Runtime Optimized Double Correlated Discrete Probability Propagation for Process Variation Character

1
Run-time Optimized Double Correlated Discrete
Probability Propagation for Process Variation
Characterization of NEMS Cantilevers

• Rasit Onur Topaloglu PhD student
  rtopalog_at_cse.ucsd.eduUniv
  ersity of California, San DiegoComputer Science
  and Engineering Department 9500
  Gilman Dr., La Jolla, CA, 92093

2
Motivation

• 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

• IBMs Millipede technology requires a matched
  array of 6464 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
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
• Its 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

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

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)
• l100 wh2
  l110 wh2
• dy3.0333e-6
  dy 4.0333e-6

9
Monte Carlo Approach in Process Estimation
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

10
FDPP Approach
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

11
Probability Discretization Theory
Discretization Operation
pdf(X)
pdf(X)
X
spdf(X)?(X)
spdf(X)
wi value of ith impulse
X

• QN band-pass filter pdf(X) and divide into bins
• Use Ngt(2/m), where m is maximum derivative of
  pdf(X), thereby obeying a bound similar to Nyquist

N in QN indicates number or bins
12
Propagation Operation

• F operator implements a function over spdfs
  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
13
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
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

15
Double Correlated FDPP Approach
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_sa W_sb Randn() where
  ?a/sqrt(a2b2)
• 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

16
Monte 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

17
Monte Carlo -DC FDPP Comparison
DC-FDPP
Compared with MC 10000 pts
??0.425 ?max1.88 ?min3.67
?3.0481e-6 max3.5993e-6 min2.61e-6

• 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
Conclusions

• 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
References

• 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