Inhomogeneous channel resistivity field effect devices

Solid-State

Electronics

Pergamon

Press

197 1. Vol. 14. pp. 1107-l

112.

Printed

in Great

Britain

INHOMOGENEOUS CHANNEL RESISTIVITY FIELD EFFECT DEVICES * T. A. DEMASSA Arizona State University, Tempe, Arizona, U.S.A. and DONALD G. GODDARD Dickson Electronics Inc., Scottsdale, Arizona, U.S.A. (Received

28 September

1970; in revisedform

6 January

1971)

Abstract - Improvement in the operation of several field-effect devices is predicted through the use of specific longitudinal inhomogeneous channel resistivity profiles with heavier doping near the source than the drain end. One device, an inhomogeneous channel field-effect transistor or ICFET, is shown to have a marked reduction in the product of channel resistance and gate to channel capacitance. Effectively, these results imply that higher frequency operation is possible without reducing the device length. General equations for this device are developed and the cut-off frequency is derived in terms of the distributed resistance and capacitance product. One of the solutions of these equations for an exponential impurity profile has shown that a frequency response improvement of six times and gain bandwidth improvement of thirty tunes is obtained when the doping level changes by a factor of 100 from source to drain. A homogeneous FET of the same length with the same drain value of doping concentration is used for the comparison, thus keeping the breakdown voltages and pinch-off voltages equivalent. Another device, called the inhomogeneous channel current limiter, is a two terminal fieldeffect device which has a larger saturation current while the pinch-off voltage and the peak source to drain operating voltage are unchanged. Equations for this device have also been developed and the solutions indicate that the saturation current can be increased by one or two orders of magnitude for a linear impurity profile with a change in doping levels by a factor of 10 or 100 between the source and drain. The comparison is again made with a homogeneous device having a doping concentration equal to that at the drain end of the inhomogeneous device. These results can also be generalized to MOSFET devices. Resume- On prevoit une amelioration dans le fonctionnement de plusieurs dispositifs a effet de champ par I’emploi de profiles de resistivite de canaux non-homogenes longitudinaux avec dopage plus prononce p&s de la source que pres de l’tvidement. On voit que pour un des dispositifs, un transistor a effet de champ a canal non-homogene, ou ICFET, il y a une reduction notable du produit de la resistance et de l’entree du canal par la capacite du canal. Effectivement, ces resultats impliquent qu’un fonctionnement a plus haute frequence est possible sans reduction de la longueur du dispositif. Des equations generales sont developpees pour ce dispositif et l’on en dtduit la frequence de coupure en fonction du produit de la resistance distribute et de la capacite. L’une des solutions de ces equations pour un profile d’impurete exponentiel montre que la reponse de frequence est amelioree par un facteur de six et la largeur de bande du gain par un facteur de trente lorsque le niveau de doping change par un facteur de 100 de la source a l’evidement. Un FET homogene de la m&me longueur avec une m&me valeur d’evidement de la concentration de doping est utilise aux fins de comparaison, gardant ainsi les tensions de rupture et les tensions de pincement equivalentes. Un autre dispositif denomme limiteur de courant de canal nonhomogene, est un dispositif a effet de champ a deux bomes qui possede un courant de saturation plus Cleve alors que la tension de pincement et la tension de c&e de fonctionnement de la source a l’evidement demeurent inchangees. Des equations pour ce dispositif ont egalement tte developpees et les solutions indiquent que le courant de saturation peut augmenter de un ou deux ordres de grandeur pour un profile d’impurete lineaire avec un changement des niveaux de doping d’un facteur de 10 ou 100 entre la source et l’evidement. On fait encore une comparaison avec un dispositif homogene ayant une concentration de doping tgale a celle a l’extremitt de I’evidement du dispositif non-homogbne. Ces r&hats peuvent egalement &tre generalises a des dispositifs MOSFET. *This research was supported by the Air Force Office of Scientific Research under the auspices of Project THEMIS Contract No. F44620-69-C-0025. 1107

