f; noise model for MOSTs biased in nonohmic region

Solid-Store Electronics Vol 23. pp. 3?5-329 Pergamon Press Ltd.. 1980. Printed in Great Bntam

l/f NOISE MODEL FOR MOSTs BIASED IN NONOHMIC REGION L. K. J. VANDAMMEand H. M. M. DE WERD Eindhoven University of Technology, Department of Electrical Engineering, Eindhoven, Netherlands

(Received 30 May 1979 in revised form 25 August 1979)

Abstract-Relations are derived for the l/f noise current and for the equivalent input noise voltage of a MOST biased in the nonlinear region. The experimentally obtained results are in agreement with the calculations. In addition the value of power spectrum of the noise current is related to that in the ohmic region. In the last region the l/f noise appears to be caused by mobility fluctuations of the charge carriers. In the nonohmic region, the noise consists of two contributions: (i) mobility fluctuations and (ii) number fluctuations in the charge carriers due to fluctuations of the effective gate voltage induced by mobility fluctuations.

INTRODUCTION

given by[6]

From a comparison between the experimental power spectrum of the equivalent input noise S,, and a derived expression for $, in MOSTs, Klaassen [ l] concluded that S,,, in saturated operation is proportional to the effective gate voltage, and inversely proportional to the gate input capacitance. In contrast to Klaassen’s results, in the majority of the MOSTs Katto et a/.[21 found S”, to be independent of the current level and thereby of the gate voltage, while $., appeared to be inversely proportional to the square of the gate input capacitance and thereby proportional to the square of the gate oxide thickness. In Ref. [l] as well as in Ref. [2] expressions for S,,, are derived from physical arguments in which a considerable part is played by the interface state density or by an effective noise trap density. Here we report on experimental results and we present a model for the l/f noise at VD > ( VG - V,)/lO. Our model considers mobility fluctuations as the source of l/f noise without trapping in surface states. The model covers all cases of S,,, n (V, - VT)a with 0 < p < 1. It also embraces S,, a (oxide thickness)’ with y equal to 1[l] or close to 2[2,3]. van der Ziel[B derived a general relation for Sveq of I/f noise caused by fluctuations SN in the number of carriers. He concluded that if the device is uniform, the noise at arbitrary drain bias VD can be expressed in terms of the noise for VD = 0. In the present paper the starting point for the relationship between the noise at VD =O and at VD > (V, - VT)/10 is our l/f noise theory for V,, -=c ( VG - Vr)/lO which is biased on mobility fluctuations[5].

S, = al (P/P,)~N Nf

(1)

where I is the d.c. current, N the total number of free carriers in the sample, f the frequency, and (Y~the dimensionless l/f noise parameter of about 2 x 10m3[7]. The mobility is given by CL.pf is the mobility if only lattice scattering were present. For homogeneous samples, l/N can easily be expressed as qpR/l*, where q is the elementary charge, R is the sample resistance between the contacts at a distance I from each other. In this way eqn (1) can be written as s, = a,(~//L,)*q~Rz*/1*f= arqPI,,RI*/1*f

(2)

where pllf is an effective l/f noise mobility. For homogeneous samples pIIf = &LI&*. Therefore, pljf I CLI pf. However, the inversion layer of a MOST is strongly inhomogeneous considering the local concentration n(x) and the local mobility p(x) as functions of the distance x from the interface. For a MOST, pllf was calculated to be 151 I

08tAx)/~#rW

dx

P”f = *I:(rol*)n(x)

(3)

where 6 is the thickness of the inversion layer. The effective or surface conductance mobility peff is defined by[8] as 0’ (~(x)/~,)n(x) dx I CL&= PIF

(4)

2.THE IlfNOISERELATIONSMIRHOMOGENEOUSSAMPLES ANDFORMO!FhAT

V,=O

When a constant current is passed through a homogeneous sample, the spectral noise power density Sr is

We find plIf epcff when local mobility p(x) is strongly reduced at the interface by surface scattering. 325

326

L.K.J.VANDAMMEand H.M. M. DE WERD

We consider an n-channel MOST. To obtain the results for a p-channel MOST, we have to replace the electron properties by those of holes. For MOSTs biased in the ohmic region the relation for S, in the drain current becomes S1 = a~qpl~rRoIZ/lZf = I’Klf

