0. Accurate Signoff with Advanced Variation Methodology
Ayhan Mutlu
Synopsys Co., Ltd.
Moonsu Kim, Eun yeung Yu, Eunbyeol Kim
Samsung Electronics Co., Ltd.
System LSI Division

1. Motivation Process & Design trends
Random variation is increasing in advanced process nodes
Design size and performance gap are also increasing, but TAT limit is almost constant
Requirement for process variation aware STA
Advanced node requires higher level of STA accuracy for better PPA within tight TAT limit
Accurate process variation will remove unnecessary pessimism and optimism at the same time
Traditional AOCV(Advanced On Chip Variation) and constraint variation modeling has limitations resulting in inherent pessimism
Need to use POCV (Parametric OCV) to resolve traditional method’s limitation using advanced modeling for delay/slew/constraint variation
In this slide, I want to introduce the motivation of this presesentation about ‘why accurate process variation aware STA is requried’
As shown in right graph, performance gap between realistic and requirement is increased linearly according to process nodes.
As shown in left graph, design size trend is similarly changed according to process advance.
Considering this trend, requirement for higher level of STA accuracy for better PPA within tight TAT limit becomes greater than before.
By this, accurate process variation will remove unnecessary pessimism and optimism at the same time.
We have traditional AOCV flow and constraint variation modeling but they have inherent pessimism.
This is a big motivation why we developed POCV flow using advanced modeling for delay/slew/constraint variation.

2. How to improve accuracy of process variation STA?
Challenge 1: slew/load impact to delay variation
In Advanced OCV (AOCV), delay variation is function of logic depth and distance, but does not consider slew/load impact
In actual design, the magnitude of process variation will increase at higher slew/load indices on advanced nodes
AOCV Variation is independent to slew/load
AOCV Variation is dependent on stage count
As shown in the left graph, In AOCV, delay variation is function of logic depth and distance but doesn’t consider slew/load impact.
But in actual design, the magnitude of process variation will increase at higher slew/load indices on advanced nodes.
Conventional STA with AOCV can’t consider slew/load impact to delay variation.
The magnitude of process variation will increase at higher slew/load indices on advanced nodes as shown in the left graph
In actual design, there are various slew/load situation.
Challenges
- How can we provide good & accurate STA?
OCV_sigma (AOCV)

3. Delay Variation Needs to Be Slew/Load Dependent
Solution: Liberty Variation Format (LVF) for slew/load-based variation
LVF includes delay variation based on actual slew/load
The cell delay variation is modeled as a function of input transition and output load per timing arc
POCV LVF delay variation σ = f (slew,load) X1 X2 X3
Propagated path
arrival time distribution
X1+ X2+X3
EXAMPLE of LVF delay variation
ocv_sigma_cell_rise ("delay_template_4x4") {
sigma_type : "late";
index_1("0.001, 0.005, 0.010, 1.00");
index_2("0.001, 0.002, 0.003, 0.10");
values( "σ11, σ12, σ13, σ14", \
"σ21, σ22, σ23, σ24", \
"σ31, σ32, σ33, σ34", \
"σ41, σ42, σ43, σ44", ); }
LVF(Liberty Variation Format) is a good solution for challenge1.
LVF includes delay variation is modeled as a function of input transition and output load per timing arc.
During POCV STA, distribution of arrival time is assumed as gaussian and calculated by propagating and accumulating each cell’s variation.
Slew
Load

4. Delay Variation Needs to Be Slew/Load Dependent
Samsung experiment results
Extracted 300 paths from real design to compare arrival times using PrimeTime POCV with MonteCarlo (MC) reference
POCV has good correlation with MC using slew/load dependent model
AOCV has large pessimism for some cases and therefore larger error spread than POCV
POCV
error
AOCV
average
-0.16%
-0.85%
stdev
1.02%
7.45%
We tried an experiment to show slew/load aware process variation impact. We extracted 300 paths from real design to compare arrival times using Primetime POCV with MC. POCV has good correlation with MC, on the other hand AOCV from lumped slew/load model has large pessimism for some cases and therefore larger error spread than POCV. POCV has smaller average error than AOCV. And we can see narrower spread of POCV error than AOCV by comparing stdev values.
AOCV(green) : lumped slew/load model
POCV(red) : slew/load aware variation model
Monte Carlo(blue): reference

