NOISE IN THE TUNNEL DIODE
BY
BARRY EARL TURNER
B.Sc., University of British Columbia, 1959
A THESIS SUBMITTED IN PARTIAL FULFILMENT OF
THE REQUIREMENTS FOR THE DEGREE OF
MASTER OF SCIENCE
In the Department
of.
PHYSICS
We accept this thesis as conforming
to the required standard
THE UNIVERSITY OF BRITISH COLUMBIA
July, 1962 In presenting this thesis in partial fulfilment of the requirements for an advanced degree at the University of
British Columbia, I agree that the Library shall make it freely available for reference and study. I further agree that permission for extensive copying of this thesis for scholarly purposes may be granted by the Head of my Department or by his representatives.
It is understood that copying or publication of this thesis for financial gain shall not be allowed without my written permission.
Department of Phy3ics
.The University of British Columbia, Vancouver 8, Canada. .
Date Augu3t 3, 1962 ii
ABSTRACT
To date, measurements of tunnel diode noise have dealt mainly with the negative conductance region, because the tunnel diode is an active circuit element only In this region. The noise has not been measured for reverse or near-forward biases due to the diffi• culties involving excessively low diode impedances in these regions.
The purpose of this thesis is to show that, from the Esaki formu• lation for the direct-tunneling currents of a tunnel diode, In the bias regions where the electronic bands overlap, a simple theory can be developed relating the power spectrum associated with the direct-tunneling current noise to the direct current passing through the diode. This theory assumes that the two oppositely- flowing direct-tunneling currents in the Esaki junction are un• corrected and that both contribute full shot noise. The theory can be critically tested only in the bias regions where the noise
Is yet unstudied, and at sufficiently high frequencies that no contaminating l/f noise exists. These conditions have been met experimentally and the noise measured quantitatively over the entire reverse and near-forward regions at a frequency of \\ Mc/s.
Impedance-transforming networks and a very low-noise preamplifier suitable to the particular source strengths and impedances pre• sented by the tunnel diode are developed for these measurements.
A noise measurement technique is chosen from among several possible ones for the high degree of accuracy and smallest dependence on a good noise figure required for the tunnel diode source. The experimental results agree with the theory and vindicate the usual assumption that the two oppositely flowing direct-tunneling electron iii currents between two bands of a degenerately-doped semiconductor are uncorrelated.
Noise measurements In the "valley" and far-forward region of the tunnel diode characteristic, where the diode current is not due to direct tunneling, do not agree with the simple two-current shot noise theory for direct-tunneling electron currents. Possible reasons for the enhanced noise measured In this region are advanced in the form of two models based on indirect-tunneling electrons via traps as the most important mechanism describing the excess or valley current. These models offer a possible explanation of the observed phenomena, but noise measurements alone appear insuffi• cient to demonstrate unambiguously the detailed mechanisms pro• ducing either the excess current or the associated enhanced noise found throughout the valley and far-forward regions. .ACKNOWLEDGMENT
I should like to thank Professor R. E. Burgess, my thesis director, for his supervision In preparing the material in this thesis.
The research was financed by the National
Research Council of Canada in the form of a Student• ship and Summer Supplement, and by the United States
Air Force Grant AFOSR 65-02l|0. iv
CONTENTS
page
CHAPTER 1. INTRODUCTION 1
1.1 Statement of the Problem 1
1.2 Summary of the Theory of Tunneling 2
1.3 Survey of the Literature 8
l.q Scope of Thesis 10
CHAPTER 2. THEORY OF TUNNEL DIODE NOISE 12
2.1 Noise Model of the Tunnel Diode 12
2.2 Noise Spectrum for Direct Tunneling Currents ll\
of Esaki
2.3 Restrictions on Esaki's Tunneling Theory 16
2»l\ Models for Noise Associated with Indirect 23
Tunneling Processes
2.1*1 Modulation in the Indirect-Tunneling 3$
Model for Valley Noise
CHAPTER 3. APPARATUS AND EXPERIMENTAL TECHNIQUES i+0
3.1 Basic Concepts and Requirements of Noise qO
Measurements
3.11 Theory and Requirements for "Low-noise" qO
Circuits
3.12 Methods of Comparison With a Standard 1+2
Noise Source
3.2 Impedance Transformations Suitable for a 4.9
Tunnel Diode Source
3.3 Development of a Low-noise Amplifier $3 V
3.31 Amplification and Noise of a Cascode 5*4
Amplifier
3.32 Cascode Circuit Designs Favoring Stability 58
3.33 Performance of the Cascode 63
3.I4 Other Apparatus and Circuitry 67
3.141 Perspective of the Overall Circuit 67
3.142 Noise Diode and Tunnel Diode Bias and 68
R.F. Circuits
3.143 Noise Diode Filament Current Supply 70
3.kk Detection of Noise Signals 72
3.5 Adopted Noise Measurement Procedure 7k
CHAPTER 1*. EXPERIMENTAL RESULTS AND INTERPRETATION 80
I4.I Reverse and Near-forward Bias Regions 80
k»2 Valley and Far-forward Bias Regions 89
CHAPTER 5. CONCLUSIONS AND OUTSTANDING PROBLEMS 93
5.1 Near-forward and Reverse Bias Regions 93
5.2 Valley and Far-forward Bias Regions 95
BIBLIOGRAPHY ' 97 vi
ILLUSTRATIONS
Figure Facing Page
1.1 Typical I - V Characteristic for a Tunnel Diode 2
1.2 Energy-band Diagram for Lightly-doped Semiconductor 3
1.3 Energy-band Diagram for Degenerately-doped Semi- 3
conductor
l,k Tunnel Diode Junction Energy-diagram at Zero Bias Ij
1.5 Tunnel Diode Junction Energy-diagram for Forward Bias i*
1.6 Tunnel Diode Junction Energy-diagram for Reverse Bias l\
1.7 Mechanisms for Indirect Tunneling in the Far-forward 7
Bias Region
2.1 Noise-equivalent Circuit for a Shot Noise Device 12
2.2 Noise-equivalent Circuit for a Tunnel Diode 13
2.3 Behavior of ^§ with Bias Voltage 15
2.1* Detailed Mechanisms Involved in Indirect Tunneling 2l|
2.5 Simplified Model- for Noise Analysis of Indirect 26
Tunneling Processes
2.6 Current Associated with "Event A" 27
2.7 Current Associated with "Event B" 29
2.8 Possible Noise Spectra for Indirect Tunneling 32
Processes
2.9 Charge Distribution Within a Tunnel Diode Junction 35
2.10 Modulation of Energy-band Diagram by Trap-involved 36
Indirect Tunneling
3.1 Schematic Circuit for Direct Measurement of a 1*0
Noise Source vii
3.2 Simplified Schematic Circuit for Direct 43
Measurement of a Noise Source
3.3 Schematic Circuit for Comparison of Unknown 44
and Calibrated Noise Sources
3.4 Schematic Noise Circuit for Attenuator and Two 46
Standard Noise Sources
3.5 Schematic Circuit for a Transformed Source 49
Coupled into a Noisy Amplifier
3.6 Noise-equivalent Circuits for a Parallel-tuned 50
Circuit
3.7 Autotransformation for Tunnel Diode Source 5l
3.8 Series-tuned Circuit Transformation for Tunnel 5l
Diode Source
3.9 Comparison of Noise Figures for Series- and Parallel- 52
tuned Circuits With Tunnel Diode Source
3.10 A.C.-equivalent Circuits of a Cascode Amplifier 5q
3.11 /.Noise-equivalent Circuits of a Cascode Amplifier 56
3.12 Typical A.C.-coupled Cascode Amplifier 59
3.13 Simplest Direct-coupled Cascode Amplifier 60
3.1i| Practical Direct-coupled Cascode Circuit With 62
Optional Cathode-follower Stage and Attenuator
3.15 - Schematic Noise Circuit for Measuring Rfl of an 64
Amplifier
3.16 Block Diagram of Complete Noise Measuring Circuit 67
3.17 Noise Diode and Tunnel Diode Bias and R.F. Circuits 68
3.18 Noise Diode Filament Current Control Circuit 71 vlil
Figure Facing Page
3.19 Circuit for Integrating Noise Signals 73
3.20 Complete Noise-equivalent Circuit for Tunnel 7k
Diode Noise Measurement
Ij.l I - V Characteristic of Sony Esaki Diode in the 80
Near-forward and Reverse Bias Regions
If.2 Theoretical and Experimental Comparison of Tunnel 85
Diode Noise in the Near-forward Bias Region k»3 Theoretical and Experimental Comparison of Tunnel 86
Diode Noise in the Reverse Bias Region
I4.4 Data for Valley and Far-forward Bias Region 89
1|.5 Dependence of Indirect Tunneling Processes on Bias 91 CHAPTER 1
INTRODUCTION
1.1 Statement of the Problem
A\ quantitative study of the noise associated with charge
transport processes In solids very often gives detailed inform•
ation about these processes which is otherwise difficult to
obtain. The process this thesis studies is quantum-mechanical
interband tunneling in degenerately-doped semiconductors, upon which the tunnel diode owes its active properties. The noise
spectrum of the diode tunneling current should be related in
principle to Esaki's theory of direct tunneling currents (form• ulated explicitly for interband tunneling in heavily-doped semi•
conductors, but applicable in broad form to tunneling in super•
conducting systems also.) Certain assumptions are unavoidable
in relating the noise spectrum to tunneling theory. In testing
these, the frequency of measurement of the spectrum must be
sufficiently high to avoid l/f noise, which cannot be related to
direct tunneling theory. The magnitude of the noise spectrum at
a single frequency is then most critically related to Esaki's
theory, in terms of measured quantities, in the near-forward and reverse bias regions of the tunnel diode I-V characteristic.
Here the diode noise signal is technically difficult to measure
due to the very low Impedance of the diode In these regions.
In the valley region of the diode characteristic, the
conduction current Is due mainly to indirect tunneling mechan•
isms which depend on energy band profiles and impurity state distributions within the forbidden gap. The noise spectrum FIGURE 1.1 TYPICAL I - V CHARACTERISTIC FOR A TUNNEL DIODE must be related to these properties in order that information on
the structure of the junction energy diagram can be obtained
through noise measurement.
The purpose of this thesis is to measure the noise spectrum
over the entire positive conductance part,of the tunnel diode
characteristic and to interpret the results in terras of Esaki's
tunneling theory where applicable.
1.2 Summary of the Theory of Tunneling
In studying diodes made from very heavily doped germanium,
Esaki (19^8) discovered negative resistance in them in the for•
ward bias direction. This he correctly interpreted as due to
interband tunneling, the properties and consequences of which we
now describe in terms of a typical I-V characteristic for a
tunnel diode, as in Figure 1.1.
Figure 1.2 shows the energy band diagram and density-of-
states profile for a modestly doped n-type semiconductor.
Shallow impurity levels (donors) are shown just below the con•
duction band, which is virtually empty so that the fermi level
lies only slightly above the middle of the forbidden gap. The
only allowed electron states in the interband gap are the impur•
ity levels, which are localized spatially.
Figure 1.3 shows the situation for a degenerately doped
semiconductor, from which tunnel diodes are made. The donor
concentration is so high that although only a small fraction of
the impurity levels at any given energy are ionized at normal
temperatures, the total number of ionized impurity sites is suffi
cient to cause the conduction band to be occupied by electrons
over an appreciable range of energies. This drives the fermi conduction band
impurity states
fermi level fermi function
valence band density of states (parabolic)
FIGURE 1.2
ENERGY-BAND DIAGRAM FOR A LIGHTLY-DOPED SEMICONDUCTOR
conduction band / fermi level
fermi function ± ± ±_ impurity states(some ionized) density of states (profile unknown in region of impurity sites)
7, valence band
FIGURE 1.3
ENERGY-BAND DIAGRAM FOR A DEGENERATELY-DOPED SEMICONDUCTOR level into the conduction hand, and also causes the density-of- states to tail off more gradually into the forbidden gap than in the lightly doped case. The density of electron-occupied states is the product of density-of-states function and fermi function
(probability-of-occupancy function), both of which are shown in
Figures 1.2 and 1,3.
A semiconductor which has nearly all states near the bottom of the conduction band filled with electrons (from ionized donors) is called an n-type degenerate semiconductor. Similarly, a degen• erately doped p-type semiconductor is one in which all the states in an appreciable energy range near the top of the valence band are empty (due to a high concentration of acceptor sites lying just above the valence band),
A tunnel diode is formed by making a p-n junction between two degenerately doped n- and p-type semiconductors. Figure l.lj shows the band structure when no bias is applied across the junction, so that the fermi levels on each side coincide in the forbidden gap.
The shaded areas represent energy levels likely occupied by elec• trons. Electrons are subject to a large potential gradient in traversing the junction due to the built-in electric field arising from fixed ionized Impurity sites of opposite charge on opposite sides of the junction.
Applying a bias across such, a junction causes one side of the energy diagram to shift vertically relative to the other side. If there were no forbidden gap, which isolates the electrons on each side of the junction, they would then travel aoross under the app• lied field. They accomplish the same transition in the presence of the gap by quantum-mechanical tunneling, if the junction is sufficiently thin. Ordinarily, this transition must conserve energy p-side n-side forbidden gap E. FIGURE l.ty
TUNNEL DIODE W/////////J/7lif)l JUNCTION ENERGY occupied DIAGRAM FOR ZERO region for BIAS electrons wmm
forbidden FIGURE 1.5 gap fermi level T/i JUNCTION ENERGY V ES ZE. DIAGRAM FOR FORWARD BIAS bias overlap '^i
///////////////
forbidden gap 1.6 EL FIGURE JUNCTION ENERGY m//////7/~/rh DIAGRAM FOR REVERSE BIAS
bias and Overlap —that is, the transition is represented by a horizontal straight line on the energy diagram. This is known as direct tunneling.
Current due to direct tunneling is proportional to the product of the probability of tunneling per electron incident on the barrier, the density of occupied states on the side from which electrons travel, and the density of unoccupied states on the other side in a region which, on the energy scale, overlaps the occupied states on the first side. For zero bias, equal and opposite currents flow
(due to the fermi function "tails" at non-zero temperatures), the net current being zero.
A forward bias causes the n-type side of the junction to rise in electron energy relative to the p-type side so that the overlap region increases at first, then decreases to zero at sufficiently increased bias. The direct tunneling current is from conduction to valence band, and rises to a peak before falling to zero at large forward bias. The "direct tunnel current" region, is shown in
Figure 1,1. At much larger forward bias, ordinary thermal p-n junction current becomes prominent, since the potential barrier of both conduction and valence bands decreases linearly with forward bias. Figure 1.5 is for a representative forward bias.
The valley current is not fully accounted for by a superposi• tion of direct tunneling and far-forward thermal p-n junction cur• rents, but also Involves a process known as indirect tunneling, to be discussed later.
A reverse bias causes the overlap between the conduction and valence bands to increase. Now, direct electron tunneling is from valence to conduction band. The number of empty states in the con• duction band which are opposite occupied states in the valence band increases indefinitely with reverse bias increase, so that the rev- erse I-V characteristic shows no maximum. Figure 1.6 shows how the n-type side of the junction is depressed relative to the p- type side for this case.
Both the energy-band model, and the I-V curve show that the tunnel diode is a voltage-controlled device, the current being a single-valued function of the applied voltage. The noise spectrum for tunneling processes will be expressed in terms of the applied voltage.
For direct tunneling, the tunneling current from conduction to valence band is denoted by I and the current flowing opposite•
ly by Ivc« Esaki's expressions are then (V=)
fc(E) c{E) P ^ " fv(E)]fv(E) ZcvdE 4^c (1.2.1)
I (V) = AP* f (E)p (E)[l - f (E)]p (E) Z dE lg V \ V vc *• c \ c vc where
D (E) = n-type conduction band density of energy states
p^(E) = p-type valence band density of energy states
f (E) = fermi function In conduction and valence bands res- c'v pectlvely
Z (E) = probability of tunneling per electron attempt In ' each direction respectively
E = lowest energy level in n-conduction band c
E^ = highest energy level in p-valence band
E = energy
The integration range depends directly on bias. and Z also
depend on bias, but less strongly. The d.c current is |l| =
11 - I I. At zero bias, I = -I . For reverse bias, I 1 cv vc cv vc cv quickly falls to a value much less than I when the bias, V, vc approaches a few kT/e of bias, T being the actual absolute temp• erature of the tunnel diode junction. For forward bias V, I becomes much less than I for the same condition. cv
The noise spectrum arising from these tunneling currents can be related to Esaki's formulation, as a function of bias voltage, most simply if it is assumed that
a) the currents I and I are uncorrelated, and cv vc
b) these currents both contribute full shot noise.