(5)

where R,, is the drain-source resistance at Vo< ( VG- VT)/10 for Vc - VT > 100mV while 1 is the channel length. K is the so-called relative l/f noise spectral density at f = 1 Hz and I the drain current. Owing to the simple relation between R. and the effective gate voltage [81 Ro= ~/[W/&&O(VG - VT)]

v=Vd(Vo-VT)andO
(lib)

Substituting eqn (10) in the relation for Nl(y) leads to NI(Y)=N,~(l-2y(v-v2/2))

(12)

where Nr is the surface concentration at the source side whereby N, = CO(VG- VT)/q. In the nonohmic region the ratio VdZ equals R and is given as follows from eqn (9) 1

‘=?=

RO

Wqp.eN,(l-v/2)=(1-v/2)

(13)

The saturation point is now defined as v = 1 for which holds R = 2Ro. The small signal impedance Z = 6VJsI equals 1 RO ’ = WqpeRN,(I - v) = (l_’

(6)

(14)

eqn (5) can be written as a~q(c1wIcLd* _ FK sr = wG( VG- VT)f - f

4. CAL.CIJLATlONOF THJZl/f NOISE

(7)

were w is the channel width and Co the gate oxide capacitance per unit area.

3. MACROSCOPICRELATIONSFOR MOSTs BIASJD WITH V, 5 V,- V,

Here we shall calculate such quantities as the surface concentration Nr(y) of free charge carriers at distance y from the source and the reverse bias voltage along the channel V(y). These relations will be used in the next section to calculate l/f noise at V, > ( VG- V,)/lO. The following assumptions are made: (i) /.L=R is no function of the drain-source voltage V, and y, but perr can be a function of Vc, (ii) a 2-dimensional conductor treatment is used with a free charge carrier concentration N,(y) (cm-*) depending on the distance y between source and drain. If V, > V0 - VT, the voltage drop d V across an elemental section dy with resistance dr is given by[8]

Idy

dv=zdR=Wqp,&(y)

(8)

where N,(y) is given in a first order approximation by Nr(y) = CO(VG- VT- V(y))/q and where V(y) is the reverse bias between the point y and the source electrode. The drain current below pinch off can be written as PI I = (wII)peffco[(vG - VTIVIJ - l/2 VD'I.

(9)

In order to obtain V(y), we integrate eqn (8) between y = 0 and y, v(Y)=(vG-

vT)[1-~(l-2Y(o-v2/2))]

Ar,, = -(A/L//J) dr

(15)

The index p denotes a change Ar in an elemental section due to a change in the mobility at y. The change A/.Lat y causes a Ar, and thus a change A V,( Y*) at Y < Y* < 1. Using eqns (8), (10) and (IS), A V,( Y*) becomes Av

(y,)=_(v-~2/~)~V~-

VT)AP

XQl-2Y(v--

Y2/2))/.L .

Ir

WI

Now a feedback process will be considered. The channel potential V(Y*) will change into V(Y*)tAV,( Y*) beyond y; AV,(Y*) changes the effective gate voltage and this results in a change of the channel resistance beyond y. A Ap increase at y causes a decrease in the resistance of an elemental section, denoted by Ar,,. In this way an increase in the effective gate voltage is introduced, resulting in an increase ANr( Y*) beyond y. Beyond y a decrease in the channel resistance Ar, will be the result. Such a positive feedback mechanism leads to a correlation between Ar,,, i.e., the fluctuation of the resistance of an elemental section at y due to mobility fluctuations, and Ar,, i.e. the resistance fluctuation in the channel beyond y which is due to a change in surface concentration. The voltage drop A Vn(Y) at the drain due to a change in N1( Y*) can be calculated as follows. ’ (v - v’I2NWcL) dY* t17j [1-2X$(v - rm)P'*

(10) which results in

where the following substitions are used Y=y/landO
lh’ THR NONOHMC REGION

We consider a MOST divided into elemental sections dy having resistances dr through which a constant current I is passed. We derive the open circuit spectral density in the l/f noise voltage S, at Y = 1. A small fluctuation in the mobility AWL in the elemental section at y will cause a small change in the resistance dr called Ar,.

(lla)

AV.(Y)=-(VG-VT) x(v- v2/2)[(~+'-(I - 2Y(v- v~/~))-"*IAP/P.

