Download 2.1

Explicit Computation of Performance as a
Function of Process Parameters
Lou Scheffer
1
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
2
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:
3
You get this:
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
4
Tau 2002
Interconnect variation
• Looking at chip cross section
• Pitch is well controlled, so spacing is not
independent
P5
P4 Pitch is well controlled
These dimensions
can vary
indpendently
P2
Width and
spacing are not
independent
P3
P1
P0
5
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 DL
• Analog designers have coped with this for years
• Mismatch is statistical and a function of distance
between two figures.
6
Tau 2002
Previous Approaches
• Worst case corners – all parameters set to 3s
- 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 3s on all parameters
- Not pessimistic enough: doesn’t handle fast M1, slow M2
7
Tau 2002
Parasitics, net delays, path delays are f(P)
• CNET= f(P0, P1, P2, …)
• DELAYNET= f(P0, P1, P2, …)
P5
P4 Pitch is well controlled
P2
P3
P1
P0
8
Tau 2002
Keeping derivatives
• We represent a value as a Taylor series
N
D  D 0   d 0  Pi
1
• Where the di describe how the value varies with
a change in process parameter Dpi
• Where Dpi itself has 2 parts Dpi = DGi + Dsi,d
- DGi is global (chip-wide variation)
- Dsi,d is the statistical variation of this value
9
Tau 2002
Design constraints map to fuzzy hyperplanes
• The difference between data and clock must be less
than the cycle
time:


D( P)  C ( P)  TMAX
• Which defines a fuzzy hyperplane in process space
ANOM  CNOM   (ai  ci )DGi   (ai sia  ci sic )  TMAX
i
i
Global
(Hyperplane)
10
Tau 2002
Statistical
(sums to distribution)
Comparison to purely statistical timing
• Two approaches are complementary
Explicit computation
Propagate functions
Statistical timing
Propagate distributions
11
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
12
optimization.
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
13
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
14
Tau 2002
The bad news – complicated functions
0.8-0.2*P1+1.0*P2
0.7+0.5*P1
2
A
B
• Shows a possible pruning
problem for a 2 input gate
1.5
1
0.5
20
19
18
17
16
15
14
13
12
11
10987
6543
210
15
a
01234
0
2
0
141516171819
5 6 7 8 9 101 1213
Tau 2002
• 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
The good news - reconvergent fanout
16
•
The classic re-convergent fanout problem
•
To avoid this, statistical timing needs to keep careful track
of common paths – can take exponential time
Tau 2002
Reconvergent fanout (continued)
• Explicit calculation gives the correct result
without common path calculations
P1 =
Plug in distribution
for P1
D0+DP1
D1+DP1
D1+DP1
17
Tau 2002
D2+DP1
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
18
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?
19
Tau 2002
Predicted variations are always smaller
• Let C = C0 + k0Dp0+ k1Dp1 , , where Dp0 has
deviation s1 and Dp1 has deviation s2 .
• Then worst corner case is: C0  3k0s 0  3k1s1
• But if Dp0 and Dp1 are independent, we have
s  (k0s 0 ) 2  (k1s 1 ) 2
• So the real 3-sigma worst case is
C0  3 (k0s 0 ) 2  (k1s 1 ) 2  C0  (3k0s 0 ) 2  (3k1s 1 ) 2
• Which is always smaller by the triangle inequality
20
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
21
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 + k0Dp0+ k1Dp1 + k2Dp2
• Since delay is dominated by C, this means delay
will also be a (near) linear function of process
variation.
22
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
23
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
24
Tau 2002