1. Rasit Onur Topaloglu University of California San Diego Computer Science and Engineering Department Ph.D. candidate www.cse.ucsd.edu/~rtopalog “Location Based On-Chip Variation”

2. Process model Motivation On-chip variation model Validation methodology Outline Experimental Results Conclusions

3. Motivation Process variations, if not considered properly, may cause chips to fail or prone designs to be impossible to attain a spec. Currently static timing analysis tools neglect cell locations Cell locations contain a valuable systematic information Increases design time and reduces yield

4. Oxide Distribution on Wafer Ref: Intel Technology Journal, Vol. 06, Issue 2, May 2002 30cm wafer, 0.13  m, SEM Oxide distribution seems to be circular & continuous

5. Process and OCV Models

6. Modeling of Process Variations : “Volcano Model” Effects such as oxide variation, threshold voltage variation, lumped as cell speed variation Cell speed Distance from center Cell speed Equ-speed circles on wafer Linearly increasing or decreasing cell speeds along radius Variation curves are circular 1.2 1.15 1.1 1.05

7. Modeling of On-chip Variation : “Angular Model” Cell speeds will be effected depending on angle wrt wafer center Assumption :  max. on-chip speed variation On-chip variation available as std. dev. only Since chips are small, circle arcs approximated to be straight lines 1 2 1 Chip may fall anywhere on wafer -model -real Process curves on chip chip1 chip2 C

8. Location of Chip Matters Equ-lines are taken to be parallel to each other and normal to the line that connects wafer center and closest corner of chip C 

9. Calculation of Speed Variation   B A |A//B| / |B| ratio is used to find process variation effect at location p p Multiply this ratio by maximum on-chip variation to find cell speed

10. Hypothesis I : Chips at Same Angle If process variation not linearly effecting cell speeds, maximum on-chip variations for chipA and chipB will differ Chip2 has more variation, simulating for it is satisfactory if equ-speed circles not evenly distributed on wafer: A B 1.2 1.15 1.05 1.1

11. Hypothesis II : “Dominant Locations” on Wafer Check a number of angles on wafer We want other dies to pass too Make sure simulating effects of process variations for dies on dominant locations is satisfactory

12. Test and Validation of Proposed Methods

13. Comparison Methodology Extract cell locations from Astro Run script that changes cell speeds of a chip at a given angle and given max. on-chip variation Compare minimum setup times and hold times with a nominal run For each dominant location angle { } Used to show that location based variations can be deteriorating as compared to worst-case runs

14. Comparison with Probabilistic Cell Speeds Run script that changes cell speeds of a chip using a uniform distribution given max. on-chip variation Compare minimum setup times and hold times with a location based deterministic run Run script that changes cell speeds of a chip using a Gaussian distribution given max. on-chip variation For each dominant location angle { } Used to show that location based variations can be deteriorating as compared to probabilistic models due to systematic variation

15. Proof I : Checking Validity of Method for Chips on Same Angle Run script that changes cell speeds of a chip at angle  Compare minimum setup times and hold times  runs For a number of variations up to max on-chip variation { } Used to show that for chips at same angle, simulating worst variation is satisfactory

16. Proof II : Checking Validity of Dominant Locations Run script that changes cell speeds of a chip given that angle Check that minimum setup or hold times are higher than found using dominant locations For a number of (angles \ dominant angles) { } Used to show that simulating for chips at dominant locations satisfactory for any location

17. Experimental Results Setup (max delay) 0.1243 0.1001 0.1206 0.1234 Uniform random Gaussian random Location based Nominal Up to 20% variation in minimum slack observed on ARM7 Or, try setting clock to 1GHz whereas your chip can run @ 800MHz on most locations on wafer 0.1242 when less variation used Hypothesis I supported

18. Where Location Based Method fits in PrimeTime? Setup (max delay) Hold (min delay) WCTYPBCBC/WCOCV max delays  paths for setup 1.74 0.53 max delays  paths 1.743.925.15 0.531.401.71 max data min clock delays for setup -1.70 -3.29 Location based falls here, more realistic than both directions underestimate overest.

19. Conclusions Location based variation fits on a more realistic scale as compared to current PrimeTime models Probabilistic models fail to be satisfactory as they neglect deterministic systematic relationship between cells Dominant locations provide a means to reduce simulation time, yet integrate more accurate process variation effects

20. Future Directions Incorporation of interconnect delay variations Proper selection of dominant locations A layout based mathematical approach