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Motivation

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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

- 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

- 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

- 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

- Between the nodes, equilibrium equation:
- It’s solution is cubic:
- Boundary conditions at ends yield four equations and four unknowns:

- Solving for x between nodes:
- where H are Hermitian shape functions:
- Following the analysis, one can find stiffness matrix using Castiglianos Theorem as:

- 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

- 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

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

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

Probability Discretization Theory: Discretization 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

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

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 :

- 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

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

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

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

- 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