The latter assumption is reasonable when we note that shot noise arises from the transport of discrete charges under an applied field, if these carriers are emitted with a Poisson distribution in time. This implies that there Is no correlation between succ• essive emissions constituting each current. Since tunneling is a very small probability process, the Poisson distribution is expected.
In determining whether or not direct tunneling currents produce pure shot noise, we shall measure the quantity
If I and I are uncorrelated and each produces full shot noise, cv vc the resultant noise will be the same as if an average current
I +1 were flowing. Hence we define 1=1 + I as the cv vc sq cv vc equivalent saturated noise current (in analogy with vacuum noise diode terminology.) The noise spectrum or noise power generated per unit bandwidth of frequency, due to direct tunneling currents, will be written in terms of the mean square of an equivalent con• stant current noise generator for the instantaneous current fluc• tuations. Experiments of Esaki and Yajima (1958) indicate that this represents more directly the physical nature of the tunneling noise than does a constant voltage generator. In analogy with shot noise arising in a temperature-saturated vacuum diode, the FIGURE 1.7
MECHANISMS FOR INDIRECT TUNNELING IN THE FAR-FORWARD BIAS REGION 7 direct tunneling noise spectrum will be specified by
= 1 corresponds to the vacuum diode case . By definition, X Q approaches Infinity as |l| approaches zero. At zero bias, ^*i? ^ must tend to a constant given by 4kT(£l/dV) evaluated at V = 0, in accordance with the thermodynamical requirement that the noise of any active system reduces to thermal noise given by a resistance equal to its value at zero bias. That this theorem holds in terms of Esaki's theory has been proved for special cases by TIemann (i960).
The Esaki formulation is inapplicable to Indirect tunneling.
This process can occur principally by way of Imperfections in the band structure, particularly in the form of localized impurity states or "traps" lying within the energy gap, and allows electrons to pass across the gap after the forward bias exceeds the value where the conduction and valence bands become "uncrossed", so that direct tunneling is impossible. Trap-involved mechanisms are de• picted by vertical and horizontal paths in Figure 1.7» The oblique path depicts phonon-assisted tunneling. In all these processes, the electrons must lose energy. These Indirect processes, which produce the valley current (Figure 1.1) are possible also when the bands are overlapped, so that the "excess" current caused by them extends well Into the negative conductance and far-forward regions on either side of the valley. Whereas direct-tunneling electrons traverse the gap very rapidly, electrons which interact with traps are captured for significant periods of time. This affects the noise. In particular, electrons which tunnel back and forth between either band, and the traps (shown by dotted arrows In Figure 1.7), can produce noise far in excess of shot noise. 8
1.3 Survey of the Literature
To date, little attempt has been made to use the noise properties of direct tunneling in tunnel diodes to check Esaki's theory, to determine the degree of correlation (if any) between
I and I , or to relate the excess current noise in the valley cv vc region to possible models for localized Impurity state distribu• tions within the forbidden gap.
Early noise measurements were at low frequencies where l/f noise dominates. This is not a property of tunneling, but arises in the bulk semiconductor surrounding the junction. Although con• taining a few features of interest to the present work, which are now summarized, these investigations are mainly irrelevant.
Esaki and Yajima (19£8) first measured noise in tunnel diodes, in the frequency range 10 to 10-* cycles/second. They qualitatively examined the difficult reverse- and near-forward bias regions
(where the diode impedance is very low) but only to determine if strong l/f noise existed. Their apparatus was not sensitive enough to detect a shot component of noise in these regions, so that they were able only to report that no strong l/f component appeared. However, their results cannot insure that a weak l/f component did not persist at 10-> cps so that much higher frequen• cies are needed to decide quantitatively if the noise in these bias regions is pure shot noise. The same authors found no strong l/f component in the negative conductance region, but in the excess current region strong l/f behavior appeared, with magnitude nearly
10^ greater than calculated shot behavior, even at 10^ cps. Their data indicate this was due mainly to excess current (later believed due to Indirect tunneling mechanisms) rather than to normal far- forward p-n junction diffusion current. Again, however, higher 9 frequencies are needed to see if the l/f behavior in the valley- region was due simply to low-frequency fluctuations of impurity trap centres or recombinations.through these centres (suggested as possible by the authors) or whether the indirect tunneling current can display only shot noise at high frequencies.
More recent measurements by M. D. Montgomery (1961) at freq• uencies from 30 to 1C-3 cps have corroborated the results of Esaki and Yajima, and have further strengthened the idea that strong l/f noise in the valley region at low frequencies Is due to indirect tunneling via a continuous distribution of impurity states.
Tiemann (I960) has made the only high-frequency noise measure• ments on tunnel diodes with a view to establishing whether a pure shot component alone is associated with tunneling currents. How• ever, he restricts himself to the negative conductance region, where he attempts to relate the expected shot component for the noise spectrum to Esaki's theory, but only In terms of a particular assumed density-of-states function, and an' assumed fermi level relative to the band edges. Though lacking generality, the numeri• cal evaluations for Esaki's integrals for these cases predict a shot noise spectrum in agreement with his data, taken at 0.5 Mc. and 100 Mc. for the restricted bias region. The noiso is not measured near the origin, so that it is not determined how the spectrum reduces to the correct thermal conductance noise at zero bias.
Theoretically, La Rosa and Wilhelmson (i960) have stated that the tunnel diode should display approximately one-half shot noise.
Assuming I and I are uncorrelated, they predict that the I cv vc vc component should produce full shot noise, due to its arising from the small-probability process of tunneling, but that the I cora- cv ponent should be greatly smoothed, due to the probability of
tunneling from conduction to valence band per electron per at•
tempt, Zqv, having a value close to 0.f>. Their reasons for this
value, and hence their conclusion of smoothed shot noise for the
tunnel diode, are erroneous.
D. Agouridis (unpublished, 1961), assuming no correlation
between I and I has related the noise spectrum to Esaki's cv vc
theory in more generality than Tiemann, but his treatment is also
somewhat specialized. He measures the noise up to 30 Mc. but also
only for the restricted biases near the peak and negative conduc•
tance regions, where the results do not so critically compare to
theory as in the higher conductance regions near zero bias.
I»l4 Scope of Thesis
To relate the noise spectrum for direct tunneling current to
Esaki's theory, assuming shot noise associated with the tunneling
components, we derive an expression for^TO which is independent of
the band structure of the diode, but depends on the bias voltage
and temperature. The spectrum Is then measured at l\ Mc. (where no
l/f component of noise persists) as a function of bias over the entire near-forward and reverse bias ranges, using suitable imped•
ance transformations for the very low diode source impedance in
this range. This is smallest in the reverse region, but the bias
is extended past three times kT/e volts to provide a range for most
critical comparison of the measured noise with theory. The tech• niques developed permit use of larger reverse biases, possibly
sufficient for avalanching in the junction accompanied by enhanced noise. Since direct tunneling current noise is uniform for all except possibly extremely high frequencies, a single measurement frequency is enough to find the spectrum magnitude, which is compared with the theory. The spectrum for indirect tunneling processes is measured in the valley region and on into the far-forward thermal-current region. Possible models are proposed in view of the experimental results. A proper comparison of the measurements and proposed models is possible only if measurements are taken in this region at several frequencies, since the predicted spectrum is in general not uniform. However, data was obtained only at q Mc. which, in conjunction with lower-frequency data by other workers in this region, is insufficient to test the models. FIGURE 2,1
NOISE-EQUIVALENT CIRCUIT FOR A SHOT NOISE DEVICE CHAPTER 2
THEORY OF TUNNEL DIODE NOISE
2.1 Noise Model of the Tunnel Diode
We define equivalent circuit generators to represent the noise in a circuit which may arise from various mechanisms, such
as the shot effect (due to the discreteness and randomness of
charge transport through part of the circuit) or thermal noise in
any resistance (due to fluctuations in thermally energetic charges.)
An equivalent generator Is assigned to each part of the circuit for which the noise arises from different mechanisms. Each generator
Is then defined to represent the fluctuations associated with the
current consisting of all electrons in motion in that part of the
circuit, as a function of time. These same electrons may influence
the current elsewhere In the circuit at the same time, but this is
accounted for by another noise generator representing fluctuations
in the latter part of the circuit. All such generators add quad•
rat ically if they are uncorrelated, as is often the case.
Any device displaying shot noise with only one component of current, is represented by a mean-square constant current generator
) of value 2el per unit bandwidth, I being the average current, across which is placed the internal impedance of the device. Thus a vacuum noise diode, operated in the temperature-saturated condi• tion, has a noise-equivalent circuit as shown in Figure 2.1. The external resistance R is taken as noiseless for simplicity. The value 2el for the generator ^i> is derived from Fourier analysis
of the pulses associated with individual electron transitsi a
short-circuit is assumed between anode and cathode for this calcu• lation. The voltage drop across R when it is included then causes <*f> ©
6
FIGURE 2.2
NOISE-EQUIVALENT CIRCUIT FOR A TUNNEL DIODE the actual fluctuating current to be
i1 = i - vOi^v) = i - i-jR/rp or
ir (r R) *1 = P/ p *
where i is the fluctuating current derived for R = 0, rp is the plate resistance of the vacuum diode, and v is the instantaneous
voltage on the anode. The result for i1 is seen to be consistent with the equivalent circuit, which shows the current i^ through R.
Similarly the thermal noise in any resistance R is represented
by a constant current generator of mean square
As a two-terminal device, the tunnel diode embodies noise due to the tunneling processes, and thermal noise in the bulk semi• conductor surrounding the junction. The noise-equivalent circuit for the tunnel diode is shown in Figure 2.2, where 2
^i^> = mean square noise current per unit bandwidth due to thermal noise in the bulk resistance Rb Rj. = resistance of the junction due to tunneling processes The total dynamic tunnel diode resistance, given by e> V/PI from the I-V characteristic, Is then Rd = R^ + R^. Experimentally, only 2 2 the composite noise spectrum due to both measured; when Rfe is known, 2 2 2 In the expression 1 1 f (E)= and f (E) = c v 1 + exp[(E - Efc)/kT] 1 + exp[(E - Efy)/kT3 where E. and E^ are the fermi levels in the conduction and val- fc fv ence bands respectively. Assuming ZCV(E,V) = ZVC(E,V) = Z(E,V) and if each component of tunneling current independently produces shot noise, then oo • I = I I |+|l I = A« p (E)p (E)ZdE ) e< 1 cvl * ' vc« J *V Ty (x + QXpj-(E . Ef )/kT] 1 2 1 + exp[(E - Efv)/kT] (l + exp[(E - Ef(J)/kT])(l + exp[(E - Efv)/kT]) r oo = A'|^exp(-Efv/kT) + exp(-Efc/kT)[ |f>c(E)pv(E)ZdE oo exp(E/kT) (1 + exp[(E - Efc)/kT])(l + exp[(E - Efv)/kT]). = A« [exp(-Efy/kT) + exp(-Efc/kT)] ^ Similarly FIGURE 2.3 BEHAVIOR OF WITH BIAS VOLTAGE 15 i|= Ii - i I I I cv vol 1 r / 1 i = A' I p (E)p (E)ZdE< A fv' J^c W (l + exp[(E-Efc)/kT] 1 + exp[(E-E )/kT]; = A'J]exp(-Efv/kT) - exp(-Efc/kT)|J £pc(E)pv(E)2dE • •J r\t> exp (E AT) (l + exp[(E - Efc)/kT])(l + exp[(E - Efv)/kT]) = At|^exp(-Efv/kT) - exp(-Efc/kT)]| ^ A' is a constant. Hence, using the definition (1,2.2) we have' E E y2 exp(-Efv/kT) + exp(-Efc/kT) ^ ^ l fc " fyl ° |exp(-Efv/kT) - exp(-Efc/kT)| 2kT Now |v), the magnitude of the applied bias, is given by |(Ef(J-Efv)|/e so that ^2 e'vl 0 = coth (2.2.1) 0 2kT It is convenient to consider only the absolute value of the bias voltage, hence also of I, so that "5Q is always positive. (However, it is consistent also to take V as negative for reverse biases, so 2 that "8 is negative also, but since I is negative in the reverse o 2 direction, Q behaves with bias as shown in Figure 2.3. VQ tends to infinity as Wl tends to zero, so that P(E,V) =pc(E)^>v(E) Z(E,V) so that it is independent of the band structure of the semi• conductors. The only requirements needed to produce (2.2.1) are: a) Z ,(E,V) = Z (E,V), i.e., tunneling reciprocity holds, and C V V c b) the occupancy functions are fermi functions. The assumed non-correlation of the current components I and I cv vc Is already assumed In the definition of }fq , equation (1.2.2). The result (2.2.1) is seen to imply the interesting relation KAJ = exp teV/kT> Finally, we note that ^ ^ 1 in general for semiconductor diodes, which are assumed to have shot noise associated with their currents, 2' whereas Yo = 1 for a vacuum diode which produces full shot noise. The distinction is that two components of current are associated with semiconductor tunneling processes, but only one component flows in vacuum diodes. 2.3 Restrictions on Esaki's Tunneling Theory In the derivation of (f which represents the noise spectrum associated with direct Esaki tunneling currents, no assumption regarding densities-of-states was required, but the occupation factor for these states was taken as the fermi function. This is a restriction on the most general statement, within the Esaki formulation, possible for direct tunneling. For example, it ex• cludes bosons from tunneling according to this formulation. The most general statement of tunneling which assumed that the density of occupied states on one side of a junction, and the density of unoccupied states on the other side, are the factors controlling the resulting tunneling currents, would be J12(E1) 6E1 = Z12(E1)n1(E1)h2(E2) dE], (2.3 J12 and J are current densities per energy increment dE^ and dE2 flowing from region 1 to 2 and region 2 to 1 respectively. E^ and Eg are any energy levels In regions 1 and 2; on an energy diagram for the overall system these have the same vertical dis• tance from the energy zero for direct tunneling, but need not In general comply with this. No explicit process, such as tunneling, need be envisioned, n-^ and n^ are densities of initial occupied states, and h^ and h2 are densities of the final unoccupied states. Hence this formulation is already too specialized to include bose particle transitions, since the occupancy per energy level Is unlimited for bosons, so that the density of unoccupied states on the side to which the particles transit, would not appear as a parameter influencing the currents. For any such generalization of the Esaki integrands as equa• tions (2.3*1)# we must satisfy the thermodynamic requirement that c>V V=0 This Is Nyquist's theorem. The average current, taken over all energy levels of the system, is dE, 1 1+eV J Z(E1,E1+eV) [n1(E1)h2(E1+eV) - n2(E )h1(E1) dE 1 where is any energy level on the "1" side of the junction, and E^ 88 E^+eV, where V Is the applied bias. We now find the most general form of the functions n. and h^ 2 such that Nyquist's theorem Is satisfied. Assuming - = Z, the conductance Is oo fr ah9(E.+eV) S>n5(E-+eV)-, Hv J Lai(Bi) ' hi(Ei} ^7 J z^Ei'VeV) dEi - CO / n S>ZJ(E-,E..+eV) 1 A + |[n1(E1)h2(E1+eV) - n2(E1+eV)h1(E1)J ~ dEx 13J/ fr ah (E.) dn.(E ) , -eo The second integral vanishes since J^^ = Jgi 88 ® when V = 0. The equivalent saturated noise current density is oO J (E E2) + Jg^E^Eg)] dEx * j[ 12 l' - eO = )h (E (E +eV) + (E +eV)h (E ) Z(E ,E +eV) dE j"fcl l 2 l n2 l l l ] l l l - OO lim J = 2 Jn1(E1)h2(E1) Z(E1,E1) dE1 eq V-K> since - J21 at V = 0, Nyquist's theorem is now written o>J(E) ' ^[J12(E) + J21(E)] V=0 2kT V=0 This relation is taken to hold for each energy level E, for at V = 0, we assume detailed-balance holds, that is, not only are the macroscopic currents (integrals of equations (2.3,1) over all energies) equal and opposite, but the components J12 and J21 are equal and opposite for every energy level E1 = E, Then Nyquist's theorem gives dh_(E) dn9(E) 1 1 n,(E) —2 -h-(E)— = — n,(E)ME) = — n(E)h(E) xdE -1 dE kT x * kT 2 1 or dh9(E) dn„(E) 1 —t - —2 = — dE h2(E) n2(E) kT where E Is any energy level. Integrating this equation between limits EQ and E (Eq arbitrary) gives hp(E) hp(E ) rl -i = —2*-—a- exp I — (E - E ) I (2.3.2) n.(E) ME) L kT °J 2 do and similarly ^(E) MEQ) rl —= = —=—— expp |— — (E- - « Ej J (2.3.2) n..(E) n (E ) LkT 0 1 l o Equations (2.3.2) are the most general relations between the func• tions n^2 (E) and h^2 (E) such that the generalized Esaki inte• grands (2.3.1) satisfy Nyquist's theorem at V = 0, If we specify n.-(E) = P (E)g(E) h,(E) = f.