mos-ak/gsa workshop at essderc/esscirc, helsinki, sept 16, 2011 a cumulative distribution...

MOS-AK/GSA Workshop at ESSDERC/ESSCIRC, Helsinki, Sept 16, 2011

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

MOS-AK/GSA Workshop at ESSDERC/ESSCIRC, Helsinki, Sept 16, 2011 2

Outline

1. Introduction

2. Introductory example

3. Cumulative-distribution function-based approach

4. Application of the CDF method for inverter and opamp design

5. Remarks on CDF-based method implementation

6. Conclusions


Importance of the yield optimization tasko Time-to-market, and cost effectiveness,o Robustness of design;

Corner (worst case – WC) methodso Fast design space exploration in terms of PVT variations,o Increase of number of corners with increase of types of devices,o Lack of correlations between different parameters sets;

Monte-Carlo (MC) methodo Time consuming, many simulations, o Does not actively manipulate / improve a design;

Worst-case distance (WCD) methodo Operates in process parameter space, not obvious from design

perspective; Cumulative distribution function (CDF) – based method

o Operates in design parameter space;

Introduction


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):

Introductory example

)LW

RRLW

R(Pmax

)RWL

RR(Pmax

)RRR(Pmax

maxsminW

maxsminW

maxminW

Rsmean,ss ,RR N


Variant IRmin 950 Rmax 1100 L 100×10-6 mRs,mean 10 /sq.Rs 0.3 /sq.

Wopt9.8×10-6 m

Variant IIRmin 900 Rmax 1100 L 100×10-6 mRs,mean 13 /sq.Rs 0.3 /sq.

Wide acceptable range of W



Variant IIIRmin 1060 Rmax 1100 L 100×10-6 mRs,mean 10 /sq.Rs 0.6 /sq.

Wopt0.93×10-6 m,but very low yield

Conclusions:1. yield depends on relation between parameter (R) constraints

and process (Rs) quality;2. cumulative distribution function (CDF) has been successfuly

used to determine optimum design.


Question: Can CDF be used for real more complex tasks ?


D - design parameter vector

M - process parameter vector

X - circuit performance vector

S – specification vector

Yield optimization task

CDF-based approach

SXDD

Pmaxargopt

XMDXX

XMDXX

XMDXX

SX

max,NNmin,N

max2,2min2,

max1,1min1,

XXX,

,

,

X=f(D, M)Partia

l yie

lds


CDF-based approach

N

1jj

j

i

,

i

ii

MM

MX,X

,X,X

MD

MD

MMDMD

nom

nom

nom

Issues:Determine Mnom

o influences performance of the nominal design

o influences performance sensitivities

Determine M

Nominal design

i-th performance

SensitivitiesProcess par. variations

Remarks:Relation to BPV method used for statistical modelling, i.e. for extraction of M

Mj – random variables


CDF-based approach

Yield optimization task may be reformulated:

Dopt should maximize joint probability:

MD

MD

MD

nom

nom

nom

,XX

NM

MX,XXP

imax,i

1i

N

1jj

j

i

,

imin,iX M

Issues:Select reliable method for yield optimization, taking into account specific features of the task

Remarks:An optimization problem has been formulated


CDF-based approach

If Mj are uncorrelated normally-distributed random variables, then

is a random variable

N

1jj

j

i

,

MM

MXY

MD nom

N

1j

2

j

i

,

2

2YY

M

jMMXwhere,,0Y

MD nom

N

I.

Sensitivities Process par. variations

Issue:Selection of uncorrelated process parameters is very important


NN

i

xx,,

1imaxi,imini,

1maxi,imini, XMDXXPXMDXXP

CDF-based approach

Due to the unavoidable correlation between performances, direct yield

and product of partial yields are not equal.

NN

i

xx,,

1imaxi,imini,

1maxi,imini, XMDXXPXMDXXP

Issues:Take into account correlations between performances.

orAssume, that

II.


CDF-based approach

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

MD

MD

MD

nom

nom

nom

,XX

NM

MX,XXP

imax,i

1i

N

1jj

j

i

,

imin,iX M

N

1inomimin,iinomimax,ii

N

1i

,XX

,XXiii

X

X nomimax,i

nomimin,i

,,

d

XXFXXF

xxf

MM DD

MD

MD

may be calculated as a product of CDFs Fi of normal distributions


0

0.1

0.2

0.3

0.4

0.5

0.6

-3 -2 -1 0 1 2 3x [arb. units]

prob

abili

ty d

ensi

ty f

unct

ion Xmax - X(D,Mnom)Xmin - X(D,Mnom)

CDF-based approach

Interpretation

If NX=1 case is considredthe task may be illustrated "geometrically"

Maximize shaded area


Calculations of standard deviations of process parameters based on

variations of performances and on performance sensitivities.

