NOVEL CALCULATIONS IN THE FIELD OF ACCURATE ANALYTICAL MOS TRANSISTOR MODELLING

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NOVEL CALCULATIONS IN THE FIELD OF ACCURATE ANALYTICAL MOS TRANSISTOR

MODELLING

L. Lauwers, K. de Meyer

To cite this version:

L. Lauwers, K. de Meyer. NOVEL CALCULATIONS IN THE FIELD OF ACCURATE ANALYTI-

CAL MOS TRANSISTOR MODELLING. Journal de Physique Colloques, 1988, 49 (C4), pp.C4-249-

C4-252. �10.1051/jphyscol:1988451�. �jpa-00227949�

NOVEL CflLCULftTIONS IN THE FIELD OF ACCURATE ANALYTICAL MOS TRANSISTOR MODELLING

L. LAUWERS and K. DE MEYER*x>

IMEC, Kapeldreef 75, B-3030 Leuven, Belgium

Résumé: Des calculs analytiques détaillées, à partir de quelques modèles physiques de base, prouvent que le courant dans un transistor à canal court peut être précisément déterminé, sans que le nombre de paramètres empiriques augmente. En plus, il est démontré que quelques unes des approximations, sur lesquelles beaucoup de modèles actuels sont basés, sont incorrectes, et que les adaptations reliées manquent de base physique.

A b s t r a c t : Extended analytical calculations, starting from basic physical models with only few assumptions made, prove that the drain-source current for a wide range of MOS transistor geometry, can be accurately determined using one single model, without enlarging the number of empirical parameters. Furthermore, it is proven, that some of the assumptions, on which lots of actual models rely, are incorrect, and the related empirical improvements commonly in use lack physical background.

1. Situation

In most circuit simulators today relatively simple analytical transistor models are being used. These models do compromise accuracy with respect to minimizing computational effort mainly because of the large amount of transistors to simulate. However, the urge for reliable simulation of complex VLSI circuits emphasizes the need for more accurate analytical device models valid over a broader range of device dimensions and operating conditions. In current models too often one still relies on approximations with very limited applicability and which even already have become invalid for a long time.

In this work detailed calculations with only a limited amount of assumptions and simplifications were made in order to obtain accurate short channel MOS model equations, suitable for medium-size digital designs, as well as analog applications and device investigation (e.g. parameter sensitivity analysis). We propose new threshold and mobility models, valid for both long and short channel devices, based on sound physical principles.

2. Model improvements

We started from the basic device physics models, which are commonly used and at this moment still prove their validity, as long as one doesn't make wrong approximations. Carrier mobility and velocity are well described as a function of the lateral field E]c and normal field E^ff at a certain point c in the channel [1][2] by:

"max and

.. MO HO -h _ CQinv(c) + Qdep(c)

* = - R ( c ) - l+0 . En e f f with Eneff- e s i

In each point between source and drain, an effective normal field, Eneff, is considered, which is typical for the mobility reduction at that point. Assuming Eneff to be the average of the surface field at the interface and the depletion field at the back of the inversion layer, £=0.5 [5]. However, we found £ to be quite lower, which reduces the weight of the inversion charge in reducing the mobility.

The sheet current JL, written as JL = Qinv(Vc).vc , can now be integrated in Schockley-style[l][8], while the mobility reduction term R(c) is now considered to be dependent on the position in the channel. Hence, the current equation depends on expressions of both inversion and depletion charge integrated from source to drain.

'^'Professor at the Katholieke Universiteit Leuven and Research Associate of the Belgian National Found for Scientific Research.

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyscol:1988451

JOURNAL DE PHYSIQUE

1 with R(Vc) = 1+ 8. (C. Qinv @dep)

0

At this point, three basic ideas do fundamentally differ from classical calculations in this area, and enable us to achieve high model accuracy, without introducing any new parameter.

First, it is emphasized that both inversion and depletion charge, given by

Qinv(c) = Cox (VGS

-

where ~ ( c ) = 2 @ ~ -VBS .tVc is the local potential drop from the surface to the bulk, do vary from source to drain, as Vc, being the channel potential, raises from OV at the source to VDS at the drain.

Furthermore, it can be seen that, as y is determined by the average doping level for the local depletion layer depth, a variable substrate doping concentration invalidates the use of a single bulkfactor y, not only for different VBS biases, but also for each applied drain-source voltage. Therefore, we developed a satisfactory relationship between y and v , which causes the bulk-factor to be dependent on both VBS and Vc:

Y(c)= Yl + Y2. (%-*I + '13. ( ~ ( c )

-

This three-parameter relationship offers an appropriate y for all depths of the local depletion layer, within the range where y1, and are fitted on data. yl keeps its physical meaning as the surface bulk-factor at the source.

Finally, classical threshold voltage models are linearized, using the Taylor series expansion for

JW(c)

the expansion outside its convergence region. This doubtful technique can simply be avoided by taking Wc =%) as new integration variable. W, then varies from wO=d=at the source, to WD=~P@F-VBS+VDS at the drain.