(E) [l - g(E)] 1 1 and 1 V1 (2.3.3) n2(E) = ^(E)g(E) h2(E) = f2(E) [l - g(E)J with p 0(E) the density-of-states functions on sides 1,2 and g(E) * i»2 the probability of occupancy of level E, we automatically insure that the occupancy function g(E) is the fermi function, since it has been assigned an upper bound of unity. This is consistent with treating g(E) as a probability function, as in the original Esaki formulation, but more important, a maximum of unity for an occupancy function arises only for fermions, due to the Pauli Exclusion Principle. "Hence substituting equations (2.3.3) into either of equations (2.3.2), whioh gives 1 - g(E) 1 - g(E ) E - E E - Eo = o_ QXp o = C(E j QXp S g(E) g(E0) kT ° kT 1 g(E) = 1 + 0(Eo) exp[(E - E )AT] serves only to check that g(E) is indeed the fermi function. (Since g(E) ^ 1, then C(E0)'^ 0. We may then put C(EQ)exp( -EQ/kT) s exp(-Ef) and identify Ef as the fermi level.) Thus the Esaki formulation for direct tunneling is applicable only to fermions, under the condition that shot noise reduce to thermal noise at V = 0. The foregoing analysis does not exclude the possibility of boss particles tunneling directly, as well as fermi particles, while satisfying the thermodynamic requirements at V = 0. The Esaki approach, in the form of equations 2.3.1 Is Inappropriate for bosons, owing to the unlimited occupancy per energy level allowed these particles. A much more general formulation alto• gether is needed in this case. It is of interest to find the most general occupancy function J J J 13 a 100 g(E) such that = (|J12| + 1 21P/U 12| " | 2lP ^c* solely of the bias and temperature (in the explicit form eV/kT) and is not dependent on the semiconductor band structure. The most general form of ^(eV/kT) is also found under this condition. We have = Y^'Jz(E1,E1+aV)| f1(E1)g(E1)p2(E1+eV)[l - g(E1+eV)] (E1+eV)g(E1+eV)f1(E1)[l - g(E1)] j d^ " lJ12l + lJ2ll = f Z(E1,E1+eV)^p1(E1)g(E1) p^E^eV^l - gtE^+eV)] + p2.(E1+eV)g(E1+eV)^1(E1)fl - g(E1)]J dE^^ For Y Q to be a function solely of eV/kT, that is, to be indepen• dent of the band structure, ^ and £2» it is required for each energy level E^ = E that (comparing the integrands of the last two equalities): *o(eV) [g(E) - g(E+eV)] = g(E) + g(E+eV) - 2g(E)g(E+eV) Here it is convenient to introduce the diraensionless variables E» = E/kT and U = eV/kT. Then * *(U) [g(E') - g(E'+U)] « g(E') + g(E'+U) - 2g(E»)g(E»+U) To.solve this functional equation for g(E'), put P(Et) = l/g(E«) - 1 Then the functional equation becomes K^) - [HE') + r(E'+U)]/fp(E») - P(E'+U)] or X = F U) T(E'+U) / p(E') = [y^(U>> ) - ll]/[*o]/f^o<(U ) + J < Hence In T(E«+U) = In F(E') + In F(u) Since, therefore, ^ln T(E'+U) }ln P(E') 5E1 d E' and since U is arbitrary, It follows that In V must be a linear function of its argument. Hence T(E') = exp (Cl + C2E») so that from the definition of P(E')» g(E') = 1 + exp (c + c E») 1 + exp (c E« - E*) 12 2> f 2 where E„ = E»kT is the fermi energy. Thus ifV is a function only f f ° o of eV/kT, g(E') is the fermi function except for an arbitrary constant The most general form of 2f,2 If It Is a function only of eV/kT Is found by combining the relation defining P(U) with the exponen• tial form of P(E') to give Y^U) - 1 *f:(u) +i 2 or >^(U) « coth (c2U/2) v2 The condition that 0Q depend only on U therefore leads to the relation ^ = coth (eV/2kT) except for the factor o^. Prom the foregoing, it is apparent that the thermodynamic requirement of Nyqulst's theorem as an added restriction, requires Cg to be unity. 23 2,4 Models for Noise Associated with Indirect Tunneling Processes In the valley region of the I - V characteristic, both diffusional p-n junction minority carrier thermal current, and current due to indirect tunneling processes contribute to the average current flowing. The indirect tunneling current dominates. In a later chapter, we report that at a frequency sufficiently high that all l/f component has disappeared from the noise in the direct tunneling regions of the I - V curve, the measured noise associated with the valley current region greatly exceeds full shot noise. This can be due to any of three causes: a) the noise associated with ordinary p-n junction thermal current may exceed shot noise. b) at frequencies sufficiently high that no l/f component exists for tunneling current noise, a l/f component may still exist for p-n junction thermal currents, since these arise from entirely different mechanisms than do tunneling currents. c) the noise associated with indirect tunneling currents may exceed shot noise. The first possibility may be quickly ruled out. Van der Ziel (19£8) has shown that the noise behavior of an ordinary p-n junc• tion diode may be represented by a current generator i in shunt with the junction, such that = qkTG - 2el per unit bandwidth, where G = 9l/^V Is the junction conductance and I is the junction current. This can be rewritten as For a normal p-n junction diode, the I - V characteristic is given by I = 1 ' 1 I FIGURE 2.U DETAILED MECHANISMS INVOLVED IN INDIRECT TUNNELING where I = constant is the "saturation" junction current. Com- s bining these relations gives . 1 + exp (-eV/kT) ~ \ld> = 2el = 2el coth (eV/2kT) S 2elfi* 1 - exp (-eV/kT) 0 The normal p-n Junction thermal current noise is therefore given by exactly the same function of bias and temperature as that due to direct tunneling currents, although the latter do not exist in the valley bias region. Since two oppositely flowing current components are associated with the ordinary junction thermal average current, both these components have full shot noise assoc• iated with them, but not more than this. The validity of the second possibility for excess noise in the valley region can be checked only by noise measurements at several frequencies. For the third possibility, we now investigate the noise spectrum for possible mechanisms involving interaction of the tunneling electrons with traps or impurity sites within the for• bidden gap. The possible paths for electrons in indirect transitions across the gap are shown in Figure 2.1|. The mechanisms are: horizontal arrows: tunneling occurs directly between bands and traps, in either direction. The densities of occupied states strongly disfavor tunnel• ing from C to A (Figure 2.4), from B to A, or from D to B^, or from D to C^. Tunneling in the opposite directions is relatively probable• vertical arrows : electrons lose energy by phonon or photon interaction(s), or by electron-electron interactions, the latter being very improb• able . oblique arrows : tunneling (horizontal component) and (phonon) absorption of the electron energy occur simult- taneously. The analysis to follow will imply that the noise arising from these processes is independent of whether electrons lose energy while transiting the gap. Thus process AD (oblique arrow) produces shot noise, as do direct-tunneling electrons. Processes AGD and AC^D also contribute only shot noise, if it is assumed (as in the liter• ature: Chynoweth et. al., i960) that these processes are rate- limited by the average rate of the horizontal transitions, and not by the vertical transitions."1" Processes ABD and AB^D will be shown to produce at most shot noise. However, processes ABA, ACA, DC-jD, and DB^D can cause greater than shot noise for the overall average indirect current, since these routes represent fluctuating currents without contributing to the average current. The relative likelihoods of each of these mechanisms Is un• known, as is the exact nature of the valley current, although it is believed due to indirect tunneling processes largely. Figure 2.4. represents a greatly simplified model. The very large number of traps undoubtedly present in the forbidden gap of degenerately doped semiconductors would allow many possible routes not shown by the extreme cases in Figure 2.4. Particularly important, from the point of view of processes producing greater than shot noise, would be the unlimited possibility for tunneling transitions back and forth between traps within the gap. These would greatly enhance the fluctuations but not the average current. Since energy losses for an electron while in the forbidden 1. If these processes were rate-controlled by the vertical transi• tion rates, a non-uniform spectrum of magnitude less than shot noise would result, due to successive electrons being correlated (since traps are occupied by at most one electron). valence band' FIGURE 2.5 SIMPLIFIED MODEL FOR NOISE ANALYSIS OF INDIRECT TUNNELING PROCESSES gap do not influence the associated noise (assuming the mechan• ism of interaction with phonons is not noisy, or can be repre• sented as thermal noise), the analysis of Indirect tunneling current noise will be based on a simplified model, as shown in Figure 2.5. Energy is conserved for all transitions, and all traps (represented by short dashed lines) are a distance rQ from . the edge of the conduction band. An electron tunneling from the conduction band has a probability p of Interacting with a trap, or a probability 1 - p of tunneling on directly to the valence band without encountering a trap. For an electron which encounters a trap, let ^> be the probability that the electron tunnels back to the conduction band sometime after capture. This will be referred to as "event B", 1 - ^ is then the probability that the electron tunnels onward to the valence band sometime after capture. This will be "event A". For either event A or B, let the probability that an electron is released by a trap in any time Interval dt after capture be constant, that is, Independent of the time. It then easily follows that the probability of an electron being in a trap at time t == t If it entered the trap at time t = 0, is P(0,t) = exp where *C ^ is the average capture period for the i**1 process (i = A or B). We assume for simplicity that the transit times for electrons between bands and traps are zero. The current associated with event A, which is defined to flow through the entire tunnel diode due to the tunneling of the electron, takes the form of two suc• cessive S-functions as shown in Figure 2.6, This arises from the in(t) t r0e/s t ts 0 FIGURE 2.6 CURRENT ASSOCIATED WITH "EVENT A" fact that a charge e at rest in a dielectric at a distance rQ from one of two conducting planes, and a distance s - r from the o other, induces charge e(s - rQ)/s and er^/s respectively on these two planes. For the case of the charge e moving at Infinite velo• city between the planes, the Induced currents are then S-functions Thus an electron tunneling from conduction band to trap at time t = 0 induces in the system a current £(rQ/s)]e S(t - 0); at time t later the current £(s - ro)/sl e^(t -K) is induced when the electron tunnels to the valence band. These currents satisfy the requirement that the integral over all time of the instantaneous current i(t) equals the total charge induced, that Is, e, at any part of the circuit. If t A Is the average capture time per elec• tron per trap for event A, the £-functions In Figure 2.6 are separated by a random variable % whose ensemble average Is T . To find the noise spectrum for events A, we assume the curr• ents due to all such electrons are mutually independent, that is, the electrons leave the conduction band randomly, and do not affect each other thereafter. With i(t) the result of a large number of Independent currents in(t) (of which Figure 2.6 is an illustration) occurring at random at the average rate fA, the noise spectrum may be found by Fourier analysis of each component in(t). In case all components in(t) have the same time constant 'E, the spectral density, or absolute value squared of the Fourier coefficient of i(t), is calculated from Carson's Theorem: 2 SA(A>) = 2^A)'7( where oo J (UJ) «J" in(t) exp (-j*>t) dt o In the present case, there is a random distribution of time con- stants f for the currents in(t) associated with the n electron undergoing process A, The required modification of Carson's Theorem is to let oo d^A = gCtOd'tf where ' J^g('C)d'tf=l be the number of events per second with a time constant between t, and X + dt. Then CO 2 sA(o>) = Z\IX„(vo)\ g(tr) where (uo) is the Fourier transform for events having time constant t . Since g(T )d't> is therefore the number of electrons per second which are captured by traps for a duration between ^ and t + dt? , we have gCOdt = i>k exp (-r/trA) di?/i?A Now 7^(60) = J^[fe = fe + e(l - f) exp(+ J^t) where f = r /s. Hence o' I? (60 )|2 = e2 (1 - 2f + 2f2) + e2 (1 - f )f 2 cos (oOU ) The noise spectrum of the current due to all electrons undergoing process A Is then oo SAov) = 2^ e2 [(1 - 2f + 2f2) A A u + f(l - f) 2cos(60*)J exp (-^/r.) dt/^, A A In(t) t rQe/s t = X t = 0 -rQe/s FIGURE 2.7 CURRENT ASSOCIATED WITH "EVENT B" (2.14.1) where ^ is the average rate of emission of electrons Interacting with traps either by process A or by process B. The average or d.c current associated with process A is where 1^ is the average current for both process A and process B combined, the latter having no average current associated with it. It is noted from equation (2.4.1) that the noise produced by currents due to process A, is less than shot noise unless f = 0, that Is, unless the traps merge with the conduction band edge. To find the noise spectrum for process B, we again assume that currents due to transitions of each electron are mutually independent. Each electron, in tunneling to a trap, being cap• tured on the average a time then tunneling back to the con- B duction band, produces the current form given in Figure 2.7. The Fourier transform COof this current is Using •2>nexp (-t/tf ) df/* B for process B, the spectral density for currents associated with process B is OO r S (60 ) B 2 2f2e2 (1 - cosM*) exp (-T/'tf ) dV/t^ B y o 2 2 2 Z k h-i>Bf e 60 ^| / (1 +0> t\ ) = !}^/e2i02^ / (l + ">2^2 ) (2.U.2) where p is the probability that an electron, after capture by a trap, tunnels back to the conduction band. The overall noise spectrum ST(60) due to all electrons which interact with traps, either by process A or by process B, is the sum of S.(oo) and S (a?) since the two processes are taken as A B mutually independent. <0) + S (60) sT(a>) = V B 2 2 2 2 » 21^Te |(1 - o )[l - 2f(l - f)a) ^ / (1 +60 ^|) + 2^f2^02rC2 / (1 + rt2t\ ) j (2.U.3) If the frequency of measurement, 40 , approaches zero, or at least becomes small compared with l/t^ and l/'tfg, the capture processes of the traps should become unimportant to the resulting noise. We should then expect shot noise to result, as is the case when elec• trons do not encounter traps. Now 2 limQ ST(4>) = 2^>Te (l -^>) = 2eIA 5 2eIT where I,p (that Is, IA) is the average current associated with trap-Involved electron transitions, process B having zero associ• ated average current. Thus shot noise is secured at low frequencies. 31 The noise due to electrons which do not encounter traps, but which tunnel directly (that is, only with phonon interactions in this model) from conduction to valence band, is easily included since It is full shot noise, that is, S.^(tv ) = 2eld, where 1^ is the average current associated with electrons transiting directly. With p the probability that an electron leaving the conduction band will encounter traps, we have IT « el>p (1 - f ) Id = ei>(l - p) where is the average rate of electrons leaving the conduction band. The noise spectrum for superimposed direct- and trap- involved processes Is, again assuming that these processes are independent, S(to ) = sT(a>) + Sd(W) and since the total average current flowing is then S(o> ) = 2el (1-p) + p(l-p) 2 (2.U.U) S 2el FIGURE 2.8 POSSIBLE NOISE SPECTRA FOR INDIRECT TUNNELING PROCESSES These relations indicate that ^2-^o© as p 1 and ^—-*1, whereas I —» 0. These are the conditions for which every electron leaving the conduction band interacts with a trap, then tunnels back to the conduction band. To decide whether greater than shot noise is possible (it certainly is not possible unless process B is operative), we examine only S^(o0), since the effect of adding S^(co ) only "dil• utes" the significant behavior of the noise spectrum. It is then convenient to define 2 tf (a>) = sT(o>) / 2eiT 2 U>2 = 1 - 2f(l-f) " + -flf ^tJ (2.11.5) l+Mj^tf i-f I+CO2?2, with use of equation (2.q.3) and the relation IT = 2>^e (l-^> ). Clearly ^ 2 1 (that Is, greater than or less than shot noise can result) depending on the various parameters involved. If no assumptions for their interrelations are made, any of the nine curves for 0 ^ versus frequency CO shown in Figure 2*8 can obtain. Since **(») • 1 - 2f(l - f) + <2(>f2)/(l - {) then £2(oo) | 1 if f/(l-f) | (l-f)/f At finite frequencies, the factors CO2 t2 / (1 + w)2/C2 ) and (A t„ / (1 + £0 ) are roughly step-functions, but in general B a ff rise rapidly toward unity at different frequencies, 1/ tA and l/tg, depending upon which of and'Cg is larger. The following table shows the correspondence between the spectrum curves and the rela- •3(3 tive sizes of the parameters involved. The entries in the table refer to the curves designated similarly In Figure 2.8. r r #7 #5 A< B #3 %=*B #1 #9 #6 #2 #8 #4 For curve #9, the completely degenerate case, "ft ^(CO ) = 1 for all 60. Only one conjecture which interrelates the parameters, can be made on physical grounds. Since l/fg is the probability per unit time of an electron In a trap leaving via event B, and similarly t we can S NCE IS for l/' 'A» assume that (1 - ^ )/^> = /^B * ^ . the probability per electron leaving the conduction band that event B will occur, and 1- ^ is the same for event A. It can then be shown from equation (2.4 »5>) that curves #2- and #5 of Figure 2.8 are physically impossible; that is, since ^0 is to be real, )i2 = 1 only for 60 = 0. No other non-restrictive assumptions can be applied, so that the model assumed for these calculations serves only to indi• cate the possibility of noise in excess of shot noise. The entire model can be somewhat generalized in-threesways: a) instead of assuming instantaneous transitions between bands and traps, the transit times can be taken as finite. (The current pulses are then rectangular, assuming no acceleration of the charge while transiting). The spectrum given In equation (2.4.3) will then Include a multiplicative faotor which causes it to fall off at frequencies of the order of l/^t and above, ^ t being the transit time. Such frequencies are very much higher than those for which the spectrum Is of interest due to Indirect tunneling mechanisms, b) instead of a o"-function distribution for trap positions between the conduction and valence bands, we can consider any spatial distribution of traps Y| (r)dr. The appropriate modifi• cation of Carson's Theorem for the spectral density is r -s S(60) ^J[^,r)^(r)dr g(t )dt o o where s is the horizontal distance between conduction and valence band (assumed independent of the energy) on the energy diagram. The spectrum profile with frequency will be different for different assumed forms of T^(r)dr so that in principle noise measurements could be related to trap distributions within the forbidden gap, as well as to capture times of the traps and relative probabilities for the various possible indirect processes. c) consider the average indirect current (valley region curr• ent) to be the difference of the averages of two indirect tunneling current components flowing in opposite directions. The component flowing from valence to conduction band will be very small, and its noise spectrum may be different to that of the opposite component considered in the