1108

T. A. DEMASSA

and DONALD

G. GODDARD

Zusammenfassung - Es wird eine Verbesserung der Funktion fir verschiedene Feldeffektbauelemente vorhergesagt bei Verwendung spezieller longitudinal-inhomogener Widerstandsprofile im Kanalbereich mit hiiherer Dotierung an der Source-Seite ais am Drain-Ende. Bei einem Feldeffekttransistor mit inhomogenen Kanal (ICFET) ergibt sich eine merkliche Erniedrigung des Produktes aus Kanalwiderstand und Gate-Kanal-Kapazitiit, woraus sich ohne Verkiirzung der Kanalliinge eine effektive ErhBhung der FrequenLgrenzen ergibt. Fiir dieses Bauelement werden allgemeine Gleichungen aufgestellt und die Grenzfrequenz als Funktion des Produktes aus verteiltem Widerstand und verteilter Kapazitat bestimmt. Die Liisung der Gleichungen fiir eine exponentielle Dotierungsverteilung mit einem Faktor 100 zwischen Source und Drain liefert eine 6-Mal hiihere Grenzfrequenz und das 30-fache Gewinn-Bandbreite-Produkt. Hierbei wurde ein homogen dotierter FET mit gleicher Kanallange und gleicher Dotierung an der Drainelektrode zum Vergleich herangezogen. soda13 Durchbruchsund pinch-off Spannung gleich waren. Ein anderes zweipoliges Feldeffekt-Bauelement, das als inhomogener Kanalstrombegrenzer bezeichnet wird, hat einen hiiheren Stittigungsstrom obwohl pinch-off-Spannung und maximale Drainspannung nicht vergndert sind. Auch fir dieses Bauelement wurden Gleichungen aufgestellt. lhre Liisung zeigt an. dal3 bei einem linearen Dotierungsprofil mit einem Faktor IO bis 100 zwischen Source und Drain der Sattigungsstrom urn eine bis zwei Gr%enordnungen erhiiht werden kann. Der Vergleich erfolgte wie vorher mit einem homogen dotierten Bauelement, dessen Dotierung dem Drainwert bei imhomogener Dotierung entspricht. Die Ergehnisse kannen such fiir MOSFET-Bauelemente verdllgemeinert werden. 1. INTRODUCTION THIS

analysis of the operation of modified field-effect devices. These devices have a longitudinal inhomogeneous channel resistivity with heavier doping near the source than the drain end. To date, all of the published theory has dealt with uniform resistivity channel devices, or with doping variations due to the gate diffusions which are perpendicular to the carrier flow in the channel and perpendicular to that proposed here [ I-31. On the other hand. there have been some patents [4-61 obtained for FET devices where a longitudinal doping profile in the channel was proposed. Those profiles proposed were intended to keep the width of the channel depletion region uniform from source to drain and not larger at the drain end. as normally is the case. The doping profile chosen for that purpose is exactly opposite to that proposed in this paper. PAPER

is

I

an

2. ANALYSIS

OF ‘THE ICFET

The idealized geometry of the inhomogeneous channel field-effect transistor called the ICFET is shown in Fig. I. This geometry is similar to that used by Shockley[7] and Dacey and Ross@. 91 in their classic developments for the homogeneous JFET. Although this is not the exact geometry of a JFET, it has been shown by Dacey and Ross that effects of geometry extrinsic to the assumed geometry may be added to the results for this geometry. The device shown in Fig. 1 is modified in that it has an inhomogeneous longitudinal channel impurity concentration. N = N(X). The main

G

S

-N------_--_____ ~_~__-_-0 __----------h P o

G

I

Fig.

I. The inhomogeneous

channel

field-effect

transistor.

advantage of the ICFET is that the product of channel resistance and gate to channel capacitance (a measure of the frequency response of the device) is reduced. Effectively, the results imply that higher frequency operation is possible without reducing the device length. The following six continuous equations are the basic relations for the ICFET[ IO]: The pinch-off

wltage V,,=;qN(L);

The built-in channel d’V, __=_ d.r2

(I)

voltage

-’

l

N(X) - 2~ sinh $

(2)

INHOMOGENEOUS

CHANNEL

The depletion region voltage

(3)

Vj=:N(X)T

The depletion region width equation dh ILJE d?r=2q2bl*.(x)N(x)2h(a-h)+qN(x)h

Ece

-hdN 2N(x)

(4) dx

The cut-ofljirequency fO = l/ (27rRC)

(9

1109

RESISTIVITY

refer to the electron mobility and impurity concentration at x = L, respectively. Solutions for three impurity profiles were obtained. Figure 2 displays the assumed impurity profiles. To determine the frequency figure of merit, fb, it is necessary to find the average RC time constant for the ICFET while it is changing from zero current to saturated current. To do this would require generating a large number of solutions to the depletion region width equation (4) and then, using those solutions, solve the RC time constant integral equation (6) several times and obtain an average. This would require considerable computer time, and it was therefore decided to average only the end values, i.e. the zero and saturated current solutions.

