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Solid-Slate Electronics, 1974. Vol. 17, pp. 1059-106 3. Pergamon Press. Printed in Great Britain

A SIMPLE THEORY TO PREDICT

THRESHOLD VOLTAGE OF

SHORT CHANNEL IGFETs

L. D. YAU

THE

Bell Laboratories, Murray Hill, New Jersey 07974, U.S.A.

(Recei ved 2 January 1974; i n revi sed f o r m 3 A p r i l 1 97 4)

Abstract-A simple expression for the threshold voltage of an IGFET is derived from a chargeconservation principle which geometrically takes into account two-dimensional edge effects. Theexpression is derived for zero drain voltage and is valid for short and long-channel lengths. Thedependence of the threshold voltage on the source and drain diffusion depth, r,, and channel length, L ,is explicitly given. In the limit, L /r, -co , the threshold voltage equation reduces to the familiarexpression for the long-channel case.

The theory is compared with the measured threshold voltages on IGFETs fabricated with 1.4, 3.8and 7.4 pm channel lengths. The dependence of the threshold voltage under backgate bias voltagesranging from zero to breakdown agrees closely with the theory.

NOTATION

free space permitivitybulk Fermi level = kT/q In ND/n,, where k is theBoltzmanns constant. T , absolute temperature, n,,intrinsic carrier concentrationsemiconductor surface potential with respect to thesubstrateoxide capacitance per cm*oxide dielectric constant

semiconductor dielectron constantchannel length (distance along the surface betweenthe junction defining the source and drain)narrower base of the trapezoidbulk dopingelectronic chargebulk charge, C/cm*junction depththreshold voltagebackgate bias voltagebackgate bias voltage at the onset of L = 0flatband voltagedepletion width under the center of the gatedepletion width at the onset of L = 0charge in the metal gate, C/cm*free carrier concentration in the channel, C/cm*fixed charge in the oxide, C/cm*

1. INTRODUCTION

Current progress in both photolithography andelectron lithography coupled with the self-alignedgate technology has led to the fabrication ofIGFETs with shorter channel lengths. As thedistance between the source and drain decreases,the influence of the source and drain on theelectrostatic potential distribution under the gate

increases. In contrast to the long-channel theory, alarge fraction of the field lines from the bulk chargeunder the gate are terminated on the source anddrain islands, causing the threshold voltage todecrease as the channel becomes shorter. Topredict the threshold voltage of short-channelIGFETs, Cheney and Kotch[l] modified thethreshold voltage expression in the case of large

substrate bias by including the effect of the depth ofthe source and drain diffusion. This resulted in acorrection at high backgate bias. To include theshort-channel effect for low backgate bias Leer21refined the model of Cheney and Ketch, and after alengthy piecewise one-dimensional analysis, heobtained a close-form expression which is verycomplicated. His theory and experiments appear toagree much better than that of Cheney and Ketch.

In this report, a very simple model is presentedwhich uses a simple geometrical approximation inconjunction with charge conservation analysis. The

result of the analysis gives a threshoid voltageexpression which has the advantage of a simpleform and at the same time retains the physicalinsight of the original charge conservation. Thetheory is compared with experiment and excellentagreement is obtained.

2. MODEL AND ANALYSIS

In the long-channel model, the threshold voltageof an IGFET is simply obtained by applying thecharge conservation principle to the region

1059

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1060 L. D.

bounded by the metal gate and bulk of thesemiconductor. This may be written as

YAU

is

W = ,/[2K,c,JqN o (24, + VB,)] (3)

Qw Qo Qs ~s=~ (1)

where QM is the charge in the metal gate, QI,includes the fixed charge in the SiO:, Qs is thecharge due to the free carrier concentration at thesemiconductor surface, and QB is the fixed chargedue to the ionized impurity in the depletion region.For a P-channel IGFET, equation (I) may beexpressed in terms of voltages as[3-S]

V, = VFB - cbs - QslC,,, (2)

where Vc is the gate voltage, VFB is the flatbandvoltage, & is the surface potential with respect tothe substrate, and CoX is the oxide intrinsiccapacitance per cm. Byusing the commonly usedcriterion for turn-on [3-51, (i.e., & = 2&, where &is the bulk Fermi level), the expression for thethreshold voltage is

(2a)

Equation (2a) is valid as long as the channel lengthof the IGFET is long compared to the junctiondepth of the source and drain. The effect of the bulkcharge QB in equation (2a) is to increase themagnitude of the threshold voltage. In the case ofshort-channel IGFETs, the full effect of QB on thethreshold voltage is decreased because near eitherend of the channel, some of the field linesoriginating from the bulk charge are terminated inthe p+ islands. Therefore, in short-channelIGFETs, Vr is lower than what is predicted byequation (2a).

In the following analysis, to obtain a simpleexpression for the threshold voltage, only theedge-effects of Qs is taken into account. This isjustified because in equation (2a). the flatbandvoltage V,, is determined by the metal-

semiconductor work function difference and thefixed charge n the oxide, and is therefore aconstant under the channel region; the variation ofthe surface potential at turn-on would also be smallcompared to & because once the surface potentialis inverted to 2&, it will practically lock at thispotential.

