Run-time Optimized Double Correlated Run-time Optimized Double Correlated Discrete Probability Propagation for Discrete Probability Propagation for

Process Variation Characterization of Process Variation Characterization of NEMS CantileversNEMS Cantilevers

Run-time Optimized Double Correlated Run-time Optimized Double Correlated Discrete Probability Propagation for Discrete Probability Propagation for

Process Variation Characterization of Process Variation Characterization of NEMS CantileversNEMS 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

MotivationMotivationMotivationMotivation

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

Applications - Atomic Force Applications - Atomic Force MicroscopyMicroscopyApplications - Atomic Force Applications - Atomic Force MicroscopyMicroscopy

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

Single Molecule SpectroscopySingle Molecule SpectroscopySingle Molecule SpectroscopySingle Molecule Spectroscopy

Cantilever deflection should be utmost accurate to measure the molecule mass

Cantilever deflection should be utmost accurate to measure the molecule mass

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 Simulating MEMS: Linear Beam Model in SugarModel in SugarSimulating MEMS: Linear Beam Simulating MEMS: Linear Beam Model in SugarModel 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:

Acquisition of Stifness MatrixAcquisition of Stifness MatrixAcquisition of Stifness MatrixAcquisition 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:

Acquisition of Mass and Damping Acquisition of Mass and Damping MatricesMatricesAcquisition of Mass and Damping Acquisition of Mass and Damping MatricesMatrices

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

Basic Sugar Input and OutputBasic Sugar Input and OutputBasic Sugar Input and OutputBasic 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

Monte Carlo Approach in Process Monte Carlo Approach in Process EstimationEstimationMonte Carlo Approach in Process Monte Carlo Approach in Process EstimationEstimation

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

WW LL hh dydy

FDPP ApproachFDPP ApproachFDPP ApproachFDPP 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

WW LL hh dydy

pdf(X)

Probability Discretization Probability Discretization Theory: Theory:

Discretization OperationDiscretization Operation

))(()( XpdfQX N

N in QN indicates number or bins

spdf(X)=(X)X

pdf(

X)

spdf

(X)

X

Ni

ii wxpX..1

)()(

wi : value of i’th impulse

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

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

Propagation OperationPropagation Operation

))(),..,(()( 1 rXXFY Xi, Y : random variables

r

r

rss

Xs

Xs

Xs

Xs wwfyppY

,..,1

1

1

1

1

1

1)),..,((..)(

pXs : probabilities of the set of all samples s belonging to X

F operator implements a function over spdf’s using deterministic sampling

F operator implements a function over spdf’s using deterministic sampling

Heights of impulses multiplied and probabilities normalized to 1 at the end

Heights of impulses multiplied and probabilities normalized to 1 at the end

Re-bin OperationRe-bin OperationRe-bin OperationRe-bin Operation

Impulses after F Resulting spdf(X)Unite into one bin

i

ii wxpX )()( ijs

ji bwstppj

.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

Correlation ModelingCorrelation ModelingCorrelation ModelingCorrelation 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

Double Correlated FDPP ApproachDouble Correlated FDPP ApproachDouble Correlated FDPP ApproachDouble 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

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

WW LL hh dydy

Monte Carlo ResultsMonte Carlo ResultsMonte Carlo ResultsMonte Carlo Results

MC 100 pts MC 1000 pts MC 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

For MC, probability density function is too noisy until high number of samples, which require high run-times, used

Monte Carlo -DC FDPP ComparisonMonte Carlo -DC FDPP ComparisonMonte Carlo -DC FDPP ComparisonMonte Carlo -DC FDPP Comparison

=3.0481e-6

max=3.5993e-6

min=2.61e-6

=0.425%

max=1.88%

min=3.67%

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

Same number of finals bins and same correlated sampling scheme used for a fair comparison

Comparable accuracy achieved using 500 times less run-time

ConclusionsConclusionsConclusionsConclusions

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

ReferencesReferencesReferencesReferences

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