327

I/f noise model for MOSTs biased in the nonohmic region

The total change in the voltage across the channel caused by a change in the mobility at y is given by the sum of AV, + AV., so by using eqns (16) and (18) one obtains for A V at the drain A V = - [(v,; - VT)(V - ?/2)/(1-

Y)I(A/.L/P). (19)

The total voltage fluctuations are obtained by adding AV at the drain for all Ap occurring between Y = 0 and Y = 1. AV consists of a stochastic part AV, and a deterministic part A V, correlated. with A VP. The AV contributions at Y = 1 caused by AP at different y are uncorrelated. From eqn (18) we see that in the ohmic region there is no contribution of changes in the surface concentration to the noise voltage at the drain. A V.( Y) + 0 for v --)0. Close to the saturation condition there is a large contribution to the noise voltage due to A V.. From eqn (16) we see that in the ohmic region A V,, at the drain is independent of Y. Close to the saturation condition there is a large difference between the contributions to the noise voltage at the drain from a Ap at the source side (Y+O) and those from the drain side (Y + 1). Owing to the lower values of N,(Y) for Y + 1 and Y+ 1 a Ap at Y + 1 results in a much higher value for A V,, than the same Ap at Y + 0. Although for v + 1 A V,, and A V,, are dependent on y, the sum A V, t A V. is independent of the place where a Aj~/p occurs. This can be seen from eqn (19). The open circuit voltage noise density at the drain is given by integrating over the whole channel length. is the so-called correlation spectral density in the mobility in space and time and is often denoted by S,,( Y, Y’, fl. In the above we have assumed that the average effective mobility in any elemental section equals the overall effective mobility P.R. Therefore, for S,,( Y, Y’,f) we use the following expression based on eqn (7)

l/f noise for VD > ( Vc - VT)/10 becomes S, KR 7=I fRo’

(23)

At the saturation point (v = 1) the relative noise in the current is just twice the relative noise in the ohmic region. To calculate the equivalent noise voltage at the gate, we write the expression for S, at v = I which we denote by SI, s,, = 2zs2ulwRo

(24

l'f

where I, denotes the current at the saturation point. Using the eqns (6) and (9) for determining I, and Ro, & becomes &

=

w?pl/f/&ET WCo(VG- VT)’ 21’f

*

(25)

If the expression for the transconductance g,,, is now used for v = 1, we write

&I =

& =(wlhL,aCoVD.

(26)

The equivalent input noise voltage at the gate S,,, = &lgm2 or s “e9

= a,q(/L,,,lPedwG 2WrCof

VT)

(27)

which is similar to the expression calculated by Klaassen[l] and van der ZieI[4] expect for the factor PI/fl~hT).

where a&~,~,/& is assumed constant along the whole channel. If there were no other scattering in the crosssection of the channel than lattice scattering, and no mobility profile, then a&r,J~.ti) would be approx. 2 x 10W3.If the mobility in the channel were no function of x, then (rl +(p,,,/pen) would be approx. 2 x 10-3(~/~,)2 as is shown in eqns (1x4). Integrating S( Y, Y’, f, over the whole channel length leads to (I/l&’ (d Y/N(Y)) = l(IN,(l - v/2)). Using this, we find an expression for S, S = (VG - v,)*v*(l- v/2)a, . (jL*,f/pc*) ” (I- v)2 WlNJ ’

(21)

From eqns (13 and 14) z,I = V,,( I- v/2)/(1- v). and S, = S./Z’. This leads to

s,

ar(p,,,l~ed _

wwdo

_

7 = IwN,( 1 - vl2)f - I*(1- v/2)f -

arw/rR l’f

. (22)

Hence, using eqns (5) and (22) we find that the relative

If the ratio ~I,f/~.* is about constant, then we expect that S,, will be proportional to ( VC - VT). Such a situation occurs with MOST transistors showing a rather constant p.ff in the G versus VC plot at low drain source voltages[5]. If the MOST shows a strong levelling off in the G vs Vo plot, demonstrating a strong dependence of p.n on VC or some series resistance, then the ratio pLI~,/~CL.R also decreases with increasing V, - VT [5]. Therefore, in such a situation S,,, is less than proportional to VG - VT. If for thinner oxide layers pL,*is low we also find a lower value of the ratio plIf/p.fl. In such cases S,,, is Q (oxide thickness)‘cv<2 as found in Ref.[2]. This might be the solution of the apparent contradiction between Refs. [I] and [2], [3] mentioned in the introduction.