foregoing. The composite spectrum for the two components will be S(iO) = SefVftAMllJ + V|(a> ) |I2| ] where the subscripts refer to the two components respectively. The noise is seen to exceed that due to one component alone, if the two components are uncorrelated. FIGURE 2.9 CHARGE DISTRIBUTION WITHIN A TUNNEL DIODE JUNCTION 35 2,kl Modulation in the Indirect-Tunneling Modal for Valley Noise The foregoing model has predicted the possibility of an enhanced spectrum over shot noise on the basis of electrons tun• neling back and forth from conduction band to traps, or from traps to valence band. Since such a process may be Infrequent relative to uni-directional tunneling transitions, which do not produce more than shot noise at any frequency, it is useful to consider possible modulation effects on the field governing the tunneling of an electron to a trap. Such effects can either enhance or decrease the noise. The tunneling of a single eleotron to a donor, or vertical transition to an acceptor changes the junction electric field, which in turn modifies the probability for successive electrons to tunnel from conduction band to donors, or from acceptors to valence band. If the transition of the Initial electron discour• ages or encourages similar transitions of successive electrons, the noise will be decreased or increased respectively for the current component arising from the particular process considered. Consider in Figure 2.1 indirect processes involving horizontal and vertical transitions. Figure 2.9 represents schematically the charge distribution in a typical tunnel diode junction, plane symmetry being assumed. The regional boundaries A, B, C correspond to those marked similarly in Figure 2J-Q. The net negative charge density on the p-side and positive charge density on the n-side is due respectively to ionized donors on the n-side and electron- occupied acceptors on the p-side of the junction area between A and C. However fixed positively charged sites exist on the p-side due to a few ionized donors there, and a few occupied acceptors on FIGURE 2.10 n-side MODULATION OF ENERGY- BAND DIAGRAM BY TRAP- INVOLVED INDIRECT TUNNELING (a) Smaller Forward Bias (b) Larger Forward Bias the n-side produce fixed negatively charged sites there. These latter "minority" sites exist due to chemical diffusion of ri- doped and p-doped material into each other during fabrication of the Junction and would not exist in an "ideal" junction. The density of charges arising from this non-ideality Is much less than that of the predominant and oppositely charged sites In each region, which determines the sign of the ^> -curve in Figure 2.9-. These "minority" sites produce an electric field smaller than, but in opposition to that produced by the majority charged sites in each region. Figure 2.10 shows the energy-band diagram for the Junction for (ai) a forward bias beyond the valley region, but small enough that the fermi level E- lies below,0 and the fermi level E. fc ' fv above, the inflection points of the conduction and valence band energy curves respectively. The solid-line diagram applies before an electron tunnels to a donor to neutralize it, while the dashed-line diagram applies after the same transition, (b) is for a larger bias, sufficient that E^c lies above, and E^v below the Inflection points of the conduction and valence band energy curves respectively. Acceptors are denoted by circles, donors by squares. Case (a); smaller biasest The diagram shows that all donors involved in the transitions considered, lie on the p-side of the junction, while all acceptors must lie on the n-side. An electron tunneling from conduction band to a donor, neutralizes the donor; this decreases the field due to the minority-charged sites on the p-side, hence increases the total junction field. (The change in field in the n-material due to the loss of a conduction electron is negligible.) The dashed-line diagram in Figure 2J0(a) shows the change in field: the solid and open squares represent donor sites before and after the transition respectively. The energy of ionization of the donors, that Is, the vertical distance from donors to edge of conduction band, does not change. The result of the transition Is then both to increase the junction field, and to decrease the tunneling gap between conduction band and donors lying in the conduction band energy range. Both effects enhance the probability of tunneling of successive electrons to donors. This may be termed a "positive" modulation since one electron encourages the future tunneling of many others. The noise is enhanced for this type of transition. However the same transition discourages electrons from fall• ing into acceptors, both because E^.Q decreases, requiring that eligible acceptors be further in the n-side of the junction, and because increases, so that some acceptors may lie below Efy, after the donor transition. This is a "negative" type of modula• tion, which tends to decrease the noise associated with acceptor- involved transitions. Next suppose the Initial electron considered falls Into a neutral acceptor Instead of neutralizing a donor. The density of negatively charged sites In the minority on the n-side is in• creased so that the junction field is decreased. As Figure 2JjO(a) Implies, this discourages successive electrons from tunneling to donors, since the gap has widened, but encourages successive electrons to transit to the valence band via acceptors, since the latter need not lie so far In the n-side of the junction to take part, and since the density of acceptors increases toward the p- side. Again there results a "positive" modulation, with enhanced noise, for the acceptor paths, but a "negative" modulation, with 38 decreased noise, for donor paths. Case (b): larger biases: Since E^c now lies above the In• flection point in the conduction band energy curve, acceptors involved in the process under consideration can lie on the p-side of the junction, and since Efv lies below the valence band inflec• tion point, donors lying in the n-side of the junction can be In• volved. Electrons will generally tunnel to n-side donors rather than to p-side donors, since the former are more numerous and lie closer to the conduction band edge. Similarly, there are more tunneling transitions to the valence band via p-side acceptors rather than n-side acceptors, assuming the process Is rate-controll• ed by horizontal and not vertical transitions. As the bias In• creases, the ratio of n-side to p-side donors involved, increases, as does the ratio of p-side to n-side acceptors. Considering n-side donors and p-side acceptors to dominate, Figure 2J0(b) shows that i) electrons tunneling to n-side donors - decrease the field and widen the tunneling gap to donors for successive electrons. Hence the neutralization of more donors is Inhibited. However the acceptor-involved processes become more likely by the decrease in field, ii) an electron falling into a p-side acceptor increases the field, hence encourages tunneling to donors by successive electrons, but inhibits successive acceptor- processes. Both of these cases produce "negative" modulation action for one process, but "positive" modulation action for the other, as is also true for case (a). For either case (a) or (b) detailed knowledge of the band picture is required to decide whether the positive or negative modulation predominates. Another important effect hitherto neglected is illustrated for donor-processes In Figure 2J0(a): donors which lie within the 39 occupied conduction band energy levels before the neutralization of one of them, may lie outside this energy range after the tran-- sition so that they can no longer be involved in the process. This tends to compensate the "positive" modulation associated with this type of transition in the foregoing. Similar considerations apply to both donors and acceptor-processes and may compensate either positive or negative modulation for either type of trap involved. Since the processes producing positive or negative modulation are opposite for case (a) and (b) respectively, a change in the noise behavior may occur at a bias roughly corresponding to that representing the crossover from case (a) to oase (b). For typical germanium tunnel diodes, the donor concentration is 1,8 x 10^ cm"-^ and the acceptor concentration is 5 x 10^ cm"*3, corresponding to Efc at 0,06 volts inside the conduction band edge and Efv at 0,23 volts Inside the valence band edge. Assuming a built-in junotion potential at zero bias of 1 volt, the biases for which E^, reaches the valence band inflection point and Efc reaches the conduction band inflection point are respectively 0,27 volts and 0*kk volts, A small change in noise may occur in a bias region centred about these values, but it may be diluted by many other compensatory mechanisms not considered. The modulation mechanisms just discussed provide an alterna• tive for enhanced noise to the mechanisms described in Seotion 2,1+, The frequency dependence of the spectrum in both theories is deter• mined mainly by the capture times of electrons in the traps, and hence should be essentially the same for both models for the enhanced noise. FIGURE 3.1 SCHEMATIC CIRCUIT FOR DIRECT MEASUREMENT OF A NOISE SOURCE CHAPTER 3 APPARATUS AND EXPERIMENTAL TECHNIQUES IJ.l Basic Concepts and Requirements of Noiae Measurements 3.11 Theory and Requirements for "Low-noise" Circuits We shall restrict discussion to circuits which can be repre• sented as q-terminal networks with clearly defined Input and output pairs. These may be active or passive. For active networks, it has been shown (I.R.E. Subcommittee on Noise, I960) that all internal (distributed) noise sources of a noisy q-terrainal network can be represented uniquely by not less than a voltage generator acting in series with any source Input voltage, and a shunt current generator acting in shunt with any input current. Figure 3*1 illustrates. A complete specification of these generators is equivalent to a complete description of the internal sources as far as their contribution to output or terminal voltages and currents is con• cerned. For noise measurement purposes, the details for specifying these generators need not exceed evaluating their mean square values. They are in general correlated statistically. For a fixed frequency, the nol3e figure for such a network is defined as total mean square noise across xx p a , = that portion of mean square noise across xx due to 2 2 2 2 = «v >+ where F as 1 represents a "perfect" network. For a 4-terminal network consisting of a vacuum tube triode input, it can be shown (I.R.E. Subcommittee on Noise, I960) that the correlation susceptance between v 05 specifying the generator ^ n^ ^kTQR whioh describes nearly all the noise. R^ is termed the noise resistance of the network, and by standard agreement is ascribed a temperature TQ of 290°K. In general other resistances in the circuit operate at about Tq or may exceed it by a few degrees. The comparison of Rjj-noise with source noise is then accurate to a few parts in 290, which is inconsequential in those cases, as here, where the R^noise is p , negligible. Physically, the generator The concept of noise figure can be extended to [[-terminal networks connected one after the other. Let the available power gain of the 1th network be G^, while it contributes noise repre- p sented at its Input terminals by a voltage 1 2 noise figure for the n stages, in terms of a source admittance Y_) connected to the first network input, which produces a mean square noise voltage <.v2> at the input, is easily shown to be P . x +<4>+ + 4^<3$ » P1 + (P2 - 1)/ 0X 4 (P3 - 1)/ G^g + For G^>> 1 for all 1, the first network is seen largely to deter• mine the overall noise figure, each following network contributing increasingly less to the degradation of P^. The principle of carefully selecting the input network to give a small value for F , with much less stringent conditions applying n to successive networks, will be demonstrated by the particular sequence of networks chosen to provide a low value of Fn for the overall circuit used to measure the tunnel diode noise signal. 3.12 Methods of Comparison With a Standard Noise Source There are in principle two ways to measure the magnitude of an unknown noise source represented as a current generator Besides methods of comparison with a standard source, described below, the magnitude of ^ig^ can °Q found "directly". To illustrate simply, the shunt noise current generator ^i2^ assoc• iated with the Input conductance, and the input conductance, are assumed negligible. The overall bandwidth is assumed limited by the circuit following the source (amplifier) and constant. £^ represents the RMS output noise power of source and amplifier for the i reading. The amplifier input is first shorted, giving FIGURE 3.2 SIMPLIFIED SCHEMATIC CIRCUIT FOR DIRECT MEASUREMENT OF A NOISE SOURCE ^1 = **2, where G ia the power gain of the amplifier. Next, the unknown source ia connected, giving £| = G2( where Rg ia the source resistance. Prom these readings, In this method, in which the equivalent circuit is shown in Figure 3.2, it has been assumed that 2 2 the "law" or response of the amplifier-rectifier system as a function of time and of input is linear and independent of the magnitude of input voltage; that is, df = G2 The method can easily be extended to an "n-laiw" amplifier, that is, one for which &^ = (G The method cannot be made independent of the amplifier law by use of an attenuator (which would insure a constant input voltage) because the schematic generator <^v^> is not accessible to attenu• ation. The method would further be complicated if the amplifier noise current generator ^i2^ were significant, since the Input voltage it produces depends on the input shunt impedance, which in general varies since the source impedance may vary. Standardiza• tion of impedances would then be required, as well as accurate specification of Most of these problems are solved by comparing the unknown source Some methods of comparison of unknown and calibrated noise sources are as follows. 1) Figure 3«3 applies. The amplifier law is assumed linear, in which case the "gain", G, has the usual meaning. The amplifier input is shorted, giving 62 = G2 convenient level such that 0^>rJ^O^t where 02 = G2( Independent of the gain. In case the amplifier law is not linear, it can be measured accurately with a variable signal generator at the Input and a detector of known and fixed law (e.g., an RMS VTVM or a square-law thermocouple). If the response obeys an n law, then & * = k'( 2/n 2 /n 2/n = 2) Besides eliminating the dependence of noise measurement on the amplifier gain, the comparison method can be modified by use of an attenuator to avoid dependence on the "law" of the amplifier. The method is particularly simple if we can assume p the amplifier noise (specified mainly by ratio (input/output) of A-^ ( <1). The response is & = k'[Af«if>z2)] °/2 b) switch the attenuator to the ratio A2 ( noise diode current until the same response © is obtained. Then , . 2 2 2 2 2 e = k'[A «i D> + ) z ]"/ The two relations give for the unknown source The result is independent of the "law" of the amplifier, the constant k^ and the shunt Impedance Z providing that it is the same for both settings of the attenuator. The attenuation ratios FIGURE 3.4 SCHEMATIC NOISE CIRCUIT FOR ATTENUATOR AND TWO STANDARD NOISE SOURCES and A2 are easily calibrated with either a signal generator or noise diode source. The bandwidth must not be limited by the attenuator, but by the amplifier. 3) If the amplifier noise cannot be neglected compared with the source noise signal, it can be calibrated and subtracted by use of an additional standard noise source. Figure 3*4 now applies. The attenuator is still connected between sources and amplifier input, as In the preceding case. Let Into the attenuator output is assumed independent of the attenuator settings, so that constant shunt resistances not Included by Rg or R are bridge- measured and their noise subtracted from that of the sources. The present method Involves switching between the unknown source ^ig> (shunted by its dynamic resistance Rg) and a calibrated resistance R, which is made equal to Rg, so that the overall shunt impedance is always the same. Since the noise associated with R is known by Nyquist's theorem, R is used to calibrate the amplifier noise as follows. a) with R switched in, and the attenuator at a ratio A-^, the res• ponse is 0 - k'[ b) the bandwidth being constant, the unknown source switched in and the attenuator set at a ratio Ag. The noise diode Is then turned up until the response is the same as in the first reading. Then 2 n/2 a = k'[ Solving these relations gives ) is used with < ig > switched in, and the ratio Ag when > Is switched in. The use of two standard noise sources for this case has permitted the noise generator < in > to be accounted for. Further, the values of A.