0033-1101/81/0504~$02.m/0 Pergamon Press Ltd.

Solid-Sore ElcWmn;cr Vol. 24. pp. 429434, 1981 Printed in Great Brikh

HOT ELECTRON NOISE AND g-r NOISE IN SHORT-CHANNEL JFETst

S. K. KM! and A. VAN DER ZIEL Electrical Engineering Department, University of Minnesota, Minneapolis, MN 55455, U.S.A.

(Received 14 July 1980; in revised form 17 September 1980)

Abstract-Hot electron noise in short-channel JFETs is measured above IWK and generation-recombination noise due to donors is measured below l20”K, where it predominates over the first. The hot electron noise increases with increasing V,, - VP for a given temperature T and increasing with decreasing T at a given Vos - V, The g-r noise in donors increases very rapidly with decreasing temperature and is found to be governed by an activation energy between I$ and 2& where I?,, is the activation energy of the donors. The theories of these effects can qualitatively explain the data; for the g-r noise it must be taken into account that the activation energy of the donors decreases with increasing electric field (Poole-Frenkel effect).

1. INTRODUCTION

In two recent papers Kim et al.[l, 21 reported noise measurements in the 1%300°K temperature regime and in the 77-150°K for high g,,, silicon JFETs. They dis- tinguished between hot electron noise observable in the high-temperature regime and donor generation-recom- bination noise (g-r noise) around 77°K; at the lower temperature the g-r noise completely masked the hot electron noise. This paper extends the measurements of these noise phenomena.

The devices were made by Dr. C. F. Hiatt, then at the Electrical Engineering Department of the University of Florida. They had a channel length of 4.1 p meter and were high-g, devices. In a typical unit the pinch-off vol- tages V,, were - 1.54 V at T = 300°K and - 1.00 V at T = 77°K. Typical saturated operating conditions at T = 300°K were ID = 2.5 mA and g,,,,, = 7.0 m mhos at Vos = - 0.70 v.

The channel length at 4.1 pm is sufficiently short so that hot electron effects begin to appear. There are two such effects involved:

(1) Due to the high fields in the channel, especially near the drain, the mobility p(E) becomes field-depen- dent and decreases with increasing field E. This lowers the drain current 1,, and the transconductance g;. Since the true drain conductance g&at zero drain bias is not altered, becauseitisalow-fieldparameter,theratiog~/gbissmaller than unity, especially for large values of (V, - VP), here g,L is the true device transconductance, V, is the gate voltage and VP the turn-on voltage of the channel.

(2) Due to the high fields in the channel the electron temperature T.(E) is larger than the lattice temperature T. The distributed noise emf in the channel has a cross- correlation spectrum [4kT,(x’)[g(x’)]-‘S(x’ - x)Aj]“’ at x, where 8(x’-x) is the Dirac delta function and g(x’) is the channel conductance for unit length at x’. As a consequence the noise is larger than the elementary theory predicts.

tsupported by AR0 contract. SMr. Kim is now with Bell Laboratories, Murray Hill, New

Jersey.

For small channel lengths the theory should be developed as a two-dimensional theory involving a field- dependent mobility. It is well known that the two- dimensional approach in itself, even if now field depen- dent mobility is involved, can already give g’,/glo< 1. We used a quasi one-dimensional theory involving g(x) and T(x); both are obtained by averaging over the channel cross section. While not fully correct, it should be reasonably accurate as long as the effects involved are relatively small and as long as the results are only used for quantitative purposes. In our devices the effects, though measurable, were still relatively small so that our approach should be acceptable.

2.DWlCE CHARACTERISTICS

Figure 1 shows the zero bias drain conductance g&= (~Z~JV~)~~=~ and the maximum transconductance g,,,., = (JZddV,,) at saturation as functions of the gate bias Vos. The operating conditions were T -300°K in both cases whereas VDs = 3 V for g,,,.,. We see that

g max -0 at VGs = - 1.54V and that gdO<gmax except near zero bias. We interpret this result as being caused by the series resistances r, and rd of the source and of the drain, probably mostly at the drain. It is easily shown[31 that if gl, and gLaX are the device parameters without series resistance effect, then the parameters g&, and g,., with series resistance effects are

That the effect shows up less strongly near zero gate bias is attributed to the fact that hot electron effects reduce the value of gl., and that this reduction is strongest when Vos - VP is largest; here VP is the pinch-off voltage of the channel.

