EE201C Final Project

Adeel MazharCharwak Apte

Problem Statement

• Need to consider reading and writing failure– Pick design point which minimizes likelihood of

failure

– (Diagrams taken from problem statement PDF file – Credited to Fang Gong.)

Exhaustive Search

• Done in 2 phases– First phase, 10 steps per design variable, 100 QMC

samples per design point– Second phase, 5 steps per design variable, 15000

QMC samples per design point– Yield was given priority over power and area– Multiple points were found with the same yield. – Sort by area, then pick lowest power @ min area

Many Points with same yieldM2vth M5vth M2leff M5leff Yield Power Area

0.425 0.2 0.095 0.095 0.999799987 3.19E-05 1.148813

0.425 0.2 0.0975 0.095 0.999799987 3.18E-05 1.14975

0.425 0.2 0.1 0.095 0.999799987 3.18E-05 1.150688

0.425 0.2 0.1025 0.095 0.999799987 3.18E-05 1.151625

0.425 0.2 0.105 0.095 0.999799987 3.17E-05 1.152563

0.5 0.2 0.095 0.095 0.999799987 3.18E-05 1.148813

0.5 0.2 0.0975 0.095 0.999799987 3.18E-05 1.14975

0.5 0.2 0.1 0.095 0.999799987 3.17E-05 1.150688

0.5 0.2 0.1025 0.095 0.999799987 3.17E-05 1.151625

0.5 0.2 0.105 0.095 0.999799987 3.16E-05 1.152563

Results of Exhaustive Search

• M2 Length=95nm• M5 Length=95nm• M2 Vth= 0.5V• M5 Vth= 0.2V• Yield= 0.999799987• Power= 3.18E-05 (nom=3.2130e-5)• Area=1.148813 (nom=1.1516)

Analytical study

• We wanted to see the effect of changing each parameter from nominal on the yield

• We found that certain parameters had a very strong and consistent effect on the yield

• We used this to allow us to reduce the solution space.

Analytical StudyParameter Effect of setting to

maxEffect of setting to min Setpoint

M2 LenRead 164.2199mV 164.3353mV

0.095 umWrite 584.4249mV 581.1114mV

M5 LenRead 164.2677mV 164.2677mV

0.095 umWrite 624.6716mV 539.8206mV

M2 VthRead 163.4407mV 165.1866mV

0.5VWrite 594.2763mV 580.3916mV

M5 VthRead 164.2677mV 164.2677mV

0.2VWrite 931.9230mV 511.0882mV

Read Rd=164.2677mV Less than 164.3mV 163.4004mV

Write Wr=582.8680mV Less than 629mV 482.6054mV

Finding Optimum Design Point• Start at a point, and use gradient descent

– Explicitly optimize for a cost function– Minimal number of wasted samples

• Leverage WLHS to quickly determine the yield of a design point

• The solution space is really only two variables, since M5 can be fixed to an optimum value, based on the yield criteria given.

• Method has to be problem specific– Gradient descent is bad if lots of local minima– Have not encountered any here

Gradient Descent

• Pick a starting point• For each parameter

– Simulate at starting point +/- that parameter– Pick best as new starting point– Repeat– If difference between old point and current point

is small, exit

Initial Results

• Used QMC for each design point• 500 points per design point• 123 spice runs – 45min

– Exhaustive is 10,000 spice runs – 4-7 hours• Yield=0.9880• M2len = 0.0941 (need to fix constraint code)• M5len = 0.100• M2vth = 0.3500• M5vth = 0.2206

Sampling Methods tried

1. Monte Carlo2. Quasi Monte Carlo3. Latin Hypercube Sampling4. Weighed Latin Hypercube Sampling5. Importance Sampling

We have talked about MC and QMC,

Next few slides look at LHS, WLHS and IS.

Latin Hypercube Sampling

• CDF is used to divide variable span into equi-probable partitions.

• The Gaussian distribution of the variable is preserved.

• LHS Samples for L1, L2, Vt1, Vt2 are obtained and randomly permuted.

• Only as good as QMC.

LHS vs QMC

1. Clusters and Voids can be identified in LHS and stratified LHS.2. Convergence rate and accuracy are lower than QMC.

Weighted LHS

• Amplification of samplesin the predicted failure zone.

• Assigning appropriate weight to the failures.

• The critical point where we start,oversampling affects the accuracy of weighted LHS.

Importance Sampling

• PDF is actually shifted to generate more samples in the predicted failure zone.

• Accuracy depends on choice of the shift vector and natureof the solution space.

• The failures are weighed: P(x)|old / P(x)|new.

Results

Technique Number of Samples

Yield Spice Runtime Spice Runtime Degradation

Wrapper Runtime

Monte Carlo 105 1.0 1.0 81x 1.0Quasi-MC 4.3x104 1.0 0.43 35x 1.2xLHS 4.7 x 104 1.0 0.47 38x 1.42x

Weighted LHS 1225 99.985% 0.01225 1.0 1.8x

Importance Sampling

1400 99.981% 0.01400 1.142x 1.7x

Convergence Rates

1 100 1000 10000 100000

QMC

LHS

MC

IS

WLHS

Conclusions

• Tested 5 different Sampling Algorithms.• WLHS and IS are the most promising.• We will leverage WLHS to find the optimum

design point.• Gradient Descent works for this problem due to

the lack of local minima in the solution space.• Monte Carlo is ~81x slower as compared to

WLHS, and 70x slower as compared to IS.