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 rtopalog@cse.ucsd.eduUniversity of California, San DiegoComputer Science and Engineering Department 9500 Gilman Dr., La Jolla, CA, 92093


Motivation
Motivation Propagation for Process Variation Characterization of NEMS Cantilevers

  • 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 Propagation for Process Variation Characterization of NEMS Cantilevers

  • 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 Propagation for Process Variation Characterization of NEMS Cantilevers

  • 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 Propagation for Process Variation Characterization of NEMS Cantilevers

  • 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 Propagation for Process Variation Characterization of NEMS Cantilevers

  • 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 Propagation for Process Variation Characterization of NEMS Cantilevers

  • 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 Propagation for Process Variation Characterization of NEMS Cantilevers

  • 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 Propagation for Process Variation Characterization of NEMS Cantilevers

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


Fdpp approach
FDPP Approach Propagation for Process Variation Characterization of NEMS Cantilevers

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


Motivation 3706182

Probability Discretization Theory: Propagation for Process Variation Characterization of NEMS CantileversDiscretization 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


Motivation 3706182

Propagation Operation Propagation for Process Variation Characterization of NEMS Cantilevers

  • 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 Propagation for Process Variation Characterization of NEMS Cantilevers

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 Propagation for Process Variation Characterization of NEMS Cantilevers

  • 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 Propagation for Process Variation Characterization of NEMS Cantilevers

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


Monte carlo results
Monte Carlo Results Propagation for Process Variation Characterization of NEMS Cantilevers

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


Monte carlo dc fdpp comparison
Monte Carlo -DC FDPP Comparison Propagation for Process Variation Characterization of NEMS Cantilevers

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


Conclusions
Conclusions Propagation for Process Variation Characterization of NEMS 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


References
References Propagation for Process Variation Characterization of NEMS 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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