2)N(NSUBln

2)P(DW

2)P(DL

2)P(UO

2)P(NSUBln

2TOX

2221

1211

2Prds

2Ngm

2NIdsat

2NVth

2Prds

2Pgm

2PIdsat

2PVth

SS

SS

Backward Propagation of Variance Method

C. C. McAndrew, “Statistical Circuit Modeling,” SISPAD 1998, pp. 288-295.

Selection of non-correlated process parameters (m)

Selection of PCM performances (e)

Measurement of the PCM data

Statistical analysis of the PCM data; estimation of PCM data variances (σe

2)

Backward propagation method used to calculate variances of model technology parameters (σm

2) Least squares algorithm Sequental quadratic programming

(SQP) method

j

i2

ij2m

2e m

eS;σSσ


CDF-based approach vs BPV method

Functional block performance variances determined experimentally

Functional block performance (PCM) sensitivities @ nominal process parameters

Process parameter variances

BPV

CDF of functional block performance variances

Functional block performance sensitivities @ nominal process parameters

Process parameter variances


CDF method - Inverter

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

pn

n,Tpnp,TDDI 1

VVVV

tt5.0t

CR1V

VV4ln

VV

V2t

CR1V

VV4ln

VV

V2t

LH,PHL,PP

outpDD

p,TDD

p,TDD

p,TLH,P

outnDD

n,TDD

n,TDD

n,THL,P

VVII IinDDmax,DD

Inverter threshold

Propagation delay


CDF method – OpAmp

4

1j j

c

ptgarcPM

gg

g

gg

gA

7o6o

6m

4o2o

2mv

SSg

gSSS 2

423

1m

3m2

22

21

2in

OpAmp performancesM.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

Equivalent input-referred noise power spectral density

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.


CDF method – Inverter, OpAmp

Monte-Carlo method 1000 samples

simple MOSFET model 0.8 m CMOS technology

o tox=20 nm,o Vthn=0.7 V, Vthp=-0.9 V

process parameters varied:o gate oxide thickness

tox,o substrate doping conc.

Nsubn, Nsubp,o carrier mobilities

on, op,o fixed charge densities

Nssn, Nssp


Contour plots of yield in design parameter space

tP y ie ld 0 .2 0 .9

ID D m a x y ie ld 0 .8 0 .4 0 .2

CDF method - Inverter

open - partial yieldsclosed - 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.


Av

yiel

d 0.

7

0.9

P M y ie ld 0 .2 0 .5 0 .7 0 .9

S in yie ld 0 .9 0 .7 0 .5 0 .2

CDF method – OpAmp

Contour plots of yield in design parameter spaceopen - partial yieldsclosed - product of partial yields (CDF)

solid - product of partial yields (CDF)dashed - product of partial yields (MC)dotted - yield (MC)


CDF-based method implementation

Objective function maximizationClose to the maximum the objective function

may exhibit a plateau;

Optimization task based on gradient approach

requires in this case 2nd order derivatives of

yield function, but…

this makes optimization based on gradient methods useless;


CDF-based method implementation

Objective function maximizationObjective function may exhibit more than one plateau or more local

maxima;

thus

a non-gradient global optimization method is required.


1. The presented CDF-based method may be used for IC block design optimizing parametric yield,

2. The method may predict parametric yield of the design,

3. The CDF-based method gives results very close to the time consuming Monte Carlo method,

4. The results of yield optimization (Yopt) based on CDF method have direct interpretation in design parameter space (problem of selection of design parameters: explicit or combined),

5. The design rules of the given IP and also discrete set of allowed solutions* may be directly used and shown in the yield plots in the design parameter space,

6. The method may be used for performances determined both analytically, as well as via Spice-like simulations (batch mode required),

Conclusions (1)

* Pierre Dautriche, "Analog Design Trends & Challenges in 28 and 20 nm CMOS Technology", ESSDERC'2011


7. Basic requirements for automatization of design task based on CDF method have been formulated,

8. If the performance constraints are mild with respect to process variability, a continuous set of design parameters, for which yield close to 100% is expected,

9. If the constraints are severe with respect to process variability, the method leads to unique solution, for which the parametric yield below 100% is expected,

10. Thus the method may be very useful for evaluation if the process is efficient enough to achieve a given yield.

11. The methodology may be used for design types (not only of ICs) taking into account statistical variability of a process and aimed at yield optimization,

12. Statistical modelling, i.e. determination of process parameter distribution is a critical issue, for MC, CDF, WCD methods of design.

Conclusions (2)


Acknowledgement

Financial support of EC within project PIAP-GA-2008-218255 ("COMON")

and

partial financial support (for presenter) of EC within project ACP7-GA-2008-212859 ("TRIADE")

are acknowledged.

mos-ak/gsa workshop at essderc/esscirc, helsinki, sept 16, 2011 a cumulative distribution...

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