NOISE IN THE TUNNEL DIODE BY BARRY EARL TURNER A THESIS ...

NOISE IN THE TUNNEL DIODE

BY

BARRY EARL TURNER

B.Sc., University of B r i t i s h 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 th i s thesis as conforming

to the required standard

THE UNIVERSITY OF BRITISH COLUMBIA

July, 1962

In presenting t h i s t h e s i s i n p a r t i a l f u l f i l m e n t o f

the requirements f o r an advanced degree a t the U n i v e r s i t y o f

B r i t i s h Columbia, I agree t h a t the L i b r a r y s h a l l make i t f r e e l y

a v a i l a b l e f o r reference and study. I f u r t h e r agree that permission

f o r extensive copying of t h i s t h e s i s f o r s c h o l a r l y purposes may be

granted by the Head o f my Department or by h i s r e p r e s e n t a t i v e s .

I t i s understood t h a t copying or p u b l i c a t i o n of t h i s t h e s i s f o r

f i n a n c i a l g a i n s h a l l not be allowed without my w r i t t e n permission.

Department of Phy3ics

.The U n i v e r s i t y of B r i t i s h Columbia, Vancouver 8, Canada. .

Date Augu3t 3, 1962

i i

ABSTRACT

To date, measurements of t u n n e l diode noise have d e a l t mainly

w i t h the negative conductance r e g i o n , because the t u n n e l diode i s

an a c t i v e c i r c u i t element only In t h i s r e g i o n . The n o i s e has not

been measured f o r r e v e r s e or near-forward b i a s e s due t o the d i f f i

c u l t i e s i n v o l v i n g e x c e s s i v e l y low diode impedances i n these r e g i o n s .

The purpose of t h i s t h e s i s i s to show t h a t , from the E s a k i formu

l a t i o n f o r the d i r e c t - t u n n e l i n g c u r r e n t s of a t u n n e l diode, In the

b i a s r e g i o n s where the e l e c t r o n i c bands o v e r l a p , a simple theory

can be developed r e l a t i n g the power spectrum a s s o c i a t e d w i t h the

d i r e c t - t u n n e l i n g c u r r e n t n o i s e to the d i r e c t c u r r e n t p a s s i n g

through the d i o d e . T h i s t h e o r y assumes t h a t the two o p p o s i t e l y -

f l o w i n g d i r e c t - t u n n e l i n g c u r r e n t s i n the E s a k i j u n c t i o n are un

c o r r e c t e d and that b o t h c o n t r i b u t e f u l l shot n o i s e . The theory

can be c r i t i c a l l y t e s t e d o n l y i n the b i a s r e g i o n s where the noise

Is yet u n s t u d i e d , and at s u f f i c i e n t l y h i g h f r e q u e n c i e s t h a t no

contaminating l / f noise e x i s t s . These c o n d i t i o n s have been met

e x p e r i m e n t a l l y and the n o i s e measured q u a n t i t a t i v e l y over the

e n t i r e r e v e r s e and near-forward r e g i o n s at a frequency of \\ Mc/s.

Impedance-transforming networks and a v e r y low-noise p r e a m p l i f i e r

s u i t a b l e t o the p a r t i c u l a r source s t r e n g t h s and impedances p r e

sented by the t u n n e l diode are developed f o r these measurements.

A noi s e measurement technique i s chosen from among s e v e r a l p o s s i b l e

ones f o r the h i g h degree of accuracy and s m a l l e s t dependence on a

good n o i s e f i g u r e r e q u i r e d f o r the t u n n e l diode source. The

experimental r e s u l t s agree w i t h the theory and v i n d i c a t e the u s u a l

assumption t h a t the two o p p o s i t e l y f l o w i n g d i r e c t - t u n n e l i n g e l e c t r o n

i i i

c u r r e n t s between two bands of a degenerately-doped semiconductor

are u n c o r r e l a t e d .

Noise measurements In the " v a l l e y " and f a r - f o r w a r d r e g i o n of

the t u n n e l diode c h a r a c t e r i s t i c , where the diode c u r r e n t i s not

due t o d i r e c t t u n n e l i n g , do not agree w i t h the simple two-current

shot n o i s e t h e o r y f o r d i r e c t - t u n n e l i n g e l e c t r o n c u r r e n t s . P o s s i b l e

reasons f o r the enhanced nois e measured In t h i s r e g i o n are advanced

i n the form of two models based on i n d i r e c t - t u n n e l i n g e l e c t r o n s v i a

t r a p s as the most important mechanism d e s c r i b i n g the excess or

v a l l e y c u r r e n t . These models o f f e r a p o s s i b l e e x p l a n a t i o n of the

observed phenomena, but n o i s e measurements alone appear i n s u f f i

c i e n t t o demonstrate unambiguously the d e t a i l e d mechanisms p r o

ducing e i t h e r the excess c u r r e n t or the a s s o c i a t e d enhanced n o i s e

found throughout the v a l l e y and f a r - f o r w a r d r e g i o n s .

.ACKNOWLEDGMENT

I should l i k e t o thank P r o f e s s o r R. E. Burgess,

my t h e s i s d i r e c t o r , f o r h i s s u p e r v i s i o n In p r e p a r i n g

the m a t e r i a l i n t h i s t h e s i s .

The r e s e a r c h was f i n a n c e d by the N a t i o n a l

Research C o u n c i l of Canada i n the form of a Student

s h i p and Summer Supplement, and by the United S t a t e s

A i r 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 T u n n e l i n g 2

1 . 3 Survey of the L i t e r a t u r e 8

l . q Scope of T h e s i s 10

CHAPTER 2 . THEORY OF TUNNEL DIODE NOISE 12

2 . 1 Noise Model of the Tunnel Diode 12

2 . 2 Noise Spectrum f o r D i r e c t T u n n e l i n g Currents ll\

of E s a k i

2 . 3 R e s t r i c t i o n s on E s a k i ' s T u n n e l i n g Theory 16

2»l\ Models f o r Noise A s s o c i a t e d w i t h I n d i r e c t 23

T u n n e l i n g Processes

2.1*1 Modulation i n the I n d i r e c t - T u n n e l i n g 3$

Model f o r V a l l e y Noise

CHAPTER 3 . APPARATUS AND EXPERIMENTAL TECHNIQUES i+0

3 . 1 B a s i c Concepts and Requirements of Noise qO

Measurements

3 . 1 1 Theory and Requirements f o r "Low-noise" qO

C i r c u i t s

3 . 1 2 Methods of Comparison With a Standard 1+2

Noise Source

3 . 2 Impedance Transformations S u i t a b l e f o r a 4.9

Tunnel Diode Source

3 . 3 Development of a Low-noise A m p l i f i e r $3

V

3 . 3 1 Amplification and Noise of a Cascode 5*4

Amplifier

3 . 3 2 Cascode C i r c u i t Designs Favoring S t a b i l i t y 58

3 . 3 3 Performance of the Cascode 63

3.I4 Other Apparatus and C i r c u i t r y 67

3.141 Perspective of the Overall C i r c u i t 67

3.142 Noise Diode and Tunnel Diode Bias and 68

R.F. C i r c u i t s

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

v i

ILLUSTRATIONS

F i g u r e F a c i n g Page

1.1 T y p i c a l I - V C h a r a c t e r i s t i c f o r a Tunnel Diode 2

1.2 Energy-band Diagram f o r L i g h t l y - d o p e d Semiconductor 3

1.3 Energy-band Diagram f o r Degenerately-doped Semi- 3

conductor

l,k Tunnel Diode J u n c t i o n Energy-diagram at Zero B i a s I j

1.5 Tunnel Diode J u n c t i o n Energy-diagram f o r Forward B i a s i*

1.6 Tunnel Diode J u n c t i o n Energy-diagram f o r Reverse B i a s l\

1.7 Mechanisms f o r I n d i r e c t T u n n e l i n g i n the Far-forward 7

Bia s Region

2.1 N o i s e - e q u i v a l e n t C i r c u i t f o r a Shot Noise Device 12

2.2 N o i s e - e q u i v a l e n t C i r c u i t f o r a Tunnel Diode 13

2.3 Behavior of ^ § w i t h B i a s Voltage 15

2.1* D e t a i l e d Mechanisms Involved i n I n d i r e c t T u n n e l i n g 2l|

2.5 S i m p l i f i e d Model- f o r Noise A n a l y s i s of I n d i r e c t 26

T u n n e l i n g Processes

2.6 Current A s s o c i a t e d w i t h "Event A" 27

2.7 Current A s s o c i a t e d w i t h "Event B" 29

2.8 P o s s i b l e Noise S p e c t r a f o r I n d i r e c t T u n n e l i n g 32

Processes

2.9 Charge D i s t r i b u t i o n W i t h i n a Tunnel Diode J u n c t i o n 35

2.10 Modulation of Energy-band Diagram by T r a p - i n v o l v e d 36

I n d i r e c t T u n n e l i n g

3.1 Schematic C i r c u i t f o r D i r e c t Measurement of a 1*0

Noise Source

v i i

3.2 S i m p l i f i e d Schematic C i r c u i t f o r D i r e c t 43

Measurement of a Noise Source

3 . 3 Schematic C i r c u i t f o r Comparison of Unknown 44

and C a l i b r a t e d Noise Sources

3 . 4 Schematic Noise C i r c u i t f o r A t t e n u a t o r and Two 46

Standard Noise Sources

3 . 5 Schematic C i r c u i t f o r a Transformed Source 49

Coupled i n t o a N o i s y A m p l i f i e r

3.6 N o i s e - e q u i v a l e n t C i r c u i t s f o r a P a r a l l e l - t u n e d 5 0

C i r c u i t

3.7 A u t o t r a n s f o r m a t i o n f o r Tunnel Diode Source 5 l

3.8 S e r i e s - t u n e d C i r c u i t T ransformation f o r Tunnel 5 l

Diode Source

3 . 9 Comparison of Noise F i g u r e s f o r S e r i e s - and P a r a l l e l - 52

tuned C i r c u i t s With Tunnel Diode Source

3.10 A.C.-equivalent C i r c u i t s of a Cascode A m p l i f i e r 5q

3.11 /.Noise-equivalent C i r c u i t s of a Cascode A m p l i f i e r 56

3.12 T y p i c a l A.C.-coupled Cascode A m p l i f i e r 59

3 . 1 3 Simplest D i r e c t - c o u p l e d Cascode A m p l i f i e r 60

3.1i| P r a c t i c a l D i r e c t - c o u p l e d Cascode C i r c u i t With 62

O p t i o n a l Cathode-follower Stage and A t t e n u a t o r

3 . 1 5 - Schematic Noise C i r c u i t f o r Measuring Rfl of an 64

A m p l i f i e r

3.16 B l o c k Diagram of Complete Noise Measuring C i r c u i t 67

3.17 Noise Diode and Tunnel Diode Bi a s and R.F. C i r c u i t s 68

3.18 Noise Diode Filament Current C o n t r o l C i r c u i t 71

v l i l

F i g u r e F a c i n g Page

3 . 1 9 C i r c u i t f o r I n t e g r a t i n g Noise S i g n a l s 73

3 . 2 0 Complete N o i s e - e q u i v a l e n t C i r c u i t f o r Tunnel 7k

Diode Noise Measurement

I j . l I - V C h a r a c t e r i s t i c of Sony E s a k i Diode i n the 80

Near-forward and Reverse B i a s Regions

If.2 T h e o r e t i c a l and E x p e r i m e n t a l Comparison of Tunnel 85

Diode Noise i n the Near-forward B i a s Region

k»3 T h e o r e t i c a l and E x p e r i m e n t a l Comparison of Tunnel 86

Diode Noise i n the Reverse Bi a s Region

I4.4 Data f o r V a l l e y and Far-forward B i a s Region 89

1|.5 Dependence of I n d i r e c t T u n n e l i n g Processes on Bi a s 91

CHAPTER 1

INTRODUCTION

1 . 1 Statement o f the Problem

A\ q u a n t i t a t i v e study of the n o i s e a s s o c i a t e d w i t h charge

t r a n s p o r t processes In s o l i d s v e r y o f t e n gives d e t a i l e d i n f o r m

a t i o n about these processes which i s otherwise d i f f i c u l t t o

o b t a i n . The process t h i s t h e s i s s t u d i e s i s quantum-mechanical

int e r b a n d t u n n e l i n g i n degenerately-doped semiconductors, upon

which the t u n n e l diode owes i t s a c t i v e p r o p e r t i e s . The n o i s e

spectrum of the diode t u n n e l i n g c u r r e n t should be r e l a t e d i n

p r i n c i p l e t o E s a k i ' s t h e o r y of d i r e c t t u n n e l i n g c u r r e n t s (form

u l a t e d e x p l i c i t l y f o r interband t u n n e l i n g i n heavily-doped semi

conductors, but a p p l i c a b l e i n broad form t o t u n n e l i n g i n super

conducting systems a l s o . ) C e r t a i n assumptions are unavoidable

i n r e l a t i n g the n o i s e spectrum t o t u n n e l i n g t h e o r y . In t e s t i n g

these, the frequency of measurement of the spectrum must be

s u f f i c i e n t l y h i g h to avoid l / f n o i s e , which cannot be r e l a t e d t o

d i r e c t t u n n e l i n g t h e o r y . The magnitude of the n o i s e spectrum at

a s i n g l e frequency i s then most c r i t i c a l l y r e l a t e d t o E s a k i ' s

theory, i n terms of measured q u a n t i t i e s , i n the near-forward and

r e v e r s e b i a s r e g i o n s of the t u n n e l diode I-V c h a r a c t e r i s t i c .

Here the diode n o i s e s i g n a l i s t e c h n i c a l l y d i f f i c u l t t o measure

due t o the v e r y low Impedance of the diode In these r e g i o n s .

In the v a l l e y r e g i o n of the diode c h a r a c t e r i s t i c , the

conduction c u r r e n t Is due mainly to i n d i r e c t t u n n e l i n g mechan

isms which depend on energy band p r o f i l e s and i m p u r i t y s t a t e

d i s t r i b u t i o n s w i t h i n the f o r b i d d e n gap. The n o i s e spectrum

FIGURE 1.1 TYPICAL I - V CHARACTERISTIC FOR A TUNNEL DIODE

must be r e l a t e d to these p r o p e r t i e s i n o r d e r t h a t i n f o r m a t i o n on

the s t r u c t u r e of the j u n c t i o n energy d iagram can be o b t a i n e d

t h r o u g h n o i s e measurement.

The purpose of t h i s t h e s i s i s t o measure the n o i s e spectrum

over the e n t i r e p o s i t i v e conductance p a r t , o f the t u n n e l d iode

c h a r a c t e r i s t i c and t o i n t e r p r e t the r e s u l t s i n terras of E s a k i ' s

t u n n e l i n g t h e o r y where a p p l i c a b l e .

1.2 Summary of the Theory of T u n n e l i n g

In s t u d y i n g d iodes made from v e r y h e a v i l y doped germanium,

E s a k i (19^8) d i s c o v e r e d n e g a t i v e r e s i s t a n c e i n them i n the f o r

ward b i a s d i r e c t i o n . T h i s he c o r r e c t l y i n t e r p r e t e d as due t o

i n t e r b a n d t u n n e l i n g , the p r o p e r t i e s and consequences o f w h i c h we

now d e s c r i b e i n terms of a t y p i c a l I-V c h a r a c t e r i s t i c f o r a

t u n n e l d i o d e , as i n F i g u r e 1.1.

F i g u r e 1.2 shows the energy band diagram and d e n s i t y - o f -

s t a t e s p r o f i l e f o r a m o d e s t l y doped n - type s e m i c o n d u c t o r .

Sha l low i m p u r i t y l e v e l s (donors ) are shown j u s t below the c o n

d u c t i o n b a n d , w h i c h i s v i r t u a l l y empty so t h a t the f e r m i l e v e l

l i e s o n l y s l i g h t l y above the midd le of the f o r b i d d e n g a p . The

o n l y a l lowed e l e c t r o n s t a t e s i n the i n t e r b a n d gap are the i m p u r

i t y l e v e l s , wh ich are l o c a l i z e d s p a t i a l l y .

F i g u r e 1.3 shows the s i t u a t i o n f o r a d e g e n e r a t e l y doped

s e m i c o n d u c t o r , f rom which t u n n e l d i o d e s are made. The donor

c o n c e n t r a t i o n i s so h i g h t h a t a l t h o u g h o n l y a s m a l l f r a c t i o n of

the i m p u r i t y l e v e l s at any g i v e n energy are i o n i z e d at normal

t e m p e r a t u r e s , the t o t a l number of i o n i z e d i m p u r i t y s i t e s i s s u f f i

c i e n t to cause the c o n d u c t i o n band t o be o c c u p i e d by e l e c t r o n s

over an a p p r e c i a b l e range of e n e r g i e s . T h i s d r i v e s the f e r m i

conduction band

impurity states

fermi level

valence band

fermi function

FIGURE 1 . 2

ENERGY-BAND DIAGRAM FOR A LIGHTLY-DOPED SEMICONDUCTOR

density of states (parabolic)

/ conduction band

fermi level

± ± ±_ impurity states(some ionized)

7, valence band

fermi function

density of states (profile unknown in region of impurity sites)

FIGURE 1 . 3

ENERGY-BAND DIAGRAM FOR A DEGENERATELY-DOPED SEMICONDUCTOR

l e v e l i n t o the conduction hand, and a l s o causes the d e n s i t y - o f -

s t a t e s to t a i l o f f more g r a d u a l l y i n t o the f o r b i d d e n gap than i n

the l i g h t l y doped case. The d e n s i t y of e l e c t r o n - o c c u p i e d s t a t e s

i s the product of d e n s i t y - o f - s t a t e s f u n c t i o n and f e r m i f u n c t i o n

( p r o b a b i l i t y - o f - o c c u p a n c y f u n c t i o n ) , b o t h of which are shown i n

F i g u r e s 1.2 and 1,3.

A semiconductor which has n e a r l y a l l s t a t e s near the bottom

of the conduction band f i l l e d w i t h e l e c t r o n s (from i o n i z e d donors)

i s c a l l e d an n-type degenerate semiconductor. S i m i l a r l y , a degen

e r a t e l y doped p-type semiconductor i s one i n which a l l the s t a t e s

i n an a p p r e c i a b l e energy range near the top of the valence band

are empty (due t o a h i g h c o n c e n t r a t i o n of a c c e p t o r s i t e s l y i n g

j u s t above the valence band),

A t u n n e l diode i s formed by making a p-n j u n c t i o n between two

d e g e n e r a t e l y doped n- and p-type semiconductors. F i g u r e l . l j shows

the band s t r u c t u r e when no b i a s i s a p p l i e d across the j u n c t i o n , so

that the f e r m i l e v e l s on each s i d e c o i n c i d e i n the f o r b i d d e n gap.

The shaded areas re p r e s e n t energy l e v e l s l i k e l y occupied by e l e c

t r o n s . E l e c t r o n s are s u b j e c t t o a l a r g e p o t e n t i a l g r a d i e n t i n

t r a v e r s i n g the j u n c t i o n due t o the b u i l t - i n e l e c t r i c f i e l d a r i s i n g

from f i x e d i o n i z e d Impurity s i t e s of opposite charge on opposite

s i d e s of the j u n c t i o n .

A p p l y i n g a b i a s across such, a j u n c t i o n causes one s i d e of the

energy diagram to s h i f t v e r t i c a l l y r e l a t i v e t o the other s i d e . I f

there were no f o r b i d d e n gap, which i s o l a t e s the e l e c t r o n s on each

s i d e of the j u n c t i o n , they would then t r a v e l aoross under the app

l i e d f i e l d . They accomplish the same t r a n s i t i o n i n the presence

of the gap by quantum-mechanical t u n n e l i n g , i f the j u n c t i o n i s

s u f f i c i e n t l y t h i n . O r d i n a r i l y , t h i s t r a n s i t i o n must conserve energy

p-side forbidden

gap

n-side

E.

W/////////J/7lif)l

occupied region for electrons

wmm

FIGURE l.ty

TUNNEL DIODE JUNCTION ENERGY DIAGRAM FOR ZERO BIAS

forbidden gap

V E S

bias overlap '^i

fermi level T/i

///////////////

Z E .

FIGURE 1.5

JUNCTION ENERGY DIAGRAM FOR FORWARD BIAS

forbidden gap

EL

m//////7/~/rh

bias and Overlap

FIGURE 1 . 6

JUNCTION ENERGY DIAGRAM FOR REVERSE BIAS

— t h a t i s , the t r a n s i t i o n i s represented by a h o r i z o n t a l s t r a i g h t

l i n e on the energy diagram. T h i s i s known as d i r e c t t u n n e l i n g .

Current due t o d i r e c t t u n n e l i n g i s p r o p o r t i o n a l t o the product

of the p r o b a b i l i t y of t u n n e l i n g per e l e c t r o n i n c i d e n t on the b a r r i e r ,

the d e n s i t y of occupied s t a t e s on the s i d e from which e l e c t r o n s

t r a v e l , and the d e n s i t y of unoccupied s t a t e s on the other s i d e i n a

r e g i o n which, on the energy s c a l e , o v e r l a p s the occupied s t a t e s on

the f i r s t s i d e . F o r zero b i a s , equal and opposite c u r r e n t s flow

(due to the f e r m i f u n c t i o n " t a i l s " at non-zero temperatures), the

net c u r r e n t b e i n g z e r o .

A forward b i a s causes the n-type s i d e of the j u n c t i o n t o r i s e

i n e l e c t r o n energy r e l a t i v e t o the p-type s i d e so t h a t the o v e r l a p

r e g i o n i n c r e a s e s at f i r s t , then decreases t o zero at s u f f i c i e n t l y

i n c r e a s e d b i a s . The d i r e c t t u n n e l i n g c u r r e n t i s from conduction to

valence band, and r i s e s t o a peak before f a l l i n g t o zero at l a r g e

forward b i a s . The " d i r e c t t u n n e l c u r r e n t " region, i s shown i n

F i g u r e 1,1. At much l a r g e r forward b i a s , o r d i n a r y thermal p-n

j u n c t i o n c u r r e n t becomes prominent, s i n c e the p o t e n t i a l b a r r i e r of

b o t h conduction and valence bands decreases l i n e a r l y w i t h forward

b i a s . F i g u r e 1.5 i s f o r a r e p r e s e n t a t i v e forward b i a s .

The v a l l e y c u r r e n t i s not f u l l y accounted f o r by a s u p e r p o s i

t i o n of d i r e c t t u n n e l i n g and f a r - f o r w a r d thermal p-n j u n c t i o n c u r

r e n t s , but a l s o Involves a process known as i n d i r e c t t u n n e l i n g , t o

be d i s c u s s e d l a t e r .

A r e v e r s e b i a s causes the o v e r l a p between the conduction and

valence bands t o i n c r e a s e . Now, d i r e c t e l e c t r o n t u n n e l i n g i s from

valence t o conduction band. The number of empty s t a t e s i n the con

d u c t i o n band which are opposite occupied s t a t e s i n the valence band

i n c r e a s e s i n d e f i n i t e l y w i t h r e v e r s e b i a s i n c r e a s e , so t h a t the r e v -

erse I-V c h a r a c t e r i s t i c shows no maximum. F i g u r e 1 . 6 shows how

the n-type s i d e of the j u n c t i o n i s depressed r e l a t i v e t o the p-

type s i d e f o r t h i s case.

Both the energy-band model, and the I-V curve show th a t the

t u n n e l diode i s a v o l t a g e - c o n t r o l l e d d e v i c e , the c u r r e n t b e i n g a

s i n g l e - v a l u e d f u n c t i o n of the a p p l i e d v o l t a g e . The n o i s e spectrum

f o r t u n n e l i n g processes w i l l be expressed i n terms of the a p p l i e d

v o l t a g e .

For d i r e c t t u n n e l i n g , the t u n n e l i n g c u r r e n t from conduction

to valence band i s denoted by I and the c u r r e n t f l o w i n g opposite

l y by I v c « E s a k i ' s expressions are then

I vc

(V) = 4 f c ( E ) P c { E ) ^ " f v ( E ) ] f v ( E ) Z c v d E

c

(V) = A P * f (E)p ( E ) [ l - f ( E ) ] p (E) Z dE lg V \ V *• c \ c vc

(1.2.1)

where

D (E) = n-type conduction band d e n s i t y of energy s t a t e s

p ^ ( E ) = p-type valence band d e n s i t y of energy s t a t e s

f (E) = f e r m i f u n c t i o n In conduction and valence bands r e s -c ' v p e c t l v e l y

Z (E) = p r o b a b i l i t y of t u n n e l i n g per e l e c t r o n attempt In ' each d i r e c t i o n r e s p e c t i v e l y

E = lowest energy l e v e l i n n-conduction band c E ^ = h i g h e s t energy l e v e l i n p-valence band

E = energy

The i n t e g r a t i o n range depends d i r e c t l y on b i a s . and Z a l s o

depend on b i a s , but l e s s s t r o n g l y . The d . c c u r r e n t i s | l | =

11 - I I. At zero b i a s , I = - I . F o r reverse b i a s , I 1 cv vc cv vc cv q u i c k l y f a l l s t o a value much l e s s than I when the b i a s , V,

vc

approaches a few kT/e of b i a s , T b e i n g the a c t u a l a b s o l u t e temp

e r a t u r e of the t u n n e l diode j u n c t i o n . F o r forward b i a s V, I

becomes much l e s s than I f o r the same c o n d i t i o n . cv

The n o i s e spectrum a r i s i n g from these t u n n e l i n g c u r r e n t s can

be r e l a t e d t o E s a k i ' s f o r m u l a t i o n , as a f u n c t i o n of b i a s v o l t a g e ,

most simply i f i t i s assumed t h a t

a) the c u r r e n t s I and I are u n c o r r e l a t e d , and cv vc

b) these c u r r e n t s b o t h c o n t r i b u t e f u l l shot n o i s e .

The l a t t e r assumption i s reasonable when we note that shot n o i s e

a r i s e s from the t r a n s p o r t of d i s c r e t e charges under an a p p l i e d

f i e l d , i f these c a r r i e r s are emitted w i t h a Poisson d i s t r i b u t i o n

i n time. T h i s i m p l i e s that there Is no c o r r e l a t i o n between succ

e s s i v e emissions c o n s t i t u t i n g each c u r r e n t . Since t u n n e l i n g i s a

v e r y s m a l l p r o b a b i l i t y p r o c e s s , the Poisson d i s t r i b u t i o n i s expected.

