-etc te oxide using BOE. Ten as resist and perform RCA clean (witout HF dip). 9. LPCVD 5 µm of polysilicon. 0. Perform potolitograpy using positive potoresist (not sown) and Mask to define te cantilever.. Dry-etc te polysilicon using reactive-ion etcing. Ten as resist and perform RCA clean (witout HF dip).. Release te cantilever by etcing te oxide wit BOE followed by super-critical freeze drying. 5 Cite as: Carol Livermore and Joel Voldman, course materi als for 6.777J Design and Fabrication of Microelectromecanical Devices, Spring 007. MIT OpenCourseWare (ttp://ocw.mit.edu/),

6 Cross sections Masks 58 µm 0 µm 5 8 Mask 8 µm 6 µm Mask 9 µm 54 µm Mask Silicon wafer Nitride Polysilicon Oxide Figure. Process flow and mask set for micromacined cantilever switc Problem 6. (7 pts): A simple MEMS resonator (a) For a doubly clamped cantilever beam, te effective spring constant is: k 6Ewt l (See p. 690 in Gere & Timosenko, Mecanics of Materials, 4 t Ed.) m actual ρ lwt Te mass is : meff 0.4m actual 8kg o 8 6Ewt g o 8Et g Te pull in voltage is V o PI 7εA cap 7ε l elec wl 7ε l elec l (b) Based on te linearized model in te text: () εa cap ε l w V C V ε l w C V ε l wv C 0 elec o o o elec o o elec, k' k k, and φ ĝ o ĝ o εa cap ĝ o ĝ o ĝ o ĝ o o 6 Cite as: Carol Livermore and Joel Voldman, course materi als for 6.777J Design and Fabrication of Microelectromecanical Devices, Spring 007. MIT OpenCourseWare (ttp://ocw.mit.edu/),

7 (c) Wen we refer te impedances to te electrical side, we multiply eac impedance on te mecanical side by (/φ), tus: V ε l w o elec k Z ρ lwtĝ 4 s ZC 4 Z ĝ / k ' o ĝ m o o Z L Ls φ (ε lelec wvo ) φ (ε l elec wv o ) s C s 4 m ρ lwtĝ o C φ k' kĝ o V o ε l elec wĝ o (ε l elec wv o ) L φ (ε l wv ) 4 elec o ĝ o is found by solving: ĝ o g o V o ε l elec w kĝ o (d) Z in Z C //(Z C + Z L ) // + sl + LC s o sc o sc (s)(c o + C + LC o C s ) s + sc 0 LC s + + L C C () Tis is a tird order system wit poles and zeros. Substituting s jω and setting te denominator and numerator to zero to find te poles and zeros respectively, we find tat: Te poles occur at s0 (ω 0) and s± jω were ω Te zeros occur at s± jω were ω LC C o + C LC o C k 4t E (e) Wen V 0 0, ω () ρ lwt l ρ Wen V 0 αv PI, ω k V o ε l elec w / ĝ o 6Ewt / l V o ε l elec w / ĝ o LC ρ lwt ρ lwt 6Ewt / l α 8Et g o ε l elec w /(7ε l elec l ĝ o ) ρ lwt 6Et α 8(g o / ĝ o ) Et / 7 ρ l 4 4t E 8α (g o / ĝ o ) ( ) (4) l ρ 7 α V PIε l ĝ o g o kĝ o elec w 4α g g o 7 o ĝ (5) o Cite as: Carol Livermore and Joel Voldman, course materi als for 6.777J Design and Fabrication of Microelectromecanical Devices, Spring 007. MIT OpenCourseWare (ttp://ocw.mit.edu/), 7

8 (f) As α in equation (4) above, V V PI, ĝ o g o, and tus: ω 4 t E ( α ) ω α l ρ At V 0 0, ω ω. At V V PI, using te above approximation: ω ω ω 0. And tus te maximum cange in ω according to tis approximation is: 00 69% Te actual cange in ω will be less tan tat calculated above because te approximate form for ω is really only accurate as α. In fact, if we solve (5) numerically, we get ĝ o g o 4t E (/ ).646t E Substituting in (4) gives: ω l ρ 7 l ρ Tus te actual maximum cange in ω will only be : 00.85% (6) 4t E (g) From (), l 7µm ω ρ π Hence l elec must be 7. µm. Use l elec 7 µm. Now from (): g o 7ε l elec V PI l 0.7µm 8Et w must be l/5 for te structure to act as a beam (rater tan a plate), ence we will pick w 0 µm. Figure below plots f ω / (π) versus V o as V o varies from 0.05V PI to 0.95V PI. Figure is a Bode plot of Z in at V o 0.05V PI and V o 0.95V PI.Te MATLAB code is attaced at te end of te solution. Cite as: Carol Livermore and Joel Voldman, course materials for 6.777J Design and Fabrication of Microelectromecanical Devices, Spring 007. MIT OpenCourseWare (ttp://ocw.mit.edu/), 8

9 Cite as: Carol Livermore and Joel Voldman, course materi als for 6.777J Design and Fabrication of Microelectromecanical Devices, Spring 007. MIT OpenCourseWare (ttp://ocw.mit.edu/), Massacusetts Institute of Tecnology. Downloaded on [DD Mont YYYY]. 9

10 % MATLAB code for prob 6. function outprob6_() g0 0.7e-6; lelec 7e-6; w 0e-6; l 7e-6; t e-6; e 8.85e-; ro 00; E 50e9; A lelec * w; k 6*(E*w*t^/l^); m t*l*w*ro; VPI sqrt(8*k*g0^/(7*e*a)); V linspace(0.05*vpi,0.99*vpi,00); for in:lengt(v) V0 V(in); G fzero(@(g) gap(g,v0,k,e,a,g0),g0); C0 e*a/g; pi C0*V0/G; kp(in) k*(-c0^*v0^/(e*a*k*g)); C pi^/kp(in); L m/pi^; w_r(in) /(*pi)*sqrt(/(l*c)); if in Z tf([/c0 0 /(L*C0*C)],[ 0 /L*(/C0+/C) 0]); end if in00 Z tf([/c0 0 /(L*C0*C)],[ 0 /L*(/C0+/C) 0]); end end w logspace(6,7,5000); bode(z,'r',z,'b',w) figure semilogy(v,w_r) function y gap(g,v0,k,e,a,g0) y G - g0 + e*a*v0^/(*k*g^); 0 Cite as: Carol Livermore and Joel Voldman, course materials for 6.777J Design and Fabrication of Microelectromecanical Devices, Spring 007. MIT OpenCourseWare (ttp://ocw.mit.edu/),