EXPERIMENTAL RESULT.9

The calculated relations presented in the chapter above will now be checked by experiments. The MOST is biased at a fixed gate voltage VG and drain voltage VD Then the power density spectrum in the short circuit current in the drain S, is measured versus frequency.

328

L. K. J.

VANDAMME

If the spectrum is purely l/f then S, at frequency 1 kHz is plotted versus I. with the same gate voltage, the drain voltage VD is adjusted to a new value and the whole procedure is repeated. This results in plots of Sr vs I with VG- VT = constant as in Fig. 1. Figure 2 shows Si vs Y for the same MOST as in Fig. 1 (3N174 of Texas Instruments). For v = 1 the ratio between S,, at VG = - 11V and Sr, at V, = - 7 V corresponds with eqn (29, demonstrating & x ( VG- VT)~.

and H. M. M. DE WERD

I

IO' f= I kHz M 100 S yq =4 kTRn

q ix

IO" IO’ L$-v,

Id”

1

I

I

I

IO

(VI

Fig. 3. Equivalent input noise resistance R, = &,/4 kT at the gate for Vp = Vo - VT,0 experimental,--calculated using eqn (27).

v,=-4.7v 3Nl74

VG=- 12.3V

Fig. 1. Experimentally obtained results for S, vs 1, compared with calculated results, x experimental,--calculated using eqn(24) or (29, f equals 1 kHz.

The measured noise is presented by crosses the calculated noise by a line. In Sections 3 and 4 we used relations which hold only for VD 5 VG - VT and RI 2R0. At voltages VD > VG - VT, SI remains about at the level of S, and we use the eqn (24) to calculate the noise in Fig. 2. For I, in eqn (24) we used the measured current in the active region (v > 1). Owing to a small increase in I with V, at a constant gate voltage, the calculated noise for v > 1 increases slightly. This trend corresponds with the experimental results. In Fig. 3 the calculated and measured values of S,,, of the Ml00 Siliconix MOST are presented versus the effective gate voltage. The equivalent input noise is expressed by an equivalent noise resistor R,, which is defined as &,V, = 4kTR. cf). If a MOST shows some decrease in the channel length with increasing V, then a mild increase in S, in comparison with SI, is observed. This effect can be expected from eqns (24),(25) and (27), where it is shown that & 0: 1-’ and S,, Q l/1. For simplicity, we used a fixed channel length in the noise calculations.

6. CONCLUSIONS

The experimental and calculated results in Figs. l-3 show a good agreement. The apparent contradiction in the literature about the dependence of S,, on C, and VG- VT has been solved. Very thin oxides go hand in hand with a decrease in peff[2] and so with the decrease in the ratio &pc,r[5]. This explains very well that S,,, is more or less inversely proportional to C&[2]. In such case S,, a (l/Co) Q d (oxide thickness) and S,,, a pl,,/pee while pl,JpL.e a d, so for thinner oxides we find Svcq0: d2 a l/CL. In the nonohmic region the l/f noise consists of two contributions, (i) mobility fluctuations, (ii) number fluctuations correlated to the mobility fluctuations. The second contribution is due to a change in the effective gate voltage along the channel caused by local mobility fluctuations.

Fig. 2. Experimentally obtained results for Sr vs Ycompared with calculated results, x experimental,~alculated. For Y> I the measured value of I was used in eqn (24).

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l/f Noise model for MOSTs biased in the nonohmic region Dev, Tokyo, (1974),Supplement of L lap. Sot. Appl. Phys. 44, 243 (1975). 3. N. R. Mantena and R. C. Lucas, Electron. I_&. 5,607 (1%9). 4. A. van der Ziel, Solid St. Electron. 21,623 (1978). 5. L. K. J. Vandamme, Solid-St. Electron. 22, 11 (1979).

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Vol. 23. No. 4-C

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6. F. N. Hooge and L. K. J. Vandamme, Phys. I!_&. A 65A, 315

(1978). 7. F. N. Hooge. Physica 83B, 14 (1976). 8. S. M. SE, Physics of Semiconductor Devices. Wiley, New York (1%9).