Solid-State Electronics Pergamon Press 1969. Vol. 12, pp. 549-555. Printed in Great Britain

THE EFFECT OF THE S U B S T R A T E UPON THE GATE A N D

D R A I N NOISE PARAMETERS OF MOSFETS*

P. S. RAOt

Department of Electrical Engineering, University of Minnesota, Minneapolis, Minnesota 55455, U.S.A.

(Received 7 October 1968; in revised form 11 December 1968)

A b s t r a c t - - T h e effect of the substrate on the gate and drain noise parameters in MOSFETs is calculated up to first order terms in j % under the assumption that the channel has thermal noise. It is shown that the substrate doping has little influence on the gate noise, it 2, and on the cross corre- lation between gate and drain noise i,ia*. The theory cannot explain Halladay and van der Ziel's data. Therefore a non-thermal noise source must be operating in the channel.

R6sum 6- -L ' e f f e t de la couche sur les param~tres de bruit de drain et de porte dans les MOS FET est calcul6 jusqu'aux termes du premier ordre de j~o, en assumant que le canal possbde du bruit thermique. On d6montre que la dope de la couche a peu d'influence sur le bruit de porte i, a et sur la corr61ation crois6e entre la porte et le bruit de drain i,ia*. La th6orie ne peut pas expliquer les donn6es de Halladay et de van der Ziel. Done, une source de bruit non-thermique devrait op6rer dans le canal.

Zusammenfassung--Der Einfluss des Substrats auf die Rauschparameter der Steuer- und Senkenelektrode eines M O S - F E T wird berechnet bis zu Termen erster Ordnung in rio. Dabei wird angenommen, dass der Leitungskanal thermisches Rauschen aufweist. Es wird gezeigt, dass die Substratbildung nur wenig Einfluss hat auf das Rauschen der Steuerelektrode i~ 2 und auf die Kreuzkorrelation zwischen Steuer- und Senkenelektrodenrauschen. Diese Theorie kann die Ergeb- nisse yon Halladay und van der Ziel nicht erkl~iren. Hieraus wird auf ein nichtthermisches Rauschen im Leitungskanal geschlossen.

NOTATION V,' L length of the channel Vr S width of the channel Vao 9 ao/2Coz, represents substrate doping and has the Ia

units of potential G(ir0) ao (2~sqNa) 1/2 e~ dielectric constant of substrate Y* N~ acceptor density in the semiconductor C~ q electronic charge C C0z E0,/d = Capacitance of the oxide layer per unit Cts

area gts d oxide thickness ~0~ dielectric constant of the oxide /~ electron drift mobility, assumed constant Ira drain voltage ~ With respect Irt gate voltage / to the Irb externally applied substrate bias source v ; IIr01 +~ ~b diffusion potential between channel and substrate

*Part ly supported by U.S. Electronics Command Contract.

I" Rosemount Engineering Company, Minneapolis, Minnesota.

off-set gate potential saturation voltage drain current conductance of the channel at a point x from the source correlation impedance Im( y~)/joJ cross correlation coefficient input capacitance input conductance for zero drain bias.

1. INTRODUCTION

JORDAN and JORDAN (1) have given the theory of drain noise in MOSFETs for negligible substrate dop ing . SAH, et al. (2> and KLAASSEN and PRINS (3)

have d i scussed the effect o f subs t ra te d o p i n g on the d ra in noise ; the la t ter have also g iven exper i - men ta l data to d e m o n s t r a t e t he effect. I n these theor ies t he noise in t he channe l is a s sumed to be t h e rma l noise. HALLADAY and VAN DER ZIEL (4) have shown, however , tha t s o m e t i m e s a wh i t e

549

550 P. S. RAO

noise source of non-thermal origin must be present to account for the observed noise.

HALLADAY and VAN O~R Z~EL (5) have also measured the gate and drain noise parameters of MOSFETs as a function of frequency and of operating conditions, and they have compared their data with the theoretical predictions based on an extension of the Jordan and Jordan theory. They have shown that the correlation capacitance C. has the wrong sign, that the correlation ad- mittance gz is much larger than can be accounted for, and that the gate noise temperature is much larger than the theory predicts. They find only agreement between theory and experiment by postulating another noise source operating in the channel.

Before such a claim can be considered valid, the effect of the substrate doping on the gate and drain noise parameters must be evaluated. This is done in the present paper. We find that substrate doping cannot explain the observations as long as the channel noise is of thermal origin. There- fore Halladay and van der Ziel's claim was justified.