The RC time constant L

”

1 h(a-h)p(x)N(x)

0

(a-h)lTx)N(x)

dx

(@

Where N(x) is the arbitrary longitudinal channel doping, Ec is the built-in channel field, /J,(X) is the electron mobility in the channel and the geometry variables are defined in Fig. 1. In the above equations, the gradual case approximation[7] was used. Basically, this approximation implies that the change in the depletion spread with a change in longitudinal position (dh/dx) is small. This was observed to be valid[ lo] for the cases chosen with a!L 4 1 and N(O)/N(L) * 102. It was also assumed that the gate regions are heavily doped P-type and that an abrupt junction exists there. The solutions presented are for an exponentially varying longitudinal impurity profile, that is, N(x) = K,exp(-&x) having a larger value near the source (x = 0) than the drain (x = L). A silicon device is assumed with typical dimensions of a = 2pm and L = 2Opm. The doping concentration at the drain end is also assumed to be lo’“/ cm3. It is further assumed that the channel doping concentration is in the range 10*5/cm3 G N(x) < 1017/cm3 so that the electron mobility in the channel is approximated by p(x)/p(L) = 1 - 0.0945 In (N(x)/N(L)) [ll]. The quantities CL(L) and N(L),

SSEVol.

14.Na.

II-D

0

x IL

I-

Fig. 2. Exponential impurity profiles for the inhomogeneous channel field-effect transistor.

This approximation to the average RC time constant is not greatly in error as was seen by observing the linearity of the I-V characteristics for the ICFET. The curves were nearly linear for a majority of the voltage swing. Therefore, the linear approximation for the RC time constant is not greatly in error. * The normalized solutions obtained for f0 with zero source-drain current, the saturation value and the average of these two are given in Table 1. This table clearly shows that an impurity distribution in the channel with higher concentration near the source than at the drain causes an improvement in f0 for the device. Specifically, the last column *This technique is similar to the method originally used by Dacey and Ross in their initial calculation off,[8].

and DONALD G. GODDARD

T. A. DEMASSA

1110

Y

shows that the distribution labeled N2 improves f0 by more than a factor of 5. The observed pinchoff voltages and also the source to drain breakdown voltages for all three devices were about equal. This was expected since the impurity concentration at the drain end [N(L)] was the same for all three devices. Table 1. Normalized cut-offfreqwncy Doping distribution

fo(10 = 0)

N = N(L) NJ/N(L) = lee-‘JX” N,/N(L) = 100e-4.6x~‘~

values

fo(lusS)

1

Jb,,,,

1.285 3.72 6.28

2.87 5.46

-x

source

1.122 3.25 5.85

N(x)

Yilb_x

A comparison of the gain of each device was obtained by observing the magnitudes of the ratios of the change in source-drain current to identical changes in gate voltage ( gm = Al,/AVg). Since this parameter is proportional to the magnitude of the source-drain current, it can also be considered to be a measure of the output power when identical source-drain voltages are applied to each device. Table 2 gives values of gm (or output power) normalized to the homogeneous case for all three impurity distributions. Also given in Table 2 is the gain bandwidth product, g& Table 2 clearly shows that the ICFET devices provide considerable improvement in these quantities over the homogeneous FET devices.

0

L

Fig. 3. The inhomogeneous

channel

current

limiter.

Table 2. Gain, power and gukbtrndwidth product values

“P Gain ( a,,) Doping distribution N = N(L)

N,/N (L) = lOe-* WI N,/N(L) = 1oOemA.6”‘L

or Power

(IDss)

I 3.04 5.4

“83, Voltage,

g”&

1 8.8 28

3. ANALYSIS OF THE ICCL

The inhomogeneous channel current limiter or ICCL is a two-terminal field-effect device with an inhomogeneously doped channel region having heavier doping near the source than the drain end. A typical current-voltage characteristic for the uniform or modified current limiter is shown in Fig. 4. Ideally, one would want to design these devices so that the pinch-off voltage is zero and

Fig.

4.

Current-voltage

V,) characteristic

of

the

semi-

conductor current limiter.

the saturation current is any desired value. Also a high peak operating voltage would be desirable. The ideal conditions, however, can be much more closely approached by the ICCL, since by keeping the impurity concentration at the drain end the same as in the homogeneous device the pinch-off voltage and source to drain breakdown voltage would remain the same. However, by increasing the impurity concentration gradually from the drain toward the source, the current can be greatly increased while the other quantities are unchanged.

