Frequency variation of the MOS-transistor VD = 0 admittance

S&State

He&on&x

Pergamon Press 1968. Vol. 11, pp. 253-260.

FREQUENCY

VARIATION

Printed in Great Britain

OF THE MOS-TRANSISTOR

vD = 0 ADMITTANCE R. F. PIERRET Departma

of Electrical Engineering and the Materials Research Laboratory, University of Illinois, Urbana, Illinois, U.S.A. (Received 8 May 1967; in

revisedform

16 June 1967)

AWraot-A lateral flow transmission line model is used to calculate the small signal equivalent capacitance and conductance of the MOS-transistor when operated in a mode in which the drain and source are grounded. The resulting theory is then compared in detail against experimental results to check its validity and, in particular, to confirm its predicted frequency dependence. It is found that the major features of the equivalent capacitance are predicted by the theory for frequencies greater than 10 kHa. The experimental conductance in the inversion region is aIso found to agree in both shape and sixe with theory. Finally, it is demonstrated that the theoretica& predicted frequency variation of conductance from u? at low frequencies to d/o at high frequencies, pecuhar to a transmission line solution, is indeed realized experimentally. R&sum6-Le module de ligne transmission g Bcoulement lateral est emp1aJrc pour c&&r la capacit6 et la conductance 6quivalente B petit signal du tram&or MOS i r6gime de source et La thtorie en resultant est ensuite cornpar& en d&ail aux rbultata exp&idrainreliesBlamasse. mentaux pour verifier sa validitc et en particulier pour con6rmer 08 d6pendance de fr4quence pr&ue. On trouve que les traits principaux de la capacite 6quivalente aont pn5vus par la th4orie pour des fr6quences supeieures B 10 kHx. La conductance experimentale dans la r&ion d’invemion eat aussi en accord avec la th&rie en ce qui conceme la forme et les dimensions. Finalement, on d&nontre que la variation de la conductance en fonction de la fr6quence prtvue thtoriquement, de oa aux basses frequences B 4~ aux hautes fr&uences, particuliere P la solution d’une ligne de transmission est vraiment realis& experimentalement. %semane&aung-Zur Berechnung der Rleinslgnal-Rapaxitgt und -Leitfghigkeit des MOS-T mnsistors, wenn Quell- und Saug-Elektroden beide gee&t sind, wird ein Modell verwendet, das den seitlichen Fluss der LadungstrHger erfasst. Die Theorie wird dann im Detail mit Experimenten verglichen um ihre Gtiltigkeit xu priifen und insbesondere die vorhergesagte Frequenxabhgngigkeit xu bestlltigen. Die Theorie sagt die wesentlichen 2X&e des Rapaaitgtsverhaltens fllr Frequenxen oberhalb 10 kHx voraus. Die experimentell bestimmte Leitfghigkeit im Inversionsgebiet stimmt ebenfalh nach G&se turd Frequenxverhalten mit der Theorie &rein. Zum Schhrss wird gexeigt, dass der theoretische Frequenxgang der Leitfghigkeit von os bei tiefen Frequenxen au d/o bei hohen Frequenxen tats&hlich experimentell gefunden wird.

INTRODUCI’ION NUCH

work has been done on the metal-oxidesemiconductor (MOS) capacitor, with the small signal capacitance voltage curve being used as the. principal experimental tool. Ideally, the theoretical C-V curve can be obtained for the low frequency and high frequency cases by considering only l Supported in part by the Air Force 05ice of Scientlfic Research (AF-AFOSR-714-67) and the Advanced Research Projects Agency (SD-131).

capacitative effects.“) In some instances it wan found, however, that the theoretical low frequency C-V curve persisted even at high experimental frequencies.(2) This phenomenon was explained in terms of a lateral a.c. current flow tranami8sion line mode1.c3) Carriers necessary to yield the ‘low’ frequency C-V curve were beieg supplied through a surface channel adjoining the MOS-capacitor. The exact solution to the lateral flow tranamiasion line model for the MOS-C, unfortunately, was

