Chapter 4b

Process Variation Modeling

Prof. Lei HeElectrical Engineering Department

University of California, Los Angeles

URL: eda.ee.ucla.eduEmail: lhe@ee.ucla.eduμx

σx

x

f (x)

Outline

Introduction to Process Variation

Spatial Correlation Modeling

As Good as Models Tell…

Our device and interconnect models have assumed:

Delays are deterministic values Everything works at the extreme case (corner case) Useful timing information is obtained

However, anything is as good as models tell Is delay really deterministic? Do all gates work at their extreme case at the same

time? Is the predicated info correlated to the measurement? …

Factors Affecting Device Delay

Manufacturing factors Channel length (gate length) Channel width Thickness of dioxide (tox) Threshold (Vth) …

Operation factors Power supply fluctuation Temperature Coupling …

Material physics: device fatigue Electron migration hot electron effects …

Lithography Manufacturing Process

As technology scales, all kinds of sources of variations Critical dimension (CD) control for minimum feature size Doping density Masking

Process Variation Trend

Keep increasing as technology scales down Absolute process variations do not scale well Relative process variations keep increasing

0.00%

15.00%

30.00%

45.00%

1997 1999 2002 2005 2006

Leff Tox Vth

Variation Impact on Delay

Extreme case: guard band [-40%, 55%] In reality, even higher as more sources of variation

Can be both pessimistic and optimistic

Sources of Variation Impact on Delay

Interconnect wiring(width/thickness/Inter layer dielectric thickness)

-10% ~ +25%

Environmental 15 %

Device fatigue 10%

Device characteristics(Vt, Tox)

5%

Variation-aware Delay Modeling

Delay can be modeled as a random variable (R.V.)

R.V. follows certain probability distribution

Some typical distributions Normal distribution Uniform distribution

Types of Process Variation Systematic vs. Random variation Systematic variation has a clear trend/pattern (deterministic variation

[Nassif, ISQED00])– Possible to correct (e.g., OPC, dummy fill)

Random variation is a stochastic phenomenon without clear patterns– Statistical nature statistical treatment of design

Inter-die vs intra-die variation Inter-die variation: same devices at different dies are manufactured

differently Intra-die (spatial) variation: same devices at different locations of the

same die are manufactured differently

Outline

Introduction to Process Variation

Spatial Correlation Modeling Robust Extraction of Valid Spatial Correlation Function Robust Extraction of Valid Spatial Correlation Matrix

Process Variation Exhibits Spatial Correlation

Correlation of device parameters depends on spatial locations

The closer devices the higher probability they are similar

Impact of spatial correlation Considering vs not considering 30% difference in timing

[Chang ICCAD03] Spatial variation is very important: 40~65% of total variation

[Nassif, ISQED00]

Modeling of Process Variation

f0 is the mean value with the systematic variation considered h0: nominal value without process variation ZD2D,sys: die-to-die systematic variation (e.g., depend on locations at wafers) ZWID,sys: within-die systematic variation (e.g., depend on layout patterns at

dies) Extracted by averaging measurements across many chips

Fr models the random variation with zero mean ZD2D,rnd: inter-chip random variation Xg

ZWID,rnd: within-chip spatial variation Xs with spatial correlation ρ Xr: Residual uncorrelated random variation

How to extract Fr key problem Simply averaging across dies will not work Assume variation is Gaussian

Process Variation Characterization via Correlation Matrix

Characterized by variance of individual component + a positive semidefinite spatial correlation matrix for M points of interests

In practice, superpose fixed grids on a chip and assume no spatial variation within a grid

Require a technique to extract a valid spatial correlation matrix Useful as most existing SSTA approaches assumed such a valid

matrix

But correlation matrix based on grids may be still too complex Spatial resolution is limited points can’t be too close (accuracy) Measurement is expensive can’t afford measurement for all points

