lEEE TRANSACTIONS ON ELECTRON DEVICES VOL. ED-13, NO. 4 APRIL, 1886

The Effects of Fixed Bulk Charge on the Thermal Noise in Metal-Oxide-Semicogductor Transistors

Abstract-The theory of the thermal noise due to channel conductance fluctuation is extended for insulated gate (MOS) field effect transistors with the gate voltage induced channel structure by including the bulk charge from the ionized impurities in the semiconductor substrate. In the saturation range of the drain characteristics, the theory shows that R,,, gms 2 213, where the equality condition corresponds to the previously obtained result for an intrinsic or chemically pure semiconductor substrate. Satisfactory correlations between theory and experimental measurements are obtained for both P-channel and N-channel silicon devices with either a thin oxide (2OOOA) or a thick oxide (6200 and 8400A) gate.

I. INTRODUCTION HE THERMAL noise due to channel conductance fluctuation in the MOS transitors have recently been the subject of several theoretical and experi-

mental investigations. It has been shown by Sah [I] that this noise is proportional to the total integrated current carrier charge in the channel. It is further shown [l] that in the simple theory of the drain characteristics [2], the product of the equivalent noise resistance referred to the gate R,, and the transconductance g, is equal to 2 / 3 and is independent of the gate voltage or drain current if the device is biased into the saturation range of the drain characteristics. However, it has been generally and widely observed in experimental measurements that the thermal noise is considerably higher than that predicted by the simple theory: R,,, = 2/3g,,. Discrepancies of a factor of two to four or more have been observed [3].

The experimental situation is best illustrated by the noise power spectra shown in Fig. 1, where the equivalent gate input noise resistance is plotted as a function of frequency for four drain currents. The discrepancy be- tween the flat portion of the noise power spectra, which is presumed to be thermal noise, and the value $gmr (gmr is determined experimentally) is quite evident from the corresponding .values labeled as A, B, C, and D on the right vertical edge of the figure. The behavior of this relatively low cutoff frequency MOS silicon transistor has also been observed in a variety of high-frequency MOS transistors of both the induced channel (channel is induced either by a gate voltage or a surface charge) and the

This paper was presented a t the 1965 IEEE Solid State Device Manuscript received July 1, 1965; revised November 1, 1965.

Research Conf., Princeton, N. J. The work reported here was supported in part through the Air Force Office of Scientific Research (AF-AFOSR-714-65) and the Advanced Research Projects Agency

and Materials Research Laboratory, University of Illinois, Urbana, The authors are with the Department of Electrical Engineering

111.

(ARPA-SD-131).

chemically built-in channel (from impurity doping of the substrate to create a junction in the bulk such as by epitaxial growth, outdiffusion of a compensated substrate, or indiffusion from an impurity source) structures.

These experimental results indicate that there are a t least three possible noise sources which may obscure the white, frequency independent channel thermal noise in the power spectra. These are the low-frequency 1/f surface noise el], the low-frequency l/fz generation noise in the depletion region near the surface (similar to that in the junction gate field effect transistor [4]), and the increase of the channel thermal noise near the transconductance cutoff frequency which is characterized by a fz dependence of noise power beyond the cutoff frequency. In addition, the gate induced noise (similar to the grid induced noise in vacuum tubes) becomes important near the transcon- ductance cutoff frequency.

In this paper, another noise contribution which would increase the white thermal noise of the channel over the 2/3g,, value is analyzed in detail. This additional thermal noise comes from the inadequacy of the simple theory of the static and low-frequency dynamic characteristics of the MOS devices [a] . It not only decreases the trans- conductance at a given gate voltage, but also increases the product R,,,g,, to a value above 2 / 3 . Its origin. rests on the uncompensated (by majority carriers) and ionized impurity in the bulk transition region near the interface which was neglected in the simple theory. The neglected charge, referred to as the bulk charge in the preceding paper [5] , is illustrated in Fig. 2(b) as & p - N a = Q B . Its effects on the static and low-frequency characteristics of the induced channel silicon MOS transistors have been studied in detail in the preceding paper.

It is evident from the charge distribution diagram given in Fig. 2(b) that if the current carrier charge Qn. is very large compared with the bulk charge QB, then the effect of bulk charge on the thermal noise would be negligible. However, for a considerable range of drain current of practical importance, the high current condition for large Qn is not attained, and the effect of bulk charge is signif- icant. An estimate of the high current condition was made in the preceding paper for the induced channel structure. The results given in (33a)-(33d) in the preceding paper show that the high current condition corresponds to B gate voltage greater than 50V,”/q5,, which may be of the order of 50 volts, a very large value for a typical transistor. Thus, we would expect that the effect of bulk charge on the thermal noise is rather important in typical device designs.

