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

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

10. FDPP Approach 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 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 WLhdy

11. pdf(X) Probability Discretization Theory: Discretization Operation N in Q N indicates number or bins spdf(X)=  (X) X pdf(X) spdf(X) X w i : value of i’th impulse Q N 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 Q N 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

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

13. Re-bin Operation Impulses after F Resulting spdf(X) Unite into one  bin where : 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 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

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 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 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(a 2 +b 2 ) 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 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(a 2 +b 2 ) 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 WLhdy

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

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

18. ConclusionsConclusions 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 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. ReferencesReferences 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 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