1. 0

2. 1 Width-dependent Statistical Leakage Modeling for Random Dopant Induced Threshold Voltage Shift Jie Gu, Sachin Sapatnekar, Chris Kim Department of Electrical and Computer Engineering University of Minnesota chriskim@umn.edu www.umn.edu/~chriskim

3. 2 Outline Introduction Conventional statistical leakage modeling Proposed statistical leakage modeling  “Microscopic” random dopant fluctuation Experimental results Application to leakage sensitive circuits  Dynamic circuits  SRAM memory bitlines Conclusions

4. 3 Motivation worst-case corner (150nm CMOS Measurements, 110°C) nominal corner 4X variation between nominal and worst-case leakage  Channel length/width variation, line edge roughness, dopant fluctuation Performance determined at nominal leakage Power/robustness determined at worst-case leakage 0 50 100 150 200 Normalized I OFF Number of dies 0123456 7

5. 4 Leakage Variation Impact: Dynamic Circuit Example Static keeper prevents the dynamic node droop Keeper has to be properly sized for sufficient noise margin Accurate leakage estimation is critical for meeting noise margin requirements 0 50 100 150 200 Pull down leakage Number of dies 0123456 7 clk... RS0 D0 RS1 D1 RS7 D7 Dyn_out keeper I leak worst-case corner Over- designed for robustness Fail to meet target robustness

6. 5 Conv. Statistical Leakage Modeling Conventional approach (square-root method): Device parameters provided by fabs: Statistically, larger devices have lesser variation W 0

7. 6 µ-RDF Induced Threshold Voltage Shift Threshold voltage depends on both  the # of dopants in the channel and  the “microscopic” random dopant placement 30mV+ V T shift has been reported by P. Wong (IEDM ’93) V T =0.78V, 130 dopants, L eff = 30nm 3D simulation results on surface potential A. Asenov, TED 1998 V T =0.56V, 130 dopants, L eff = 30nm Evenly distributed dopants Dominant leakage path Unevenly distributed dopants S D S D

8. 7 Golden Statistical Leakage Modeling Golden approach: Need to sum the leakage dist. of the sub-devices Device parameters provided by fabs: W 0

9. 8 Conventional versus Golden Method Leakage model shows 32% discrepancy btwn 3σ values  µ(V T ) reduces when adding lognormal distributions Delay model matches well with golden results  µ(V T ) does not change when adding normal distributions 32nm PTM Leakage distributionDelay distribution 32nm PTM

10. 9 Statistical Leakage Estimation Effective V T concept introduced to model width-dependency inaccurate V Ti W 0 Golden Conventional Proposed W=nW 0 V Ti+1 Given reference device parameter: W 0, 0T V  0T V  ),,W(f)(V ),,W(f) eWI 0T0T 0T0T Teff VV VV /mkTqV leak       00 VT VT /mkTqV leak LW LW /)(V ) eWI 0Tsq 0T T       0T 0T Ti V V n 1i /mkTqV 0leak )(V ) eWI       (V Ti varies due to RDF)

11. 10 Previous Work and Our Contribution To the best of our knowledge, this is the first work to model the V T dependency on device width  Previous work proposed by Ananthan (DAC06), Chang (DAC05), Narendra (ISLPED02) did not consider this Simple closed-form expression derived that can handle continuous width case Previous work Proposed

12. 11 Given a reference device with W x, μ 0, σ 0 W y =nW x  Discrete width multiplication  Directly apply Wilkinson’s method W y =αW x (α=n/m)  Continuous width multiplication (α is any rational number)  Extend Wilkinson’s method to handle continuous integration Calculation of Effective V T

13. 12 Wilkinson’s Method First moment: Second moment: Moment matching results: Sum of lognormals can be approximated as a single lognormal with a calculable mean and standard deviation y is the new Gaussian variable calculated by moments matching A. Abu-Dayya, IEEE Vehicular Technology Conference, 1994

14. 13 Discrete Width Multiplication (W y =nW x ) where Both the mean and sigma of V T decreases with larger device width The mean and sigma of V T also decreases with smaller r x Effective V T Sub-device V T

15. 14 Spatial Correlation Coefficient r y goes up as width increases (r x ≤ r y ≤1 and r y =1 iff r x =1)

16. 15 Reference Device Size Independence Same results can be obtained independent of the reference device size Given μ y, σ y of a large device with W y, we can reverse the calculation to find out μ x, σ x of a smaller device with W x

17. 16 Assume there exists a small virtual device that satisfies W x =mW 0, W y =nW 0. Applying Wilkinson’s for both we get, Continuous Width Multiplication (W y =αW x ) Expression identical to the discrete width multiplication case where Effective V T Sub-device V T

18. 17 Width Dependent V T Statistics 21mV difference in V T mean between conventional and golden method No significant difference in V T sigma between golden, square-root, and proposed method 32nm PTM 21mV

19. 18 Leakage Distribution Comparison: Different Widths Leakage estimation error (3σ point) reduced from 10.5% to 0.8% using the proposed model

20. 19 Leakage Distribution Comparison: Different σ/µ’s Error of conventional approach increases to 45.5% for larger σ/µ due to larger variation between the sub-devices Proposed approach exhibits a smaller leakage estimation error (<12.0%) limited by the accuracy of the Wilkinson’s formula

21. 20 Leakage Distribution Comparison: Different Correlation Coefficients Conventional model does not consider spatial correlation (assumes that sub-devices are uncorrelated) Proposed work’s estimation error is small for a wide- range of V T correlation coefficients

22. 21 Design Example: Dynamic Circuit Keeper Sizing Conventional approach underestimates the pull-down leakage misguiding the designer to use a smaller keeper This work shows that a 30% keeper size up is required to meet the target noise margin

23. 22 Design Example: SRAM Bitline Actual bitline delay is 3-12% longer than expected when using the conventional model due to underestimated bitline leakage

24. 23 Conclusions “Microscopic” RDF leads to width-dependent V T Conventional statistical V T model is inaccurate  Only capable of modeling on current (linear function of V T )  Fails to model leakage current (exponential function of V T )  Exhibits as much as 45% error in 3σ leakage value Proposed width-dependent V T model  Simple closed-form expression with than 5% estimation error  Can be expanded to general sources of within-device variation  Handles both uncorrelated and correlated process variables  Useful for leakage-sensitive circuit designs such as dynamic circuits, SRAM bitlines, and subthreshold logic