Noise in junction- and MOS-FET's at high temperatures

Solid-,State Electronics Pergamon Press 1969. Vol. 12, pp. 861-866.

Printed in Great Britain

NOISE IN JUNCTION- AND MOS-FET'S AT HIGH TEMPERATURES A. VAN DER ZIEL* Electrical Engineering Department, University of Florida, Gainesville, Fla., 32601, U.S.A. (Received 6 February 1969; in revised form 28 March 1969) &lmtract--At high temperatures a leakage current flows to the gate of a junction-FET and to the substrate of an MOS-FET. It is shown that in a junction-FET there is shot noise of the gate current and a strongly correlated noise component in the drain current. In an MOS-FET the effect of the leakage current is less significant, but there is a shot noise component at the drain. Leakage currents seriously effect the low-noise operation of junction-FET's where.as the effect in MOS-FET's is relatively small. R 6 s u m 6 - - A de hautes temp6ratures, un courant de fuite s'6coule h la porte d'une jonction-FET et/t la couche d'un FET-MOS. On d6montre que dans une jonction-FET,il y a u n bruit shot d'tm courant de porte et une composante fortement corr61~e du bruit clans le courant d'6puisement. Dans les FET-MOS, l'effet du c.ourant de fuite est moins significatif, mais il y a une composante du bruit shot au drain. Les courants de fuite affectent s6rieusement l'op6ration h bruit r~duit des FETs de jonctions oh l'effet des FETs MOS est relativement faible. Z u s a m t n e ~ f ~ g - - B e i hohen Temperaturen fliesst ein Leckstrom zur Steuerelecktrode eines FET mit pn-t3bergang und zum Substrat eines MOS-FET. Es wird gezeigt, dass im ersten Fall ein Schrotrausehen im Steuerstrom auftritt und eine stark korrelierte Rauschkomponente im Senkenstrom. Beim MOS-FET ist die Wirkung des Leckstromes weniger bedeutend; es tritt aber eine Schrotrauschkomponente im Senkenstrom auf. Die Leckstr6me bei FET's mit pn-t2bergang beeintlussen ernstlich den Betrieb bei kleinen Rauschwerten, wtihrend ihr Effekt bei MOS-FET's verh~ltnismiissig gering ist. 1. INTRODUCTION AT HIGH temperatures a leakage current flows to the gate of a j u n c t i o n - F E T and to the substrate of an M O S - F E T . We investigate in this paper how this affects the noise behavior in these devices. It is well-known from the early work on j u n c t i o n F E T ' s C1) that the leakage current flowing to the gate shows full shot noise and that it has a detrimental effect on the noise figure of the device. It will be shown here that there is also a corresp o n d i n g noise component at the drain that is strongly correlated with the gate noise. I n M O S - F E T ' s the effect of the leakage current

is less significant, since no leakage current flows to the gate. Nevertheless, there is a noise component at the drain, and this causes some deterioration in the noise figure at high temperatures. T h e effect is much less pronounced than in j u n c t i o n F E T ' s , however. I n m o d e m F E T ' s the effect is insignificant at room temperature, b u t at elevated temperatures it determines the high-temperature limit of lownoise operation of the device, and for that reason an evaluation of this noise source is significant. Since the two types of devices are closely related, we can solve both problems simultaneously. W e do so for the j u n c t i o n - F E T and apply it to the MOS-FET. T h e results to be proved i n this paper are * Permanent address: Electrical Engineering Department, University of Minnesota, Minneapolis, Minn. accurate below pinch-off. But, as is usual in F E T theory, the results in the limit of pinch-off remai 55455, U.S.A. 861

862

A. VAN DER Z I E L s

Is (

ig~

d

AXo

g

L ~ ig; x = Xo

x= o

\

x:L

X

L_jg(X)WAXo

Fzo. 1. H a l f o f a j u n c t i o n - F E T s h o w i n g t h e d.c. c u r r e n t flow a n d t h e noise c u r r e n t flow w h e n t h e d r a i n is o p e n for a.c.

a good approximation beyond pinch-off, so that this is no real limitation to the theory.

where g(V) is the conductance per unit length at a distance x from the source. For the d.c. characteristic we put V(x)= Vo(x), where Vo(x) is the d.c. potential in the channel, so that

2. DIRECT CURRENT CBARACT]BRISTIC OF THE ]XlNCaXON.F]Er AT m o l l ~ T I J R ~ Let the device under consideration have an n-type channel, let the gate be slightly back-biased, and let the direction of positive current flow be from drain to source (Fig. 1). l e t Js(x) be the current density of the leakage current flowing to the gate, and let to be the width of the conducting channel. Then, if I(x) is the current in the channel dI

I(x)= r, +tofA(u)du,

(3)

f Io(u)du = fg(y)dy 0

i,

=

tof A(u)du.

r(.) = g(v)~--f,

Oa)

(2)

0

(4)

