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R 642 Philips Res. f-epts 22, 505-514, 1967 THERMAL NOISE OF MOS TRANSISTORS by F. M. KLAASSEN and J. PRINS Abstract A theory is given for the thermal noise of MOS transistors which takes into account the influence of substrate doping. It is shown that the noise resistance is a simple function of the transconductance, the effective gate voltage and the pinch-off voltage. In practical transistors the value of the noise resistance may be a factor of 10 higher than the value predicted by the simple theory (zero substrate dope). Measurements are discussed that support the theory. 1. Introduetion The influence of the substrate doping upon the DC characteristics of silicon MOS transistors has recently been the subject of several theoretical and experimental investigations. Within the scope of this paper we report the results for the transconductance obtained by Sah and Pao 1) 'and independently by Van , Nielen and Memelink 2). They have shown that only in the saturation range does the substrate doping have a marked effect upon the value of the transconductance gmO' In this case gmO is no longer proportional to the gate-source voltage Vg, as it is for a MOS transistor with zero substrate doping, but is proportional to a certain voltage Vp ~ Vg, called pinch-off voltage, which is the channel potential for which the channel charge-at the drain becomes zero. So where the proportion constant f3 has been determined by the gate capacitance and the channel geometry. The value of Vp is decreased by increasing the doping or the oxide thickness (cf. ref. 2). Besides on the DC characteristic the substrate doping also has some influence upon the thermal noise of the channel. Sah et al. 4) have analysed the noise for small values of doping, but the mathematics was rather involved and restricted to very small or very large drain voltages. However, in the case of heavy doping it is not so difficult to 'understand the influence upon the noise. To begin with we expect that, analogous to a normal resistor, the noise will be proportional to the total integrated current charge in the channel. When the substrate doping is large the potential in the channel everywhere is much smaller than the gate-source voltage, because of the space charge in the substrate depletion layer; so we expect that this integral will be nearly proportional to the voltage Vg. Thus we obtain the simple result 506 F. M. KLÀASSEN and J. PRINS where Vp and gmO are dependent on the substrate doping and (3 is a proportion constant for gmO' This expression differs by a factor of Vg/Vp from the result valid for a MOS transistor without substrate 3). For moderate values of doping the integral is not so simple and a careful analysis is necessary. In this paper a closed-form expression for the noise is given without restrictions for the drain voltage, and the results of measurements are' reported for MOS transistors with substrate doping in the range from 1014 to 1017 cmr ". 2. Theoretical analysis When we assume a constant surface-charge density CoxVT, which is present irrespective of the potential difference Vg __:_Vex), the mobile-charge density in the n-type channel of a MOS transistor will be given by 2) (see fig. 1) am(x) = -Cox [Vg - VT - + a[V(x) + Va]1/2; Vex)] . capacitance of the oxide layer per unit area, applied gate potential, Vg offset gate potential, VT potential in the channel at location x, VeX) total induced surface charge, Cox[Vg - VT- V(x)]: depletion charge of the p-type substrate, a[V(x) + VaJl/2 Cox Ol: 8 (2 eqNA)1/2, dielectric constant of silicon, electron charge, acceptor density of the substrate, diffusion potential between channel and bulk. +\iI +Var -.x _ Subsfraie' L = contact Fig. 1. Schematic representation , I "I of an n-channel MOS transistor. (1) THERMAL NOISE OF MOS TRANSISTORS 507 The channel current I at position x and time t will be I(x,t) dV -p,WGm(x) . dx = + h(x,t), (2) where W is the width of the channel, p, 'the mobility of electrons and h(x,t) a random noise-current source at location x. The preceding equation is used as a starting point for the derivation of the noise spectrum. Assuming that the fluctuations of all quantities appearing in (2) are small, we may split up all time-dependent quantities into a large steadystate value and a