Computer Aided Design of Micro-Electro-Mechanical Systems

Transcription

1 Computer Aided Design of Micro-Electro-Mechanical Systems From Energy Losses to Dick Tracy Watches D. Bindel Courant Institute for Mathematical Sciences New York University Temple University, 7 Nov 2007

2 The Computational Science Picture Application modeling filter Disk Beam Shear ring,... Mathematical analysis Physical modeling and finite element technology Structured eigenproblems and reduced-order Parameter-dependent eigenproblems Software engineering HiQLab SUGAR FEAPMEX / MATFEAP

3 The Computational Science Picture Application modeling filter Disk Beam Shear ring,... Mathematical analysis Physical modeling and finite element technology Structured eigenproblems and reduced-order Parameter-dependent eigenproblems Software engineering HiQLab SUGAR FEAPMEX / MATFEAP

4 Outline Anchor disk 4 5

5 Outline Anchor disk 4 5

6 What are MEMS?

7 MEMS Basics Micro-Electro-Mechanical Systems Chemical, fluid, thermal, optical (MECFTOMS?) Applications: Sensors (inertial, chemical, pressure) Ink jet printers, biolab chips Radio devices: cell phones, inventory tags, pico radio Use integrated circuit (IC) fabrication technology Tiny, but still classical physics

8 RF MEMS Microguitars from Cornell University (1997 and 2003) MHz-GHz mechanical Favorite application: radio on chip Close second: really high-pitch guitars

9 The Mechanical Cell Phone RF amplifier / preselector Mixer IF amplifier / filter... Tuning control Local Oscillator Your cell phone has many moving parts! What if we replace them with integrated MEMS?

10 Ultimate Success Calling Dick Tracy!

11 Disk Resonator V in AC DC V out

12 Disk Resonator V in AC L x C 0 C x R x V out

13 Electromechanical Model Kirchoff s current law and balance of linear momentum: d dt (C(u)V ) + GV = I external ( ) 1 Mu tt + Ku u 2 V C(u)V = F external Linearize about static equilibium (V 0, u 0 ): C(u 0 ) δv t + G δv + ( u C(u 0 ) δu t ) V 0 = δi external M δu tt + K δu + u (V0 C(u 0) δv ) = δf external where 2 K = K 1 2 u 2 (V 0 C(u 0)V 0 )

14 Electromechanical Model Assume time-harmonic steady state, no external forces: [ ] [ ] [ ] iωc + G iωb δ ˆV δ Î B T K ω 2 = external M δû 0 Eliminate the mechanical terms: Y (ω) δ ˆV = δîexternal Y (ω) = iωc + G + iωh(ω) H(ω) = B T ( K ω 2 M) 1 B Goal: Understand electromechanical piece (iωh(ω)). As a function of geometry and operating point Preferably as a simple circuit

15 Damping and Q Designers want high quality of resonance (Q) Dimensionless damping in a one-dof system d 2 u du + Q 1 dt2 dt + u = F (t) For a resonant mode with frequency ω C: Q := ω 2 Im(ω) = Stored energy Energy loss per radian To understand Q, we need damping!

16 The Designer s Dream Ideally, would like Simple for behavioral simulation Parameterized for design optimization Including all relevant physics With reasonably fast and accurate set-up We aren t there yet.

17 Outline Anchor disk 4 5

18 Resonator D+ D D+ D S S+ S+ Anchored at outside corners Excited at northwest corner Sensed at southeast corner Surfaces move only a few nanometers S

19 Model Reduction Finite element model: N = 2154 Expensive to solve for every H(ω) evaluation! Build a reduced-order model to approximate behavior Reduced system of 80 to 100 vectors Evaluate H(ω) in milliseconds instead of seconds Without damping: standard Arnoldi projection With damping: Second-Order ARnoldi (SOAR)

20 SOAR and ODE structure Damped second-order system: Mu + Cu + Ku = Pφ y = V T u. Projection basis Q n with Second Order ARnoldi (SOAR): M n u n + C n u n + K n u n = P n φ y = Vn T u where P n = Q T n P, V n = Q T n V, M n = Q T n MQ n,...