0

If V a is the drain voltage, then V o = V a at x = L, so that the source current I 8 and the drain current I~ may be written Va

Is =

dV

Vo

v

'sx +wfdv f J.(u)du = f g(y)dy. 0

If V(x) is the potential in the channel at a distance x from the source, then we have also

0

or, by substituting (1)

0

where I s is the current at the source. If L is the length of the channel, the total gate current is

;

Vo

z

x

(1)

dx

or

z

= J,(x)to;

dr0

&(x) = g(Vo)

L

g(y)dy0

G = &+&.

v

dv 0

u)du; 0

(s)

I f nowZJs(x ) is independent of x, we have

N O I S E I N J U N C T I O N - A N D M O S - F E T ' S AT H I G H T E M P E R A T U R E S

Jr(x) = It/(wL), so that

863

whereas without fluctuation z

1

dVo

z, +~of A(u)du = g(Vo) d---x"

z, = -zf g(y)dy-V,;

(7a)

0

(5a)

Subtracting the two equations yields

1

~r, = _ z f g(y)dy +½z,.

8it=d[g(Vo)AV]

0

OXL

In that case the drain conductance ga may be written

ga = OVa = OVa = --~g(Va),

forO
o.

In the same way we find (9)

(6)

where g(Va) is the channel conductance at the drain. This result should be true if the device is non-saturated. I f Jr(x) depends somewhat on the voltage V(x), as it well might, I t and the second term in (5) may depend somewhat upon V a. However, this dependence will not be very strong and hence little error is made by ignoring it, unless the leakage current is comparable to the device current. We may therefore assume that (6) is reasonably correct even if Jr(x) depends somewhat on V a. I f one replaces the word "gate" by "substrate" the above treatment also holds for the M O S - F E T . T h e results should also remain reasonably accurate beyond pinch-off.

(8)

J

In this derivation it was again assumed that the dependence of Jr(x) upon Vo(x ) could be neglected. Integrating (8) and (9), and bearing in mind that AV = 0 at x = 0 yields and

xoSis = g(Vo) A VI x- z.

(10)

g ( V o ) a Vl z o L = g( Vo)a Vl ~ . x.

so that

xoSi, = g(Va)AV(L).

(11)

I f we now short-circuit the output, the shortcircuit current in the drain due to the source 8i: is

8i a = AV(L)g a = 8 i : ? ,

(12)

1.,

3.

S H O T N O I S E OF T H E LEAJLAGE CURRENT A N D ITS EFFECTS

Let a fluctuating current 8is flow into the channel in a small section between x o and x o + A x o. Let the drain first be open for a.c., so that it always carries the current I a. Then 8it flows from the section ~uco to the source and the current in the section between Ax o and the drain is not altered (see Fig. 1). If V = V o + A V , we have for 0 < x < x o with fluctuation

x

x, +~of A(u)du +8i,

A S s , ( f ) = 2qJ,(xo)wAxo;

~ s ~ b o = 2qJ,(xo),~

axo;

(13)

Xo ~kStd(f) -~- 2qJg(Xo)W ( L t ~ o •

0

--- gCVo + a r O

whereas the fluctuating current flowing into the gate is equal to 8it. T h e current 8/a flows out of the drain. I f 8/, and 8i a have self spectral densities A S u ( f ) and ASaa(f ) and a cross spectral density ASta(.f), we have, since 8/t shows full shot noise when the gate is slightly back-biased

d(Vo-I-AV) dx

(7) By integrating over the length of the channel,

A. VAN DER Z I E L

864

we find for the total spectral densities Ssr(f), See(] ) and S s a(f) r.

all of it is contributed to the drain current fluctuations. The spectral density of the drain noise may therefore be written

Sss(f) = 2qwfJs(xo)dx o = 2qls;

Saa(,f ) = 2q[(5)Ics + Ia, ] = 2q~lsu b L

S~a(f) = 2qw

#

(Xo)

dx 0

(14)

0

L Xo

0

In the case that Js(Xo) is independent of x o this yields, since J,(xo) = Is/(trL )

s , b0 = 2 # , ;

sea(f) = 5.2qr,;

(15)

= 5.2qx,. These equations should remain reasonably accurate when Jg(x) is a slow function of Vo(x). They should also remain reasonably accurate beyond pinch-off. The gate current thus shows full shot noise of the leakage current I s , the drain current shows 5 of full shot noise of I t and the drain current and the gate current are strongly correlated. The correlation coefficient is e .