253

254

R. F. PIERRET

complicated by two factors. One was that the MOS-C typically possesses cylindrical geometry and hence the transmission, line solution involved cylindrical bessel, functions. The second was that the physical situation outside of the MOS-C was ill-defined. For these reasons, the exact theory and experiment were not amenable to a close comparison. A physical situation very similar to the MOS-C with lateral current flow exists in the MOST. In the latter case, however, the geometry is typically linear or approximately linear. Further, the existence of the highly doped drain and source islands at the ends of the region under the gate leads to a well-de&d problem. In this paper we propose to apply a somewhat modified small-signal lateral-flow transmission-line model to the linear MOS-transistor. The analysis is restricted to the case where the drain and source are grounded (Vn = 0). The resulting theory for the V, = 0 admittance of the MOST is carefully examined and compared with experiment. Of particular interest is the frequency dependence of both the equivslent parallel conductance and equivalent parallel capacitance of the structure. We will find that the equivalent parallel conductance has a. predicted. and observed. frequency dependence which is peculiar to a transmission line solution. In addition we will exhibit that the variation of the equivalent parallel capacitance with frequency, sometimes blamed solely on the interaction of surface states,(*) is explained, in essence, by the theory developed herein.

THEORY Derivation of the equivaht chit Although’ it is possible to derive the V, = 0 MOST equivalent circuit from the governing control equations (continuity equations, current equations, etc.), it is easier and more satisfying to obtain the equivalent circuit from a physical argument. Let the x-coordinate be measured from the semiconductor surface; let the y-coordinate be parallel to the surface, witby = 0 taken at the channel side of the drain. If we now look at an arbitrary point y along the semiconductor surface channel we observe the following (a) The semiconductor surface is connected to the gate through the oxide capacitance, C,.

(b) The a.c. surface potential, v8, is controlled in part by the motion of the bulk majority carriers to and away from the surface in response to the small signal gate potential, v,. Since the surface channel itself is essentially depleted of bulk majority carriers (surface inversion assumed), this majority carrier fluctuation takes place at a distance W from the surface, where W equals the width of the channel. Consequently, the surface is coupled to the bulk by a capacitance, C,, = &es/ W, which is identical to the high frequency semiconductor capacitance of the MOS-capacitor.(l) (c) The a.c. surface potential is also influenced by the bulk minority carriers that enter through the drain and source and flow down the surface channel. At the point y, some of these carriers stop, whiie others proceed further down the channel. The pile-up of these carriers at the point y corresponds to a capacitance I$‘,.The path along which the carriers flow can be represented by resistors in and out of the pointy.

FIG.1. Small signal transmkion-line model. L = length of the channel; Z = width of the channel; C,, = oxide capacitance/unit area; C,, = high frequency semiconductor capacitance/unit area; C, = channel capacitance/unit area; gd = drain to source conductance.

In view of the preceding discussion, we can therefore represent a small cross-sectional area, ZAy, about the point y, by the model of Fig. 1. Note that all of the well known features of the MOS-transistor are included in this model. For example, the source and drain are connected by a pure resistive path of total conductance g,+ At very high frequencies, little charge can enter through the source and drain (v M v8) and CMOST M CoCH/(Co+CH) which, as desired, is identical to the high frequency capacitance of an MOS-capacitor. (Large subscripts on the capacitances imply pure capacitative units (pF); small subscripts imply capacitance/unit area.) At low

FREQUENCY

VARIATION

frquenck, u at all points along the channel is appdy zero and CMOEJTW CO(CC+GIY (Co+ Cc+ C,), which should equal the low frquency capacitance of an MOS capacitor. It should be obvious from this last statement that c, = cr.-c,

(1)

where CI is the low frequency semiconductor capacitance of an MOS-C.‘l’ The proposed model, although specifically derived for the c88e of an inverted surface, is valid for all attainable gate voltages because of the fact that g, --f oo and C, + C, as the surface heads towards accumulation. It should be noted that the model does not take into account the recombination-generation of carriers or the interaction of surface states. Formal solution At any point along the channel we can write two node equations:

The boundary conditions imposed by the physical situation require that v = 0 at y = 0 and that the current flowing in the y-direction at L/2 must vanish or (b/e) = 0 at y 3: L/2. With these boundary conditi~ the particular solution obtained for v is:

w(Co+CH)Cc

j Y

= [

(2) 0 = ‘$[v(y-Ay,2)-u(y)]

+joCJAy(v8-v).

ioxide= Y= -

Equation (2) can be solved for v, to yield, cowl + c,u va = c,+c,+c,’

(4)

(7b)

1

j&J,-

LIP s 0

%W

-vd

0%

dr.

