EE C247B ME C218 Introduction to MEMS Design Spring 2016
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1 EE C47B ME C18 Introduction to MEMS Design Spring 016 Prof. Clark T.-C. Nguyen Dept. of Electrical Engineering & Computer Sciences University of California at Berkeley Berkeley, CA 9470 Lecture EE C45: Introduction to MEMS Design LecM 1 C. Nguyen 11/18/08 1 Lecture Outline Reading: Senturia, Chpt. 5, Chpt. 6 Lecture Topics: Energy Conserving Transducers Charge Control Voltage Control Parallel-Plate Capacitive Transducers Linearizing Capacitive Actuators Electrical Stiffness Electrostatic Comb-Drive 1 st Order Analysis nd Order Analysis EE C45: Introduction to MEMS Design LecM 1 C. Nguyen 11/18/08 11
2 Basic Physics of Electrostatic Actuation Goal: Determine gap spacing g as a function of input variables First, need to determine the energy of the system Two ways to change the energy: Change the charge q Change the separation g DW(q,g) = VDq + F e Dg dw = Vdq + F e dg Note: We assume that the plates are supported elastically, so they don t collapse EE C45: Introduction to MEMS Design LecM 1 C. Nguyen 11/18/08 3 Here, the stored energy is the work done in increasing the gap after charging capacitor at zero gap Stored Energy EE C45: Introduction to MEMS Design LecM 1 C. Nguyen 11/18/08 4
3 Having found stored energy, we can now find the force acting on the plates and the voltage across them: Charge-Control Case + - V EE C45: Introduction to MEMS Design LecM 1 C. Nguyen 11/18/08 5 Voltage-Control Case Practical situation: We control V Charge control on the typical sub-pf MEMS actuation capacitor is difficult Need to find F e as a partial derivative of the stored energy W = W(V,g) with respect to g with V held constant? But can t do this with present W(q,g) formula Solution: Apply Legendre transformation and define the co-energy W (V,g) EE C45: Introduction to MEMS Design LecM 1 C. Nguyen 11/18/
4 Co-Energy Formulation For our present problem (i.e., movable capacitive plates), the co-energy formulation becomes EE C45: Introduction to MEMS Design LecM 1 C. Nguyen 11/18/08 7 Electrostatic Force (Voltage Control) Find co-energy in terms of voltage Variation of co-energy with respect to gap yields electrostatic force: Variation of co-energy with respect to voltage yields charge: EE C45: Introduction to MEMS Design LecM 1 C. Nguyen 11/18/
5 Spring-Suspended Capacitive Plate EE C45: Introduction to MEMS Design LecM 1 C. Nguyen 11/18/08 9 Charge Control of a Spring-Suspended C I EE C45: Introduction to MEMS Design LecM 1 C. Nguyen 11/18/
6 Voltage Control of a Spring-Suspended C EE C45: Introduction to MEMS Design LecM 1 C. Nguyen 11/18/08 11 Stability Analysis Net attractive force on the plate: An increment in gap dg leads to an increment in net attractive force df net EE C45: Introduction to MEMS Design LecM 1 C. Nguyen 11/18/
7 Pull-In Voltage V PI V PI = voltage at which the plates collapse The plate goes unstable when (1) and () Substituting (1) into (): When a gap is driven by a voltage to (/3) its original spacing, collapse will occur! EE C45: Introduction to MEMS Design LecM 1 C. Nguyen 11/18/08 13 Voltage-Controlled Plate Stability Graph Below: Plot of normalized electrostatic and spring forces vs. normalized displacement 1-(g/g o ) Spring Force Forces Electrical Forces Stable Equilibrium Points Increasing V Normalized Displacement EE C45: Introduction to MEMS Design LecM 1 C. Nguyen 11/18/