½(2qI,) . . . . [(2qI,). 5(2qi,)] 1/2

½ C~t

V'3/2 ~ 0.87

(16)

where fl is a factor somewhat less than unity. I f Iss and I a , are comparable, fl should lie between ½ and 5- It is here assumed that the substrate is slightly back-biased. Besides the noise of the leakage current, the channel also shows thermal noise in each case. I f the device is kept at the temperature T, the thermal noise density at the short-circuited drain is S~(f) = y . 4k Tgao (17) Here gno is the value of the drain conductance ga at zero drain voltage and y is a parameter depending on the operating conditions and on the device under consideration. The parameter y = 1 at zero drain voltage; at saturation ~, lies between ½ and } for the junction-FET (1) and near } for the M O S - F E T . There is, of course one obvious effect of the temperature on the noise and that is that the temperature T of the device occurs in (17). However, that is a relatively weak dependence on T. On the other hand Isu b and I s vary as const.exp(-qEs/kT ) where E s is the gap width of the material on which the devices are made; therefore this effect will predominate at high temperatures.

(15a)

Having solved the junction-FET problem, we have at the same time solved the M O S - F E T problem. We must here separate the substrate current/sub into three parts: (a) A leakage current l u flowing from source to substrate. This current shows full shot noise but it gives no contribution to the drain current fluctuations. (b) A leakage current lcs flowing from channel to substrate. This current shows full shot noise and it gives a contribution ] . 2Xllcs to the spectral density of the drain noise. (c) A leakage current Ia, flowing from drain to substrate. This current shows full shot noise, and

4.

NOISE FIGURE A T ELEVATED TE~PMTU~S

We now calculate the noise figure for intermediate frequencies, where on the one hand the 1If noise is negligible, whereas on the other hand high-frequency effects such as induced gate noise are still insignificant. We must here set the criterion for the limit of usefulness of the device at higher temperatures. Since the F E T is essentially a high-impedance device, we set the value of the source conductance gs, somewhat arbitrarily, at gs = 10-4U. Moreover, we set the limiting noise figure for that value Ofgs at F = 2 ( = 3 db); this is again somewhat arbitrary, but, because of the strong dependence

N O I S E IN J U N C T I O N - AND M O S - F E T ' S AT H I G H T E M P E R A T U R E S

865

+

T

igs vg

lout

1 FIG. 2. Equivalent noise circuit of the M O S - F E T . Here i. 2 = 4 k T o g , A f , ia a = [i. 2 q l s u b A f ; it~ a ~ Y . 4 k T g a o A f .

of Ig upon T this hardly affects the high-temperature limit of the device. We first turn to the MOS-FET, which has its equivalent circuit shown in Fig. 2. As can be seen by inspection, we have for the noise figure F, if T o is room temperature (290°K),

channel to the substrate hardly ever affects the lownoise operation of the device. We now turn to the junction-FET, which has its equivalent circuit shown in Fig. 3. As seen from this figure

Zou~ = h-F=

1 +fl ( - - - - ~ I'ubgs \ 2 k T o] g m 2

+ v - Tgaogs --T O gra2

-

_

F=I+

Substituting ~ = ½, (q/2kTo) _~ 20, g8 = 10-4U and g m = 2 x l 0 - S U yields Isu b = 4m.A. Since it is very unlikely that one would ever operate an M O S - F E T at so high a leakage current, we conclude: In an M O S - F E T the leakage current from the

is

it

gs

(19)

+Za

lisgm/g, +in[ 2 + ith2 i,2(gm/g,) 2

= i+ (2___~o)it"[

gs

1 Ig.~21

(18a)

1.

gm 2

~

-

we have, if T Ois room temperature

We now observe that the last term in (18) is relatively small since gm ~- gao is of the order of a few thousand/xU and g, is 10-4U. The limit of usefulness is therefore determined by the condition -

g8

(18)

+ V-

T gaogs

o g J"

(20)

W e again observefor gs= 10 -4 U and gin=2 x 10 -3 U thatthe lastterm in (20) isrelativelysmall. Moreover, l+gs/gm+½gJgm 2 is close to unity under

igs

lout

FxG. 3. Equivalent noise circuit for the junction-FET. Here it2 = ~ I t A f ;

itia = i. ~IrtAf;

ia 2 = i. ~ I t A f ;

i,a = 4 k T o g , A f ; i,~ 2 = Y . 4 k T g a o A f •

A. VAN DER Z I E L

866

that condition. Therefore the limit of usefulness of the device may be expressed by the condition. q i, - - _ 1.

(2Oa)

2k To g~ Substituting

q/2kTo~_20 and

between it and i n does not have a beneficial effect on the noise figure. The final conclusion is therefore that the noise of the gate leakage current gives a major contribution to the noise figure of junction-FET's. In M O S - F E T ' s this leakage current is negligible, and hence the noise figure is much less temperature dependent.

I g = 5v,A.

Acknowledgement--The author is indebted to Dr. L. E. GmlqTim, Dean of the University of Florida Graduate School, for support and to Dr. E. R. CHXNETTE,E.E., Department of the University of Florida, for his kind interest in the work.

This indicates that the drcuit under the above condition ceases to be useful for relatively small gate currents. Moreover, we see that the correlation

~ C E I. Compare e.g.A. VXN DER ZmL, Proc. !nstn Radio Engrs 50, 1808 (1962).

gs=10-4U yields