An analysis and derivation of the noise figure of a tunnel-diode microwave amplifier are presented. The agreement between the measured noise figure and the theoretical results is an indirect check on the existence of full shot noise in germanium tunnel diodes at microwave frequencies. The limiting noise temperature of the amplifier is eI0R/2k, and can be approached by using diodes with small (RC) products in which the extreme overcoupling (load mismatch) and high gain can be achieved simultaneously

Cook, J. S.

Yariv, A.

English

Caltech Authors - Main

PROCEEDINGS OF THE IRE
A Noise Investigation of Tunnel-Diode
Microwave Amplifiers*
A. YARIVt, MEMBER, IRE, AND J. S. COOKt, MEMBER, IRE
Summary-An analysis and derivation of the noise figure of a
tunnel-diode microwave amplifier are presented. The agreement be-
tween the measured noise figure and the theoretical results is an in-
direct check on the existence of full shot noise in germanium tunnel
diodes at microwave frequencies. The limiting noise temperature of
the amplifier is elIR/2k, and can be approached by using diodes with
small (RC) products in which the extreme overcoupling (load mis-
match) and high gain can be achieved simultaneously.
INTRODUCTION
A MPLIFIERS utilizing tunniel diodes were first re-
ported by Chang' at frequencies up to 80 Mc.
He also suggested that the noise conitribution of
the diode was due to the shot noise accompanying the
tunneling process. This suggestion was con1firmed by
Lee and MAIonitgomery,2 who showed by measurements
at 5 Mc that full shot noise was inideed genierated by the
tunnel diode. Tiemaini3 has reported results similar to
those of Lee and Mllontgomery, obtained from noise
measurements at 500 kc on germaniium diodes.
Approximate noise analyses of tunniel-diode ampli-
fiers were given by Somnmlers, et al.,4 Tiemann, and
Hines and Anderson.5 The simiplified equivalent circuits
used were sufficient to derive the limiting nioise behavior
for very high gain and extremle overcoupling. Yariv,
et al.,6 reported the operationi of a tunnel-diode amplifier
at microwave frequencies anid measured its noise figure.
In this paper, the details are giveni of the noise analy-
sis which led to the noise temperature formula stated by
Yariv, et al.6 This anialysis is based on an "exact" equiva-
lent circuit, and its results are useful in the intermediate
range of small or medium gain as well as in the limiting
case described above. The theoretical results are com-
pared with the nioise measuremiieints.
I. THE PHYSICAL SOURCES OF NOISE
In additioni to the omnipresent noise generated by the
ohmic losses in the spreading and contact resistance of
the diode and the surrouniding mlicrowave structure, we
* Received by the IRE, October 3, 1960.
t Bell Telephone Labs., Inc., Murray Hill, N. J.
I K. K. N. Chang, "Low-noise tunnel-diode amplifier," PROC.
IRE (Correspondence), vol. 47, pp. 1268-1269; July, 1959.
2 C. A. Lee and H. C. Monitgomlery, "Determination of forward
and reverse tunneling cuirrents in Esaki diodes by shot noise measure-
ments," Bull. Amn. Phys. Soc., vol. 3, p. 160; March 21, 1960.
3 J. J. Tieman, "Shot noise in tulnnlel diode amplifiers," PROC. IRE,
vol. 48, pp. 1418-1423; Augtust, 1960.
4 H. S. Sommners, Jr., et al., "Tuinnel diodes for low noise amplifi-
cation," 1959 IRE WESCON CONVENTION RECORD, pt. 3, pp. 3-8.
" M. E. Hines and W. W. Anderson, "Noise performance theory
of Esaki (tunnel) diode amplifiers," PROC. IRE (Correspondence),
vol. 48, p. 789; April, 1960.
