Eigenproblems in Resonant MEMS Design David Bindel UC Berkeley, CS Division Eigenproblems inResonant MEMS Design – p.1/21 What are MEMS? Eigenproblems inResonant MEMS Design – p.2/21 RF MEMS Microguitars from Cornell University (1997 and 2003) MHz-GHz mechanical resonators Uses: RF signal processing (better cell phones) Sensing elements (e.g. chemical sensors) Really high-pitch guitars Eigenproblems inResonant MEMS Design – p.3/21 Micromechanical filters Radio signal Mechanical filter Filtered signal Capacitive drive Capacitive sense Your cell phone is already mechanical! Uses a quartz surface-acoustic wave (SAW) filter Can do better using MEMS MEMS filters can be placed on-chip Versus SAWs: smaller, lower power Success =⇒ “Calling Dick Tracy!” Eigenproblems inResonant MEMS Design – p.4/21 Damping Want to minimize damping Measure by “quality of resonance” |ω| Q= Im(ω) Electronic filters have too much Understanding of damping in MEMS is lacking Several sources of damping Anchor loss Thermoelastic damping Fluid damping Material losses Eigenproblems inResonant MEMS Design – p.5/21 Damping Want to minimize damping Measure by “quality of resonance” |ω| Q= Im(ω) Electronic filters have too much Understanding of damping in MEMS is lacking Several sources of damping Anchor loss Thermoelastic damping Fluid damping Material losses Eigenproblems inResonant MEMS Design – p.5/21 Example: Disk anchor loss Electrode Disk V− Wafer V+ V+ SiGe disk resonators built by E. Quévy Axisymmetric model with bicubic mesh, about 10K nodal points Eigenproblems inResonant MEMS Design – p.6/21 Perfectly matched layers Model half-space with a perfectly matched layer Complex coordinate change x 7→ z(x; ω) Apply a complex coordinate transformation Generates a non-physical absorbing layer Idea works with general linear wave equations First applied to Maxwell’s equations (Berengér 95) Similar idea introduced earlier in quantum mechanics (exterior complex scaling, Simon 79) Eigenproblems inResonant MEMS Design – p.7/21 Scalar wave example −c2 uzz − ω 2 u = 0 Outgoing wave exp(−iz) Incoming wave exp(iz) 1 1 0.5 0.5 0 0 −0.5 −0.5 −1 0 5 10 15 −1 20 0 5 10 15 20 Transformed coordinate z = x + iy 0 −1 −2 −3 −4 −5 0 2 4 6 8 10 12 14 16 18 Eigenproblems inResonant MEMS Design – p.8/21 Scalar wave example −c2 uzz − ω 2 u = 0 Outgoing wave exp(−iz) Incoming wave exp(iz) 1 3 2 0.5 1 0 0 −0.5 −1 −1 0 5 10 15 −2 20 0 5 10 15 20 Transformed coordinate z = x + iy 0 −1 −2 −3 −4 −5 0 2 4 6 8 10 12 14 16 18 Eigenproblems inResonant MEMS Design – p.8/21 Scalar wave example −c2 uzz − ω 2 u = 0 Outgoing wave exp(−iz) Incoming wave exp(iz) 1 6 4 0.5 2 0 0 −0.5 −1 −2 0 5 10 15 −4 20 0 5 10 15 20 Transformed coordinate z = x + iy 0 −1 −2 −3 −4 −5 0 2 4 6 8 10 12 14 16 18 Eigenproblems inResonant MEMS Design – p.8/21 Scalar wave example −c2 uzz − ω 2 u = 0 Outgoing wave exp(−iz) Incoming wave exp(iz) 1 15 10 0.5 5 0 0 −0.5 −1 −5 0 5 10 15 20 −10 0 5 10 15 20 Transformed coordinate z = x + iy 0 −1 −2 −3 −4 −5 0 2 4 6 8 10 12 14 16 18 Eigenproblems inResonant MEMS Design – p.8/21 Scalar wave example −c2 uzz − ω 2 u = 0 Outgoing wave exp(−iz) Incoming wave exp(iz) 1 40 30 0.5 20 0 10 0 −0.5 −10 −1 0 5 10 15 20 −20 0 5 10 15 20 Transformed coordinate z = x + iy 0 −1 −2 −3 −4 −5 0 2 4 6 8 10 12 14 16 18 Eigenproblems inResonant MEMS Design – p.8/21 Scalar wave example −c2 uzz − ω 2 u = 0 Outgoing wave exp(−iz) Incoming wave exp(iz) 1 100 80 0.5 60 40 0 20 0 −0.5 −20 −1 0 5 10 15 20 −40 0 5 10 15 20 Transformed coordinate z = x + iy 0 −1 −2 −3 −4 −5 0 2 4 6 8 10 12 14 16 18 Clamp solution at transformed end to isolate outgoing wave. Eigenproblems inResonant MEMS Design – p.8/21 Choice of transformations Generally z depends nontrivially on ω Needed for frequency-independent attenuation Common choice is dz = 1 − σ(x)/k dx What if we use a fixed transformation? Can choose to absorb well over finite ω range Solve a linear eigenvalue problem Amounts to rational approx of true radiation condition (in discrete case) Eigenproblems inResonant MEMS Design – p.9/21 Behavior with fixed transformations 0 5 10 15 20 25 30 0 5 10 15 20 25 30 Start with (K − ω 2 M )u = e1 Eigenproblems inResonant MEMS Design – p.10/21 Behavior with fixed transformations 0 5 10 15 20 25 30 0 5 10 15 20 25 30 Schur complement to eliminate PML unknowns Eigenproblems inResonant MEMS Design – p.10/21 Behavior with fixed transformations −1 Relative error in boundary coefficient 10 −2 10 −3 10 −4 10 20 30 40 50 60 70 80 Elements per wavelength 90 100 Compare last coefficient with exact (discrete) BC Eigenproblems inResonant MEMS Design – p.10/21 Complex symmetry Finite element equations (forced vibration) are −ω 2 M u + Ku = F where M and K are complex symmetric. Row and column eigenvectors are transposes Second-order accuracy with modified Rayleigh quotient: θ(v) = (v T Kv)/(v T M v) Can have v T M v ≈ 0 Propagating modes (continuous spectrum) Not the modes of interest for resonators Eigenproblems inResonant MEMS Design – p.11/21 Q variation 8 10 6 Q 10 4 10 2 10 0 10 1.2 1.3 1.4 1.5 1.6 Film thickness (µm) 1.7 1.8 Small geometry variation =⇒ large damping variation Solid line is simulated; dots are measured Eigenproblems inResonant MEMS Design – p.12/21 Effect of varying film thickness 0.25 e a b d Imaginary frequency (MHz) c 0.2 0.15 0.1 0.05 0 46 a = 1.51µm b = 1.52µm c = 1.53µm d = 1.54µm d = 1.55µm 46.5 d c e b a 47 Real frequency (MHz) 47.5 48 Sudden dip in Q comes from an interaction between a (mostly) bending mode and a (mostly) radial mode Eigenproblems inResonant MEMS Design – p.13/21 Model reduction Would like a reduced model which Preserves second-order accuracy for converged eigs Keeps at least Arnoldi’s accuracy otherwise Is physically meaningful Idea: Build an Arnoldi basis V Double the size: W = orth([Re(V ), Im(V )]) Use W as a projection basis Resulting system is still a Galerkin approximation with real shape functions for the continuum PML equations Eigenproblems inResonant MEMS Design – p.14/21 Example: Disk resonator response Transfer (dB) 0 -20 -40 -60 -80 47.2 47.25 47.3 Frequency (MHz) Phase (degrees) 200 100 0 47.2 47.25 47.3 Frequency (MHz) Eigenproblems inResonant MEMS Design – p.15/21 Example: Disk resonator response |H(ω) − Hreduced(ω)|/H(ω)| 10−2 10−4 10−6 Structure-preserving ROM Arnoldi ROM 45 46 47 48 49 50 Frequency (MHz) Eigenproblems inResonant MEMS Design – p.16/21 Thermoelastic damping (TED) u is displacement, T = T0 + θ is temperature σ = C−βθ1 ρutt = ∇ · σ ρcv θt = ∇2 θ−βT0 tr(t ) Second-order mechanical + first-order thermal equation Eigenproblems inResonant MEMS Design – p.17/21 Thermoelastic damping (TED) u is displacement, T = T0 + θ is temperature σ = C−βθ1 ρutt = ∇ · σ ρcv θt = ∇2 θ−βT0 tr(t ) Second-order mechanical + first-order thermal equation Temperature change causes stress (thermal expansion) Volumetric strain rate causes thermal fluctuations Eigenproblems inResonant MEMS Design – p.17/21 Thermoelastic damping (TED) Non-dimensionalized equation: σ = Ĉ − ξθ1 utt = ∇ · σ θt = η∇2 θ − tr(t ) Typical MEMS scales: ξ and η small Perturbation about ξ = 0 is effective Eigenproblems inResonant MEMS Design – p.18/21 Perturbation computation Discrete time-harmonic equations: −ω 2 Muu u + Kuu u + Kut θ = 0 iωDtt θ + Ktt θ + iωDtu u = 0 Approximate ω by perturbation about Kuθ = 0: −ω02 Muu u0 + Kuu u0 = 0 iω0 Dθθ θ0 + Kθθ θ0 + iω0 Dtu u0 = 0 Choose v : v T u0 6= 0 and compute " #" # " # (−ω02 Muu + Kuu ) −2ω0 Muu u0 δu −Kuθ θ0 = T v 0 δω 0 Eigenproblems inResonant MEMS Design – p.19/21 Comparison to Zener’s model −4 Thermoelastic Damping Q −1 10 −5 10 Zener’s Formula −6 HiQlab Results 10 −7 10 5 10 6 10 7 10 8 10 9 10 10 10 Frequency f(Hz) Good match to Zener’s approximation for TED in beams Real and imaginary parts after first-order correction agree to about three digits with Arnoldi Eigenproblems inResonant MEMS Design – p.20/21 Conclusions MEMS resonator simulations give interesting problems Damped resonators =⇒ nonlinear eigenproblems Introduce auxiliary variables to get exact or approximate linear problem There’s still useful structure in non-Hermitian problems! References: Bindel and Govindjee. “Elastic PMLs for Resonator Anchor Loss Simulation.” (IJNME, to appear) HiQLab home page: www.cs.berkeley.edu/d̃bindel/hiqlab/ Eigenproblems inResonant MEMS Design – p.21/21

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