Thermal noise in field-effect devices

Transcription

1 Thermal noise in field-effect devices J. W. Haslett, M.Sc, and F. N. Trofimenkoff, Ph.D. Abstract Thermal-noise calculations for both junction-gate and m.o.s. field-effect transistors are performed using a straightforward circuit-analysis technique based on the equivalent circuit. Expressions are obtained for i%, / and the correlation coefficient of i g and i d at moderately high frequencies. A comparison with van der Ziel's results for the bulk f.e.t. shows that the expression for the gate noise can be simplified considerably and that the expression for the correlation coefficient c is in error. The correct expression for c is given in terms of van der Ziel's equations and is also presented in simplified form. Approximate expressions for the noise factor of both bulk and m.o.s.f.e.t. amplifiers are presented in a form convenient for comparison with experimental measurements. It is found that, for the model under consideration, at higher frequencies the thermal-noise performance of the m.o.s.f.e.t. is similar to or better than that of the bulk f.e.t., for comparable gate capacitances and transconductances. List of symbols For junction-gate f.e.t. a = halfchannel height, defined in Fig. 1 c = correlation coefficient, defined by eqn. 25 Cj = magnitude of correlation coefficient C x gate-source capacitance of intrinsic device, defined by eqn. 4 C 2 = gate-drain capacitance of intrinsic device, defined by eqn. 5 C/, C,. = capacitances, defined by eqn. 22 C gs = total gate-source capacitance, including strays C gd = total gate-drain capacitance, including strays / = cyclical frequency /o = break frequency of the spot-noise factor, defined by eqn. 38 f\(y,z),f2{y,z),fi(y,z) = functions of the direct-bias conditions, defined by eqns. 4 and 5 F = noise factor, defined by eqn. 37 F min = minimum noise factor, defined by eqn. 40 F ch contribution to noise factor due to channel noise, defined by eqn. 36 %j,(y,z) = function of bias conditions, defined by eqn. 23 g 0 output conductance, defined by eqn. 2 g m = transconductance, defined by eqn. 3 g mo = maximum transconductance, defined by eqn. 7 i gr = noise current into gate resulting from thermal noise of R g ig = gate-noise current resulting from thermal noise in channel i d = thermal-noise current from the conducting channel i dl = total noise current in short-circuited drain 'dtr = noise current at drain due to thermal noise of R g I d = direct drain current, defined by eqn. I k = Boltzmann's constant L = active channel length N c = ionised-impurity density in channel p = function of bias conditions, defined by eqn. 23 P(f) = power spectrum, defined by eqn. 30 PTW) P R (f) = power spectrum of total output-noise current = power spectrum of drain-noise current due to the signal source conductance q = magnitude of electronic charge Q(l,z) = function of gate bias, defined by eqn. 16 /?/, R r = resistances at the point x in the channel, defined by eqns. 10 and 11 R n = equivalent noise resistance, defined by eqn. 18 R g gate resistance Rg O p, = gate resistance which defined by eqn. 39 S(f) Fourier spectrum of defined by eqn. 29 minimises noise factor, the noise current /(/) Paper 5929 E, first received 23rd January and in revised form 21st May 1969 Mr. Haslett and Dr. Trofimenkoff are with the Department of Electrical Engineering, University of Calgary, Calgary, Alta., Canada PROC. IEE, Vol. 116, No. II, NOVEMBER 1969 / = channel breadth T = absolute temperature, K V gs = direct gate-source voltage V ds = direct drain-source voltage dv n (x) = thermal-noise voltage generated by a length of channel dx at x W = gate-channel potential at a point x in the channel W d = gate-channel potential at drain W s = gate-channel potential at source = pinchoff potential x = distance along channel from the source