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Width-dependent Statistical Leakage Modeling for Random Dopant Induced Threshold Voltage Shift

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. Outline. Introduction

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Width-dependent Statistical Leakage Modeling for Random Dopant Induced Threshold Voltage Shift

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  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

  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

  3. 200 150 Number of dies 100 50 0 0 1 2 3 4 5 6 7 Normalized IOFF Motivation (150nm CMOS Measurements, 110°C) nominal corner worst-case 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

  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 200 keeper Over-designed for robustness Fail to meet target robustness clk 150 Dyn_out Number of dies 100 worst-case corner RS0 RS1 RS7 50 ... D1 D7 D0 0 Ileak Ileak Ileak 0 1 2 3 4 5 6 7 Pull down leakage

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

  6. µ-RDF Induced Threshold Voltage Shift D D S S Dominant leakage path Evenly distributed dopants Unevenly distributed dopants VT=0.78V, 130 dopants, Leff = 30nm VT=0.56V, 130 dopants, Leff = 30nm 3D simulation results on surface potential A. Asenov, TED 1998 • Threshold voltage depends on both • the # of dopants in the channel and • the “microscopic” random dopant placement • 30mV+ VT shift has been reported by P. Wong (IEDM ’93)

  7. W 0 G D S V W Ti 0 W = nW 0 V Ti + 1 Golden Statistical Leakage Modeling Device parameters provided by fabs: Golden approach: • Need to sum the leakage dist. of the sub-devices

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

  9. m s V V T 0 T 0 n - qV /mkT å - µ qV /mkT I W e µ T - I W e qV /mkT sq Teff µ I W e Ti leak leak leak 0 m = m (V ) m = m s = (V ) f ( W , , ) i 1 T V m sq T 0 Teff V V m = m (V ) T 0 T 0 Ti V × W L s = m s (V ) f ( W , , ) T 0 s = s (V ) / s Teff V V s = s (V ) T V T 0 T 0 × W L sq T 0 Ti V T 0 0 0 Statistical Leakage Estimation Proposed Conventional Golden W V 0 Ti W = nW 0 V Ti + 1 ( V varies due to RDF ) Ti Given reference device parameter : , , W 0 inaccurate • Effective VT concept introduced to model width-dependency

  10. Previous Work and Our Contribution • To the best of our knowledge, this is the first work to model the VT 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 Proposed Previous work

  11. Calculation of Effective VT Given a reference device with Wx, μ0, σ0 • Wy=nWx • Discrete width multiplication • Directly apply Wilkinson’s method • Wy=αWx (α=n/m) • Continuous width multiplication (α is any rational number) • Extend Wilkinson’s method to handlecontinuous integration

  12. Wilkinson’s Method • 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 First moment: Second moment: Moment matching results: A. Abu-Dayya, IEEE Vehicular Technology Conference, 1994

  13. Discrete Width Multiplication(Wy=nWx) Effective VT Sub-device VT where • Both the mean and sigma of VT decreases with larger device width • The mean and sigma of VT also decreases with smaller rx

  14. Spatial Correlation Coefficient • ry goes up as width increases (rx≤ ry ≤1 and ry=1 iff rx=1)

  15. Reference Device Size Independence • Same results can be obtained independent of the reference device size • Given μy, σyof a large device with Wy, we can reverse the calculation to find out μx, σx of a smaller device with Wx

  16. Continuous Width Multiplication(Wy=αWx) Assume there exists a small virtual device that satisfies Wx=mW0, Wy=nW0. Applying Wilkinson’s for both we get, Effective VT Sub-device VT where • Expression identical to the discrete width multiplication case

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

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

  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

  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 VT correlation coefficients

  21. keeper clk Dyn_out RS0 RS1 RS7 ... D1 D7 D0 Ileak Ileak Ileak 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

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

  23. Conclusions • “Microscopic” RDF leads to width-dependent VT • Conventional statistical VT model is inaccurate • Only capable of modeling on current (linear function of VT) • Fails to model leakage current (exponential function of VT) • Exhibits as much as 45% error in 3σ leakage value • Proposed width-dependent VT 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

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