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 23: Electrical l Stiffness EE C245: Introduction to MEMS Design Lecture 23 C. Nguyen 11/20/08 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 2 nd Order Analysis EE C245: Introduction to MEMS Design Lecture 23 C. Nguyen 11/20/08 2
Linearizing the Voltage-to-Force Transfer Function EE C245: Introduction to MEMS Design Lecture 23 C. Nguyen 11/20/08 3
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 C245: Introduction to MEMS Design Lecture 23 C. Nguyen 11/20/08 4
Differential Capacitive Transducer The net force on the suspended d center electrode is EE C245: Introduction to MEMS Design Lecture 23 C. Nguyen 11/20/08 5
Remaining Nonlinearity (Electrical l Stiffness) EE C245: Introduction to MEMS Design Lecture 23 C. Nguyen 11/20/08 6
Parallel-Plate Capacitive Nonlinearity Example: clamped-clamped laterally driven beam with balanced electrodes Nomenclature: V a or v A v a = v a cosωt Electrode Conductive Structure k m V A d 1 V a or v A = V A + v a t m x AC or Signal Component Total Value (lower case variable; v 1 DC Component lower case subscript) (upper case variable; V 1 upper case subscript) V P F d1 EE C245: Introduction to MEMS Design Lecture 23 C. Nguyen 11/20/08 7
Parallel-Plate Capacitive Nonlinearity Example: clamped-clamped laterally driven beam with balanced electrodes Expression for C/ x: Electrode Conductive Structure k m d 1 m x v 1 F d1 V 1 V P EE C245: Introduction to MEMS Design Lecture 23 C. Nguyen 11/20/08 8
Parallel-Plate Capacitive Nonlinearity Thus, the expression for force from the left side becomes: Electrode Conductive Structure k m d 1 m x v 1 F d1 V 1 V P EE C245: Introduction to MEMS Design Lecture 23 C. Nguyen 11/20/08 9
Parallel-Plate Capacitive Nonlinearity Retaining only terms at the drive frequency: F d C o 1 2 C o 1 1 = VP v cosωot VP x sin ωot ω 1 1 + 1 o 2 d d 1 Drive force arising from the Proportional to input excitation voltage at displacement the frequency of this voltage 90 o phase-shifted from drive, so in phase with displacement These two together th mean that t this force acts against the spring restoring force! A negative spring constant Since it derives from V P, we call it Co1 2 ε the electrical stiffness, given by: ke = VP1 = V 2 P1 3 1 2 A EE C245: Introduction to MEMS Design Lecture 23 C. Nguyen 11/20/08 10 d 1 d 1
Electrical Stiffness, k e The electrical stiffness k e behaves like any other stiffness It affects resonance frequency: k k m k = = m m Electrode e ωo d 1 k = m k 1 m k e m 2 V = 1 1 P εa ω o ω o 3 k d m 1 Frequency is now a function of dc-bias V P1 1 2 1 2 v 1 V 1 V P Conductive Structure k m m F d1 x EE C245: Introduction to MEMS Design Lecture 23 C. Nguyen 11/20/08 11
Voltage-Controllable Center Frequency Gap EE C245: Introduction to MEMS Design Lecture 23 C. Nguyen 11/20/08 12
Microresonator Thermal Stability 1.7ppm/ o C Poly-Si μresonator - 17ppm/ o C Thermal stability of poly-si micromechanical resonator is 10X worse than the worst case of AT-cut quartz crystal EE C245: Introduction to MEMS Design Lecture 23 C. Nguyen 11/20/08 13
Geometric-Stress Compensation Use a temperature dependent mechanical stiffness to null frequency shifts due to Young s modulus thermal dep. [Hsu et al, IEDM 00] [W.-T. Hsu, et al., IEDM 00] Problems: stress relaxation compromised design flexibility [Hsu et al IEDM 2000] EE C245: Introduction to MEMS Design Lecture 23 C. Nguyen 11/20/08 14
Voltage-Controllable Center Frequency Gap EE C245: Introduction to MEMS Design Lecture 23 C. Nguyen 11/20/08 15
Excellent Temperature Stability Resonator Top Metal Electrode [Hsu et al MEMS 02] Top Electrode-to- Resonator Gap Elect. Stiffness: k e ~ 1/d 3 Frequency: f o ~ (k m - k e ) 0.5 1.7ppm/ o C Elect.-Stiffness Compensation 0.24ppm/ o C AT-cut Quartz Crystal at Various Cut Angles Counteracts reduction in frequency due to Young s modulus temp. dependence Uncompensated μresonator 100 [Ref: Hafner] On par with quartz! EE C245: Introduction to MEMS Design Lecture 23 C. Nguyen 11/20/08 16