6 A. Yariv, et al., "Operation of an Esaki diode microwave ampli-
fier," PROC. IRE (Correspondence), vol. 48, p. 1155; June, 1960.
must consider the shot noise accompaniying the tunniel-
ing process.7 There are two separate and independenit
tunineling currents, one consisting of electrons tunnelinlg
from the N to the P side of the junction, which we de-
inote by ii, and a second curreit, i2, made up of electrons
crossing the junctioni in the opposite directioni. The niet
measurable dc current is Io-= ii -| i2 . Since the cur-
renits ii and i2 are uncorrelated, the total shot noisel is
represented by a current generator of mean-square
amplitude,
j2 = 2e( 1iI + i2 I )B, (1)
where e is the electronic charge and B is the bandwidth
in cycles per seconid in which the noise is conisidered.
When the diode is biased so as to operate in the niegative
resistance region (see Fig. 1), the currenit i2 is negligibly
small and the approximation
Io I ill + I i
nmay be used.
Fig. 1-A typical voltage-current characteristic
curve of an Esaki diode.
II. THE EQUIVALENT CIRCUIT
The amplifier constructed for the noise experiment
consists of a germanium diode mounited in a sinigle port
cavity which is coupled to the load anid the source by a
low-loss circulator as shown in Fig. 2, the coupling being
conitinuously variable. The equivalent circuit of this
anmplifier is shown in Fig. 3. The diode is represented by
a negative resistance -R, shunted by a capacitance C,
7 L. Esaki, "New phenomenon in narrow germanian p-n junc-
tions," Phys. Rev., vol. 109, pp. 602-603; June, 1958.
8 See, for instance, J. R. Pierce, "Physical sources of noise,"
PROC. IRE, vol. 44, pp. 601-608; May, 1956.
1961 739
PROCEEDINGS OF TIHE IREJ
f - 4500MC
SOURCE
CAVITY -
Fig. 2-A schematic view of the Esaki diode amplifier.
)VN2:=4K TCRCB
,>RC >RL
2eR, s
R~ ~~~~~c
Fig. 3-The equivalent circuit of the Esaki diode amplifier.
which accounts for the barrier capacitance. The spread-
inig and contact resistance of the diode are lumped
together with the ohmic losses of the cavity in R,. The
noise generated by the three sources of loss listed above
is provided by a Johnson noise generator of meani-square
voltage amplitude,
Fig. 4-The series equivalent circuit of the Esaki diode amplifier.
Series resonance occurs at
(2)Wres -OIo/1 - --1W02R2C2
where wo2= 1/LC. For resonianice to occur at real fre-
queincies, we must fulfill the condition
cooRC > I (3)
for w02R2C2>>A, wres Woo.
A necessary aind sufficienit conditioni for positive gain
is that the real part of the total series resistanice, seeni
lookinig back into the amplifier terminals, be negative;
or from Fig. 4,
R
> Re.1 + W2&R2C2 (4)
The highest frequency at which this canl take place is
called the cutoff frequency, JC,:
V2= 4kTCRCB,
where k is the Boltzmiann constant and Te is the ambi-
ent temperature of the cavity. L stands for the total
inductance (parasitic and inteintional). The shot noise
genierator
i2= 2eJoB
was discussed above. The load resistance is RL.
For the purpose of analysis, it is more convenient to
transform the circuit to one in which all the elements are
either in series or in parallel. The series equivalent cir-
cuit and the new expressions for the equivalent series
circuit elemenets are showni in Fig. 4.
N/R/Rc- 1
fco .= R2r,RC (5)
an-d represenits the highest useful frequetncy for a given
diode and circuit.
