1 Tau 2002

Explicit Computation of Performance as a Function of Process Parameters

Lou Scheffer

2 Tau 2002

What’s the problem?

• Chip manufacturing not perfect, so….

• Each chip is different

• Designers want as many chips as possible to work

• We consider 3 kinds of variation

‑ Inter-chip

‑ Intra-chip

‣Deterministic

‣Statistical

3 Tau 2002

Intra-chip Deterministic Variation (not considered further in this presentation)

• Optical Proximity Effects

• Metal Density Effects

• Center vs corner focus

You draw this: You get this:

4 Tau 2002

Inter-chip variation

• Many of the sources of variation affect all objects on the same layer of the same chip.

• Examples:

‑ Metal or dielectric layers might be thicker/thinner

‑ Each exposure could be over/under exposed

‑ Each layer could be over/under etched

5 Tau 2002

Interconnect variation

• Looking at chip cross section

• Pitch is well controlled, so spacing is not independent

These dimensions can vary indpendently

Pitch is well controlled

P1

P2

P3

P4

P0

P5

Width and spacing are not independent

6 Tau 2002

Intra-chip statistical variation

• Even within a single chip, not all parameters track:

‑ Gradients

‑ Non-flat wafers

‑ Statistical variation

‣Particularly apparent for transistors and therefore gates

‣Small devices increase the role of variation in number of dopant atoms and L

• Analog designers have coped with this for years

• Mismatch is statistical and a function of distance between two figures.

7 Tau 2002

Previous Approaches

• Worst case corners – all parameters set to 3

‑ Does not handle intra-chip variation at all

• 6 corner analysis

‑ Classify each signal/gate as clock or data

‑ Cases: both clock and data maximally slow, clock maximally slow and data almost as slow, etc.

• Problems with these approaches

‑ Too pessimistic: very unlikely to get 3 on all parameters

‑ Not pessimistic enough: doesn’t handle fast M1, slow M2

8 Tau 2002

Parasitics, net delays, path delays are f(P)

• CNET= f(P0, P1, P2, …)

• DELAYNET= f(P0, P1, P2, …)

Pitch is well controlled

P1

P2

P3

P4

P0

P5

9 Tau 2002

Keeping derivatives

• We represent a value as a Taylor series

• Where the di describe how the value varies with a change in process parameter pi

• Where pi itself has 2 parts pi = Gi + si,d

‑ Gi is global (chip-wide variation)

‑ si,d is the statistical variation of this value

N

iPdDD1

00

10 Tau 2002

Design constraints map to fuzzy hyperplanes

• The difference between data and clock must be less than the cycle time:

• Which defines a fuzzy hyperplane in process space

Global Statistical

(Hyperplane) (sums to distribution)

MAXTPCPD )()(

MAXNOMNOM TscsaGcaCAi

iciiaii

iii )()(

11 Tau 2002

Comparison to purely statistical timing• Two approaches are complementary

Propagate functions

Propagate distributions

Explicit computation

Statistical timing

12 Tau 2002

Similarities in timing analysis

• Extraction and delay reduction are straightforward, timing is not

• Latest arriving signal is now poorly defined

• If a significant probability for more than one signal to be last, both must be kept (or some approximate bound applied).

• Pruning threshold will determine accuracy/size tradeoff.

• Must compute an estimate of parametric yield at the end.

• Provide a probability of failure per path for optimization.

13 Tau 2002

Differences

• Propagate functions instead of distributions

• Distributions of process parameters are used at different times

‑ Statistical timing needs process parameters to do timing analysis

‑ Explicit computation does timing analysis first, then plugs in process distributions to get timing distributions.

‣Can evaluate different distributions without re-doing timing analysis

14 Tau 2002

Pruning

• In statistical timing

‑ Prune if one signal is ‘almost always’ earlier

‑ Need to consider correlation because of shared input cones

‑ Result is a distribution of delays

• In this explicit computation of timing

‑ Prune if one is earlier under ‘almost all’ process conditions

‑ Result is a function of process parameters

‑ Bad news – an exact answer could require (exponentially) complex functions

‑ Good news - no problem with correlation

15 Tau 2002

The bad news – complicated functions

a

01234567

891011121314151617181920

01234567 8910111213141516171819200

0.5

1

1.5

2 • Shows a possible pruning problem for a 2 input gate

• Bottom axes are two process parameters; vertical is MAX(A,B)

• Can keep it as an explicit function and prune when it gets too expensive

•Can cover with one (conservative) plane

0.8-0.2*P1+1.0*P2

0.7+0.5*P1

A

B

16 Tau 2002

The good news - reconvergent fanout

• The classic re-convergent fanout problem

• To avoid this, statistical timing needs to keep careful track of common paths – can take exponential time

17 Tau 2002

Reconvergent fanout (continued)

• Explicit calculation gives the correct result without common path calculations

D0+P1

P1 =

D1+P1D2+P1

D1+P1

Plug in distribution for P1

18 Tau 2002

Real situation is a combination of both

• Gate delays are somewhat correlated but have a big statistical component

• Wire delays (particularly close wires) are very highly correlated but have a small random component.

• Delays consist of two parts that combine differently

Distribution of statistical part is also a function of process variation

19 Tau 2002

So what’s the point of explicit computation?

• Not so worst case timing predictions

‑ Users have complained for years about timing pessimism

‑ Could be about 10% better (see experimental results)

‑ Could save months by eliminating unneeded tuning

• Will catch errors that are currently missed

‑ Fast/slow combinations are not currently verified

• Can predict parametric yield

‑ What’s the timing yield?

‑ How much will it help to get rid of a non-critical path?

20 Tau 2002

Predicted variations are always smaller

• Let C = C0 + k0p0+ k1p1 , , where p0 has deviation and p1 has deviation .

• Then worst corner case is:

• But if p0 and p1 are independent, we have

• So the real 3-sigma worst case is

• Which is always smaller by the triangle inequality

211

200 )()( kk

211

2000

211

2000 )3()3()()(3 kkCkkC

11000 33 kkC

21 Tau 2002

Won’t this be big and slow?

• Naively, adds an N element float vector to all values

• But, an x% change in a process parameter generally results in <x% change in value

‑ Can use a byte value with 1% accuracy

• A given R or C usually depends on a subset

‑ Just the properties of that layer(s)

• Net result – about 6 extra bytes per value

• Some compute overhead, but avoids multiple runs

22 Tau 2002

Experimental results for explicit part only

• Start with a 0.18 micron, 5LM, 144K net design

• First – is the linear approximation OK?

‑ Generated 35 cases with –20%,0,+20% variation of three most relevant parameters for metal-2 layer

‑ For each lumped C value did coeffgen, then HyperExtract, then a least-squares fit

‑ Less than 1% error for C = C0 + k0p0+ k1p1 + k2p2

• Since delay is dominated by C, this means delay will also be a (near) linear function of process variation.

23 Tau 2002

More Experimental Results

• Next, how much does it help?

‑ Varied each parameter (of 17) individually

‑ Compared to a worst case corner (3 sigma everywhere)

‑ Average 7% improvement in prediction of C

• Will expect a bigger improvement for timing

‑ Since it depends on more parameters, triangle inequality is (usually) stronger

24 Tau 2002

Conclusions

• Outlined a possible approach for handling process variation

‑ Handles explicit and statistical variation

‑ Theory straightforward in general

‣Pruning is the hardest part, but there are many alternatives

‑ Experiments back up assumptions needed

‑ Memory and compute time should be acceptable