^ and are arbitrary, although they must be known, so that fixed values for these ratios may be chosen independent of the source levels. As is seen in the next method, the use of two standard sources along with an attenuator over- specifies the problem; the use for a standard resistance noise source, R, in conjunction with an attenuator, is for certain cases where a noise diode may not be used as a standard noise source (e.g., at frequencies under ^ 100 cps, flicker or l/f noise may override the shot noise in the diode.) In such cases, the values of A^ and Ag are not arbitrary, but must be adjusted so that the amplifier response is independent of whether R or k) When the noise diode is an acceptable standard source, its use along with a standard resistance noise source R, obviates P the need for an attenuator. The sources , R, and the noise s * diode are now connected directly to the amplifier input. Assume e - k'[ 2 Next, the unknown source ^i >, with associated resistance Rs equal to R, Is switched in and the noise diode turned up until the amplifier response & is regained: 2 n 2 / 2 e> » k [ Solving, 2 2 2 =- n n A + applies in this result if Method 3) is not appreciably more inaccurate than method 4), since the attenuator ratios can be accurately calibrated. However use of an attenuator arranged as in Figure 3»3 degrades the noise figure, since it attenuates sources but not amplifier noise. (It is possible, however, to place such attenuation between amplifier noise generators and amplifier, effectively, so as to attenuate amplifier noise also. This is done by using a preamplifier with gain sufficient that its noise completely overrides the noise of another amplifier or receiver which follows it. The attenuator is between preamplifier and receiver so that it attenuates preamplifier noise, while the receiver noise is neglected altogether.) Method q) is advantageous mainly in that the experimental arrangement measures directly the difference ^i^^ - ^*JJD ^ • 2 2 2 If «, then both < i2 > and < i ^ > are large quanti• ties, each measurable to an accuracy limited by recording a fluc• tuating quantity 6 . Method 3) measures < i2 > and < I2^ > separ• ately so that the uncertainty In their difference, obtained alje- braically, far exceeds that of method 4). A modified version of method 4) will be used to measure tunnel diode noise in this thesis. n n @ AAAA/SA- noisy transforming G network FIGURE 3.5 SCHEMATIC CIRCUIT FOR A TRANSFORMED SOURCE COUPLED INTO A NOISY AMPLIFIER 1*9 3.2 Impedance Transformations Suitable for a Tunnel Diode Source The foregoing section indicates that it is unneccessary for ,i the voltage at the amplifier input due to the noise sources to override that due to the equivalent noise generators of the ampli• fier. However for some sources, such as the tunnel diode biased 2 2 2 R anywhere in the reverse or near-forward regions, would be overridden by the noise of even the very low-noise ampli• fiers to such an extent that measurement of the tunnel diode noise would be impossible if the diode were connected directly to the amplifier input. That is, the response $ would always be the P 2 same In method q) whether it would be determined entirely (within experimental accuracy of recording a fluctuating response) by the amplifier noise itself. The excessively low value of the wide range of diode Impedances encountered over the bias range of Interest are the two main difficulties In measuring tunnel diode noise in the near-forward and reverse bias regions. A network is required to transform the tunnel diode source impedance so that It a) amplifies the small tunnel diode noise to a level which over• rides the noise of any following network, and b) accomplishes this without adding appreciable noise itself (that Is, thermal noise of resistances in the network any of which may override the tunnel diode source noise; the network will exclude active elements which are too noisy.) These properties are summarized by requiring a satisfactory noise figure in terms of the source and amplifier noise voltages appearing across terminals xx In Figure 3.5. The gain of the preamplifier following xx is sufficiently high that the noise of circuitWhen sR followinrepresentg sth eth epreamplifie total tunner mal ydiod be e neglectedresistanc. e R, at FIGURE 3.6 NOISE-EQUIVALENT CIRCUITS FOR A PARALLEL-TUNED CIRCUIT biases-where Rd 500 ohms, a suitable "transforming" network is simply a parallel-tuned circuit across the diode. The noise- representation for this network is in Figure 3.6. = < i^. > + IjkTr per unit bandwidth. It can be shown that Figure 3.6(a) is equivalent to Figure 3.6(b) where R Ci Q2r (for Q>> 1) has thermal noise current generator l*kT/R associated with It. R^ represents the noise of the preamplifier which follows (assuming the equi• valent shunt noise current generator representing grid-circuit loading is negligible at frequencies of interest). The noise figure of the circuit in Figure 3.6(b) is then, with Z the total shunt impedance, 2 2 2 2 F = 1 + (Z + l4kTQRn) / 2 where TQ = 290°K. is the standard temperature assigned to Rp. To is 2 estimate the magnitude of F, = 2e(Jlcv| • l!vcl) assumed to be a shot noise generator for the tunnel diode. Then l|kT r 1 1 + d 1 + 2el Q r eq L The approximate form holds for R^ small compared to Q r 1+0 kohms typically). The temperature of RR has been taken as that of the overall circuit. At T = 290°K, i|kT/2e ^ 1/20 volt. A good low- noise amplifier has Rn £i 300 ohms or less. Since 2e(|lcv| + llvc|) = UkT/R^lvno and aince R^ evaluated at zero bias is typically 15 TnBSe ohms for germanium tunnel diodes, then |lov| + l^vc^ ~ ^ values lead to the excellent noise figure of F £2. 1 + .005. However it is seen that F increases rapidly as R^ decreases. FIGURE 3.7 AUT OTRANSFORMATION FOR A TUNNEL DIODE SOURCE FIGURE 3.8 SERIES-TUNED CIRCUIT TRANSFORMATION FOR A TUNNEL DIODE SOURCE For small values of R^, such as are encountered in the reverse and near-forward bias regions, a transforming network must be used to step up the tunnel diode noise source voltage a) autotransformer: As shown In Fig. 3*7 the output is tuned to Improve the noise figure. By adjusting'the tapping ratio, the autotransformer can couple the tunnel diode with the high-impedance amplifier continuously from very small values of R^, to very large values where the autotransformer becomes a parallel tuned circuit. If losses (chiefly due to coil resistance) are included it is difficult to analyse an autotransformer in terms of noise figure, or even of voltage gain. Experimentally It may also prove unsatis• factory at frequencies higher than a few Mc/s due to the fact that an idealized analysis (losses neglected) Indicates the voltage step-up to be proportional to the total inductance of the coil and to its coefficient of coupling, rather than simply to the turns ratio. Self-resonance effects place an upper limit on the achiev• able value of the total mutual Inductance at higher frequencies; the resulting limited voltage step-up may represent an Inferior noise figure. b) series-tuned circuit: This circuit, .shown In Figure 3.8, has voltage step-up dependent only on the Q, and not on the Induc• tance, of the coil. A satisfactory noise figure is easily obtained despite the fact that a "match" of tunnel diode and amplifier Input 2 impedances is obtained only for r = R^ and l/(A»C) (Rd + r) equal I p 1.5 4 parallel-tuned circuit series-tuned circuit R • 0 15 ohm 1/6O0 FIGURE 3.9 COMPARISON OF NOISE FIGURES FOR SERIES- AND PARALLEL- TUNED CIRCUITS WITH TUNNEL DIODE SOURCE to the amplifier Input Impedance. The noise figure for the circuit In terms of source and unwanted noise voltages appearing across the input terminals xx of a noiseless amplifier is IjkT P = 1 + % + R (0>C)2(l + — \ <12> 2' where Cid>= 2e()lcv| + |lvc|), and has the same temperature as the overall circuit. The effective Q. determining the step-up of source and coil noise is l/(Rd + r)(£OC), less than the coil Q due to the damping of R,. a For the series-tuned circuit, F deteriorates as Rd decreases, as for the parallel-tuned circuit. However a good noise figure is obtained as long as R^ does not become small compared with r, which may be made very small. Assuming the use of a coil with Q ^- 100, 2 2 so that (60C) (Rd + r) is small, the effects of the noisy amplifier may be very largely suppressed with the series-tuned circuit, due to its voltage step-up; this is impossible when the parallel-tuned circuit is used with small R,. The smaller is R,, the better the a d series-tuned circuit suppresses amplifier noise, so that the noise figure is limited chiefly by inductor noise competing with the tunnel diode source noise. It is seen that R^ damps the series-tuned circuit. When R^ becomes ^ l/(£OC), the damping becomes excessive such that the parallel-tuned circuit assumes a better noise figure for the same value of R^ than does the series-tuned circuit. A comparison of noise figures as a function of Rd for the series- and parallel- tuned circuits is shown in Figure 3»9» (In assessing networks suitable for tunnel diode impedance transformation, it is to be noted that the criterion for maximum voltage across the output terminals of the arbitrary network is not necessarily equivalent to the transformed impedance at the output terminals of the network being matched to the input imped• ance of the circuit following. Similarly, the criterion for opti• mum power transfer from source to output of the transforming net• work may be specified only for a given network, since both the output voltage and the transformed impedance at the network output will be functions of some impedance associated with the network itself, e.g., the series resistance of the coil In tuned circuits.) 3.3 Development of a Low-noise Amplifier The noise figures of the foregoing source-transforming net• works are satisfactory for small values of amplifier noise resis• tance RJJ, which at frequencies not over 30 Mc/s specifies the noise due to vacuum tube circuits following the transformed tunnel diode source. This noise is usually due almost entirely to the first tube, if its associated stage has gain much greater than unity. For pentodes or tetrodes, which suffer partition noise, RN is typically 2.5 kohms or more, a value which is seriously detri• mental to the noise figure for the circuits discussed. Triodes, with RN typically 500 ohms or less, give a satisfactory noise figure. At frequencies high enough for the present work, a single triode Input stage suffers excessive input admittance in the amplifying grounded-cathode configuration due to the Miller effect which depends on the large grid-anode Interelectrode capacitance of triodes. The input admittance Is largely capacitive, but this seriously impairs the Q of the series-tuned circuit coupling the A.C.-EQUIVALENT CIRCUITS OF A CASCODE AMPLIFIER source Into this stage, and the noise figure suffers. The cascode or Wallman circuit (Wallman, et. al., 191+8) overcomes the Miller effect, which is proportional to the gain of the triode stage involved, by using a grounded-cathode triode Input stage followed by a grounded-grid stage; the latter acts as a low Impedance plate load for the grounded-cathode stage whose gain Is then low (usually about 1). The Miller effect is virtually inoperative under this condition so that a low input admittance is obtainable.' The overall gain of the two stages can be comparable to that of a single pentode stage, while the overall noise can be very little above that of a single triode stage. The conditions under which these properties can be realized are now given in detail. 3.31 Amplification and Noise of a Cascode Amplifier The essentials of the cascode circuit are as in Figure 3.10. (a) is the a.c. circuit and (b) is the Norton equivalent of (a>). To understand the cascode operation, the transconductance, plate resistance, and gain factor of a single tube which would be equivalent, electronically, to the cascode are calculated, g^ 2» rpl,2» 5113(3 ^1 2 ar9 resPeotiVQiy tne transconductances, plate resistances, and amplification factors of the l9t and 2nd tubes. To calculate the overall transconductance of the circuit, defined as the a.c. current I flowing in the plate circuit of tube #2 per unit input voltage on the grid of tube #1, the output terminals AB are imagined shorted. Then solving (- i + g^r^ • (- i + gxe)rpl = 0 with vn = (g..e - i)r gives r gl(l +Ai2) ni g x i/e = .-± 5_Ei ^ g, rp2+(l+/u2)rpl > > ar where the latter approximation applies when - ^ 1 *d rp^ £± rp2 (e.g. similar high-gain tubes used). The equivalent plate resistance of the cascode, defined as the change in voltage at the output plate per unit change of tube current is found by open-circuiting terminals AB. If V is the voltage across AB, then V = g V r + g er g er (1 2 l P2 l pl = l pl where v.. = g, er _ when 1=0. The effective plate resistance J- pi is then r V A r + U + ,P p = - / = p2 *2 Pl The overall gain factor is simply = v/e = ^x(l + ju2) Hence for the overall circuit, p. = gr • The high value of r causes the circuit to behave like a single pentode stage; with an arbitrary load across terminals AB it is easily shown that the circuit gain, A, is given by pAl + p0) Z pZL A = 1 k = h— gz (1+ + r + Z r + Z ^2^pl p2 L P L for r >> Z , as is usual for pentodes. P L The input impedance of the grounded grid stage is (rp2 + Z^) / (u2 + 1) which is required to be small, so that the FIGURE 3«11 NOISE-EQUIVALENT CIRCUITS OF A CASCODE AMPLIFIER gain of the first stage, and hence the Miller effect, will be small. Thus for overall large gain, as well as for small input admittance, it is required that p.^ and p.^ be large and that r^2 be small, that is, g^ be large. Under these conditions the resistance looking to the right at points xx (Figure 3.10(a)) is approximately l/gg, when the real part of Z^ is much less than as would be the case in wide band amplifiers. The resistance looking to the left at points xx is which typically is much larger than l/g2« This combination of low resistance to the right, and high resis• tance to the left, is the crucial property of the cascode circuit, with respect both to stability and noise figure. The gain of the first stage is approximately g-^/gg under these conditions, and this ratio is near unity. Such low gain makes the first stage stable, that is, have an acceptably low input admittance. Since the overall gain is approximately g^Z^ with 2' the output load, the cascode circuit is equivalent to a pentode of transconductance very nearly g^. The analysis shows the overall gain to be independent of g2 which is nevertheless chosen as large as possible to obtain as small gain as possible for the first stage. The criteria for low noise associated with the cascode circuit are now considered. Neglecting noise of circuit resistances com• pared to tube noise, the noise-equivalent circuit for the cascode is shown in Figure 3»H» The generators i^ and i2 denote the noise sources for the tubes. The Input grid is short-circuited, the noise of the system then being specified completely by the short-circuit noise current i in the plate circuit of the second n tube. With terminals AB short-circuited, 57 i = (1 +^2)rpl h + rp2 *2 rp2 + U '^pl The denominator of this result is r , the overall plate resistance P of the cascode. The noise due to the first tube already is seen to dominate. In order to represent the cascode noise as an equi• valent thermal noise resistance Rn in series with the grid circuit of the first tube, one forms 11 ) : *^\t l> * D2< l> r [rp2 + (1 + ^2> pl] where the noise generators i^ and ig are assumed uncorrelated. Now it is well-known that the output noise in the plate circuit of a single tube can be represented by the amplified thermal noise of a resistance Rn appearing at the grid of a noiseless tube, by the relation Thus AtR . (1 2 r : [rp2+ d+/.2)rpl]' The equivalent noise resistance Rn for the complete cascode cir• cuit will be defined as R = f n Then 58 (1 + )2 g r f R + *p| Af R R. /*2 l pl ^o^ nl *2 ^o n2 n 2 l4kT0Af [gl(l+ ^2) r ] 2 L _ + | ^ -l R^ ~ R, + R n / u? nl n2 nl n2 L^(i+/i2)J ^ In the usual way, we put R £^ €/ &i o (se9> ^or instance, hi, 2 Van der Ziel, Noise, 195>3> P» 102), This relation applies for a triode operated in any condition. € is a numerical constant, of value approximately 3* Thus R n Hence the main condition for a low-noise cascode circuit is a large value for g^. It has been stressed that the equivalent amplifier noise current generator representing induced grid noise and noise associated with conductance due to feedback, transit- time loading, or to input circuitry, can always be made unimportant at the frequencies of Interest in this thesis. Thus the cascode noise is to a very good approximation specified solely by R^, 3.32 Cascode Circuit Designs Favoring Stability Besides minimizing the value of Rn, the choice of tubes with large g^ and g2 has been shown as the main criterion for large gain for the cascode, since for a fixed output load Z , the gain ii Is close to g ZL. The tube type 4I7A (581|2) was selected for both stages of the cascode used to measure tunnel diode noise. The value of g is 27 ramhos, of plate resistance is 1600 ohms, and of amplification FIGURE 3.12 TYPICAL A.C.-COUPLED CASCODE AMPLIFIER factor is 44. The problem of stability in an amplifier using tubes of super-high transconductance becomes difficult. A high-g tube acts as a large current generator which favors feedback both parasiti- cally and electronically, especially by magnetic coupling. Most commercial cascode circuits avoid use of super-high transconductance tubes, and use a.c.-coupled stages in preference to direct-coupled stages. A typical a.c.