THE STATIC AND DYNAMIC
CHARACTERISTICS
TUNNEL DIODES AND THEIR APPLICATIONS
OF SERIES-CONNECTED
IN DIGITAL CIRCUITS
by
CLEMENT ANDRE TEWFIK SALAMA
B.A.Sc, University
of B r i t i s h
C o l u m b i a , 1961
A THESIS SUBMITTED IN PARTIAL FULFILMENT
OF
THE REQUIREMENTS FOR THE DEGREE OF
MASTER OF APPLIED
SCIENCE
i n t h e Department
of
Electrical
We a c c e p t
this
Engineering
thesis
required
THE UNIVERSITY
as c o n f o r m i n g
to the
standard
OF BRITISH COLUMBIA
December, 1962
In presenting this thesis in partial fulfilment of
the requirements for an advanced degree at the University of
British Columbia, I agree that the Library shall make i t freely
available for reference and study.
I further agree that permission
for extensive copying of this thesis for scholarly purposes may be
granted by the Head of my Department or by his representatives.
It is understood that copying or publication of this thesis for
financial gain shall not be allowed without my written permission.
Department of
EUc^/cg.j
Engineering
The University of British Columbia,
Vancouver 8, Canada.
Date
y**\uoAy
ro ^
1
y
S7*3
ABSTRACT
A m u l t i s t a b l e composite volt-ampere
realized using
states
a number o f t u n n e l
c a n be o b t a i n e d
connected
i n series.
investigate
diodes.
using n suitably
and dynamic
P r e l i m i n a r y work d e a l s w i t h
tunnel
diode
figure
of merit
device
the
The
of the s t a t i c
characteristics
circuit
and t h e i r
characteristics
The
The t e m p e r a t u r e
Experimental
o f the composite
multistate
circuit.
diode
dependence o f t h e
results
speed o f 12.5 ns f o r a f o u r
are presented
state
circuit
diodes.
versatility
of the composite
the i n h e r e n t h i g h
speed o f t h e t u n n e l
the
multistate device
u s e f u l i n high-speed
counting.
as a
f o r the generation of
diode
and
such a s :
time on t h e
Additional conditions
o f a two t u n n e l
ratio.
a v a i l a b l e tunnel
of a
o p e r a t i o n are d e r i v e d from the study o f
i s also investigated.
showing an o p e r a t i n g
using
of such a c i r c u i t .
circuit.
c o n d i t i o n s d e r i v e d i n v o l v e the tunnel
capacitances
i s to
T h i s work s e r v e s
of the m u l t i s t a t e
to ensure proper
dynamic
diodes
the switching behaviour
r e q u i r e d number o f s t a b l e s t a t e s .
dynamic
study
characteristics
determines the c o n d i t i o n s necessary
necessary
stable
chosen t u n n e l
and t h e c u r r e n t o v e r d r i v e .
study
A maximum o f 2
and t h e dependence o f t h e s w i t c h i n g
background to the study
The
c a n be
The main p u r p o s e o f t h i s
the s t a t i c
single
characteristic
characteristic
diodes
obtained,
combine t o make
digital applications
b i n a r y a d d i t i o n , a n a l o g - t o - d i g i t a l c o n v e r s i o n .and
These a p p l i c a t i o n s a r e d i s c u s s e d
briefly.
ACKNOWLEDGEMENT
The a u t h o r i s i n d e b t e d
supervising
t o D r . M.P. Beddoes, t h e
professor
of this
project, f o r h i s help
throughout the course
of t h i s
research.
Grateful
Electric
the
Company
National
acknowledgement
Limited f o r a Fellowship
Research Council
The work d e s c r i b e d
National
i s given
to the Northern
awarded
f o r a Studentship
i n this
and guidance
i n 1961, and t o
awarded
t h e s i s was s u p p o r t e d
Research C o u n c i l under Grant
(BT<-68).
i n 1962.
by t h e
TABLE OF CONTENTS
Page
List
of I l l u s t r a t i o n s
List
of Tables
v
viii
Acknowledgement
List
of Special
ix
Symbols and Terms
x
1.
Introduction
2.
The T u n n e l D i o d e
as a N e g a t i v e R e s i s t a n c e D e v i c e .
2.1
Volt-Ampere
Diode
Characteristic
Temperature
Parameters
Dependence o f T u n n e l D i o d e
2.2
2.3
3.
3.2
3.3
4.
of a Tunnel
.
3
5
D-C
10
Tunnel Diode E q u i v a l e n t
Single
3.1
1
Tunnel Diode
Circuit
IT
Switching Behaviour
.........
13
Dependence o f t h e F i g u r e o f M e r i t on t h e
E l e c t r i c a l P a r a m e t e r s o f t h e J u n c t i o n ......
14
Tunnel Diode
16
S w i t c h i n g Speed
Case
I:
Current Bias
18
Case
I I : Voltage Bias
23
Approximate Formula f o r the Rise
Tunnel Diode
Time o f a
23
3.4
Experimental Results
25
3.5
Summary
27
S t a t i c and Dynamic C h a r a c t e r i s t i c s
Connected Tunnel Diodes
4.1
Static Characteristic
Connected i n S e r i e s
of S e r i e s 28
o f Two T u n n e l D i o d e s
C h a r a c t e r i s t i c s of n Tunnel Diodes
28
4.2
Static
...
33
4.3
Experimental Results
34
4.4
Dynamic C h a r a c t e r i s t i c s
36
Page
4.5
Two T u n n e l D i o d e s M u l t i s t a t e . C i r c u i t
4.6
Switching Behaviour
4.7
Effect
o f C a p a c i t a n c e on T r a n s i e n t B e h a v i o u r
4.8
Effect
o f t h e D-C P a r a m e t e r s
Diodes
on t h e S w i t c h i n g B e h a v i o u r
Effect
of Inductance
4.9
. ....... .
............oo........a...
39
.
............
on T r a n s i e n t B e h a v i o u r
48
on t h e C i r c u i t O p e r a t i o n
51
4.11 E f f e c t
o f Temperature
Tunnel Diodes
i nSeries
...-
4.13 E x p e r i m e n t a l R e s u l t s
5.
6
•
47
...
o f I n p u t P u l s e R i s e Time on T r a n s i e n t
4.14 Summary
43
of the Tunnel
4.10 E f f e c t
4.12 E x t e n s i o n t o Three
37
52
53
...,•«.«»»'••..*...».
. . o . o . o o . . . . . . . . . .
56
Applications
57
5.1
Full
57
5.2
Analog-to-Digital
3*3
C o"u.n."t G
G one
B i n a r y Adder
1 u s i o UH
Appendix
I
i*
«•-«-««
#«
Converter
59
* * * # « » » » o « » » * « « « o f t » » o » o « « o « e « » #
• « • • . • « , * • • * . .oo«<«
* • « • « • •
o * o o o o « » a * « « » * o « 0 «
Measurement o f T u n n e l D i o d e
P a r a m e t e r s -'•••»•...
AI.1
Bias C i r c u i t
and S t a b i l i t y
AI.2
Tunnel Diode
T e s t Mount
AI.3
E x p e r i m e n t a l Measurements and R e s u l t s
Appendix
II
..................