superimposed small time-dependent signal I(x,t) = 10 V(x,t) = + i(x,t), Vo(x) + v(x,t), (3) dG Gm(x,t) = Gm(Vo) + -- dVo v(x,t). By making use of the relation (2) we obtain for the steady-state part 10: (4) Assuming that Vo(x) is a monotonically increasing function of x! relation (4) is a transformation formula between the variables x and Voo Moreover, this relation yields an expression for the current 10 if one integrates from the source to the drain: (5) where L is the source-drain distance and the subscript d refers to the drain. Substituting (3) into (2) we obtain for the fluctuation d i(x,t) = -p,W . - [G,nCVo)v(x,t)] dx . + h(x,t), (6) if we neglect second-order terms. Usually the noise current of the device is measured when the output is shortcircuited for a.c. voltages. Thus for the short-circuit current noise of the channel we have L L J i(x,t)dx = J h(x,t)d~. o' . 0 . (7) , 508 F. M. KLAASSEN and J. PRINS Since we are more interested in the spectral density of the noise current we write i(x,t) ~ ~ + jOOl,(x) + ...l exp (joot),) (8) = hex) exp (jwt), h(x,t) in which we assume that the low-frequency spectrum io is independent of x, if the spectrum of the source hex) is white in the frequency range to be considered. Proof of this assumption is beyond the scope of this paper and will be given elsewhere 5). Consequently by equating terms with equal power in w we obtain from (7) for the spectral density of the short-circuit noise current at the drain L <fa. fa*) L = L-2 J dx J < h(x). o h*(x'» dx'. (9) 0 If we restrict ourselves to the frequency range in which thermal noise is the dominating noise' source we have <hex) . h*(x'» = 4kTf-t WO'III(x)5(x - x'), (IQ) where 5 is the Dirac delta function. Substituting this expression into (9) and using expression (4) in order to eliminate the variable x we obtain VOd f 0'1II2(Vo)dVo _ _ 4kTp,W 0 <t«. io*) = -----L VOd f O'III(Vo)dVo o The integrals appearing is a simple function of Usually the channel resistance at the input, (11) in (11) can be computed immediately, because O'm(VO) the stationary voltage Vo. noise is transferred. to the thermal noise of a certain called the noise-equivalent resistance Rn: .(12) where (13) Elaborating the integrals in (11) and using (1), (1.2)and (13) we finally obtain (14) THERMAL NOISE OF MOS TRANSISTORS in which the dimensionless factor a 509 is given by a= Vg'2 -!(a/Cox) Vg' VOd'l/2 + t,(a2/Cox2 -2V9')Vod' where a = (2 eqNA)1/2, Vg' Vo/ = = + ~(a/CoJVo/3/2 + tVOd'2 (15) Vg- VT + Va), VOd + Va. Afterwards we shall see that his factor varies between tand 10 in practical situations. Because the expression for a is not very clear some special cases will be considered: (a) Transistor operating far below saturation. In this case we have and a V'9 ././ " -C V.Od '3/2 , ox from which assumption we obtain ., (16) I (0) Transistor with low substrate doping. In this case we may neglect all terms with the factor a, which results in V'2 9 V.Od 'V'9 + ~v. '2 "3" Od (17) •If this transistor also operates in saturation we have VOd' ~ Vg' and we obtain the well-known result 3) (18) (c) Of special interest is the general case when the transistor operates in satu- iation. Following Sah 1) and Van Nielen 2) we then have .......,.•, and I. 510 F •.M. KLAASSEN and J PRINS Consequently R; may be approximated by R R:! n 1__ V.' 0 ( 2 Vp' +_1) 6 g-l (19) m in which and This is indeed a closed-form expression for a MOS transistor operating in saturation and it is valid for all possible values of substrate doping and oxide thickness. As a general result we may conclude that the noise-equiv- _ alent resistance is a monotonically increasing function of the substrate dope. and the oxide thickness, because the pinch-off voltage is a monotonically decreasing function of the same parameters 2) (cf. also figs 4a and b). Besides the noise of the channel current it is also possible to compute the f noise that is induced in the gate circuit at higher frequencies. However, since' we have found that the substrate has little influence upon the gate noise, this ~ subject will be left out of consideration here. In the next section the expressions for the noise resistance are compared with. the results of experiments. . ~ 3. Comparison with experimental results Although silicon MOS transistors also produce flicker noise, it is still possible, . applying a proper technology and choosing suitable dimensions, to make tran- ; sistors in which thermal noise predominates in a wide frequency range : (f> 10 kc/s). In fig. 2 we have illustrated this by plotting the power spectrum .