y= w d iw p z= W s \ a conductivity of channel material i/> 0 = equilibrium contact potential 1 (1 + 7Z 1 ' 2 ) y = c 5 (1 + 3z'/ 2 ) For m.o.s.f.e.t. c' = correlation coefficient, defined by eqn. 56 c'i = magnitude of correlation coefficient C[ gate-source capacitance of intrinsic device defined by eqn. 41 C' 2 = gate-drain capacitance of intrinsic device, defined by eqn. 42 C' gs, C' gd total gate-source and gate-drain capacitances, respectively C ox oxide capacitance per unit length of channel W ^ gs + ^gd / 0 = break frequency of the noise factor, defined by F' '' min So m onto L' g igopt eqn. 60 noise factor, defined by eqn. 59 minimum noise factor, defined by eqn. 62 contribution to noise factor due to channel noise, defined by eqn. 58 output conductance, defined by eqn. 43 transconductance, defined by eqn. 44 transconductance at pinchoff, defined by eqn. 48 function of bias conditions, defined by eqn. 54 thermal noise due to conducting channel gate noise due to capacitive coupling with channel active channel length equivalent noise resistance, defined by eqn. 50 gate resistance which minimises noise factor, defined by eqn. 61 T ox = thickness of oxide insulating layer v s gate-channel potential at source v d = gate-channel potential at drain V' gs = direct gate-source voltage V' ds = direct drain-source voltage V T = threshold voltage Z = device breadth e ox permittivity of oxide insulating layer [x carrier mobility in channel 1863

2 1 Introduction Thermal-noise calculations for both junction- and insulated-gatefield-effecttransistors have been presented by a number of authors. 1 " 5 Since that time, additional work on the small-signal equivalent circuits of these devices has been published The purpose of the paper is to demonstrate that thermal-noise calculations can be performed simply by using an equivalent-circuit approach rather than by reverting to fundamental considerations of device operation. Results are presented in a compact form for both junction- and insulated-gate devices. 2 Junction-gate f.e.t. 2.1 Equivalent circuit A cross-section of an f.e.t. is shown in Fig. 1. The device can be divided into an 'intrinsic' portion involving the active channel and an 'extrinsic' portion, which accounts for stray capacitances and parasitic resistances as illustrated. Reddy and Trofimenkoff 6 have derived a small-signal equivalent circuit for the active device, which takes the form shown in Fig. 2. For frequencies well below the cutoff frequency of the f.e.t., the resistances r { and r 2 are negligible compared with the reactances of C { and C 2, and / 0 is very small, so that the active channel may be represented to a good approximation by the circuit shown in Fig. 3. Shockley 7 has shown that the direct drain current I d is given by /,, = loat _(«:y"l«r m where a = conductivity of the channel material. Utilising eqn. 1, it is easy to show that the output conductance g o is given by 80 L \ and the transconductance g m is 8m L \\ It should be noted that w d = «Ao + v gt - y ds where V gs and V ds are the gate-source and drain-source voltages, respectively, and </r 0 is the equilibrium contact potential. Van der Ziel 2 has derived expressions for the gate-source capacitance C x and the gate-drain capacitance C 2 in terms of device dimensions and bias conditions as follows: _ 2qN caltf 2 (y,z) where Cl _ 2qN c altmy,z) 23/2) _ (4) extrinsic capacitances f 2 (y,z) -y m ) - y) V/////&V/7///X//Y//, I type material parasitic parasitic resistance(r s ) resistance^ substrat< Fig. 1 Bulk f.e.t with bias voltages applied A(y,z) = (y - z) -\{y m - z" 2 ) and y=w d \ z= W s \ At pinchoff, W d = tv p, so that y = 1 and (6) (7) Fig. 2 General form of intrinsic equivalent circuit _ 3qN calt Tt can be shown that, at moderately high frequencies, the effects of the parasitic resistances R s and R d are negligible, so that stray capacitances from gate to source and gate to drain appear directly in parallel with C { and C 2, yielding a modified equivalent circuit for the complete device, as shown in Fig. 4. The capacitances C gs and C gd represent the total gate-source and gate-drain capacitances, respectively. (9) gd - d fls Fig. 3 Simplified equivalent circuit for the active channel 1864 Fig. 4 Modified equivalent circuit including stray capacitances PROC. IEE, Vol. 116, No. 11, NOVEMBER 1969