410

SAH ET At. : FIXED BULK CHARGE AND THERMAL NOISE 411

and 6VD = 0. In this integration, let us consider the noise current flowing in the drain lead due to a voltage fluctua- tion located at a position between y and y + Ay. Then,

~ I D V J - Y - AYI = lIbg 61, dY

= -lIAv p J d ( Q , SV>

=: +cc,ZQ,(y C A Y ) S V Y 4- AY) (3) and

~ I D ( Y - 0) /* 610 dY = -l* P,W(Q,~V)

= - P,tX!n(Y) 6 V(Y) * (4)

From the Nyquist Theorem for resistance noise, we have

A m = A[SV(y + Ay) - 6V(y)]' = 4kTARAf

Fig. 1. A typical noise power spectrum of a P-channel silicon MOS transistor.

(b) Fig. 2. (a) Silicon induced channel MOS transistor structure and

(b) the charge variation.

11. THEORETICAL ANALYSIS AND NUMERICAL CALCULATIONS

The channel thermal noise can be calculated by taking {SI the fluctjuation of the drain current given by [5]

ID = pnZ(-QJ(dV/dY) (1) which gives

61, = -pmZ[6Qn(dV/dy) + &n(d6V/d~) l

== - EL,^ [( 6 V ) (dQJd V (d V / ~ Y > + Q, (d 6 V/dy ) l

-pnZ(d/dy) CQnS VI. (2) The second step in the above calculation is obtained from the fact that Qn is an explicit function of the channel voltage V. This equation may be integrated for the condi- tion of short circuit between the drain and the source and between the gate and the source so that 6Va = 0

= 4kTAfAV/I , . ( 5 )

Substituting (3) and (4 ) into ( 5 ) and noting that noise voltages at different locations in the channel are not correlated, we obtain

A Z == -4kTAf(p,/L2)Q,AyZ (6)

for the short circuit mean square noise current in the drain due to the thermal voltage fluctuation across the slice of channeI of thickness Ay located at y. In this analysis, use is made of (1) in ( 5 ) to eliminate AV.

The total short circuit noise current in the drain is then the integral of (6) over the entire length of the channel and is given by

__ SI; = -4kTAf(p,,/La)QC (7)

where Qc is defined by

The equivalent noise resistance referred to the oxide gate is then

R,, = 6Vi /4kTAf ~

~

= 61;/gm4kTAf

= - Q c / C O V D ~ ~ (9)

where use is made of (7) for Tc and (25) in the preceding paper for gm. The dependence of $.so on V , in gm is neg- lected. Thus, the general relationship of the product Rpngm is given by

R u n g , = -Qc/CoVD ( 10)

where C, = K0~,LZ/xo is the total capacitance of the oxide gate.

The explicit expression for the total current carrier charge in the channel Qc can be obtained after performing

412 SEEE TRANSACTIONS ON ELECTRON DEVICES "APR,PL

the integration of (8) using Q, and Q B defined in the preceding paper. [See (14) and (4) of the preceding paper.] The complete expression is given by

-Qc = (2cd3) ((VG - V T + VB)3

- ( I r , - Ti , + V B - V D I 3 + 3V",D(1 f VD/4+F)

- 8 9 ~ V B ( V G V T + V B + 29F)[(1 + VD/@F)"'" - 11 $- (48/5)4:J',[(1 + vD/24F)"/" - 11 1 f {Z( V,J - V, + 1 7 B ) V D -

- (8/3)9FVB[(1 + v D / 2 9 F ) 3 ' 2 - I]} (1 1)

As an illustration of the behavior of the thermal noise as a function of bias points, this long expression for Qc is computed numerically for xa = 2000 A and N a = 101'/cm3 in a silicon-oxide (KO = 4) silicon ( K , = 12) structure. The dependence of R,,g, on the drain voltage VD with VG - V T as a parameter is plotted in Fig. 3 both for the saturation range and the nonsaturation range of the drain characteristics. In the saturation range, V D is set equal to VDs, which is a function of V , only and is independent of V,. Thus, the thermal noise in the saturation range becomes independent of the drain voltage in the saturation range as illustrated in this figure.