(8)

U8ing equation8 (4) and (7), and integrating, we obtain Co% CO(CH+ Cc) c,+c,+c,-(c,+c,+c,)(c,+c,) x

(3)

I

[The solution for L/2 Q y < L is obtained by reg?Yy by L-y in equation (7).] D = 0 adnuttance of the MOST can now be easily calculated from

.y =jw

&2L + -+~fy+L?rPb-WI

l”

2(Co + c, + C,)g,

%

- v#) - jo&ZAyv,

(74

Co%/(Co+ Cd

VO =

0 = jwC,,ZAy(v, - v,) + jwCAy(v

255

OF THE MOST

[l-S]).

(9)

Separating out the capacitative and conductive parts of Y, defining AC = Co - Im( Y), and using equation (1) we finally obtain: AC = Co-C

=

co2 co+c,

Taking the limit as Ay + 0 in equation (3), we obtain, o gd L &’ = ---+jwCc(o,-u). (5) 2 zay= Combining the last two equations we obtain the differential equation to be solved.

1iw

co'(cL-cd +(co+cLwo+c,)

w4

x 11 -F&l co2(cL-GfJ

G=w

F&l

WW

(co+cL)(co+Gf~

sinh e cash [ + sin 6 cos f

F& =.

2&cos’t + sinhan

sinh f coah f - sine toe f Fa(R -

q(cosy

+ sinha

(104

VW

R. F. PIERRET

256

Co Ct Cp gr

= capacitance ‘of the oxide over the channel = low frequency semiconductor capacitaqce = high frequency eemiconductor capacitance = d.c. conductance of the channel

Exa?ninrirg the solution

Although the solution for the admittance of the MOST (with the drain and source grounded) may appear foreboding at first glance, a careful examination exhibits its inherent simplicity. In

FSG.2. F&

Fii. 2 we have plotted P&) and Fo(# vs. f. They arc een to be smooth functions of t; F&) starts from unity and deeaya slowly toward xero; F,(t) atartafromxero,reachesamaximumandthen alao decaya & xero. Asymptotically,

WEI = F&J =

$

F,(& = 1-ikf4;

F&f) = #p...f 5 0.3. (12)

. ..[ x 3

(11)

From theseasymptotic sdutiops we see that AC+

Cog

me-to

(w+O)

( 13)

co+c,

AC-+ &0

asf4ao H’

(w+ao).

(14)

This is not too surprising since we have more or less arranged things so that the capacitance would be limited by a low and high frequency capacitance identical to that of the MOS-capacitor. For w between 0 and co, the second term in the AC expression [equation (lOa)] gives the relative departure of the capacitance from the low frequency curve toward the high frequency curve. Of more interest, however, is the fact that G

end F&)

--f

wa

1 CoYG- GfY [ 6 (Co+ Cd% 1

as

(40

(15)

vs. 1.

)Ias

I‘-+co.(16)

Since the bracketed quantities in relationships (15) and (16) are independent of w, if the d.c. gate voltage (V,> is held constant while w is increased from xero, the conductance will first increase as wa, pass through a tram&ion region, and then vary as dw. The dw dependence can be expected at relatively low frequencies near turnon because of the small sixe of g, (and therefore large sixe of #. Far past turn-on, the large sixe of gd dictates that the 40 dependence can be expetted only at very high frequencies, i.e. the

FREQUENCY

VARIATION

OF

THE

257

MOST

can be ascertained from a simple ‘lumped’ model of the device. The 40 dependence, however, is characteristic of the distributed nature of the channel. The somewhat simple frequency dependence of the conductance lends itself very nicely to experimental verification. Of particular interest is the ratio of G at a given frequency to G,,r, the conductance at an arbitrary reference frequency. Theoretically, G

--_=Gref {

=

f

FG[Sref

frer

dtf/frer)l

(17)

F&red

w/2?r, Vo held constant. G/Gref vs.

f

with

ref as a parameter constitute a family of curves.