-coupled circuit Is shown in Figure 3»12. L^ provides a d.c. return path to ground for the grounded-grid stage and at high frequencies (e.g. above 30 Mc/s) is made resonant with Ogp, the grid-anode tube capacity, of the grounded-cathode stage to prevent excessive grid-anode coupling. Also the noise figure is often slightly improved by tuning C at higher frequencies. Lg tunes stray capacity between cathode of second tube and ground, although very broadly because of the heavy input loading of the grounded grid stage. Tuning is shown for both Input and output circuits although this may be undesirable at lower frequencies, where it is unnecessary, because of bandwidth res• triction. At frequencies as low as 4 Mc/s, used in the present study, the grid-anode feedback impedance l/WC of the first stage cannot be significantly increased by tuning C , nor can Interstage strays SP compare with loading of the grounded-grid stage. Neither L^ nor Lg are therefore useful to the signal operation, whereas along with the inductive strays of the wiring associated with the several Interstage components, they enhance magnetic parasitic feedback. With use of the super-high transconductance ql7A tube, it was found that even such precautions as mounting all coils with mutu• ally perpendicular axes, and use of extensive shielding does not FIGURE 3.13 SIMPLEST DIRECT-COUPLED CASCODE AMPLIFIER 60 guarantee a stable amplifier. Instability can also arise through coupling of stages through the high-voltage supply in this circuit. Even small or heavily-damped oscillations are manifested by sharp increase in the input conductance, due probably to grid current In the first stage, or to the onset of grid current in any circuit following the cascode so that a large output conductance arises, which is electronically fed back to the Input. (One advantage of the indirect-coupled circuit of Figure 3*12 at very high frequencies is that the grid of the second stage is at d.c. as well as a.c. ground. This allows a short, non-reactive connection to ground, whereas in the direct-coupled circuit, to be described next, the grid must be capacitively coupled to ground. It may be shown that any stray inductance In the grid lead can produce not only instability but also negative Input conductance for a grounded-grid stage, especially at higher frequencies.) Rather than achieve stability either by use of neutralizing feedback networks (which guarantee stability only over a narrow frequency band) or by biasing the tubes to decrease the transcon• ductance (thereby losing the advantage of high transconductance which minimizes amplifier noise arising from the tubes), a simpler circuit using direct-coupled stages is preferable. A simplified version is shown in Figure 3»13» The tendency for magnetic para• sitic feedback is reduced by the use of at most only one coil, and by the minimal number of interstage components which permits direct point-to-point wiring, thus reducing spurious inductive coupling between stages. This form of coupling is particularly troublesome when high-g tubes are involved, whereas capacitive feedback is likely harder to control when high-71 tubes are used. For d.c. operation, each tube in Figure 3*13 carries the same current so that assuming the tubes are identical (and neglecting the drop across R^)» the bias of the grounded-grid stage auto• matically adjusts itself by cathode follower action so that the current through each tube is the same. The bias on each tube Is also approximately the same. The d.c. potential of the second stage grid is fixed at about half the B+ voltage by the potential divider (the setting of which is not critical: the self-regulation action of the tubes maintains very closely the same bias over a wide range of potential divider settings). A practical embodiment of the direct-coupled circuit of Figure 3.13 includes the following: a) the heaters of the second stage must operate at close to the d.c. potential of the cathode, to prevent excessive 60 cycle injection from heaters to cathode, as well as arcing. b) the components in the first stage cathode and second stage grid circuits must be dressed close to the chassis, to minimize Inductive loops. c) all heater and B supply circuits must be carefully de• coupled or filtered at the operating frequency of 4 Mc/s. d) in contrast to a grounded-cathode stage, where any reflec• tion of the output load to the input is due solely to parasitic coupling of the grid and anode, in a grounded-grid stage there is direct coupling (electronically) of the output load at the input of the stage. This property is Independent of parasitics. For this reason the input impedance of a cascode circuit is not as well isolated from the output as it is in a 2-stage cascaded amplifier (this lack of isolation Is further enhanced in the use of i|17A tubes due to their large interelectrode capacities). If changes in the output load are unavoidable (e.g., If a multi-ratio >v 417A II 0.1 vwv 1 Meg. / 1+.7 kohra 2.2 kohra 0.1 100 kohra 100 kohm rVWVV-rWWV 0.1 0.1 FIGURE 3.Ill PRACTICAL DIRECT-COUPLED CASCODE CIRCUI WITH OPTIONAL CATHODE-FOLLOWER STAGE AND TWO-POSITION ATTENUATOR I 6922 (all capacitors are in yuFd. unless otherwise marked) Heater Decoupling Arrangement o attenuator with only moderately constant input impedance forms the cascode output load), but if strictly constant cascode Input impedance Is required, as when the Input impedance loads a series- tuned tunnel diode coupling circuit, then a cathode-follower stage Is useful in isolating the cascode from varying output loads. Accordingly the circuit shown In Figure 3«lfy was adopted for noise measurements, with the cathode-follower stage optional. For two reasons, the circuit is designed to be broad-band (the resis• tance in the output tuned plate load heavily damps the tuning): a) in order for both tunnel diode signal and cascode noise to override the noise of the high-gain receiver which follows, the limiting bandwidth for the overall system should be imposed by the receiver rather than by the cascode; the worst condition is when the source itself limits the bandwidth. b) even in the direct-coupled cascode, which greatly reduces the tendency of feedback by magnetic coupling, instability tends to occur by tuned-grid tuned-plate action, the grid-anode capacity of the first stage providing the coupling. The amplifier may tend to oscillate when a series-tuned circuit Is connected to the input grid, although it may otherwise be stable. Such action is dis• couraged by damping the output tuned circuit, while Increasing the bandwidth. The gain of the cascode can still be large, regardless of the high load conductance, since the overall transconductance is very large. The gain of the cathode-follower stage Is g Rk / (1 + g R^) where R^ Is the total cathode resistance to ground, g R^ is made large for a gain close to unity, and also to minimize the effective Input capacity of the stage, which is given by Cg^ / (1 + g R^) added to C . A suitable tube with small C _ and C , but large g gp SP ia the 6922 or E 88 CC. The cathode resistor is split as in Figure 3.1i|. to maximize R^ while maintaining correct bias. It is desirable at higher frequencies to tune the cathode circuit of the cathode-follower, since excessive capacity can be shown to produce negative conductance at the input of the stage, although this may be small. The noise of the cathode follower stage is equivalent to approximately a f>00 ohm thermal source, since the 6922 is a triode. This adds directly to the 2.5 kohra equivalent noise source of the receiver; the overall noise figure of the system depends negligibly on inclusion of the cathode-follower stage,' since the cascode gain is sufficient that all noise following it is overridden. 3.33 Performance of the Cascode This discussion excludes the cathode-follower stage. Since the gain of the cascode depends critically on the impedance in the plate circuit of the grounded-grid stage, the gain should be measured with the cascode connected and tuned with the input of the receiver which is to follow it during noise measurements. Using a l^OO-cycle modulated signal generator with calibrated r.f. output voltage, one applies a convenient signal to the cascode Input which overrides the circuit noise, and notes the receiver response, using an RMS a.c. VTVM as a recorder. The signal gener• ator is then connected directly to the receiver input and the signal level increased until the same receiver response is regained. One insures that the bias of the first receiver stage does not change when the signal generator, which has very low d.c. internal resistance, is connected. Since the receiver gain is therefore constant, the ratio of the two signal generator levels gives the FIGURE 3.15 SCHEMATIC NOISE CIRCUIT FOR MEASURING Rn OF AN AMPLIFIER 61» voltage gain of the cascode. With an output load of typically 4.7 kohras, and the 4l7A!s biased for a g of 25 mrahos, the gain was typically 100 for the circuit used. This is sufficient to override the receiver noise. The equivalent noise resistance, Rn, of the cascode, can be measured several ways, most of which utilize the schematic circuit of Figure 3»l5« Rjj is assumed closely to represent all of the amplifier noise, and the cascode input conductance is assumed negligible compared to l/R , where R Is the r.f. plate load of a P P noise diode standard source. The amplifier input being tuned, R P is about 1.5 kohms. For input levels which do not overload it, the cascode response can safely be taken as a linear function of the input voltage. An attenuator separates the cascode from the receiver, whose response may not be assumed linear. Then Rfl for the cascode is measured as follows. 1) For the attenuator set at a voltage reduction ratio A^, and the noise diode circuit connected to the cascode input with the noise diode current zero, the receiver RMS output voltage is n 2 0 = k'[A2 4kT0(Rn + Rp)] / if T is the temperature of both R and R . With the attenuator P ratio now at A2 ( 2 n/2 £ = k[A^qkT0(Rn + Rp) + 2el R , }] where I is the noise diode current. Solving for RR gives 65 2eIR,g2 2™0 DR2: 2 R 2 R ^ • IjkT^A^) - l] " P " (Al/A2y - 1 " P An external bridge measurement of Rp is required to great accuracy, since R depends oritically on R . To avoid this disadvantage, the method may be extended as follows, 2) The attenuator first is bypassed and the amplifier input is shorted, giving a receiver output n 2 B =* k[hkToRj / With Rp connected to the input, the attenuator is adjusted to a factor A^ which produces the same receiver output: 2 n/2 e = k.[A itw^ + Rp)] Finally, the attenuator is set at A2 and the noise diode turned up to regain the same receiver response: 2 n/2 6 = k.[4 {llkT^ + Rp) + 2el R1] These relations combine to give 2 Rp (1/201)(Af - A ) / (1 - A§) n ~ (1/Aj) - 1 " (1/A2) - 1 3) If the law of the receiver is closely linear, the above method may be used without an attenuator, 0Q is then taken as the receiver output when the cascode input is shorted, 0^ as the receiver response when the noise diode circuit, that is, Rp, is connected to the cascode input, and &~y as the receiver output when the noise diode is turned on to any convenient current I, Rn Is then given by R„ = (1/201)(0f - ef) / te2 - eg) 4) A much simpler, but less accurate method of finding RR is to connect a resistance across the cascode Input of value such that the noise power, as detected at the receiver output, is doubled over that due to the amplifier with shorted input. The value of such a resistance then equals Rfi. Generally Rfi Is small enough so that the noise generated by tuned circuits in shunt (which insure that the substitution resistance is not shunted by the ex• cessive input capacity of the cascode) can be neglected. By the above methods, and at a frequency of 4 Mc/s, R^ was measured as 50 ohms ( + 20 ohms) for the cascode circuit with 4I7A tubes biased to a transconductance of about 25 ramhos. This value of Rfi is lower than expected by the foregoing noise analysis by a factor of three (negative feedback of an uncontrolled nature may be involved In the discrepancy). If Q as 100, r = 3 ohms for the coll of the series-tuned cir• cuit which couples the tunnel diode into the cascode, then the equivalent thermal noise source connected to the cascode is about 4 kohms when the tunnel diode assumes its zero bias resistance of about 20 ohms. The cascode noise is negligible in comparison, so that the overall noise figure of the circuit for measuring tunnel diode noise is determined mainly by the coil noise of the series- tuned circuit. series- or low- atten• high-gain integrator Esterline- parallel- noise uator narrow or Angus tuned cascode for band smoothing recorder circuit -| pre• pro• receiver -i circuit for amplifier visional with noise use two levels I.P. frequencies RMS a.c. VTVM for calibration T levels -0*0- tunnel calibrated noise diode resistance diode bias sources bias and r.f. and r.f. circuit circuits FIGURE 3.16 BLOCK DIAGRAM OF COMPLETE NOISE noise MEASURING CIRCUIT diode 1|00 cps filament modulated control r.f. signal circuit generator for calibration 67 3»U Other Apparatus and Circuitry 3.ql Perspective of the Overall Circuit Figure 3«l6 shows the relation between the major sections of the complete circuit for the measurement of noise in the tunnel diode. The Impedance-transformed tunnel diode source, as well as the noise diode source, which must always be confronted with the same Impedances and transformations for a valid comparative noise measurement, is shown coupled to the low-noise preamplifier. The attenuator which follows may be bypassed; its use in measuring the equivalent noise resistance of the cascode has been explained; it will later be shown to be unnecessary in comparing the tunnel diode and noise diode sources. The use of the standard or calibrated resistance sources, and the role of the signal generator are explained in detail shortly, in connection with the particular method adopted to compare tunnel diode and noise diode sources. Briefly, calibrated resistors, of known (thermal) noise generation, are switched into the tunnel diode position to calibrate the preamplifier and other unspecified noise sources, Including those of the series-tuned circuit, and other resistive shunts. The tunnel diode is then switched in and biased until Its resistance equals the calibrated resistance, so that the overall impedance conditions remain unchanged, as is necessary for the comparison of tunnel diode and noise diode sources. The signal generator provides the reference signal by which the tunnel diode resistance is made equal to the calibrated resistance. This signal appears on the a.c. VTVM at the receiver output, where• as the much smaller noise signals appear on the more sensitive to series-tuned circuit input to filament control circuit FIGURE 3.17 NOISE DIODE AND TUNNEL DIODE BIAS AND R.F. CIRCUITS (all capacitors are in jiFd. except "C"). 68 Esterline-Angus recorder, after suitable integration. 3.1*2 Noise Diode and Tunnel Diode Bias and R.F. Circuits Figure 3.17 shows the circuitry associated with the tunnel diode and noise diode sources. Besides the series-tuned circuit input impedance, the r.f. load for the noise diode and the equivalent noise current generator of the tunnel diode is the dynamic tunnel diode conductance, which for the bias region principally under study, has a minimum value of about 1/120 mho. Due to the very large voltage division imposed on the equivalent noise generators of the j? kohm and 10 kohra resistors, the noise contributed by them at the series-tuned circuit input is negligible compared to that of the tunnel diode and noise diode sources. (The combined conductance of the two £ kohm resistors, the 10 kohm resistor, and the parallel-tuned circuit never exceeds l/l800 mho). All resistors in the r.f. circuit which carry d.c. current are wire-wound to avoid current noise. All 0.1..pFd. blocking or bypass capacitors are ceramic or mica, to Insure non- inductive behavior at the high frequency. The d.c. load line for the tunnel diode I - V characteristic is set by the $ kohm d.c. feed resistor in its bias circuit. This prevents the tunnel diode from operating in the negative conductance region, since the diode switches along the load line from any un• stable operating point in that region. All operating points in the positive conductance regions are stable. Very fine bias control for the tunnel diode is obtained with a 1 kohm 48-turn helipot connected as a potential divider for a 22-volt battery which pro• vides a more noise-free current than would a vacuum tube power supply. The bias voltage is measured with a calibrated high- impedance d.c. VTVM, suitably r.f.-decoupled from the tunnel diode as is the battery supply. Switch S (Figure 3«17) is closed when• ever the bias is read. The d.c. current through the tunnel diode is measured with a moving coll milllammeter which has been cali• brated against a standard \% Weston ammeter, as has the moving-coil milliammeter in the noise diode plate circuit. The B+ for the noise diode Is decoupled for both low and r.f. frequencies, the former to prevent super-position of 60 cycle with the r.f. signal at the noise diode anode. The parallel-tuned circuit serves fur• ther to short-circuit any superimposed 60 cycle or its harmonics, which could otherwise pass through the r.f. cascode circuit to modulate the r.f. in the receiver, or overload the first stage. (The parallel-tuned circuit coil is also useful In preventing charging transients arising from any rapid variation in the noise diode B+ from appearing across the tunnel diode.) The i|00 cycle modulated r.f. signal generator injects through either R, the calibrated resistance, or the tunnel diode, a signal into the series-tuned circuit, whose response will critically depend on the value of the damping resistance R. The tunnel diode resistance can be made very accurately equal to the known calibrated resistance by adjusting its bias until the series-tuned circuit response is the same whether R or the tunnel diode provides the damping. The way in which circuit noise excluding that of the tunnel diode is accounted for by this technique Is presented in the following section. The 10-ohm internal resistance of the signal generator renders the calibration technique insensitive when the tunnel diode resistance or R becomes less than 10 ohms; in that case, a 1-ohm shunt placed across the signal generator output restores the sensitivity. Since the input impedance of the series-tuned circuit is only about 3 ohms at resonance, significant voltage division of the noise diode and tunnel diode signals may occur if there is any impedance in series with these sources. For a valid comparison of these sources, the O.ljaFd. blocking capacitors in series with each of these sources must present closely similar impedance. Precautions are taken in the circuit wiring to introduce a minimum of unspecified inductive strays which would act in series with the tunnel diode and noise diode. The tunnel diode and output of the noise diode circuit are built close together and to the input of the series-tuned circuit to further improve the high-frequency characteristics. The noise diode filament circuit is decoupled and shielded with copper plate partitions from the rest of the noise diode circuit, and the entire noise diode circuit is shielded from the tunnel diode circuitry and from the output parallel-tuned circuit. This prevents r.f. or l.f. coupling between the different circuits. All circuitry Is shielded from the surroundings by completely enclosing metal chassis. 3«43 Noise Diode Filament Current Supply At noise measurement frequencies much above 60 cycles, the filaments of the 5722 noise diode can operate on 60-cycle power, the thermal time constant of the filaments being long enough that no high harmonics of 120-cycle modulation of the emission results. The Sylvania 5722 noise diode operates with filament current be• tween 1 and 2 Amperes, corresponding to anode currents up to 35 mA. in the temperature-limited condition. The anode current (and hence the noise power output of the noise diode) depends very sensitively (exponentially) on the filament current, and since the to noise diode filaments filament transformer 100 kohnf 6CL6 100 kohm helipot 1> W\AA/ 220 ohms power 0,5 uFd. transformer regulated T 110 volt mains FIGURE 3.18 NOISE DIODE FILAMENT CURRENT CONTROL CIRCUIT 71 noise power must be finely adjustable and stable at any arbitrary level during noise measurements, the filament current must be con• trollable and stable to a degree beyond the capability of a potentiometer which could carry the large current required. The circuit of Figure 3 •18 solves the problem. The plate-to-plate impedance of two triode-connected 6CL6 power-pentodes in push-pull operation is stepped down by a 3:1 turns-ratio power transformer and Inserted into the primary of a filament transformer which powers the noise diode filaments. The 6CL6's are push-pull connected to cancel the d.c. current in the power transformer secondary which would otherwise saturate the core, giving Improper impedance transformation as a result. The resistor network between the -300 volt supply and ground Is select• ed to allow variation of the 6CL6 grid bias from -50 volts to -25 volts (the 1 kohm power resistor across the helipot reduces the power-dissipating requirements of the helipot). The transformed plate-to-plate impedance inserted into the filament transformer primary then varies over a range which causes the filament current of the noise diode to be finely continuous over the desired range. The 220-ohra resistor improves the linearity of current control over the helipot range. The 0.5 ^iFd. capacitor inhibits the tendency for slow oscillations of the control circuit when first turned on. The 60-cycle supply feeding the filament transformer primary must be carefully regulated: an ordinary Sorenson regulator is sufficient. 