2. THEORETICAL ANALYSIS

The drain current in an n-channel MOSFET may be expressed as,

V d

dV o ~ L-1 f h = G(Vo)~2- ~ or, I a = G(Vo) dVa, (1)

0

where, V o is the channel voltage at location 'x', L is the length of the channel, Va is the drain voltage, and G(Vo) is the channel conductance per unit length at the point 'x'.

The transconductance of the device is,

gra-- ~v~

where gg, is the gate voltage; gm reaches its maximum value at saturation.

The drain conductance is, by Leibniz's rule,

o h go = = G(Va) /L (la)

ev~

which has a maximum value gao = G(O)/L for V a = 0, and zero value in saturation, i.e., G(Vao) = 0; where Vao is the saturation drain voltage.

According to J. A. VAN NIELEN and O. W. ~IEMELINK, (6) the channel conductance is given by,

C(Vo) = /zSCo,[( V, ' - Vo) - 2~:"2( V 0 + V b'):/2]

(2)

where S is the width of the channel, Coz is the gate-channel capacitance per unit area, Vg' is the effective gate voltage, V~' is the effective sub- strate bias, and

q~ = (2%qNa)l/2/ZCx. (2a)

The first term in (2) results from the total charge (mobile and depletion charge) induced by the gate, whereas the second term represents the immobile depletion charge consisting of ionized acceptors.

The thermal noise arising from the random motion of the carriers in the channel gives rise to drain noise and induced gate noise.

In order to calculate (7) the noise up to first order terms in j~o, we assume a fluctuating voltage AVzo in the channel between x 0 and xo+Ax o (Fig. 1). This voltage gives rise to a noise current in the short-circuited output which produces a distributed fluctuating voltage AV(x) along the channel.

For a fluctuating voltage AV(x) we have,

v(x, t) = Vo(X)+±v(x , t)

Ia(t ) = [oa--AIa(t) = G(V) d V / d x

×:o - - - . . . j__LV, o ~:, Xo X,~.Xo

(B)

FIG. 1. (a) Cross sect ion and bias d i ag ram of M O S transistor, (b) Potential distribution due to an e.m.f.

AV=o between x0 and Xo+AXo.

D R A I N N O I S E P A R A M E T E R S O F M O S F E T S

o r

A/a(t) = - d [ a ( V o ) A V(x, t)]/dx, (3)

where V o and Ioa are the d.c. values and AI a flows out of the drain. We have, here, used the fact that Ia(t ) is independent of x.

Solving (3), with the boundary conditions AV(O) = AV(L) = 0 for a short-circuited device (Fig. 1), we obtain,

G(Vo)AV(x) = -- /~Iax; for 0 < x < x o

(4) and

G( Vo)A V(x ) = - A I a ( x - L);

f o r x 0 + A x o < x < L.

Since there is a j ump AVz0 in potential between x o and x 0 + Axo, one easily finds,

AIa(t ) = [a(Vo)/L]AVxo. (4a)

T h e fluctuating voltage &V(x) in the channel, being eapacitively coupled to the gate, induces a fluctuating charge AQe(t ) at the gate, such that

d[AQg(t)] = - SCoz dxA V( x).

The total charge AQg(t) and the short-circui t gate current AIg(t) are therefore,

and

L

AQe(t ) = - SCo~ f AV(x) dx, 0

(5)

Ale(t ) = d[AQg(t)]/dt (5a)

where AI e flows into the gate. We now make a Four ie r Analysis of AI a and

AIg, not ing that AVz0 has a spectral intensi ty 4kT Ax/G(Vo). In t roducing the Four ie r com- ponents Ai a and Aig of the drain and gate noise, (Ai e up to the first order te rm in joJ) we can calculate

Aia 2, Aie2 , and AieAia*.

Summing over all the incremental sections Ax from source to drain, which amounts to an inte-

gration, the drain noise iaS; the gate noise ig 2, and the cross correlation ieia* , can be evaluated. T h e calculation is tedious, and therefore only the results, in saturation, are given.

551

3. R E S U L T S I f we now introduce the following functions

(with the simplifying notation, a = Vb'/Vg', and

b = ef/Ve' ).