~ of the channel noise as a function of frequency for various drain currents. At. ; all times the transistor operated in saturation . . In order to test our equation (19) we first consider the thermal noise of several MOS transistors with different substrate doping (O·08-30Qcm material) and the same oxide layer. In fig. 3 we have plotted the experimental values of ot: as a function of the acceptor density for an effective gate voltage between 5 and 10 V. Moreover we compare the experimental values with the theoretical THERMAL 511 MOSlr E3 III esuIJstraW =30.a cm-:- 5 <, 1'oci>l1> <, <, r-, 2 I' "~ "f'..i' 10* r-. -- , Id-:,f5pA 1 0""""'" r~~ 5 2 1iT 1 . NOISE OF MOS TRANSISTORS 5 2 10 2 - 100pA • K I 0·5mA ~ 3·0mA 5 102 5 1r1 -f(kc/s) 2 . -"- o 5 2 Fig. 2. Noise spectrum of an n-channel MOS transistor. 14 ~gm~ 12 T MOST E3-8 hox= 0·2p l{fo>Vp oV =5V .V =fOV 10 vg",,5~V· g g 8 / 6 y/ 4 2 o ,Iq. 10 ~ 2 5 ,7~ 10 2 5 0 ~1-fOV- ~ p-- ~16 10 2 5 ,11 -NA Fig. 3. Thermal noise vs substrate doping. Drawn curve represents theory. values (drawn curves). These curves were obtained by substituting the appropriate values of Vp in eq. (19). The pinch-off voltage as a function of the substrate doping and the oxide thickness is given by 2) F. M. KLAASSEN 512 (a. ~ " 1 Vp - Vg - - 2 Cox and J. PRINS )2{( Cox)'1/2 - 1}., 4Vg' 1+ a where and In figs 4a and b we have plotted the values of Vp, which we have used. Inspecting .. 18 16 \f:>CII. '4 I Ç)l/ h=0.2e I~ V ~" .:V / 12 lr \" /' V 10 / 8 / ./ [7/ V 6 ~ ~ V/V /' VV l~ V »: r~ ~ ~ 6 2 I--- I--8 10 v - I--~ 12 ~ 14 V \1> /V Y V V V V I~/ 16 =007 14 ./ a V ,óV ~ _.,. ~ VV I-- I-16 18 __.. 20 -- id'(_.,.- p0- V 22 24 IY -Vr(V) Nr1018 26 4'J ~:~=+==X=~==4=~==~~Mll-~ ~~ NA=3.1d%m3 __ ~~~=~0~.7~V-+ __ -+__~ __~~~~ Vp r 5 b ro'~--+---+-~~--+-~+---~--~~ 20 25 ---~ 30 -Vr 35 4() Fig. 4. The pinch-off voltage vs the 'effective gate voltage; doping; (b) for different oxide-layer thicknesses. (a): for different substrate THERMAL 5.13 NOISE OF MOS TRANSISTORS fig. 3 we may conclude that the agreement between theory and experiments is quite good. Moreover we observe that ais only a very weak function ofthe gate voltage. . Besides the above experiments we also studied the thermal noise of several MOS transistors with oxide layers of different thickness. In fig. 5 we have plotted Rn gmo t 10 / 8 ~/ / 6 V 42 L MOST E12-16 = 3.1015cm-3 Vs/=5-1SV Vod>Vp -r-- NA V I 1-- ~ °° ()'2 (}4. (J.6 OB 1.0 1-2 1·4 1-6 ---hox(/1) . Fig. 5. Thermal noise vs oxide-layer thickness. Drawn curve represents theory. the experimental values of a vs the oxide thickness with constant substrate doping. In this case too, the agreement with the theoretical values (drawn curve) is quite good. Finally we give a plot of the thermal noise vs the drain voltage (fig. 6). In 0·5 f?ogJno t Q.4- {)O3 , \ MOST £13 NA=3.1015 1\ fIox=0'2p Vp"'" O.6Vg' .\ . " 04 00 ()'2 ().4 ().6 . ().8 1.0 1·2 1.1;. 1.6 --(~) '9 Fig. 6. Thermal noise vs the drain voltage. Drawn curve represents theory. the same figure we have plotted the theoretical value of a due to eq. (15). I~ this case the experimental values deviate somewhat from the theoretical values. These deviations may partly be explained by the fact that the noise resistance is difficult to measure below saturation (low gain and large output conductance). However, this range is not of practical interest. 514 F. M. KLAASSEN and J. PRINS Thus, as a general result we may conclude that the thermal noise of a MOS transistor is well described by a theory based on the simple current-voltage relation (eq. (2)). Acknowledgement Our thanks are due to Dr F. C. Eversteyn and Mr H. L. Peek for preparing the MOS transistors. Eindhoven, June 1967 RBFERENCES 1) 2) 3) 4) 5) C. T. Sah and R. C. Pao, IEEE Trans. EI. Dev. ED-13, 393-409, 1966. J. A. van Nielen and O. W. MemeIink, Phil. Res. Repts 22, 55-71, 1967. M. Shoji, IEEE Trans. EI. Dev. ED-13, 520-524, 1966. C. T. Sah, S. Y. Wu and F. H. Hielscher, IEEE Trans. EI. Dev. ED-13, 410-414, 1966. F. M. Klaassen, to be published.

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