EE C245 ME C218 Introduction to MEMS Design

EE C245 ME C218 Introduction to MEMS Design Fall 2007 Prof. Clark T.-C. Nguyen Dept. of Electrical Engineering & Computer Sciences University of California at Berkeley Berkeley, CA 94720 Lecture 20: Equivalent Circuits it I & Project EE C245: Introduction to MEMS Design Lecture 20 C. Nguyen 11/6/08 1

Lecture Outline Reading: Senturia, Chpt. 5, Chpt. 6, Chpt. 21 Lecture Topics: Lumped Mechanical Equivalent Circuits Project: Gyroscopes Deep Reactive-Ion Etching (revisited) Lossless Transducers EE C245: Introduction to MEMS Design Lecture 20 C. Nguyen 11/6/08 2

Free-Free Beam Frequency (cont) Applying B.C. s, get A=B and B=D, and (3) Setting the determinant = 0 yields Which has roots at Substituting (2) into (1) finally yields: Free-Free Beam Frequency Equation EE C245: Introduction to MEMS Design Lecture 19 C. Nguyen 11/4/08 3

Higher Order Free-Free Beam Modes More than 10x increase Fundamental Mode (n=1) 1 st Harmonic (n=2) 2 nd Harmonic (n=3) EE C245: Introduction to MEMS Design Lecture 19 C. Nguyen 11/4/08 4

Mode Shape Expression The mode shape expression can be obtained by using the fact that A=B and C=D into (2), yielding Get the amplitude ratio by expanding (3) [the matrix] and solving, which yields Then just substitute the roots for each mode to get the expression for mode shape Fundamental Mode (n=1) [Substitute ] EE C245: Introduction to MEMS Design Lecture 19 C. Nguyen 11/4/08 5

Lumped Parameter Mechanical Equivalent Circuit it EE C245: Introduction to MEMS Design Lecture 19 C. Nguyen 11/4/08 6

Equivalent Dynamic Mass Once the mode shape is known, the lumped parameter equivalent circuit can then be specified Determine the equivalent mass at a specific location x using knowledge of kinetic energy and velocity z Location x W h Maximum Kinetic Energy Density Equivalent Mass = Maximum Velocity @ location x Maximum Velocity Function EE C245: Introduction to MEMS Design Lecture 19 C. Nguyen 11/4/08 7

Equivalent Dynamic Mass z Location x W h We know the mode shape, so we can write expressions for displacement and velocity at resonance EE C245: Introduction to MEMS Design Lecture 20 C. Nguyen 11/6/08 8

Equivalent Dynamic Stiffness & Damping z Location x W h Stiffness then follows directly from knowledge of mass and resonance frequency And damping also follows readily EE C245: Introduction to MEMS Design Lecture 20 C. Nguyen 11/6/08 9

Equivalent Lumped Mechanical Circuit z Location x W h K eq (x) C eq (x) K M eq (x) 2 eq ( x ) = ω o M eq M eq ( x) ( x ) = ρ ωo Meq( x) C eq ( x ) = Q 2 A l [ u ( x )] dx o 2 [ u( x)] EE C245: Introduction to MEMS Design Lecture 20 C. Nguyen 11/6/08 10

Equivalent Lumped Mechanical Circuit z Example: Polysilicon w/ l=14.9μm, W=6μm, h=2μm 70 MHz W h K eq (0) = 19,927927 N/m M eq (0) = 1.03x10-13 kg K eq (node) = M eq (node) = C eq (node) = K eq (l/2) = 53,938 N/m M eq (l/2) = 2.78x10-13 kg C eq (0) = 5.66x10-9 kg/s C eq (l/2) = 1.53x10-8 kg/s EE C245: Introduction to MEMS Design Lecture 20 C. Nguyen 11/6/08 11

3CC 3λ/4 Bridged μmechanical Filter Performance: f o =9MHz, BW=20kHz, PBW=0.2% IL=2 I.L.=2.79dB, Stop. Rej.=51dB 20dB S.F.=1.95, 40dB S.F.=6.45 V P In Out 0 Transmis ssion [db B] -10-20 -30-40 -50-60 P in =-20dBm [S.-S. Li, Nguyen, FCS 05] Sharper roll-off Loss Pole 8.7 8.9 9.1 9.3 [Li, et al., UFFCS 04] Frequency [MHz] Design: L r =40μm W r =6.5μm h r =2μm L c =3.5μm 35 L b =1.6μm V P =10.47V P=-5dBm R Qi =R Qo =12kΩ EE C245: Introduction to MEMS Design Lecture 20 C. Nguyen 11/6/08 12

Electromechanical Analogies k eq l x c x r x m eq c eq EE C245: Introduction to MEMS Design Lecture 20 C. Nguyen 11/6/08 13

Electromechanical Analogies (cont) Mechanical-to-electrical correspondence in the current analogy: EE C245: Introduction to MEMS Design Lecture 20 C. Nguyen 11/6/08 14

Bandpass Biquad Transfer Function k eq m eq c eq EE C245: Introduction to MEMS Design Lecture 20 C. Nguyen 11/6/08 15