5. How to improve accuracy of process variation STA?
Challenge 2: Common point optimism reduction for Min Pulse Width(MPW) and half-cycle path
Different transition at common point  need to keep process variation at common point
In AOCV, process variation can’t be exactly calculated by statistical sum of rising and falling paths and it can’t be separated with other variation models  Process variation will be removed from slack computation C A B
cell1
cell2
A B C
Launch clock : F  R  F
Capture clock: R  F  R
The other challenge #2 is common point optimism in the min pulse width and half-cycle path.
In those types of timing paths, there are different transitions at common point and this means actually launch path clock has different transistor  with capture path  In the AOCV based variation at common point C can be optimistic as it may be removed by CRPR. But Actual process variation at Point C should be RSS between Launch and capture paths’ variations.
AOCV-based variation at point C
= max_derate – min_derate (CRPR)
= 0 (max_derate = min_derate)
Actual process variation at Point C
= sqrt( σcell1,rise^2+σcell2,fall^2+σcell1,fall^2+σcell2,rise^2 )
v.s.
AOCV-based variation can be optimistic as it may be removed by CRPR

6. Common Point Optimism Reduction in MPW
Solution: Accurate process variation for MPW using POCV
Using POCV, accurate process variation for MPW and half cycle’s slack computation will remove optimism
Samsung experiment results
For extracted 10K paths from real design, average 3.3% optimism reduction can be reflected into MPW slack (or half cycle path) computation by POCV model
POCV process variation at Point C (same as actual)
= sqrt( σcell1,rise^2+σcell2,fall^2+σcell1,fall^2+σcell2,rise^2 ) C A B
cell1
cell2
A B C
Launch clock : F  R  F
Capture clock: R  F  R
avg
stdev
MPW Delta
-3.32%
0.79%
Solution for challenge #2 can be resolved by POCV because Process variation’s are modeled in LVF per each timing arc(rising/falling).
By Using POCV, we can remove optimism for this case by computing accurate process variation for MPW and hal cycle CRPR.
We tested this with real design and average 3.3% optimism reduction from 10K paths in min pulse with slack.
< MPW Delta: Optimism Reduction>
Delta = (new_pw – old_pw) / old_pw
new_pw: pulse width considering process variation at common point using POCV
old_pw: pulse width not considering process variation (traditional)

7. How to improve accuracy of process variation STA?
Challenge 3: Delay variation is highly correlated with output slew variation
Output slew variation will affect delay variation in next stage
Output slew variation has an increased impact at lower voltage
MonteCarlo Simulation for Inverter Cell
? Dσcell2
? Sσcell2
Highly correlated
Dσcell1
Sσcell1
X axis: cell delay
Y axis: output slew
Highly correlated
ρ = 0.78
Sin,cell2 ^
cell1
cell2
Dσcell1: cell1’s delay variation
Sσcell1 : cell1’s output slew variation
S in,cell1 : cell2’s input slew
Challenge #3 is how to deal slew variation impact on delay variation?
Actually, delay variation is highly correlated with output slew variation like below right side figures.
And output slew variation will be propagated to next cell’s input.
Therefore we can guess slew variation at cell1 will affect cell2’s variation also.