In determining whether or not d i r e c t t u n n e l i n g c u r r e n t s produce

pure shot n o i s e , we s h a l l measure the q u a n t i t y

I f I and I are u n c o r r e l a t e d and each produces f u l l shot n o i s e , cv vc the r e s u l t a n t n o i s e w i l l be the same as i f an average c u r r e n t

I +1 were f l o w i n g . Hence we d e f i n e 1 = 1 + I as the cv vc sq cv vc e q u i v a l e n t s a t u r a t e d noise c u r r e n t ( i n analogy w i t h vacuum nois e

diode terminology.) The noise spectrum or noise power generated

per u n i t bandwidth of frequency, due t o d i r e c t t u n n e l i n g c u r r e n t s ,

w i l l be w r i t t e n i n terms of the mean square of an e q u i v a l e n t con

s t a n t c u r r e n t n o i s e generator f o r the instantaneous c u r r e n t f l u c

t u a t i o n s . Experiments of E s a k i and Yajima ( 1 9 5 8 ) i n d i c a t e t h a t

t h i s r e p r e s e n t s more d i r e c t l y the p h y s i c a l nature o f the t u n n e l i n g

n o i s e than does a constant v o l t a g e generator. In analogy w i t h

shot n o i s e a r i s i n g i n a temperature-saturated vacuum diode, the

FIGURE 1.7

MECHANISMS FOR INDIRECT TUNNELING IN THE FAR-FORWARD BIAS REGION

7 d i r e c t t u n n e l i n g n o i s e spectrum w i l l be s p e c i f i e d by

 = 2 e | l | K ^ f (1.2.3)

= 1 corresponds t o the vacuum diode case

approaches I n f i n i t y as | l | approaches z e r o .

e. By d e f i n i t i o n , X Q

. At zero b i a s , ^ * i ? ^

must tend t o a constant given by 4kT(£l/dV) evaluated at V = 0, i n

accordance w i t h the thermodynamical requirement t h a t the noise of

any a c t i v e system reduces t o thermal noise given by a r e s i s t a n c e

equal to i t s value at zero b i a s . That t h i s theorem holds i n terms

of E s a k i ' s theory has been proved f o r s p e c i a l cases by TIemann (i960). The E s a k i f o r m u l a t i o n i s i n a p p l i c a b l e to I n d i r e c t t u n n e l i n g .

T h i s process can occur p r i n c i p a l l y by way of Imperfections i n the

band s t r u c t u r e , p a r t i c u l a r l y i n the form of l o c a l i z e d i m p u r i t y

s t a t e s or " t r a p s " l y i n g w i t h i n the energy gap, and allows e l e c t r o n s

t o pass across the gap a f t e r the forward b i a s exceeds the value

where the conduction and valence bands become "uncrossed", so t h a t

d i r e c t t u n n e l i n g i s i m p o s s i b l e . T r a p - i n v o l v e d mechanisms are de

p i c t e d by v e r t i c a l and h o r i z o n t a l paths i n F i g u r e 1.7» The o b l i q u e

path d e p i c t s phonon-assisted t u n n e l i n g . In a l l these p r o c e s s e s ,

the e l e c t r o n s must l o s e energy. These I n d i r e c t p r o c e s s e s , which

produce the v a l l e y c u r r e n t ( F i g u r e 1.1) are p o s s i b l e a l s o when the

bands are overlapped, so t h a t the "excess" c u r r e n t caused by them

extends w e l l Into the negative conductance and f a r - f o r w a r d regions

on e i t h e r s i d e of the v a l l e y . Whereas d i r e c t - t u n n e l i n g e l e c t r o n s

t r a v e r s e the gap v e r y r a p i d l y , e l e c t r o n s which i n t e r a c t w i t h t r a p s

are captured f o r s i g n i f i c a n t p e r i o d s of time. T h i s a f f e c t s the

n o i s e . In p a r t i c u l a r , e l e c t r o n s which t u n n e l back and f o r t h between

e i t h e r band, and the t r a p s (shown by dotted arrows In F i g u r e 1.7),

can produce noi s e f a r i n excess of shot n o i s e .

8

1.3 Survey of the L i t e r a t u r e

To date, l i t t l e attempt has been made t o use the n o i s e

p r o p e r t i e s of d i r e c t t u n n e l i n g i n t u n n e l diodes t o check E s a k i ' s

t h e o r y , t o determine the degree of c o r r e l a t i o n ( i f any) between

I and I , or t o r e l a t e the excess c u r r e n t n o i s e i n the v a l l e y cv vc r e g i o n t o p o s s i b l e models f o r l o c a l i z e d Impurity s t a t e d i s t r i b u

t i o n s w i t h i n the f o r b i d d e n gap.

E a r l y n o i s e measurements were at low f r e q u e n c i e s where l / f

n o i s e dominates. T h i s i s not a p r o p e r t y of t u n n e l i n g , but a r i s e s

i n the bulk semiconductor surrounding the j u n c t i o n . Although con

t a i n i n g a few f e a t u r e s o f i n t e r e s t t o the present work, which are

now summarized, these i n v e s t i g a t i o n s are mainly i r r e l e v a n t .

E s a k i and Yajima (19£8) f i r s t measured nois e i n t u n n e l d i o d e s ,

i n the frequency range 10 t o 10-* c y c l e s / s e c o n d . They q u a l i t a t i v e l y

examined the d i f f i c u l t r e v e r s e - and near-forward b i a s regions

(where the diode impedance i s v e r y low) but o n l y t o determine i f

s t r o n g l / f n o i s e e x i s t e d . T h e i r apparatus was not s e n s i t i v e

enough t o d e t e c t a shot component of noise i n these r e g i o n s , so

t h a t they were able o n l y t o r e p o r t that no s t r o n g l / f component

appeared. However, t h e i r r e s u l t s cannot i n s u r e that a weak l / f

component d i d not p e r s i s t at 10-> cps so t h a t much h i g h e r f r e q u e n

c i e s are needed t o decide q u a n t i t a t i v e l y i f the noise i n these b i a s

r e g i o n s i s pure shot n o i s e . The same authors found no s t r o n g l / f

component i n the negative conductance r e g i o n , but i n the excess

c u r r e n t r e g i o n s t r o n g l / f b e h a v i o r appeared, w i t h magnitude n e a r l y

10^ g r e a t e r than c a l c u l a t e d shot b e h a v i o r , even at 10^ c p s . T h e i r

data i n d i c a t e t h i s was due mainly t o excess c u r r e n t ( l a t e r b e l i e v e d

due t o I n d i r e c t t u n n e l i n g mechanisms) r a t h e r than t o normal f a r -

forward p-n j u n c t i o n d i f f u s i o n c u r r e n t . Again, however, h i g h e r

9

f r e q u e n c i e s are needed t o see i f the l / f behavior i n the valley-

r e g i o n was due simply t o low-frequency f l u c t u a t i o n s of i m p u r i t y

t r a p c e n t r e s or recombinations.through these c e n t r e s (suggested as

p o s s i b l e by the authors) or whether the i n d i r e c t t u n n e l i n g c u r r e n t

can d i s p l a y o n l y shot n o i s e at h i g h f r e q u e n c i e s .

More re c e n t measurements by M. D. Montgomery (1961) at f r e q

u e ncies from 30 t o 1C-3 cps have c o r r o b o r a t e d the r e s u l t s of E s a k i

and Yajima, and have f u r t h e r strengthened the i d e a t h a t s t r o n g l / f

n o i s e i n the v a l l e y r e g i o n at low f r e q u e n c i e s Is due t o i n d i r e c t

t u n n e l i n g v i a a continuous d i s t r i b u t i o n of i m p u r i t y s t a t e s .

Tiemann (I960) has made the o n l y high-frequency n o i s e measure

ments on t u n n e l diodes w i t h a view t o e s t a b l i s h i n g whether a pure

shot component alone i s a s s o c i a t e d w i t h t u n n e l i n g c u r r e n t s . How

ever, he r e s t r i c t s h i m s e l f t o the negative conductance r e g i o n ,

where he attempts t o r e l a t e the expected shot component f o r the

n o i s e spectrum t o E s a k i ' s theory, but o n l y In terms of a p a r t i c u l a r

assumed d e n s i t y - o f - s t a t e s f u n c t i o n , and an' assumed f e r m i l e v e l

r e l a t i v e t o the band edges. Though l a c k i n g g e n e r a l i t y , the numeri

c a l e v a l u a t i o n s f o r E s a k i ' s i n t e g r a l s f o r these cases p r e d i c t a

shot n o i s e spectrum i n agreement w i t h h i s d a t a , taken at 0.5 Mc.

and 100 Mc. f o r the r e s t r i c t e d b i a s r e g i o n . The noiso i s not

measured near the o r i g i n , so t h a t i t i s not determined how the

spectrum reduces t o the c o r r e c t thermal conductance noi s e at zero

b i a s .

T h e o r e t i c a l l y , La Rosa and Wilhelmson (i960) have s t a t e d t h a t

the t u n n e l diode should d i s p l a y approximately one-half shot n o i s e .

Assuming I and I are u n c o r r e l a t e d , they p r e d i c t t h a t the I cv vc vc

component should produce f u l l shot n o i s e , due t o i t s a r i s i n g from

the s m a l l - p r o b a b i l i t y process of t u n n e l i n g , but t h a t the I cora-cv

ponent should be g r e a t l y smoothed, due t o the p r o b a b i l i t y of

t u n n e l i n g from conduction t o valence band per e l e c t r o n per a t

tempt, Z q v , having a value c l o s e to 0.f>. T h e i r reasons f o r t h i s

v a l u e , and hence t h e i r c o n c l u s i o n of smoothed shot n o i s e f o r the

t u n n e l diode, are erroneous.

D. A g o u r i d i s (unpublished, 1961), assuming no c o r r e l a t i o n

between I and I has r e l a t e d the n o i s e spectrum t o E s a k i ' s cv vc t h e o r y i n more g e n e r a l i t y than Tiemann, but h i s treatment i s a l s o

somewhat s p e c i a l i z e d . He measures the n o i s e up t o 30 Mc. but a l s o

o n l y f o r the r e s t r i c t e d b i a s e s near the peak and n e g a t i v e conduc

tance r e g i o n s , where the r e s u l t s do not so c r i t i c a l l y compare t o

theory as i n the h i g h e r conductance r e g i o n s near zero b i a s .

I»l4 Scope of T h e s i s

To r e l a t e the n o i s e spectrum f o r d i r e c t t u n n e l i n g c u r r e n t t o

E s a k i ' s theory, assuming shot n o i s e a s s o c i a t e d w i t h the t u n n e l i n g

components, we d e r i v e an e x p r e s s i o n f o r ^ T O which i s independent of

the band s t r u c t u r e of the diode, but depends on the b i a s v o l t a g e

and temperature. The spectrum Is then measured at l\ Mc. (where no

l / f component of noise p e r s i s t s ) as a f u n c t i o n of b i a s over the

e n t i r e near-forward and r e v e r s e b i a s ranges, u s i n g s u i t a b l e imped

ance t r a n s f o r m a t i o n s f o r the v e r y low diode source impedance i n

t h i s range. T h i s i s s m a l l e s t i n the reverse r e g i o n , but the b i a s

i s extended past three times kT/e v o l t s t o provide a range f o r most

c r i t i c a l comparison of the measured nois e w i t h t h e o r y . The t e c h

niques developed permit use of l a r g e r r e verse b i a s e s , p o s s i b l y

s u f f i c i e n t f o r a v a l a n c h i n g i n the j u n c t i o n accompanied by enhanced

n o i s e . Since d i r e c t t u n n e l i n g c u r r e n t n o i s e i s uniform f o r a l l

except p o s s i b l y extremely h i g h f r e q u e n c i e s , a s i n g l e 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 i f 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 d e f i n e e q u i v a l e n t c i r c u i t generators t o re p r e s e n t the

noise i n a c i r c u i t which may a r i s e from v a r i o u s mechanisms, such

as the shot e f f e c t (due t o the d i s c r e t e n e s s and randomness of

charge t r a n s p o r t through p a r t of the c i r c u i t ) o r thermal noi s e i n

any r e s i s t a n c e (due to f l u c t u a t i o n s i n t h e r m a l l y e n e r g e t i c charges.)

An e q u i v a l e n t generator Is assigned t o each p a r t of the c i r c u i t f o r

which the noise a r i s e s from d i f f e r e n t mechanisms. Each generator

Is then d e f i n e d t o repr e s e n t the f l u c t u a t i o n s a s s o c i a t e d w i t h the

c u r r e n t c o n s i s t i n g of a l l e l e c t r o n s i n motion i n t h a t p a r t of the

c i r c u i t , as a f u n c t i o n of time. These same e l e c t r o n s may i n f l u e n c e

the c u r r e n t elsewhere In the c i r c u i t at the same time, but t h i s i s

accounted f o r by another n o i s e generator r e p r e s e n t i n g f l u c t u a t i o n s

i n the l a t t e r p a r t of the c i r c u i t . A l l such generators add quad

r a t i c a l l y i f they are u n c o r r e l a t e d , as i s o f t e n the ca s e .

Any device d i s p l a y i n g shot n o i s e w i t h o n l y one component of

c u r r e n t , i s represented by a mean-square constant c u r r e n t g e n e r a t o r

) of value 2el per u n i t bandwidth, I b e i n g the average c u r r e n t ,

across which i s pl a c e d the i n t e r n a l impedance of the d e v i c e . Thus

a vacuum noise diode, operated i n the temperature-saturated c o n d i

t i o n , has a n o i s e - e q u i v a l e n t c i r c u i t as shown i n F i g u r e 2.1. The

e x t e r n a l r e s i s t a n c e R i s taken as n o i s e l e s s f o r s i m p l i c i t y . The

value 2el f o r the generator ^ i > i s d e r i v e d from F o u r i e r a n a l y s i s

of the pu l s e s a s s o c i a t e d w i t h i n d i v i d u a l e l e c t r o n t r a n s i t s i a

s h o r t - c i r c u i t i s assumed between anode and cathode f o r t h i s c a l c u

l a t i o n . The v o l t a g e drop across R when i t i s i n c l u d e d then causes

< * f > ©

6

FIGURE 2.2

NOISE-EQUIVALENT CIRCUIT FOR A TUNNEL DIODE

the a c t u a l f l u c t u a t i n g c u r r e n t t o be

i 1 = i - v O i ^ v ) = i - i-jR/rp

o r

*1 = i rP / ( r p * R )

where i i s the f l u c t u a t i n g c u r r e n t d e r i v e d f o r R = 0, r p i s the

p l a t e r e s i s t a n c e of the vacuum diode, and v i s the instantaneous

v o l t a g e on the anode. The r e s u l t f o r i 1 i s seen t o be c o n s i s t e n t

w i t h the e q u i v a l e n t c i r c u i t , which shows the c u r r e n t i ^ through R.

S i m i l a r l y the thermal n o i s e i n any r e s i s t a n c e R i s represented

by a constant c u r r e n t g e n e r a t o r of mean square = l|kT/R per

u n i t bandwidth.

As a two-terminal d e v i c e , the t u n n e l diode embodies nois e due

t o the t u n n e l i n g p r o c e s s e s , and thermal noise i n the b u l k semi

conductor surrounding the j u n c t i o n . The n o i s e - e q u i v a l e n t c i r c u i t

f o r the t u n n e l diode i s shown i n F i g u r e 2.2, where 2

< l t ^ = mean square n o i s e c u r r e n t per u n i t bandwidth due t o t u n n e l i n g c u r r e n t s

^ i ^ > = mean square n o i s e c u r r e n t per u n i t bandwidth due t o thermal noise i n the b u l k r e s i s t a n c e R b

Rj. = r e s i s t a n c e of the j u n c t i o n due t o t u n n e l i n g processes

The t o t a l dynamic t u n n e l diode r e s i s t a n c e , given by e> V/PI from the

I - V c h a r a c t e r i s t i c , Is then R d = R^ + R^. E x p e r i m e n t a l l y , o n l y 2 2

the composite n o i s e spectrum due t o b o t h and can be 2

measured; when Rfe i s known, can e a s i l y be found a l j e b r a i o a l l y

u s i n g the present noise model.

14 2.2 Noise Spectrum f o r Direct Tunneling Currents of Esaki

2 2 2 In the expression <i.> = 2eTtf Af# we relate If to the Esaki b O °

formulation f o r dir e c t tunneling currents, equations (1.2.1), using

the d e f i n i t i o n (1.2.2). The l i m i t s f o r the Esaki integrals can be

extended to - o o and +o© without changing the values of the i n t e

g r a l s . The fermi functions are written e x p l i c i t l y as

1 1 f (E)= and f (E) =

c 1 + exp[(E - E f c)/kT] v 1 + exp[(E - E f y)/kT3

where E. and E^ are the fermi levels i n the conduction and v a l -fc fv ence bands respectively. Assuming

Z C V(E,V) = Z V C(E,V) = Z(E,V)

and i f each component of tunneling current independently produces

shot noise, then oo •

I = I I | + | l I = A« p (E)p (E)ZdE ) e <* 1 c v l ' vc« J *V T y ( x + Q X p j - ( E . E f )/kT]

1 2 1 + exp[(E - E f v ) / k T ] ( l + exp[(E - E f ( J)/kT])(l + exp[(E - E f v)/kT])

r oo

= A'|^exp(-E f v/kT) + exp(-E f c/kT)[ |f>c(E)pv(E)ZdE oo

exp(E/kT)

(1 + exp[(E - E f c ) / k T ] ) ( l + exp[(E - E f v)/kT]).

= A« [exp(-E f y/kT) + exp(-E f c/kT)] ^

S i m i l a r l y

FIGURE 2.3

BEHAVIOR OF WITH BIAS VOLTAGE

15

i | = I i - i I I I cv vol

r / 1

1 i = A' I p (E)p (E)ZdE< A

J ^ c W ( l + exp[(E-E f c)/kT] 1 + exp[(E-E )/kT]; fv'

= A 'J]exp(-E f v/kT) - exp(-E f c/kT)|J £ p c ( E ) p v ( E ) 2 d E •

•J r\t>

exp (E A T )

(l + exp[(E - E f c)/kT])(l + exp[(E - E f v)/kT])

= At|^exp(-E f v/kT) - exp(-E f c/kT)]|

A' is a constant. Hence, using the definition (1,2.2) we have'

y 2 exp(-E f v/kT) + exp(-E f c/kT) ^ ^ l E

f c " E f y l ° |exp(-E f v/kT) - exp(-E f c/kT)| 2kT

Now | v ) , the magnitude of the applied bias, is given by | ( E f ( J - E f v ) | / e

so that ^2 e ' v l 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, i t 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, <i.>remains positive as is physically necessary.)

2 2 Q behaves with bias as shown in Figure 2 . 3 . V Q tends to

i n f i n i t y as Wl tends to zero, so that

<i?> = 2e|l|fr 2 > hkT H t 0 dV V=0

compatible with Nyquist's theorem, as must be the case.

The r e s u l t ( 2 . 2 . 1 ) i s o b t a i n a b l e f o r any f u n c t i o n

P(E,V) = p c(E)^> v(E) Z(E,V)

so t h a t i t i s independent of the band s t r u c t u r e of the semi

conductors. The o n l y requirements needed t o produce ( 2 . 2 . 1 ) a r e :

a) Z ,(E,V) = Z (E,V), i . e . , t u n n e l i n g r e c i p r o c i t y h o l d s , and C V V c

b) the occupancy f u n c t i o n s are f e r m i f u n c t i o n s .

The assumed n o n - c o r r e l a t i o n of the c u r r e n t components I and I cv vc

Is a l r e a d y assumed In the d e f i n i t i o n of }f q, equation ( 1 . 2 . 2 ) .

The r e s u l t ( 2 . 2 . 1 ) i s seen t o imply the i n t e r e s t i n g r e l a t i o n

K A J = e x p teV/kT> F i n a l l y , we note t h a t ^ ^ 1 i n g e n e r a l f o r semiconductor d i o d e s ,

which are assumed t o have shot n o i s e a s s o c i a t e d w i t h t h e i r c u r r e n t s , 2'

whereas Y o = 1 f o r a vacuum diode which produces f u l l shot n o i s e .

The d i s t i n c t i o n i s th a t two components of c u r r e n t are a s s o c i a t e d

w i t h semiconductor t u n n e l i n g p r o c e s s e s , but on l y one component

flows i n vacuum d i o d e s .

2.3 R e s t r i c t i o n s on E s a k i ' s T u n n e l i n g Theory

In the d e r i v a t i o n of (f which r e p r e s e n t s the noise spectrum

a s s o c i a t e d w i t h d i r e c t E s a k i t u n n e l i n g c u r r e n t s , no assumption

r e g a r d i n g d e n s i t i e s - o f - s t a t e s was r e q u i r e d , but the occupation

f a c t o r f o r these s t a t e s was taken as the f e r m i f u n c t i o n . T h i s i s

a r e s t r i c t i o n on the most g e n e r a l statement, w i t h i n the E s a k i

f o r m u l a t i o n , p o s s i b l e f o r d i r e c t t u n n e l i n g . F o r example, i t ex

cludes bosons from t u n n e l i n g a c c o r d i n g t o t h i s f o r m u l a t i o n .

The most g e n e r a l statement of t u n n e l i n g which assumed t h a t

the d e n s i t y of occupied s t a t e s on one s i d e of a j u n c t i o n , and the

density of unoccupied states on the other side, are the factors controlling the resulting tunneling currents, would be

J 1 2 and J are current densities per energy increment dE^ and dE 2 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 distance 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 i n i t i a l occupied states, and h^ and h 2 are densities of the f i n a l 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 equations (2.3*1)# we must satisfy the thermodynamic requirement that

This Is Nyquist's theorem. The average current, taken over a l l energy levels of the system, is

J 1 2 ( E 1 ) 6E1 = Z 1 2 ( E 1 ) n 1 ( E 1 ) h 2 ( E 2 ) dE], (2.3

c>V V=0

Z(E1,E1+eV) [n 1(E 1)h 2(E 1+eV) - n 2(E 1 + e V)h 1(E 1) J

dE, 1

dE 1

where i s any energy l e v e l on the " 1 " s i d e o f the j u n c t i o n , and

E ^ 8 8 E^+eV, where V Is the a p p l i e d b i a s .

We now f i n d the most g e n e r a l form of the f u n c t i o n s n. and

h^ 2 such t h a t Nyquist's theorem Is s a t i s f i e d . Assuming -

= Z, the conductance Is

oo

Hv f r ah9(E.+eV) S>n 5(E-+eV)-,

J L a i ( B i ) ' h i ( E i } ^7 J z ^ E i ' V e V ) d E i - CO /

n S>ZJ(E-,E..+eV)

+ |[n 1(E 1)h 2(E 1+eV) - n2(E1+eV)h1(E1)J ~ 1 A d E x

13J/ f r ah (E.) dn.(E ) ,

-eo

The second i n t e g r a l vanishes s i n c e J ^ ^ = J g i 8 8 ® w h e n V = 0 .

The e q u i v a l e n t s a t u r a t e d n o i s e c u r r e n t d e n s i t y i s

oO

* j [ J 1 2 ( E l ' E 2 ) + J g ^ E ^ E g ) ] d E x

- eO

= j " f c l ( E l ) h 2 ( E l + e V ) + n 2( E l + e V ) h l ( E l ) ] Z ( E l , E l + e V ) d E l

- OO

l i m J V-K> eq = 2 J n 1 ( E 1 ) h 2 ( E 1 ) Z ( E 1 , E 1 ) d E 1

s i n c e - J21 a t V = 0 , Nyquist's theorem i s now w r i t t e n

o>J(E) ' ^ [ J 1 2 ( E ) + J 2 1 ( E ) ] V=0 2kT V=0

T h i s r e l a t i o n i s taken t o hold f o r each energy l e v e l E, f o r at

V = 0 , we assume d e t a i l e d - b a l a n c e h o l d s , t h a t i s , not o n l y are

the macroscopic currents (integrals of equations (2.3,1) over a l l energies) equal and opposite, but the components J 1 2 and J 2 1 are equal and opposite for every energy level E 1 = E, Then Nyquist's theorem gives

dh_(E) dn9(E) 1 1 n,(E) — 2 - h - ( E ) — = — n,(E)ME) = — n(E)h(E)

x d E -1 dE kT x * kT 2 1 o r dh9(E) dn„(E) 1

—t - — 2 = — dE h2(E) n 2(E) kT

where E Is any energy level. Integrating this equation between limits E Q and E ( E q arbitrary) gives

hp(E) h p(E ) r l -i = —2*-—a- exp I — (E - E ) I (2.3.2) n.(E) M E ) L kT °J 2 do

and similarly ^(E) M E Q ) r l —= = —=—— exp — - « n..(E) n (E ) LkT 0

1 l o exp | — ( E - E j J (2.3.2)

Equations (2.3.2) are the most general relations between the functions n^ 2(E) and h^ 2(E) such that the generalized Esaki integrands (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 V 1 (2.3.3) n2(E) = ^(E)g(E) h2(E) = f 2(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 i t 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

f u n c t i o n a r i s e s o n l y f o r fermions, due t o the P a u l i E x c l u s i o n

P r i n c i p l e . "Hence s u b s t i t u t i n g equations (2.3.3) i n t o e i t h e r of

equations (2.3.2), whioh g i v e s

1 - g(E) 1 - g(E ) E - E E - E o = o_ Q X p o = C ( E j Q X p S

g(E) g ( E 0 ) kT ° kT

1 g(E) =

1 + 0(E o) e x p[(E - E ) A T ]

serves o n l y t o check t h a t g(E) i s indeed the f e r m i f u n c t i o n .

(Since g(E) 1, then C ( E 0 ) ' ^ 0. We may then put C(E Q)exp( -E Q/kT)

s e x p ( - E f ) and i d e n t i f y E f as the f e r m i l e v e l . )

Thus the E s a k i f o r m u l a t i o n f o r d i r e c t t u n n e l i n g i s a p p l i c a b l e

o n l y t o fermions, under the c o n d i t i o n t h a t shot n o i s e reduce t o

thermal n o i s e at V = 0.

The f o r e g o i n g a n a l y s i s does not exclude the p o s s i b i l i t y of

boss p a r t i c l e s t u n n e l i n g d i r e c t l y , as w e l l as f e r m i p a r t i c l e s ,

while s a t i s f y i n g the thermodynamic requirements at V = 0. The

E s a k i approach, i n the form of equations 2.3.1 Is I n a p p r o p r i a t e

f o r bosons, owing t o the u n l i m i t e d occupancy per energy l e v e l

allowed these p a r t i c l e s . A much more g e n e r a l f o r m u l a t i o n a l t o

gether i s needed i n t h i s case.