FIG. 3. Sample G/Gr.f vs.f plot with &,r as a parameter. fr.i = loo kHz.

conductance is expected to vary as wa over most of the measurable frequency spectrum. We might mention that the wa dependence at large g, values

-v,

All data points, taken at different frequencies but with Vo held constant, should lie along a given member of this family. In this way it is possible to verify the predicted frequency dependence without any knowledge of the transistor parameters. A typical family of theoretical G/GreI vs. f curves with fief at 100 kHz is shown in Fig. 3. EXPERIMENT The data to be presented herein, characteristic of the many devices examined, was obtained with an annular (approximately linear) long charmel MOS-transistor fabricated on an n-type silicon substrate of 1.Ox 1016/cm3 doping. The gate oxide

(VOLTS)

AC (FF)

FIG. 4. Observed variation of capacitance with voltage at various small-signal test frequencies for the transistor described in the text. .5

258

R.

F.

PIERRET

AC (WI

FIG. 5. Theoretical variation of capacitance with voltage corresponding to the experimental curves of Fig. 4.

FIG. 6. Observed variation of conductance with voltage at various small-signal test frequencies for the transistor described in the text.

was thermally grown, at llOo”C, in dry oxy The transistor parameters were: oxide thickr x0, equal to 0.223 p; channel width, 2, (circ ference of the annular gate ring), equal to 0 cm; channel length, L, equal to 76 CL.The subs1 doping and oxide thickness were deduced from C-V characteristics of an MOS-capacitor fa cated adjacent to the transistor. The cha length and width were obtained by direct meas ment. The approximately linear annular strut was used to eliminate edge effects. The Vn = 0 C-V and G-V curves obta: with the aforementioned transistor for frequer ranging from 10 kHz to 10 MHz are presente Figs. 4 and 6 respectively. These are to be c pared with the corresponding theoretical cu presented in Figs. 5 and 7. It should be noted in calculating the theoretical curves it is neces to know the effective mobility, pelf, of the can in the channel (since g, is proportional to the ej tive mobility). For the case of moderate and sti turn-on, this mobility can be deducedc5) from d.c. (or low frequency) g, data and was foun be approximately constant at one half the 1 mobility, pa. In calculating the theoretical t and G-V curves it was therefore assumed pefr = t~g/2 for all V,. In comparing Figs. 4 and 5 we see that the served frequency dependence is explained ra handily by the theory. The experiment is of co

FREQUENCY

Ml

14

I

I

I

I

,3

,z

1,

10

VARIATION

1

I

1

1

9

8

7

6

5

4

3

-v,’ (VOIts) FIG. 7. Theoretical variation of conductance with voltage corresponding to the experimental curvea of Fig. 6.

shifted to the left, due to charges in the oxide, and somewhat spread out, probably due to surface states at the Si-SiO, interface. The theoretical C-V curves are seen to dip somewhat lower than their experimental counterparts possibly due to small errors in the device parameters. In any case it is obvious that the variation of the C-V curve with frequency, at frequencies above 10 kHz, drises principally from a frequency effect associated with the lateral flow of carriers into and out of the channel from the adjoining source and drain. In comparing Figs. 6 and 7 we see that there is excellent agreement between the experimental and theoretical curves in both shape and magnitude for frequencies of 100 kHz or greater. However, below 100 kHz the observed conductance curves tend to peak at a higher magnitude and at a lower 1V,l value than that predicted