INHOMOGENEOUS

CHANNEL

1111

RESISTIVITY

Calculations of the source-drain current (In) are made from a modified form of equation (4) given by N’N (a - h) (V/2) + N%h’ (a - h) = &,,/q2p,,b (7) where the functional variation with x has been omitted and N’ = dN/dx and h’ = dhldx. Equation (7) neglects the built-in field term and contains an additional factor of 2 in the last term to account for the slight difference in geometry. The above equation is a first order nonlinear differential equation in h and is solved for various values of source-drain current, I,, by digital computer techniques. When the impurity profile, N(x), is a known quantity the computations are initiated by assuming a small value of ID. Then ID is increased by a small increment and h = h(x) is calculated again. Each computation is checked to determine whether or not h at x = L is equal to a. If this is not true the current ID is increased again and the computation of h = h(x) is repeated. Finally, when h at x = L is equal to a, the value of source-drain current is the saturation current and the computations cease. For simplicity, N(x) was chosen here to be a linear function of x with a maximum value at the source [N(O)] and a minimum value at the drain [N(L)]. The solutions are obtained by dividing the channel into II regions in the x direction and solving equations (3) and (7) in each of these regions. To initiate the solution the depletion width at the source [h(O)] is calculated from the known doping concentration and the known zero potential there [h(O) depends only upon the built-in junction potential]. Equations (3) and (7) are then solved iteratively in every other region using the solutions for the previous region. Figures 5 and 6 display the calculations for the current vs. voltage in the ICCL. Figure 5 has an impurity concentration at the drain end of 1Ol5 atoms/cm3, while Fig. 6 is for 1OB6atoms/cm. The source end impurity concentration is the parameter in these curves and it is varied over two orders of magnitude. These figures indicate that the current levels in these devices can be considerably altered by utilizing a nonuniform longitudinal impurity profile. It should be observed that the current magnitudes are increased by about two orders of magnitude

Fig. 5. Source-drain

current vs. voltage.

Fig. 6. Source-drain

current vs. voltage.

above the uniform channel device in both examples for the best case. Also, since the peak source-drain operating voltage is essentially unaltered, high

1112

T. A. DEMASSA

current and high voltage operation becomes compatible with one another in the ICCL.

and DONALD

REFERENCES

more

4. CONCLLISIONS The solutions described above have shown that considerable improvement in the operation ot field-effect devices is possible when an inhomogeneous longitudinal impurity profile exists in the channel. For the case of the ICFET. the results have shown that significant improvement in frequency performance, device gain, output power and gain bandwidth product is possible. For the case of the ICCL, it was shown that the saturation current could be increased considerably while maintaining about the same pinchoff voltage and peak source drain operating voltage as the homogeneous device. The results obtained, however. are theoretical only. Because these results appear promising the fabrication of these devices should be conducted. Ion implantation might be the best method of fabricating the nonuniform channel. Also, lateral diffusion under the oxide mask might be advantageous. Extensions of this theory to MOSFET devices have been initiated and improvement is expected in this case also.

Cr. GODDARD

I. R. R. Bockemuehl. 2. 3. 4.

5 6.

7. 8.

9. IO. Il.

12.

IEEE

Truns

Eluctro~~

f)ecict,.c.

ED-IO. 3 1 (I 963). R. S. C:. Cobbold. fEEE TI-ci,r.\. E/cc.t~/~ Dec>ic.er. ED-12. 302 (1965). F. A. Lindholm and P. R. Graq. IEEE 7rcrm. Ektron L)coic~.s. ED-13. 8 I9 ( 1966). F. P. Heiman and K. H. Zainiger. S~~?ic.o,zdrr~,r~~r Dctiice F~~bricutiorr. U.S. Pat. No. 3.411.199 Nov. ( 1968). R. Muller, Unipulur ‘Trunsisror,fiw High E~P~IIPN~~~.\. U.S. Pat. No. 3.449.645 June (1969). R. Noyce, Field E&t Tnrnsktor. U.S. Pat No. 2869,055 Jan. (1969). W. Shockley. Proc. Instrz. Radio Engrs. 40. 1365 (1952); ibid40. 1289. (1952). G. C. Dacey and I. M. Ross. Proc. Iwrn. Rtrdio Etqrs. 41,970 ( 1953). G. C. Dacey and 1. M. Ross. Bell Sysf. tech. J. 34. II49 (1955). D. G. Goddard. The DrQi. F jr/d-E&/ Trcrmisfot Arizona State University Engineering Report, May (1970). J. Lindmayer and C. Wrigley. F‘~rr&~,nr,~rtr/.\ (t/ Srnziconductor Drvkcs. p. 26 I. Van Nostrand. New York ( 1965 ). R. M. Warner. Jr.. W. H. Jackson. E. I. Doucette. H. A. Stone. Jr.. A Sernkomktor Current Limitrr. BeI/ Sysf. tech. J. Publications. Monograph 3087 (1958).