I t i s of i n t e r e s t t o f i n d the most g e n e r a l occupancy f u n c t i o n

g(E) such t h a t = ( | J 1 2 | + 1 J2 1P/U J

1 2| " | J2 l P 1 3 a ^ c * 1 0 0

s o l e l y of the b i a s and temperature ( i n the e x p l i c i t form eV/kT)

and i s not dependent on the semiconductor band s t r u c t u r e . The most

g e n e r a l form of ( e V / k T ) i s a l s o found under t h i s c o n d i t i o n . We

have

= Y^ ' J z(E 1,E 1+aV)| f 1 ( E 1 ) g ( E 1 ) p 2 ( E 1 + e V ) [ l - g ( E 1 + e V ) ]

( E 1 + e V ) g ( E 1 + e V ) f 1 ( E 1 ) [ l - g ( E 1 ) ] j d ^

" l J12l + l J 2 l l

= f Z ( E 1 , E 1 + e V ) ^ p 1 ( E 1 ) g ( E 1 ) p ^ E ^ e V ^ l - gtE^+eV)]

+ p 2 . ( E 1 + e V ) g ( E 1 + e V ) ^ 1 ( E 1 ) f l - g ( E 1 ) ] J dE^^

Fo r Y Q t o be a f u n c t i o n s o l e l y o f eV/kT, t h a t i s , t o be indepen

dent of the band s t r u c t u r e , ^ and £ 2 » i t i s r e q u i r e d f o r each

energy l e v e l E ^ = E t h a t (comparing the integrands of the l a s t two

e q u a l i t i e s ) :

* o ( e V ) [ g ( E ) - g(E+eV)] = g(E) + g(E+eV) - 2g(E)g(E+eV)

Here i t i s convenient t o i n t r o d u c e the diraensionless v a r i a b l e s

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 t h i s f u n c t i o n a l equation f o r g(E'), put

P ( E t ) = l / g ( E « ) - 1

Then the f u n c t i o n a l equation becomes

K ^ ) - [ H E ' ) + r(E'+U)]/fp(E») - P(E'+U)] or

> > - l ] / [ * o < Hence

T(E'+U) / p(E') = [y^(U) - l ] / f ^ o ( U ) + X J = F < U )

In T(E«+U) = In F(E') + In F (u)

S i n c e , t h e r e f o r e ,

^ l n T(E'+U) } l n P(E')

5 E 1 d E'

and s i n c e U i s a r b i t r a r y , I t f o l l o w s t h a t In V must be a l i n e a r

f u n c t i o n of i t s argument. Hence

T(E') = exp ( C l + C 2 E » )

so t h a t from the d e f i n i t i o n of P(E')»

g(E') = 1 + exp (c + c E») 1 + exp (c E« - E*) 1 2 2> f

2 where E„ = E»kT i s the f e r m i energy. Thus ifV i s a f u n c t i o n o n l y f f ° o of eV/kT, g(E') i s the f e r m i f u n c t i o n except f o r an a r b i t r a r y

constant

The most g e n e r a l form of 2f2, I f I t Is a f u n c t i o n o n l y of eV/kT

Is found by combining the r e l a t i o n d e f i n i n g P(U) w i t h the exponen

t i a l form of P ( E ' ) t o give

Y ^ U ) - 1

*f:(u) + i 2

or >^(U) « c o t h (c 2U/2)

v 2 The c o n d i t i o n t h a t 0 Q depend o n l y on U t h e r e f o r e leads t o the

r e l a t i o n ^ = c o t h (eV/2kT) except f o r the f a c t o r o^. Prom the

f o r e g o i n g , i t i s apparent t h a t the thermodynamic requirement of

Nyqulst's theorem as an added r e s t r i c t i o n , r e q u i r e s Cg t o be u n i t y .

23

2,4 Models f o r Noise A s s o c i a t e d w i t h I n d i r e c t T u n n e l i n g Processes

In the v a l l e y r e g i o n of the I - V c h a r a c t e r i s t i c , b o t h

d i f f u s i o n a l p-n j u n c t i o n m i n o r i t y c a r r i e r thermal c u r r e n t , and

c u r r e n t due t o i n d i r e c t t u n n e l i n g processes c o n t r i b u t e t o the

average c u r r e n t f l o w i n g . The i n d i r e c t t u n n e l i n g c u r r e n t dominates.

In a l a t e r c hapter, we r e p o r t t h a t at a frequency s u f f i c i e n t l y h i g h

t h a t a l l l / f component has disappeared from the nois e i n the d i r e c t

t u n n e l i n g r e g i o n s of the I - V curve, the measured nois e a s s o c i a t e d

w i t h the v a l l e y c u r r e n t r e g i o n g r e a t l y exceeds f u l l shot n o i s e .

T h i s can be due to any of three causes:

a) the n o i s e a s s o c i a t e d w i t h o r d i n a r y p-n j u n c t i o n thermal c u r r e n t may exceed shot n o i s e .

b) at f r e q u e n c i e s s u f f i c i e n t l y h i g h t h a t no l / f component e x i s t s f o r t u n n e l i n g c u r r e n t n o i s e , a l / f component may s t i l l e x i s t f o r p-n j u n c t i o n thermal c u r r e n t s , s i n c e these a r i s e from e n t i r e l y d i f f e r e n t mechanisms than do t u n n e l i n g c u r r e n t s .

c) the n o i s e a s s o c i a t e d w i t h i n d i r e c t t u n n e l i n g c u r r e n t s may exceed shot n o i s e .

The f i r s t p o s s i b i l i t y may be q u i c k l y r u l e d out. Van der Z i e l

( 1 9 £ 8 ) has shown that the nois e b e h a v i o r o f an o r d i n a r y p-n junc

t i o n diode may be represented by a c u r r e n t generator i i n shunt

w i t h the j u n c t i o n , such t h a t = qkTG - 2 e l per u n i t bandwidth,

where G = 9 l / ^ V Is the j u n c t i o n conductance and I i s the j u n c t i o n

c u r r e n t . T h i s can be r e w r i t t e n as

F o r a normal p-n j u n c t i o n diode, the I - V c h a r a c t e r i s t i c i s given

by

I =

1 ' 1 I

FIGURE 2.U

DETAILED MECHANISMS INVOLVED IN INDIRECT TUNNELING

where I = constant i s the " s a t u r a t i o n " j u n c t i o n c u r r e n t . Com-s b i n i n g these r e l a t i o n s g i v e s

. 1 + exp (-eV/kT) ~ \ l d > = 2el = 2el c o t h (eV/2kT) S 2elfi*

1 - exp (-eV/kT) 0

The normal p-n J u n c t i o n thermal c u r r e n t n o i s e i s t h e r e f o r e given

by e x a c t l y the same f u n c t i o n of b i a s and temperature as t h a t due

t o d i r e c t t u n n e l i n g c u r r e n t s , a l t h o u g h the l a t t e r do not e x i s t i n

the v a l l e y b i a s r e g i o n . Since two o p p o s i t e l y f l o w i n g c u r r e n t

components are a s s o c i a t e d w i t h the o r d i n a r y j u n c t i o n thermal

average c u r r e n t , b o t h these components have f u l l shot n o i s e assoc

i a t e d w i t h them, but not more than t h i s .

The v a l i d i t y of the second p o s s i b i l i t y f o r excess n o i s e i n

the v a l l e y r e g i o n can be checked o n l y by n o i s e measurements at

s e v e r a l f r e q u e n c i e s .

F o r the t h i r d p o s s i b i l i t y , we now i n v e s t i g a t e the n o i s e

spectrum f o r p o s s i b l e mechanisms i n v o l v i n g i n t e r a c t i o n of the

t u n n e l i n g e l e c t r o n s w i t h t r a p s or i m p u r i t y s i t e s w i t h i n the f o r

bidden gap.

The p o s s i b l e paths f o r e l e c t r o n s i n i n d i r e c t t r a n s i t i o n s

across the gap are shown i n F i g u r e 2.1|. The mechanisms a r e :

h o r i z o n t a l arrows: t u n n e l i n g occurs d i r e c t l y between bands and t r a p s , i n e i t h e r d i r e c t i o n . The d e n s i t i e s of occupied s t a t e s s t r o n g l y d i s f a v o r t u n n e l i n g from C t o A ( F i g u r e 2.4), from B t o A, or from D t o B^, or from D to C^. T u n n e l i n g i n the opposite d i r e c t i o n s i s r e l a t i v e l y p robable•

v e r t i c a l arrows : e l e c t r o n s l o s e energy by phonon or photon i n t e r a c t i o n ( s ) , o r by e l e c t r o n - e l e c t r o n i n t e r a c t i o n s , the l a t t e r b e i n g v e r y improbable .

o b l i q u e arrows : t u n n e l i n g ( h o r i z o n t a l component) and (phonon) a b s o r p t i o n of the e l e c t r o n energy occur s i m u l t -t a n e o u s l y .

The a n a l y s i s t o f o l l o w w i l l imply that the n o i s e a r i s i n g from these

processes i s independent of whether e l e c t r o n s l o s e energy while

t r a n s i t i n g the gap. Thus process AD ( o b l i q u e arrow) produces shot

n o i s e , as do d i r e c t - t u n n e l i n g e l e c t r o n s . Processes AGD and AC^D

a l s o c o n t r i b u t e o n l y shot n o i s e , i f i t i s assumed (as i n the l i t e r

a t u r e : Chynoweth e t . a l . , i960) t h a t these processes are r a t e -

l i m i t e d by the average r a t e of the h o r i z o n t a l t r a n s i t i o n s , and not

by the v e r t i c a l t r a n s i t i o n s . " 1 " Processes ABD and AB^D w i l l be shown

t o produce at most shot n o i s e . However, processes ABA, ACA, DC-jD,

and DB^D can cause g r e a t e r than shot noise f o r the o v e r a l l average

i n d i r e c t c u r r e n t , s i n c e these routes r e p r e s e n t f l u c t u a t i n g c u r r e n t s

without c o n t r i b u t i n g t o the average c u r r e n t .

The r e l a t i v e l i k e l i h o o d s of each of these mechanisms Is un

known, as i s the exact nature of the v a l l e y c u r r e n t , although i t

i s b e l i e v e d due to i n d i r e c t t u n n e l i n g processes l a r g e l y . F i g u r e

2.4. r e p r e s e n t s a g r e a t l y s i m p l i f i e d model. The v e r y l a r g e number

of t r a p s undoubtedly present i n the f o r b i d d e n gap of d e g e n e r a t e l y

doped semiconductors would a l l o w many p o s s i b l e routes not shown by

the extreme cases i n F i g u r e 2 . 4 . P a r t i c u l a r l y important, from the

p o i n t of view of processes producing g r e a t e r than shot n o i s e , would

be the u n l i m i t e d p o s s i b i l i t y f o r t u n n e l i n g t r a n s i t i o n s back and

f o r t h between t r a p s w i t h i n the gap. These would g r e a t l y enhance

the f l u c t u a t i o n s but not the average c u r r e n t .

Since energy l o s s e s f o r an e l e c t r o n while i n the f o r b i d d e n

1. I f these processes were r a t e - c o n t r o l l e d by the v e r t i c a l t r a n s i t i o n r a t e s , a non-uniform spectrum of magnitude l e s s than shot n o i s e would r e s u l t , due to s u c c e s s i v e e l e c t r o n s b e i n g c o r r e l a t e d ( s i n c e t r a p s are occupied by at most one e l e c t r o n ) .

valence band'

FIGURE 2.5

SIMPLIFIED MODEL FOR NOISE ANALYSIS OF INDIRECT TUNNELING PROCESSES

gap do not i n f l u e n c e the a s s o c i a t e d n o i s e (assuming the mechan

ism of i n t e r a c t i o n w i t h phonons i s not n o i s y , or can be r e p r e

sented as thermal n o i s e ) , the a n a l y s i s of I n d i r e c t t u n n e l i n g

c u r r e n t noise w i l l be based on a s i m p l i f i e d model, as shown i n

F i g u r e 2 .5 . Energy i s conserved f o r a l l t r a n s i t i o n s , and a l l

t r a p s ( r e p r e s e n t e d by s h o r t dashed l i n e s ) are a d i s t a n c e r Q from .

the edge of the conduction band. An e l e c t r o n t u n n e l i n g from the

conduction band has a p r o b a b i l i t y p of I n t e r a c t i n g w i t h a t r a p ,

or a p r o b a b i l i t y 1 - p of t u n n e l i n g on d i r e c t l y t o the valence

band without e n c o u n t e r i n g a t r a p . F o r an e l e c t r o n which encounters

a t r a p , l e t > be the p r o b a b i l i t y t h a t the e l e c t r o n tunnels back t o

the conduction band sometime a f t e r c a p t u r e . T h i s w i l l be r e f e r r e d

t o as "event B", 1 - ^ i s then the p r o b a b i l i t y t h a t the e l e c t r o n

tunnels onward t o the valence band sometime a f t e r c a p t u r e . T h i s

w i l l be "event A".

For e i t h e r event A or B, l e t the p r o b a b i l i t y t h a t an e l e c t r o n

i s r e l e a s e d by a t r a p i n any time I n t e r v a l dt a f t e r capture be

c o n s t a n t , t h a t i s , Independent of the time. I t then e a s i l y f o l l o w s

t h a t the p r o b a b i l i t y of an e l e c t r o n b e i n g i n a t r a p at time t == t

I f i t entered the t r a p at time t = 0 , i s

P ( 0,t) = exp

where *C ^ i s the average capture p e r i o d f o r the i * * 1 process ( i =

A or B ) .

We assume f o r s i m p l i c i t y t h a t the t r a n s i t times f o r e l e c t r o n s

between bands and t r a p s are z e r o . The c u r r e n t a s s o c i a t e d w i t h

event A, which i s d e f i n e d t o flow through the e n t i r e t u n n e l diode

due t o the t u n n e l i n g of the e l e c t r o n , takes the form of two suc

c e s s i v e S-functions as shown i n F i g u r e 2 . 6 , T h i s a r i s e s from the

i n ( t ) t

r 0 e/s

t ts 0

FIGURE 2.6 CURRENT ASSOCIATED WITH "EVENT A"

f a c t that a charge e at rest i n a d i e l e c t r i c at a distance r Q

from one of two conducting planes, and a distance s - r from the o

other, induces charge e(s - r Q ) / s and er^/s respectively on these

two planes. For the case of the charge e moving at I n f i n i t e velo

c i t y 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

l a t e r the current £(s - r o ) / s l e ^ ( t -K) i s induced when the

electron tunnels to the valence band. These currents s a t i s f y the

requirement that the i n t e g r a l over a l l time of the instantaneous

current i ( t ) equals the t o t a l charge induced, that Is, e, at any

part of the c i r c u i t . I f t A Is the average capture time per el e c

tron per trap f o r event A, the £-functions In Figure 2.6 are

separated by a random variable % whose ensemble average Is T .

To f i n d the noise spectrum f o r events A, we assume the curr

ents due to a l l such electrons are mutually independent, that i s ,

the electrons leave the conduction band randomly, and do not

affe c t each other thereafter. With i ( t ) the res u l t of a large

number of Independent currents i n ( t ) (of which Figure 2.6 i s an

i l l u s t r a t i o n ) occurring at random at the average rate f A , the

noise spectrum may be found by Fourier analysis of each component

i n ( t ) . In case a l l components i n ( t ) have the same time constant

'E, the spectral density, or absolute value squared of the Fourier

c o e f f i c i e n t of i ( t ) , i s calculated from Carson's Theorem:

S A(A>) = 2 ^ A ) ' 7 ( < o ) l 2

where oo

J (UJ) «J" i n ( t ) exp (-j*>t) dt o

In the present case, there i s a random d i s t r i b u t i o n of time con-

s t a n t s f f o r the c u r r e n t s i n ( t ) a s s o c i a t e d w i t h the n e l e c t r o n

undergoing process A, The r e q u i r e d m o d i f i c a t i o n of Carson's

Theorem i s t o l e t oo

d ^ A = gCtOd'tf where ' J ^ g ( ' C ) d ' t f = l

be the number of events per second w i t h a time constant between

t, and X + d t . Then CO

sA(o>) = Z\IX„(vo)\2 g ( t r )

where (u o ) i s the F o u r i e r t r a n s f o r m f o r events having time

constant t . Since g ( T )d't> i s t h e r e f o r e the number of e l e c t r o n s

p e r second which are captured by t r a p s f o r a d u r a t i o n between ^

and t + dt? , we have

g C O d t = i>k exp ( - r / t r A ) d i ? / i ? A

Now

7^(60) = J^[fe <f(t - 0) + (1 - f ) e 6"(t exp(-JO>t) dt o

= f e + e ( l - f ) exp(+ J ^ t )

where f = r / s . Hence o'

I? (60 )| 2 = e 2 (1 - 2f + 2 f 2 ) + e 2 (1 - f ) f 2 cos (oOU )

The n o i s e spectrum of the c u r r e n t due t o a l l e l e c t r o n s undergoing

process A Is then oo

SAov) = 2 ^ e 2

A A u [ ( 1 - 2f + 2 f 2 )

+ f ( l - f) 2cos(60*)J exp ( - ^ / r . ) d t / ^ , A A

I n ( t ) t

r Qe/s

t = X

t = 0

-r Qe/s

FIGURE 2 . 7

CURRENT ASSOCIATED WITH "EVENT B"

(2.14.1) where ^ i s the average r a t e of emission of e l e c t r o n s I n t e r a c t i n g

w i t h t r a p s e i t h e r by process A or by process B. The average or d.c

c u r r e n t a s s o c i a t e d w i t h process A i s

where 1 i s the average c u r r e n t f o r b o t h process A and process B

combined, the l a t t e r having no average c u r r e n t a s s o c i a t e d w i t h i t .

I t i s noted from equation (2.4.1) t h a t the n o i s e produced by

c u r r e n t s due t o process A, i s l e s s than shot n o i s e u n l e s s f = 0,

that I s , u n l e s s the t r a p s merge w i t h the conduction band edge.

To f i n d the n o i s e spectrum f o r process B, we again assume

t h a t c u r r e n t s due t o t r a n s i t i o n s of each e l e c t r o n are m u t u a l l y

independent. Each e l e c t r o n , i n t u n n e l i n g t o a t r a p , b e i n g cap

tured on the average a time then t u n n e l i n g back t o the con-B

d u c t i o n band, produces the c u r r e n t form given i n F i g u r e 2 . 7 . The

F o u r i e r t r a n s f o r m of t h i s c u r r e n t i s CO

U s i n g

•2> n exp ( - t / t f ) d f / * B

f o r process B, the s p e c t r a l d e n s i t y f o r c u r r e n t s a s s o c i a t e d w i t h

process B i s

S (60 ) B 2 B y

OO r 2 f 2 e 2 (1 - cosM*) exp (-T/'tf ) dV/t^

o

k h - i > B f 2 e 2 6 0 2 ^ | / (1 +0> Z t \ )

= ! } ^ / e 2 i 0 2 ^ / ( l + " > 2 ^ 2 ) (2.U.2)

where p i s the p r o b a b i l i t y t h a t an e l e c t r o n , a f t e r capture by a

t r a p , t u n n e l s back t o the conduction band.

The o v e r a l l n o i s e spectrum S T ( 6 0 ) due t o a l l e l e c t r o n s which

i n t e r a c t w i t h t r a p s , e i t h e r by process A or by process B, i s the

sum of S.(oo) and S (a?) s i n c e the two processes are taken as A B m u t u a l l y independent.

sT(a>) = V < 0 ) + S B ( 6 0 )

» 2 1 ^ T e 2 | ( 1 - o ) [ l - 2 f ( l - f ) a ) 2 ^ 2 / (1 + 6 0 2 ^ | )

+ 2 ^ f 2 ^ 0 2 r C 2 / (1 + rt2t\ ) j

(2.U.3) I f the frequency of measurement, 40 , approaches z e r o , o r at l e a s t

becomes s m a l l compared w i t h l/t^ and l / ' t f g , the capture processes

of the t r a p s should become unimportant t o the r e s u l t i n g n o i s e . We

should then expect shot n o i s e t o r e s u l t , as i s the case when e l e c

t r o n s do not encounter t r a p s . Now

l i m Q S T(4>) = 2 ^ > T e 2 ( l -^>) = 2 e I A 5 2 e I T

where I,p ( t h a t I s , I A ) i s the average c u r r e n t a s s o c i a t e d w i t h

t r a p - I n v o l v e d e l e c t r o n t r a n s i t i o n s , process B having zero a s s o c i

ated average c u r r e n t . Thus shot n o i s e i s secured at low f r e q u e n c i e s .

31 The noise due to electrons which do not encounter traps, but

which tunnel directly (that i s , only with phonon interactions in this model) from conduction to valence band, is easily included since It is f u l l shot noise, that is, S.^(tv ) = 2el d, 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

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,

I T « el>p (1 - f ) I d = ei>(l - p)

S(to ) = sT(a>) + S d(W)

and since the total average current flowing is

then

S(o> ) = 2el (1-p) + p(l-p)

S 2el 2 (2.U.U)

FIGURE 2 . 8

POSSIBLE NOISE SPECTRA FOR INDIRECT TUNNELING PROCESSES

These r e l a t i o n s i n d i c a t e t h a t ^ 2 - ^ o © as p 1 and — - * 1 , whereas

I — » 0 . These are the c o n d i t i o n s f o r which every e l e c t r o n l e a v i n g

the conduction band i n t e r a c t s w i t h a t r a p , then tunnels back t o the

conduction band.

To decide whether g r e a t e r than shot n o i s e i s p o s s i b l e ( i t

c e r t a i n l y i s not p o s s i b l e u n l e s s process B i s o p e r a t i v e ) , we

examine o n l y S^(o0), s i n c e the e f f e c t of adding S^(co ) on l y " d i l

u t e s " the s i g n i f i c a n t b e h a v i o r o f the noise spectrum. I t i s then

convenient t o d e f i n e

tf2(a>) = s T(o>) / 2 e i T

= 1 - 2 f ( l - f ) " + - f l f 2 U > 2 ^ t J (2.11.5)

l + M j ^ t f i - f I+CO 2? 2,

w i t h use of equation (2.q .3) and the r e l a t i o n I T = 2>^e (l-^> ).

C l e a r l y ^ 2 1 ( t h a t I s , g r e a t e r than or l e s s than shot n o i s e

can r e s u l t ) depending on the v a r i o u s parameters i n v o l v e d . I f no

assumptions f o r t h e i r i n t e r r e l a t i o n s are made, any of the nine

curves f o r 0 versus frequency CO shown i n F i g u r e 2*8 can o b t a i n .

Since

* * ( » ) • 1 - 2 f ( l - f ) + < 2 ( > f 2 ) / ( l - { )

then

£ 2 ( o o ) | 1 i f f / ( l - f ) | ( l - f ) / f

At f i n i t e f r e q u e n c i e s , the f a c t o r s CO2 t 2 / (1 + w ) 2 / C 2 ) and

(A t„ / (1 + £0 ) are rou g h l y s t e p - f u n c t i o n s , but i n g e n e r a l B a

r i s e r a p i d l y toward u n i t y at d i f f e r e n t f r e q u e n c i e s , 1/fftA and l / t g ,

depending upon which of and'Cg i s l a r g e r . The f o l l o w i n g t a b l e

shows the correspondence between the spectrum curves and the r e l a -

•3(3 t i v e sizes of the parameters involved. The entries i n the table

r e f e r to the curves designated s i m i l a r l y In Figure 2.8.

r A < r B #3 #7 #5

% = * B #1 #9 #6

#2 #8 #4

For curve #9, the completely degenerate case, "ft (CO ) = 1 f o r a l l

60.

Only one conjecture which i n t e r r e l a t e s the parameters, can be

made on physical grounds. Since l / f g i s the p r o b a b i l i t y per unit

time of an electron In a trap leaving v i a event B, and s i m i l a r l y

f o r l/'t'A» w e c a n assume that (1 - ^ )/ > = /^B S * N C E ^ I S

. the p r o b a b i l i t y per electron leaving the conduction band that event

B w i l l occur, and 1- ^ i s the same f o r event A. It can then be

shown from equation (2.4 »5>) that curves #2- and #5 of Figure 2.8 are

p h y s i c a l l y impossible; that i s , since ^0 i s to be r e a l , )i 2 = 1 only

f o r 60 = 0. No other non-restrictive assumptions can be applied, so

that the model assumed f o r these calculations serves only to i n d i

cate the p o s s i b i l i t y of noise in excess of shot noise.

The entire model can be somewhat generalized in-threesways:

a) instead of assuming instantaneous t r a n s i t i o n s between

bands and traps, the t r a n s i t times can be taken as f i n i t e . (The

current pulses are then rectangular, assuming no acceleration of

the charge while t r a n s i t i n g ) . The spectrum given In equation

(2.4.3) w i l l then Include a m u l t i p l i c a t i v e faotor which causes i t

t o f a l l o f f at f r e q u e n c i e s of the order of l / ^ t and above, ^ t

b e i n g the t r a n s i t time. Such f r e q u e n c i e s are v e r y much h i g h e r

than those f o r which the spectrum Is of i n t e r e s t due to I n d i r e c t

t u n n e l i n g mechanisms,

b) i n s t e a d of a o"-function d i s t r i b u t i o n f o r t r a p p o s i t i o n s

between the conduction and valence bands, we can c o n s i d e r any

s p a t i a l d i s t r i b u t i o n of t r a p s Y | ( r ) d r . The a p p r o p r i a t e m o d i f i

c a t i o n of Carson's Theorem f o r the s p e c t r a l d e n s i t y i s

r - s

S(60) o o

^ J [ ^ , r ) ^ ( r ) d r g ( t ) d t

where s i s the h o r i z o n t a l d i s t a n c e between conduction and valence

band (assumed independent of the energy) on the energy diagram.

The spectrum p r o f i l e w i t h frequency w i l l be d i f f e r e n t f o r d i f f e r e n t

assumed forms of T ^ ( r ) d r so t h a t i n p r i n c i p l e n o i s e measurements

could be r e l a t e d t o t r a p d i s t r i b u t i o n s w i t h i n the f o r b i d d e n gap, as

w e l l as to capture times of the t r a p s and r e l a t i v e p r o b a b i l i t i e s

f o r the v a r i o u s p o s s i b l e i n d i r e c t p r o c e s s e s .

c) c o n s i d e r the average i n d i r e c t c u r r e n t ( v a l l e y r e g i o n c u r r

ent) to be the d i f f e r e n c e of the averages of two i n d i r e c t t u n n e l i n g

c u r r e n t components f l o w i n g i n opposite d i r e c t i o n s . The component

f l o w i n g from valence t o conduction band w i l l be v e r y s m a l l , and i t s

n o i s e spectrum may be d i f f e r e n t t o t h a t of the opposite component

considered i n the f o r e g o i n g . The composite spectrum f o r the two

components w i l l be

S ( i O ) = S e f V f t A M l l J + V|(a> ) | I 2 | ]

where the s u b s c r i p t s r e f e r t o the two components r e s p e c t i v e l y . The

n o i s e i s seen t o exceed t h a t due t o one component alone, i f the two

components are u n c o r r e l a t e d .