8 Advantages of Electrostatic Actuators Easy to manufacture in micromachining processes, since conductors and air gaps are all that s needed low cost! Energy conserving only parasitic energy loss through I R losses in conductors and interconnects Variety of geometries available that allow tailoring of the relationships between voltage, force, and displacement Electrostatic forces can become very large when dimensions shrink electrostatics scales well! Same capacitive structures can be used for both drive and sense of velocity or displacement Simplicity of transducer greatly reduces mechanical energy losses, allowing the highest Q s for resonant structures EE C45: Introduction to MEMS Design LecM 1 C. Nguyen 11/18/08 15 Problems With Electrostatic Actuators Nonlinear voltage-to-force transfer function Relatively weak compared with other transducers (e.g., piezoelectric), but things get better as dimensions scale EE C45: Introduction to MEMS Design LecM 1 C. Nguyen 11/18/
9 Linearizing the Voltage-to-Force Transfer Function EE C45: Introduction to MEMS Design LecM 1 C. Nguyen 11/18/08 17 Linearizing the Voltage-to-Force T.F. Apply a DC bias (or polarization) voltage V P together with the intended input (or drive) voltage v i (t), where V P >> v i (t) v(t) = V P + v i (t) EE C45: Introduction to MEMS Design LecM 1 C. Nguyen 11/18/
10 Differential Capacitive Transducer The net force on the suspended center electrode is EE C45: Introduction to MEMS Design LecM 1 C. Nguyen 11/18/08 19 Remaining Nonlinearity (Electrical Stiffness) EE C45: Introduction to MEMS Design LecM 1 C. Nguyen 11/18/
11 Parallel-Plate Capacitive Nonlinearity Eample: clamped-clamped laterally driven beam with balanced electrodes Electrode Nomenclature: V a or v A v a = v a cost Conductive Structure k m V A d 1 V a or v A = V A + v a t m Total Value AC or Signal Component (lower case variable; v 1 DC Component lower case subscript) (upper case variable; upper case subscript) V 1 V P F d1 EE C45: Introduction to MEMS Design LecM 1 C. Nguyen 11/18/08 1 Parallel-Plate Capacitive Nonlinearity Eample: clamped-clamped laterally driven beam with balanced electrodes Electrode Epression for C/: Conductive Structure k m d 1 m v 1 F d1 V 1 V P EE C45: Introduction to MEMS Design LecM 1 C. Nguyen 11/18/
12 Parallel-Plate Capacitive Nonlinearity Thus, the epression for force from the left side becomes: Electrode Conductive Structure k m d 1 m v 1 F d1 V 1 V P EE C45: Introduction to MEMS Design LecM 1 C. Nguyen 11/18/08 3 Parallel-Plate Capacitive Nonlinearity Retaining only terms at the drive frequency: Co1 Co1 Fd 1 VP v cosot V 1 1 P1 o d d Drive force arising from the input ecitation voltage at the frequency of this voltage 1 sin t EE C45: Introduction to MEMS Design LecM 1 C. Nguyen 11/18/ Proportional to displacement 90 o phase-shifted from drive, so in phase with displacement These two together mean that this force acts against the spring restoring force! A negative spring constant Since it derives from V P, we call it Co1 A the electrical stiffness, given by: ke VP1 V P1 3 d 1 o d 1 1 1
13 Electrical Stiffness, k e The electrical stiffness k e behaves like any other stiffness It affects resonance frequency: o k m V o o 1 - k k km k 1 - m k P1 m m - k m e m e 1 A 3 d 1 1 Electrode v 1 d 1 Conductive Structure k m m F d1 Frequency is now a function of dc-bias V P1 V 1 V P EE C45: Introduction to MEMS Design LecM 1 C. Nguyen 11/18/08 5 Voltage-Controllable Center Frequency Gap EE C45: Introduction to MEMS Design LecM 1 C. Nguyen 11/18/