Methods o f A p p r o x i m a t i n g 'Tunnel
.......
64
66
70
A l l . 2 Two Term E x p o n e n t i a l A p p r o x i m a t i o n
Appendix I I I F a c t o r s I n f l u e n c i n g the Choice of the
Load L i n e R e s i s t a n c e R i n a M u l t i s t a t e
References
64
Diode
Polynomial Approximations
C X I* C U. 1 ~b
62
66
Curves
AII.l
6 0 -
e » « o » e - o * e » o « « o o » o » o A « » « o « f r o o o o o » *
............................................
71
73
73
77
L I S T OP
ILLUSTRATIONS
Figure
2-1.
Page
Energy-band
at
Thermal
Scheme a t J u n c t i o n
Equilibrium
Characteristic
Diode
..<,....
2-2.
D-C
2-3.
Components o f t h e V o l t - A m p e r e
a Tunnel Diode
V a r i a t i o n o f T u n n e l D i o d e D-C
2-4.
of a Tunnel
of a Tunnel Diode
5
Characteristic
of
8
Parameters
with
Temperature
10
2- 5.
Equivalent
Circuits
3- 1.
a)
Tunnel Diode E q u i v a l e n t
Single
4
of Tunnel Diodes
11
Circuit
.....
b)
3-2.
3-3.
3-4.
3-5.
3-6.
Tunnel Diode C h a r a c t e r i s t i c I l l u s t r a t i n g
S w i t c h i n g Load Line
Switching Transient v ( t ) f o r Various Overdrive
Factors
16
21
N o r m a l i z e d S w i t c h i n g and D e l a y Time V e r s u s
Overdrive
21
T i m i n g D e l a y as a F u n c t i o n o f T u n n e l
C a p a c i t a n c e and Peak C u r r e n t
22
Diode
Dynamic v - i T r a n s i e n t B e h a v i o u r f o r L = 1,
10
100 nh
24
3-7.
Tunnel Diode
3- 8.
E x p e r i m e n t a l S w i t c h i n g Waveforms
4- 1.
Linearized
Switching Test C i r c u i t
Characteristic
Curve
25
26
of a Tunnel
Diode
4-3.
20
S w i t c h i n g Time, D e l a y Time and R i s e Time V e r s u s
Overdrive
...
and
4-2.
16
30
a)
Tunnel Diodes
b)
Composite
a)
Experimental; C h a r a c t e r i s t i c s of the.Negative
R e s i s t a n c e Elements
Two E l e m e n t s Composite C h a r a c t e r i s t i c
b)
Individual
Characteristics
...
Characteristic
30
30
35
35
Figure
,
Page
c)
Three
Elements Composite C h a r a c t e r i s t i c
d)
Four Elements Composite C h a r a c t e r i s t i c
4-4.
Two
Tunnel
4-5.
a)
Dynamic v - i C h a r a c t e r i s t i c s
Diode{j M u l t i s t a t e
Multistate
4-6.
Circuit (00—
and
Voltage
a)
Dynamic v - i C h a r a c t e r i s t i c s
C i r c u i t (00 —
and
a)
Dynamic v - i C h a r a c t e r i s t i c s
b)
Voltage
of
of
38
the
(00 —
Ol)
...
40
the
(00—10)
...
42
the
11)
44
C u r r e n t Waveforms
of Capacitance
..........
42
C u r r e n t Waveforms
and
35
10)
Voltage
Circuit (00—
....
40
C u r r e n t Waveforms
b)
Multistate
of
35
Ol)
b)
Multi/state
4-7.
Circuit
...
(00—
l l ) ...
on T r a n s i e n t B e h a v i o u r
44
4-8.
Effect
..
46
4-9.
4-10.
4-11.
Dynamic v - i C h a r a c t e r i s t i c f o r (5yl
= 0.32 ma .
E f f e c t of I n d u c t a n c e on T r a n s i e n t B e h a v i o u r ...
E x p e r i m e n t a l V o l t a g e Waveforms f o r a Two
Tunnel Diode C i r c u i t
48
49
4- 12. E x p e r i m e n t a l V o l t a g e Waveforms f o r a
Tunnel
Diode
Circuit
5- 1.
Full
AI-1.
a)
Tunnel
b)
Equivalent C i r c u i t
Three
.
B i n a r y Adder
AI-2.
Tunnel
AI-3.
Diode
Diode
54
55
58
Test C i r c u i t
65
65
C o a x i a l Mount
....................
67
a)
D i a g r a m o f t h e T e s t C i r c u i t ...............
67
b)
Bridge E x t e r n a l Connections
...............
- 67
AI-4.
A d m i t t a n c e C h a r a c t e r i s t i c s of a IN2939 D i o d e
as a F u n c t i o n o f V o l t a g e . . . . . . . . . . . . . . . . . . . . . .
A I - 5 . C a p a c i t a n c e V a r i a t i o n as a F u n c t i o n o f
Voltage
69
A I I - 1 . T1925 Germanium T u n n e l
72
Diode
69
Figure
Page
a)
Actual
and C a l c u l a t e d
Characteristics
b)
P e r c e n t E r r o r Between t h e A c t u a l
Calculated Characteristics
......
72
and
72
L I S T OF TABLES
Table
2.1
.
Page
Properties
Materials
of Tunnel Diodes
Semiconductor:
Used i n T h e i r F a b r i c a t i o n
4.1
D-C
4.2
Temperature C o e f f i c i e n t s
for
and
Parameters
6
o f t h e Two D i o d e s
o f t h e D-C
39
Parameters
t h e Ge and GaAs T u n n e l D i o d e s
4.3
Stable
S t a t e s f o r Three
5.1
Truth Table of F u l l
AII.l
Polynomial Approximations f o r
All.2
Exponential Approximations
52
Tunnel Diodes Device
..
Binary Addition
Tunnel Diodes
53
58
....
f o r Tunnel Diodes
..
72
74
L I S T OF SPECIAL SYMBOLS AND TERMS
Symbol
First
Defined
i n Section
V
p
=
D i o d e Peak V o l t a g e
2.1
V
y
=
Diode V a l l e y V o l t a g e
2.1
V^ =
Diode Forward V o l t a g e
2.1
Ip =
D i o d e Peak C u r r e n t
2.1
ly
-
Diode V a l l e y
m
=
Reduced
P
Current
...................
E l e c t r o n mass
(relative)
2.1
ina
S e m i c o n d u c t o r M a t e r i a l .................
2.1
=
E n e r g y Gap i n a S e m i c o n d u c t o r M a t e r i a l .
2.1
=
Relative Dielectric
2.1
=
Tunneling P r o b a b i l i t y
2.1
f(v) =
Diode v - i C h a r a c t e r i s t i c
2.3
C(v) =
Diode C a p a c i t a n c e
E
e
Constant
...........
r
Z
Cy =
|r|
C/lp
=
-
C|r| =
T
=
2
Diode V a l l e y
...... 2.3
Capacitance
Magnitude o f the Diode
Resistance
2.3
Negative
3.
