"WCAP" : WORST CASE ANALYSIS PROGRAM : A TOOL FOR STATISTICAL CIRCUIT SIMULATION

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”WCAP” : WORST CASE ANALYSIS PROGRAM : A TOOL FOR STATISTICAL CIRCUIT SIMULATION

N. Ballay, B. Baylac

To cite this version:

N. Ballay, B. Baylac. ”WCAP” : WORST CASE ANALYSIS PROGRAM : A TOOL FOR STATIS- TICAL CIRCUIT SIMULATION. Journal de Physique Colloques, 1988, 49 (C4), pp.C4-269-C4-273.

�10.1051/jphyscol:1988456�. �jpa-00227954�

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JOURNAL DE PHYSIQUE

Collogue C4, supplement au n°9. Tome 49, septembre 1988 C4-269

"WCAP" : WORST CASE ANALYSIS PROGRAM : A TOOL FOR STATISTICAL CIRCUIT SIMULATION

N. BALLAY and B. BAYLAC

SGS-Thomson Microelectronics, Central R and D/CIS/TDMC, BP 217, F-38019 Grenoble Cedex, France

Résumé - Le but du programme 'WCAP' est d'extraire automatiquement les dispersions des paramètres du premier ordre d'un modèle CAO, directement depuis plusieurs histogrammes d'indicateurs classiques (tension de seuil, courants statiques, ...) usuellement mesurés sur des testeurs industriels.

Abstract - The goal of 'WCAP' Program is to extract automatically first order CAD model parameter spreads directly from various histograms of classical indicators (threshold voltage, static currents, ...) usually measured on industrial parametric testers.

INTRODUCTION

Usually, these spreads of first order parameters were either directly measured on sophisti- cated R & D parametric testers for new processes, or "estimated" by process/design/model people for industrial processes. In both cases, the estimated spreads of first order model parameters let to overestimated and unrealistic spreads of electric quantities (current, access time, ...) mainly because 3-sigma variations on each first order parameter induce 5 to 8 sigma variations on current or access time which are complex functions of ALL the first order model parameters.

In 'WCAP' program, the indicators used to control the process are considered as linear functions of the first order model parameters whose sensitivity coefficients are calculated using multivariable regression procedure. Chapter 1 will show how at this point, their theoritical histograms may be easily calculated analytically. The fit with experimental distributions thus leading to REALISTIC minimum and maximum values for the first order model parameters is described in chapter 2. A real application is then developed.

1 - MATHEMATICAL MODEL OF PROBABILITIES

The fundamental basis of the program is the exact calculation of probabilities for linear functions of variables. This calculation is rather easy in the case of 2 or 3 variables when each of them varies uniformaly over a given range and figure 1 summarizes one particular case with 3 variables. Larger number of variables should induce triple, quadruple, ...

integrals and complex distribution (Gaussian instead of Uniform) lead to non-analytical integrals. That is why we assumed the 2 following keypoints :

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyscol:1988456

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i) each indicator can be considered as a linear function of the first order model parameters.

ii) the distribution of each first order model parameters is uniform.

Fig. 1

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Theoritical histogram of a linear function of 3 variables uniformaly distributed

The previous assumptions are valid when the indicators are properly chosen. In the case of NOS devices, the fol lowing indicators should be selected :

i) threshold voltage on a long and wide HOST (may be different substrate biases) ii) static current on a long and wide NOST (may be different biases)

iii) static current on a short and wide b!OST (may be different biases) iv) static current on a long and narrow MOST (nay be different biases)

A multivariable regression applied on each indicator versus the different model parameters (TOX,

NB,

UO, DL, DW) can be easily performed and the agreement is fair. Rote that the other second order model parameters have been measured on typical wafers and are kept constant. This remains correct as soon as there is no correlation among parameters. That is why we used a specific home-made model for lqOS devices whose parameters are independent /I/,

/2/

and translation to other simple models are then performed.

The simple linear formula used to represent t'he indicators allow an easy and fast calculation of their distribution, assuming an uniform distribution of every model parameter. At this step, we use Marquartd's algorithm /3/ to fit actual distributions by modifying the window of first order model parameters.