Sinice our main initerest is in ani amplifier operatinig at
reson-iance, we evaluate the elemenits of the equivalent
circuit of Fig. 4 for the coniditioin co rs The result is
shown in Fig. 5. The effective niegative resistance is now
represenited by the series resistanice
-R
R - R(6)W02R2C2
740 . I pril
Fariv and Cook: A Noise Investigzation of Tunnel-Diode Microwave Amplifiers
V22 : 4k TC RC B
zR2 2
CO)~ R C
RL
-. 2ei0BR2
v2y= -2 22RVNi~(A) R C2
Fig. 5-The series equivalent circuit of the Esaki
diode amplilier at resonianice.
while the shot noise is niow provided by a voltage gen-
erator of meall-squLiare amplitude
2eIoBR2
VN12 = 0RCwo2RC2C
The physical importance attached to the elements of
the equivalent circuit of Fig. 5 is of paramount im-
portance and contains a number of pitfalls, so that a few
pertinent remarks are in order. Since the resistanices are
all in series, they are proportional to power, with the
result that onie may wronigly inifer, for instanice, that
the power generated by the diode is proportionial to 1
and, therefore, that for large gain or large power output,
diodes with large R are superior. In actuality, the geni-
erated power is proportioiial to /IR, xvhich for a given
diode material is proportionial to the junictioIn area.
Physical significanice caii only be attached to ratios of
aiiy pair of resistaInces in Fig. 5. The quanitity R'/IRL is
equal to the ratio of power genierated by the diode to
that delivered to the load, wvhile RL/Rc, for inistanice,
gives the ratio of the power consumed by the load to
that expenided in the cavity. RL/RC is ofteni referred to
as the couplinig ratio of the cavity. (Ratio of cavity
"Q" to external Q"'.)
III. THE GAIN-BANDWIDTH PRODUCT
The power gaini G of the amplifier is definied as the
ratio of the power delivered to the load (the reflected
power in our case) to the power available fromn the
source (inicidenit) anid is thus equal to the squared mag-
niitude of the reflectioni coefficient measured at the
amplifier terminials. Usinig Fig. 5, we have
FRL - (Ra- R' ) 2G = (C(7)
LRL + (R, - R')
The bandwidth Af is the distance, in cps, betweeni the
half-power frequenicies. It is derived by replacing R, - R'
in (7) by the total series impedanice away fromn resonanice
as given by Fig. 4. The result is
RL-+RC-R'
Af -
2irL I1\iL( W02R2C2)
yielding for the voltage gain-baiidwidth product
(A/Grs\f->/Gf _ RL + R- RC ,(VG/Z)res,Af v'G/;AJ
2irL - )
\ o2R2C2J
(8)
Eq. (8) takes a meaninlgful formii upoIn substitution of
the high gain condition9
RI ; RG + RL (9)
and the definiition of R' as giveln by (6). The result is
(10)(\GAf)high gain = 1 -
irRC I---RC
CO'"2RC2/
in which form it is ameniable to experimiien-tal veri-
ficationi.
To increase the gain-bandwidth product, we have to
use a diode with a small (RC). This happenis, niot onily
through the dependenice of V/GA/ on (RC) as giveni by
(10), but also through a less obvious decrease of R,/1'.
If we assume that the total iniductanice has been made
as smiiall as possible, operation at one frequenicy meanis
usinig the same C. If one niow goes to a diode imiaterial
with an initrinsically smaller (RC) product, it becomes
possible to operate at the sanie frequenicy with a diode
having a smaller R,10 i.e., a diode generatinig more
power, which corresponids to a smaller R,CR' ratio.
IV. THE NOISE TEMPERATURE OF THE AM.PLIFIER
The nioise temperature Tie of the amplifier," wlhich is
somnetimes called the effective iniput nloise tem11perature,
cani be definied as the inicrease in the source temiiperature
which is required to keep the noise power output a coni-
stant if the amplifier, hypothetically, were renidered
noiseless. It is fouind by equatinig that part of the total
noise output originiatinig withini the amlplifier, VA, to
kTeGB, i.e.,
NA = kTeGB,
' This conidition, when taken with an equality sign, makes the
gain infinite Isee (7)] alnd becomes the "start-oscillation" conditioln.10 The niew diode will have the samiie area as the old onie, but will
possess a larger current denisity anid, conisequently, a larger currelnt.Te is related to the noise figure F by
F= 1+ 290
1961 741
PROCEEDINGS OF THE IRE
or using the equivalent circuit of Fig. 5 and the gain
definitioII of (7),
kTeB (R - R. + R')2
(RL + R, - R')2
R+RL
4kTB(RL + R, - R)
2eIOBRP'RL
(RL + RC- R')2
where T, is the temperature of the cavity and diode.