FIGURE 2 .9

CHARGE DISTRIBUTION WITHIN A TUNNEL DIODE JUNCTION

35

2,kl Modulation i n the I n d i r e c t - T u n n e l i n g Modal f o r

V a l l e y Noise

The f o r e g o i n g model has p r e d i c t e d the p o s s i b i l i t y of an

enhanced spectrum over shot n o i s e on the b a s i s of e l e c t r o n s t u n

n e l i n g back and f o r t h from conduction band t o t r a p s , or from t r a p s

t o valence band. Since such a process may be Infrequent r e l a t i v e

t o u n i - d i r e c t i o n a l t u n n e l i n g t r a n s i t i o n s , which do not produce

more than shot n o i s e at any frequency, i t i s u s e f u l t o c o n s i d e r

p o s s i b l e modulation e f f e c t s on the f i e l d governing the t u n n e l i n g

of an e l e c t r o n to a t r a p . Such e f f e c t s can e i t h e r enhance or

decrease the n o i s e .

The t u n n e l i n g of a s i n g l e e l e o t r o n to a donor, or v e r t i c a l

t r a n s i t i o n t o an a c c e p t o r changes the j u n c t i o n e l e c t r i c f i e l d ,

which i n t u r n m o d i f i e s the p r o b a b i l i t y f o r s u c c e s s i v e e l e c t r o n s

t o t u n n e l from conduction band t o donors, or from a c c e p t o r s t o

valence band. I f the t r a n s i t i o n of the I n i t i a l e l e c t r o n d i s c o u r

ages or encourages s i m i l a r t r a n s i t i o n s of s u c c e s s i v e e l e c t r o n s ,

the n o i s e w i l l be decreased or i n c r e a s e d r e s p e c t i v e l y f o r the

c u r r e n t component a r i s i n g from the p a r t i c u l a r process c o n s i d e r e d .

C o n s i d e r i n F i g u r e 2.1 i n d i r e c t processes i n v o l v i n g h o r i z o n t a l

and v e r t i c a l t r a n s i t i o n s . F i g u r e 2.9 r e p r e s e n t s s c h e m a t i c a l l y the

charge d i s t r i b u t i o n i n a t y p i c a l t u n n e l diode j u n c t i o n , plane

symmetry b e i n g assumed. The r e g i o n a l boundaries A, B, C correspond

t o those marked s i m i l a r l y i n F i g u r e 2J-Q. The net negative charge

d e n s i t y on the p - s i d e and p o s i t i v e charge d e n s i t y on the n-side i s

due r e s p e c t i v e l y t o i o n i z e d donors on the n-side and e l e c t r o n -

occupied acceptors on the p - s i d e of the j u n c t i o n area between A

and C. However f i x e d p o s i t i v e l y charged s i t e s e x i s t on the p - s i d e

due t o a few i o n i z e d donors t h e r e , and a few occupied acceptors on

n-side FIGURE 2 . 1 0

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 s i t e s there. These

l a t t e r "minority" s i t e s exist due to chemical d i f f u s i o n of ri-

doped and p-doped material into each other during f a b r i c a t i o n of

the Junction and would not exist in an " i d e a l " junction. The

density of charges a r i s i n g from th i s non-ideality Is much less

than that of the predominant and oppositely charged s i t e s In each

region, which determines the sign of the > -curve i n Figure 2.9-.

These "minority" s i t e s produce an e l e c t r i c f i e l d smaller than,

but i n opposition to that produced by the majority charged s i t e s

in each region.

Figure 2.10 shows the energy-band diagram f o r the Junction

f o r (ai) a forward bias beyond the v a l l e y region, but small enough

that the fermi l e v e l E- l i e s below,0 and the fermi l e v e l E . fc ' fv

above, the i n f l e c t i o n points of the conduction and valence band

energy curves respectively. The s o l i d - l i n e diagram applies

before an electron tunnels to a donor to neutralize i t , while

the dashed-line diagram applies a f t e r the same t r a n s i t i o n , (b)

i s f o r a larger bias, s u f f i c i e n t that E ^ c l i e s above, and E ^ v

below the I n f l e c t i o n points of the conduction and valence band

energy curves respectively. Acceptors are denoted by c i r c l e s ,

donors by squares.

Case (a); smaller biasest The diagram shows that a l l donors

involved i n the t r a n s i t i o n s considered, l i e on the p-side of the

junction, while a l l acceptors must l i e on the n-side. An electron

tunneling from conduction band to a donor, neutralizes the donor;

t h i s decreases the f i e l d due to the minority-charged s i t e s on the

p-side, hence increases the t o t a l junction f i e l d . (The change i n

f i e l d in the n-material due to the loss of a conduction electron

i s n e g l i g i b l e . ) The dashed-line diagram i n Figure 2J0(a) shows

the change i n f i e l d : the s o l i d and open squares r e p r e s e n t donor

s i t e s b e f o r e and a f t e r the t r a n s i t i o n r e s p e c t i v e l y . The energy

of i o n i z a t i o n o f the donors, t h a t I s , the v e r t i c a l d i s t a n c e from

donors t o edge of conduction band, does not change. The r e s u l t

of the t r a n s i t i o n Is then b o t h t o i n c r e a s e the j u n c t i o n f i e l d , and

t o decrease the t u n n e l i n g gap between conduction band and donors

l y i n g i n the conduction band energy range. Both e f f e c t s enhance

the p r o b a b i l i t y of t u n n e l i n g o f s u c c e s s i v e e l e c t r o n s t o donors.

T h i s may be termed a " p o s i t i v e " modulation s i n c e one e l e c t r o n

encourages the f u t u r e t u n n e l i n g o f many o t h e r s . The n o i s e i s

enhanced f o r t h i s type of t r a n s i t i o n .

However the same t r a n s i t i o n discourages e l e c t r o n s from f a l l

i n g i n t o a c c e p t o r s , b o t h because E^.Q decreases, r e q u i r i n g t h a t

e l i g i b l e a c c e p t o r s be f u r t h e r i n the n-side o f the j u n c t i o n , and

because i n c r e a s e s , so t h a t some acceptors may l i e below E f y ,

a f t e r the donor t r a n s i t i o n . T h i s i s a " n e g a t i v e " type o f modula

t i o n , which tends t o decrease the nois e a s s o c i a t e d w i t h a c c e p t o r -

i n v o l v e d t r a n s i t i o n s .

Next suppose the I n i t i a l e l e c t r o n c o n s i d e r e d f a l l s Into a

n e u t r a l a c c e p t o r Instead of n e u t r a l i z i n g a donor. The d e n s i t y

of n e g a t i v e l y charged s i t e s In the m i n o r i t y on the n-side i s i n

creased so t h a t the j u n c t i o n f i e l d i s decreased. As F i g u r e 2JjO(a)

I m p l i e s , t h i s discourages s u c c e s s i v e e l e c t r o n s from t u n n e l i n g t o

donors, s i n c e the gap has widened, but encourages s u c c e s s i v e

e l e c t r o n s t o t r a n s i t t o the valence band v i a a c c e p t o r s , s i n c e the

l a t t e r need not l i e so f a r In the n-side of the j u n c t i o n t o take

p a r t , and s i n c e the d e n s i t y o f acceptors i n c r e a s e s toward the p-

s i d e . Again there r e s u l t s a " p o s i t i v e " modulation, w i t h enhanced

n o i s e , f o r the a c c e p t o r paths, but a " n e g a t i v e " modulation, w i t h

38

decreased n o i s e , f o r donor p a t h s .

Case ( b ) : l a r g e r b i a s e s : Since E ^ c now l i e s above the I n

f l e c t i o n p o i n t i n the conduction band energy curve, acceptors

i n v o l v e d i n the process under c o n s i d e r a t i o n can l i e on the p - s i d e

of the j u n c t i o n , and s i n c e E f v l i e s below the valence band i n f l e c

t i o n p o i n t , donors l y i n g i n the n-side of the j u n c t i o n can be I n

v o l v e d . E l e c t r o n s w i l l g e n e r a l l y t u n n e l t o n - s i d e donors r a t h e r

than t o p - s i d e donors, s i n c e the former are more numerous and l i e

c l o s e r t o the conduction band edge. S i m i l a r l y , there are more

t u n n e l i n g t r a n s i t i o n s t o the valence band v i a p - s i d e acceptors

r a t h e r than n-side a c c e p t o r s , assuming the process Is r a t e - c o n t r o l l

ed by h o r i z o n t a l and not v e r t i c a l t r a n s i t i o n s . As the b i a s I n

c r e a s e s , the r a t i o of n-side t o p - s i d e donors i n v o l v e d , i n c r e a s e s ,

as does the r a t i o of p - s i d e t o n-side a c c e p t o r s .

C o n s i d e r i n g n-side donors and p - s i d e acceptors t o dominate,

F i g u r e 2J0(b) shows t h a t i ) e l e c t r o n s t u n n e l i n g t o n-side donors -

decrease the f i e l d and widen the t u n n e l i n g gap t o donors f o r

s u c c e s s i v e e l e c t r o n s . Hence the n e u t r a l i z a t i o n of more donors i s

I n h i b i t e d . However the a c c e p t o r - i n v o l v e d processes become more

l i k e l y by the decrease i n f i e l d , i i ) an e l e c t r o n f a l l i n g i n t o a

p - s i d e a c c e p t o r i n c r e a s e s the f i e l d , hence encourages t u n n e l i n g t o

donors by s u c c e s s i v e e l e c t r o n s , but i n h i b i t s s u c c e s s i v e a c c e p t o r -

p r o c e s s e s . Both of these cases produce " n e g a t i v e " modulation a c t i o n

f o r one p r o c e s s , but " p o s i t i v e " modulation a c t i o n f o r the o t h e r , as

i s a l s o t r u e f o r case ( a ) . F o r e i t h e r case (a) o r (b) d e t a i l e d

knowledge of the band p i c t u r e i s r e q u i r e d t o decide whether the

p o s i t i v e o r negative modulation predominates.

Another important e f f e c t h i t h e r t o n e g l e c t e d i s i l l u s t r a t e d

f o r donor-processes In F i g u r e 2J0(a): donors which l i e w i t h i n the

39 occupied conduction band energy l e v e l s before the n e u t r a l i z a t i o n

of one of them, may l i e o u t s i d e t h i s energy range a f t e r the t r a n - -

s i t i o n so that they can no l o n g e r be i n v o l v e d i n the p r o c e s s .

T h i s tends t o compensate the " p o s i t i v e " modulation a s s o c i a t e d w i t h

t h i s type of t r a n s i t i o n i n the f o r e g o i n g . S i m i l a r c o n s i d e r a t i o n s

a p p l y t o bot h donors and acce p t o r - p r o c e s s e s and may compensate

e i t h e r p o s i t i v e or negative modulation f o r e i t h e r type o f t r a p

i n v o l v e d .

Since the processes producing p o s i t i v e o r negative modulation

are opposite f o r case (a) and (b) r e s p e c t i v e l y , a change i n the

noise b e h a v i o r may occur at a b i a s r o u g h l y c o r r e s p o n d i n g t o t h a t

r e p r e s e n t i n g the c r o s s o v e r from case (a) t o oase ( b ) . F o r t y p i c a l

germanium t u n n e l d i o d e s , the donor c o n c e n t r a t i o n i s 1,8 x 10^ cm"-

and the a c c e p t o r c o n c e n t r a t i o n i s 5 x 10^ cm"*3, c o r r e s p o n d i n g t o E f c at 0,06 v o l t s i n s i d e the conduction band edge and E f v at 0,23

v o l t s I n s i d e the valence band edge. Assuming a b u i l t - i n j u n o t i o n

p o t e n t i a l at zero b i a s of 1 v o l t , the b i a s e s f o r which E^, reaches

the valence band i n f l e c t i o n p o i n t and E f c reaches the conduction

band i n f l e c t i o n p o i n t are r e s p e c t i v e l y 0,27 v o l t s and 0*kk v o l t s ,

A s m a l l change i n n o i s e may occur i n a b i a s r e g i o n c e n t r e d about

these v a l u e s , but i t may be d i l u t e d by many o t h e r compensatory

mechanisms not c o n s i d e r e d .

The modulation mechanisms j u s t d i s c u s s e d p r o v i d e an a l t e r n a

t i v e f o r enhanced noi s e t o the mechanisms d e s c r i b e d i n S e o t i o n 2,1+,

The frequency dependence of the spectrum i n b o t h t h e o r i e s i s d e t e r

mined mainly by the capture times of e l e c t r o n s i n the t r a p s , and

hence should be e s s e n t i a l l y the same f o r bot h models f o r the

enhanced n o i s e .

FIGURE 3.1

SCHEMATIC CIRCUIT FOR DIRECT MEASUREMENT OF A NOISE SOURCE

CHAPTER 3

APPARATUS AND EXPERIMENTAL TECHNIQUES

IJ.l B a s i c Concepts and Requirements of Noiae Measurements

3.11 Theory and Requirements f o r "Low-noise" C i r c u i t s

We s h a l l r e s t r i c t d i s c u s s i o n t o c i r c u i t s which can be r e p r e

sented as q - t e r m i n a l networks w i t h c l e a r l y d e f i n e d Input and output

p a i r s . These may be a c t i v e o r p a s s i v e . F o r a c t i v e networks, i t

has been shown (I.R.E. Subcommittee on Noise, I960) t h a t a l l

i n t e r n a l ( d i s t r i b u t e d ) n o i s e sources o f a n o i s y q - t e r r a i n a l network

can be represented u n i q u e l y by not l e s s than a v o l t a g e generator

a c t i n g i n s e r i e s w i t h any source Input v o l t a g e , and a shunt c u r r e n t

generator a c t i n g i n shunt w i t h any input c u r r e n t . F i g u r e 3*1

i l l u s t r a t e s .

A complete s p e c i f i c a t i o n o f these generators i s e q u i v a l e n t t o

a complete d e s c r i p t i o n of the i n t e r n a l sources as f a r as t h e i r

c o n t r i b u t i o n t o output o r t e r m i n a l v o l t a g e s and c u r r e n t s i s con

cerned. F o r nois e measurement purposes, the d e t a i l s f o r s p e c i f y i n g

these generators need not exceed e v a l u a t i n g t h e i r mean square

v a l u e s . They are i n g e n e r a l c o r r e l a t e d s t a t i s t i c a l l y . F o r a f i x e d

frequency, the nol3e f i g u r e f o r such a network i s d e f i n e d as

t o t a l mean square n o i s e across xx p a , =

t h a t p o r t i o n of mean square n o i s e across xx due t o 

= « v 2 > + < v 2t > ) / < v 2 > * 1 + < v 2

t > / < v § >

where < v 2 ^ a r i s e s from b o t h generators <V2 > and ^ i ^ ^ which, w i t h

the terminals xx are shown i n Figure 3.1. <vf > i s s p e c i f i c a l l y

defined as that part of the mean square t o t a l voltage across xx

a r i s i n g from the source generator <Cig $ which i s connected to

the input of the noisy I j-terminal device. The noise figure i s an

o v e r a l l "figure of merit" f o r the s u i t a b i l i t y of the network to

operate on a s i g n a l without excessive d i s t o r t i o n by i t s own noise.

F as 1 represents a "perfect" network.

For a 4-terminal network consisting of a vacuum tube triode

input, i t can be shown (I.R.E. Subcommittee on Noise, I960) that

the c o r r e l a t i o n susceptance between < v £ > and I s n e g l i g i

ble at frequencies whose periods are much less than the t r a n s i t

time i n the tube, and that both c o r r e l a t i o n and non-correlation

conductance (that i s , t o t a l input conductance) i s n e g l i g i b l e when

there i s l i t t l e grid loading. The current generator ^ l 2 ^ * s

then unimportant. Since these are the conditions under which our

c i r c u i t s operate i n the present work, we concentrate hereafter on

specifying the generator ^ vn ^ 0 5 ^kT QR whioh describes nearly

a l l the noise. R^ i s termed the noise resistance of the network,

and by standard agreement i s ascribed a temperature T Q of 290°K.

In general other resistances i n the c i r c u i t operate at about T q or

may exceed i t by a few degrees. The comparison of Rjj-noise with

source noise i s then accurate to a few parts i n 290, which i s

inconsequential i n those cases, as here, where the R^noise i s p ,

n e g l i g i b l e . Physically, the generator <Cv> represents the shot-

e f f e c t i n the tubes of the network.

The concept of noise figure can be extended to [[-terminal

networks connected one a f t e r the other. Let the available power

gain of the 1 t h network be G^, while i t contributes noise repre-p sented at i t s Input terminals by a voltage <v . . • The o v e r a l l

1*2 1 2

n o i s e f i g u r e f o r the n s t a g e s , i n terms of a source ( w i t h

admittance Y_) connected t o the f i r s t network i n p u t , which produces

a mean square noi s e v o l t a g e < . v 2 > at the i n p u t , i s e a s i l y shown t o be

P . x +<4>+ + 4^<3$ < v s > G l < v s > G l G 2 < V s >

» P 1 + (P 2 - 1 ) / 0 X 4 (P 3 - 1 ) / G^g + F o r G^>> 1 f o r a l l 1 , the f i r s t network i s seen l a r g e l y t o d e t e r

mine the o v e r a l l n o i s e f i g u r e , each f o l l o w i n g network c o n t r i b u t i n g

i n c r e a s i n g l y l e s s t o the degradation of P^.

The p r i n c i p l e of c a r e f u l l y s e l e c t i n g the input network t o give

a s m a l l value f o r F , w i t h much l e s s s t r i n g e n t c o n d i t i o n s a p p l y i n g n

t o s u c c e s s i v e networks, w i l l be demonstrated by the p a r t i c u l a r

sequence of networks chosen to p r o v i d e a low value of F n f o r the

o v e r a l l c i r c u i t used t o measure the t u n n e l diode n o i s e s i g n a l .

3 . 1 2 Methods of Comparison With a Standard Noise Source

There are i n p r i n c i p l e two ways t o measure the magnitude of

an unknown nois e source represented as a c u r r e n t generator < i g ^ .

Besides methods of comparison w i t h a standard source, d e s c r i b e d

below, the magnitude of ^ i g ^ c a n °Q found " d i r e c t l y " . To

i l l u s t r a t e simply, the shunt noi s e c u r r e n t generator ^ i 2 ^ a s s o c

i a t e d w i t h the Input conductance, and the i n p u t conductance, are

assumed n e g l i g i b l e . The o v e r a l l bandwidth i s assumed l i m i t e d by

the c i r c u i t f o l l o w i n g the source ( a m p l i f i e r ) and c o n s t a n t . £ ^

r e p r e s e n t s the RMS output n o i s e power of source and a m p l i f i e r f o r

the i r e a d i n g . The a m p l i f i e r i n p u t i s f i r s t s h o r t e d , g i v i n g

FIGURE 3 . 2

SIMPLIFIED SCHEMATIC CIRCUIT FOR DIRECT MEASUREMENT OF A NOISE SOURCE

^ 1 = **2<!v2'>, where G i a the power gain of the a m p l i f i e r .

Next, the unknown source i a connected, g i v i n g £ | = G 2(<v 2>+<ia>Rg)

where R g i a the source r e s i s t a n c e . Prom these r e a d i n g s ,

In t h i s method, i n which the e q u i v a l e n t c i r c u i t i s shown i n F i g u r e

3.2, i t has been assumed t h a t 2 2 < v g > and < v n > are u n c o r r e l a t e d

the "law" or response of the a m p l i f i e r - r e c t i f i e r system as a f u n c t i o n o f time and of in p u t i s l i n e a r and independent of the magnitude of in p u t v o l t a g e ; t h a t i s ,

df = G 2 < v 2 > f o r a l l i n p u t s < v 2 >

The method can e a s i l y be extended t o an "n-laiw" a m p l i f i e r , t h a t i s ,

one f o r which &^ = (G < v ^ > ) ; but n must be measured,

i n v o l v i n g Inaccuracy and the f a c t t h a t few dev i c e s have a s i n g l e

value of n f o r a l l u s e f u l Input v o l t a g e ranges. Inaccuracy a l s o

a r i s e s In measuring the g a i n , G, of the a m p l i f i e r , and p a r t i c u l a r l y

the requirement t h a t G be oonstant i s not e a s i l y met i n p r a c t i c e .

The method cannot be made independent o f the a m p l i f i e r law by use

of an a t t e n u a t o r (which would i n s u r e a constant input v o l t a g e )

because the schematic generator <^v^> i s not a c c e s s i b l e t o a t t e n u

a t i o n . The method would f u r t h e r be complicated i f the a m p l i f i e r

n o i s e c u r r e n t generator ^ i 2 ^ were s i g n i f i c a n t , s i n c e the Input

v o l t a g e i t produces depends on the input shunt impedance, which i n

ge n e r a l v a r i e s s i n c e the source impedance may v a r y . S t a n d a r d i z a

t i o n of impedances would then be r e q u i r e d , as w e l l as accurate

s p e c i f i c a t i o n of .

Most of these problems are so l v e d by comparing the unknown

source <Cig > w i t h a standard n o i s e source. The l a t t e r i s u s u a l l y

FIGURE 3.3

SCHEMATIC CIRCUIT FOR COMPARISON OF UNKNOWN AND CALIBRATED NOISE SOURCES

a vacuum nois e diode operated i n the t e m p e r a t u r e - l i m i t e d c o n d i t i o n ,

or a standard r e s i s t a n c e producing thermal n o i s e . F i g u r e 3»3

i l l u s t r a t e s the b a s i c comparison teohnique f o r a n o i s e diode source

represented by ^ j ^ ^ • T h i s type of standard source i s p a r t i c u

l a r l y convenient beoause of i t s v e r y h i g h impedance i n the temper

a t u r e - s a t u r a t e d c o n d i t i o n . I t can t h e r e f o r e i n j e c t a c o n t i n u o u s l y

v a r i a b l e n o i s e c u r r e n t I n t o the c i r c u i t without changing the . 2

impedance c o n d i t i o n s , so t h a t the vo l t a g e s i g n a l due t o C i _ > w i l l i 3

be unchanged d u r i n g the comparison. (Any s i g n i f i c a n t shunt imped

ances a s s o c i a t e d w i t h the noise diode or oth e r c i r c u i t components

are c o n s t a n t , c o n t r i b u t e a constant thermal n o i s e , and may be

measured d i r e c t l y by br i d g e and s u b t r a c t e d . )

Some methods of comparison o f unknown and c a l i b r a t e d n o i s e

sources are as f o l l o w s .

1) F i g u r e 3«3 a p p l i e s . The a m p l i f i e r law i s assumed l i n e a r ,

i n which case the " g a i n " , G, has the u s u a l meaning. The a m p l i f i e r

input i s s h o r t e d , g i v i n g 62 = G 2 < v 2 > . The unknown source Is

then connected, g i v i n g £ 2 = G 2( < v 2 > + R 2 ) . F i n a l l y the x n S 3

n o i s e diode i s connected and i t s n o i s e s i g n a l i n c r e a s e d t o any

convenient l e v e l such t h a t 0^>rJ^O^t where 0 2 = G 2 ( < v 2 > + R 2 + R 2 ) . S o l v i n g these r e l a -d n s s WD s

t i o n s f o r gives

<i2> = <4><*2 - * 2 > / 0 i > Independent of the g a i n . In case the a m p l i f i e r law i s not l i n e a r ,

i t can be measured a c c u r a t e l y w i t h a v a r i a b l e s i g n a l generator at

the Input and a d e t e c t o r o f known and f i x e d law (e.g., an RMS

VTVM or a square-law thermocouple). I f the response obeys an n

law, then & * = k'(<vf » n / 2 where k ' i s a p r o p o r t i o n a l i t y constant

but Is not r e f e r r e d t o as the " g a i n " . The unknown noise source

i s then given by

 = < i | D x ^ f / n - e 2 / n ) / ( e 2 / n - e p )

which i s Independent of the magnitude of the a m p l i f i e r response,

i f the n-law i s independent of the magnitude of the input n o i s e .

2) Besides e l i m i n a t i n g the dependence of n o i s e measurement

on the a m p l i f i e r g a i n , the comparison method can be m o d i f i e d by

use of an a t t e n u a t o r t o avoid dependence on the "law" of the

a m p l i f i e r . The method i s p a r t i c u l a r l y simple i f we can assume p

the a m p l i f i e r n o i s e ( s p e c i f i e d mainly by < v n > a t lower f r e q u e n c i e s )

i s n e g l i g i b l e r e l a t i v e t o the source n o i s e v o l t a g e . (Very o f t e n ,

e i t h e r by c a r e f u l design of the a m p l i f i e r , o r by s u i t a b l e impedance

t r a n s f o r m a t i o n s f o r the source, t h i s c o n d i t i o n can be met.) The

method i s then

a) connect the unknown source and s e t the a t t e n u a t o r at a v o l t a g e

r a t i o ( i n p u t / o u t p u t ) of A- ( <1). The response i s & = k ' [ A f « i f > z 2 ) ] ° / 2

b) s w i t c h the a t t e n u a t o r t o the r a t i o A 2 (<A^) and t u r n up the

noise diode c u r r e n t u n t i l the same response © i s o b t a i n e d .

Then , .

e = k ' [ A 2 « i 2D > + ) z 2 ] " / 2

The two r e l a t i o n s give f o r the unknown source

The r e s u l t i s independent of the "law" of the a m p l i f i e r , the

constant k^ and the shunt Impedance Z p r o v i d i n g t h a t i t i s the

same f o r bot h s e t t i n g s of the a t t e n u a t o r . The a t t e n u a t i o n r a t i o s

FIGURE 3 .4

SCHEMATIC NOISE CIRCUIT FOR ATTENUATOR AND TWO STANDARD NOISE SOURCES

and A 2 are e a s i l y c a l i b r a t e d w i t h e i t h e r a s i g n a l generator or

noise diode source. The bandwidth must not be l i m i t e d by the

a t t e n u a t o r , but by the a m p l i f i e r .

3) I f the a m p l i f i e r noise cannot be n e g l e c t e d compared w i t h

the source n o i s e s i g n a l , i t can be c a l i b r a t e d and s u b t r a c t e d by

use of an a d d i t i o n a l standard n o i s e source. F i g u r e 3*4 now a p p l i e s .

