Motivation

1. Run-time Optimized Double Correlated Discrete Probability Propagation for Process Variation Characterization of NEMS Cantilevers Rasit Onur Topaloglu PhD student rtopalog@cse.ucsd.eduUniversity of California, San DiegoComputer Science and Engineering Department 9500 Gilman Dr., La Jolla, CA, 92093

2. Motivation • Cantilevers are fundamental structures used extensively in novel applications such as atomic force microscopy or molecular diagnostics, all of which require utmost precision • Such aggressive applications require nano-cantilevers • Manufacturing steps for nano-structures bring a burden to uniformity between cantilevers designed alike • These process variations should be able to be estimated to account for and correct for the proper working of the application

3. Applications - Atomic Force Microscopy • IBM’s Millipede technology requires a matched array of 64*64 cantilevers • Aggressive bits/inch targets drive cantilever sizes to nano-scales • Process variations might incur noise to measurements hence degrade SNR of the disk • Correct estimation will enable a safe choice of device dimension : optimization

4. Single Molecule Spectroscopy • Cantilever deflection should be utmost accurate to measure the molecule mass

5. Simulating MEMS: Linear Beam Model in Sugar • Each node has 3 degrees of freedom v(x) : transverse deflection u(x) : axial deflection (x) : angle of rotation • Between the nodes, equilibrium equation: • It’s solution is cubic: • Boundary conditions at ends yield four equations and four unknowns:

6. Acquisition of Stifness Matrix • Solving for x between nodes: • where H are Hermitian shape functions: • Following the analysis, one can find stiffness matrix using Castiglianos Theorem as:

7. Acquisition of Mass and Damping Matrices • Equating internal and external work and using Coutte flow model, mass and damping matrices found: • Hence familiar dynamics equation found: • where displacements are and the force vector is • W, L , H can be identified as most influential

8. Basic Sugar Input and Output • mfanchor {_n("substrate"); material = p1, l = 10u, w = 10u} • mfbeam3d {_n("substrate"), _n("tip"); material = p1, l = a, w = b, h = c} • mff3d {_n("tip"); F = 2u, oz = (pi)/(2) l=100 w=h=2 l=110 w=h=2 dy=3.0333e-6 dy = 4.0333e-6

9. Monte Carlo Approach in Process Estimation W L h dy • Pick a set of numbers according to the distributions and simulate : this is one MC run • Repeat the previous step for 10000 times • Bin the results to get final distribution

10. FDPP Approach W L h dy • Discretize the distributions • Take all combinations of samples : each run gives a result with a probability that is a multiple of individual samples • Re-bin the acquired samples to get the final distribution • Interpolate the samples for a continuous distribution

11. Probability Discretization Theory: Discretization Operation pdf(X) • QN band-pass filter pdf(X) and divide into bins • Use N>(2/m), where m is maximum derivative of pdf(X), thereby obeying a bound similar to Nyquist pdf(X) X spdf(X)=(X) spdf(X) wi : value of i’th impulse X N in QN indicates number or bins

12. Propagation Operation • F operator implements a function over spdf’s using deterministic sampling Xi, Y : random variables • Heights of impulses multiplied and probabilities normalized to 1 at the end pXs : probabilities of the set of all samples s belonging to X

13. Re-bin Operation Resulting spdf(X) Unite into one  bin Impulses after F • Samples falling into the same bin congregated in one • Without R, Q-1 would result in a noisy graph which is not a pdf as samples would not be equally separated where :

14. Correlation Modeling • Width and length depend on the same mask, hence they are assumed to be highly correlated ~=0.9 • Height depends on the release step, hence is weakly correlated to width and length ~=0.1

15. Double Correlated FDPP Approach W L h dy • Instead of using all samples exhaustively, since samples are correlated, create other samples using the sample of one parameter (e.g.W as reference): ex. L_s=a W_s+b Randn() where=a/sqrt(a2+b2) • Do this twice, one for (+) one for (-) correlation so that the randomness in the system is also accounted for towards both sides of the initial value; hence double-correlated

16. Monte Carlo Results MC 100 pts MC 1000 pts MC 10000 pts • For MC, probability density function is too noisy until high number of samples, which require high run-times, used =3.0409-6 =3.0407e-6 =3.0352e-6

17. Monte Carlo -DC FDPP Comparison DC-FDPP Compared with MC 10000 pts • Same number of finals bins and same correlated sampling scheme used for a fair comparison • Comparable accuracy achieved using 500 times less run-time =0.425% max=1.88% min=3.67% =3.0481e-6 max=3.5993e-6 min=2.61e-6

18. Conclusions • Monte Carlo methods are time consuming • A computational method presented for 500 times faster speed with reasonable accuracy trade-off • The method has been successfully integrated into the Sugar framework using Matlab and Perl scripts • Such methods can be used while designing and optimizing nano-scale cantilevers and characterizing process variations amongst matched cantilevers

19. References • Cantilever-Based Biosensors in CMOS Technology, K.-U. Kirstein et al. DATE 2005 • High Sensitive Piezoresistive Cantilever Design and Optimization for Analyte-Receptor Binding, M. Yang, X. Zhang, K. Vafai and C. S. Ozkan, Journal of Micromechanics and Microengineering, 2003 • MEMS Simulation using Sugar v0.5, J. V. Clark, N. Zhou and K. S. J. Pister, in Proceedings of Solid-State Sensors and Actuators Workshop, 1998 • Forward Discrete Probability Propagation for Device Performance Characterization under Process Variations, R. O. Topaloglu and A. Orailoglu, ASPDAC, 2005