Figure 2 shows gdo and g,,,., at 77°K as functions of VGs; for the case of g,,,., we used Vm = 3 V. We see that g,$O is much smaller than g,,,. Part of this is again caused by series resistance effeqts in gdo, whereas the remainder is attributed to carrier freeze-out in the chan- nel when V, =OV.

SSE Vol. 24. No. 5-D 429

430 S. K. KIM and A. VAN DER ZIEL

7-

2-

I

Fig. I. g,,, and g,,, vs V,, at T = 300°K. FD 110 No. 1.

Ocvlce: F 1,Oel; JFET

T-77 ‘K Vos’3V for gm

%..

x -0 2 -0.4 -06 -0.6 -I’0 - \ 2 -14 ”

V GS

Fig. 2. g,,, and g,,,, vs VGs at 7’ = 77°K. FD 110 No. 1.

The freeze-out affects g,, more than g,,, because the degree of ionization is larger at large fields (Poole- Frenkel effect[4]. On the other hand the dependence of the mobility on the field (hot electron effect) reduces

g n-lax and this reduction is most pronounced when Vos - VP is larger; that is near zero gate bias. We note that the pinch-off voltage VP at T = 77°K is about - 1.00 V.

3. NOISE MEASUREMENTS

We measured Z&J = S,Jj)/Zq as a function of frequency at different bias and temperature; here S,(j) is the spectrum of the drain current and q is the electron charge. We noticed that a limiting value of Z, was reached at frequencies above a few hundred kHz; this limiting noise is here studied in detail.

At low frequencies we measured generation-recom- bination (g-r) noise, probably due to hole-electron generation in the space charge region[5]. This well- known effect does not concern us here. In the units tested the limiting noise was reached well below 1 MHz at all bias voltages and temperatures. To be on the safe side we did all our measurements at 4 MHz.

Figure 3 shows the parameter asat, defined by

2qI,, = a,,, .4 kTg,ax or (Y,., = A A 2 kT g,,,

(2)

as a function of (V,,- VP) at Vm=3V, with the temperature 7’ as a parameter. It can be seen that at a given temperature asat increases with increasing Vos -

V,, whereas at a given value of V. - V, the value of asat increases with decreasing temperature.

We attribute this dependence of asa, upon T and VGs - VP to hot electron effects. It should be obvious that at a given T the hot electron effects become more pronounced at larger fields, i.e. at larger V,, - VP,

whereas it is also well known that hot electron effects set in at lower fields when the temperature is lowered. The hot-electron interpretation can thus qualitatively explain the data (see Section 4).

Kim et al. [ 11 also plotted

asat( T] * (T/30@ (3)

as a function of Vcs - VP and found that the curves were much closer together. For that reason they ap- proximated asa,(T) as

O,,t( T) = a,.,(300)(300/ T) (3a)

where a,,,((300) is the value of asat at T = 300°K.

16.

06

0.6 -

04.

02.

Ocwce FO I IO*1

L&’ t3v

f -4MHz

0’ 1 0 02 04 06 06 10 12 14 16 ”

vGS- vP

Fig. 3. Q, vs (V,, - VP) with temperature as a parameter (hot electron noise). FD 110 No. 1.

Hot electron noise and g-r noise in short-channel JFETs 431

Figure 4 shows that this is not a good approximation. Here a.,(T) is plotted as a function of the absolute temperature T at Vos = -0.7 V and Vm = 3 V. Also shown are two curves

o,.,(T) = cr,.*(300)(300/ TY (4)

for S = 1.0 and S = 0.42, respectively. The first one gives a rather poor fit, whereas the second gives only a good fit at the highest temperatures. A value S = 0.6 would have given a reasonable fit down to about 150°K.