Figure of merit
3.
F i g u r e o f m e r i t *.
Precursor Pulse Duration
3.
4.6
r
()I
V
=
D i f f e r e n c e i n the V a l l e y
the Two D i o d e s .. •.
Term
Percent Overdrive:
R i s e Time:
D e l a y Time:
S w i t c h i n g Time:
Precursor Pulse:
Currents of
4.8
3.2
3.2
3.2
3.2
4.6
1
THE
STATIC AND
DYNAMIC CHARACTERISTICS OF
TUNNEL DIODES AND
THEIR APPLICATIONS IN DIGITAL CIRCUITS
1.
A great v a r i e t y
made f r o m
SERIES-CONNECTED
INTRODUCTION
of n e g a t i v e
r e s i s t a n c e d e v i c e s have b e e n
semiconductor m a t e r i a l s ; a recent
one
i s the
tunnel
diode.^
I n the
shown i n the
reason
past
use
for this
of t u n n e l
interest
advantages:
high
simplicity,
small
environmental
s e v e r a l years, widespread
diodes
lies
i n the
s p e e d , low
size,
high
and
the
use
advent of the
tunnel
of the volt-ampere
characteristics.
composite
characteristics
the mode o f c o n n e c t i o n .
l e a d to a host
performance
combination
was
found
individual
The
The
inherent,
power d i s s i p a t i o n ,
stability
with
device
changes i n
and
nuclear
i n o b t a i n i n g g a i n and
r e v i v e d the
characteristics
The
radiation.
direction-
of s e v e r a l
form and
the
2
n
of
the
devices
and
complex c h a r a c t e r i s t i c s
of d i g i t a l
logic
stable states,
main p u r p o s e o f t h e
generated
resistance devices
functions.
resistance devices
obey c e r t a i n
in
inter-
complexity
individual
of ways of u s i n g n e g a t i v e
to generate
interest
t o generate- m u l t i s t a b l e
depend on the
The
of n n e g a t i v e
devices
diode
resistance devices
composite
i n the
been
device.
connected negative
can
has
circuits.
diode's
c o n d i t i o n s s u c h as t e m p e r a t u r e
u s i n g the
The
tunnel
switching
There a r e , however, d i f f i c u l t i e s
ality
in digital
interest
One
interesting
connected i n
(2)
' provided
series
the
rules.
f o l l o w i n g study
i s to i n v e s t i g a t e
2
the
static
using
and
tunnel
The
tunnel
device.
these
and
second
The
circuit
consists
element.
diode
negative
and
The
w i t h the
operation
section deals
solutions.
characteristics
First,
s u p p o r t e d by
that
the
of the
satisfy
the
non-linear
investigation
the
analysis
Next, the
circuit
a
negative
as
i t s volt-ampere
discussed
l i m i t a t i o n s as
a
by
computer
w i t h the
o f the
system, and
derived
briefly.
single
switching
s o l u t i o n of
used to v e r i f y
i n the
provides
n e x t two
static
and
tunnel
diode
results, establishes
sections.
dynamic
circuit.
the
i n d i v i d u a l tunnel
required
relationship
d i o d e s must
multistable
composite
dynamic b e h a v i o u r o f a t u n n e l
verified
by
a computer
static
relations
analysis
diode
simulation
experiment.
some a d d i t i o n a l
from the
the
composite c h a r a c t e r i s t i c ,
i s i n v e s t i g a t e d by
establishes
conditions
first
a
discussion
i n o r d e r to g e n e r a t e the
multistate
are
r e s u l t s are
p a r a m e t e r s o f the
characteristic.
and
diode
The
behaviour of
series-connected
experimental
static
elements.
investigation in this section
section deals
a graphical
combination
device,
circuit
i s c a r r i e d out
a n e c e s s a r y b a c k g r o u n d t o the
third
the
w i t h the
Experimental
The
tunnel
of
its possibilities
equations.
The
resistance
i t s equivalent
analysis
such a
o f f o u r main s e c t i o n s .
section deals
characteristic,
The
c h a r a c t e r i s t i c s of
d i o d e s as
thesis
introductory
resistance
dynamic
to
of
This
complementing
ensure
proper
operation.
The
tions
fourth
of the
digital
and
last
multistate
functions
conversion,and
section deals
with possible
composite d e v i c e s
s u c h as b i n a r y - a d d i t i o n ,
counting.
i n the
applica-
performance
analog-to-digital
of
3
2.
THE
The
TUNNEL DIODE AS
tunnel
diode
made o f v e r y h i g h l y
a negative
voltage
A NEGATIVE RESISTANCE DEVICE
i s e s s e n t i a l l y a narrow p-n
doped s e m i c o n d u c t o r m a t e r i a l ,
resistance
over a l i m i t e d
range when b i a s e d
i n the
(several
junction,
which
tenths
exhibits
of a v o l t )
forward d i r e c t i o n .
The
(3)
negative
resistance
f r o m the
so
i s of the
non-parametric
called "tunneling"
type
and
mechanism, a s t r i c t l y
arises
quantum
mechanical e f f e c t .
I n a normal r e c t i f y i n g
p-n
diode,
conduction i n
f o r w a r d d i r e c t i o n o c c u r s p r e d o m i n a n t l y by
minority
consist
and
c a r r i e r s across
the
p-n
w h i c h t r a v e l i n the
contributions,
because of
charge, c o n s t i t u t e
the
Angstroms) w h i c h i s h e a v i l y
F e r m i > l e v e l Ep,
gap,
as
f o r the
p -
the
shown i n F i g u r e
i t gives
called."tunnel
the
within
2-1•
by
This
rise
e f f e c t " or
(4)
Zener
v
.
to
vice versa
forbidden
u n d e r the
gap
same
( o f the
the
a new
overlap
Electrons
Both
sign.
valence
the
A of
n the
emission
a c t i o n of
empty or
a large
band
side
band
the
energy s t a t e
in
"tunnel"
c o n d u c t i o n band
electric
so
first
changing t h e i r e n e r g y s t a t e ,
i n t o the
150
forbidden
c o n d u c t i o n mechanism:
i n a given
the
impurities,
c o n d u c t i o n band on
c a u s e s an
side,
o r d e r of
the
within
internal field
v a l e n c e band c a n , w i t h o u t
t h r o u g h the
of the
of f a l l i n g w i t h i n
and
as
considered
instead
n -
s i g n of
doped w i t h c o n t r o l l e d
diode, f a l l s
the
e d g e s , and
side
junction
to the
direction.
i n the
current
rectifying
on
these c a r r i e r s
opposite
a conventional
d i f f u s i o n of
p - side
difference
However, i n a n a r r o w p-n
the
junction:
o f h o l e s w h i c h d i f f u s e from the
electrons
the
the
field.
and
4
C o n d u c t i o n Band
Electron
Energy
cn
p Valence
tunneling
the
band o v e r l a p
the
two
must be
this
A.
are
subtracted
only
Since
(due
to
a p p l i e d , the
net
a forward voltage
case the
On
the
in
charges)
In p a r t i c u l a r , w i t h
zero;
no
therefore
t h o u g h e a c h component
hand, the
components are
net
the
also
current
overlap,
zero.
in
involved,
o p p o s i t e l y moving
which j u s t destroys
individual
s i g n are
c u r r e n t must be
other
involved
i n states included
c h a r g e s o f one
from each o t h e r .
i s non-zero.
electrons
those f a l l i n g
c a n c e l l a t i o n must t a k e p l a c e ,
itself
for
process
type
E n e r g y - b a n d Scheme a t J u n c t i o n
o f a T u n n e l D i o d e a t Thermal
Equilibrium.
components o f c u r r e n t
external bias
exact
2-1,
m e n t i o n e d above, the
the
-
Transition
Region
Band
Figure
As
n
type-
For
by
vanishes
but
in
5
intermediate v o l t a g e s , a net forward
rates
c u r r e n t flows because the
o f change o f t h e s e two components w i t h b i a s v o l t a g e
differ.