3 2.2 Thermal-noise calculations Trofimenkoff 8 has presented a method of evaluating the short-circuit drain-noise current by splitting the transistor Equating eqn. 15 and eqn. 17 yields (18) where Q(l, z) is of the order of 0 6 for normal bias conditions. 2.3 Gate noise An extension of the split-transistor technique may be used to calculate the gate noise which results from capacitive coupling with the channel at moderately high frequencies. The intrinsic capacitances are included in the 2-transistor equivalent circuit, as shown in Fig. 6. It should be noted that s,=s d,=d Fig. 6 2-transistor representation for gate-noise calculations b Fig. 5 Split-transistor technique a Schematic of bulk f.e.t. b 2-transistor representation of f.e.t. g at a point x in the channel, as shown in Fig. 5. The resistances R, and R r are given by _J laat) R,~ x \ 1 l^lu-t L-x\ \ (ID The thermal noise associated with the resistance of a length of channel dx at x is given by 2aat<\-(^ 4kTAfdx W n '/ 2 (12) where A/= equivalent noise bandwidth of the system. The contribution to the drain-noise current may then be expressed as (R, + R r ) 2 (13) Substitution for dx from eqn. 1, for dv n (x) 1 from eqn. 12, and integration from the source (W = W s ) to the drain {W = W d ) yield the mean-square drain-noise current in a bandwidth A/: in agreement with van der Ziel's 1 results. At pinchoff,.... (14) /J = 4kTAfg mo Q(l, z) (15) 1 (1 + 3z l/2 ) where Q(l, 2 ) = IL± I r/5 j (,«) The equivalent noise resistance at the input may be obtained by writing lf l = 4kTR n g 2 m0af (17) PROC. IEE, Vol. 116, No. II, NOVEMBER 1969 the source and drain terminals are a.c. short-circuited to the gate, eliminating the effects of stray capacitance. If it is assumed that capacitive currents flowing to the gate are small in comparison with conductive currents flowing through the channel, the channel current can be assumed to be constant. If this is the case, the voltage at d t is given by dv { = R,di d. (19) and the voltage at s 2 is dv 2 = - R r di d (20) dv n (x) where di,, = (R, + Rr ) (20 Using eqns. 19 and 20, we can show that the incremental gate current di g is W W.. (22) f W W \ The function A ( -, -~) is obtained by replacing y by V J (177 ) m ec l n - 5 > anc * h \ TTT ' 777 ) ' s obtained by replacing z \ WpJ \ J Substitution of appropriate values into eqn. 22 and integration from source to drain yields where '1 = 4A:7A/oi2 C 2 (l - z l/2 ) ) = [P 2 {(y ~ z) + i (y 2 - z 2 ) - f (y 3 ' 2 - z 3 ' 2 )} - lp& (y z 3 ' 2 ) - (y 2 - z 2 ) + i (y 5 ' 2 z 5 ' + {i (y 2 - z 2 ) - f (y 5 ' 2 - z 5 / 2 ) + i (y 3 - z 3 )}] $ (y z 3 ' 2 ) - i (y 2 - z 2 ) and p = A(y, again in agreement with van der Ziel's results. 2 At pinchoff, g 3 (y,z) may be simplified to yield (23) / 1 (1 + 7z'/ 2 ) 4kTR n A/oj 2 C 2 ~ 5 (1 + 3z" 2 (24) ) ' ' " This result is much simpler than that presented by van der 1865