The a t t e n u a t o r i s s t i l l connected between sources and a m p l i f i e r

i n p u t , as In the p r e c e d i n g case. Let < v & > be the i n p u t v o l t a g e

t o the n o i s e l e s s a m p l i f i e r due t o the e f f e c t i v e a m p l i f i e r n o i s e

generators <(v 2 )> and . The Impedance l o o k i n g t o the l e f t

I nto the a t t e n u a t o r output i s assumed independent of the a t t e n u a t o r

s e t t i n g s , so t h a t < v f t > Is c o n s t a n t . On the source s i d e o f the

a t t e n u a t o r , as b e f o r e , reactances are assumed tuned out, and a l l

constant shunt r e s i s t a n c e s not Included by R g or R are b r i d g e -

measured and t h e i r n o i s e s u b t r a c t e d from t h a t of the s o u r c e s . The

present method Involves s w i t c h i n g between the unknown source ^ i g >

(shunted by i t s dynamic r e s i s t a n c e R g) and a c a l i b r a t e d r e s i s t a n c e

R, which i s made equal t o R g, so t h a t the o v e r a l l shunt impedance

i s always the same. Since the n o i s e a s s o c i a t e d w i t h R i s known by

Nyquist's theorem, R i s used t o c a l i b r a t e the a m p l i f i e r noise as

f o l l o w s .

a) w i t h R switched i n , and the a t t e n u a t o r at a r a t i o A- , the r e s

ponse i s

0 - k'[<v 2> + A 2 ( Z 2 ) ] n / 2

p

b) the bandwidth b e i n g c o n s t a n t , the unknown source i s now

switched i n and the a t t e n u a t o r s e t at a r a t i o Ag. The n o i s e

diode Is then turned up u n t i l the response i s the same as i n

the f i r s t r e a d i n g . Then

a = k'[<vf> + A | (<±f>• <i|D>") z 2 J n / 2

S o l v i n g these r e l a t i o n s g i v e s

 = <4><4/4) - <4> / 2 2 which i s independent of k, n, and Z. I t i s noted t h a t <^ip>

3 It

has been assumed. I f the converse i s t r u e , the a t t e n u a t o r r a t i o A^

i s used w i t h switched i n , and the r a t i o Ag when > Is

switched i n . The use of two standard n o i s e sources f o r t h i s case

has p e r m i t t e d the noise generator t o be accounted f o r .

F u r t h e r , the values of A. and are a r b i t r a r y , a l t h o u g h they must

be known, so t h a t f i x e d v a l u e s f o r these r a t i o s may be chosen

independent of the source l e v e l s . As i s seen i n the next method,

the use of two standard sources along w i t h an a t t e n u a t o r over-

s p e c i f i e s the problem; the use f o r a standard r e s i s t a n c e n o i s e

source, R, i n c o n j u n c t i o n w i t h an a t t e n u a t o r , i s f o r c e r t a i n cases

where a no i s e diode may not be used as a standard noise source

(e.g., at f r e q u e n c i e s under ^ 100 cps, f l i c k e r o r l / f n o i s e may

o v e r r i d e the shot n o i s e i n the diode.) In such c a s e s , the values

of A^ and Ag are not a r b i t r a r y , but must be ad j u s t e d so t h a t the

a m p l i f i e r response i s independent o f whether R or i s . s

switched i n .

k) When the noise diode i s an acc e p t a b l e standard source,

i t s use along w i t h a standard r e s i s t a n c e n o i s e source R, obv i a t e s P

the need f o r an a t t e n u a t o r . The sources , R, and the no i s e s *

diode are now connected d i r e c t l y t o the a m p l i f i e r i n p u t . Assume

 < . The r e a d i n g & i s f i r s t obtained w i t h the c a l i -

b r a ted r e s i s t a n c e R switched i n :

e - k'[<vf > + z 2]«/2

1*8

N e x t , the unknown source ^ i 2 > , w i t h a s s o c i a t e d r e s i s t a n c e R s

e q u a l t o R, Is sw i tched i n and the n o i s e d iode tu rned up u n t i l

the a m p l i f i e r response & i s r e g a i n e d :

e> » k / [ < v f > + « i 2 > + ) z 2 ] n / 2

S o l v i n g ,

 = - 

A n+n a p p l i e s i n t h i s r e s u l t i f > .

Method 3 ) i s not a p p r e c i a b l y more i n a c c u r a t e than method 4 ) ,

s i n c e the a t t e n u a t o r r a t i o s can be a c c u r a t e l y c a l i b r a t e d . However

use o f an a t t e n u a t o r a r ranged as i n F i g u r e 3 » 3 degrades the n o i s e

f i g u r e , s i n c e i t a t t e n u a t e s s o u r c e s but not a m p l i f i e r n o i s e . ( I t

i s p o s s i b l e , however , t o p l a c e s u c h a t t e n u a t i o n between a m p l i f i e r

n o i s e g e n e r a t o r s and a m p l i f i e r , e f f e c t i v e l y , so as t o a t t e n u a t e

a m p l i f i e r n o i s e a l s o . T h i s i s done by u s i n g a p r e a m p l i f i e r w i t h

g a i n s u f f i c i e n t t h a t i t s n o i s e c o m p l e t e l y o v e r r i d e s the n o i s e o f

a n o t h e r a m p l i f i e r o r r e c e i v e r wh ich f o l l o w s i t . The a t t e n u a t o r i s

between p r e a m p l i f i e r and r e c e i v e r so t h a t i t a t t e n u a t e s p r e a m p l i f i e r

n o i s e , w h i l e the r e c e i v e r n o i s e i s n e g l e c t e d a l t o g e t h e r . )

Method q) i s advantageous m a i n l y i n t h a t the e x p e r i m e n t a l

arrangement measures d i r e c t l y the d i f f e r e n c e ^ i ^ ^ - ^ * J J D ^ •

I f « < ! 2 > , then b o t h and are l a r g e q u a n t i

t i e s , e a c h measurable t o an a c c u r a c y l i m i t e d by r e c o r d i n g a f l u c

t u a t i n g q u a n t i t y 6 . Method 3) measures and < I2^ > s e p a r

a t e l y so t h a t the u n c e r t a i n t y In t h e i r d i f f e r e n c e , o b t a i n e d a l j e -

b r a i c a l l y , f a r exceeds t h a t o f method 4 ) .

A m o d i f i e d v e r s i o n o f method 4) w i l l be used t o measure t u n n e l

d iode n o i s e i n t h i s t h e s i s .

noisy transforming network

n n @ AAAA/SA-

<t>(b G in

FIGURE 3.5

SCHEMATIC CIRCUIT FOR A TRANSFORMED SOURCE COUPLED INTO A NOISY AMPLIFIER

1*9 3 . 2 Impedance Transformations S u i t a b l e f o r a Tunnel Diode Source

The f o r e g o i n g s e c t i o n i n d i c a t e s t h a t i t i s unneccessary f o r

,i the v o l t a g e at the a m p l i f i e r i n p u t due to the n o i s e sources to

o v e r r i d e t h a t due t o the e q u i v a l e n t noise generators of the a m p l i

f i e r . However f o r some sources, such as the t u n n e l diode b i a s e d 2 2 2

anywhere i n the reverse or near-forward r e g i o n s , < v g > = ^ * s ^ R s

would be o v e r r i d d e n by the noise of even the v e r y low-noise a m p l i

f i e r s t o such an extent that measurement of the t u n n e l diode noise

would be i m p o s s i b l e i f the diode were connected d i r e c t l y t o the

a m p l i f i e r i n p u t . That i s , the response $ would always be the P 2

same In method q) whether <C 15: > or were switched i n , s i n c e ti 3

i t would be determined e n t i r e l y ( w i t h i n experimental accuracy of

r e c o r d i n g a f l u c t u a t i n g response) by the a m p l i f i e r n o i s e i t s e l f .

The e x c e s s i v e l y low value of < v g > f o r the t u n n e l diode, and

the wide range of diode Impedances encountered over the b i a s range

of I n t e r e s t are the two main d i f f i c u l t i e s In measuring t u n n e l diode

n o i s e i n the near-forward and r e v e r s e b i a s r e g i o n s . A network i s

r e q u i r e d t o t r a n s f o r m the t u n n e l diode source impedance so t h a t I t

a) a m p l i f i e s the s m a l l t u n n e l diode n o i s e t o a l e v e l which over

r i d e s the n o i s e of any f o l l o w i n g network, and b) accomplishes t h i s

without adding a p p r e c i a b l e noise i t s e l f ( t h a t I s , thermal n o i s e of

r e s i s t a n c e s i n the network any o f which may o v e r r i d e the t u n n e l

diode source n o i s e ; the network w i l l exclude a c t i v e elements which

are too n o i s y . ) These p r o p e r t i e s are summarized by r e q u i r i n g a

s a t i s f a c t o r y n o i s e f i g u r e i n terms of the source and a m p l i f i e r n o i s e

v o l t a g e s appearing across t e r m i n a l s xx In F i g u r e 3 . 5 . The gain of

the p r e a m p l i f i e r f o l l o w i n g xx i s s u f f i c i e n t l y h i g h that the n o i s e

of c i r c u i t s f o l l o w i n g the p r e a m p l i f i e r may be n e g l e c t e d . When R r e p r e s e n t s the t o t a l t u n n e l diode r e s i s t a n c e R, at

F I G U R E 3.6

N O I S E - E Q U I V A L E N T C I R C U I T S FOR A PARALLEL-TUNED C I R C U I T

biases-where R d 5 0 0 ohms, a s u i t a b l e " t r a n s f o r m i n g " network i s

simply a p a r a l l e l - t u n e d c i r c u i t a cross the d i o d e . The n o i s e -

r e p r e s e n t a t i o n f o r t h i s network i s i n F i g u r e 3 . 6 . = < i^. >

+ i s the t o t a l n o i s e generated by the 2 - t e r m i n a l t u n n e l

d i o d e , r i s the o o i l r e s i s t a n c e w i t h mean square n o i s e v o l t a g e o f

IjkTr p e r u n i t bandwidth. I t can be shown t h a t F i g u r e 3 . 6 ( a ) i s

e q u i v a l e n t t o F i g u r e 3 . 6(b) where R Ci Q 2 r ( f o r Q>> 1) has thermal

n o i s e c u r r e n t generator l*kT/R a s s o c i a t e d w i t h I t . R^ r e p r e s e n t s

the noise of the p r e a m p l i f i e r which f o l l o w s (assuming the e q u i

v a l e n t shunt noise c u r r e n t generator r e p r e s e n t i n g g r i d - c i r c u i t

l o a d i n g i s n e g l i g i b l e at f r e q u e n c i e s of i n t e r e s t ) . The nois e

f i g u r e of the c i r c u i t i n F i g u r e 3 . 6(b) i s then, w i t h Z the t o t a l

shunt impedance,

F = 1 + ( Z 2 + l4kT QR n) / 2 2

where T Q = 290°K. i s the standard temperature assigned t o Rp. To

estimate the magnitude of F, = 2 e ( J l c v | • l! v c l) i s

assumed t o be a shot n o i s e generator f o r the t u n n e l d i o d e . Then

l |kT 1 +

r 1

2 e l eq L

Q r d 1 +

The approximate form holds f o r R^ s m a l l compared t o Q r 1+0 kohms

t y p i c a l l y ) . The temperature of R R has been taken as t h a t of the

o v e r a l l c i r c u i t . At T = 290°K, i|kT/2e ^ 1/20 v o l t . A good low-

no i s e a m p l i f i e r has R n £i 300 ohms or l e s s . S ince 2 e ( | l c v | + l l v c | )

= UkT/R^lvno a n d a i n c e R ^ eval u a t e d at zero b i a s i s t y p i c a l l y 15

ohms f o r germanium t u n n e l diodes, then | l o v | + l^vc^ ~ ^ T n B S e

values l e a d t o the e x c e l l e n t n o i s e f i g u r e o f F £2. 1 + . 0 0 5 . However

i t i s seen t h a t F i n c r e a s e s r a p i d l y 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

F o r s m a l l v a l u e s o f R^, s u c h as are encountered i n the

r e v e r s e and n e a r - f o r w a r d b i a s r e g i o n s , a t r a n s f o r m i n g network

must be used t o s t e p up the t u n n e l d iode n o i s e source v o l t a g e

< V f / = <Ti_>Rf wh ich i s e x c e s s i v e l y s m a l l due t o R o r R, b e i n g a a a s d

as s m a l l as 4 ohms i n the f a r - r e v e r s e b i a s r e g i o n . In terms o f a

good n o i s e f i g u r e , two p o s s i b i l i t i e s f o r c o u p l i n g the t u n n e l d iode

source i n t o a high- Impedance a m p l i f i e r a r i s e :

a) a u t o t r a n s f o r m e r : As shown In F i g . 3*7 the output i s tuned

t o Improve the n o i s e f i g u r e . By a d j u s t i n g ' t h e t a p p i n g r a t i o , the

a u t o t r a n s f o r m e r can c o u p l e the t u n n e l d iode w i t h the h igh- impedance

a m p l i f i e r c o n t i n u o u s l y f rom v e r y s m a l l v a l u e s o f R^, t o v e r y l a r g e

v a l u e s where the a u t o t r a n s f o r m e r becomes a p a r a l l e l tuned c i r c u i t .

I f l o s s e s ( c h i e f l y due t o c o i l r e s i s t a n c e ) are i n c l u d e d i t i s

d i f f i c u l t t o a n a l y s e an a u t o t r a n s f o r m e r i n terms o f n o i s e f i g u r e ,

o r even o f v o l t a g e g a i n . E x p e r i m e n t a l l y I t may a l s o prove u n s a t i s

f a c t o r y at f r e q u e n c i e s h i g h e r than a few Mc/s due t o the f a c t t h a t

an i d e a l i z e d a n a l y s i s ( l o s s e s n e g l e c t e d ) I n d i c a t e s the v o l t a g e

s t e p - u p t o be p r o p o r t i o n a l t o the t o t a l i n d u c t a n c e o f the c o i l and

t o i t s c o e f f i c i e n t o f c o u p l i n g , r a t h e r than s i m p l y t o the t u r n s

r a t i o . S e l f - r e s o n a n c e e f f e c t s p l a c e an upper l i m i t on the a c h i e v

a b l e v a l u e o f the t o t a l mutual Inductance a t h i g h e r f r e q u e n c i e s ;

the r e s u l t i n g l i m i t e d v o l t a g e s t e p - u p may r e p r e s e n t an I n f e r i o r

n o i s e f i g u r e .

b) s e r i e s - t u n e d c i r c u i t : T h i s c i r c u i t , .shown In F i g u r e 3.8,

has v o l t a g e s t e p - u p dependent o n l y on the Q, and not on the I n d u c

t a n c e , o f the c o i l . A s a t i s f a c t o r y n o i s e f i g u r e i s e a s i l y o b t a i n e d

d e s p i t e the f a c t t h a t a "match" o f t u n n e l d iode and a m p l i f i e r Input

impedances i s o b t a i n e d o n l y f o r r = R^ and l / ( A » C ) 2 ( R d + r ) e q u a l

I

p

1 .5 4

parallel-tuned circuit

series-tuned circuit

R • 0 15 ohm 1 / 6 O 0

F I G U R E 3.9

COMPARISON OF N O I S E F I G U R E S FOR S E R I E S - AND P A R A L L E L -TUNED C I R C U I T S W I T H TUNNEL D I O D E SOURCE

t o the a m p l i f i e r Input Impedance. The n o i s e f i g u r e f o r the c i r c u i t

In terms of source and unwanted n o i s e v o l t a g e s appearing across the

i n p u t t e r m i n a l s xx of a n o i s e l e s s a m p l i f i e r i s

IjkT P = 1 +

<12> % + R ( 0 > C ) 2 ( l + — \

2'

where Ci d>= 2 e ( ) l c v | + | l v c | ) , and has the same temperature as

the o v e r a l l c i r c u i t . The e f f e c t i v e Q. d e t e r m i n i n g the step-up of

source and c o i l noise i s l / ( R d + r)(£OC), l e s s than the c o i l Q due

t o the damping of R,. a

F o r the s e r i e s - t u n e d c i r c u i t , F d e t e r i o r a t e s as R d decreases,

as f o r the p a r a l l e l - t u n e d c i r c u i t . However a good nois e f i g u r e i s

obtained as l o n g as R^ does not become s m a l l compared w i t h r , which

may be made v e r y s m a l l . Assuming the use of a c o i l w i t h Q ^- 1 0 0 ,

so t h a t ( 6 0 C ) 2 ( R d + r ) 2 i s s m a l l , the e f f e c t s of the n o i s y a m p l i f i e r

may be v e r y l a r g e l y suppressed w i t h the s e r i e s - t u n e d c i r c u i t , due t o

i t s v o l t a g e step-up; t h i s i s i m p o s s i b l e when the p a r a l l e l - t u n e d

c i r c u i t i s used w i t h s m a l l R,. The s m a l l e r i s R,, the b e t t e r the a d

s e r i e s - t u n e d c i r c u i t suppresses a m p l i f i e r n o i s e , so t h a t the n o i s e

f i g u r e i s l i m i t e d c h i e f l y by i n d u c t o r n o i s e competing w i t h the

t u n n e l diode source n o i s e .

I t i s seen that R^ damps the s e r i e s - t u n e d c i r c u i t . When R^

becomes ^ l/(£OC), the damping becomes e x c e s s i v e such t h a t the

p a r a l l e l - t u n e d c i r c u i t assumes a b e t t e r n o i s e f i g u r e f o r the same

value of R^ than does the s e r i e s - t u n e d c i r c u i t . A comparison of

n o i s e f i g u r e s as a f u n c t i o n of R d f o r the s e r i e s - and p a r a l l e l -

tuned c i r c u i t s i s shown i n F i g u r e 3»9» < C i d> has been assumed

constant (Chapter I4 shows t h i s t o be a f a i r a pproximation), and a

t y p i c a l c o i l w i t h Q « 100 and r = 1+ ohms i s assumed.

53

(In a s s e s s i n g networks s u i t a b l e f o r t u n n e l diode impedance

t r a n s f o r m a t i o n , i t i s to be noted t h a t the c r i t e r i o n f o r maximum

vo l t a g e across the output t e r m i n a l s of the a r b i t r a r y network i s

not n e c e s s a r i l y e q u i v a l e n t t o the transformed impedance at the

output t e r m i n a l s o f the network b e i n g matched t o the input imped

ance of the c i r c u i t f o l l o w i n g . S i m i l a r l y , the c r i t e r i o n f o r o p t i

mum power t r a n s f e r from source t o output of the t r a n s f o r m i n g n e t

work may be s p e c i f i e d only f o r a given network, s i n c e both the

output v o l t a g e and the transformed impedance at the network output

w i l l be f u n c t i o n s of some impedance a s s o c i a t e d w i t h the network

i t s e l f , e.g., the s e r i e s r e s i s t a n c e of the c o i l In tuned c i r c u i t s . )

3 . 3 Development of a Low-noise A m p l i f i e r

The n o i s e f i g u r e s of the f o r e g o i n g s o u r c e - t r a n s f o r m i n g n e t

works are s a t i s f a c t o r y f o r s m a l l values o f a m p l i f i e r n o i s e r e s i s

tance RJJ, which at f r e q u e n c i e s not over 30 Mc/s s p e c i f i e s the

noise due to vacuum tube c i r c u i t s f o l l o w i n g the transformed t u n n e l

diode source. T h i s n o i s e i s u s u a l l y due almost e n t i r e l y t o the

f i r s t tube, i f i t s a s s o c i a t e d stage has gain much g r e a t e r than

u n i t y . F o r pentodes or t e t r o d e s , which s u f f e r p a r t i t i o n n o i s e , R N

i s t y p i c a l l y 2 .5 kohms o r more, a value which i s s e r i o u s l y d e t r i

mental t o the noise f i g u r e f o r the c i r c u i t s d i s c u s s e d . T r i o d e s ,

w i t h R N t y p i c a l l y 500 ohms or l e s s , give a s a t i s f a c t o r y n o i s e

f i g u r e . At f r e q u e n c i e s h i g h enough f o r the present work, a s i n g l e

t r i o d e Input stage s u f f e r s e x c e s s i v e input admittance i n the

a m p l i f y i n g grounded-cathode c o n f i g u r a t i o n due t o the M i l l e r e f f e c t

which depends on the l a r g e grid-anode I n t e r e l e c t r o d e c a p a c i t a n c e

of t r i o d e s . The in p u t admittance Is l a r g e l y c a p a c i t i v e , but t h i s

s e r i o u s l y impairs the Q of the s e r i e s - t u n e d c i r c u i t c o u p l i n g the

A.C.-EQUIVALENT CIRCUITS OF A CASCODE AMPLIFIER

source Into t h i s stage, and the noise figure s u f f e r s .

The cascode or Wallman c i r c u i t (Wallman, e t . a l . , 191+8)

overcomes the M i l l e r e f f e c t , which i s proportional to the gain of

the triode stage involved, by using a grounded-cathode triode

Input stage followed by a grounded-grid stage; the l a t t e r acts as

a low Impedance plate load f o r the grounded-cathode stage whose

gain Is then low (usually about 1 ) . The M i l l e r e f f e c t i s v i r t u a l l y

inoperative under this condition so that a low input admittance i s

obtainable.' The o v e r a l l gain of the two stages can be comparable

to that of a single pentode stage, while the o v e r a l l noise can be

very l i t t l e above that of a single triode stage. The conditions

under which these properties can be realized are now given in d e t a i l .

3.31 Amplification and Noise of a Cascode Amplifier

The essentials of the cascode c i r c u i t are as i n Figure 3 . 1 0 .

(a) i s the a.c. c i r c u i t and (b) i s 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, e l e c t r o n i c a l l y , to the cascode are calculated, g^ 2» r p l , 2 » 5 1 1 3 ( 3 ^1 2 a r 9 r e s P e o t i V Q i y t n e transconductances, plate resistances, and amplification factors of the l 9 t and 2 n d tubes.

To calculate the o v e r a l l transconductance of the c i r c u i t ,

defined as the a.c. current I flowing i n the plate c i r c u i t 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 + g x e ) r p l = 0

with v n = (g..e - i ) r gives

g l ( l + A i 2 ) r n i g x i/e = .-± 5 _ E i ^ g, r p 2 + ( l + / u 2 ) r p l

where the l a t t e r approximation a p p l i e s when - >^ > 1 a r*d r p ^ £± r p 2

(e.g. s i m i l a r h i g h - g a i n tubes u s e d ) .

The e q u i v a l e n t p l a t e r e s i s t a n c e of the cascode, d e f i n e d as the

change i n v o l t a g e at the output p l a t e p e r u n i t change of tube

c u r r e n t i s found by o p e n - c i r c u i t i n g t e r m i n a l s AB. I f V i s the

v o l t a g e across AB, then

V = g 2 V l rP 2 + g l e r p l = g l e r p l ( 1

where v.. = g, e r _ when 1 = 0 . The e f f e c t i v e p l a t e r e s i s t a n c e J- p i

i s then

r p = - V/ A = rp 2

+ U + * 2 , P P l

The o v e r a l l gain f a c t o r i s simply

= v/e = ^ x ( l + ju 2)

Hence f o r the o v e r a l l c i r c u i t , p. = g r • The h i g h value o f r

causes the c i r c u i t t o behave l i k e a s i n g l e pentode s t a g e ; w i t h an

a r b i t r a r y load across t e r m i n a l s AB i t i s e a s i l y shown that the

c i r c u i t g a i n , A, i s given by

pAl + p0) Z pZL A = 1 k = h— gz

( 1 + ^ 2 ^ p l + rp2 + Z L rP + Z L

f o r r >> Z , as i s u s u a l f o r pentodes. P L

The input impedance of the grounded g r i d stage i s

( r p 2 + Z^) / ( u 2 + 1) which i s r e q u i r e d t o be s m a l l , so t h a t the

FIGURE 3 « 1 1

NOISE-EQUIVALENT CIRCUITS OF A CASCODE AMPLIFIER

gain of the f i r s t stage, and hence the Miller effect, w i l l be small. Thus for overall large gain, as well as for small input admittance, i t is required that p.^ and p.^ be large and that r ^ 2 be small, that i s , 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 l e f t at points xx is which typically is much larger than l/g2« This combination of low resistance to the right, and high resistance to the l e f t , is the crucial property of the cascode c i r c u i t , with respect both to s t a b i l i t y and noise figure. The gain of the f i r s t stage is approximately g- /gg under these conditions, and this ratio is near unity. Such low gain makes the f i r s t stage stable, that i s , 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 g 2 which is nevertheless chosen as large as possible to obtain as small gain as possible for the f i r s t stage.

The c r i t e r i a for low noise associated with the cascode circuit are now considered. Neglecting noise of circuit resistances compared to tube noise, the noise-equivalent circuit for the cascode is shown in Figure 3»H» The generators i ^ and i 2 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,

5 7

i = ( 1 + ^ 2 ) r p l h + r p 2 * 2

r p 2 + U ' ^ p l

The denominator of thi s r e s u l t i s r , the ov e r a l l plate resistance P

of the cascode. The noise due to the f i r s t tube already i s seen

to dominate. In order to represent the cascode noise as an equi

valent thermal noise resistance R n i n series with the grid c i r c u i t

of the f i r s t tube, one forms

 : 11 *^)\t<il> * rD 2 

[rp2 + ( 1 + 2> rpl] where the noise generators i ^ and i g are assumed uncorrelated.

Now i t i s well-known that the output noise i n the plate c i r c u i t

of a single tube can be represented by the amplified thermal noise

of a resistance R n appearing at the grid of a noiseless tube, by

the r e l a t i o n

 * , 2 „ = J U n 2 „ UkT R . . A f 1 , 2 ' p i , 2 " 1 , 2 H o n l , 2

where subscripts 1 , 2 , r e f e r to f i r s t and second tubes respectively.

Thus

 . (1 ^ 2 > 2 g f r p l ^ o ^ ^ l + 4*vt ^oAtRr,z [ r p 2 + d + / . 2 ) r p l ] ' :

The equivalent noise resistance R n f o r the complete cascode c i r

c u i t w i l l be defined as

R n = / ^ 0 ^ f g

where g = p. / r i s the o v e r a l l transconductance of the cascode.

Then

58

R. ( 1 +/*2 ) 2 g l r p l ^ o ^ f R n l + *2 * p | ^ o A f Rn2

n l 4 k T 0 A f [ g l ( l + ^ 2 ) r ] 2

L _ + | ^ - l 2 R ^ ~ R , + R n / u ? n l L (i+/i2)J n 2 n l n 2 ^

In the u s u a l way, we put R h i , 2 £^ € / &i o ( s e 9 > ^ o r i n s t a n c e ,

Van der Z i e l , N o i se, 195>3> P» 102), T h i s r e l a t i o n a p p l i e s f o r a

t r i o d e operated i n any c o n d i t i o n . € i s a nume r i c a l c o n s t a n t , o f

value approximately 3* Thus

Hence the main c o n d i t i o n f o r a low-noise cascode c i r c u i t i s a

l a r g e value f o r g^. I t has been s t r e s s e d t h a t the e q u i v a l e n t

a m p l i f i e r n o i s e c u r r e n t generator r e p r e s e n t i n g induced g r i d n o i s e

and n o i s e a s s o c i a t e d w i t h conductance due t o feedback, t r a n s i t -

time l o a d i n g , o r t o in p u t c i r c u i t r y , can always be made unimportant

at the f r e q u e n c i e s of I n t e r e s t i n t h i s t h e s i s . Thus the cascode

nois e i s t o a ve r y good approximation s p e c i f i e d s o l e l y by R^,

3.32 Cascode C i r c u i t Designs F a v o r i n g S t a b i l i t y

Besides m i n i m i z i n g the value of R n, the choice o f tubes w i t h

l a r g e g^ and g 2 has been shown as the main c r i t e r i o n f o r l a r g e

gain f o r the cascode, s i n c e f o r a f i x e d output l o a d Z , the gain ii

Is c l o s e t o g Z L .