21 Simulation x Frequency (Hz) x x Frequency (Hz) x 10 7 Amplitude (db) Phase (rad) 2

22 Measurement S. Bhave, MEMS 05

23 Outline Anchor disk 4 5

24 Damping Mechanisms Possible loss mechanisms: Fluid damping Material losses damping Anchor loss Model substrate as semi-infinite with a Perfectly Matched Layer (PML).

25 Perfectly Matched Layers Complex coordinate transformation Generates a perfectly matched absorbing layer Idea works with general linear wave equations Electromagnetics (Berengér, 1994) Quantum mechanics exterior complex scaling (Simon, 1979) Elasticity in standard finite element framework (Basu and Chopra, 2003)

26 Model Problem Domain: x [0, ) Governing eq: Fourier transform: Solution: 2 u x u c 2 t 2 = 0 d 2 û dx 2 + k 2 û = 0 û = c out e ikx + c in e ikx

27 Model with Perfectly Matched Layer σ Regular domain x Transformed domain d x dx = λ(x) where λ(s) = 1 iσ(s) d 2 û d x 2 + k 2 û = 0 û = c out e ik x ik x + c in e

28 Model with Perfectly Matched Layer σ Regular domain x Transformed domain d x dx = λ(x) where λ(s) = 1 iσ(s), 1 d λ dx ( 1 λ ) dû + k 2 û = 0 dx û = c out e ikx kσ(x) + c in e ikx+kσ(x) Σ(x) = x 0 σ(s) ds

29 Model with Perfectly Matched Layer σ Regular domain If solution clamped at x = L then x Transformed domain c in c out = O(e kγ ) where γ = Σ(L) = L 0 σ(s) ds

30 Model Problem Illustrated 1 Outgoing exp( i x) 1 Incoming exp(i x) Transformed coordinate Re( x)

31 Model Problem Illustrated 1 Outgoing exp( i x) 3 Incoming exp(i x) Transformed coordinate Re( x)

32 Model Problem Illustrated 1 Outgoing exp( i x) 6 Incoming exp(i x) Transformed coordinate Re( x)

33 Model Problem Illustrated 1 Outgoing exp( i x) 15 Incoming exp(i x) Transformed coordinate Re( x)

34 Model Problem Illustrated 1 Outgoing exp( i x) 40 Incoming exp(i x) Transformed coordinate Re( x)

35 Model Problem Illustrated 1 Outgoing exp( i x) 100 Incoming exp(i x) Transformed coordinate Re( x)

36 Finite Element Implementation ξ 2 x 2 x 2 Ω ξ 1 x(ξ) Ω e x(x) Ω e Combine PML and isoparametric mappings k e = BT D B J dω Ω m e = ρn T N J dω Ω Matrices are complex symmetric x 1 x 1

37 Eigenvalues and Model Reduction Want to know about the transfer function H(ω): Can either H(ω) = B T (K ω 2 M) 1 B Locate poles of H (eigenvalues of (K, M)) Plot H in a frequency range (Bode plot) Usual tactic: subspace projection Build an Arnoldi basis V for a Krylov subspace K n Compute with much smaller V KV and V MV Can we do better?

38 Variational Principles Variational form for complex symmetric eigenproblems: Hermitian (Rayleigh quotient): ρ(v) = v Kv v Mv Complex symmetric (modified Rayleigh quotient): θ(v) = v T Kv v T Mv First-order accurate eigenvectors = Second-order accurate eigenvalues. Key: relation between left and right eigenvectors.