Considerably simplified expressions for the equivalent gate input noise resistance can be obtained in the saturation range of the drain characteristics for small and large drain saturation voltage VDs. These asymptotic expressions can be obtained directly by a brute force power series expansion of (11). However, considerable saving in algebra can be made if the series expansion of the integrand of (8) is made first before integration. Proceeding in either way, the following asymptotic ex- pressions for the noise resistance are obtained:

R s n s g m s = (2/3)[1 + ( V , / 4 4 ~ ) ] V D S I $ ~ / 5 (12)

= (2/3)[1 + (11/15)(V~/d-)l

VDs 2 206F. (13) The additional factor of 1 + (VB/44F) from bulk charge

in (12) comes from VD8 = ( V , - VT)/(l + 'VB/44,) [see (32a) of preceding paper], and Qc is not affected by the presence of bulk charge for small channels. On the other hand, for large V,,, the correction factor of (11/15) comes entirely from Qc. Both of these asymptotic results show explicitly the effect of the fixed bulk charge, They point out that the bulk charge not only decreases the trans- conductance as we have shown in the preceding paper, and hence, increases R,,, but also gives an additional increase expressed by the factor in [ of (12) and (13).

The asymptotic solution for smaI1 V , or V,, given by (12) is computed and graphed in Fig. 4 as a function of the impurity concentration in the bulk for the induced channel structure with silicon oxide as the insulator and silicon as the substrate. The curves in this figure show that a considerable increase of R,,,gme is expected for a

Silicon

X,frnOA N ~ = I ~ % ~ ?

23OC

Fig. 3 . The theoretical noise resist,ance referred to the gate input as a function of drain voltage for a, silicon NOS transistor with gate voltage as the parameter.

3 c

Fig. 4. The asymptotic low current value of the equivalent input

with oxide thickness as the parameter. noise resistance as a function of the bulk impurity ooncentratmn

typical device design with x. = 2000 A and N n = 1016/cma. The exact expression for R,,,g,, obtained from (10)

using (11) is computed for two practical situations a8 ep,

function of the gate voltage V o - V T . These are shown in Figs. 5 and 6. I n Fig. 5 a family is shown with x. = 2000 A and N A as the parameter, and in Fig. 6 with N A = 1016/cm3 and x. as the parammeter.

These families of graphs show that at a normal operating condition such as Va - VT = 5 volts in a practical device design of x. = 2000 A and N A = 5 X 1016/~ma, the product R,,,g,, is about two times greater than that predicted by the zero bulk charge theory of (2/3).

1966 SAH ET AL. : FIXED BULK CHARGE AND THERMAL NOXSB 413

v, - v, volts

Fig. 5 . The equivalent gate input noise resistance times the trans- conductance in the saturation range of the drain characteristics of a silicon MOS transistor with the bulk impurity concentration as the parameter.

0 -1 I I I

0 2 4 6 8 10 12 14 16 18 x) V,-V, volts

Fig. 6. The equivalent gate input noise resistance times the trans- oonductance m the saturation range of the drain characteristics of a silioon MQS transistor with the oxide thickness as the parameter.

111. COMPARISONS WITH EXPERIMENTS

Noise nleasurements are made on some of the silicon induced channel devices reported in the preceding paper. The material and geometrical parameters are shown in Table I of the preceding paper.

I n Fig. 7, the experimental measurements of the noise resistance as a function of frequency in the high-frequency range are graphed for a P-channel structure which has an oxide thickness of 6200 A (Unit 631011C-26 listed in Table I of the preceding paper). The corresponding value of 2/3g,. is computed from the measured g,, and labeled on the right vertical axis in Fig. 7 . I6 is evident that Rona is considerably higher than the corresponding 2/3gm9 for these curves. The product of R,,,gma, taking the plateau region of R,,, for each drain current level, is graphed in Fig. 8, where a family of theoretical curves for the oxide thickness of 6200 A is also shown. The noise data cor- respond closely with theory using either the starting

impurity Concentration of 5 X 1014/cm3 or the measured impurity concentration of 6.4 x lOI4/cm3 from the C-V plot of the gate. (See Table I of the preceding paper.) Two sets of data corresponding to two drain voltages are shown in this figure. The circles are for VD = 20 volts, while the solid dots which are lower are for VD = 10 volts. The slight difference between the data of the two drain voltages indicates additional noise arising from the de- pleted and high field region near the drain junction.

The effect of bulk charge on the thermal noise is con- siderably smaller for thin oxides, The unit P631011-9, a P-channel silicon device with 2100 A oxide thickness, is measured, and the data are shown in Fig. 9. The data compare very well with the predicted noise using a sub- strate concentration of N D = 1.5 X 1015/cm3. The starting value is 5.2 x 1014/cm3, and the value measured from the gate C-V plot is 1.56 X 1015/cm3, as indicated in Table I of the preceding paper.