Solving for 7'e yields
4T,R,RL 2eIoRR'RL
(RL - R, + R')2 k(RL - RC + R')2
which can be transformiied to
G I RL) 2k (RL)] (1
in which form only ratios of resistance appear. This form
is particularly useful, since in it the dependenice of the
noise temperature on the gain is separated from its de-
pendenice on the coupling ratios. This bracketing is
justified operationally, since the same value of gain can
be obtained with different combinationls of coupling
ratios.
V. EXP'ERIMENTAL RESULTS
The pertinent characteristics of the phosphorus-
(loped germlaniiumii diode were:
Ipeak = 1 ma,
IoR = 8.6 X 10-2 at the biasing point,
RC 3.5 X 10-10 + 10 per cent,
R = 175 ohms.
The diode was mounited between the ridge and top
wall of a short-termiiniated waveguide, the distance be-
tween the diode and the short being adjustable for
tuninlg purposes. The ridge waveguide tapered gradually
into a 2-inch X 1-inch input waveguide which was
coupled to the input and output via a low-loss circulator.
'l'he diode coupling was controlled by two screws placed,
respectively, 8 anid 8 guide wavelengths in front of the
diode. Fig. 6 shows a photograph of the ridge waveguide
cross section at the diode position, while Fig. 7 shows,
with considerable magnificationi, the immediate area
surroundinig the diode.
The measured gain-banidwidth product at 4500 Mc
was 3.6 X 108, and was checked at a numiiber of gain
settinlgs betweenl 17 db and 25 db, where it was found to
be a constanit, in agreemenit with (10). The gain-
bandwidth product deteriorated for lower gain set-
tilngs.
The measured noise temperature was 1740°K and was
taken with the amplifier set for a gain of 25 db. To
compare this result with that predictable from (8), it is
necessary to know the power ratios RC/RL and R'IRL.
This was accomplished by usinig the measured values of
Fig. 6-Cross section of the ridge waveguide at
the diode position.
Fig. 7-An enlarged view of the diode anid its surroundinigs. The
diode is motunted between the two pins at the center of the photo-
graph.
V\GAf and (RC) to solve for R,IR in (10), and then
by using the high gain conditioni (9) to solve for the re-
quired ratios. With the numiierical data given above, the
ratios are:
R 0
-= 0.6,Rt
R'
-= 2.5,
RL
RC
-= 1.5.
RL
Substitution in (11) yields
(Te)calcuIated = 18900K + 10 per cent,
which is to be compared with the experimenital value of
1740°K. The main source of error is the uncertainty in
(RC). The satisfactory agreemenit between the experi-
mental and theoretical results serves as an indirect
check on the existence of full shot noise at microwave
frequencies in tunniel diodes. A more recent version of
the same amplifier yielded a noise temperature of
12000K.
VI. CONCLUDING REMARKS
A number of factors which are evident fromn (11)
affect the noise temperature. We shall treat them sepa-
rately.
742 A pril
Yariv and Cook: A Noise Investigation of Tunnel-Diode Microwave Amplifiers
The depenidenice of Te on the gain G is contained in
the factor (\'G+ 1)2/G, which decreases asymptotically
toward unity with increasing gain so that high gain
operation is imperative.
The shot noise contributioii enters through the term
IoR. It can be minimized by optimum biasing of the
diode in such a maniner that the product IoR is a minii-
mum. In practice, it was found impractical to operate
the diode for anly considerable distance beyond the mid-
range of the negative resistance, because the increasinig
value of R, due to the curvature of the V-I curve, more
thani offset the decrease in io. The problem of the exact
shape of the V-I curve is initimately related to that of the
excess current'2'3 anid has niot been settled yet.