Thus f o r i n c r e a s i n g b i a s v o l t a g e , t h e c u r r e n t i n c r e a s e s , r e a c h e s
a peak*" and t h e n d e c r e a s e s
producing
a negative
incremental
resistance.
2.1
Volt-Ampere
The
diode
Characteristic
low f r e q u e n c y
junction
It differs
i n that
from
i texhibits
the n e g a t i v e r e s i s t a n c e .
currents
r a n g i n g from
a g i v e n semiconductor
of a t y p i c a l
to a large
device,
that
o f any o t h e r t y p e
tunnel
o f p-n
a "hump" o f c u r r e n t g i v i n g
Tunnel
material,
However, f o r
t h e v o l t a g e s c a l e must
e x t e n t , and i t i s d e t e r m i n e d
Hence, t h e t u n n e l d i o d e
t h e power range
r i s e to
d i o d e s have been made w i t h peak
10 microamperes t o 10 amperes.
of t h e s e m i c o n d u c t o r .
its
characteristic
Diode.
i s shown i n F i g u r e 2-2 t o g e t h e r w i t h t h e i m p o r t a n t d-c
parameters.
fixed
of a Tunnel
remain
by t h e e n e r g y gap
i s a low v o l t a g e
o f w h i c h c a n be e x t e n d e d
o n l y by i n c r e a s i n g
current.
V
j
p
= Peak V o l t a g e
V„ = Valley
1
Voltage
P
Figure
2-2.
D—C C h a r a c t e r i s t i c
of a Tunnel
Diode
Tunnel
conductor
d i o d e s have b e e n made from
materials.
Table
2.1
lists
some of t h e i r p r o p e r t i e s . ^ ^'
a number of
five
of them t o g e t h e r
with
' ^^
P r o p e r t i e s of T u n n e l
Diodes ( T y p i c a l V a l u e s )
P r o p e r t i e s of S e m i c o n d u c t o r
Materials
*
m
E (ev.)
e
g
r
Type of
semi-
I /I
r
P
V
1
GaAs
1.35
0.13
11.1
40
Si
1.11
0.78
11.2
4
Ge
0.67
0.44
16
15
0.48
InAs
0.33
0.051
11.7
12
0.25
InSb
0.18
0.028
15.9
10
0.14
m^. = r e d u c e d
E
= energy
e l e c t r o n mass
0.7
(relative)
gap
o
e
= relative
Table
2.1
dielectric
P r o p e r t i e s of T u n n e l D i o d e s
Used i n T h e i r F a b r i c a t i o n
Tunneling,as
treated
constant
and
a quantum m e c h a n i c a l
i n considerable d e t a i l
Semiconductor
p r o c e s s , has
Materials
been
i n the l i t e r a t u r e , ^ ^ '
'
'
(9)
and
details
voltage,
only a b r i e f
discussion will
o f the v o l t - a m p e r e
a h i g h c u r r e n t flows through
is
injection.
unexpectedly
First
has
curve.
the d i o d e
to e x p l a i n
A t low
due
to
two
v o l t a g e ranges,
been c a l l e d
the
the
forward
band-to-
high v o l t a g e , current flows
Between t h e s e
h i g h and
g i v e n here
characteristic
band t u n n e l i n g ; a t s u f f i c i e n t l y
forward
be
by
the c u r r e n t
"excess" c u r r e n t .
c o n s i d e r t h e t u n n e l i n g c u r r e n t ; ^ ^ as
stated
p r e v i o u s l y , t h i s c u r r e n t c o n s i s t s of two components f l o w i n g
a c r o s s the j u n c t i o n i n o p p o s i t e d i r e c t i o n s ; I
from the v a l e n c e
1
t o t h e c o n d u c t i o n band, and I
t o the v a l e n c e band.
proportional
the
The c u r r e n t
band.
f (E)
from the c o n d u c t i o n band
a t any e n e r g y l e v e l
E is
i n t h e c o n d u c t i o n band,
i n t h e v a l e n c e band, and t h e
p r o b a b i l i t y from t h e c o n d u c t i o n band t o t h e v a l e n c e
Letting p
densities
I
t o t h e number o f e l e c t r o n s
number o f a v a i l a b l e s t a t e s
tunneling
flowing
(E) and p
(E) r e p r e s e n t
the energy
i n t h e c o n d u c t i o n and v a l e n c e bands
state
respectively,
and f ( E ) t h e c o r r e s p o n d i n g F e r m i d i s t r i b u t i o n f u n c t i o n s
c
y
d e n o t i n g the p r o b a b i l i t y t h a t
a given
and
p r o b a b i l i t i e s i n the two d i r e c t i o n s ,
Z
and Z
v c
c v
the t u n n e l i n g
Then f (E) P (E) = d e n s i t y
c
c
of conduction-band e l e c t r o n
occupied
(l
- f
(E)) P
(E) = d e n s i t y
energy s t a t e i s occupied,
states
i n dE,
of • -^al-easce$'.-band e l e c t r o n
states unoccupied
i n dE.
Thus
E
I
cv
= A
I
E
Z
c y
p ( E ) f ( E ) ( l - f (E)]P (E) dE
c
c
cn
where A i s t h e j u n c t i o n a r e a ;
v
v
...(2-1)
and t h e i n t e r v a l o f i n t e g r a t i o n i s
from t h e c o n d u c t i o n - b a n d edge o f t h e n - s i d e E cn', t o t h e
v a l e n c e band edge o f t h e p - s i d e E
overlap
(Figure
2-1).