Gyroscopes EE C245: Introduction to MEMS Design Lecture 20 C. Nguyen 11/6/08 16

Classic Spinning Gyroscope A gyroscope measures rotation rate, which then gives orientation very important, of course, for navigation Principle i of operation based on conservation of momentum Example: classic spinning gyroscope Rotor will preserve its angular momentum (i.e., will maintain its axis of spin) despite rotation of its gimbled chassis EE C245: Introduction to MEMS Design Lecture 20 C. Nguyen 11/6/08 17

Vibratory Gyroscopes Generate momentum by vibrating structures Again, conservation of momentum leads to mechanisms for measuring rotation ti rate and orientation ti Example: vibrating mass in a rotating frame y Driven into vibration along the y-axis Mass at rest y Get an x component x x of motion C(t) Rotate 30 o C(t 2 ) > C(t 1 ) C(t 1 ) C(t 2 ) y-displaced mass Capacitance between mass and frame = constant EE C245: Introduction to MEMS Design Lecture 20 C. Nguyen 11/6/08 18

Basic Vibratory Gyroscope Operation Principle of Operation Tuning Fork Gyroscope: Input Rotation a r c Coriolis () Response z Ω r Driven Vibration @ f o v r Coriolis Torque x z y EE C245: Introduction to MEMS Design Lecture 20 C. Nguyen 11/6/08 19

Basic Vibratory Gyroscope Operation Principle of Operation Tuning Fork Gyroscope: Input Rotation a r c Coriolis () Response Coriolis Torque z Ω r Driven Vibration @ f o x v r Am mplitude Coriolis Acceleration Coriolis Force Drive/ Response Spectra: Drive Response r x Driven Velocity r r r a c = 2v Ω r F k r ma f o (@ T 1 ) c c c z 2 k ω r y Coriolis Displacement = = Beam Stiffness = Response Rotation Rate r a Beam Mass ω Frequency EE C245: Introduction to MEMS Design Lecture 20 C. Nguyen 11/6/08 20

Vibratory Gyroscope Performance Principle of Operation Tuning Fork Gyroscope: Input Rotation z Driven Vibration @ f o r x = r F k c = Beam Mass r ma k c r a = ω Beam Stiffness c 2 r Frequency r r r a c = 2v Ω Driven Velocity Ω r a r v r To maximize the output signal x, c need: Large sense-axis mass Small sense-axis stiffness p (Above together mean low resonance frequency) Large drive amplitude for large driven velocity (so use comb- drive) z If can match drive freq. to y sense freq., then can amplify output by Q times Coriolis () Response Coriolis Torque x EE C245: Introduction to MEMS Design Lecture 20 C. Nguyen 11/6/08 21

MEMS-Based Gyroscopes Vibrating Ring Gyroscope Tuning Fork Gyroscope [Ayazi, GA Tech.] Tuning Fork Gyroscope [Draper Labs.] 3.2 mm Nuclear Magnetic Resonance Gyro [NIST] [Najafi, Michigan] Laser Polarizer 1 mm Rb/Xe Cell Photodiode 1 mm θ & t EE C245: Introduction to MEMS Design Lecture 20 C. Nguyen 11/6/08 22

MEMS-Based Tuning Fork Gyroscope Drive Electrode Tuning Tuning Drive Electrode Di Drive Mode Md Quadrature Cancellation In-plane pan drive and sense modes pick pc up z-axis rotations Mode-matching for maximum output sensitivity From [Zaman, Ayazi, et al, MEMS 06] Mode EE C245: Introduction to MEMS Design Lecture 20 C. Nguyen 11/6/08 23

MEMS-Based Tuning Fork Gyroscope Ω r z Tuning Drive Voltage Signal (-) Output Current (+) Output Current Tuning Drive Electrode Drive [Zaman, Ayazi, et al, MEMS 06] Drive Oscillation Sustaining Amplifier Differential TransR Amplifier EE C245: Introduction to MEMS Design Lecture 20 C. Nguyen 11/6/08 24

MEMS-Based Tuning Fork Gyroscope Drive and sense axes must be stable or at least track one another to avoid output drift Drive Electrode Tuning Quadrature Cancellation Need: small or matched drive and sense axis temperature coefficients to suppress drift Amplitu ude Tuning Drive Electrode Drive Drive Response Problem: if drive frequency changes relative to sense frequency, output changes bias drift T 1 Response EE C245: Introduction to MEMS Design Lecture 20 C. Nguyen 11/6/08 25 T 2 f o (@ T 1 ) f o (@ T 2 ) ω

Mode Matching for Higher Resolution For higher resolution, can try to match drive and sense axis resonance frequencies and benefit from Q amplification Drive Electrode Tuning Problem: mismatch between drive and sense frequencies even larger drift! Tuning Drive Electrode Drive T 1 Response Quadrature Cancellation Need: small or matched drive and sense axis temperature coefficients to make this work Amplitu ude Drive Response EE C245: Introduction to MEMS Design Lecture 20 C. Nguyen 11/6/08 26 T 2 f o (@ T 1 ) f o (@ T 2 ) ω