It is quite clear from Fig. 4 that a new noise phenomenon sets in below 15O”K. We shall give evidence that this noise is caused by generation-recombination (g-r) noise involving donors. Figure 5 shows anpt vs V,, - VP with the device temperature T as a parameter. There is a very rapid increase in CY~.~ with decreasing temperature; (Y,.~ increases by much more than one order of magnitude.

Figure 6 shows the equivalent noise resistance R,, = asa,/gmnx as a function of elkT at Vm = 3.0 V, V,, = - 0.5 V. When correcting for a hot electron noise resis- tance of 75 0 at higher temperatures, the corrected curve R, vs e/kT becomes exponential with an activation energy of 0.061 eV. This activation energy increases when the operating point comes closer to pinch-off.

IO

Device, FD IlO*

v,,.-0 IV. V#.5’3V

f.4MHz

011 ’ ” j ” 1 j ” I, * A, * 70 90 II0 130 IS0 I70 190 210 230 250 270

Temperature in *K

Fig. 4. o&T) vs temperature “K at Vos = - 0.7 V and V, = 3 V (hot electron noise). FD 1 IO No. I.

loo Dewce FD I IOW 2 71°K

f=4Mtiz r/

v,,=3v

Fig. 5. a-,(T) vs (V os - VP) with temperature as a parameter (g-r noise). FD 110 No. 2.

Dewce: FD IlO*

v,,. t3v

I. 4MHZ

50 60 70 80 90 100 II0 120 130 140 W“)

e/&T

Fig. 6. R. vs e/kT at Vos = - 0.5 V (g-r noise). FD 110 No. 1.

For example, Fig. 7 shows R. = asat/gmDx as a function of e/kT at Vm = 3.OV and VGs = - l.OV where the device is very nearly cut off. When correcting for a hot electron noise resistance of 210 fi at higher temperatures, the corrected plot again becomes exponential with an activation energy of 0.087 eV. See, however, Table I and the discussion in Section 5.

Other units showed similar behavior. It will be shown in Section 5 that the g-r noise caused by donors should indeed have an activation energy of that order of mag- nitude and that this activation energy should depend on bias in the manner observed. The hot electron noise discussed earlier should not have such an activation energy.

Dewce FD llO#l

v,,= t3v

f = 4MHz

V,,=lV

orrected for bockground

/ E0 =O 087V

IO' 50 60 70 80 90 IO0 II0 120 130 W’)

e/&T

Fig. 7. Same but at V,, = - 1.0 V (near cut-off).

432 S. K. KIM and A. VAN DER ZIEL

Table 1. Values of E, for different conditions and devices

Device V 8s .E. from R. I?, from aSa,

(in V) (in meV) (in meV)

F 110 No. 1 - 0.5 61 52 F 1lONo. 1 - 0.7 58 52 F 1lONo. I - 1.0 87 F 110 No. 2 -0.5 74

The two remaining values of E, in the table could not be evaluated from the available data.

3 t -

------l.q,~*,ma,ot l&e*

----- - I,,h.rmo, at V&Z0

Corrected for thermal noise

Fig. 8. INI vs V, with Vo.s as a parameter at 77°K. FD 110 No. 2.

Figure 8 shows Z, vs VDs with Vos as a parameter at T = 77°K and f = 4 MHz. At low values of Vos the noise corresponds to thermal noise of the drain conductance g,,. At higher values of VDs the g-r noise sets in and it varies as V&. Finally the value of C, saturates when the device characteristic saturates. We shall see that in g-r noise involving donors I., varies as V’,, as long as V,, is not too large.

4. DlSClJSSlON OF THE HOT ELECTRON EFFECTS

In this case we have to take the effects of the series resistances r, and rd on the source and drain side of the channel into account. To simplify matters it will be assumed that in the noise measurements the device is saturated, this eliminates the effeCt Of rd [3].