72 3»kk Detection of Noise Signals Following the cascode-amplifled noise signals, a high-gain receiver (Airmec Type C86I4) with two Intermediate frequencies and conventional crystal diode detector Is used to convert the 4 Mc/s noise signals to audio frequencies. The receiver, with a gain typically of 10^, is modified for noise measurements in the following ways. a) the AVC is disconnected. b) the front-end antenna tuned circuits are disconnected from the first r.f. stage, to which the cascode output is directly connected. A reasonable impedance match is then obtained. The receiver is now tunable solely with the first local oscillator, so that care is taken to insure that it is always tuned to the same frequency as the cascode and not to an image frequency. If methods of noise measurement were used which require knowing the receiver response law, the law could be standardized by connecting a square-law detector such as a thermocouple to the output of the second I.F. amplifier of the receiver, or to an additional I.F. amplification stage if needed. This Is a common procedure which is avoided by the method used in this thesis. The signal level of the receiver output is adjusted by two sensitivity controls: the "L.F. Gain" controls only the signal level entering the audio stages from the detector, while the "H.F. Gain" controls the I.F. signal before the detector, which for this receiver is the vulnerable point to overload. During noise measurements, the H.F. Gain is adjusted to Insure no detector overloading, while the L.F. Gain is set at maximum. To measure accurately the audio mean square noise power IS 3k 6 Esterline- R< Angus Recorder O a FIGURE 3.19 CIRCUIT FOR INTEGRATING NOISE SIGNALS • 73 i output of the receiver (which Is proportional to the mean square r.f. noise power entering the receiver), integration is required. The audio noise signal is rectified and filtered with a long time- constant circuit as shown in Figure 3.19. R-j^ and R2 provide the d.c. circuit for a lN3q germanium diode. blocks the high voltage of the receiver audio output tube, the integrator circuit being connected to the primary rather than to the secondary of the audio output transformer to derive larger signal strength. Gfe (0.03> ^uFd.) and R^ (2.2 kohms) are also designed to attenuate heavily the low frequency components of the noise signal before they reach the germanium rectifier. The low frequency components are difficult to Integrate, since their associated autocorrelation function has significant value for time intervals of the order of the time constant of the R^ - C filter circuit, which was about 0.25 seconds. Thus the fluctuations in the recorded integrated output are reduced when the - R^ network attenuates frequencies under about 200 cycles, while the overall bandwidth of the noise appearing at the rectifier circuit is very little decreased, so that the output strength does not suffer. The small fluctuations that remain are recorded over at least one minute on an Esterline- Angus paper recorder which permits an accurate comparison of two small-fluctuation signals, since all of the averaging Information entering the recorder over a conveniently long period can be utilized in forming the final average by eye. FIGURE 3.20 COMPLETE NOISE-EQUIVALENT CIRCUIT FOR TUNNEL DIODE NOISE MEASUREMENT 7k 3«5 Adopted Noise Measurement Procedure The method to be described avoids the use both of an attenu• ator, which must be calibrated, and the dependence upon the law of a receiver. The complete equivalent noise circuit for the r.f. noise measuring system is shown in Figure 3»20. R' represents conductances associated with the tunnel diode bias circuit, which are always In shunt in the tunnel diode branch. The conductance In the noise diode r.f. circuit, which includes the parallel-tuned circuit (see Figure 3*17) is repre- sented by Rp. "C^g^^ specifies noise in the series-tuned cir• cuit. R is the calibrated resistance with which the tunnel diode is compared. Let ^i^> represent the total noise generated between the terminals of the tunnel diode (that Is, both tunneling current noise and thermal noise for the bulk resistance are included). The procedure for finding currents. Then putting V"T = 2kT/e, where T is the actual absolute temperature of the tunnel diode, and noting that the I-V characteristic for the tunnel diode has 5 I /<5 V < i/V for all V = 0 in the near-forward bias region, we have, for all V = 0: 2 2 = 2eITDcoth(V/VT)> 2eITD(V,IA) > 2eVT(3l^V) 5 l*kTG = That is, <1a>><1R> in the near-forward bias region. However in the reverse bias region, the I-V characteristic shows that 31 /5 V > i/V. It 75 is therefore impossible to decide from the I - V characteristic which of ^i2^ and K, i2 > Is larger In the reverse region. 2 2 Experimentally it Is found that region. These inequalities are used in the following procedure: a) with R switched in and the qOO-cycle modulated signal generator delivering a fixed Mc/s signal of level sufficient to override circuit noise, the receiver output is noted on the a.c. VTVM. (At frequencies such as i\ Mc/s it is advantageous to use a signal generator rather than the noise diode signal for equalizing tunnel diode and calibrated resistances, if, as here, it can be assumed that the conductance of both tunnel diode and calibrated resistance is frequency-independent over a frequency interval equal to the bandwidth of the overall circuit. This is because the sig• nal generator can produce a non-fluctuating output if it overrides circuit noise, whereas a noise diode signal causes small error in equalizing calibrated and tunnel diode resistances, due to small fluctuations in the integrated output. However a noise diode must be used at frequencies of 30 Mc/s and higher.) The tunnel diode is then switched in and the bias (forward or reverse) adjusted until the VTVM reads as before for the receiver output. Bias current and voltage for the tunnel diode are recorded. For biases in the valley region and far-forward region, the bias voltage Is increased until the diode current reaches the peak point, then it switches over to the upper positive part of the I - V curve. From there the desired bias is reached by decreasing or Increasing the bias voltage. Excessive decrease will cause the diode to switch to the near-forward region. b) whichever of ^i,^ and Assuming momentarily that Cl2^ » the analysis for the method Is as follows. Let K, v2 'y represent the mean square noise voltage appearing on the grid of a noiseless amplifier-receiver combination due to the noise generators < iR) > , < , < 1^ > ', > , as well as receiver noise (though negligible) which may be represented v by an additional noise generator at the cascode input, ^ a ^ will be constant since the total Impedance across the equivalent noise current generators is always the same. For the forward bias regions, and when the tunnel diode is switched in, 6 » k«[ 2 where Z is the total Impedance across the generator < i^ > , hence is dominated by the tunnel diode conductance. The same output 2 noise level is now obtained when < iD /> is switched in and the n noise diode turned up: 2 2 e - 1I.[ These relations give 2 2 2 Q 2 2 2 2 2 n 2 0 = k'[ 2 2 2 =- <1n d> Thus for any bias, o p p + for forward bias The method is seen to be Independent of assumptions regarding whether ^in^ can be neglected at low frequencies, or whether the cascode gain is sufficient to neglect receiver noise. A reasonable noise figure is needed, however, due to the limit of sensitivity of variation of the integrated signal with small differences in source. The generator < iR(> is negligible in the reverse bias region P since R£ is entirely swamped by the tunnel diode conductance, but may contribute slightly in the forward bias region near peak or valley points of the I - V curve, where R = R^ becomes a few per• cent of R£, which by external bridge measurement is about q kohms. Since "^i?.. > always appears additively with ^ i2 > then P For Li i_l • s 2eITD 2e [lTD H I R«J " Ifc, where I„,n and Iwn are the average currents In the tunnel diode and noise diode respectively. ^ 2 rather than Y 2, the parameter representing tunneling current noise alone, is the only directly measurable quantity, since the bulk thermal noise can never be separated from the tunneling current noise by any noise experi• ment. Algebraically, If2 is found by use of the noise model of the tunnel diode, given In Figure 2.2. The equivalent circuit shown there gives 2 2 2elm„ 2elmn TD TD Combining the latter two expressions for & 2 gives i*kT Rb Y 2 = (Hf \— —l1 2e - 1\R + ^ 29 R< ° UJ L W J " JTD ^TD t (3.5.1) where R = Rd + R^ and T = 300°K. is the actual temperature of the circuit, for which l+kT/2e has the value l/l9»4 volts. ^ is of course dimensionless. Rb, the bulk series resistance, is specified by the tunnel diode manufacturers (Sony Corporation) to be 1.5 ohms. In carrying out the experiment, one selects values of R corresponding to bias points on the tunnel diode I-V curve evenly spaced along the voltage coordinate (V being the signifi• cant bias parameter for a voltage-controlled device such as a tunnel diode, and which appears in the theoretical relation, equation (2.2.1) for "Jf2 ). The values of the calibrated resistors R are found by accurate bridge measurement at 4 Mc/s. To compare the experimental values of ^2 given by equation (3»5«1) with the theoretically expected values given by ^2 = coth (eV/2kT), the latter must be modified to account for the fact that V is the voltage across the tunneling junction, whereas we measure a voltage V» across junction and bulk resistance lumped together. The experimental result given by equation (3«5.1) is therefore compared with the theoretically expected relation 2 V = coth (eV/2kT) = coth [e(V» - ITDRb)/2kTj in which the voltage drop across the bulk resistance of the tunnel diode is subtracted. ITD(mA.) 130 ohm 8$ ohm 37 ohm 2b, ohm 18 ohm 21 ohm V'(mV.) —J— —t— -\ ?*• -60 -20 20 40 60 16 ohm -1 )12.3 ohm -2 -3 FIGURE 4.I 9.6 ohm CURRENT-VOLTAGE CHARACTER• ISTIC FOR SONY ESAKI DIODE IN NEAR-FORWARD AND REVERSE BIAS REGIONS (dloda resistances are Indicated) -5 7.7 ohm -6 -7 f?.f? ohm --8 80 CHAPTER k EXPERIMENTAL RESULTS AND INTERPRETATION q.l Reverse and Near-forward Bias Regions The I - V characteristic for these bias regions for the Sony Germanium Esaki diode under study, is presented in Figure 4.1. Typical voltages, currents and resistances are shown. In the near-forward and reverse regions the noise voltage at the output of the series-tuned circuit due to the noise diode is always equalized to the absolute difference of the voltage at the 2 2 2 same point fR/(R + r)] (l/0>C) 2eITE)y Af due to the tunnel diode, and the voltage [R/(R + r)]2 (1/iOC)2 (I|kT/R)A,f due to the calibration resistance R. The ratio of tunnel diode shot noise current generator to thermal noise generator at the same conductance should be given by v/vT IA tanh (V/VT) dl/^V with,VT =s 2kT/e. The first factor increases, but the second factor decreases with Increasing bias V. Experimentally it Is found that ^i^> - ^i^^ increases quite linearly with V in the near-forward region, and also that the tunnel diode noise signal voltage is quite constant over this bias range. The latter indi• 2 cates that 2eITDV Af decreases as V Increases at about the same rate that [R/(R + r)]2 increases (the noise of the series- tuned coil being quite small throughout). In the reverse region, the mean-square voltage at the series- tuned circuit output increases slowly with reverse bias. This is due both to the shape of the I-V characteristic which shows ITD Increases rapidly for small increase In V (that Is, small decrease in V2), and to the decrease in series-tuned circuit damping as R decreases in the reverse region, allowing the coil noise step-up to increase. Since R does not vary rapidly with bias in the rev• erse region, the noise output voltage behavior of the series-tuned network indicates the behavior of 2*1^*1 with bias fairly closely. As in the near-forward region, the noise diode output must be increased as tunnel diode bias increases, that is, Ci^> - In the equation (3«5>«1): ,2 T hkT ND ° llU 1 I L2e ITTDD\ "R "R'p 'j TD J 2e ITDR| from which the experimental values of $2 are computed, with I always the absolute magnitude of the average tunnel diode current, the data obtained show that as R decreases steadily from its peak value of about 130 ohms to a value of about $ ohms in the far- reverse region, the following behavior occurs: a) the term including (l/R - l/R£) increases steadily from an Insignificant contribution to $2 near the peak, to the dominant term In the far reverse region, b) the term Including Ijjj)/lTD almost entirely determines V near the peak region, but Its relative importance decreases steadily until in the reverse region it Is insignificant, c) the bulk resistance term forms a negligible percentage of at peak biases but Increases in Importance as R decreases, until in the far-reverse region (that is, V < - 2kT/e volts) it contributes up to 2$% of H Here the acouracy, as well as the definition of R^, in terms of the value specified by the manu• facturer, becomes important. In assessing uncertainties to be assigned to the experimental terms in equation (3.5.1) the following empirical behavior Is relevant. The difference in the receiver output 0 for < i2 > and response Is found to be very insensitive to the value of IND in this region. However the term involving %D/1TD in equation (3.5.1) is insignificant in the reverse region, hence so is the error in 2 uncertainty in Imn« For accuracy, all measurements in the reverse range were repeated and the resulting values of 1^ measured for a given value of R were averaged. The deviations were small. Excluding bulk thermal noise in the tunnel diode, if it is assumed that near the origin < i2 > = 2e 1^ }$2 C± q.kT/R, then 2 ^ d i|kT/2eITDR which, neglecting the noise of R£ , Is the dominant term in equation (3.5*1)• (With bulk thermal noise included as in Figure 2.2, there results under the present approx• 2 2 imation (R/Rt) (4kT/2eITDR) - (l|kT/2e) (Rb/lTDR ); this form according to the approximation holds best near the origin). It therefore follows that In the reverse bias region and for small forward bias, tf2 can be found quite accurately even If the noise experiment is omitted, since to a good approximation the only Information needed In equation (3»5«1) is R as a function of ITD for the tunnel diode. This implies that for the reverse bias region, the I - V characteristic determines the noise, or converse ly, if the device displays shot noise, the Information obtained from a noise experiment is only a minor correction to X ^. This approximation is increasingly inaccurate for increasing reverse biases, although even for biases approaching the power-handling capability of the tunnel diode, the I - V characteristic alone predicts the noise to within 10% accuracy in ^ 2. The inaccuracy Increases much more rapidly In the forward region with increasing bias, since R increases rapidly. If we assume, for forward or reverse biases much less than 2kT/e, that the noise of the tunneling currents closely approxi• mates thermal noise for a resistance of the same value as that of the tunnel diode, then 2 = 2eITD coth(eV/2kT) ^ 4kT ( 3 $ V) (where the component of ^ijij ^ describing bulk thermal noise in the diode is included and improves the approximation). With VT = 2kT/e, the differential equation S dITD/lTD Ci [coth(VAT)]dVAT coth (V) dV« results, assuming temperature is constant, with solution ITD ~. I0 sinh (VAT) For V << V^, this relation describes the I-V curve near the origin; the form Is concluded solely from the noise properties near the origin since they verify that llrt„/l,r_l = exp (eV/kT), The possibility of using this result with the Esaki integrals for I s li - I I to determine the functional forms for Z or for TD ' cv vc1 cv F(E,V) = Z(E,V) £ (E) f> (E + V) is Impossible since for biases near the origin, the Esaki integrals cannot be approximated. In summary, the comparison of < if> , or to a good u t approximation, with K I > as a function of V shows that: i+kT(<)l/<5 V) < n ljkT(c>lA> V)| and < 2el-, "tf 2 X 0 v1,i1>o ]v=o for forward bias 4kT(dl/<>V) > l|kT(dl/c> V)l and > 2el-, K2 A 0 v1,i1 for reverse bias. I = ITD is the average tunnel diode current and T is the actual diode temperature. If the noise temperature Tn of the tunnel diode is defined as T 2eI n " TD*o / i4k(^)lTD/aV) then for constant temperature T, Tn < T for reverse biases, and ^ Arc tanh (l/tf 2) peak voltage •1.0 = 66 mV. FIGURE 1|.2 THEORETICAL AND EXPERIMENTAL COMPARISON FOR TUNNEL DIODE NOISE IN THE NEAR-FORWARD BIAS REGION •0.8 (RMS deviation of experimental points from theoretical straight line is 2.0 % ) •0.6 theoretical curve: slope = e/2kT = 0*0194 mV."1 for T = 300°K. ••0.4 --0.2 0 10 20 30 ko V (mV. .T > T for forward biases. The ratio Tn/T cannot be evaluated nor limits assigned to it without a relation for I,pD as a function 2 of V. Whether I|kT(^Imn/ ^V) I is less than 2eI(T,T.> for forward IV=0 iD 0 bias, or greater than 2el^^%^ for reverse bias also depends on the I - V characteristic. That the magnitude of the shot noise 2eITDy2 and indeed the fact that it may exceed or be less than thermal noise for the same resistance depends on the bias and I - V characteristic, Is due to the noise dependence on the two currents Icv and Ivc flowing in the tunnel diode. The most significant results of this thesis are contained in the graphs of Figures q.2 and I4.3. The theoretical relation 2 If = coth e(V - ITDRb)/2kT is displayed by plotting 2 arc tanh (l/tf ) against the junction bias V = V - ITDRb» The result is a straight line of slope 2kT/e. Experimental values ofV| are computed from equation (3«5»1) and plotted in the same way, for points corresponding to biases at which noise results were obtained. The RMS deviation of the experimental points from the theoretical line is 2,0% for the near-forward region, and the slope of the best straight line through the experimental points equals the theoretical slope, for which T = 300° K. was assumed. In the reverse region the RMS deviation from the theoretical line for 300° K. is systematic but is 3,5%. The best straight line through the experimental points has a slope corresponding to T = 3l4»6° K rather than the measured 300° K.; the RMS deviation from this line is under 2%, In graph• ing arc tanh (l/^f2) as the ordinate rather than V2 , a constant percent discrepancy with bias of the experimental values of ^ 2 from coth(eV/2kT) manifests itself as an increasing mean square A Arc tanh • - 1.2 FIGURE 4.3 THEORETICAL AND EXPERIMENTAL COMPARISON FOR TUNNEL 4- 1.0 DIODE NOISE IN THE REVERSE BIAS REGION (RMS deviation of experimental points from best fit is 1.9 % ) f 0.8 experimental fit: slope = a/2kT for T = 314,6° K. theoretical line: slope = e/2kT 4- 0.6 = 0.0194 mV.