Comparisons between theory and experiments are also made for N-channel devices. The d a b of the noise re- sistances in the saturation range are plotted as a function of the gate voltage in Figs. 10 and 11 for a thick oxide (8400 A) and a thin oxide (2000 A) unit, respectively. The data tend to fall below the theoretical curves at low gate voltages. This is due to the outdiffusion of the p-type impurity during the high temperature cycles, which re- duces the impurity concentration at the surface below the bulk value. Thus, for small gate voltage or small channel, the bulk charge comes mainly from the surface region where N A is smaller due to outdiffusion and, hence, the bulk charge effect on the thermal noise becomes smaller.

The effect due to the outdiffusion of the p-type impurity is particularly clear for the thick oxide unit 631014-7 shown in Fig. 10. Thus, for large channel or V o - V,, the thermal noise corresponds to a substrate doping of about 2 X 101'/cm3, which compares favorably with the starting resistivity value of 1.4 x 101'/cm3 and the gate C-Y value of 3.2 X 101'/cm3, listed in Table I of the preceding paper. As the gate voltage becomes smaller, the thermal noise data correspond to 1016/~m3, showing smaller bulk charge effect as indicated in Fig. 10.

Surface mobility reduction at large channel or large V , - V , would aIso tend to increase R,,, above its theoretical value (or decrease its value at low Vff - VT below the theory). However, for the range of V G - VT. of these measurements, which corresponds to a maximum of I,, = 1 mA, the mobility reduction is not too important. A rough estimate from Fig. l3(b) of the preceding paper for the unit N631014-7 shows a reduction of only 13 per- cent at V o = - 5 volts or Vff - V , = - 19 volts, in- dicating a negligible correction at the highest gate voltage shown in Fig. 10.

The effect of outdiffusion of the bulk impurity an the thermal noise is also evident, in the data of the thin oxide (2000 A) unit N641119-4 shown in Fig. 11.

414 IEEE TRANSACTIONS ON ELECTRON DEVICES APRIL

VD.2OV I ’ “ ‘ I ,

Id1 105 2 4 ; 6 ’ 8 & 6 2 3 4 ;A frequency C/S

FIG. 7. Experimental data of the equivalent gate input noise resistance as a function of frequency in the saturation range of a P-channel, thick oxide (6200A) MOS silicon transistor.

Fig. 8. Rensqma vs. -(Vo - V,) data for two drain voltages compared with theory for the P-channel, thick oxide (6200A) MOS transistor.

‘ 7 ’ ’ ’ ” I &=2000A

0.6 2 4 6 I

-(vG -VJ volts

FIG. 9. Comparison between experimental data and theory of Rqnegmme vs. -(VC. - Vp) for a thin oxide (2100A), P-channel slllcon MOS translstor.

”’ 2 4 6 6 10 22 14 (VG - vr.’ volts

Fig. 10. Comparison between experiments and theory of R,,, vs. (VQ - Vp) for a thick oxide (8400A) N-channel silicon MOS transistor.

vG- Y, volts

Fig. 11. Comparison between experiments and theory of R,, vs. (V, - V,) for a thin oxide (2000A) N-channel silicon MOS transistor.

REFERENCES [l] C. T. Sah, “Noise in metal-oxide-semiconductor transistor,”

Paper 6(a), presented at the 1964 Symp. on Fluctuation Phe-

sota, Minneapolis; “Theory and experiments on the S/f surface nomena in Solids and Solid-state Devices, University of Minne-

presented a t the 1964 Solid State Device Research Conf., noise of MOS insulated-gate field-effect transistors,” Paper V-2,

Boulder, Colo.; abstracted in IEEE Trans. on Electron Devices, vol. ED-11, pp. 534-535, November 1964; this analysis is also published subsequently by A. G. Jordan and N. A. Jordan,

Trans. on Electron Devices, vol. ED-12, pp. 148-156, i\iIarch 1965; “Theory of noise m metal oxide semiconductor devices,” IEEE

[2] C. T. Sah, “Characteristm of the metal-oxide-semiconductor see particularly equations (42) and (43).

transistors,” IEEE Trans. on Electron Devices, vol. ED-11,

[31 A. van der Ziel, private communication at the 1964 Symp. on Fluctuation Phenomena in Solids and Solid-state Devices,

1141 C. T. Sah, “Theory of low-frequency generation noise in junction- University of Minnesota, Minneapolis.

gate field-effect transistors,” Proc. IEEE, vol. 52, pp. 795-814, July 1964.

[5] C. T. Sah and H. C. Pao, “The effects of fixed bulk charge on the characteristics of metal-oxide-semiconductor transistors.”

pp. 324-345, July 1964.

this issue, page 393.

Proc. IRE, vol. 50, pp. 1808-1812, August 1964. [6] A. van der Ziel, “Thermal noise in field-effect transistors,”