We first consider the device without series resistances and denote the device parameters by primes. We then have for the spectrum of the drain current fluctuations[6]

&cf) = rL’4kTgbo = Q:,, * 4mL (5)

where gh,, and g&, have the same meaning as before. The parameter y:.* is defined as

Here T is room temperature, T,(> T) the electron tem- perature, g(V,,) the channel conductance for unit length

at the position x in the channel where the potential is V,, g(0) the channel conductance for unit length at V,ss =0 and VDso the drain voltage at pinch-off. If hot electron effects are present the mobility p is field dependent and this affects g( Vo). The drain current ID is

Id =; vmg(V&dV,, (5b)

so that g:o =g(O)/L, and gk,, is affected by the field dependent mobility. It may then be shown that ga,,< gd0.

According to (5)

, asat = Y:at&do/gLx (6)

so that a:., > ~1.~. All these parameters can in principle bt calculated. If r, = 7 (no hot electron effects) then Y,,~ is about 2/3 and g,,, = gLo. When T, > T, ~1.~ can be larger and gLaX < gio, so that a:,, is still larger. Since T,(E) increases with increasing field strength E and the mobility Jo decreases with increasing field strength, (Y:.~ at a given temperature T will increase with ( VGs - VP). Also, since at lower temperatures hot electron effects set in at lower fields, a:., at a given ( VGs - VP) will increase with decreasing temperature. This can therefore qualita- tively explain Fig. 3. There should be no reason, however, why eqn (3a) should be valid.

The ratio gLo/gL,, is > 1, not only because of hot electron effects, but two-dimensional effects may also play a part. Since both effects are evaluated simul- taneously, it is difficult to separate them. For qualitative discussions it is sufficient to know that gb,/g&,, > 1 and that this parameter increases with increasing V, - VP and with decreasing T.

If one now takes series resistance effects into account, and one writes at saturation for the spectrum of the short-circuited drain current [3]

s,dU) = &at ~4mnax (7)

where g,,, is given by (I), then it is found that

If gL,,r, is relatively small, a,,* = (~1,~ so that the above qualitative discussion remains valid.

5.DJBCUSSION OFTHE GENERATION-RECOMBINATION

NOIBE DDE TO DONORSIQ]

We now turn to the generation-recombination noise due to donors[4]. If no electric field is present in the channel, the activation energy of the donors is E. (-44 mV), but if a large field is present the activation energy becomes E. - A V, where

AV = (qE)“*/(m,c,)“* = 7.58 x 10-5(E/#2 (9)

(Poole-Frenkel effect). Here E is the field strength in

Hot electron noise and g-r noise in short-channel JFETs 433

V/m, eS the relative dielectric constant of the semicon- ductor and c0 the dielectric constant of free space.

The equations governing the generation-recombination processes between electrons and donors are

lo-” set and that the measurements are performed at 4MH2, the term Wan* may be neglected. The total noise is therefore

g(n) = l9(Nd -n), r(n) = pn* (10)

where Nd is the donor and n the electron concentration. Here 0 and p are constants and

B = $ exp (eA V/kT) (lOa)

where &, is the value of 0 at low fields. If we now denote the equilibrium concentration at the

field E by no and introduce the parameter u = JNa, which has the value u. at E = 0, and apply the equili- brium condition

e( Nd - no) = pno* (11)

then we obtain the relations

(114

Since u. can be evaluated as a function of the tem- perature T, we can evaluate u as a function of the field strength E and of the temperature T.

Furthermore the time constant T of the generation- recombination process is given by[6]

i=(-g+g),=eo(2+)exp($!) (12)

where the value 7. of T for low fields is given by

1ho = eo(2 - u~)/u~. (124

If AN is the number of carriers in a section Ax and SAN its fluctuation, then the parameter p is defined by

It is easily shown that

p = (1 - u)/(2 - u) (13a)

which has the value

Bo = (1 - uoN2 - uo) (13b)

at low fields. /3 and T are functions of the field E and decrease with increasing field. Also, the mobility p(E) of the carriers decreases with increasing field.