U o = VaojVg '= 1+26-~/[462+46(1+a)] (6)

F 1 = 1 - (1 - Us) 2 - (8/3)bl/2[( U o + a) 3/2 - a 3j2]

(7)

24 F 2 = 1 - (1 - Us) 3 +~-61J2[( U o + a) 5 /2 - a 5/2]

+ 66[( U o + a) 2 - a 2] - 86112(1 + a)

X [( U 0 + a) aj2 -- a 3/2] (S)

16 F s = ( 1 - Us) 3 - 1 - --bl l2[(Uo+a)5/Z-a s12 ]

5

+ (3 + 86112a3/2) U o (9)

32 F4 = 1 - (1 - Us) 4 - --6112[( U o + a) w2 - a w2]

7

4 - -~(1 - U s ) [ 1 - ( 1 - Uo) a]

16 - - -~-b[( U o + a) 3 + a 3]

32 + ~-bl/2( 1 + U o + 2a)[( U o + a) 5/2 - a s12]

+ 8b(U o + a) [( U o + a) 2 - a 2]

32 - - -61J2(a+ a 2 + Us+ Usa)

3

x [( Uo+ a) 3 Iu- a3/2], (lo)

and, 1 1

F s = (UoF4/2) - ~Uo2F2+ ~Uo s

+ [(Us + a) 9/2- aS/2] 6112

1 - Uo + [( Uo +

24 8 ) 1 × -- __ab112_ bl/2

7 -7 + ? U°3

552 p.

/16 24 \ + [( Uo + a) 5,2- aS/Z]~-~-abl/2 + -~a2b 1/2)

-F t( ~o-l-a)3lZ--aa/2](1-I-a)(-- ~a2bl/2),

(11)

we may sum up the results of the calculations as follows. For a device operat ing in saturation,

Drain current:

C ~2~ 2 I a =(l~ gsVg / 2 L ) F 1. (12)

Drain noise:

ia 2 = (2/3)(4kTAf)(IxC~sV//L2)(F2/F~) (13)

Gate noise:

= 48krAfggs(1- 2al/261/2 )

x [(S/27)F2Fa2 + SFsF12- ~F1FaF4] /F15.

(14)

Cross correlation:

i~ia* = 4kTAfj~Cgs[(4/9)F2F a - F1F4]/F1 a (15)

S . R A O

Correlation admit tance:

Yz = gz +J~°Cz = (igia*/iao2)gm

g . = 0; and

C z = Cg s ~ ;o [ (2 [3)F2Fa- (3/2)F~F4]/F2F~ 2.

(16) The result gz = 0, is a consequence of the fact that we neglected terms of the order (joJ) 2 in Aig.

The correlation coefficient is,

C = (igia*)*/(ig2ia2) 1/2

= -j[(4/9)F2F a - F1F4]/[(2Fz/3 Uo) / 8

x (1 - 2al/2bl12) [~F2F32 + 8F12F~

4 \~,,'2 -F1FaF~) ~ . (17) 3

Where Cgs = SCozL, is the input capacitance, and ggs =-- (co2Cgsi)/12gdo, is the input conduc- tance for zero drain bias?, and Vao = 0 at 4ab = 1 (i.e., there is no channel if 4ab = 1).

These expressions are reasonably accurate over a wide frequency range and are valid for all possible values of substrate doping, such that 4ab < 1.

1.2

_ _ i i 1.0

0'9!) I

1 A ~ O - -

I L

.03 .05 .07 O.I 0.5 0.5 0.7 1.0 3,0 B D

FIG. 2. igia*/i~oiao* as a function of b with a as a parameter.

50

DRAIN NOISE PARAMETERS OF MOSFETS 553

19

I . E

13

I.E J

A = '/k o

.01 .03 .05 .07 0.1 0 3

A=V4 - -

0.5 0.7 1.0

S =~

FIC. 3. i,z/i~o 2 as a f u n c t i o n o f b w i t h a as a p a r a m e t e r .

7 - 5.0 5.0

7 ~ The normalized values iga* , lg 2 and C z are

graphically represented in Figs. 2 4 , as a function of the substrate doping parameter, b, for various values of the substrate bias parameter, a. The normalization is carried out with respect to the condition of zero substrate conductivity (b = 0);

the condition b = 0 is indicated by the subscript "zero". Figure 2 gives ig ia*/ igoiao* as a function of b with a as a parameter. Except for the condition

* T h i s d i f f e r s f r o m t h e i n p u t c o n d u c t a n c e a t s a t u r a - t i o n w h i c h is m o r e c o m m o n l y u s e d .