To include the source and drain junction depth inthe expression for VT, w e assume a cylindrical edgewith a radius, r,, equal to the depth of the p _ slandsas shown in Fig. 1. Directly under the middle of thegate, the width of the bulk space charge at turn-on

assuming a uniform N-type substrate. Withoutgoing through a two-dimensional analysis, the fieldlines arising from the bulk charge can be intuitivelyapproximated as drawn in Fig. 1. The field linesoriginating from the fixed charges inside thetrapezoidal region are terminated within the chan-nel length L, whereas the field lines from the fixedcharge outside the trapezoidal region are termi-nated in the p + islands. Based on this geometricalapproximation, the total bulk charge inside thetrapezoid is

QAL = q i W(y)& (4)

Thus Qh is the average charge per unit area in achannel of length I, and by straightforwardtrigonometric analysis,

The trapezoidal bulk charge causes a nonuniformsurface potential along the channel. This makes itimpossible to define uniquely a surface potential forthe threshold voltage. To circumvent this difficulty,we make an arbitrary smoothing approximationthat the effect of Qh on the surface potential isuniform along the channel at turn-on, and therefore

k

SlO2

I P METAL

IFig. 1. Model to calculate the threshold voltage.

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Threshold voltage of IGFETs 1061

equation (2) becomes

(6)

As we will see below, this equation gives a goodprediction of VT for IGFETs with channel lengthsfrom 1.4 to 7.4 pm.

Equation (6) has the same form and physicalinterpretation as equation (2a) in which QB isreplaced by an effective charge, QL. The charge ++ ++conservation principle is applied in a lumped n-SUBSTRATEmanner, to the region which includes the gate, theintervening oxide and the bulk charge inside thetrapezoid. This idea could be extended tononuniform bulk charge by numerically integratingQh in the trapezoid. In the derivation of equation(6) the fringing field around the edges of the metal

gate is ignored because the metal-gate overlap withFig. 2. IGFET with infinite junction depth, illustrating the

the source and drain is normally larger than themodel of Fig. 1 as L + 0.

oxide thickness.For IGFETs under zero backgate bias, equation

when the backgate bias exceeds the limiting

(6) is applicable to all practical doping levels andcondition in equation (7), or

channel lengths. However, under sufficiently largebackgate bias, the trapezoid will approach atriangular shape as L+O. Under this conditionequation (5) can be solved for W, and at the onsetof L = O ,

where equation (6) no longer applies. For long-channel IGFETs this point is seldom reached

W L' +O) = n++ i

L(7)

before pn junction breakdown. For short channelIGFETs the backgate bias will help to suppresspunch-through at L -+ 0 , and V s c > V k does not

Equations (3) and (7) can be used to solve for imply punch-through [2]. Therefore it is legitimateV& = V s c L + O ) at this onset condition. The to project what model one might use beyond thisfollowing asymptotic limits may be deduced from point. From geometrical considerations, we canequation (7): project the triangular area enclosed by QL as equal

to WL/2 for W > W. Geometrically this appearsW + m for v,=O (8) to be somewhat higher than the upper limit of Qh.

and The lower limit is to assume that QA saturates for

W+ for r , = a . (9)W > W as can be envisioned for r i = 5 0 n Fig. 2.Therefore for an arbitrary r , and L , Q B in equation(2) is replaced by QB(u, where (Y is bounded by the

In the case of equation (8), the diffused region is two limits above. This givesinfinitesimally thin, thus the field lines from thebulk are parallel, and they intersect the semicon- Q h = q N Wa , W > W 10)ductor surface uniformly at right angles. Thereforefor thin gate-oxide and r , = 0, the source and drain where W/2 W < a < l/2. Note that if the upperjunctions will have no two-dimensional edge-effect limit is used, i.e. CK l/2, one could still expect anon the threshold voltage provided no d.c. voltage is apparent substrate doping four-times lower thanapplied between the source and drain. In equation the actual substrate doping. The 4-fold reduction in(9), the diffused region forms an infinite wall, and as the apparent substrate doping comes from the factL + 0, the two slant sides of the trapezoid intersect that W a l/d(ND), hence ND x l/a.to enclose the effective bulk charge QL inside a 45 Experimentally, the slopes of V T v s d/(21 ~+triangle as illustrated in Fig. 2. A problem arises vBc) curves appear constant for VBG > V&. To fit

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1062 L. D.

the data analytically beyond V& by assumingdVr/d[v(2& + V,C)] = constant, equation (6)(after some algebra) becomes

VT= V,,-g.-~[w +2a (W-W )1 (11)0 1

where

1

(y = 4(1+ 2r,/L)(12)

W and W are given by equations (3) and (7),respectively. Note that 01 in equation (12) variesfrom 0 to I/4. In section 4 where theory iscompared with experiment in the region Ve, > Vkc,equation (11) is used.

3. THEORETICAL CURVES

Although equation (6) is relatively simple to use,

a few curves are plotted to demonstrate explicitlythe threshold voltage dependence on the junctiondepth, channel length, substrate doping and back-gate bias. The dependence on the oxide thickness issimple and explicit in equation (6), and therefore noillustration is given on the oxide thickness depen-dence.

