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 [email protected] 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