-1 for T = 300°K. V i o.k •f 0.2 4- 10 20 30 40 5o 60 V(raV.) 86 deviation with bias of the experimental values of arc tanh (l/lf2) from the expected straight line relation; that is, a line with different slope results. This behavior appears in Figure i|.3» where the percent error in % | is not bias dependent. In the near-forward plot, the experimental points fit the theoretical curve within experimental uncertainty. The results thus vindicate the assumptions made in developing an expression for two-current shot noise from the Esaki integrals, and also confirm, for at least the band-independent aspects, Esaki»s formulation of tunneling. Further, the results mean that indirect tunneling mechanisms, If they produce greater than shot noise, and which in principle can operate in the negative conductance and near-forward regions, are not present sufficiently to enhance the direct-tunneling shot noise by a measureable amount. (The likelihood that the experimental and theoretical agreement shown in Figure U•2 can be due to a combination of smoothed direct- tunneling current shot noise (that is, that correlation between the tunneling currents I and I exists) and excess noise due ° cv vc to indirect tunneling processes persisting in this region, is very small even for a spot agreement at a single bias. Agreement over the entire bias region rules out such compensating effects entirely. The ratio of direct to indirect tunneling current should increase with increased overlap of the conduction and valence bands. In the reverse bias region the ratio is largest so that if the measured noise is consistent with direct tunneling only in the near-forward region, it must similarly be due only to direct tunneling processes in the reverse range, within experimental accuracy. Thus the tendency, shown in Figure I4.3 for the reverse region, for the experimental values of Y2 to exceed the theor• etically predicted ones (the percent discrepancy is bias-indepen• dent) is not likely due to fundamental processes which enhance the noise in this region only. More likely causes are: a) Incorrect value assumed for tunnel diode bulk resistance Rb. This possibility Is easily ruled out as follows. The discrepancy between experimental and theoretical values of Y 2 Is reduced if the experimental points move to the left along the V - axis, which corresponds to an increase In over the assumed value. However, treating R and I as constants, and In the << reverse region putting IND/lTD (UkT/2eITD)(l/R - l/Rp) in equation (3«5.1) gives 2 Y £ia/(R - Rb)2 .bRb /(R . Rfe)2 where a and b are constants and a/b ££. R. Hence 2f2 increases as 2 R^ is made larger (assuming Rb < R always), or arc tanh (l/iT ) decreases independent of bias for Increase in R^. The experi• mental points thus move in the direction of the arrow shown In Figure i|.3 for one of the points, that Is, parallel to the theor• etical curve, so that uncertainties in R, cannot explain the small D but consistent discrepancy. b) the systematic discrepancy Is more likely due to systematic errors in measurement, particularly in the calibration technique for equating two resistances. If the tunnel diode resistance is not exactly matched with the calibration resistance, then not only do the impedances change during a noise measurement, invalidating the comparison with the standard noise diode source, but also an erroneous value of 1^ is used in equation (3»5»1)« Due to the large value of 2 mA/ohm for dITD/dR In the far reverse region, and the reduction in sensitivity of the method due to the series- tuned coil resistance and signal generator internal resistance, a 1% uncertainty in the calibration integrated signal can produce a 10% uncertainty in ITD when R is 7 ohms. This Is larger than the 3,5% RMS scatter of points in Figure 3.1\ And exceeds the disagreement in slopes of the best experimental and theoretical lines. Such a small systematic error implied in the calibration could not appear in the near-forward region results, since R is much larger, and dl^jy^dR much smaller there. The agreement of theory and experiment throughout the bias region dominated by direct tunneling is interpreted to mean that: a) Esaki's formulation for direct tunneling applies in a generalized form in which F(E,V) = Z(E,V) £ (E) ^>(E+V) is any function. b) the tunneling currents I and I„„ are Independent and in cv vc the ratio exp(eV/kT) in absolute value. This also implies that tunneling reciprocity holds, that is, ZCV(E,V) = ZVC(E,V). c) indirect tunneling processes, if they enhance the noise, and which dominate in the valley and far-forward regions while persisting into the negative conductance region, are immeasureably small In the near-forward and reverse regions, even at the peak. d) processes such as avalanching, which would enhance the noise over that due to direct tunneling, do not occur in the .far reverse region for biases within the power capabilities of the Sony tunnel diode under test. (Avalanching can occur If the junction width is greater than the mean free path of electrons in the gap while tunneling, so that they can ionize lattice sites in the gap through collision: this enhances the current as well as the noise, although dI™/dV for tunneling is so large In the far —I —I 1 1 1 1 1 1 1 220 2U0 260 280 300 320 340 360 380 bias voltage V» (mV.) reverse region that avalanching would not change the character• istic noticeably, but only the noise. Typical Esaki junctions probably do not exceed in width the mean free electron path so that no avalanche noise would be expected.) 4.2 hValley and Far-forward Bias Regions The data for this region show that the noise temperature of the tunnel diode greatly exceeds that corresponding to thermal noise of a resistance R of the same value. If ^ 2 were given by the direct-tunneling relation coth(eV/2kT), it would have the value unity throughout this bias range, whereas the data give a minimum value of 4.8 for ft2 near the valley. The tunnel diode resistance typically exceeds 100 ohms in this region, so that the series-tuned coil noise is largely suppressed by the damping, whereas the equivalent excess noise current generator of the tunnel diode is large enough completely to override all other noise. A parallel-tuned circuit would therefore suffice to couple the tunnel diode into the cascode amplifier, and the low noise figure resulting should allow high accuracy. The values of ^f2, measured In the same way as in the reverse and near-forward regions, are plotted against bias voltage in 2 lrp Figure 4.4. For comparison, 1^ and I = 7f D are also graph• ed. The range of I ' for which data is obtained Is too small for an unambiguous relation between bias and I to be determined, so that the significance of this data is solely that it finds excess noise to persist up to the measuring frequency of 4 Mc/s. The results of M. D. Montgomery (19&1) in this region, at 1 kc/s, indicated an exponential relation between Ieq and the bias voltage, over a range from 100 raV. (in the negative conductance region) to 700 mv*., corresponding to a range of I from 10 mA. to 1.7 A. The present data at l\ Mc/s does not appear at variance with an exponential relation, although I is found to have much smaller magnitude at k Mc/s than at 1 kc/s. This is expected only if the commonly recognized mechanisms producing l/f noise were contaminating Montgomery's results. (These would involve surface- states In the bulk germanium and low-frequency fluctuations of potential barriers and trap positions). Since noise associated with ordinary thermal diffusion current in the far-forward region is described by 2 = coth (eV/2kT) which has value unity in this region (see Section 2»k)» the excessive *62 values of o obtained for currents well into the far-forward region (where the tunnel diode impedance has decreased to less than 30 ohms) suggests the possibility that ITD in this region is due predominantly to Indirect tunneling processes. Esaki and Yajima (1958) have obtained a closely linearly decreasing behavior for X0 with frequency up to 100 kc/s in this bias region, which on extrapolation shows # 2 should fall to unity at about 1 Mc/s if their relation persists at higher frequencies than 100 kc/s. The present results show that it does not, and suggest that the excess noise measured at 4 Mc/s does not involve the usual l/f mechanisms which might have accounted for their results at lower frequencies. On the other hand, if noise is enhanced proportional to the indirect-tunneling current magnitude, then the data indicate that indirect-tunneling current increases with bias at least up to 2 mA. into the far-forward region, as the bands become further separated. The reason for this Is not at present clear. Figure indicates the band picture for a forward bias near the valley (solid-line diagram) and a larger bias (dashed-line diagram). If, for instance, FIGURE DEPENDENCE OF INDIRECT TUNNELING PROCESSES ON BIAS indirect processes involve only direct tunneling to donors followed by vertical transitions into the valence band, or verti• cal transitions to acceptors followed by direct tunneling to the valence band, then two compensating effects occur as bias is increased: a) the number of donors which can project vertically onto the area representing unoccupied states In the valence band in• creases with bias; this enhances the Indirect process via donors as bias increases. b) since the energy between donors and conduction band, for a fixed spatial coordinate, must be independent of bias, the number of donors which lie in the energy range between E„ and E , X c c and hence which can be involved in the process discussed, decreases with bias: this inhibits the indirect process via donors as bias is increased. Similar considerations apply for acceptor-involved transitions. Figure shows that at zero absolute temperature, only those donors positioned in the singly-hatched area at the lower bias, but the doubly-hatched area at the higher bias, may aid the current in the latter case, but not in the former--that is, cause an Increase in the excess current with bias. Clearly very few donors satisfy ' these conditions, but any that do, cause current to increase. Two similar (but on the diagram, narrower) ranges for acceptors are depicted. Again the narrow singly-hatched range represents acceptors which,cannot be involved at the lower bias, whereas the narrow doubly-hatched area represents acceptors which can be in• volved at the higher bias. Similarly, areas can be drawn for both donors and acceptors for which these sites can aid the indirect processes at the lower bias, but not at the higher bias. A single acceptor (circle) and a single donor (square) which are in this category, are shown (solid figures for lower bias, open figures for higher bias). These correspond to decrease In excess current with bias. Since the diagrams are not significantly modified at non-zero tempera• tures, these statements show that it is not clear that excess current should increase with bias for the indirect processes discussed. 93 CHAPTER 5 CONCLUSIONS AND OUTSTANDING PROBLEMS 5>«1 Near-forward and Reverse Bias Regions The agreement of the noise measurements with the predicted coth (eV/2kT) relation oyer the entire region dominated by direct tunneling excludes beyond reasonable doubt all possibilities except that Esaki's tunneling formulation for direct transitions is applicable at least In all aspects not dependent on the band structure. The assumptions that the direct tunneling currents I and I are uncorrelated, that tunneling reciprocity holds, and V c that lloy^vc^ s exp (QVAT) also are vindicated. No useful Information can be gained by extending the measure• ments in the reverse region until avalanche occurs, accompanied by enhanced noise. (Little information for the forbidden gap width would be obtained from the bias at which avalanching set In, which is not obtained more unambiguously by measurement of the capacitance associated with the junction.) The reverse current is augmented with increased reverse bias not only by increasing overlap of the bands, but also by the Increasing field in the junction which augments the penetration factor associated with direct tunneling. The independence of the two currents I and Iyo should In no way depend on bias, however, so that for biases insufficient to cause avalanching, the noise should not be enhanced over shot noise. Noise measurements up to the peak current in the near-forward region show that if excess noise In the valley region arises from Indirect tunneling processes, then these processes are negligible or in some way fail to enhance the noise above shot noise in the overlapped band regions. An asymmetry of "excess" or indirect tunneling current with bias on either side of the valley region is possible since the excess noise persists Into the very far-forward bias region, while disappearing altogether somewhere In the nega• tive conductance region. The tunnel diode noise temperature is found to be less than Its actual temperature In the reverse bias region, equal to its actual temperature at zero bias, and greater than its actual temperature for all forward biases. Extension of the noise measurements in near-forward or reverse regions either to different frequencies, or to lower temperatures, serves little useful purpose. At lower temperatures, kT/e is less, so that the bias at which V2 should become nearly unity* decreases. This could act as a temperature-dependent check on the coth(eV/2kT) relation. Perhaps biases could be extended further in both direc• tions at lower temperatures without over-heating the tunnel diode. Accuracy required for the bulk resistance R^ Is less at lower temperatures since the bulk noise is reduced, while the tunneling current noise should not be. However these are advantages in practice only. Similarly the measurement of noise in other types of tunnel diodes, such as GaAs types with higher peak-to-valley current ratios, are unlikely to give further insight into direct- tunneling processes. 95 5.2 Valley and Far-forward Bias Regions Many problems remain, or are generated in this section. The noise data indicate only that at 4 Mc/s excess noise, that is, noise greater than shot noise, exists. This result has been ob• tained before, at various temperatures of the tunnel diode, and at lower frequencies. The excess current has been found largely temperature independent, which suggests that it arises from a tunneling process, and supports the present noise measurements in that they show that the current in the region measured is not thermal or diffusion p-n junction current, since this also must obey the coth (eV/2kT) relation. The importance of extending noise measurements in the far- forward region to greatly increased biases, and over a wide range of frequencies above and below 4 Mc/s is clear. Extension into the negative conductance region at various frequencies would also be instructive. If the noise spectrum as a function of frequency resembles any of the curves in Figure 2.8 an interpretation may exist in terms of the unmodulated stepping-stone model given in Section 2.4 although it is doubtful If very close resemblance can be expected due to the simplified model assumed for the Indirect tunneling processes. The interpretation of the bias-dependence of the excess noise (over an extended range) in terras of the modulated stepping-stone model is admittedly difficult, again due to the qualitative and simplified model discussed. The Indicated exponen• tial relation of excess noise with bias apparently has no associa• tion with the modulated stepping-stone model unless the enhanced noise is proportional to the magnitude of indirect tunneling cur• rent which itself may Increase exponentially with bias. It must be emphasized however that not only the results of the present noise measurements, but likely those at other frequen• cies and temperatures, do not demonstrate that the excess current or the noise associated with it arises from a tunneling process. Further, the question why indirect tunneling, if it can be shown responsible for the enhanced noise in the far-forward region, should fail to produce excess noise in the near-forward and reverse regions also, has not been answered. Neither of the mechanisms which has been considered in Chapter 2 for enhanced noise should operate solely in the valley and far-forward region, but not elsewhere. Nor should the indirect tunneling current itself. Similarly the usual causes of enhanced noise commonly referred to as "l/f" (namely, fluctuations in trap charge densities or positions or interactions of electrons with surface states in the bulk mater• ial) should not favor the far-forward bias region while completely disappearing in the near-forward and reverse regions. The independence of the valley current on temperature, surface etching of the bulk material, chemical surroundings, etc., which has been observed by Esaki and Yajima (1958) among others, Indicates that some form of tunneling is most likely responsible for the exoeas-current• A model based on this, and consistent over the entire I-V characteristic, which can predict enhanced noise of bias dependence according to Figure l+.lj (this dependence agrees with that found by other workers), but which predicts only shot noise for reverse and near-forward biases, probably Includes many basic processes and mechanisms too complicated to discuss in this thesis. 97 BIBLIOGRAPHY Agouridis, D. 1961. Study of Noise in Semiconductors and Semiconductor Devices (Elec. Eng. Dept., U. of Minnesota, Inst, of Technology, Third Report). Chynoweth, A.G. et. al. 19&1. Phys. Rev... 121^,6,84. Esaki, L. 1958. Phys. Rev. 109, 603. Esaki, L. and Yajima, T. 1958. J. Phys. Soc. Japan, 1^, 1281. I.R.E. Subcommittee on Noise i960. Proc. I.R.E. 1^8, 60. La Rosa, R. and Wilhelmsen, C.R. i960. Proc. I.R.E. (Correspondence) 1+8, 1903. Montgomery, M.D. 1961. J. Appl. Phys. ^2, 21+08. Tiemann, J.J. i960. Proc. I.R.E. l|8, 11+18. van der Ziel, A. 1958. Proc. I.R.E. 1+JD, 589* van der Ziel, A. 1954. Noise, Prentice-Hall, Inc., New York. Wallman, H. et. al. 1948. Proc. I.R.E. ^6, 700.