, i . e . , t h r o u g h the band
Similarly
E
I
=
V
r
A
P
/
vc
Z
y c
P (E) f ( E ) [ l - f (E)]P (E) dE
v
y
c
c
J E cn
The
general
net
tunneling
...(2-2)
shape of the
6
c u r r e n t ' (I
two
cv
tunneling
- I
vc
) are
components as w e l l
shown i n F i g u r e
5
as
the
2-3.
rtcultc.nt
~T
'1
(1) dependence of the
voltage
Tunnel Current
on
(2) dependence o f the
voltage
D i f f u s i o n Current
(3) dependence o f the E x c e s s C u r r e n t
voltage
Figure
Next
forward
2-3.
consider
voltages
i n the
d e g e n e r a t e donor l e v e l s
energies
the
greater
p - side.
on
oh
Components of the V o l t - A m p e r e
C h a r a c t e r i s t i c of a T u n n e l D i o d e .
the
excess c u r r e n t which occurs
range where the
i n the
than those
at
e l e c t r o n s i n the
n - s i d e have been r a i s e d t o
o f the
Ideally, tunneling
degenerate acceptor
of e l e c t r o n s from the
levels
in
conduction
to
the v a l e n c e
should
the
t h e n be
forward
practice
at
band, i n a s i n g l e e n e r g y - c o n s e r v i n g
impossible
injection
current:
make up
and
the
A
interband
on
t h a t o f the
the
n o t e d , the
The
since
flow.
In
actual
normal
current
diffusion
excess c u r r e n t i s
i t s behaviour
peak c u r r e n t ;
the
peak and
same dependence on p r e s s u r e ,
d o n o r and
acceptor
to
excess
on
concentrations
that
j u n c t i o n . ^
few
h y p o t h e s e s have been p r o p o s e d
tunneling
transitions
i s now
excess c u r r e n t
i s c a u s e d by
t h r o u g h the
but
localized
e n e r g y gap.
subsequently,
i n the
o n l y p a r t o f the way,
imperfection
T h i s mechanism was
o f i t , by
can
be
content
o f the
excess current
energy l e v e l s
first
'(l^)
the
range;
that
e l e c t r o n s t u n n e l i n g not
s e v e r a l authors
current
to e x p l a i n
a g r o w i n g amount of e v i d e n c e
e n e r g y gap,
more or l e s s
confirmation
c u r r e n t due
should
i n e x c e s s o f the
t u n n e l i n g process
however, t h e r e
the
carriers
first
e x h i b i t much o f the
temperature,
diffusion
term excess c u r r e n t .
t o the
generally parallels
currents
Esaki
i s considerably
hence the
p r i m a r i l y due
o n l y the
of m i n o r i t y
as Y a j i m a and
such b i a s e s
and
transition,
completely
making use
present
have o b t a i n e d
crystal,
d e l i b e r a t e l y c h a n g i n g the
e i t h e r by
of
in
s u g g e s t e d by E s a k i
and,
strong
showing t h a t the magnitude of the
a l t e r e d by
the
excess
imperfection
s u i t a b l e d o p i n g or by r a d i a t i o n
damage.
B a s e d on
curve
(3) F i g u r e
a theoretical
2-3
represents
a n a l y s i s of
an
s u c h a mechanism,
approximation adopted
by
(9)
Kane
f o r the
applied
t o the
dependence o f the
tunnel
diode.
excess
current
on
the
voltage
10
2.2
Temperature
The
Figure
2-4
temperature
Dependence o f T u n n e l D i o d e
t u n n e l d i o d e d-c p a r a m e t e r s
shows t h e v a r i a t i o n
D-C
are temperature
o f t h e d-c
parameters
f o r a T1925 Germanium t u n n e l d i o d e .
peak c u r r e n t d e c r e a s e s by 4?o as the t e m p e r a t u r e
room t e m p e r a t u r e
to-6D°C.
temperatures.
(13)
'
The
i s lowered
is
The
mv/°C and
the
valley
i n energy
gap
lowered,and
out of the Fermi f u n c t i o n a t h i g h
peak v o l t a g e Vp
and
forward voltage
decrease n e a r l y l i n e a r l y w i t h i n c r e a s i n g temperature
about 0.06
from
temperature.
o f the i n c r e a s e
and d e c r e a s e i n t u n n e l i n g as t h e t e m p e r a t u r e
of the smearing
with
shown t h e
temperature.
as shown, i n c r e a s e s m o n o t o n i c a l l y w i t h
T h i s b e h a v i o u r i s expected because
because
As
dependent.
S l i g h t l y b e y o n d room t e m p e r a t u r e ,
peak c u r r e n t d e c r e a s e s w i t h i n c r e a s i n g
current,
Parameters.
1.0
mv/°C r e s p e c t i v e l y .
at rates
of
v ( m t
\
11
2.3
Tunnel Diode E q u i v a l e n t
The
resistance
tunnel
device
diode
negative
bulk
tunnel
diode
and
2-5(a).
negative
a p p l i c a t i o n s , the
c h a r a c t e r i z e d by
i t s housing
elements
of F i g u r e
in series:
and
generator;
no
delay
and
its differential
an
C(v)
tunnel
diode.
effects
are
j u n c t i o n capacitance
mechanism i n t h i s
(8)
the
The
large
c o n s i s t s of
a resistance r,
d
s h u n t e d by
non-linear
The
shown i n
thesis,
2-5(b),
L,,
d'
series
s w i t c h i n g mode
extended.
inductance
C,
as
L
of t h i s
shown i n F i g u r e
f ( v ) r e p r e s e n t i n g the
of t h e
the
storage
2-5(a) was
dependant c a p a c i t a n c e
characteristic
fast
inductance
t h e main c o n c e r n
equivalent c i r c u i t ,
current generator
the
signal
However, f o r l a r g e s i g n a l
circuit
a voltage
very
small
controlled
resistance ( - r ) , i t s j u n c t i o n capacitance
equivalent
three
For
is satisfactorily
a p p l i c a t i o n s , which are
signal
i s a voltage
(VCNR).
r e s i s t a n c e r , and
Figure
Circuit.
a
non-linear
d-c
t u n n e l i n g phenomenon i s
i n c l u d e d i n the
current
i s assumed t o r e p r e s e n t a l l
device.
1
(a)
Small
Signal
Figure
2-5.
(b)
Equivalent Circuits
Large
of T u n n e l
Signal
Diodes,
12
Measurements o f t h e
tunnel diode
p a r a m e t e r s have been d e a l t w i t h
(14),(15),(16)
o n
iy
a
e x t e n s i v e l y i n the
variation
results
are
=
C
be
and V
literature;
the
i n Appendix
Theoretically,
approximated
I
the
by:
-n
n=-2-
Q
s
where CQ i s t h e
i s considered
included).
o f C w i t h v o l t a g e may
C(v)
circuit
method aimed a t m e a s u r i n g
capacitance v a r i a t i o n with voltage
(experimental
equivalent
capacitance
i s the v o l t a g e
gap
of the
...(2-3)
tunnel diode
or the v o l t a g e
at zero v o l t a g e ,
f o r which
the
O
c a p a c i t a n c e would t h e o r e t i c a l l y
results
indicate
It w i l l
be
g i v e n by
the
The
= o
c
the v a l l e y
v -
_ _
—
experimental
and
capacitance
1
n- 2
p a r a m e t e r i n the
capacitance,
Cy
0.5.
as
.... (2-4)
equivalent
i s the n o n - l i n e a r c u r r e n t
i characteristic.
theoretical^'^
of a n a l y t i c a l
Appendix I I .