OF

THE

MOST

by theory. Also, the right hand slopes of theoretical cures are seen to be much steeper t that observed experimentally. This difference slope might again be attributed to the spreac effect due to surface states. It is unlikely, howe that the peak values can be seriously affected by surface state interaction. Another possible explr tion is that the effective mobility, which cam be accurately measured in the region very 40s turn-on, decreases below the constant v8 assumed in calculating the theory. A decreas the effective mobility near turn-on could exp not only the observed lower frequency peak heil and positions, but would also tend to decrease right hand slopes of the theoretical GUI (Theoretically, a decrease in the effective mob is not to be expected. An apparent decrc however, may result from a variation of the t sistor parameters or oxide charge along the lej of the channel.) As was mentioned earlier, a parameter-h pendent test of the theory can be achieved by t strutting a G/Grer vs. frequency plot. Such a for a representative number of Vo values is played in Fig. 8 for data taken with the MC previously described. Note that all G/Grer va corresponding to a given V, value do indeed within experimental accuracy, along a g member of the trer theoretical family. In each the frequency dependence changes, as expec from UP at low frequencies to d6~ at hi: frequencies. It is interesting to interject that, u the frer determined from the G/Grer vs. f : it is possible to estimate both the surface pote and effective mobility corresponding to a g Vo value. The effective mobility found in way using the data of Fig. 8 was approxim: &2 for 1V,l x 6 V. However, pelf for 1VGl 5 was estimated to be considerably less than / hence supporting the earlier hypothesis that effective mobility decreases near turn-on. It should be mentioned that a theoret experimental comparison is best achieved bulk dopings 2 1015/cm3. For bulk doF < 1015/cm3 the surface state conductance hun (which is not shown in Fig. 6 but occurs at a V, = -2V) merges with the frequency e hump and therefore impedes a clear observ: of the effect discussed herein. In some de possessing a high surface state density, a set

R.

260

CO1

.Ol

.I

1

10

F.

PIERRET

100

f (MHZ)

FIG. 8. G/&r va.f for various Vc values.f,,r = 100 kI-Iz. Data taken employing the MOST described in the text. The dashed lines represent theoretical &,,I curves.

small surface state conductance bump was also observed in the moderate inversion region. Experimental verification of the theory is also limited to frequencies greater than N 10 kHz, for which the surface state capacitance is negligible, and to frequencies less than N 50 MHz (depending on the bulk doping and bulk width), for which the bulk resistance is negligible. CONCLUSIONS

A smah signal lateral flow transmission line model has been applied to the MOS-transistor. Because of the device’s typical linear or approxi-

mately linear geometry, and because of the welldefined boundaries of the surface channel, it was possible, using the lateral flow transmission line model, to develop a relatively simple and easily verifiable theory for the V, = 0 capacitance and conductance of the structure. In comparing the theory with experiment it was found that the variation of the equivalent capacitance with frequency (f 2 10 kHz), sometimes blamed solely on the interaction of surface states, arose, in fact, principally from the impeding effect of the channel resistance on the lateral flow of carriers into and out of the channel. Further, the theoretical G-V curves were found to compare quite favorably in both shape and size to the experimental curves over a large range of frequencies. Some low frequency disagreement was postulated as arising from a decrease in the effective mobility near turn-on which was not taken into account in calculating the theoretical curves. Finally, it was demonstrated that the theoretically predicted frequency variation of the conductance from w2 at low frequencies to 2/w at high frequencies, peculiar to a transmission line solution, was indeed realized experimentally. Acknowledgments-The author sincerely acknowledges the support, in both facilities and guidance, provided by Dr. C. T. SAH and thanks Mr. R. kiNDlmSON for his assistance in obtaining the experimental data.

REFERENCES 1. A. S. GROVE, B. E. DEAL, E. H. SNOW and C. T. SAH, Solid&. electron. 8, 145 (1965). 2. S. R. HOFSTEIN, K. H. ZAININGBRand G. WARFIELD, PYOC. IEEE (Corresp.) 52, 971 (1964). 3. E. H. NICOLLIANand A. GOETZBERGER,IEEE Trans. electron Devices 12. 108 (1965). 4. L. L. ROSIER, IEEE Trans. klectron Devices 13, 260 (1966). 5. 0. LEISTIKO, JR., A. S. GROVBand C. T. S.~H, IEEE Trans. electron Devices 12, 248 (1965). 6. E. H. NICOLLIAN and A. GOETZBERGER, Appl. Pkys. Lett. 7, 216 (1965).