4 Ziel, 2 and it can be shown that the two expressions are identical. 2.4 Correlation coefficient The degree of correlation between the gate and drain noise is expressed in terms of a correlation coefficient defined by l S l d / 25 \ A nodal analysis of the circuit yields. _., (i g + igr)xg(gmol dt l d + + jcor g (C gs +C gd ) (32) On replacing currents by equivalent current spectra in eqn. 32, g c a d Taking the product of eqn. 21 with the complex conjugate of eqn. 22 and integrating from source to drain: -4kTkfjaj2c/N c alt r l *' = I'M**) -z 3/2 )-i(/-z 2 )},. 2 - z 3 ' 2 ) (26) Substitution of eqns. 26, 23 and 14 into eqn. 25 and setting y = I yields l/go, showing thermal-noise sourcesc = - z)" 2 {5(l + 3z'/ 2 )(l + 7z'/ 2 )}'/ 2 (27) ' Van der Ziel has obtained an expression for c which is more complicated and which does not agree with the numerical data. 2 The expression in its correct form is given by Fig. 7 Simple f.e.t. amplifier with taking the product with the complex conjugate and performing the averaging procedure and noting that i g and i d are correlated, it can be shown that C gd )Y C gd ) {corjc gs + C gd )Y (33) l0.12 (1 +2z'/ 2 ) c = 3z'/ 2 ) z'/ 2 ) 15 2z'<' 2 ) 2 10 and it can be shown that this reduces to eqn Figure of merit The spot-noise factor is the most commonly used figure of merit for evaluating the noise performance of an amplifier and may be expressed as total output noise power output thermal noise power due to the source conductance.... (28) The noise power is usually described in terms of a power spectrum, which gives the mean-square noise voltage or current per unit bandwidth. In order to obtain the power spectrum of a random stationary process such as thermal noise, a transformation from the time domain to the frequency domain and a suitable averaging procedure are required. Montgomery 9 shows that it is possible to replace a steady-state noise current /(/) by a Fourier spectrum defined by 1 r' +T. S(f) = - ^^ i{t)e-^'dt (29) where T is any finite time interval. The power spectrum is then obtained by noting that "TT 00 P(f) = -T-,= \ E{S*(f)S(g)}dg (30) L±J J 00 where denotes the expected value, and the asterisk denotes the complex conjugate. The noise factor may then be written as c PTU) PR(/) (31) where P T (f) = power spectrum of the total output noise current PRU) ~ power spectrum of the noise at the output due to the signal source conductance only. The equivalent circuit of a simple amplifier including noise sources is shown in Fig. 7, where the symbols are as defined in the list of symbols z } 1/2 Similarly, the mean-square contribution to the output current from the gate resistance is given by <" R 1 + {a>r g (C gs + C gd )Y Substitution of eqns. 33 and 34 into eqn. 31 yields oj 2 C}Q(l,z)yR g where C, = C gc + C 7 gi " " ^gd ~ 5 (1 + 3z" 2 ) Smo Smo (34) ^ Smo.... (35) The first term is the contribution due to channel noise, the second is due to gate noise, and the third is a result of correlation between i g and i d. It should be noted that C ( is the gate-drain capacitance for the intrinsic portion of the f.e.t. only, while C gd represents the total gate-drain capacitance, including strays. At moderately high frequencies, o> <^ g mo IC gd, and, if R g > V<?mo» tne term ucgdlimo is mucn smaller than ojr g C,, so that the channel-noise contribution is approximately given by Smo "-g {1 + (cvr.c,) 2 } (36) A comparison of eqn. 36 with the contributions due to gate noise and correlation shows that the channel-noise term dominates, provided that C x < C,, which is always the case, so that ~ 1 + Q0,z) ' + ' (37) PROC. IEE, Vol. 116, No. II, NOVEMBER 1969

5 (38) The gate resistance that will minimise the noise factor is found by differentiating eqn. 35 with respect to R g and equating the result to zero. Again, at moderately high frequencies and for values of R g > l/# mo, and the minimum noise factor is given by (39) At pinchoff, J. 2 1 and R n = - -r- J mo The mean-square value of the gate-noise current is given by (51) (52) '7 = 9 ' 7 V 2 _ V2\2t v \ v ) Z( V s> V d) (53) /-,, ~ l ^ ' (40) where C o = L,'C 0X and Since the optimum gate resistor R g opt is frequency-dependent, it is not possible to minimise/'over a large range of frequencies using a fixed R g. 3 Metal-oxide-semiconductor f.e.t. 3.1 Equivalent circuit The equivalent circuit for a metal-oxide-semiconductor f.e.t. takes the same form as that for the junctiongate f.e.t., as shown in Figs. 3 and 4. In order to avoid confusion, primed values will be used to distinguish the m.o.s.f.e.t. parameters from those of the bulk f.e.t. Using the transmission-line equations derived by Candler and Jordan, 10 a small-signal equivalent circuit has been derived for the device, and the components have been expressed, in terms of bias conditions and device dimensions as follows: c;-!c.