14 Microresonator Thermal Stability -1.7ppm/ o C Poly-Si mresonator - 17ppm/ o C Thermal stability of poly-si micromechanical resonator is 10X worse than the worst case of AT-cut quartz crystal EE C45: Introduction to MEMS Design LecM 1 C. Nguyen 11/18/08 7 Use a temperature dependent mechanical stiffness to null frequency shifts due to Young s modulus thermal dep. [Hsu et al, IEDM 00] Geometric-Stress Compensation [W.-T. Hsu, et al., IEDM 00] Problems: stress relaation compromised design fleibility [Hsu et al IEDM 000] EE C45: Introduction to MEMS Design LecM 1 C. Nguyen 11/18/
15 Voltage-Controllable Center Frequency Gap EE C45: Introduction to MEMS Design LecM 1 C. Nguyen 11/18/08 9 Ecellent Temperature Stability Resonator Top Metal Electrode Top Electrode-to- Resonator Gap Elect. Stiffness: k e ~ 1/d 3 Frequency: f o ~ (k m - k e ) 0.5 Counteracts reduction in frequency due to Young s modulus temp. dependence -1.7ppm/ o C Uncompensated mresonator [Hsu et al MEMS 0] Elect.-Stiffness Compensation -0.4ppm/ o C AT-cut Quartz Crystal at Various Cut Angles On par with quartz! 100 EE C45: Introduction to MEMS Design LecM 1 C. Nguyen [Ref: Hafner] 11/18/
16 Measured Df/f vs. T for k e - Compensated mresonators 1500 V P -V C 16V V 500 1V 10V 9V 0 8V 7V V 4V Design/Performance: V f o =10MHz, Q=4, V V P =8V, h e =4mm -0.4ppm/ o C CC d o =1000Å, h=mm W r =8mm, L r =40mm [Hsu et al MEMS 0] Temperature [K] Slits help to release the stress generated by lateral thermal epansion linear TC f curves 0.4ppm/ o C!!! EE C45: Introduction to MEMS Design LecM 1 C. Nguyen 11/18/08 31 Can One Cancel k e w/ Two Electrodes? What if we don t like the dependence of frequency on V P? Can we cancel k e via a differential input electrode configuration? If we do a similar analysis for F d at Electrode : F d o -V Subtracts from the F d1 term, as epected P V P C d o C d o v cos t o sin t Adds to the quadrature term k e s add, no matter the electrode configuration! o EE C45: Introduction to MEMS Design LecM 1 C. Nguyen 11/18/08 3 v 1 V 1 F d F d1 Electrode 1 d 1 d V P m k m Electrode v V 16 16
17 Problems With Parallel-Plate C Drive Nonlinear voltage-to-force transfer function Resonance frequency becomes dependent on parameters (e.g., bias voltage V P ) Output current will also take on nonlinear characteristics as amplitude grows (i.e., as approaches d o ) Noise can alias due to nonlinearity Range of motion is small For larger motion, need larger gap but larger gap weakens the electrostatic force Large motion is often needed (e.g., by gyroscopes, vibromotors, optical MEMS) F d F d1 Electrode 1 v 1 V 1 V P d 1 d m k m Electrode v V EE C45: Introduction to MEMS Design LecM 1 C. Nguyen 11/18/08 33 Electrostatic Comb Drive EE C45: Introduction to MEMS Design LecM 1 C. Nguyen 11/18/
18 Electrostatic Comb Drive Use of comb-capacitive tranducers brings many benefits Linearizes voltage-generated input forces (Ideally) eliminates dependence of frequency on dc-bias Allows a large range of motion y Stator Rotor z Comb-Driven Folded Beam Actuator EE C45: Introduction to MEMS Design LecM 1 C. Nguyen 11/18/08 35 Comb-Drive Force Equation (1 st Pass) Top View y z L f Side View EE C45: Introduction to MEMS Design LecM 1 C. Nguyen 11/18/