Design/Performance: f o =10MHz, Q=4,000 V P =8V, h e =4μm d o =1000Å, h=2μm W r =8μm, L r =40μm [Hsu et al MEMS 02] Measured Δf/f vs. T for k e - Compensated μresonators 1500 1000 500 0-500 -1000-1500 0.24ppm/ o C 300 320 340 360 380 Temperature [K] V P -V C Slits help to release the stress generated by lateral l thermal expansion linear TC f curves 0.24ppm/ o C!!! 16V 14V 12V 10V 9V 8V 7V 6V 4V 2V 0V CC EE C245: Introduction to MEMS Design Lecture 23 C. Nguyen 11/20/08 17
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 i input electrode configuration? F d2 F d1 k If we do a similar analysis for F d2 at Electrode 2: d 1 d 2 Subtracts from the F d1 term, as expected Co2 Fd 2 = VP v cos ωo t ω 2 2 o d + V 2 P2 2 C d o 2 2 2 x sinω t o V Adds to the quadrature term k e s add, 1 V V 2 P no matter the electrode configuration! EE C245: Introduction to MEMS Design Lecture 23 C. Nguyen 11/20/08 18 v 1 Ele ectrode e 1 m x k m Electrod de 2 v 2
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 x 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, V vibromotors, optical MEMS) v 1 F d2 F d1 Ele ectrode e 1 d 1 d 2 m x k k m Electrod de 2 v 2 V 1 V V 2 P EE C245: Introduction to MEMS Design Lecture 23 C. Nguyen 11/20/08 19
Electrostatic Comb Drive EE C245: Introduction to MEMS Design Lecture 23 C. Nguyen 11/20/08 20
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 x Comb-Driven Folded Beam Actuator EE C245: Introduction to MEMS Design Lecture 23 C. Nguyen 11/20/08 21
Comb-Drive Force Equation (1 st Pass) Top View y x z x L f Side View EE C245: Introduction to MEMS Design Lecture 23 C. Nguyen 11/20/08 22
Lateral Comb-Drive Electrical Stiffness Top View y x z x L f Side View Again: C ( x ) = 2Nεhx C 2Nεh = d x d No ( C/ x) x-dependence no electrical stiffness: k e = 0! Frequency immune to changes in V P or gap spacing! EE C245: Introduction to MEMS Design Lecture 23 C. Nguyen 11/20/08 23
Typical Drive & Sense Configuration y z x EE C245: Introduction to MEMS Design Lecture 23 C. Nguyen 11/20/08 24
Comb-Drive Force Equation (2 nd Pass) In our 1 st pass, we neglected Fringing fields Parallel-plate l l capacitance between stator and rotor Capacitance to the substrate All of these capacitors must be included when evaluating the energy expression! Stator Rotor Ground Plane EE C245: Introduction to MEMS Design Lecture 23 C. Nguyen 11/20/08 25
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 i energy [Gary Fedder, Ph.D., UC Berkeley, 1994] EE C245: Introduction to MEMS Design Lecture 23 C. Nguyen 11/20/08 26
Case: V r = V P = 0V C sp depends on whether or not fingers are engaged Capacitance Expressions Capacitance per unit length Region 2 Region 3 [Gary Fedder, Ph.D., UC Berkeley, 1994] EE C245: Introduction to MEMS Design Lecture 23 C. Nguyen 11/20/08 27
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 i energy [Gary Fedder, Ph.D., UC Berkeley, 1994] EE C245: Introduction to MEMS Design Lecture 23 C. Nguyen 11/20/08 28
Simulate to Get Capacitors Force Below: 2D finite element simulation 20-40% reduction of F e,x EE C245: Introduction to MEMS Design Lecture 23 C. Nguyen 11/20/08 29
Vertical Force (Levitation) W 1 dcsp 2 1 dcrp 2 1 dcrs F e, z = = Vs + Vr z 2 dz 2 dz + 2 dz d C sp, e + C Fe, z = Nx 2 dz For V r = 0V (as shown): ( V V ) 2 ( ) rs 2 1 rs EE C245: Introduction to MEMS Design Lecture 23 C. Nguyen 11/20/08 30 s V s r
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 F z γ V z 2 P ( z z ) o z o = k e ( z z) o Electrical Stiffness EE C245: Introduction to MEMS Design Lecture 23 C. Nguyen 11/20/08 31
Vertical Resonance Frequency ω z /ω zo Vertical resonance frequency Vertical resonance frequency at V P = 0V k + k ω z z e = where z k V 2 e zo k z z ω = Lateral resonance frequency = γ z o Signs of electrical stiffnesses in MEMS: Comb (x-axis) k e = 0 Comb (z-axis) k e > 0 Parallel Plate k e < 0 EE C245: Introduction to MEMS Design Lecture 23 C. Nguyen 11/20/08 32
Suppressing Levitation Pattern ground plane polysilicon into differentially excited electrodes to minimize field lines terminating on top of comb Penalty: x-axis force is reduced EE C245: Introduction to MEMS Design Lecture 23 C. Nguyen 11/20/08 33
Force of Comb-Drive vs. Parallel-Plate Comb drive (x-direction) V 1 = V 2 = V S = 1V Differential Parallel-Plate (y-direction) V 1 = 0V, V 2 = 1V Parallel-plate generates a much larger force; but at the cost of linearity EE C245: Introduction to MEMS Design Lecture 23 C. Nguyen 11/20/08 34