The tube type 4 I 7 A (581|2) was s e l e c t e d f o r b o t h stages of the

cascode used t o measure t u n n e l diode n o i s e . The value of g i s

27 ramhos, of p l a t e r e s i s t a n c e i s 1600 ohms, and of a m p l i f i c a t i o n

R n

FIGURE 3.12

TYPICAL A.C.-COUPLED CASCODE AMPLIFIER

f a c t o r i s 44 .

The problem of s t a b i l i t y i n an a m p l i f i e r u s i n g tubes of

super-high transconductance becomes d i f f i c u l t . A high-g tube a c t s

as a l a r g e c u r r e n t generator which f a v o r s feedback b o t h p a r a s i t i -

c a l l y and e l e c t r o n i c a l l y , e s p e c i a l l y by magnetic c o u p l i n g .

Most commercial cascode c i r c u i t s avoid use of s u p e r - h i g h

transconductance tubes, and use a.c.-coupled stages i n p r e f e r e n c e

t o d i r e c t - c o u p l e d s t a g e s . A t y p i c a l a.c.-coupled c i r c u i t Is shown

i n F i g u r e 3»12. L^ p r o v i d e s a d.c. r e t u r n p a t h t o ground f o r the

grounded-grid stage and at h i g h f r e q u e n c i e s (e.g. above 30 Mc/s)

i s made resonant w i t h Ogp, the grid-anode tube c a p a c i t y , of the

grounded-cathode stage t o prevent e x c e s s i v e grid-anode c o u p l i n g .

A l s o the n o i s e f i g u r e i s o f t e n s l i g h t l y improved by t u n i n g C at

h i g h e r f r e q u e n c i e s . Lg tunes s t r a y c a p a c i t y between cathode of

second tube and ground, a l t h o u g h v e r y b r o a d l y because of the heavy

input l o a d i n g of the grounded g r i d s t a g e . Tuning i s shown f o r both

Input and output c i r c u i t s a l t h o u g h t h i s may be u n d e s i r a b l e at lower

f r e q u e n c i e s , where i t i s unnecessary, because of bandwidth r e s

t r i c t i o n .

At f r e q u e n c i e s as low as 4 Mc/s, used i n the present study,

the grid-anode feedback impedance l/WC of the f i r s t stage cannot

be s i g n i f i c a n t l y i n c r e a s e d by t u n i n g C , nor can I n t e r s t a g e s t r a y s SP

compare w i t h l o a d i n g of the grounded-grid s t a g e . N e i t h e r L^ nor Lg

are t h e r e f o r e u s e f u l t o the s i g n a l o p e r a t i o n , whereas a l o n g w i t h

the i n d u c t i v e s t r a y s of the w i r i n g a s s o c i a t e d w i t h the s e v e r a l

I n t e r s t a g e components, they enhance magnetic p a r a s i t i c feedback.

With use of the s u p e r - h i g h transconductance ql7A tube, i t was

found t h a t even such p r e c a u t i o n s as mounting a l l c o i l s w i t h mutu

a l l y p e r p e n d i c u l a r axes, and use of e x t e n s i v e s h i e l d i n g does not

FIGURE 3.13 SIMPLEST DIRECT-COUPLED CASCODE AMPLIFIER

60

guarantee a s t a b l e a m p l i f i e r . I n s t a b i l i t y can a l s o a r i s e through

c o u p l i n g of stages through the h i g h - v o l t a g e supply i n t h i s c i r c u i t .

Even s m a l l or heavily-damped o s c i l l a t i o n s are manifested by sharp

i n c r e a s e i n the i n p u t conductance, due p r o b a b l y t o g r i d c u r r e n t In

the f i r s t stage, or t o the onset of g r i d c u r r e n t i n any c i r c u i t

f o l l o w i n g the cascode so t h a t a l a r g e output conductance a r i s e s ,

which i s e l e c t r o n i c a l l y f ed back t o the Input.

(One advantage of the i n d i r e c t - c o u p l e d c i r c u i t of F i g u r e 3*12

at v e r y h i g h f r e q u e n c i e s i s t h a t the g r i d of the second stage i s at

d.c. as w e l l as a.c. ground. T h i s allows a s h o r t , n o n - r e a c t i v e

connection t o ground, whereas i n the d i r e c t - c o u p l e d c i r c u i t , t o be

d e s c r i b e d next, the g r i d must be c a p a c i t i v e l y coupled to ground.

I t may be shown that any s t r a y inductance In the g r i d l e a d can

produce not o n l y i n s t a b i l i t y but a l s o n e g ative Input conductance

f o r a grounded-grid stage, e s p e c i a l l y at h i g h e r f r e q u e n c i e s . )

Rather than achieve s t a b i l i t y e i t h e r by use of n e u t r a l i z i n g

feedback networks (which guarantee s t a b i l i t y o n l y over a narrow

frequency band) or by b i a s i n g the tubes to decrease the t r a n s c o n

ductance ( t h e r e b y l o s i n g the advantage of h i g h transconductance

which minimizes a m p l i f i e r n o i s e a r i s i n g from the t u b e s ) , a s i m p l e r

c i r c u i t u s i n g d i r e c t - c o u p l e d stages i s p r e f e r a b l e . A s i m p l i f i e d

v e r s i o n i s shown i n F i g u r e 3»13» The tendency f o r magnetic p a r a

s i t i c feedback i s reduced by the use of at most o n l y one c o i l ,

and by the minimal number of i n t e r s t a g e components which permits

d i r e c t p o i n t - t o - p o i n t w i r i n g , thus r e d u c i n g s p u r i o u s i n d u c t i v e

c o u p l i n g between s t a g e s . T h i s form of c o u p l i n g i s p a r t i c u l a r l y

troublesome when high-g tubes are i n v o l v e d , whereas c a p a c i t i v e

feedback i s l i k e l y harder t o c o n t r o l when high-71 tubes are used.

For d.c. o p e r a t i o n , each tube i n F i g u r e 3*13 c a r r i e s the same

current so that assuming the tubes are identical (and neglecting the drop across R^)» the bias of the grounded-grid stage automatically 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 c r i t i c a l : 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) a l l heater and B supply circuits must be carefully decoupled or filtered at the operating frequency of 4 Mc/s.

d) in contrast to a grounded-cathode stage, where any reflection 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 i t 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

v w v 1 Meg.

100 kohra r V W V V - r W W V

0.1

II 0.1

/

2.2 kohra

1+.7 kohra

100 kohm

0.1

0.1

FIGURE 3.Ill

I 6922

PRACTICAL DIRECT-COUPLED CASCODE CIRCUI WITH OPTIONAL CATHODE-FOLLOWER STAGE AND TWO-POSITION ATTENUATOR

( a l l c a p a c i t o r s are i n yuFd. u n l e s s otherwise marked)

Heater D e c o u p l i n g

Arrangement

o

attenuator with only moderately constant input impedance forms the cascode output load), but i f 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 resistance 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 i t may otherwise be stable. Such action is discouraged by damping the output tuned circuit, while Increasing the bandwidth. The gain of the cascode can s t i l l 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

i a the 6922 or E 88 CC. The cathode r e s i s t o r i s s p l i t as i n

F i g u r e 3.1i|. t o maximize R^ while m a i n t a i n i n g c o r r e c t b i a s . I t

i s d e s i r a b l e at h i g h e r f r e q u e n c i e s to tune the cathode c i r c u i t o f

the c a t h o d e - f o l l o w e r , s i n c e e x c e s s i v e c a p a c i t y can be shown t o

produce negative conductance at the i n p u t of the stage, a l t h o u g h

t h i s may be s m a l l .

The n o i s e of the cathode f o l l o w e r stage i s e q u i v a l e n t to

approximately a f>00 ohm thermal source, s i n c e the 6922 i s a t r i o d e .

T h i s adds d i r e c t l y t o the 2.5 kohra e q u i v a l e n t n o i s e source of the

r e c e i v e r ; the o v e r a l l n o i s e f i g u r e of the system depends n e g l i g i b l y

on i n c l u s i o n of the c a t h o d e - f o l l o w e r stage,' s i n c e the cascode gain

i s s u f f i c i e n t t h a t a l l n o i s e f o l l o w i n g i t i s o v e r r i d d e n .

3.33 Performance of the Cascode

T h i s d i s c u s s i o n excludes the c a t h o d e - f o l l o w e r s t a g e . Since

the gain of the cascode depends c r i t i c a l l y on the impedance i n the

p l a t e c i r c u i t of the grounded-grid stage, the gain should be

measured w i t h the cascode connected and tuned w i t h the input of

the r e c e i v e r which i s t o f o l l o w i t d u r i n g n o i s e measurements.

U s i n g a l^OO-cycle modulated s i g n a l generator w i t h c a l i b r a t e d r . f .

output v o l t a g e , one a p p l i e s a convenient s i g n a l t o the cascode

Input which o v e r r i d e s the c i r c u i t n o i s e , and notes the r e c e i v e r

response, u s i n g an RMS a.c. VTVM as a r e c o r d e r . The s i g n a l gener

a t o r i s then connected d i r e c t l y t o the r e c e i v e r i n p u t and the

s i g n a l l e v e l i n c r e a s e d u n t i l the same r e c e i v e r response i s r e g a i n e d .

One i n s u r e s t h a t the b i a s of the f i r s t r e c e i v e r stage does not

change when the s i g n a l generator, which has v e r y low d.c. i n t e r n a l

r e s i s t a n c e , i s connected. Since the r e c e i v e r gain i s t h e r e f o r e

c o n s t a n t , the r a t i o of the two s i g n a l generator l e v e l s g i v e s the

FIGURE 3.15 SCHEMATIC NOISE CIRCUIT FOR MEASURING R n OF AN AMPLIFIER

61» v o l t a g e gain of the cascode. With an output l o a d o f t y p i c a l l y

4.7 kohras, and the 4l7A !s b i a s e d f o r a g of 25 mrahos, the gain

was t y p i c a l l y 100 f o r the c i r c u i t used. T h i s i s s u f f i c i e n t t o

o v e r r i d e the r e c e i v e r n o i s e .

The e q u i v a l e n t n o i s e r e s i s t a n c e , R n, of the cascode, can be

measured s e v e r a l ways, most of which u t i l i z e the schematic c i r c u i t

of F i g u r e 3»l5« Rjj i s assumed c l o s e l y t o r e p r e s e n t a l l of the

a m p l i f i e r n o i s e , and the cascode input conductance i s assumed

n e g l i g i b l e compared to l/R , where R Is the r . f . p l a t e l o a d of a P P

n o i s e diode standard source. The a m p l i f i e r i n p u t b e i n g tuned, R P

i s about 1.5 kohms. F o r i n p u t l e v e l s which do not overload i t ,

the cascode response can s a f e l y be taken as a l i n e a r f u n c t i o n of

the input v o l t a g e . An a t t e n u a t o r separates the cascode from the

r e c e i v e r , whose response may not be assumed l i n e a r . Then Rfl f o r

the cascode i s measured as f o l l o w s .

1) For the a t t e n u a t o r s e t at a v o l t a g e r e d u c t i o n r a t i o A^,

and the n o i s e diode c i r c u i t connected t o the cascode input w i t h

the n o i s e diode c u r r e n t zero, the r e c e i v e r RMS output v o l t a g e i s

0 = k'[ A2 4 k T 0 ( R n + R p ) ] n / 2

i f T i s the temperature of b o t h R and R . With the a t t e n u a t o r P

r a t i o now at A 2 (<A^) and the n o i s e diode c u r r e n t i n c r e a s e d u n t i l

the same r e c e i v e r response i s r e g a i n e d , then

£ = k [ A ^ q k T 0 ( R n + R p) + 2el R2, }] n / 2

where I i s the n o i s e diode c u r r e n t . S o l v i n g f o r R R g i v e s

65 ,2 ™ D2 2eIRg 20 R:

^ • I j k T ^ A ^ ) 2 - l ] " R P " ( A l / A 2 y 2 - 1 " R P

An e x t e r n a l b r i d g e measurement of R p i s r e q u i r e d t o great accuracy,

s i n c e R depends o r i t i c a l l y on R . To avoid t h i s disadvantage, the

method may be extended as f o l l o w s ,

2) The a t t e n u a t o r f i r s t i s bypassed and the a m p l i f i e r i n p u t

i s s h o r t e d , g i v i n g a r e c e i v e r output

B =* k [ h k T o R j n / 2

With Rp connected t o the i n p u t , the a t t e n u a t o r i s adjusted t o a

f a c t o r A^ which produces the same r e c e i v e r output:

e = k . [ A 2 i t w ^ + R p ) ] n / 2

F i n a l l y , the a t t e n u a t o r i s set at A 2 and the nois e diode turned

up t o r e g a i n the same r e c e i v e r response:

6 = k.[4 { l l k T ^ + R p ) + 2el R21] n / 2

These r e l a t i o n s combine t o give

R p (1/201)(Af - A 2 ) / (1 - A § )

n ~ (1/Aj) - 1 " (1/A 2) - 1

3) I f the law of the r e c e i v e r i s c l o s e l y l i n e a r , the above

method may be used without an a t t e n u a t o r , 0 Q i s then taken as the

r e c e i v e r output when the cascode input i s s h o r t e d , 0^ as the

r e c e i v e r response when the nois e diode c i r c u i t , t h a t i s , Rp, i s

connected t o the cascode i n p u t , and &~y a s the r e c e i v e r output when

the n o i s e diode i s turned on t o any convenient c u r r e n t I , R n Is

then given by

R„ = (1 / 2 0 1 ) ( 0 f - ef) / te2 - eg)

4 ) A much s i m p l e r , but l e s s accurate method of f i n d i n g R R i s

to connect a r e s i s t a n c e across the cascode Input of value such t h a t

the n o i s e power, as detected at the r e c e i v e r output, i s doubled

over t h a t due t o the a m p l i f i e r w i t h shorted i n p u t . The value of

such a r e s i s t a n c e then equals Rfi. G e n e r a l l y Rfi Is s m a l l enough

so t h a t the nois e generated by tuned c i r c u i t s i n shunt (which

i n s u r e t h a t the s u b s t i t u t i o n r e s i s t a n c e i s not shunted by the ex

c e s s i v e input c a p a c i t y o f the cascode) can be n e g l e c t e d .

By the above methods, and at a frequency o f 4 Mc/s, R^ was

measured as 50 ohms ( + 20 ohms) f o r the cascode c i r c u i t w i t h 4 I 7 A

tubes b i a s e d t o a transconductance o f about 25 ramhos. T h i s value

of Rfi i s lower than expected by the f o r e g o i n g n o i s e a n a l y s i s by a

f a c t o r of three ( n e g a t i v e feedback of an u n c o n t r o l l e d nature may be

i n v o l v e d In the d i s c r e p a n c y ) .

I f Q as 100, r = 3 ohms f o r the c o l l of the s e r i e s - t u n e d c i r

c u i t which couples the t u n n e l diode i n t o the cascode, then the

e q u i v a l e n t thermal n o i s e source connected t o the cascode i s about

4 kohms when the t u n n e l diode assumes i t s ze r o b i a s r e s i s t a n c e o f

about 20 ohms. The cascode n o i s e i s n e g l i g i b l e i n comparison, so

tha t the o v e r a l l n o i s e f i g u r e of the c i r c u i t f o r measuring t u n n e l

diode n o i s e i s determined mainly by the c o i l n o i s e of the s e r i e s -

tuned c i r c u i t .

s e r i e s - o r p a r a l l e l -tuned c i r c u i t

low-n o i s e

cascode -| p r e

a m p l i f i e r

a t t e n u a t o r f o r

p r o v i s i o n a l

use

t u n n e l diode b i a s

and r . f . c i r c u i t

T -0*0-

c a l i b r a t e d r e s i s t a n c e sources

n o i s e diode b i a s

and r . f . c i r c u i t s

1|00 cps modulated r . f . s i g n a l g e n e r a t o r

f o r c a l i b r a t i o n

n o i s e diode

f i l a m e n t c o n t r o l c i r c u i t

h i g h - g a i n i n t e g r a t o r E s t e r l i n e -narrow or Angus band smoothing r e c o r d e r

r e c e i v e r -i c i r c u i t f o r w i t h noise two l e v e l s

I.P. f r e q u e n c i e s

RMS a.c. VTVM f o r

c a l i b r a t i o n l e v e l s

FIGURE 3.16

BLOCK DIAGRAM OF COMPLETE NOISE MEASURING CIRCUIT

67

3»U Other Apparatus and C i r c u i t r y

3 . q l P e r s p e c t i v e of the O v e r a l l C i r c u i t

F i g u r e 3 « l 6 shows the r e l a t i o n between the major s e c t i o n s of

the complete c i r c u i t f o r the measurement of n o i s e i n the t u n n e l

diode.

The Impedance-transformed t u n n e l diode source, as w e l l as the

n o i s e diode source, which must always be confronted w i t h the same

Impedances and t r a n s f o r m a t i o n s f o r a v a l i d comparative n o i s e

measurement, i s shown coupled t o the low-noise p r e a m p l i f i e r . The

a t t e n u a t o r which f o l l o w s may be bypassed; i t s use i n measuring the

e q u i v a l e n t n o i s e r e s i s t a n c e of the cascode has been e x p l a i n e d ; i t

w i l l l a t e r be shown to be unnecessary i n comparing the t u n n e l diode

and n o i s e diode s o u r c e s .

The use of the standard or c a l i b r a t e d r e s i s t a n c e sources, and

the r o l e of the s i g n a l generator are e x p l a i n e d i n d e t a i l s h o r t l y ,

i n connection w i t h the p a r t i c u l a r method adopted to compare t u n n e l

diode and n o i s e diode s o u r c e s . B r i e f l y , c a l i b r a t e d r e s i s t o r s , of

known (thermal) n o i s e g e n e r a t i o n , are switched i n t o the t u n n e l

diode p o s i t i o n t o c a l i b r a t e the p r e a m p l i f i e r and other u n s p e c i f i e d

n o i s e sources, I n c l u d i n g those of the s e r i e s - t u n e d c i r c u i t , and

other r e s i s t i v e shunts. The t u n n e l diode i s then switched i n and

b i a s e d u n t i l I t s r e s i s t a n c e equals the c a l i b r a t e d r e s i s t a n c e , so

that the o v e r a l l impedance c o n d i t i o n s remain unchanged, as i s

n e c e s s a r y f o r the comparison of t u n n e l diode and n o i s e diode s o u r c e s .

The s i g n a l generator p r o v i d e s the r e f e r e n c e s i g n a l by which the

t u n n e l diode r e s i s t a n c e i s made equal t o the c a l i b r a t e d r e s i s t a n c e .

T h i s s i g n a l appears on the a.c. VTVM at the r e c e i v e r output, where

as the much s m a l l e r n o i s e s i g n a l s appear on the more s e n s i t i v e

to series-tuned c i r c u i t input

to filament control c i r c u i t

FIGURE 3.17 NOISE DIODE AND TUNNEL DIODE BIAS AND R.F. CIRCUITS ( a l l capacitors are i n jiFd. except "C").

68

E s t e r l i n e - A n g u s r e c o r d e r , a f t e r s u i t a b l e i n t e g r a t i o n .

3.1*2 Noise Diode and Tunnel Diode Bias and R.F. C i r c u i t s

F i g u r e 3 . 1 7 shows the c i r c u i t r y a s s o c i a t e d w i t h the t u n n e l

diode and noise diode s o u r c e s .

Besides the s e r i e s - t u n e d c i r c u i t i n put impedance, the r . f .

l o a d f o r the n o i s e diode and the e q u i v a l e n t n o i s e c u r r e n t generator

of the t u n n e l diode i s the dynamic t u n n e l diode conductance, which

f o r the b i a s r e g i o n p r i n c i p a l l y under study, has a minimum value of

about 1/120 mho. Due to the v e r y l a r g e v o l t a g e d i v i s i o n imposed on

the e q u i v a l e n t n o i s e generators of the j? kohm and 10 kohra r e s i s t o r s ,

the noise c o n t r i b u t e d by them at the s e r i e s - t u n e d c i r c u i t i n put i s

n e g l i g i b l e compared to t h a t of the t u n n e l diode and n o i s e diode

s o u r c e s . (The combined conductance of the two £ kohm r e s i s t o r s ,

the 10 kohm r e s i s t o r , and the p a r a l l e l - t u n e d c i r c u i t never exceeds

l / l 8 0 0 mho). A l l r e s i s t o r s i n the r . f . c i r c u i t which c a r r y d.c.

c u r r e n t are wire-wound t o avoid c u r r e n t n o i s e . A l l 0.1..pFd.

b l o c k i n g or bypass c a p a c i t o r s are ceramic or mica, t o Insure non-

i n d u c t i v e b e h a v i o r at the h i g h frequency.

The d.c. l o a d l i n e f o r the t u n n e l diode I - V c h a r a c t e r i s t i c

i s set by the $ kohm d.c. feed r e s i s t o r i n i t s b i a s c i r c u i t . T h i s

prevents the t u n n e l diode from o p e r a t i n g i n the negative conductance

r e g i o n , s i n c e the diode switches a l o n g the l o a d l i n e from any un

s t a b l e o p e r a t i n g p o i n t i n t h a t r e g i o n . A l l o p e r a t i n g p o i n t s i n the

p o s i t i v e conductance r e g i o n s are s t a b l e . Very f i n e b i a s c o n t r o l

f o r the t u n n e l diode i s obtained w i t h a 1 kohm 4 8-turn h e l i p o t

connected as a p o t e n t i a l d i v i d e r f o r a 2 2 - v o l t b a t t e r y which pro

v i d e s a more n o i s e - f r e e c u r r e n t than would a vacuum tube power

supp l y . The b i a s v o l t a g e i s measured w i t h a c a l i b r a t e d h i g h -

impedance d.c. VTVM, s u i t a b l y r . f . - d e c o u p l e d from the t u n n e l diode

as i s the b a t t e r y supply. S w i t c h S ( F i g u r e 3 « 1 7 ) i s c l o s e d when

ever the b i a s i s rea d . The d.c. c u r r e n t through the t u n n e l diode

i s measured w i t h a moving c o l l milllammeter which has been c a l i

b r a t e d a g a i n s t a standard \% Weston ammeter, as has the m o v i n g - c o i l

milliammeter i n the noise diode p l a t e c i r c u i t . The B + f o r the

no i s e diode Is decoupled f o r both low and r . f . f r e q u e n c i e s , the

former t o prevent s u p e r - p o s i t i o n o f 60 c y c l e w i t h the r . f . s i g n a l

at the nois e diode anode. The p a r a l l e l - t u n e d c i r c u i t serves f u r

t h e r t o s h o r t - c i r c u i t any superimposed 60 c y c l e or i t s harmonics,

which c o u l d otherwise pass through the r . f . cascode c i r c u i t t o

modulate the r . f . i n the r e c e i v e r , o r overload the f i r s t s t a g e .

(The p a r a l l e l - t u n e d c i r c u i t c o i l i s a l s o u s e f u l In p r e v e n t i n g

c h a r g i n g t r a n s i e n t s a r i s i n g from any r a p i d v a r i a t i o n i n the noise

diode B + from appearing across the t u n n e l diode.)

The i|00 c y c l e modulated r . f . s i g n a l generator i n j e c t s through

e i t h e r R, the c a l i b r a t e d r e s i s t a n c e , o r the t u n n e l diode, a s i g n a l

i n t o the s e r i e s - t u n e d c i r c u i t , whose response w i l l c r i t i c a l l y

depend on the value o f the damping r e s i s t a n c e R. The t u n n e l diode

r e s i s t a n c e can be made v e r y a c c u r a t e l y equal t o the known c a l i b r a t e d

r e s i s t a n c e by a d j u s t i n g i t s b i a s u n t i l the s e r i e s - t u n e d c i r c u i t

response i s the same whether R or the t u n n e l diode p r o v i d e s the

damping. The way i n which c i r c u i t n o i s e e x c l u d i n g that o f the

t u n n e l diode i s accounted f o r by t h i s technique Is presented i n the

f o l l o w i n g s e c t i o n . The 10-ohm i n t e r n a l r e s i s t a n c e o f the s i g n a l

g enerator renders the c a l i b r a t i o n technique i n s e n s i t i v e when the

t u n n e l diode r e s i s t a n c e o r R becomes l e s s than 10 ohms; i n t h a t

case, a 1-ohm shunt placed across the s i g n a l generator output

r e s t o r e s the s e n s i t i v i t y .

Since the i n p u t impedance of the s e r i e s - t u n e d c i r c u i t i s o n l y

about 3 ohms at resonance, s i g n i f i c a n t v o l t a g e d i v i s i o n of the

n o i s e diode and t u n n e l diode s i g n a l s may occur i f there i s any

impedance i n s e r i e s w i t h these s o u r c e s . F o r a v a l i d comparison of

these sources, the O . l j a F d . b l o c k i n g c a p a c i t o r s i n s e r i e s w i t h

each of these sources must present c l o s e l y s i m i l a r impedance.

Pr e c a u t i o n s are taken i n the c i r c u i t w i r i n g t o i n t r o d u c e a minimum

of u n s p e c i f i e d i n d u c t i v e s t r a y s which would act i n s e r i e s w i t h the

t u n n e l diode and n o i s e diode. The t u n n e l diode and output of the

n o i s e diode c i r c u i t are b u i l t c l o s e t o g e t h e r and t o the i n p u t of

the s e r i e s - t u n e d c i r c u i t t o f u r t h e r improve the high-frequency

c h a r a c t e r i s t i c s . The noise diode f i l a m e n t c i r c u i t i s decoupled

and s h i e l d e d w i t h copper p l a t e p a r t i t i o n s from the r e s t of the

n o i s e diode c i r c u i t , and the e n t i r e noise diode c i r c u i t i s s h i e l d e d

from the t u n n e l diode c i r c u i t r y and from the output p a r a l l e l - t u n e d

c i r c u i t . T h i s prevents r . f . or l . f . c o u p l i n g between the d i f f e r e n t

c i r c u i t s . A l l c i r c u i t r y Is s h i e l d e d from the surroundings by

completely e n c l o s i n g metal c h a s s i s .

3«43 Noise Diode Filament Current Supply

At n o i s e measurement f r e q u e n c i e s much above 60 c y c l e s , the

f i l a m e n t s of the 5722 n o i s e diode can operate on 6 0 - c y c l e power,

the thermal time constant of the f i l a m e n t s b e i n g l o n g enough that

no h i g h harmonics of 1 2 0-cycle modulation of the emission r e s u l t s .

