Chebyshev Affine Arithmetic Based Parametric Yield Prediction Under Limited Descriptions of Uncertainty

Transcription

1 Chebyshev Affine Arithmetic Based Parametric Yield Prediction Under Limited Descritions of Uncertainty Dr. Janet Wang ECE Deartment The University of Arizona

2 Power Leakage Exonential rise of IC ower dissiation Device dimension scales down. Threshold voltage shrinks. Great ortion of total ower consmtion May accont for 50% of the total delay Will be frther aggravated Significant imact on arametric yield 2

3 Parameter Variations Strong deendence of leakage on both rocess and environmental variations. Case a large sread in leakage crrent. 2 kinds of variations are considered. Process arameters Leff, Vth, and Tox Environmental arameters Vdd, and T 3

4 Parametric Yield Prediction Parameter variations redces the yield of designs. Yield rediction methods are reqired to model the deendency. Limited by fndamental featres of IC design Incomlete rocess characterization data. Large ncertainty in statistic metrics. Correlation between arameters. 4

5 Main Proses Uncertainty reresentations Probability reresentations Consider both rocess variations and environme ntal ncertainty Consider correlation between arameters Provide reliable robability bonds for leakage c rrent 5

6 Chebyshev Affine Arithmetic Affine form: xˆ x x ε x ε L x n ε n ε [,] [ ] 0 and i x 0 E ε i, central vale ε, noise symbols x i, artial deviations i 6

7 Chebyshev Affine Oerations Can be easily exanded 3 cases: xˆ ± yˆ x0 y0 x y ε L x n y n ε n αxˆ αx αx ε L 0 α x n ε n xˆ ± ζ x ζ x ε L 0 x n ε n Still in affine form 7

8 8 Non-affine Oerations, is not affine Aroximations reqired reresents aroximation error Retrns an Affine form * f,, ˆ ˆ, * n f y x f z ε ε L k k n n n a z z z z f z ε ε ε ε ε L L 0,, ˆ k k z ε,,,,,, * * n a n n f f e ε ε ε ε ε ε L L L

9 Chebyshev Aroximations in the form of affine combinations : α x βy ζ Otimal: minimizes the maximm absolte error Geometric illstration zˆ α xˆ ζ δε k Cmlative Probability x x x 9

10 Leakage Model An emirical model Obtained from SPICE simlation Model the deendency on arameter variations Leff: qadratic exonential deendency Vth: exonential deendency Tox: exonential deendency Vdd: exonential deendency T: ser linear deendency aroximated as exo nential 0

11 Analytical Eqations Mathematical reresentations of leakage model Sbthreshold leakage model I sb sb, nom e Gate leakage model I I aδl gate I gate, nom bδl cδv Total leakage is the smmation 2 e I I total sb hδt I ox gate th kδvdd dδv dd eδt

12 Parameter Decomosition Parameter variations frther decomosed into two comonents. ΔP global ΔP ΔP global ΔP local, the global inter-chi variations ΔP local, the local intra-chi variations Assmed to be indeendent and normal Reslt in also normal distribtion ΔP 2

13 Imroved Leakage Model Sbthreshold leakage model I sb I sb, nom e aδl 2 l 2aΔLg b ΔL cδv l th, l dδvdd ΔT e aδl 2 g bδl g cδv th, g Gate leakage model I gate I gate, nom e hδt ox, l kδvdd e hδt ox, g They are correlated 3

14 Isses with New Technology Nodes Parameters are difficlt to extract: ncertainty in robability distribtions 70 nm and below We se a set of CDs consisting of a left and a right bond x x x x Cmlative Probability x i x i x i x 4

15 Chebyshev vs. Discretized Method x x x x x x x x x x x x 5

16 PLPB Reresentation Linearization on CD: Piece-wise Linear Probability Bonds PLPB Comtation on Parameters CD fnctions x x 6

17 Piece-wise Linear CD with its inverse x P x x P x 7

18 8 Deendency Bonds of ZY Uer bond Lower bond ] max[ ] [0, if if Y Y Y ] min[,] [ if if Y Y x Y

19 Dataflow of Comting Y x Y y Y Y g Y Y min g [,] 9

20 20 Y - Y g Y min g Z

21 2 Deendency bonds of -Y Uer bond Lower bond 0 ] max[ ] [0, if if Y Y x Y ] min[,] [ if if Y Y x Y

22 Yield Prediction Procedre 22

23 Yield Prediction Procedre contined zˆ xˆ ± ζ zˆ α xˆ zˆ xˆ ± yˆ 23

24 Exeriment Environments 65nm Technology node PTM model. Leff 24.5nm Coefficients extracted by SPICE simlations. Parameter variations Modeled as trncated Gassian distribtions. Can be well handled if non-gassian. Leff: 20% variation, Vth: 0% variation, Tox: 8% variat ion Vdd: 0% variation, T: 0 variation. 24

25 Comarison with MC simlation and inter val analysis: I sb Imrovements: 50% ercentile 3.3% 95% ercentile 27.% Mean vale 25.%.674->

26 Comarison with MC simlation and inter val analysis: I gate Imrovements: 50% ercentile 5.5% 95% ercentile 7.6% Mean vale 5.3%.42->.96 26

27 Comarison with MC simlation and inter val analysis: I total Imrovements: 50% ercentile 0.9% 95% ercentile 23.6% Mean vale 2.7%.566->

28 Contors for inter-chi L Variation Shorter channel len gth cases more si gnificant variation of leakage crrent. 28

29 Conclsion Based on Chebyshev affine arithmetic Handle ncertainty of distribtions Deal with correlations among variations Efficient and reliable yield rediction 29