1 -
most i m p o r t a n t
r e p r e s e n t i n g the
the use
The
following equation:
a p a r t from the
satisfactory
to i n f i n i t y .
exponent n v a r i e s between 0.4
u s e f u l to d e f i n e here
M V
f(v)
t h a t the
go
1 7
^'^
1 8
approximations
^
The
circuit,
generator
l a c k of a
e x p r e s s i o n f o r f ( v ) l e d to
which are d e a l t w i t h
in
13
SINGLE TUNNEL DIODE SWITCHING BEHAVIOUR
3.
As
a first
the p e r f o r m a n c e
possibilities
is
developed
static
s t e p , i t was t h o u g h t w o r t h w h i l e t o a n a l y s e
of a s i n g l e
tunnel diode
as a s w i t c h i n g d e v i c e .
Against
t h e main s u b j e c t o f t h e t h e s i s :
and dynamic b e h a v i o u r
series-connected
tunnel
S t a r t i n g with
Experimental
one t u n n e l
p r e d i c t e d b y t h e computer
for
diode,
a treatment
system was c a r r i e d o u t .
solutions.
the general
expressed
as e i t h e r
of merit
the C/lp
where C i s t h e c a p a c i t a n c e ,
negative
/
r e s i s t a n c e and I
a useful rule
The
,
v
' of the diode
or C | r | constant
of merit
p a r a m e t e r s s u c h as d o p i n g ,
first;
without
f i g u r e of
i s usually
of the diode;
| r | i s t h e magnitude o f t h e a v e r a g e
i s t h e peak c u r r e n t .
by t h e f o l l o w i n g e q u a t i o n
figure
o f thumb
(22)
r e l a t i o n between t h e C | r | and C/l-p
approximately
behaviour
The m a t h e m a t i c a l a n a l y s i s
o p e r a t i o n and p r o v i d e s
The f i g u r e
of
computer
t h e s w i t c h i n g t i m e as a f u n c t i o n o f t h e d e v i c e ' s
merit.
of the
combination
a digital
to v e r i f y
(2l)
the
background
diodes.
r e s u l t s were f o u n d
diode
this
o f an i n t e r e s t i n g
s i m u l a t i o n of the p e r t i n e n t n o n - l i n e a r
clarifies
and i n v e s t i g a t e i t s
Empirically,
constants i s g i v e n
( f o r a Ge t u n n e l
diode):
and i t s dependence on v a r i o u s
e n e r g y gap e t c . , w i l l
the subsequent a n a l y s i s w i l l
be p u r e l y
be
considered
mathematical,
any p h y s i c a l c o n s i d e r a t i o n o f t h e d e t a i l e d mechanism o f
14
tunneling.
T h i s m a t h e m a t i c a l a p p r o a c h may
b e c a u s e the
time constant
of t u n n e l i n g
(23 )
dielectric
t h a n the
r e l a x a t i o n time
equivalent
circuit
of
f r e q u e n c y range of
3.1
(Figure
2-1)
justified
the
sec.)
i s much
is valid
smaller
In other
o v e r the
words
the
wide
interest.
tunneling
i n any
F i g u r e of M e r i t
Junction.
current
on
the
Electrical
t h r o u g h a narrow p-n
semiconductor i s p r o p o r t i o n a l
probability Z = Z
tunneling
d e t e r m i n e d by
(l0~^ sec).
s e c t i o n 2.3
Dependence of t h e
Parameters of the
The
entirely
-13
(10
C | r | time c o n s t a n t
be
= Z
V c
(see
equation
junction
to
the
2-1).
The
cv
(9)
p r o b a b i l i t y Z has
tunneling
been e v a l u a t e d
e l e c t r o n t r a n s i t i o n s across
terms o f F,
the
e n e r g y gap
electron m
and
i s given
Z =
a region
E
and
v
' for individual
of constant
the
field
r e d u c e d mass o f
F,
in
the
by:
TC
exp
m
*4
e h
E
3/2
n
...(3-1)
F
where
h = Planck's
constant,
e = e l e c t r o n i c charge.
Defining
abrupt
an
average F by:
junction i s given
W
=
We
F = E
, where the w i d t h W
by:' "*'
2
N + P
NP
2 e
E
of
an
15
e being
the d i e l e c t r i c
constant,
and N , P t h e d o p i n g
t h e n and p s i d e s o f t h e j u n c t i o n r e s p e c t i v e l y .
for F i n equation
3-1
eh
NP
0
I
is
stands
p
e
~
A
W
a
are g i v e n by:
...(3-3)
'
= A Z
( e q s . 2-1,
f o r the t u n n e l i n g i n t e g r a l
the j u n c t i o n area.
...(3-2)
g
C and t h e peak c u r r e n t lp
r
/
Substituting
2
The j u n c t i o n c a p a c i t a n c e
where
on
we g e t :
VI u
exp
levels
2-2)
and A
Therefore:
e(NP)
(N + P) E
C/Ip-
exp
i3(m
a (
*
N + Pa „
e —Np-) E,
2
g
...(3-4)
E
where a and P a r e c o n s t a n t s .
For high l e v e l s
of doping,
becomes t h e dominant f a c t o r
a given
the
lower the f i g u r e
For
resistance
of m e r i t
concentration:
of merit
a given doping
factor
i n t h e above e x p r e s s i o n .
semiconductor the f i g u r e
depends on t h e d o p i n g
the exponential
Thus f o r
i s not a constant but
the higher
and t h e f a s t e r
the c o n c e n t r a t i o n
the s w i t c h i n g .
c o n c e n t r a t i o n , the negative
(-r) i s i n v e r s e l y p r o p o r t i o n a l t o t h e t u n n e l i n g
16
probability
Z and
depends o n l y t o a s m a l l
of s t a t e s .
Thus e q u a t i o n s
3-2
and
3-3
s m a l l band g a p s , s m a l l d i e l e c t r i c
masses have t h e
For
Table
2.1
which l i s t s
conductors,
the
3.2
purpose
lowest
we
can
f i g u r e s of
(Figure
diode
the
the
diode
tunnel
lead
semi-
the m a t e r i a l s
with
Speed.
by
to the v a l u e s
total
of s e v e r a l
InAs a r e
the
equivalent
an a m p l i f i e r ,
e x t e r n a l r e s i s t a n c e B,
the
effective
merit.
3 - 1 ( a ) ) c o u l d be
represents
small
with
constants.
t h a t InSb and
represented
element, a c c o r d i n g
density
of comparison, r e f e r r i n g back to
Tunnel Diode S w i t c h i n g
The
and
the m a t e r i a l c o n s t a n t s
see
on t h e
show t h a t m a t e r i a l s
constants,
s m a l l e s t C | r j time
the
extent
series
and
of the
the
oscillator
d-c
or
source
inductance
inductance
circuit
of the
L.
switching
voltage V Q ,
This
inductance
circuit including
inductance.
: \
*low^_
voltage
state
(b).