The S y l v a n i a 5722 n o i s e diode operates w i t h f i l a m e n t c u r r e n t be

tween 1 and 2 Amperes, co r r e s p o n d i n g t o anode c u r r e n t s up t o 35

mA. i n the t e m p e r a t u r e - l i m i t e d c o n d i t i o n . The anode c u r r e n t (and

hence the n o i s e power output of the noise diode) depends very

s e n s i t i v e l y ( e x p o n e n t i a l l y ) on the f i l a m e n t c u r r e n t , and s i n c e the

to noise diode filaments

filament transformer

power transformer

regulated 110 volt mains

100 kohnf

6CL6

1> W \ A A / 220 ohms

100 kohm helipot

T 0,5 uFd.

FIGURE 3.18

NOISE DIODE FILAMENT CURRENT CONTROL CIRCUIT

71

n o i s e power must be f i n e l y a d j u s t a b l e and s t a b l e at any a r b i t r a r y

l e v e l d u r i n g noise measurements, the f i l a m e n t c u r r e n t must be con

t r o l l a b l e and s t a b l e t o a degree beyond the c a p a b i l i t y of a

potentiometer which could c a r r y the l a r g e c u r r e n t r e q u i r e d . The

c i r c u i t of Fig u r e 3 •18 s o l v e s the problem.

The p l a t e - t o - p l a t e impedance of two t r i o d e - c o n n e c t e d 6CL6 power-pentodes i n p u s h - p u l l o p e r a t i o n i s stepped down by a 3 : 1

t u r n s - r a t i o power tr a n s f o r m e r and I n s e r t e d i n t o the primary o f a

f i l a m e n t t r a n s f o r m e r which powers the n o i s e diode f i l a m e n t s . The

6CL6's are p u s h - p u l l connected t o c a n c e l the d.c. c u r r e n t i n the

power tra n s f o r m e r secondary which would otherwise s a t u r a t e the

co r e , g i v i n g Improper impedance t r a n s f o r m a t i o n as a r e s u l t . The

r e s i s t o r network between the - 3 0 0 v o l t s u p p l y and ground Is s e l e c t

ed t o a l l o w v a r i a t i o n of the 6 C L 6 g r i d b i a s from - 5 0 v o l t s t o - 2 5

v o l t s (the 1 kohm power r e s i s t o r across the h e l i p o t reduces the

p o w e r - d i s s i p a t i n g requirements of the h e l i p o t ) . The transformed

p l a t e - t o - p l a t e impedance i n s e r t e d i n t o the f i l a m e n t t r a n s f o r m e r

primary then v a r i e s over a range which causes the f i l a m e n t c u r r e n t

of the noise diode to be f i n e l y continuous over the d e s i r e d range.

The 220-ohra r e s i s t o r improves the l i n e a r i t y of c u r r e n t c o n t r o l over

the h e l i p o t range. The 0 . 5 ^iFd. c a p a c i t o r i n h i b i t s the tendency

f o r slow o s c i l l a t i o n s of the c o n t r o l c i r c u i t when f i r s t turned on.

The 6 0 - c y c l e s u p p l y f e e d i n g the f i l a m e n t t r a n s f o r m e r primary must be

c a r e f u l l y r e g u l a t e d : an o r d i n a r y Sorenson r e g u l a t o r i s s u f f i c i e n t .

72

3»kk D e t e c t i o n of Noise S i g n a l s

F o l l o w i n g the c a s c o d e - a m p l i f l e d n o i s e s i g n a l s , a hi g h - g a i n

r e c e i v e r (Airmec Type C86I4) w i t h two Intermediate f r e q u e n c i e s

and c o n v e n t i o n a l c r y s t a l diode d e t e c t o r Is used t o convert the

4 Mc/s noise s i g n a l s t o audio f r e q u e n c i e s . The r e c e i v e r , w i t h a

gain t y p i c a l l y of 10^, i s mo d i f i e d f o r n o i s e measurements i n the

f o l l o w i n g ways.

a) the AVC i s d i s c o n n e c t e d .

b) the fr o n t - e n d antenna tuned c i r c u i t s are disconnected

from the f i r s t r . f . stage, t o which the cascode output i s d i r e c t l y

connected. A reasonable impedance match i s then o b t a i n e d . The

r e c e i v e r i s now tunable s o l e l y w i t h the f i r s t l o c a l o s c i l l a t o r , so

th a t care i s taken t o i n s u r e t h a t i t i s always tuned t o the same

frequency as the cascode and not t o an image frequency.

I f methods of noise measurement were used which r e q u i r e

knowing the r e c e i v e r response law, the law cou l d be s t a n d a r d i z e d

by c o n n e c t i n g a square-law d e t e c t o r such as a thermocouple t o the

output of the second I.F. a m p l i f i e r of the r e c e i v e r , o r t o an

a d d i t i o n a l I.F. a m p l i f i c a t i o n stage i f needed. T h i s Is a common

procedure which i s avoided by the method used i n t h i s t h e s i s .

The s i g n a l l e v e l of the r e c e i v e r output i s adjus t e d by two

s e n s i t i v i t y c o n t r o l s : the "L.F. Gain" c o n t r o l s o n l y the s i g n a l

l e v e l e n t e r i n g the audio stages from the d e t e c t o r , while the

"H.F. Gain" c o n t r o l s the I.F. s i g n a l before the d e t e c t o r , which

f o r t h i s r e c e i v e r i s the v u l n e r a b l e p o i n t t o o v e r l o a d . During

n o i s e measurements, the H.F. Gain i s adjusted t o Insure no d e t e c t o r

o v e r l o a d i n g , while the L.F. Gain i s s e t at maximum.

To measure a c c u r a t e l y the audio mean square n o i s e power

IS 3k

R< 6

Esterline-Angus Recorder

O

a

FIGURE 3.19

CIRCUIT FOR INTEGRATING NOISE SIGNALS

• 7 3 i

output of the r e c e i v e r (which Is p r o p o r t i o n a l t o the mean square

r . f . n o i s e power e n t e r i n g the r e c e i v e r ) , i n t e g r a t i o n i s r e q u i r e d .

The audio n o i s e s i g n a l i s r e c t i f i e d and f i l t e r e d w i t h a l o n g time-

constant c i r c u i t as shown i n F i g u r e 3 . 1 9 . R-j and R 2 p r o v i d e the

d.c. c i r c u i t f o r a lN3q germanium d i o d e . b l o c k s the h i g h

v o l t a g e of the r e c e i v e r audio output tube, the i n t e g r a t o r c i r c u i t

b e i n g connected to the primary r a t h e r than t o the secondary of

the audio output t r a n s f o r m e r t o d e r i v e l a r g e r s i g n a l s t r e n g t h .

Gfe (0.03> uFd.) and R^ ( 2 . 2 kohms) are a l s o designed t o attenuate

h e a v i l y the low frequency components of the n o i s e s i g n a l before

they r e a c h the germanium r e c t i f i e r . The low frequency components

are d i f f i c u l t t o I n t e g r a t e , s i n c e t h e i r a s s o c i a t e d a u t o c o r r e l a t i o n

f u n c t i o n has s i g n i f i c a n t value f o r time i n t e r v a l s of the o r d e r of

the time constant of the R^ - C f i l t e r c i r c u i t , which was about

0 . 2 5 seconds. Thus the f l u c t u a t i o n s i n the recorded i n t e g r a t e d

output are reduced when the - R^ network attenuates f r e q u e n c i e s

under about 200 c y c l e s , while the o v e r a l l bandwidth of the n o i s e

appearing at the r e c t i f i e r c i r c u i t i s very l i t t l e decreased, so

t h a t the output s t r e n g t h does not s u f f e r . The s m a l l f l u c t u a t i o n s

t h a t remain are recorded over at l e a s t one minute on an E s t e r l i n e -

Angus paper r e c o r d e r which permits an accurate comparison of two

s m a l l - f l u c t u a t i o n s i g n a l s , s i n c e a l l of the a v e r a g i n g Information

e n t e r i n g the r e c o r d e r over a c o n v e n i e n t l y l o n g p e r i o d can be

u t i l i z e d i n forming the f i n a l average by eye.

FIGURE 3 . 2 0

COMPLETE NOISE-EQUIVALENT CIRCUIT FOR TUNNEL DIODE NOISE MEASUREMENT

7k

3«5 Adopted Noise Measurement Procedure

The method to be d e s c r i b e d avoids the use b o t h of an a t t e n u

a t o r , which must be c a l i b r a t e d , and the dependence upon the law

of a r e c e i v e r . The complete e q u i v a l e n t n o i s e c i r c u i t f o r the r . f .

noise measuring system i s shown i n F i g u r e 3»20.

R' r e p r e s e n t s conductances a s s o c i a t e d w i t h the t u n n e l diode

b i a s c i r c u i t , which are always In shunt i n the t u n n e l diode

branch. The conductance In the n o i s e diode r . f . c i r c u i t , which

i n c l u d e s the p a r a l l e l - t u n e d c i r c u i t (see F i g u r e 3*17) i s r e p r e -

sented by R p. " C^g^^ s p e c i f i e s n o i s e i n the s e r i e s - t u n e d c i r

c u i t . R i s the c a l i b r a t e d r e s i s t a n c e w i t h which the t u n n e l diode

i s compared.

L e t ^ i ^ > r e p r e s e n t the t o t a l n o i s e generated between the

t e r m i n a l s of the t u n n e l diode ( t h a t I s , b o t h t u n n e l i n g c u r r e n t

noise and thermal noise f o r the b u l k r e s i s t a n c e are i n c l u d e d ) .

The procedure f o r f i n d i n g i s based on the f a c t t h a t 2 2

K i j ^ =* C i r j ^ , i n g e n e r a l . F o r assume t h a t thermal n o i s e f o r d n the b u l k r e s i s t a n c e i s much l e s s than shot n o i s e due t o t u n n e l i n g

c u r r e n t s . Then p u t t i n g V"T = 2kT/e, where T i s the a c t u a l

absolute temperature of the t u n n e l diode, and n o t i n g t h a t the

I - V c h a r a c t e r i s t i c f o r the t u n n e l diode has 5 I / < 5 V = 2 e I T D c o t h ( V / V T ) > 2 e I T D ( V , I A ) > 2 e V T ( 3 l ^ V ) 5 l*kTG = 

That i s ,

< 1 a > > < 1 R >

i n the near-forward b i a s r e g i o n . However i n the r e v e r s e b i a s

r e g i o n , the I - V c h a r a c t e r i s t i c shows t h a t 3 1 / 5 V > i / V . I t

75

i s t h e r e f o r e i m p o s s i b l e t o decide from the I - V c h a r a c t e r i s t i c

which of ^ i 2 ^ and K, i 2 > Is l a r g e r In the re v e r s e r e g i o n . 2 2

E x p e r i m e n t a l l y i t Is found t h a t ^ 1 ^ > i n the re v e r s e b i a s

r e g i o n .

These i n e q u a l i t i e s are used i n the f o l l o w i n g procedure:

a) w i t h R switched i n and the qOO-cycle modulated s i g n a l

g enerator d e l i v e r i n g a f i x e d Mc/s s i g n a l of l e v e l s u f f i c i e n t t o

o v e r r i d e c i r c u i t n o i s e , the r e c e i v e r output i s noted on the a.c.

VTVM. (At f r e q u e n c i e s such as i\ Mc/s i t i s advantageous t o use a

s i g n a l generator r a t h e r than the nois e diode s i g n a l f o r e q u a l i z i n g

t u n n e l diode and c a l i b r a t e d r e s i s t a n c e s , i f , as here, i t can be

assumed t h a t the conductance o f bot h t u n n e l diode and c a l i b r a t e d

r e s i s t a n c e i s frequency-independent over a frequency i n t e r v a l e q u a l

t o the bandwidth of the o v e r a l l c i r c u i t . T h i s i s because the s i g

n a l generator can produce a n o n - f l u c t u a t i n g output i f i t o v e r r i d e s

c i r c u i t n o i s e , whereas a nois e diode s i g n a l causes s m a l l e r r o r i n

e q u a l i z i n g c a l i b r a t e d and t u n n e l diode r e s i s t a n c e s , due t o s m a l l

f l u c t u a t i o n s i n the i n t e g r a t e d output. However a nois e diode must

be used at f r e q u e n c i e s of 30 Mc/s and hi g h e r . ) The t u n n e l diode

i s then switched i n and the b i a s (forward or r e v e r s e ) adjusted

u n t i l the VTVM reads as before f o r the r e c e i v e r output. B i a s

c u r r e n t and v o l t a g e f o r the t u n n e l diode are r e c o r d e d . F o r b i a s e s

i n the v a l l e y r e g i o n and f a r - f o r w a r d r e g i o n , the b i a s v o l t a g e Is

i n c r e a s e d u n t i l the diode c u r r e n t reaches the peak p o i n t , then i t

switches over t o the upper p o s i t i v e p a r t of the I - V c u r v e . From

there the d e s i r e d b i a s i s reached by d e c r e a s i n g o r I n c r e a s i n g the

b i a s v o l t a g e . E x c e s s i v e decrease w i l l cause the diode t o s w i t c h t o

the near-forward r e g i o n .

b) whichever of ^ i , ^ and produces more n o i s e power

for a given value of Rd = R is switched in and the noise power output of the receiver is recorded as a smoothed constant mean square level on the recorder,

o) whichever of C i , and produces less noise power a R — —

is switched in and the noise diode current turned up until the same level (averaged for a considerable time Interval on the recorder) Is obtained as In b).

Assuming momentarily that Is negligible compared to C l 2 ^ » the analysis for the method Is as follows.

Let K, v 2 'y represent the mean square noise voltage appearing on the grid of a noiseless amplifier-receiver combination due to the noise generators , < , < 1 > ', > , as well as receiver noise (though negligible) which may be represented by an additional noise generator at the cascode input, ^ v

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«[<vf> + Z 2 ] " / 2

2 where Z is the total Impedance across the generator , hence is dominated by the tunnel diode conductance. The same output

2 noise level is now obtained when is switched in and the n

noise diode turned up:

e - 1I.[<vf> + (<i2> + )z2jn/2

These relations give = + 

d R ND Similarly In the reverse bias region, K i | is switched in alone,

f o l l o w e d by ^ i ^ ^ and "> switched i n and a d j u s t e d t o

e q u a l i z e the l e v e l , «£i > b e i n g removed. Then R

0 = k'[<v 2> + Z 2 ] Q / 2 = k . [ <v 2> + « i 2 > + = - < 1 n2d>

Thus f o r any b i a s ,

o p p + f o r forward b i a s = j 

a « -ND _ f o r r e v e r s e b i a s

The method i s seen to be Independent of assumptions r e g a r d i n g

whether ^ i n ^ can be n e g l e c t e d at low f r e q u e n c i e s , or whether the

cascode gain i s s u f f i c i e n t t o n e g l e c t r e c e i v e r n o i s e . A reasonable

n o i s e f i g u r e i s needed, however, due t o the l i m i t of s e n s i t i v i t y of

v a r i a t i o n of the i n t e g r a t e d s i g n a l w i t h s m a l l d i f f e r e n c e s i n source.

The g e n e r a t o r i s n e g l i g i b l e i n the r e v e r s e b i a s r e g i o n P

s i n c e R£ i s e n t i r e l y swamped by the t u n n e l diode conductance, but

may c o n t r i b u t e s l i g h t l y i n the forward b i a s r e g i o n near peak or

v a l l e y p o i n t s of the I - V curve, where R = R^ becomes a few p e r

cent of R£, which by e x t e r n a l b r i d g e measurement i s about q kohms.

Since "^i?.. > always appears a d d i t i v e l y w i t h ^ i 2 > then P

 = i < £ > - 

F o r , which i n c l u d e s b o t h b u l k - r e s i s t a n c e thermal

n o i s e and the t u n n e l i n g c u r r e n t n o i s e , we may d e f i n e

L i i_ l • s 2eI T D 2e [ l T D H I R«J " Ifc,

where I„,n and I w n are the average c u r r e n t s In the t u n n e l diode and

n o i s e diode r e s p e c t i v e l y . ^ 2 r a t h e r than Y 2 , the parameter

r e p r e s e n t i n g t u n n e l i n g c u r r e n t n o i s e alone, i s the o n l y d i r e c t l y

measurable q u a n t i t y , s i n c e the b u l k thermal n o i s e can never be

separated from the t u n n e l i n g c u r r e n t n o i s e by any n o i s e e x p e r i

ment. A l g e b r a i c a l l y , If2 i s found by use of the noise model of

the t u n n e l diode, given In F i g u r e 2 . 2 . The e q u i v a l e n t c i r c u i t

shown there gives

2 2 < C i J >

2 e l m „ 2 e l m n TD TD

Combining the l a t t e r two ex p r e s s i o n s f o r & 2 g i v e s

i*kT R b

Y 2 = (Hf \— —l1 - 1\ + ° UJ L 2 e

W R J " JTD 2 9 ^TD R <t

( 3 . 5 . 1 )

where R = R d + R^ and T = 3 0 0°K. i s the a c t u a l temperature of

the c i r c u i t , f o r which l+kT/2e has the value l / l9»4 v o l t s . ^ i s

of course d i m e n s i o n l e s s . R b, the bulk s e r i e s r e s i s t a n c e , i s

s p e c i f i e d by the t u n n e l diode manufacturers (Sony C o r p o r a t i o n ) t o

be 1 . 5 ohms.

In c a r r y i n g out the experiment, one s e l e c t s v a l u e s of R

co r r e s p o n d i n g t o b i a s p o i n t s on the t u n n e l diode I - V curve

e v e n l y spaced along the v o l t a g e c o o r d i n a t e (V b e i n g the s i g n i f i

cant b i a s parameter f o r a v o l t a g e - c o n t r o l l e d device such as a

t u n n e l diode, and which appears i n the t h e o r e t i c a l r e l a t i o n ,

equation ( 2 . 2 . 1 ) f o r "Jf2 ). The value s of the c a l i b r a t e d r e s i s t o r s

R are found by accurate b r i d g e measurement at 4 Mc/s.

To compare the experimental values of ^ 2 given by equation

(3»5«1) w i t h the t h e o r e t i c a l l y expected values given by ^ 2 =

c o t h (eV/2kT), the l a t t e r must be m o d i f i e d t o account f o r the f a c t

t h a t V i s the v o l t a g e across the t u n n e l i n g j u n c t i o n , whereas we

measure a v o l t a g e V» across j u n c t i o n and b u l k r e s i s t a n c e lumped

t o g e t h e r . The experimental r e s u l t given by equation (3«5.1) i s

t h e r e f o r e compared w i t h the t h e o r e t i c a l l y expected r e l a t i o n

V 2 = c o t h (eV/2kT) = c o t h [e(V» - I T DR b)/2kTj

i n which the v o l t a g e drop across the bul k r e s i s t a n c e of the t u n n e l

diode i s s u b t r a c t e d .

— J — -60

I T D(mA.)

8$ ohm 130 ohm

37 ohm

18 ohm 2b, ohm

21 ohm — t —

V'(mV.) -\ ?*•

-20

16 ohm

)12.3 ohm

9.6 ohm

7.7 ohm

-1

-2

-3

-5

-6

- 7

20 40 60

FIGURE 4 . I

CURRENT-VOLTAGE CHARACTERISTIC FOR SONY ESAKI DIODE IN NEAR-FORWARD AND REVERSE BIAS REGIONS

(dloda resistances are Indicated)

f?.f? ohm --8

80

CHAPTER k

EXPERIMENTAL RESULTS AND INTERPRETATION

q . l Reverse and Near-forward Bi a s Regions

The I - V c h a r a c t e r i s t i c f o r these b i a s r e g i o n s f o r the Sony

Germanium E s a k i diode under study, i s presented i n F i g u r e 4 . 1 .

T y p i c a l v o l t a g e s , c u r r e n t s and r e s i s t a n c e s are shown.

In the near-forward and r e v e r s e regions the nois e v o l t a g e at

the output o f the s e r i e s - t u n e d c i r c u i t due t o the n o i s e diode i s

always e q u a l i z e d t o the absolute d i f f e r e n c e of the v o l t a g e at the

same p o i n t f R / ( R + r ) ] 2 ( l / 0 > C ) 2 2 e I T E ) y 2 A f due t o the t u n n e l

diode, and the v o l t a g e [ R / ( R + r ) ] 2 ( 1 / i O C ) 2 (I|kT/R)A,f due t o

the c a l i b r a t i o n r e s i s t a n c e R. The r a t i o of t u n n e l diode shot

n o i s e c u r r e n t generator t o thermal n o i s e generator at the same

conductance should be given by

w i t h , V T =s 2kT/e. The f i r s t f a c t o r i n c r e a s e s , but the second

f a c t o r decreases w i t h I n c r e a s i n g b i a s V. E x p e r i m e n t a l l y i t Is

found t h a t ^ i ^ > - ^ i ^ ^ i n c r e a s e s q u i t e l i n e a r l y w i t h V i n the

near-forward r e g i o n , and a l s o t h a t the t u n n e l diode n o i s e s i g n a l

v o l t a g e i s q u i t e constant over t h i s b i a s range. The l a t t e r i n d i

c a tes t h a t 2 e I T D V 2 A f decreases as V Increases at about the

same r a t e t h a t [ R / ( R + r)] 2 i n c r e a s e s (the n o i s e of the s e r i e s -

tuned c o i l b e i n g q u i t e s m a l l throughout).

In the re v e r s e r e g i o n , the mean-square v o l t a g e at the s e r i e s -

tuned c i r c u i t output i n c r e a s e s s l o w l y w i t h r e v e r s e b i a s . T h i s i s

ta n h (V/V T) d l / ^ V

v/v T I A

due both to the shape of the I - V characteristic which shows I T D

Increases rapidly for small increase In V (that Is, small decrease in V 2 ) , 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 reverse 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^> - Increases with bias.

In the equation (3«5>«1): ,2

° l l U L2e I T D\R R'j

T

ND TD 1 " "p ' ITD J

hkT 2e I T DR|

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)/l T D almost entirely determines V

near the peak region, but Its relative importance decreases steadily until in the reverse region i t 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 i s , V < - 2kT/e volts) i t contributes up to 2$% of H Here the acouracy, as well as the

definition of R^, in terms of the value specified by the manufacturer, 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 and increases away from zero for forward or reverse bias, because the series-tuned circuit noise becomes significant as the damping decreases in the reverse region, tending to override the difference In sources.) The large O - difference in the integrated output for biases near the peak, and the fact that O varies sensitively with noise diode current In this area, allows an accurate measurement of the noise diode current needed to equalize the two sources. However the term in 1 ^ dominates in equation ( 3 . 5 . 1 ) for *6 2.

This behavior is distinguished from that of the reverse region, where the 0 - difference is so small that i t is almost obscured In the fluctuations of the integrated receiver response, even when a careful long-time average of the recorded output is made by eye. The percent uncertainty In measured I to equalize 0 - responses for the two sources is therefore very large, since also the 0-

response Is found to be very insensitive to the value of I N D in this region. However the term involving % D / 1 T D in equation ( 3 . 5 . 1 )

is insignificant in the reverse region, hence so is the error in <J2 due to this term. (Due to the large value of 3 l T D / 3 R in the reverse region, a small uncertainty in equalizing the tunnel diode resistance with the calibrated resistance R yields a magnified uncertainty in Im n « For accuracy, a l l measurements in the reverse

range were repeated and the r e s u l t i n g values of 1^ measured f o r

a given value of R were averaged. The d e v i a t i o n s were s m a l l .

E x c l u d i n g b u l k thermal n o i s e i n the t u n n e l diode, i f i t i s

assumed t h a t near the o r i g i n = 2e 1^ }$2 C± q.kT/R, then

^ 2 d i|kT/2eI T DR which, n e g l e c t i n g the n o i s e of R£ , Is the

dominant term i n equation (3 .5*1)• (With b u l k thermal n o i s e

i n c l u d e d as i n F i g u r e 2.2, there r e s u l t s under the present approx

imation ( R / R t ) 2 ( 4 k T / 2 e I T D R ) - (l|kT/2e) ( R b / l T D R 2 ) ; t h i s form

a c c o r d i n g t o the approximation holds b e s t near the o r i g i n ) .

I t t h e r e f o r e f o l l o w s t h a t In the r e v e r s e b i a s r e g i o n and f o r

s m a l l forward b i a s , tf2 can be found q u i t e a c c u r a t e l y even I f the

n o i s e experiment i s omitted, s i n c e t o a good approximation the

o n l y Information needed In equation (3»5«1) i s R as a f u n c t i o n of

I T D f o r the t u n n e l d i o d e . T h i s i m p l i e s t h a t f o r the r e v e r s e b i a s

r e g i o n , the I - V c h a r a c t e r i s t i c determines the n o i s e , or converse

l y , i f the device d i s p l a y s shot n o i s e , the Information obtained

from a n o i s e experiment i s o n l y a minor c o r r e c t i o n t o X ^. T h i s

approximation i s i n c r e a s i n g l y i n a c c u r a t e f o r i n c r e a s i n g r e v e r s e

b i a s e s , a l t h o u g h even f o r b i a s e s approaching the power-handling

c a p a b i l i t y of the t u n n e l diode, the I - V c h a r a c t e r i s t i c alone

p r e d i c t s the n o i s e t o w i t h i n 10% accuracy i n ^ 2 . The i n a c c u r a c y

Increases much more r a p i d l y In the forward r e g i o n w i t h i n c r e a s i n g

b i a s , s i n c e R i n c r e a s e s r a p i d l y .

I f we assume, f o r forward or r e v e r s e b i a s e s much l e s s than

2kT/e, th a t the n o i s e of the t u n n e l i n g c u r r e n t s c l o s e l y a p p r o x i

mates thermal n o i s e f o r a r e s i s t a n c e of the same value as t h a t of

the t u n n e l diode, then

 = 2eI T D coth(eV/2kT) ^ 4 k T ( 3 $ V)

(where the component o f ^i j i j ^ d e s c r i b i n g b u l k t h e r m a l n o i s e i n

the d iode i s i n c l u d e d and improves the a p p r o x i m a t i o n ) . W i t h

V T = 2kT/e, the d i f f e r e n t i a l e q u a t i o n

d I T D / l T D C i [ co th (VA T ) ] dVA T S c o t h ( V ) dV«

r e s u l t s , assuming temperature i s c o n s t a n t , w i t h s o l u t i o n

I T D ~ . I 0 s i n h (VA T )

F o r V << V ^ , t h i s r e l a t i o n d e s c r i b e s the I - V curve n e a r the

o r i g i n ; the form Is conc luded s o l e l y f rom the n o i s e p r o p e r t i e s

near the o r i g i n s i n c e t h e y v e r i f y t h a t l l r t „ / l , r _ l = exp ( e V / k T ) ,

The p o s s i b i l i t y o f u s i n g t h i s r e s u l t w i t h the E s a k i i n t e g r a l s f o r

I s l i - I I t o determine the f u n c t i o n a l forms f o r Z o r f o r TD ' cv v c 1 cv

F ( E , V ) = Z ( E , V ) £ (E) f> (E + V) i s I m p o s s i b l e s i n c e f o r b i a s e s near

the o r i g i n , the E s a k i i n t e g r a l s cannot be a p p r o x i m a t e d .