In this notation the contribution of the section Ax to the output noise spectrum 5&(f) is[6]

$,t_f, = y w A Vo. (14)

If we now bear in mind that r(E) is of the order of

= y V,,p(E)B(EME). (15)

This result is valid as long as the device is not saturated ( VD 5 Vmo). The weighted average in the second half of (IS) is taken over the channel length. The noise resis- tance at saturation is thus found to be

R = S”(l) n qzDvDs p(E)P(E)@) (16) 4kTg;., = m

so that #)P(E)T(E) can be evaluated from the measurements.

For an accurate comparison between theory and experiment one should evaluate the integral in (15) numerically. However, since this is difficult, we give instead a somewhat rougher estimate.

The reason why R. has an activation energy is that p(E)@) has an activation energy that is about 2Eo at low fields, and locally 2(Eo- AV) at high fields. Hence &should have an activation energy 2(Eo-TV), where AV is a properly weighted average. In Fig. 6 we found an activation energy of 61 mV. This yields hV= 13.4mV, which corresponds to a weighted field of 3000- 4000 V/cm. This is a very reasonable value for our short- channel devices.

If the device is nearly cut off, as in Fig. 7,the field in the channel is much smaller and hence AV is cor- respondingly smaller. This explains why the measured activation energy of R. is so close to the limiting value 2Eo = 88 mV in Fig. 7.

Table 1 shows the activation energies for two devices at different gate bias VGs and evaluated from different noise parameters. Some values of E, were measured from the plot R. vs e/kT, whereas others were measured from the plot a,,, vs e/kT. There is a considerable spread in the data, indicating that our measurements of E,, are not very accurate.

There are several reasons for this: (1) R, and asat contain different parameters that are

functions of T. The dependence of R, and aoac should therefore not be accurately exponential in e/kT. Since these parameters were not taken into account here, and are difficult to take into account properly, the values of E, from R. and from asat need not fully agree.

(2) We corrected for hot electron noise by subtracting an appropriate constant value of R, or of asat from the data. The hot electron noise is a function of temperature, however, and for that reason there is a certain un- certainty in the slope of In R, or In asat as a function of elkT.

(3) VP depends somewhat on T and this was difficult to take into account properly. This effect is most pronounced when the channel is near cut-off (at Vos =

434 S. K. KIM and A. VAN DER ZIEL

- 1.0 V) and should result in a value of E. that is somewhat too large.

(4) According to the above theory the activation energy-of the donors is E,,- A V locally, and hence E,, - A V for a weighted average over the channel. Kim et a1.[2] neglected the Poole-Frenkel effect and found that in that case /3r had an activation energy of 2E, for relatively high temperatures and E. for temperatures far below 77°K. If the Poole-Frenkel effect is taken into account the activation energy of R. or (Y should be c(EO-rn, where e = 2 for sufficiently large tem- peratures and E = 1 for very low temperatures. In the above discussion we took e = 2; this might be a reason- able approximation as long as the plot In R, vs e/kT is reasonably close to a straight line.

At low values of Vm, p(E)j3(EME)=p&c, is in- dependent of VDs, where po, PO, r. are the low-field values. In that case S,,(f) varies as LV,, or as V’,,. This explains the results of Fig. 8. For larger values of

Vm the integral in (15) should be evaluated and the result compared with experiment. The two remaining values of E. in the table could not be evaluated from the available data.

Acknowledgement-The authors are indebted to Dr. C. F. Hiatt for supplying the short-channel JFETs.

REFERENCES

1. S. K. Kim, A. van der Ziel and L. M. Rucker, So/id-St. Electron. 21, 1259 (1978).

2. S. K. Kim, A. van der Ziel and L. M. Rucker, So/id-St. Electron. 21, 1099 (1978).

3. W. C. Bruncke, Proc. IEEE 51,378 (1963). 4. A. van der Ziel, R. Jindal, S. K. Kim, H. S. Park and J. P.

Nougier, So/id-St. Electron. 22, 177 (1979). 5. C. T. Sah, Proc. IEEE 52,795 (1964). 6. A. van der Ziel, Noise Sources, Characterization. Measure-

ment. Prentice Hall, New Jersey (1970).