A method for CMOS IC design towards yield optimization · X XDM X X S N,min N N,max 2,min 2 2,max 1,min 1 1,max ... The design rules of the given IP and also discrete set of allowed

MOS-AK/GSA Workshop Fraunhofer IIS, Dresden, April 26-27, 2012

A method for CMOS IC design towards yield optimization

D. Tomaszewski a) , M. Yakupov b) a) Institute of Electron Technology, Warsaw, Poland

b) MunEDA GmbH, Munich, Germany

MOS-AK/GSA Workshop, Fraunhofer IIS, Dresden, April 26-27, 2012 2

Outline

1.  Introduction

2.  Introductory example (1D, 2D)

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.  Conclusion, further work


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

v  Corner (worst case – WC) methods o  Fast design space exploration in terms of process and environment

condition (PVT) variations, o  Increase of number of corners; some worst-case corners may be

missed, o  Lack of correlations between different parameters sets;

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

v  Worst-case distance (WCD) method o  Operates in process parameter space;

v  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

⋅≤≤⋅

=≤⋅≤

=≤≤

( )σ∝ Rsmean,ss ,RNR


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.

Introductory example Rmin (Ω) 980 925 800 940 940

Rmax (Ω) 1020 1075 1120 1060 1060

L (m) 1.20E-4 1.20E-4 1.20E-4 1.20E-4 1.20E-4

RS,mean (Ω/sq) 10 10 10 10 13 σRs (Ω/sq) 0.3 0.3 0.3 0.1 0.1


Given: sheet resistance Rs, resistor narrowing ΔW being uncorrelated random variables of normal distribution:

Task: Design a resistor, which fulfills the condition: Rmin< R <Rmax;

Find W, L (design):

Introductory example (2D case)

)RRR(Pmax maxminL,W

≤≤

( )σ∝ Rsmean,ss ,RNR ( )σΔΔ∝Δ Wmean ,WNW

⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜

⎝

⎛

Δ+

⋅Δ+

⋅−

≤Δ+Δδ

−δ

≤

Δ+

⋅Δ+

⋅−

≈≤Δ+

⋅≤=≤≤

WWLRWW

LRR

WWW

RR

WWLRWW

LRR

Pmax

)RWWL

RR(Pmax)RRR(Pmax

mean

mean,smean

mean,smax

meanmean,s

s

mean

mean,smean

mean,smin

L,W

maxsminL,W

maxminL,W

( )σ∝δ

δ+=

Rss

smean,ss

,0NRRRR

( )σΔ∝Δδ

Δδ+Δ=Δ

W

mean,0NW

WWW


Introductory example (2D)

length

wid

th

Rmin (Ω) 980 940 940 Rmax (Ω) 1020 1060 1060 Rs,mean (Ω/sq) 10 10 10 σRs (Ω/sq) 0.3 0.3 0.1 ΔWmean (m) -2.0E-7 -2.0E-7 -2.0E-7 σΔW (m) 3.0E-8 3.0E-8 1.0E-8


v  D - design parameter vector v  M - process parameter vector v  X - circuit performance vector v  S – specification vector

Yield optimization task

CDF-based approach

( )SXDD

∈= Pmaxargopt

( )( )

( )⎪⎪

⎩

⎪⎪

⎨

⎧

≤≤

≤≤

≤≤

≡∈

XMDXX

XMDXXXMDXX

SX

max,NNmin,N

max2,2min2,

max1,1min1,

XXX ,

,,

X=f(D, M) Partial yields


CDF-based approach ( ) ( ) ( )

( )∑=

δ⋅∂∂

+≈δ+=N

1jj

j

i

,iii

MM

MX,X,X,X

MDMDMMDMD

nom

nomnom

Issues: v  To determine Mnom ⇒  performances of the nom. design ⇒  performance sensitivities v  To determine δM

Nominal design i-th performance Sensitivities Process par. variations

Remarks: v Relation to BPV method used for statistical modelling, i.e. for extraction of δM – random var.

Dopt should maximize joint probability:

( )( )

( )⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛−≤δ⋅

∂∂

≤−= =

∑N

,XXMMX,XXP

X M

1iimax,i

N

1jj

j

i

,imin,i MDMD

MDnomnom

nomIssues: v  Select reliable method for yield

optimization

Remarks: v  An optimization problem has

been formulated


CDF-based approach

If Mj are uncorrelated normally-distributed random variables, then

( )( )

( )∑σσ∑==

σ⋅⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛

∂∂

=∝δ⋅∂∂

=N

1j

2

j

i

,

2

2YY

N

1jj

j

i

,

M

j

M

MMX,,0NM

MXY

MDMD nomnom

I.