^.&±g> (40 At pinchoff, v d = 0 64 and ig 2 = 135 g m (54) (55) The correlation coefficient is independent of bias conditions for the pinchoff case and has a value (56) Utilising the expressions for the thermal-noise generators given by eqns. 49 and 53, the excess noise factor for a simple amplifier similar to that shown in Fig. 7 may be written in exact form as where C ox = It should be noted that (43) (44) 2 a> 2 R g C 0 C' t \ g' mo R 8 C' t + ( 7~) &mo ' (57) where C,' = C' g, + C gd The first term is the contribution due to channel noise, the second is that due to gate noise, and the third is that due to correlation. As for the bulk f.e.t. at moderately high frequencies, co < g m olc' g d> so tnat tne channel-noise contribution is approximately given by At pinchoff, v d = 0 and C[ = i C OX L' (45) C 2 = 0 (46) g'o = 0 (47) For R g > llg mo, it is found that the channel noise dominates, provided that C\ > C o, which is usually the case, so that the noise factor F' is (59) tonto /_' where /Q = 2-nRgC, (60) 3.2 Thermal noise The procedure involved in calculating the noise cpmponents for the m.o.s.f.e.t. is identical to that for the bulk f.e.t. just described. In order to avoid repetition, only the final results are quoted. The thermal noise of the conducting channel may be written as (v * + v ' v " + v ' d) Using a procedure similar to that for the bulk f.e.t., it is found that the optimum R g which will minimise the noise factor is given by yielding a minimum noise factor (62) yielding an equivalent noise resistance R' - 2 ' " ~ 377' ( v - v*) mo \ s d> PROC. 1EE, Vol. 116, No. II, NOVEMBER Comparison of results A comparison of eqn. 62 with eqn. 40 shows that the minimum noise factor for the m.o.s.f.e.t. is nearly the same 1867

6 as that for the bulk f.e.t., provided that the transconductances and input capacitances are comparable. It is possible to obtain m.o.s.f.e.t.s with high transconductance and low input capacitance, so that, for the model under consideration, the thermal-noise performance of the m.o.s.f.e.t. at higher frequencies is expected to be as good as or better than that of the bulk f.e.t. 5 References 1 VAN DER ZIEL, A.: 'Thermal noise in FET's', Proc. Inst. Radio Engrs., 1962, 50, pp VAN DER ZIEL, A.: 'Gate noise in FET's at moderately high frequencies', Proc. Inst. Elect. Electron. Engrs., 1963, 51, pp BRUNKE, w. c, and VAN DER ZIEL, A.: 'Thermal noise in junctiongate FET's', IEEE Trans., 1966, ED-13, (3), pp SHOJI, M. : 'Analysis of high-frequency thermal noise of enhancement mode MOS field-effect transistors', ibid., 1966, ED-13, (6), pp JORDAN, A. c, and JORDAN, N. A. : 'Theory of noise in MOS devices', ibid., 1965, ED-12, (3), pp REDDY, B., and TROFIMENKOFF, F. N. : 'F.E.T. high-frequency analysis', Proc. IEE, 1966, 113, (11), pp SHOCKLEY, w.: 'A unipolar field-effect transistor', Proc. Inst. Radio Engrs., 1952, 40, pp TROFIMENKOFF, F. N.: 'Thermal noise in field-effect transistors', Proc. Inst. Elect. Electron. Engrs., 1965, 53, (9), pp MONTGOMERY, H. c.: 'Transistor noise in circuit applications', Proc. Inst. Radio Engrs., 1952, 40, pp CANDLER, D. p., and JORDAN, A. c: 'A Small-signal, high-frequency analysis of the insulated-gate FET', Internal. J. Electron., 1965, 19, (2), pp PROC. IEE, Vol. 116, No. 11, NOVEMBER 1969