8. Consider Correlation between Slew and Delay variation
Solution: POCV considers correlation between slew and delay variation
Better accuracy for lower voltages by reflecting correlation between slew and delay variation in STA
Samsung experiment results
About 1% accuracy improvement by correlated slew variation modeling for 150 extracted paths
<POCV> Correlated delay and slew variation
Dσcell2
Dσcell2(Sσcell1)
Sσcell2
Sσcell2 (Sσcell1)
Dσcell1
Sσcell1
Dσcell1: cell1’s delay variation w/o input slew variation
Sσcell1 : cell1’s output slew variation
Dσcell2 : cell2’s delay variation w/o input slew variation
Dσcell2(Sσcell1) : cell2’s delay variation induced by input slew variation
Sσcell2 : cell2’s output slew variation w/o input slew variation
Sσcell2 (Sσcell1) : cell2’s output slew variation induced by input slew variation ^
cell1
cell2
In POCV, this slew variation can be considered with its correlation with delay variation.
In this figure, cell2’s variation is partially correlated with cell1 and
this partial correlation impact can be separated from pure independent random variation (Dsigma_cell2)
and modeled as function of Ssigma_cell1 which is Dsigmacell2(Ssigmacell1).
PT does same way to consider cell1’ slew variation impact on cell2’s slew variation which is Ssigmacell2 function of Ssigma cell1.
By considering this partial correlation induced by slew variation, we have 1% accuracy improvement comparing with MC simulation.
This table shows there cases of POCV error comparing with Montecalro simulation. In the delay variation only case, we have average 1.6% error but this will be reduced to 0.84% by considering slew variation. But this case2 doesn’t consider partial correlation relation between slew and delay. Like third case which consider this correlation, we have 0.56% average error.
Delay variation only
Delay variation + slew
variation (no correlation)
variation (correlated)
Error(%)
= (STA –MC)/MC
Mean =1.55%
Stdev = 0.70%
Mean =0.84%
Stdev = 0.56%
Mean = 0.56%
Stdev = 0.55%

9. How to improve accuracy of process variation STA?
Challenge 4: Traditionally, Flop’s constraint variation is added to nominal constraint value as margin
 Delay variation and constraint variation in STA becomes linear sum
However, delay variation and constraint variation is independent and not correlated
 Delay variation and constraint variation should be statistical sum
By separating constraint variation with nominal constraint value  theoretically maximum 40% pessimism reduction
A) Traditional linear sum: total variation = σpath + σH
B) POCV Statistical sum: total variation = SQRT ( σpath^2 + σH^2)
where σpath = SQRT ( σL^2 + σC^2)
Max difference between linear and statistical sum is when σpath= σH
 Max { (A–B) / B } = ( ) / = ~40%
 This is the pessimism from linear sum method
By adopting separate constraint variation model, maximum 40% pessimism of slack variation can be reduced σL σC σH

10. Statistical Modeling of Constraint Variation
Solution: POCV Support of Constraint Variation
Uses statistical sum of delay and constraint variation
Theoretical maximum of 40% pessimism reduction than traditional method
Samsung experiment results
Extracted 1000 paths from real design to evaluate real world pessimism reduction when using separate modeling of FF constraint variation
Linear sum pessimism = (linear sum slack – statistical sum slack)/linear sum slack
Max ~40% pessimism reduction
Majority of paths achieve close to 40% pessimism reduction in total variation
Average 37% pessimism reduction of 1000 paths
avg
stdev
Linearsum
Pessimism
-0.37
0.04

11. Summary
Review potentially opportunities to improve accuracy challenges caused by limitation of traditional variation (AOCV) and review solutions using advanced method of PrimeTime POCV
Slew/load aware delay variation modeling
Accuracy improvement  more opportunity of both robust and optimized design based on PBA accuracy enhancement
Common point optimism removal for Half-cycle and Min Pulse Width
Average 3.3% optimism reduction  robust design for MPW and half cycle path
Correlated slew variation impact on accuracy
Average 1% mean accuracy improvement by correlated modeling of delay and slew variation
Statistical constraint variation modeling
Average 37% pessimism reduction compared to traditional linear sum of delay variation and constraint variation
POCV based accurate analysis is necessary to meet tight PPA requirement with limited resources for advanced nodes
Results shown in this presentation based on Samsung Advanced node design using Synopsys PrimeTime POCV Analysis
From this accuracy enhancement, we have more chance to optimize design.