39 Accurate Model Reduction Build new projection basis from V : W = orth[re(v ), Im(V )] span(w ) contains both K n and K n = double digits correct vs. projection with V W is a real-valued basis = projected system is complex symmetric

40 Disk Resonator Simulations

41 Disk Resonator Mesh Resonating disk Electrode Wafer (unmodeled) 2 x PML region x 10 5 Axisymmetric model with bicubic mesh About 10K nodal points in converged calculation

42 Mesh Convergence 7000 Computed Q Linear Quadratic Cubic Mesh density Cubic elements converge with reasonable mesh density

43 Model Reduction Accuracy Transfer (db) Phase (degrees) Frequency (MHz) Frequency (MHz) Results from ROM (solid and dotted lines) nearly indistinguishable from full model (crosses)

44 Model Reduction Accuracy H(ω) Hreduced(ω) /H(ω) Frequency (MHz) Structure-preserving ROM Arnoldi ROM Preserve structure = get twice the correct digits

45 Response of the Disk Resonator

46 Variation in Quality of Resonance 10 8 Q Film thickness (µm) Simulation and lab measurements vs. disk thickness

47 Explanation of Q Variation Imaginary frequency (MHz) 0.25 e a b d c a = 1.51 µm b = 1.52 µm c = 1.53 µm 0.05 d = 1.54 µm e = 1.55 µm d c e a b Real frequency (MHz) Interaction of two nearby eigenmodes

48 Outline Anchor disk 4 5

49 Damping (TED)

50 Damping (TED) u is displacement and T = T 0 + θ is temperature σ = Cɛ βθ1 ρü = σ ρc v θ = (κ θ) βt0 tr( ɛ) Coupling between temperature and volumetric strain: Compression and expansion = heating and cooling Heat diffusion = mechanical damping Not often an important factor at the macro scale Recognized source of damping in micro Zener: semi-analytical approximation for TED in s We consider the fully coupled system

51 Nondimensionalized Equations Continuum equations: Discrete equations: σ = Ĉɛ ξθ1 ü = σ θ = η 2 θ tr( ɛ) M uu ü + K uu u = ξk uθ θ + f C θθ θ + ηkθθ θ = C θu u Micron-scale poly-si devices: ξ and η are Linearize about ξ = 0

52 Perturbative Mode Calculation Discretized mode equation: ( ω 2 M uu + K uu )u = ξk uθ θ (iωc θθ + ηk θθ )θ = iωc θu u First approximation about ξ = 0: ( ω0 2 M uu + K uu )u 0 = 0 (iω 0 C θθ + ηk θθ )θ 0 = iω 0 C θu u 0 First-order correction in ξ: δ(ω 2 )M uu u 0 + ( ω0 2 M uu + K uu )δu = ξk uθ θ 0 Multiply by u0 T : ( ) u δ(ω 2 T ) = ξ 0 K uθ θ 0 u0 T M uuu 0

53 Zener s Model 1 Clarence Zener investigated TED in late 30s-early 40s. 2 Model for s common in MEMS literature. 3 Method of orthogonal thermodynamic potentials == perturbation method + a variational method.

54 Comparison to Zener s Model Damping Q HiQlab Results Frequency f(hz) Zener s Formula Comparison of fully coupled simulation to Zener approximation over a range of frequencies Real and imaginary parts after first-order correction agree to about three digits with Arnoldi

55 Outline Anchor disk 4 5

56 Onward! What about: Modeling more geometrically complex devices? Modeling general dependence on geometry? Modeling general dependence on operating point? Computing nonlinear dynamics? Digesting all this to help designers?

57 Future Work Code development Structural elements and elements for different physics Design and implementation of parallelized version Theoretical analysis More damping mechanisms Sensitivity analysis and variational model reduction Application collaborations Use of nonlinear effects (quasi-static and dynamic) New designs (e.g. internal dielectric drives) Continued experimental comparisons

58 s RF MEMS are a great source of problems Interesting applications Interesting physics (and not altogether understood) Interesting computing challenges

59 Concluding Thoughts The difference between art and science is that science is what we understand well enough to explain to a computer. Art is everything else. Donald Knuth The purpose of computing is insight, not numbers. Richard Hamming

60 Resonator Nonlinear eigenvalue perturbation Electromechanical model Hello world! Reflection Analysis HiQLab D+ D D+ D S S+ S+ Anchored at outside corners Excited at northwest corner Sensed at southeast corner Surfaces move only a few nanometers S