The (IoR) product is il)herenltly smaller for semli-
conductors with small eniergy gaps (InlSb-0.17 v, InAs-
0.33 v, Ge-0.72 v) which shouldl therefore be preferred
for low noise applicationis. This advantage is offset to
sonme extent by the fact that, assuminilg the same (RC)
product, the high-energy-gap diodes genierate more
power and thus imiake it easier to overcouple the cavity
anid still achieve high gain.
T'he shot noise conitributioni to the nioise temperature
is proportionial to R'/RL, which, accordinlg to (9), is
always larger thani unity. Inl the limit wheni the cavity
losses are very si)]all comparedl to the genierated power
anid the load power, i.e., wNhenl R,1R'; JR,RL<<1, the
ratio R'l L approaches unity. As pointed out at the
conclusion of Sectioni III, this situationi is approached,
for a diode operatinig at a fixed frequency, by usinlg
diodes with smiiall values of (RC). The shot nioise coni-
tributioii to the amplifier's nioise temuperature is tlhus
always larger, but hopefully niot much larger, than
eIoR/2k. Iin ouir amplifier this was -'5000K.
The same coniditionis that minimliize the shot nioise
contributioni also cause the ratio R,/RL to be a minii-
mum, thus miniimizinig the conitributioin of the cavity
losses at T, [see (II)].
"2R. A. Logan anid A. G. Chynioweth, "Effect of radiation damage
on excess current in Esaki diodes," Bull. Am. Phys. Soc., vol. 5,
series II, p. 375; Jitne 15, 1960.
13 T. Yajima and L. Esaki, "Excess noise in narrow germaniuin
p-n jninctions," J. Phys. Soc. Japan, vol. 13, pp. 1281-1287; Noveni-
ber, 1958.
The reduction of (RC) has a beneficial effect on both
the gain-bandwidth product and the nioise temperature
of the amplifier; it also creates a number of problems.
The first is one of stability. For stable operationi we
must satisfy the conidition'4
L(w) < R8(o()RC
at all frequenicies. Where L(Z) and R8(w) are, respec-
tively, the total inductance and resistance in series with
the diode, they are shown as functions of freqluenicy
sinice stable operationi has to obtain for all frequenicies.
The difficulty of satisfying the stability conditioni in-
creases with smaller (RC) product.
A smnall value of RC makes R smlaller, at a given
frequenicy, and enitails usinlg a mnicrowave struicture with
low characteristic impedanices. In the diode used in our
experimenit, R-175 ohms and RC-3.5X10-0. Had we
used inistead a uniit with RC'-1011, which is available,
the niegative resistance R would have been -5 ohmiis,
which is a very low impedanice level ini microwave cir-
cuitry anid creates a gamut of problemiis, such as miiatch-
ing to standard componients with miiuch higher im-
pedanices and the high attenuationi of low-impedanice
waveguides.
It is Inot likely that tuinnel diode will ever threateni
the mlaser anid paramiietric amplifiers as a low-noise
amplifier. It imay still finld applicationi in cases where its
extreme simplicity anid econiomiiy may be traded for in-
creased noise, especially wheni tunniiel-diode amnplifiers
with nloise figures substanitially lower thani the oine re-
ported above will be made. This improvemiienit is imiainily
conitingeint on1 the utilization aiid taminig of diodes with
small (RC) products anid oni the use of low-energy-gap
(lio(les-with smiiall (IoR) products.
ACKNOWLEDGMENT
The authors wish to express their inidebtedniess to
P. E. Butzieni for his able participationi inl the experi-
menital work, anid to E. Dickteni for fabricating anid
mounitinig the diodes used inl the experimenit.
14 R. L. Wallace, unipuiblished report.
7431961


A Noise Investigation of Tunnel-Diode Microwave Amplifiers

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A Noise Investigation of Tunnel-Diode Microwave Amplifiers

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