In summary, the compar ison o f , o r t o a good u t

a p p r o x i m a t i o n , w i t h K I > as a f u n c t i o n o f V shows t h a t : n i+kT(<)l/<5 V) < ljkT(c>lA> V ) | and < 2el-, "tf 2

v 1 , i 1 > o ]v=o X 0

f o r fo rward b i a s

4kT (d l /<>V) > l |kT(dl /c> V ) l and > 2el-, K2

v 1 , i 1 < o IV=0 A 0

f o r r e v e r s e b i a s .

I = I T D i s the average t u n n e l d iode c u r r e n t and T i s the a c t u a l

d iode t e m p e r a t u r e . I f the n o i s e temperature T n o f the t u n n e l d iode

i s d e f i n e d as

T n " 2 e I T D * o / i4k(^)l T D/aV)

then f o r c o n s t a n t temperature T , T n < T f o r r e v e r s e b i a s e s , and

^ Arc tanh (l/tf 2)

• 1 . 0

•0.8

•0.6

••0.4

- - 0 . 2

0

FIGURE 1|.2

THEORETICAL AND EXPERIMENTAL COMPARISON FOR TUNNEL DIODE NOISE IN THE NEAR-FORWARD BIAS REGION

(RMS deviation of experimental points from theoretical straight line is 2 . 0 % )

10 20 30

peak voltage = 66 mV.

theoretical curve: slope = e/2kT = 0*0194 mV."1

for T = 300°K.

ko V (mV.

.T > T for forward biases. The ratio Tn/T cannot be evaluated nor limits assigned to i t without a relation for I,pD as a function of V. Whether I|kT(^Im n / V) I is less than 2eI(T,T.>2 for forward

IV=0 i D 0

bias, or greater than 2el^^%^ for reverse bias also depends on the I - V characteristic. That the magnitude of the shot noise 2 e I T D y 2 and indeed the fact that i t 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 I c v and I v c 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 If 2 = coth e(V - I T DR b)/2kT is displayed by plotting arc tanh ( l / t f 2 ) against the junction bias V = V - I T DR b» The result is a straight line of slope 2kT/e.

Experimental values o f V | 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 graphing arc tanh ( l / ^ f 2 ) as the ordinate rather than V 2 , a constant percent discrepancy with bias of the experimental values of 2

from coth(eV/2kT) manifests i t s e l f as an increasing mean square

A Arc tanh

• - 1.2

4- 1.0

f 0 . 8

4- 0.6

i o.k

•f 0.2

FIGURE 4.3

THEORETICAL AND EXPERIMENTAL COMPARISON FOR TUNNEL DIODE NOISE IN THE REVERSE BIAS REGION

(RMS deviation of experimental points from best f i t i s 1.9 % )

V

experimental f i t : slope = a/2kT f o r T = 314,6° K.

the o r e t i c a l l i n e : slope = e/2kT = 0.0194 mV.-1

f o r T = 300°K.

4-10 20 30 40 5o 60 V(raV.)

86

d e v i a t i o n w i t h b i a s of the experimental values o f arc tanh (l/lf 2)

from the expected s t r a i g h t l i n e r e l a t i o n ; t h a t i s , a l i n e w i t h

d i f f e r e n t slope r e s u l t s . T h i s b e h a v i o r appears i n F i g u r e i| . 3 »

where the percent e r r o r i n % | i s not b i a s dependent.

In the near-forward p l o t , the experimental p o i n t s f i t the

t h e o r e t i c a l curve w i t h i n experimental u n c e r t a i n t y . The r e s u l t s

thus v i n d i c a t e the assumptions made i n d e v e l o p i n g an e x p r e s s i o n

f o r two-current shot n o i s e from the E s a k i i n t e g r a l s , and a l s o

c o n f i r m , f o r at l e a s t the band-independent a s p e c t s , Esaki»s

f o r m u l a t i o n of t u n n e l i n g . F u r t h e r , the r e s u l t s mean that i n d i r e c t

t u n n e l i n g mechanisms, I f they produce g r e a t e r than shot n o i s e ,

and which i n p r i n c i p l e can operate i n the negative conductance

and near-forward r e g i o n s , are not present s u f f i c i e n t l y t o enhance

the d i r e c t - t u n n e l i n g shot noise by a measureable amount. (The

l i k e l i h o o d t h a t the experimental and t h e o r e t i c a l agreement shown

i n F i g u r e U • 2 can be due t o a combination o f smoothed d i r e c t -

t u n n e l i n g c u r r e n t shot noise ( t h a t i s , that c o r r e l a t i o n between

the t u n n e l i n g c u r r e n t s I and I e x i s t s ) and excess noi s e due ° cv vc t o i n d i r e c t t u n n e l i n g processes p e r s i s t i n g i n t h i s r e g i o n , i s

ve r y s m a l l even f o r a spot agreement at a s i n g l e b i a s . Agreement

over the e n t i r e b i a s r e g i o n r u l e s out such compensating e f f e c t s

e n t i r e l y .

The r a t i o o f d i r e c t t o i n d i r e c t t u n n e l i n g c u r r e n t should

i n c r e a s e w i t h i n c r e a s e d o v e r l a p of the conduction and valence

bands. In the rev e r s e b i a s r e g i o n the r a t i o i s l a r g e s t so t h a t

i f the measured n o i s e i s c o n s i s t e n t w i t h d i r e c t t u n n e l i n g o n l y i n

the near-forward r e g i o n , i t must s i m i l a r l y be due o n l y t o d i r e c t

t u n n e l i n g processes i n the r e v e r s e range, w i t h i n experimental

accuracy.

Thus the tendency, shown i n F i g u r e I4.3 f o r the re v e r s e

r e g i o n , f o r the experimental values o f Y 2 to exceed the t h e o r

e t i c a l l y p r e d i c t e d ones (the percent d i s c r e p a n c y i s b i a s - i n d e p e n

dent) i s not l i k e l y due t o fundamental processes which enhance

the n o i s e i n t h i s r e g i o n o n l y . More l i k e l y causes a r e :

a) I n c o r r e c t value assumed f o r t u n n e l diode b u l k r e s i s t a n c e

R b. T h i s p o s s i b i l i t y Is e a s i l y r u l e d out as f o l l o w s . The

d i s c r e p a n c y between experimental and t h e o r e t i c a l values of Y 2

Is reduced i f the experimental p o i n t s move t o the l e f t a l o n g the

V - a x i s , which corresponds t o an i n c r e a s e In over the assumed

v a l u e . However, t r e a t i n g R and I as c o n s t a n t s , and In the

rev e r s e r e g i o n p u t t i n g I N D / l T D < < (UkT/2eI T D)(l/ R - l / R p ) i n

equation (3«5.1) gives

Y 2 £ i a / ( R - R b)2 . b R b / ( R . R f e )2

where a and b are constants and a/b ££. R . Hence 2f 2 i n c r e a s e s as

R^ i s made l a r g e r (assuming R b < R always), o r a r c tanh ( l / i T 2 )

decreases independent of b i a s f o r Increase i n R^. The e x p e r i

mental p o i n t s thus move i n the d i r e c t i o n of the arrow shown In

F i g u r e i | . 3 f o r one of the p o i n t s , that I s , p a r a l l e l t o the t h e o r

e t i c a l curve, so t h a t u n c e r t a i n t i e s i n R, cannot e x p l a i n the s m a l l D

but c o n s i s t e n t d i s c r e p a n c y .

b) the syste m a t i c d i s c r e p a n c y Is more l i k e l y due t o syste m a t i c

e r r o r s i n measurement, p a r t i c u l a r l y i n the c a l i b r a t i o n technique

f o r e q u a t i n g two r e s i s t a n c e s . I f the t u n n e l diode r e s i s t a n c e i s

not e x a c t l y matched w i t h the c a l i b r a t i o n r e s i s t a n c e , then not o n l y

do the impedances change d u r i n g a noise measurement, i n v a l i d a t i n g

the comparison w i t h the standard n o i s e diode source, but a l s o an

erroneous value of 1 ^ i s used i n equation (3»5»1)« Due t o the

l a r g e value of 2 mA/ohm f o r d I T D / d R In the f a r r e v e r s e r e g i o n ,

and the r e d u c t i o n i n s e n s i t i v i t y of the method due to the s e r i e s -

tuned c o i l r e s i s t a n c e and s i g n a l generator i n t e r n a l r e s i s t a n c e ,

a 1% u n c e r t a i n t y i n the c a l i b r a t i o n i n t e g r a t e d s i g n a l can produce

a 10% u n c e r t a i n t y i n I T D when R i s 7 ohms. T h i s Is l a r g e r than

the 3,5% RMS s c a t t e r o f p o i n t s i n F i g u r e 3.1\ And exceeds the

disagreement i n slopes of the best experimental and t h e o r e t i c a l

l i n e s . Such a s m a l l s y s t e m a t i c e r r o r i m p l i e d i n the c a l i b r a t i o n

c o u l d not appear i n the near-forward r e g i o n r e s u l t s , s i n c e R i s

much l a r g e r , and dl^jy^dR much s m a l l e r t h e r e .

The agreement of the o r y and experiment throughout the b i a s

r e g i o n dominated by d i r e c t t u n n e l i n g i s i n t e r p r e t e d t o mean t h a t :

a) E s a k i ' s f o r m u l a t i o n f o r d i r e c t t u n n e l i n g a p p l i e s i n a

g e n e r a l i z e d form i n which F(E,V) = Z(E,V) £ (E) ^>(E+V) i s any

f u n c t i o n .

b) the t u n n e l i n g c u r r e n t s I and I„„ are Independent and i n cv vc

the r a t i o exp(eV/kT) i n absolute v a l u e . T h i s a l s o i m p l i e s t h a t

t u n n e l i n g r e c i p r o c i t y h o l d s , t h a t i s , Z C V ( E , V ) = Z V C ( E , V ) .

c) i n d i r e c t t u n n e l i n g p r o c e s s e s , i f they enhance the n o i s e ,

and which dominate i n the v a l l e y and f a r - f o r w a r d r e g i o n s while

p e r s i s t i n g i n t o the negative conductance r e g i o n , are immeasureably

s m a l l In the near-forward and r e v e r s e r e g i o n s , even at the peak.

d) processes such as a v a l a n c h i n g , which would enhance the

noise over t h a t due to d i r e c t t u n n e l i n g , do not occur i n the .far

r e v e r s e r e g i o n f o r b i a s e s w i t h i n the power c a p a b i l i t i e s o f the

Sony t u n n e l diode under t e s t . (Avalanching can occur I f the

j u n c t i o n w i d t h i s g r e a t e r than the mean f r e e p a t h o f e l e c t r o n s i n

the gap while t u n n e l i n g , so t h a t they can i o n i z e l a t t i c e s i t e s i n

the gap through c o l l i s i o n : t h i s enhances the c u r r e n t as w e l l as

the n o i s e , a l t h o u g h d I ™ / d V f o r t u n n e l i n g i s so l a r g e In the f a r

—I —I 1 1 1 1 1 1 1 220 2U0 2 6 0 2 8 0 300 320 340 3 6 0 3 8 0

bias voltage V» (mV.)

r e v e r s e r e g i o n that a v a l a n c h i n g would not change the c h a r a c t e r

i s t i c n o t i c e a b l y , but o n l y the n o i s e . T y p i c a l E s a k i j u n c t i o n s

p r o b a b l y do not exceed i n w i d t h the mean f r e e e l e c t r o n p a t h so

t h a t no avalanche n o i s e would be expected.)

4.2 h V a l l e y and Far-forward Bias Regions

The data f o r t h i s r e g i o n show that the n o i s e temperature of

the t u n n e l diode g r e a t l y exceeds t h a t c o r r e s p o n d i n g t o thermal

n o i s e of a r e s i s t a n c e R of the same v a l u e . I f ^ 2 were given by

the d i r e c t - t u n n e l i n g r e l a t i o n coth(eV/2kT), i t would have the

value u n i t y throughout t h i s b i a s range, whereas the data give a

minimum value of 4.8 f o r ft2 near the v a l l e y . The t u n n e l diode

r e s i s t a n c e t y p i c a l l y exceeds 100 ohms i n t h i s r e g i o n , so t h a t the

s e r i e s - t u n e d c o i l n o i s e i s l a r g e l y suppressed by the damping,

whereas the e q u i v a l e n t excess n o i s e c u r r e n t generator of the

t u n n e l diode i s l a r g e enough completely t o o v e r r i d e a l l other

n o i s e . A p a r a l l e l - t u n e d c i r c u i t would t h e r e f o r e s u f f i c e t o couple

the t u n n e l diode i n t o the cascode a m p l i f i e r , and the low noise

f i g u r e r e s u l t i n g should allow h i g h a c c u r a c y .

The values of ^ f 2 , measured In the same way as i n the r e v e r s e

and near-forward r e g i o n s , are p l o t t e d a g a i n s t b i a s v o l t a g e i n

F i g u r e 4.4. F o r comparison, 1^ and I = 7f2 l r p D are a l s o graph

ed. The range of I ' f o r which data i s obtained Is too s m a l l f o r

an unambiguous r e l a t i o n between b i a s and I t o be determined, so

t h a t the s i g n i f i c a n c e of t h i s data i s s o l e l y t h a t i t f i n d s excess

n o i s e to p e r s i s t up t o the measuring frequency of 4 Mc/s.

The r e s u l t s of M. D. Montgomery (19&1) i n t h i s r e g i o n , at

1 k c / s , i n d i c a t e d an e x p o n e n t i a l r e l a t i o n between I e q and the b i a s

v o l t a g e , over a range from 100 raV. ( i n 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 i f 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 values of *6 2 obtained for currents well into the far-forward o region (where the tunnel diode impedance has decreased to less than 30 ohms) suggests the possibility that I T D in this region is due predominantly to Indirect tunneling processes. Esaki and Yajima (1958) have obtained a closely linearly decreasing behavior for X 0 with frequency up to 100 kc/s in this bias region, which on extrapolation shows # 2 should f a l l to unity at about 1 Mc/s i f their relation persists at higher frequencies than 100 kc/s. The present results show that i t 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, i f 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

i n d i r e c t processes i n v o l v e o n l y d i r e c t t u n n e l i n g to donors

f o l l o w e d by v e r t i c a l t r a n s i t i o n s i n t o the valence band, or v e r t i

c a l t r a n s i t i o n s t o a c c e p t o r s f o l l o w e d by d i r e c t t u n n e l i n g to the

valence band, then two compensating e f f e c t s occur as b i a s i s

i n c r e a s e d :

a) the number of donors which can p r o j e c t v e r t i c a l l y onto

the area r e p r e s e n t i n g unoccupied s t a t e s In the valence band i n

c r e a s e s w i t h b i a s ; t h i s enhances the I n d i r e c t process v i a donors

as b i a s i n c r e a s e s .

b) s i n c e the energy between donors and conduction band, f o r

a f i x e d s p a t i a l c o o r d i n a t e , must be independent of b i a s , the

number of donors which l i e i n the energy range between E„ and E , X c c

and hence which can be i n v o l v e d i n the process d i s c u s s e d , decreases

w i t h b i a s : t h i s i n h i b i t s the i n d i r e c t process v i a donors as b i a s i s

i n c r e a s e d .

S i m i l a r c o n s i d e r a t i o n s apply f o r a c c e p t o r - i n v o l v e d t r a n s i t i o n s .

F i g u r e shows t h a t at z e r o a b s o l u t e temperature, o n l y those

donors p o s i t i o n e d i n the s i n g l y - h a t c h e d area at the lower b i a s , but

the doubly-hatched area at the h i g h e r b i a s , may a i d the c u r r e n t i n

the l a t t e r case, but not i n the former--that i s , cause an Increase

i n the excess c u r r e n t w i t h b i a s . C l e a r l y v e r y few donors s a t i s f y '

these c o n d i t i o n s , but any t h a t do, cause c u r r e n t to i n c r e a s e . Two

s i m i l a r (but on the diagram, narrower) ranges f o r acceptors are

d e p i c t e d . Again the narrow s i n g l y - h a t c h e d range r e p r e s e n t s

acceptors which,cannot be i n v o l v e d at the lower b i a s , whereas the

narrow doubly-hatched area r e p r e s e n t s acceptors which can be i n

v o l v e d at the h i g h e r b i a s .

S i m i l a r l y , areas can be drawn f o r b o t h donors and acceptors

f o r which these s i t e s can a i d the i n d i r e c t 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 temperatures, these statements show that i t is not clear that excess current should increase with bias for the indirect processes discussed.

93

C H A P T E R 5

CONCLUSIONS AND OUTSTANDING PROBLEMS

5>«1 Near-forward and Reverse B i a s Regions

The agreement of the noise measurements w i t h the p r e d i c t e d

c o t h (eV/2kT) r e l a t i o n oyer the e n t i r e r e g i o n dominated by d i r e c t

t u n n e l i n g excludes beyond reasonable doubt a l l p o s s i b i l i t i e s

except t h a t E s a k i ' s t u n n e l i n g f o r m u l a t i o n f o r d i r e c t t r a n s i t i o n s

i s a p p l i c a b l e at l e a s t In a l l aspects not dependent on the band

s t r u c t u r e . The assumptions t h a t the d i r e c t t u n n e l i n g c u r r e n t s I

and I are u n c o r r e l a t e d , t h a t t u n n e l i n g r e c i p r o c i t y h o l d s , and V c

t h a t l l o y ^ v c ^ s e x p ( Q V A T ) a l s o are v i n d i c a t e d .

No u s e f u l Information can be gained by extendi n g the measure

ments i n the re v e r s e r e g i o n u n t i l avalanche o c c u r s , accompanied by

enhanced n o i s e . ( L i t t l e i n f o r m a t i o n f o r the f o r b i d d e n gap wid t h

would be obtained from the b i a s at which a v a l a n c h i n g s e t In, which

i s not obtained more unambiguously by measurement of the cap a c i t a n c e

a s s o c i a t e d w i t h the j u n c t i o n . ) The re v e r s e c u r r e n t i s augmented

w i t h i n c r e a s e d r e v e r s e b i a s not onl y by i n c r e a s i n g o v e r l a p of the

bands, but a l s o by the I n c r e a s i n g f i e l d i n the j u n c t i o n which

augments the p e n e t r a t i o n f a c t o r a s s o c i a t e d w i t h d i r e c t t u n n e l i n g .

The independence of the two c u r r e n t s I and I y o should In no way

depend on b i a s , however, so t h a t f o r b i a s e s i n s u f f i c i e n t t o cause

a v a l a n c h i n g , the noise should not be enhanced over shot n o i s e .

Noise measurements up t o the peak c u r r e n t i n the near-forward

r e g i o n show that i f excess n o i s e In the v a l l e y r e g i o n a r i s e s from

I n d i r e c t t u n n e l i n g p r o c e s s e s , then these processes are n e g l i g i b l e

or i n some way f a i l t o enhance the n o i s e above shot n o i s e i n the

overlapped band r e g i o n s . An asymmetry of "excess" or i n d i r e c t

t u n n e l i n g c u r r e n t w i t h b i a s on e i t h e r s i d e of the v a l l e y r e g i o n i s

p o s s i b l e s i n c e the excess n o i s e p e r s i s t s Into the v e r y f a r - f o r w a r d

b i a s r e g i o n , while d i s a p p e a r i n g a l t o g e t h e r somewhere In the nega

t i v e conductance r e g i o n .

The t u n n e l diode n o i s e temperature i s found to be l e s s than

I t s a c t u a l temperature In the r e v e r s e b i a s r e g i o n , equal t o i t s

a c t u a l temperature at zero b i a s , and g r e a t e r than i t s a c t u a l

temperature f o r a l l forward b i a s e s .

E x t e n s i o n of the n o i s e measurements i n near-forward or r e v e r s e

r e g i o n s e i t h e r t o d i f f e r e n t f r e q u e n c i e s , or t o lower temperatures,

serves l i t t l e u s e f u l purpose. At lower temperatures, kT/e i s l e s s ,

so t h a t the b i a s at which V 2 should become n e a r l y u n i t y * decreases.

T h i s could a c t as a temperature-dependent check on the coth(eV/2kT)

r e l a t i o n . Perhaps b i a s e s could be extended f u r t h e r i n b o t h d i r e c

t i o n s at lower temperatures without o v e r - h e a t i n g the t u n n e l d i o d e .

Accuracy r e q u i r e d f o r the b u l k r e s i s t a n c e R^ Is l e s s at lower

temperatures s i n c e the b u l k n o i s e i s reduced, while the t u n n e l i n g

c u r r e n t n o i s e should not be. However these are advantages i n

p r a c t i c e o n l y . S i m i l a r l y the measurement o f n o i s e i n other types

of t u n n e l diodes, such as GaAs types w i t h h i g h e r p e a k - t o - v a l l e y

c u r r e n t r a t i o s , are u n l i k e l y t o give f u r t h e r i n s i g h t i n t o d i r e c t -

t u n n e l i n g p r o c e s s e s .

95

5 . 2 V a l l e y and Far-forward B i a s Regions

Many problems remain, or are generated i n t h i s s e c t i o n . The

noise data i n d i c a t e o n l y t h a t at 4 Mc/s excess n o i s e , t h a t i s ,

n o i s e g r e a t e r than shot n o i s e , e x i s t s . T h i s r e s u l t has been ob

t a i n e d b e f o r e , at v a r i o u s temperatures of the t u n n e l diode, and at

lower f r e q u e n c i e s . The excess c u r r e n t has been found l a r g e l y

temperature independent, which suggests t h a t i t a r i s e s from a

t u n n e l i n g p r o c e s s , and supports the present n o i s e measurements i n

that they show t h a t the c u r r e n t i n the r e g i o n measured i s not

thermal or d i f f u s i o n p-n j u n c t i o n c u r r e n t , s i n c e t h i s a l s o must

obey the c o t h ( e V / 2 k T ) r e l a t i o n .

The importance of extending n o i s e measurements i n the f a r -

forward r e g i o n t o g r e a t l y i n c r e a s e d b i a s e s , and over a wide range

of f r e q u e n c i e s above and below 4 Mc/s i s c l e a r . E x t e n s i o n i n t o the

negative conductance r e g i o n at v a r i o u s f r e q u e n c i e s would a l s o be

i n s t r u c t i v e . I f the n o i s e spectrum as a f u n c t i o n o f frequency

resembles any of the curves i n F i g u r e 2.8 an i n t e r p r e t a t i o n may

e x i s t i n terms of the unmodulated stepping-stone model given i n

S e c t i o n 2 . 4 a l t h o u g h i t i s d o u b t f u l I f ve r y c l o s e resemblance can

be expected due t o the s i m p l i f i e d model assumed f o r the I n d i r e c t

t u n n e l i n g p r o c e s s e s . The i n t e r p r e t a t i o n of the bias-dependence of

the excess n o i s e (over an extended range) i n terras o f the modulated

stepping-stone model i s a d m i t t e d l y d i f f i c u l t , again due t o the

q u a l i t a t i v e and s i m p l i f i e d model d i s c u s s e d . The I n d i c a t e d exponen

t i a l r e l a t i o n of excess n o i s e w i t h b i a s a p p a r e n t l y has no a s s o c i a

t i o n w i t h the modulated stepping-stone model u n l e s s the enhanced

n o i s e i s p r o p o r t i o n a l t o the magnitude of i n d i r e c t t u n n e l i n g c u r

r e n t which i t s e l f may Increase e x p o n e n t i a l l y w i t h b i a s .

I t must be emphasized however t h a t not o n l y the r e s u l t s of

the present n o i s e measurements, but l i k e l y those at o t h e r f r e q u e n

c i e s and temperatures, do not demonstrate t h a t the excess c u r r e n t

or the n o i s e a s s o c i a t e d w i t h i t a r i s e s from a t u n n e l i n g p r o c e s s .

F u r t h e r , the q u e s t i o n why i n d i r e c t t u n n e l i n g , i f i t can be shown

r e s p o n s i b l e f o r the enhanced nois e i n the f a r - f o r w a r d r e g i o n ,

should f a i l t o produce excess n o i s e i n the near-forward and r e v e r s e

r e g i o n s a l s o , has not been answered. N e i t h e r of the mechanisms

which has been considered i n Chapter 2 f o r enhanced nois e should

operate s o l e l y i n the v a l l e y and f a r - f o r w a r d r e g i o n , but not

elsewhere. Nor should the i n d i r e c t t u n n e l i n g c u r r e n t i t s e l f .

S i m i l a r l y the u s u a l causes of enhanced n o i s e commonly r e f e r r e d t o

as " l / f " (namely, f l u c t u a t i o n s i n t r a p charge d e n s i t i e s or p o s i t i o n s

or i n t e r a c t i o n s of e l e c t r o n s w i t h s u r f a c e s t a t e s i n the b u l k mater

i a l ) should not f a v o r the f a r - f o r w a r d b i a s r e g i o n while completely

d i s a p p e a r i n g i n the near-forward and reverse r e g i o n s .

The independence of the v a l l e y c u r r e n t on temperature, s u r f a c e

e t c h i n g of the b u l k m a t e r i a l , chemical surroundings, e t c . , which

has been observed by E s a k i and Yajima ( 1958) among o t h e r s , I n d i c a t e s

t h a t some form of t u n n e l i n g i s most l i k e l y r e s p o n s i b l e f o r the

exoeas-current• A model based on t h i s , and c o n s i s t e n t over the

e n t i r e I - V c h a r a c t e r i s t i c , which can p r e d i c t enhanced n o i s e of

b i a s dependence a c c o r d i n g to F i g u r e l+.lj ( t h i s dependence agrees

w i t h t h a t found by o t h e r workers), but which p r e d i c t s o n l y shot

n o i s e f o r r e v e r s e and near-forward b i a s e s , p r o b a b l y Includes many

b a s i c processes and mechanisms too complicated t o d i s c u s s i n t h i s

t h e s i s .

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. a l . 19&1 . Phys. Rev... 121^,6,84.

Esaki, L. 1 9 5 8 . Phys. Rev. 1 0 9 , 603. Esaki, L. and Yajima, T. 1 9 5 8 . J. Phys. Soc. Japan, 1^, 1 2 8 1 .

I.R.E. Subcommittee on Noise i 9 6 0 . Proc. I.R.E. 1^8, 60. La Rosa, R. and Wilhelmsen, C.R. i 9 6 0 . Proc. I.R.E.

(Correspondence) 1+8, 1 9 0 3 .

Montgomery, M.D. 1961. J. Appl. Phys. ^ 2 , 21+08.

Tiemann, J.J. i 9 6 0 . Proc. I.R.E. l | 8 , 11+18. van der Z i e l , A. 1 9 5 8 . Proc. I.R.E. 1+JD, 5 8 9 *

van der Z i e l , A. 1 9 5 4 . Noise, Prentice-Hall, Inc., New York. Wallman, H. et. a l . 1 9 4 8 . Proc. I.R.E. 6, 700.

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