Tunnel Diode C h a r a c t e r i s t i c
S w i t c h i n g Load L i n e .
v.
high
voltage
state
Illustrating
t—r
17
The
the
stability
negative
the
switch
tunnel
diode
switching
characteristic.
i n three
one i s u n s t a b l e
d i c t a t e that
diode
total
circuit
current
tunnel
the v o l t a g e
The l o a d l i n e
i , f ( v ) and i
diode
i n t e r s e c t s the
(x,y)
are stable
as shown i n
d e s c r i b i n g the
circuit
l a r g e s i g n a l '..
are:
...(3-5)
L ~ = E - B i - v
...(3-6)
c
...(3-7)
= i - f(v)
terminal voltage
drop across
E = V + v ;
0
s'
n
s u p e r i m p o s e d on
C f | = i - f(v)
i
where v = the
load line
tunnel
^^l^min*
region provided
p o i n t s , two o f w h i c h
3-1 ( a ) ; t h e e q u a t i o n s
b e h a v i o u r o f the
C o n s e q u e n t l y the
(z).
D e f i n i n g the
Figure
tunnel
i f the
diode.
through t h i s
3 - 1 ( b ) shows the
characteristic
and
region
r e s i s t a n c e o f the
d'iode c a n o n l y
the
i s an u n s t a b l e
circuit
r e s i s t a n c e i s l a r g e r t h a n the minimum magnitude o f
negative
Figure
o f the
r e s i s t a n c e p o r t i o n o f the
characteristic
positive
(16)
criteria
the
o f an i d e a l
diode's
= d-c
diode
excluding
series resistance,
source
voltage,
= t r i g g e r i n g source
voltage,
18
f(v)
= a seventh
order polynomial
A I I - l ) which c l o s e l y
characteristic
C = tunnel diode
equal
It
fits
the observed v - i
o f a 1 ma Ge t u n n e l d i o d e
capacitance
t o the v a l l e y
(T1925),
assumed c o n s t a n t and
capacitance.
s h o u l d be m e n t i o n e d t h a t e q u a t i o n s
the
( g i v e n by e q u a t i o n
3-5, 3-6 a r e i n v a r i a n t t o
following substitutions:
C
= aC
and
or
L' = a
;
i '= ai
c
c
Q
•••
,
\J—O)
i '=ai
C
= bC
L ' = b L and t ' = b t
where a and b a r e n u m e r i c a l
with
R' = a
a diode
constants.
...(3-9)
T h e r e f o r e we may d e a l
f o r w h i c h C = 10 p f and l p = 1 ma w i t h o u t
loss of
generality.
In
in
the f o l l o w i n g a n a l y s i s ,
two d i f f e r e n t
modes:
t h e t u n n e l d i o d e was b i a s e d
a c u r r e n t mode R
c
(horizontal
= R » | rl
'
'mm
y
l o a d l i n e ) , and a v o l t a g e mode R = R "> I r I . .
»
e
V
/\ \
m
switching
i n both
analyzing
the r e s u l t s
*
of
c a s e s was i n v e s t i g a t e d .
i t will
associated with
o v e r d r i v e i s the excess
as a p e r c e n t
Case I :
Current
The
source
of the diode
be u s e f u l
i
The
n
However, b e f o r e
to d e f i n e the " o v e r d r i v e "
a given input signal.
The p e r c e n t
i n c u r r e n t a t the diode
peak v o l t a g e
peak c u r r e n t .
Bias
t u n n e l d i o d e was b i a s e d by a c o n s t a n t d-c c u r r e n t
at approximately
half
t h e peak c u r r e n t
.
Prior to
19
switching,
was
the
diode
a c c o m p l i s h e d by
current,
defined
parallel
with
the
When the
"small"
source
the
was
the
here
i n i t s low
a step-function current
power
diode
overall
describing
the
of the
does n o t
circuit.
In t h i s
n e t w o r k may
be
C
Switching
trigger
source
tunnel
affect
case
diode
the
(L=0,
expressed
source,
where
c
the
i s swamped by
dynamic b e h a v i o u r
R=R )>
C
i n the
the
the
of
equations
simplified
= i - f(v)
i
in
supply.
i s d r i v e n from a c u r r e n t
s e r i e s inductance
r e s i s t a n c e and
state.
a p p l i c a t i o n of a p o s i t i v e
as
d-c
voltage
form:
...(3-10)
= i - f (v)
i = I„ + i ; 1^ = d-c
0
s'
0
i
bias
= trigger
current,
'
current.
(28)
Equation
method u s i n g
the
waveforms of the
3-10
IBM
was
s o l v e d by
1620
computer.
voltage
across
the
the
Runge-Kutta-Gill
Figure
diode
3-2
shows
for various
the
overdrive
factors.
The
for
change
voltage
o f the
time
time
i s customarily
e i t h e r the v o l t a g e - i o r c u r r e n t - c h a n g e
total
the
switching
expected.
switching
switching
can
be
For
time
the
a
to r e a c h
as
90$
the
of
case under c o n s i d e r a t i o n
is significant.
transients i n Figure
divided into
defined
"delay
3-2
time",
A
time
the
only
closer investigation
shows t h a t the
w h i c h i s the
switching
time
20
f o r a 0 t o 10% change i n o u t p u t
required
voltage;
and a
"rise
t i m e " , w h i c h i s t h e time r e q u i r e d f o r a 10% t o 90% change i n
output
voltage.
time, r i s e
time
Figure
3-3
and d e l a y
T.D. V o l t a g e
shows t h e v a r i a t i o n
time w i t h
of
switching
overdrive.
(v)
#
1
' 2
3
4
5
6
5
Overd.
3.6%
S.6%
13.6%
18.6%
23„6ti
33.6%
10
6
Nanoseconds
Figure
3-2.
It
Switching Transient v ( t ) f o r Various
Overdrive
F a c t o r s . (C •= 10 p f , I
1 ma, B = B )
'
c
i s observed t h a t the v a r i a t i o n s i n o v e r d r i v e
apparent e f f e c t
of changing the d e l a y
of the output
have t h e
waveform t o
(oo)
a much g r e a t e r
the
fact
extent
that f o r small
s m a l l n e a r the d i o d e
rate
than the r i s e
overdrive
To make t h e c u r v e s
applicable,
respect
the switching
and t h i s
i n the i n i t i a l
of Figure
and d e l a y
to the c h a r a c t e r i s t i c
.
v
This
i s due t o
the c a p a c i t i v e c u r r e n t i s
peak v o l t a g e ,
o f change o f v o l t a g e
time
results
stage
i n a slow
o f the t r a n s i e n t .
3-3 more g e n e r a l l y
t i m e s were n o r m a l i z e d
C | r | time
constant
with
and p l o t t e d
iTime
(ns)
t k Normalized
C Irl
61
IO'/-
207% Current
Figure
3-3.
3 0 /
46'/-
Overdrive
S w i t c h i n g Time, D e l a y Time and
R i s e Time v e r s u s O v e r d r i v e
(C = 10 p f , I = 1 ma, R = R,)
p
Time
22
versus
overdrive
on a l o g - l o g
C | r | time c o n s t a n t
its
The i m p o r t a n c e
delay
3-4.
G i v e n the
c a n deduce f r o m F i g u r e
f o r a given overdrive
of the d e l a y
switching
circuits
time f o r 10fo o v e r d r i v e
and peak c u r r e n t .
has a d e l a y
of 2 ns.
factor.
i s evident.