Issues: v  Select uncorrelated process parameters v  Process parameters not normally distributed

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

II.

( )( ) ( )( )∏==

≤≤=⎟⎟⎠

⎞⎜⎜⎝

⎛≤≤

NN

i

xx,,

1imaxi,imini,

1maxi,imini, XMDXXPXMDXXP

Issue: Account for correlations between performances, or assume, that


CDF-based approach

Based on assumptions I, II a joint probability (parametric yield) may be calculated as a product of CDFs Fi of normal distributions

( )( )

( )

( )( ) ( )( )[ ]∏

∏ ∫

=

=

−

−

−−−=

=

N

1inomimin,iinomimax,ii

N

1i

,XX

,XXiii

X

X nomimax,i

nomimin,i

,,

d

XXFXXF

xxf

MM DD

MD

MD

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

ensit

y fu

nctio

n Xmax - X(D,Mnom)Xmin - X(D,Mnom)Interpretation If NX=1 case is considred the task may be illustrated "geometrically"

Maximize shaded area


Calculations of standard deviations of process parameters based on variations of performances and on performance sensitivities.

( )

( )

( )

( )

( )

( )

( )

( )

⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢

⎣

⎡

σ

σ

σ

σ

σ

σ

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡=

⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢

⎣

⎡

σ

σ

σ

σ

σ

σ

σ

σ

…

…………

…

…

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

SSSS

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 ⎟⎟⎠

⎞⎜⎜⎝

⎛

∂∂

==ji2

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

ββ+

⋅ββ+−=

pn

n,Tpnp,TDDI 1

VVVV

( )tt5.0t

CR1VVV

4lnVV

V2t

CR1VVV4ln

VVV2

t

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

∑= ⎟

⎟

⎠

⎞

⎜⎜

⎝

⎛ω−Π=4

1j j

cp

tgarcPM

⎟⎟⎠

⎞⎜⎜⎝

⎛

+⋅⎟⎟⎠

⎞⎜⎜⎝

⎛

+=

ggg

ggg

A7o6o

6m

4o2o

2mv

( )SSgg

SSS 24

23

1m

3m2

22

21

2in +⋅⎟⎟

⎠

⎞⎜⎜⎝

⎛++=

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

Equivalent input-referred noise power spectral density

•  gmi, goi - input and output conductances of i-th transistor,

•  ωc 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 µm 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

µon, µop, o  fixed charge densities

Nssn, Nssp


Contour plots of yield in design parameter space

tP yield 0.2 0.9

IDDmax yield 0.8 0.4 0.2

CDF method - Inverter

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.


Av y

ield

0.

7 0.9

PM yield 0.2 0.5 0.7 0.9

Sin yield 0.9 0.7 0.5 0.2

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)


CDF-based method implementation

Objective function maximization v Close 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 maximization v  Objective function may exhibit more than one plateau or more local

maxima;

thus

v  a non-gradient global optimization method is required.

MOS-AK/GSA Workshop, Fraunhofer IIS, Dresden, April 26-27, 2012 21 21

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 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,

4.  The method may be very useful for evaluation if the process is efficient enough to achieve a given yield,

5.  The CDF-based method gives results, which are very close to the Monte Carlo method,

6.  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),

Conclusions, further work (1)

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

MOS-AK/GSA Workshop, Fraunhofer IIS, Dresden, April 26-27, 2012 22 22

7.  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,

8.  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,

9.  The method may be used for performances determined both analytically, as well as via Spice-like simulations (batch mode) – to be done,

10.  Method should be developed to take into account performance correlations and non-gaussian distributions – to be done,

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.

Conclusions, further work (2)


Acknowledgement

Financial support of EC within project PIAP-GA-2008-218255 ("COMON") partial financial support (for presenter) of EC within project ACP7-GA-2008-212859 ("TRIADE") Assistance of Dr.Grzegorz Gluszko (ITE) at the measurements are acknowledged.

Documents

A method for CMOS IC design towards yield optimization · X XDM X X S N,min N N,max 2,min 2 2,max 1,min 1 1,max ... The design rules of the given IP and also discrete set of allowed