Issue: Zero Rate Bias Error Imbalances in the system can lead to zero rate bias error Quadrature Cancellation Di Drive Electrode Tuning Tuning Drive Electrode Mass imbalance off-axis motion of the proof mass Quadrature Cancellation Drive imbalance off-axis motion of the proof mass Output signal in phase with the Coriolis acceleration Quadrature output signal that can be confused with the Coriolis acceleration EE C245: Introduction to MEMS Design Lecture 20 C. Nguyen 11/6/08 27

Nuclear Magnetic Res. Gyroscope The ultimate in miniaturized spinning gyroscopes? from CSAC, we may now have the technology to do this -20º 0º Better if this is a noble gas nucleus (rather than e-), since nuclei are heavier less susceptible to B field 20º Soln: Spin polarize Xe 129 nuclei by first polarizing e- of Rb 87 (a la CSAC), then allowing spin exchange Atoms Aligned Nuclear Spins -20º 0º 20º 3.2 mm Challenge: suppressing the effects of B field Laser Polarizer Rb/Xe Cell Photodiode 1 mm θ & t 1 mm EE C245: Introduction to MEMS Design Lecture 20 C. Nguyen 11/6/08 28

Deep Reactive-Ion Etching EE C245: Introduction to MEMS Design Lecture 20 C. Nguyen 11/6/08 30

Deep Reactive-Ion Etching (DRIE) The Bosch process: Inductively-coupled plasma Etch Rate: 1.5-4 μm/min Two main cycles in the etch: Etch cycle (5-15 s): SF 6 (SF x+ ) etches Si Deposition cycle: (5-15 s): C 4 F 8 deposits fluorocarbon protective polymer (CF 2- ) n Etch mask selectivity: SiO 2 ~ 200:1 Photoresist ~ 100:1 Issue: finite sidewall roughness scalloping < 50 nm Sidewall angle: 90 o ±2 o EE C245: Introduction to MEMS Design Lecture 20 C. Nguyen 11/6/08 31

DRIE Issues: Etch Rate Variance Etch rate is diffusion-limited and drops for narrow trenches Adjust mask layout to eliminate large disparities Adjust process parameters (slow down the etch rate to that governed by the slowest feature) Etch rate decreases with trench width EE C245: Introduction to MEMS Design Lecture 20 C. Nguyen 11/6/08 32

DRIE Issues: Footing Etch depth precision Etch stop: buried layer of SiO 2 Due to 200:1 selectivity, the (vertical) etch practically just stops when it reaches SiO 2 Problem: Lateral undercut at Si/SiO 2 interface footing Caused by charge accumulation at the insulator Poor charge relaxation and lack of neutralization of e - s at insulator Ion flux into substrate bt t bild builds up (+) potential Charging-induced potential perturbs the E-field Distorts the ion trajectory Result: strong and localized damage to the structure at Si-SiO 2 interface footing EE C245: Introduction to MEMS Design Lecture 20 C. Nguyen 11/6/08 33

Recipe-Based Suppression of Footing Use higher process pressure to reduce ion charging [Nozawa] High operating pressure concentration of (-) ions increases and can neutralize (+) surface charge Issue: must introduce as a separate recipe when the etch reaches the Si-insulator interface, so must be able to very accurately predict the time needed for etching Adjust etch recipe to reduce overetching [Schmidt] Change C 4F 8 flow rate, pressure, etc., to enhance passivation and reduce overetching Issue: Difficult to simultaneously control footing in a narrow trench and prevent grass in wide trenches Use lower frequency plasma to avoid surface charging [Morioka] Low frequency more ions with low directionality and kinetic energy neutralizes (-) potential barrier at trench entrance Allows e - s to reach the trench base and neutralize (+) charge maintain charge balance inside the trench EE C245: Introduction to MEMS Design Lecture 20 C. Nguyen 11/6/08 34

Metal Interlayer to Prevent Footing (a) Photolithography 1 (f) Silicon Thinning (sacrificial) (b) Preparatory trenches (g) Photolithography 2 (c) Metal interlayer deposition (h) DRIE Pre-defined metal interlayer grounded to substrate supplies e s to neutralize (+) charge and prevent charge accumulation at the Si-insulator interface (d) Lift-off (remove PR) (i) Remove metal interlayer (e) Anodic Bonding (i) Metallize EE C245: Introduction to MEMS Design Lecture 20 C. Nguyen 11/6/08 35

Footing Prevention (cont.) Below: DRIE footing over an oxide stop layer Right: efficacy of the metal interlayer footing prevention approach [Kim, Stanford] No metal interlayer Footing No footing With metal interlayer [Kim, Seoul Nat. Univ.] EE C245: Introduction to MEMS Design Lecture 20 C. Nguyen 11/6/08 36

DRIE Examples High aspectratio gear Tunable Capacitor [Yao, Rockwell] Microgripper [Keller, MEMS Precision Instruments] EE C245: Introduction to MEMS Design Lecture 20 C. Nguyen 11/6/08 37