61 Model Reduction Finite element model: N = 2154 Expensive to solve for every H(ω) evaluation! Build a reduced-order model to approximate behavior Reduced system of 80 to 100 vectors Evaluate H(ω) in milliseconds instead of seconds Without damping: standard Arnoldi projection With damping: Second-Order ARnoldi (SOAR) Nonlinear eigenvalue perturbation Electromechanical model Hello world! Reflection Analysis HiQLab

62 Simulation x 10 Nonlinear eigenvalue perturbation Electromechanical model Hello world! Reflection Analysis HiQLab Frequency (Hz) x x Frequency (Hz) x 10 7 Amplitude (db) Phase (rad) 2

63 Measurement Nonlinear eigenvalue perturbation Electromechanical model Hello world! S. Bhave, MEMS 05 Reflection Analysis HiQLab

64 Contributions Built predictive model used to design checkerboard Used model reduction to get thousand-fold speedup fast enough for interactive use Nonlinear eigenvalue perturbation Electromechanical model Hello world! Reflection Analysis HiQLab

65 General Picture If w A = 0 and Av = 0 then Nonlinear eigenvalue perturbation Electromechanical model δ(w Av) = w (δa)v This implies If A = A(λ) and w = w(v), have w (v)a(ρ(v))v = 0. ρ stationary when (ρ(v), v) is a nonlinear eigenpair. If A(λ, ξ) and w0 and v 0 are null vectors for A(λ 0, ξ 0 ), w0 (A λδλ + A ξ δξ)v 0 = 0. Hello world! Reflection Analysis HiQLab

66 Electromechanical Model Nonlinear eigenvalue perturbation Electromechanical model Hello world! Reflection Analysis HiQLab Kirchoff s current law and balance of linear momentum: d dt (C(u)V ) + GV = I external ( ) 1 Mu tt + Ku u 2 V C(u)V = F external Linearize about static equilibium (V 0, u 0 ): C(u 0 ) δv t + G δv + ( u C(u 0 ) δu t ) V 0 = δi external M δu tt + K δu + u (V0 C(u 0) δv ) = δf external where 2 K = K 1 2 u 2 (V 0 C(u 0)V 0 )

67 4 HiQLab s Hello World 2 Nonlinear eigenvalue perturbation Electromechanical model Hello world! Reflection Analysis HiQLab 0 2 mesh = Mesh:new(2) mat = 4make_material( silicon2, planestrain ) mesh:blocks2d( { 0, l }, { -w/2.0, w/2.0 }, mat ) mesh:set_bc(function(x,y) x 10 6 if x == 0 then return uu, 0, 0; end end)

68 HiQLab s Hello World Nonlinear eigenvalue perturbation Electromechanical model Hello world! >> mesh = Mesh_load( mesh.lua ); >> [M,K] = Mesh_assemble_mk(mesh); >> [V,D] = eigs(k,m, 5, sm ); >> opt.axequal = 1; opt.deform = 1; >> Mesh_scale_u(mesh, V(:,1), 2, 1e-6); >> plotfield2d(mesh, opt); Reflection Analysis HiQLab

69 Continuum 2D model problem Nonlinear eigenvalue perturbation Electromechanical model Hello world! Reflection Analysis HiQLab k L { 1 iβ x L λ(x) = p, x > L 1 x L. ( ) 1 1 u + 2 u λ x λ x y 2 + k 2 u = 0

70 Continuum 2D model problem Nonlinear eigenvalue perturbation Electromechanical model Hello world! Reflection Analysis HiQLab k L { 1 iβ x L λ(x) = p, x > L 1 x L. ( ) 1 1 u ky 2 u + k 2 u = 0 λ x λ x

71 Continuum 2D model problem k L Nonlinear eigenvalue perturbation Electromechanical model Hello world! Reflection Analysis HiQLab { 1 iβ x L λ(x) = p, x > L 1 x L. ( ) 1 1 u + kx 2 u = 0 λ x λ x 1D problem, reflection of O(e kx γ )

72 Discrete 2D model problem Nonlinear eigenvalue perturbation Electromechanical model Hello world! Discrete Fourier transform in y Solve numerically in x k L Project solution onto infinite space traveling modes Extension of Collino and Monk (1998) Reflection Analysis HiQLab