19 Lateral Comb-Drive Electrical Stiffness Top View y z L f Side View Again: C Nh C Nh d d No (C/) -dependence no electrical stiffness: k e = 0! Frequency immune to changes in V P or gap spacing! EE C45: Introduction to MEMS Design LecM 1 C. Nguyen 11/18/08 37 Typical Drive & Sense Configuration y z EE C45: Introduction to MEMS Design LecM 1 C. Nguyen 11/18/
20 Comb-Drive Force Equation ( nd Pass) In our 1 st pass, we accounted for Parallel-plate capacitance between stator and rotor but neglected: Fringing fields Capacitance to the substrate All of these capacitors must be included when evaluating the energy epression! Stator Rotor Ground Plane EE C45: Introduction to MEMS Design LecM 1 C. Nguyen 11/18/08 39 Comb-Drive Force With Ground Plane Correction Finger displacement changes not only the capacitance between stator and rotor, but also between these structures and the ground plane modifies the capacitive energy [Gary Fedder, Ph.D., UC Berkeley, 1994] EE C45: Introduction to MEMS Design LecM 1 C. Nguyen 11/18/
21 Case: V r = V P = 0V C sp depends on whether or not fingers are engaged Capacitance Epressions Capacitance per unit length Region Region 3 [Gary Fedder, Ph.D., UC Berkeley, 1994] EE C45: Introduction to MEMS Design LecM 1 C. Nguyen 11/18/08 41 Comb-Drive Force With Ground Plane Correction Finger displacement changes not only the capacitance between stator and rotor, but also between these structures and the ground plane modifies the capacitive energy [Gary Fedder, Ph.D., UC Berkeley, 1994] EE C45: Introduction to MEMS Design LecM 1 C. Nguyen 11/18/
22 Simulate to Get Capacitors Force Below: D finite element simulation 0-40% reduction of F e, EE C45: Introduction to MEMS Design LecM 1 C. Nguyen 11/18/08 43 Vertical Force (Levitation) W 1 dcsp 1 dcrp 1 dcrs Fe, z Vs Vr s - z dz dz dz 1 d Csp, e Crs Fe, z N V dz For V r = 0V (as shown): V V EE C45: Introduction to MEMS Design LecM 1 C. Nguyen 11/18/08 44 s r
23 Simulated Levitation Force Below: simulated vertical force F z vs. z at different V P s [f/ Bill Tang Ph.D., UCB, 1990] See that F z is roughly proportional to z for z less than z o it s like an electrical stiffness that adds to the mechanical stiffness zo - z Fz zvp kezo - z o Electrical Stiffness z EE C45: Introduction to MEMS Design LecM 1 C. Nguyen 11/18/08 45 z / zo Vertical resonance frequency Vertical Resonance Frequency Vertical resonance frequency at V P = 0V k k z z e z k e V zo k where z zo = Lateral resonance frequency Signs of electrical stiffnesses in MEMS: Comb (-ais) k e = 0 Comb (z-ais) k e > 0 Parallel Plate k e < 0 EE C45: Introduction to MEMS Design LecM 1 C. Nguyen 11/18/
24 Suppressing Levitation Pattern ground plane polysilicon into differentially ecited electrodes to minimize field lines terminating on top of comb Penalty: -ais force is reduced EE C45: Introduction to MEMS Design LecM 1 C. Nguyen 11/18/08 47 Force of Comb-Drive vs. Parallel-Plate L f L d Gap = d o = 1 mm Thickness = h = mm Finger Length = L f = 100 mm Finger Overlap = L d = 75 mm Comb drive (-direction) V 1 = V = V S = 1V F e, Differential Parallel-Plate (y-direction) V 1 = 0V, V = 1V F F F e, y e, e, y Parallel-plate generates a much larger force; but at the cost of linearity EE C45: Introduction to MEMS Design LecM 1 C. Nguyen 11/18/ oh V d o hl s o d do 1 hl 1 o d do oh V d o V s V L d d o 4 4
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