Taking above consicLerations rigourously into account, leads to the following formulation of the current:

where T(W) and ~ ( w ) are polynomials in wc:

wo

T(Wc) = TA -TBWC - T ~ w ~ ~ - T D W ~ ~ w c

=-

R ( w ~ ) = RA + R B W ~ + R ~ w ~ ~ dvc = 2wc dwc

In both numerator and denominator polynomial, only the first terms, TA and RA, are dominant, and depend on the applied gate- and bulk- voltage. The higher order coefficients only depend on the model parameters:

TA=VGS -VFB -2@F -Yld%+ RA= I+ BCTA

TB=YI - ~ 2 5 - ~2 6 ~ 3 RB= B(1-<)TB

Tc= 1

+

TD=

n

The analytical solving of this current integral, which is not explained here, leads to:

PWo)

IDS = cby.ll%- +cwl.(?V~-Wo) + c ~ ~ . ( ~ D ~ - w ~ ~ ) +cW3.(wD3-wO3 ) (WD)

with:

3. Implementation in a CAD-model and remarks on data-fits

After introduction of some classically modelled small geometry effects, e.g. drain. induced barrier lowering and the influence of the bird's beak [3], this equation is implemented in a full CAD-model, where, for device- investigation purposes, several options for the saturation region and -voltage are provided [3]. The saturation voltage is determined to obtain continuous slope in the transition region.

Fittings have been carried out on several I-V data sets by our parameter extraction program SIMPAR [6]. In the saturation region, also a slope fitting is performed, which is adequate for analog applications.

IDS-VGS. IDS-VDS fittings for the new model are shown on figs.1 to 6, where the simulated curves (dotted) are compared with the measured data (full lines). The relative RMS-residuals are never above 1%, for short channel as well as for long channel devices, both for nMOS and PMOS.

Fig. 1,2: IDS-VGS data fitting with the new model for long channel PMOS (left) and nMOS (right); to,=23n; Wm=20p, Lm=20p

Fig. 3,4: IDS-VDS data fitting in the triode region with the new model for long channel PMOS (left) and nMOS (right);

bx=23n; Wm=20p, Lm=20p

JOURNAL DE PHYSIQUE

Fig. 5,6: 1 ~ s - v ~ ~ data fitting in the triode region with the new model for short channel PMOS (left) and nMOS (right);

tox=23n; Wmz20p, k f ~ 1 . 3 ~

As we compve this formula with other models, it can first be said that some physical parameters, e.g.

mobility and threshold voltage, loose their unique device definition, as their values and effects have been locally taken into account. Furthermore, in view of the obtained accuracy in a wide range of data, applied voltages and device geometries, one can conclude that earlier commonly made approximations, were too rough, or restricted to a small validity range. Therefore, it is not surprising that parameters that resulted from these approximations, f.i. FB, a,.

. .

It is remarked that, besides by the series resistance and the saturation velocity, the current in the whole triode region is only influenced by two small geometry parameters, modeling drain induced barrier lowering and the bird's beak. No other parameters are involved.

4 . Implementation in a circuit simulator

The full CAD model has been integrated for testing purposes in the circuit simulator ELDO [7]. Simulations are now being done, and some results will be presented at the conference.

5. Conclusions

The fitting results on several data sets are very accurate. The major importance of these detailed calculations is that we can now rely on a closed current formula, which performs unseen accuracy in analytical modelling. One can take this formula as a starting and reference point for making ones own approximations, possibly deriving a simplier expression the accuracy of which could be acceptable for certain operation conditions.

Acknowledgments

Our thanks are due to W.Maes, L.Dupas, E.Machiels and W.Magnus for various support to this work. We l i e to thank the CNET, France, for assistance with the implementation of the CAD model in ELDO. Part of this work has been supported by the Everest-Esprit project 962.

References

[I] B.T.Murphy: A unijiedfield-effect transistor theory including velocity saturation, IEEE-JSSC, SC-15, jun1980, pp325-327 [2] S.C.Sun,J.D.Plummer: Electron mobility in Inversion and Accumulation layers on thermally oxidized silicon surfaces,

IEEE-JSSC, SC-15, aug1980, pp562-573

[3] S.Liu: A unified Mosjet CAD-model, Berkeley, 1981

[4] G.Mercke1, J.Borei, N.Z.Cupcea: An accurate large-signal MOS-transistor model for use in CAD, IEEE-TED, ED-19,1972, pp681-690

[5] G.T.Wright: Physical and CALLmodels for the implanted channel VLSIMosjet, IEEE-TED, ED-34, apr1987. pp823-833 [6] W.Maes, K. De Meyer, L.Dupas: SIMPAR: A versatile technology independent parameter extraction ppogram using a new

optimizedfit strategy, IEEE-TCAD, CAD-5, apr1986, pp320-325

[7] B.Hennion, P.Senn: ELDO: A new third generation circuit simulator using the one step relaxation method, proc.ISCAS, 1985, ~ ~ 1 0 6 5 - 1 0 6 8

[8] S.M.Sze: Physics of semiconductor devices, Murray-Hill, Wiley