Run-time Optimized Double Correlated Discrete Probability Propagation for Process Variation Characte...
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Run-time Optimized Double Correlated Discrete Probability Propagation for Process Variation Characterization of NEMS Cantilevers.

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Motivation 3706182

Run-time Optimized Double Correlated Discrete Probability Propagation for Process Variation Characterization of NEMS Cantilevers

Rasit Onur Topaloglu PhD student [email protected] of California, San DiegoComputer Science and Engineering Department 9500 Gilman Dr., La Jolla, CA, 92093



  • 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

Applications atomic force microscopy

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

Single molecule spectroscopy

Single Molecule Spectroscopy

  • Cantilever deflection should be utmost accurate to measure the molecule mass

Simulating mems linear beam model in sugar

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:

Acquisition of stifness matrix

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:

Acquisition of mass and damping matrices

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

Basic sugar input and output

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

Monte carlo approach in process estimation

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

Fdpp approach

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

Motivation 3706182

Probability Discretization Theory: Discretization Operation


  • 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





wi : value of i’th impulse


N in QN indicates number or bins

Motivation 3706182

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

Re bin operation

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 :

Correlation modeling

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

Double correlated fdpp approach

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(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

Monte carlo results

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




Monte carlo dc fdpp comparison

Monte Carlo -DC FDPP Comparison


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









  • 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



  • 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

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