2 2 2 2F G S R

1,

1,

1

1

M

M

Overall varianceGlobal variance

Spatial variance

Random varianceSpatial correlation matrix

Process Variation Characterization via Correlation Function

A more flexible model is through a correlation function If variation follows a homogeneous and isotropic random (HIR) field spatial correlation described by a valid correlation function (v) – Dependent on their distance only– Independent of directions and absolute locations – Correlation matrices generated from (v) are always positive

semidefinite Suitable for a matured manufacturing process

2 2cov( , ) ( )i j G SF F v Spatial covariance

1

1

1

d1

d1

d1

2

3

2 2

2 2 2

cov( , ) ( )i j G Sv

i j G S R

F F v

Overall process correlation

Outline

Process Variation

Spatial Correlation Modeling Robust Extraction of Valid Spatial Correlation Function Robust Extraction of Valid Spatial Correlation Matrix

Robust Extraction of Spatial Correlation Function

Given: noisy measurement data for the parameter of interest with possible inconsistencyExtract: global variance G

2, spatial variance S2, random

variance R2, and spatial correlation function (v)

Such that: G2, S

2, R2 capture the underlying variation model,

and (v) is always valid

N sample chips

M measurement sites

1

1

M

2 2 2 2F G S R

Global variance

Spatial variance

Random variance

( )v Valid spatial correlation function

2

…

fk,i: measurement at chip k and location i

i

k

How to design test circuits and place them are not addressed in this work

Variance of the overall chip variation

Variance of the global variation

Spatial covariance

We obtain the product of spatial variance S2 and spatial

correlation function (v)

Need to separately extract S2 and (v)

(v) has to be a valid spatial correlation function

Extraction Individual Variation Components

Unbiased Sample Variance

Robust Extraction of Spatial Correlation

Solved by forming a constrained non-linear optimization problem

Difficult to solve impossible to enumerate all possible valid functions

In practice, we can narrow (v) down to a subset of functions Versatile enough for the purpose of modeling

One such a function family is given by

K is the modified Bessel function of the second kind is the gamma function Real numbers b and s are two parameters for the function family

More tractable enumerate all possible values for b and s

11

1( ) 2 ( ) ( 1)2

s

s

b vv K b v s

2

2 2

, ( )min : ( ) cov( )s

s gv

v v

Robust Extraction of Spatial Correlation

Reformulate another constrained non-linear optimization problem

2

212 1 2

1, ,

min : 2 ( ) ( 1) cov( )2s

s

s s gb s

b vK b v s v

2 2. . : s fcs t

Different choices of b and s different shapes of the function each function is a valid spatial correlation function

Outline

Process Variation

Spatial Correlation Modeling Robust Extraction of Valid Spatial Correlation Function Robust Extraction of Valid Spatial Correlation Matrix

Robust Extraction of Spatial Correlation Matrix

Given: noisy measurement data at M number of points on a chip

Extract: the valid correlation matrix that is always positive semidefinite

Useful when spatial correlation cannot be modeled as a HIR field

Spatial correlation function does not exist SSTA based on PCA requires to be valid for EVD

N sample chips

M measurement sites

1

1

M

1

1

1

0

1

M

M

Valid correlation matrix

2

…i

k

fk,i: measurement at chip k and location i

Extract Correlation Matrix from Measurement

Spatial covariance between two locations

Variance of measurement at each location

Measured spatial correlation

Assemble all ij into one measured spatial correlation matrix A But A may not be a valid because of inevitable

measurement noise

ijA

Robust Extraction of Correlation Matrix

Find a closest correlation matrix to the measured matrix A Convex optimization problem Solved via an alternative projection algorithm

Details in the paper

Reference

Sani R. Nassif, “Design for Variability in DSM Technologies”. ISQED 2000: 451-454

Jinjun Xiong, Vladimir Zolotov, Lei He, "Robust Extraction of Spatial Correlation," ( Best Paper Award ) IEEE/ACM International Symposium on Physical Design , 2006.