Figure 3 shows the effect of the junction depth on(V, - VFR for varying channel length. 1VT - VFB isused as the ordinate for the theoretical curvesbecause the constant, VFB, depends on the type ofmetal gate used and the processing procedure. Abackgate bias of 5.0 V is used. As L + 00, all the

ot 2 3 4 5 6L microns1

Fig. 3. Theoretical curves of the threshold voltage as afunction of channel length, for various junction depths.

01 1 I 1 1 I0 1 2 3 4 5 6

L mIcronsI

Fig. 4. Theoretical threshold voltage as a function ofchannel length for various substrate dopings.

O

15.

,O-

,5-

O-

5-

O-

5-

o-

5-0

- THEORY

EXPERIMENT

0 L=74pm c 04

L=38pm ? 0 4

0 L=l.4pm +_ 02

Fig. 5. Comparison of theory and experiment. (T = 297K,T,, = 520 A, r, = 0.5 pm, N, = 1.6 x lOI cm-, V,, =

0.30 V).

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Threshold voltage of IGFETs 1063

curves approach the rj = 0 line as expected. Figure4 illustrates the effect of the substrate doping on thethreshold voltage for ri = 0.5 pm. Again a backgatebias is used to increase the short-channel effect.The theoretical curves in Fig. 5 illustrate the effectof the backgate bias on ] VT - VFsl for variouschannel lengths. This type of plot has been

popularly used to calculate the substrate dopingdensity[6]. It is clear from this figure that thedoping density obtained from the slope of the VT vsd VBG) is no longer correct for short channellengths.

4. COMPARISON WITH EXPERIMENTS

The experimental threshold voltages were meas-ured by a linear extrapolation of the channelconductance in a plot of the conductance vs VG [S].This was carried out by using a 10 mV a.c. signalacross the source and drain which were at zero d.c.

potential. The channel conductance was plottedagainst Vc using an x-y recorder, and thethreshold voltage was determined by extrapolatingthe linear region of the plot to the zero drainconductance point.

The devices used in this experimental compari-son were IGFETs on the same chip processed withelectron beam lithography. Self-aligned tungsten-gate technology was used to define the gates. Thesource and drain were implanted with 5 x lO/cmboron ions at 50 keV. From the IGFET processingprogram of Poon[7], the final junction depth was

calculated to be 0.5 pm. The actual channel lengthis determined by the geometric length minus theeffect of source and drain diffusion under the gate.Probable errors in the channel lengths are +-0.2 pm for the 1.4 pm gate and i 0.4 pm for the3.8 pm and the 7.4 Frn gates. The bulk doping,oxide thickness and the flatband voltage weremeasured with the aid of large MOS capacitor(MOSC) pads on the same chip.

The theoretical curves in Fig. 5 were drawnaccording to equation (6) using the independentlymeasured MOSC data for No, VFB and T,,. For the1.4 pm channel length, the range of validity ofequation (6) is only up to 16.3 V. Beyond thisvoltage, equation (11) was used to extend the

theoretical curve. The discrepancy between theexperimental points and the theoretical curves iswithin the error bounds due to the indeterminacy ofthe actual active channel length, L. The errorbounds in L are given in Fig. 5. Secondly, theflatband voltage variations from device to device isnot taken into account. C-V measurements on the

large MOSC pads on the same wafer varied byapproximately kO.05 V. Thirdly, in view of theimperfect geometrical approximation in derivingequation (6), perfect agreement is not to beexpected. On the other hand, it should be noted thatthe backgate bias dependence of VT is a sensitivetest for the theory, and it is evident that theagreement between theory and experiment is close.

5. CONCLUSION

A simple expression for the threshold voltage isderived from the charge conservation principle

which geometrically includes the two-dimensionaledge effects. The model takes into account thesource and drain junction depth, channel length,and substrate doping which are shown to substan-tially affect the threshold voltage. The finalexpression is valid for short channel as well as longchannel IGFETs. A comparison of the theory withexperiments on various short channel IGFETsshows a good agreement.

Acknowledgements-The author would like to thank L. R.Thibault for the measurements. The discussions with H.

C. Poon, and the use of his IMPFET program were veryhelpful. He is grateful to R. C. Henderson for pointing outthis experimental phenomena to the authors attention,and finally, to R. F. W. Pease for his continuous support.

REFERENCES

1. G. T. Cheney and R. A. Ketch Proc. IEEE 56, 8871968).

2. H. S. Lee, ECS Semiconductor Silicon, p. 791 (1973).3. C. T. Sah and H. C. Pao. IEEE Trans. Electron

Devices, ED-U, 393 1966).4. S. M. Sze, Physics of Semiconductory Devices, chapter

10, Wiley, New York (1969).5. A. S. Grove, Physics and Technology of Semiconduc-

tors Devices, chapter 9, Wiley, New York (1967).6. R. H. Crawford, MOSFET in Circuit Design, p. 52,McGraw-Hill, New York (1967).

7. H. C. Poon, private communication.