Lo. ' ' i -___/ 2 I i , .7 ~ : ' A .. . . I / I

.5 I I I T t ~ , * ' / , J I ' b ' I

,:,.3 F I I " ~ ' ~ [ . ~ "

~ . 2

.05

.03

.02

.01 .OI . 02 .05 .05 .07 0.I 0 .2 0 .5 0.5 0.7 1.0 2.0 5.0

B

FIG. 4. Y=/Y=o as a f u n c t i o n o f b w i t h a as a p a r a m e t e r .

5.0

554 P . S .

a < 1/10 the value of i j a * depends very little upon the substrate doping, so that little error is made by using the approximation b = 0 (zero

substrate conductivity). Figure 3 gives ig2/igo 2 as a function of b with a as a parameter. Except for

a < 1/10 the value of tg 2 depends very little on substrate doping, so that little error is made by using the approximation b = 0. Since there is always a contact potential difference between source and substrate, the condition a < 1/10 is not very likely to occur in practice.

Figure 4 gives Cz/Czo as a function of b with a as a parameter. I t is seen that Cz/Czo never changes sign, though it decreases monotonically with in- creasing b. Therefore the effect of the substrate can never explain the sign of C z found by Halladay and van der Ziel. The dependence of Cz/C~o must

be attributed to the dependence of ia 2 and gm upon b.

0 5

0 4 !

t 0.3

"U"

02

0.1

0 .01

i i

i i

.03 .05 .07 OI

FIG. 5. ICI as a function

Figure 5 gives the absolute value of ]C] of the correlation coefficient C as a function of b with a as a parameter. It is seen that ]C] decreases monotonically with increasing value of b but never reaches zero; this dependence of ICI upon b must be attributed to the dependence of ia 2 upon b.

We thus conclude that the effect of substrate doping on the thermal noise of the channel cannot explain H A L L A D A ¥ and V A N D E R Z I E L ' S

experimental data. (s) A noise source of non-

RAO

thermal origin must thus have been operating in their units.

Acknowledgement--I wish to express my sincere thanks to Dr. VAN DEe ZIEL for his continued guidance and inspiration. I also thank WILLIAM L. MURRAY of Rosemount Engineering Company, for his encourage- ment.

REFERENCES

1. A. G. JORDAN and N. A. JORDAN, IEEE Trans. Electron Devices ED-12, 323 (1965).

2. C. T. SAH and H. C. PAO, IEEE Trans. Electron Devices ED-13, 393 (1966) ;

C. T. SAH, S. Y. Wu and F. M. HIELSCHEe, IEEE Trans. Electron Devices ED-13, 410 (1966).

3. F. M. KLAASSEN and J. PAINS, Philips Res. Re]). 22, 505 (1967).

4. H. E. HALLADAY and A. VAN DEe ZIEL, Electron. Lett. 4, (17) 366 (1968).

5. H. E. HALL&DAY and A. VAN DER ZIEL, Solid-St. Electron. 12, 161 (1969).

.'Ao~ A= I/z . . . . . . . . . . . . . . . . . .

0.3 0.5 0.7 I0

B

of b with a as a parameter.

A , 0

3 . 0 I

5 .0

6. J. A. VAN NIELEN and O. W. MEMELINK, Philips Res. Rep. 22, 55 (1967).

7. A. VAN DER ZIEL, Proc. 1EEE 51, (3) (1963). For a further explanation see Appendix.

APPENDIX The evaluation of Yz up to first order terms in j (o

If Y= is the h.f. transconductance, the general defini- tion of Y2 is

igia* Y z - - - Ym = P(j~o) + Q(jo~) 2 +

ia 2

DRAIN NOISE PARAMETERS OF MOSFETS 555

Using the Taylor expansions:

i j a * = AjoJ + B ( i w ) 2 +

Ym = C + D ( j ~ ) +

ia ~ = iaid* = E + Y(joJ)2 +

yields A C B C + A D

P = - - ; Q = E E

Note that C and E are tow frequency values, so that the first order term of Y~ only requires calculation of the coefficient A. This coefficient is evaluated in this paper. The calculation does not require an evaluation of the terms B, D and F, so that it is indeed allowed to consider AIa(t) to be independent of x for the evaluation of A.