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

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slide1

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

slide2

Outline

  • Introduction
  • Introductoryexample
  • Cumulative-distribution function-based approach
  • Application of the CDF method for inverter and opamp design
  • Remarks on CDF-basedmethodimplementation
  • Conclusions
slide3

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;
slide4

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

slide5

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

slide6

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 ?

slide7

CDF-based approach

  • D - design parameter vector
  • M- process parametervector
  • X - circuit performancevector
  • S – specificationvector
  • Yieldoptimizationtask

X=f(D,M)

Partial yields

slide8

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
slide9

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
slide10

CDF-based approach

IfMjareuncorrelatednormally-distributedrandomvariables, then

is a randomvariable

I.

 N

Sensitivities

Process par. variations

  • Issue:
  • Selection of uncorrelatedprocessparametersisveryimportant
slide11

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
slide12

CDF-based approach

Based on assumptions I, II a joint probability (parametric yield)

may be calculated as a product of CDFsFiof normal distributions

slide13

CDF-based approach

Interpretation

If NX=1 case is considred

the task may be illustrated "geometrically"

Maximize shaded area

slide14

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.

slide15

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

slide16

CDF method - Inverter

Inverter performancesJ.P.Uemura, "CMOS Logic Circuit Design", Kluwer, 2002

Inverter threshold

Propagation delay

slide17

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

slide18

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
slide19

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.

slide20

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)

slide21

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;

slide22

CDF-based method implementation

  • Objectivefunctionmaximization
  • Objectivefunctionmayexhibitmorethan one plateau ormorelocalmaxima;
  • thus
  • a non-gradient globaloptimizationmethodisrequired.
slide23

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

slide24

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

slide25

Acknowledgement

Financial support of EC withinprojectPIAP-GA-2008-218255 ("COMON")

and

partialfinancialsupport (for presenter) of EC withinproject ACP7-GA-2008-212859 ("TRIADE")

areacknowledged.