73 Nondimensionalization k L λ(x) = { 1 iβ x L p, x > L 1 x L. Nonlinear eigenvalue perturbation Electromechanical model Hello world! Reflection Analysis HiQLab Rate of stretching: Elements per wave: Elements through the PML: βh p (k x h) 1 and (k y h) 1 N

74 Nondimensionalization k L λ(x) = { 1 iβ x L p, x > L 1 x L. Nonlinear eigenvalue perturbation Electromechanical model Hello world! Reflection Analysis HiQLab Rate of stretching: Elements per wave: Elements through the PML: βh p (k x h) 1 and (k y h) 1 N

75 Discrete reflection behavior Nonlinear eigenvalue perturbation Electromechanical model Hello world! Reflection Analysis HiQLab log 10 (βh) log 10 (r) at (kh) 1 = Number of PML elements Quadratic elements, p = 1, (k x h) 1 = 10 2

76 Discrete reflection decomposition Model discrete reflection as two parts: Far-end reflection (clamping reflection) Approximated well by continuum calculation Grows as (k x h) 1 grows Interface reflection Discrete effect: mesh does not resolve decay Does not depend on N Grows as (k x h) 1 shrinks Nonlinear eigenvalue perturbation Electromechanical model Hello world! Reflection Analysis HiQLab

77 Discrete reflection behavior Nonlinear eigenvalue perturbation Electromechanical model Hello world! Reflection Analysis HiQLab log 10 (βh) log 10 (r) at (kh) 1 = Number of PML elements log 10 (βh) log 10 (r interface + r nominal) at (kh) 1 = Number of PML elements Quadratic elements, p = 1, (k x h) 1 = 10 Model does well at predicting actual reflection Similar picture for other wavelengths, element types, stretch functions

78 Choosing PML parameters Discrete reflection dominated by Interface reflection when k x large Far-end reflection when k x small Heuristic for PML parameter choice Choose an acceptable reflection level Choose β based on interface reflection at kx max Choose length based on far-end reflection at kx min Nonlinear eigenvalue perturbation Electromechanical model Hello world! Reflection Analysis HiQLab

79 Enter HiQLab Existing codes do not compute quality factors... and awkward to prototype new solvers... and awkward to programmatically define meshes So I wrote a new finite element code: HiQLab Nonlinear eigenvalue perturbation Electromechanical model Hello world! Reflection Analysis HiQLab

80 Heritage of HiQLab Nonlinear eigenvalue perturbation SUGAR: SPICE for the MEMS world System-level simulation using modified nodal analysis Flexible device description language C core with MATLAB interfaces and numerical routines FEAPMEX: MATLAB + a finite element code MATLAB interfaces for steering, testing solvers, running parameter studies Time-tested finite element architecture But old F77, brittle in places Electromechanical model Hello world! Reflection Analysis HiQLab

81 Other Ingredients Nonlinear eigenvalue perturbation Electromechanical model Hello world! Reflection Analysis Lesser artists borrow. Great artists steal. Picasso, Dali, Stravinsky? Lua: Evolved from simulator data languages (DEL and SOL) Pascal-like syntax fits on one page; complete language description is 21 pages Fast, freely available, widely used in game design MATLAB: The Language of Technical Computing Good sparse matrix support Star-P: Standard numerical libraries: ARPACK, UMFPACK HiQLab

82 HiQLab Structure Nonlinear eigenvalue perturbation Electromechanical model Hello world! Reflection Analysis HiQLab User interfaces (MATLAB, Lua) Problem description (Lua) Core libraries (C++) Solver library (C, C++, Fortran, MATLAB) Element library (C++) Standard finite element structures + some new ideas Full scripting language for mesh input Callbacks for boundary conditions, material properties MATLAB interface for quick algorithm prototyping Cross-language bindings are automatically generated

83 Contributions Wrote a new code, HiQLab, to study damping HiQLab is based on my earlier simulators: SUGAR, for system-level MEMS simulation FEAPMEX (now MATFEAP), for scripting parameter studies Nonlinear eigenvalue perturbation Electromechanical model Hello world! Reflection Analysis HiQLab