A cumulative distribution function-based method for yield optimization of CMOS ICs

1. A cumulative distribution function-based method for yield optimizationof CMOS ICs M. Yakupov*, D. Tomaszewski Division of Silicon Microsystem and Nanostructure Technology,Institute of Electron Technology, Warsaw, Poland * Currently MunEDA GmbH, Munich, Germany

2. Outline • Introduction • Introductoryexample • Cumulative-distribution function-based approach • Application of the CDF method for inverter and opamp design • Remarks on CDF-basedmethodimplementation • Conclusions

3. Introduction • Importanceof the yieldoptimizationtask • Time-to-market, and cost effectiveness, • Robustness of design; • Corner (worstcase – WC)methods • Fast design space exploration in terms of PVT variations, • Increase of number of corners with increase of types of devices, • Lack of correlations between different parameters sets; • Monte-Carlo (MC) method • Time consuming, many simulations, • Does not activelymanipulate / improve a design; • Worst-casedistance (WCD) method • Operates in processparameterspace, not obvious from design perspective; • Cumulativedistributionfunction (CDF) – basedmethod • Operates in design parameterspace;

4. Introductory example Given:length L (design), sheet resistance Rs (model) being a random variable of normal distribution: Task: Design a resistor, which fulfills the condition: Rmin<R<Rmax; Find W (design): N

5. Introductory example Variant I Rmin 950  Rmax 1100  L 100×10-6 m Rs,mean 10 /sq. Rs 0.3 /sq. Wopt9.8×10-6 m Variant II Rmin 900  Rmax 1100  L 100×10-6 m Rs,mean 13 /sq. Rs 0.3 /sq. Wide acceptable range of W

6. Introductory example Variant III Rmin 1060  Rmax 1100  L 100×10-6 m Rs,mean 10 /sq. Rs 0.6 /sq. Wopt0.93×10-6 m, but very low yield Conclusions: yield depends on relation between parameter (R) constraints and process (Rs) quality; cumulative distribution function (CDF) has been successfuly used to determine optimum design. Question: Can CDF be used for real more complex tasks ?

7. CDF-based approach • D - design parameter vector • M- process parametervector • X - circuit performancevector • S – specificationvector • Yieldoptimizationtask X=f(D,M) Partial yields

8. CDF-based approach Process par. variations Nominal design i-th performance Sensitivities • Issues: • DetermineMnom • influences performance of the nominal design • influences performance sensitivities • DetermineM • Remarks: • Relation to BPVmethodused for statisticalmodelling, i.e. for extraction of M • Mj – randomvariables

9. CDF-based approach Yield optimization task may be reformulated: Dopt should maximize joint probability: • Issues: • Select reliablemethod for yieldoptimization, takingintoaccountspecificfeatures of the task • Remarks: • Anoptimization problem hasbeenformulated

10. CDF-based approach IfMjareuncorrelatednormally-distributedrandomvariables, then is a randomvariable I.  N Sensitivities Process par. variations • Issue: • Selection of uncorrelatedprocessparametersisveryimportant

11. CDF-based approach Due to the unavoidablecorrelation between performances, direct yield and product of partial yields are not equal. II. • Issues: • Take intoaccountcorrelationsbetweenperformances. • or • Assume, that

12. CDF-based approach Based on assumptions I, II a joint probability (parametric yield) may be calculated as a product of CDFsFiof normal distributions

13. CDF-based approach Interpretation If NX=1 case is considred the task may be illustrated "geometrically" Maximize shaded area

14. Backward Propagation of Variance Method Calculations of standard deviations of process parameters based on variations of performances and on performance sensitivities. C. C. McAndrew, “Statistical Circuit Modeling,” SISPAD 1998, pp. 288-295.

15. CDF-based approach vs BPV method BPV Functionalblock performance (PCM) sensitivities @ nominalprocessparameters Process parameter variances Functional block performance variances determined experimentally Functionalblock performance sensitivities @ nominalprocessparameters CDF of functional block performance variances Process parameter variances

16. CDF method - Inverter Inverter performancesJ.P.Uemura, "CMOS Logic Circuit Design", Kluwer, 2002 Inverter threshold Propagation delay

17. CDF method – OpAmp OpAmp performances M.Hershenson, et al.,"Optimal Design of a CMOS Op-Amp via Geometric Programming", IEEE Trans. CAD ICAS, Vol.20, No.1 Low-frequency gain Phase margin • gmi, goi- input and output conductances of i-th transistor, • c is a unity-gain bandwidth, • pj- j-th pole of the circuit, • Sk- input-referred noise power spectral densities, consisting of thermal and 1/f components. Equivalent input-referred noise power spectral density

18. CDF method – Inverter, OpAmp • Monte-Carlo method 1000 samples • simple MOSFET model • 0.8 m CMOS technology • tox=20 nm, • Vthn=0.7 V, Vthp=-0.9 V • process parameters varied: • gate oxide thickness tox, • substrate doping conc. Nsubn, Nsubp, • carrier mobilitieson, op, • fixed charge densities Nssn, Nssp

19. CDF method - Inverter Contour plots of yield in design parameter space open - partial yields closed - product of partial yields (CDF) solid - product of partial yields (CDF) dashed - product of partial yields (MC) dotted - yield (MC) A "valley" results from the specification of tP,min constrain.

20. CDF method – OpAmp Contour plots of yield in design parameter space open - partial yields closed - product of partial yields (CDF) solid - product of partial yields (CDF) dashed - product of partial yields (MC) dotted - yield (MC)

21. CDF-based method implementation • Objectivefunctionmaximization • Close to the maximum the objectivefunctionmayexhibit a plateau; • Optimizationtaskbased on gradient approachrequires in thiscase 2nd order derivatives of yieldfunction, but… this makes optimization based on gradient methods useless;

22. CDF-based method implementation • Objectivefunctionmaximization • Objectivefunctionmayexhibitmorethan one plateau ormorelocalmaxima; • thus • a non-gradient globaloptimizationmethodisrequired.

23. Conclusions (1) The presented CDF-basedmethodmay be used for IC block design optimizingparametricyield, The methodmaypredictparametricyield of the design, The CDF-based method gives results very close to the time consuming Monte Carlo method, The results of yield optimization (Yopt) based on CDF method have direct interpretation in designparameterspace (problem of selection of design parameters: explicitorcombined), The design rules of the given IP and alsodiscrete set of allowedsolutions* may be directly used and shown in the yield plots in the design parameter space, The methodmay be used for performancesdeterminedbothanalytically, as well as via Spice-likesimulations (batchmoderequired), * Pierre Dautriche, "Analog Design Trends & Challenges in 28 and 20 nm CMOS Technology", ESSDERC'2011 23

24. Conclusions (2) Basic requirements for automatization of design taskbased on CDF methodhavebeenformulated, If the performance constraintsare mild with respect to processvariability,a continuous set of design parameters, for which yield close to 100% is expected, If the constraints are severe with respect to processvariability, the method leads to unique solution, for which the parametric yield below 100% is expected, Thus the methodmay be veryuseful for evaluationif the processisefficientenough to achieve a givenyield. The methodologymay be used for design types (not only of ICs) takingintoaccountstatisticalvariabilityof a process and aimedatyieldoptimization, Statistical modelling, i.e. determination of processparameterdistributionis a criticalissue, for MC, CDF, WCD methods of design. 24

25. Acknowledgement Financial support of EC withinprojectPIAP-GA-2008-218255 ("COMON") and partialfinancialsupport (for presenter) of EC withinproject ACP7-GA-2008-212859 ("TRIADE") areacknowledged.