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3

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APPLICATION TO CMOS 1.2 U PROCESS

We v a l i d a t e t h i s program on a 1.2 u Ci@S process and we now present t h e c a s e of t h e N-channel iblOSFET family. The chosen i n d i c a t o r s were i n t h a t c a s e :

i) threshold voltage a t VBS=O f o r W/L=25/25

i i ) s t a t i c c u r r e n t f o r W/L=25/25 and VDS=O. 1, VGS=5, VBS=O i i i ) s t a t i c c u r r e n t f o r W/L=25/25 and VDS=5.0, VGS=5, VBS=O i v ) s t a t i s c u r r e n t f o r W/L=25/1.2 and VDS=O.l, VGS=5, VBS=O

v ) s t a t i c c u r r e n t f o r 91/~:25/1.2 and ViJS=5.0, VGS=5, VBS=O v i ) s t a t i c c u r r e n t f o r W/L=1.2/25 and VDS=O.l, VGS=5, VBS=O v i i ) s t a t i c c u r r e n t f o r W/L=1.2/25 and VDS=5.0, VGS=5, VBS=O

The model parameters whose spread were f i t t e d a r e :

i ) oxide t h i c k n e s s i i channel doping l e v e l i i i ) mobility

i v ) d i f f e r e n c e between drawn and e l e c t r i c a l poly length V ) d i f f e r e n c e between drawn and e l e c t r i c a l poly width

Figures 2 t o 4 a r e output examples pointing out t h e accuracy of t h i s analysis. The d i f f e r e n t assumptions a r e s o r e a l i s t i c and, moreover, t h e r e i s a strong r e l a t i o n between the f i n a l spread of each model parameter and i t s r e a l one t h a t could have been measured ( s e e f i g u r e 5 ) .

Dashed histograms : experiment Dotted histograms : theory

6 . 2 0 S . 1 O 6 . 6 0 6 . 8 0 7 . 0 7 . 2 0 7 . 4 7 . 6 0

V T O r 1 0 . - - I

Fig. 2

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D i s t r i b u t i o n of threshold voltage f o r a long and wide MOST

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Fig. 3

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Distribution of saturation current for a long and wide MOST Dashed histograms : experiment

Dotted histograms : theory

Dashed histograns : experiment Dotted histograms : theory

NNOST

2 5 / 2 5

< _ 8 0 5 . 0 0 5 2 0

I D < 5 , 5 , 0 > * l o . . - 4

N N O S T 25/1

. 2

Fig. 4

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Distribution of saturation for a short and wide MOST

Histogram 2.000; lo*[$]

Stars : experiment

Cont : theory ^* ^t ^{* *}

* * * * * * *

+ *

-1.000 1.000 3.000 5. OO!

DELTAL ()I&$ -I*

Fig. 5

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Actual and extracted by 'WCAP' histograns for DELTAL model paramter

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4

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CONCLUSION

This program i s an e f f i c i e n t t o o l t o evaluate accurate spreads o f model parameters needed f o r c i r c u i t simulation d i r e c t l y from histograms b u i l t w i t h data from production l i n e .

Once the spread o f each model parameter i s known, actual worst case parameter sets can be b u i l t . The same k i n d o f analysis might be applied on r e s u l t s o f SPICE simulations t o know the t h e o r i t i c a l d i s t r i b u t i o n o f desired variables (access time, gain, bandwidth,

...

^1. Once t h i s i s done, the design y i e l d o f a c i r c u i t can be evaluated. To do so, some software must be w r i t t e n t o generate SPICE i n p u t f i l e s and t o e x t r a c t automatically i t s r e s u l t s o f simulation. This i s the key t o achieve proper simulations o f analog c i r c u i t s when worst cases cannot be e a s i l y defined.

The methods used i n 'WCAP' program can be applied t o every model, whose i n p u t parameters are independent, f o r i4OS technologies as w e l l as BIPOLAR ones.

REFERENCES

/1/ N. Ballay, B. Baylac, PI. Rauber, "Threshold Voltage I~lodel f o r Complex MOS Structures o f VLSI Technologies", ESSDERC Tech. Dig., 1984

/2/ N. Ballay, B. Baylac, N. Rauber, "3-D Analytical tilode1 o f N and P i;lOSFET1s f o r Enhance- ment and Depletion Mode Devices i n the Submicron Range", ESSDERC Tech. Dig., 1984

/3/ D.W. Marquardt, "An A l g o r i thin for Least-Squares Estimation o f Nonlinear Parameters", J. Soc. Indust. Appli

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Math., Vol. 11, no.2, Jun 1963