Figure
F o r example
3-5
shows t h e
diode
a 10 p f , 1 ma
(ns = n a n o s e c o n d = 10 ^ s e c . )
T>dofj hm* (ns) .
i
/
—
—
Jr i".
9,1
A
— -
0,61
--jA-
— /
/
0,00/
—
-
10
_.
/oe
Capacitance
Figure
3-5.
3-4,
time i n t h e o p e r a t i o n o f
as a f u n c t i o n of t h e
capacitance
10
i n Figure
of a Ge d i o d e , we
s w i t c h i n g and d e l a y t i m e s
tunnel diode
scale
T i m i n g D e l a y as a F u n c t i o n o f T u n n e l
D i o d e C a p a c i t a n c e and Peak C u r r e n t .
diode
23
Case I I :
Voltage
In t h i s
Bias
case,
R and
were
solved for various values
equations
3-5
and
Switching
was
accomplished
trigger voltage
(0.5
ns
rise
supply.
The
the
time and,
the
by
time) v o l t a g e
source
behaviour
found
as
exhibit
c u r r e n t - b i a s case.
The
rise
3-6
inductance
i n c r e a s e the
t i m e by
an
appreciable
It
s h o u l d be
throughout
i t was
a c c o u n t had
found
f o l l o w s the
henry).
delay
The
t h a t the
analysis.
effect
approximate formula
d e r i v e d under the
o f L.
load l i n e
time and,
t h a t t a k i n g the
negligible
Figure
shows
I t i s seen
only f o r L = 1
larger values
t h e r e f o r e , the
of
switching
amount.
noted
the
time.
f o r various values
capacitance
was
This i s a v a l i d
capacitance
on t h e
Approximate Formula f o r the
2.
to
the
(nh = n a n o h e n r y = 10~^
1.
power
to a smaller extent,
dynamic v - i c u r v e
An
found
d-c
delay
nh.
3.3
was
the
s w i t c h i n g time and
the
switching
Rise
f o r the
assumed
assumption
variation
time.
Time of a T u n n e l
rise
into
time
can be
Diode.
easily
f o l l o w i n g assumptions:
The
load line
the
peak o f the
The
v - i characteristic
i s assumed h o r i z o n t a l and
tangent
to
characteristic.
of the
L.
positive
the
that
since
diode
f o r the
of R and
step-function
i n series with
the
and
to a f f e c t
dynamic v - i b e h a v i o u r
constant
a p p l i c a t i o n of a
a modified
voltage v ( t ) across
was
L were assumed f i n i t e
the
d e f i n e d h e r e as
same g e n e r a l
inductance
3-6
both
diode
is
linearized.
,.Current
(ma)
• +-^10<?o o f F i n a l
one
Input V o l t a g e v
Voltage
=0.35^
s
Bias Voltage V
Figure
3-6.
n
= 0.44v
Dynamic v - i T r a n s i e n t B e h a v i o u r f o r L = 1,
10 and 100 h h . (C = 10 p f , I = 1 ma,
R
= 500 ohms).
p
From e q u a t i o n
3-10 we haves
i c =
for
s w i t c h i n g we have:
i
and
edf tt
i
c
AV
= C --jr
c
= I_ - I
P
V
where % i s t h e r i s e
T r
time,
25
AV ^ V
r
- V
p
c_
= c
- i
d p
v
F -
V
I,
)
p
P
V
...(3-11)
1
IT,
This
the
figure
current
3.4
equation
shows t h e dependence o f t h e r i s e
of- m e r i t , t h e f o r w a r d
time
on
v o l t a g e , and t h e peak t o v a l l e y
ratio.
Experimental
The
Results.
experimental
of tunnel diodes
used to t e s t
i s shown i n F i g u r e
were h i g h - f r e q u e n c y
was b u i l t
circuit
non-inductive
i n a coaxial
TEKTRONIX
3-7.
the s w i t c h i n g
time
A l l the r e s i s t o r s
used
resistors
circuit
form.
-vvw-
-
r
v
IK
W
( fe*,5X)
50"--
TYPE 111
PULSE
and t h e whole
TEKTRONIX
TYPE N
SAMPLING
GEN.
PLUG-IN
P u l s e R i s e Time = 0 . 5 n s Z
T.D.
P u l s e D u r a t i o n = 20ns
under
Test
P l u g - i n Rise
= 0. 6ns-
Time =
:
Figure
3-7.
Figure
3-8
( l p = 1 ma,
circuit.
Tunnel
Diode S w i t c h i n g
Test
shows t h e waveforms o b t a i n e d
Cy = 10 p f ) t u n n e l d i o d e
To i l l u s t r a t e
of the i n p u t s i g n a l s
Circuit.
f o r a T1925
u s i n g the s w i t c h i n g
test
t h e d e l a y between i n p u t and o u t p u t ;
(waveform l ) as s e e n a c r o s s
the
output
one
26
terminals with
output
the diode
o p e n - c i r c u i t e d , was s u p e r i m p o s e d
waveforms 2 and 3 o b t a i n e d w i t h t h e d i o d e
on t h e
i n the c i r c u i t .
These o u t p u t ..wavef orms were o b t a i n e d f o r c u r r e n t o v e r d r i v e s o f
20$ and 40$ r e s p e c t i v e l y ,
and were d i s p l a y e d u s i n g a T e k t r o n i x
type
The i n p u t s i g n a l
N sampling
plug-in.
20$ o v e r d r i v e c a s e .
The r i s e
0.6 ns and s h o u l d be a c c o u n t e d
total
the
rise
time
i s the square
scope and c i r c u i t r i s e
Figure
3-8.
(1)
(2)
(3)
The
This agrees
computer
formula
3.5
time
corresponds
of the sampling
t o the
plug-in.is
f o r i n t h e measurement, s i n c e t h e
r o o t o f t h e sum o f t h e s q u a r e s o f
times.
E x p e r i m e n t a l S w i t c h i n g . Waveforms
( l ns/div h o r i z o n t a l , 10 mv/dxr
vertical)
I n p u t V o l t a g e , f o r 20$ O v e r d r i v e
Output V o l t a g e , f o r 20$ O v e r d r i v e
O u t p u t V o l t a g e , f o r 40$ O v e r d r i v e .
displayed rise
times
are of the order
o f 3.5 ns
f a i r l y w e l l w i t h the v a l u e s p r e d i c t e d from.the
solutions,
and t h e v a l u e g i v e n by t h e a p p r o x i m a t e
i n e q u a t i o n s 3-11.
Summary,
lo
The s w i t c h i n g time
of the tunnel diode
c o n s i s t s of
27
a delay
time and
inductance,
and
on
figure
the
the
delay
of m e r i t
delay
A
and,
3.
For
t h e b e t t e r the
4.
to a s m a l l e r
rise
effect
the
of
series
figure
of
merit
t i m e i s d e p e n d e n t on
extent,
inductance
changes t h e
a given
doping
figure
For
d e p e n d e n t on the
of
a given
on
the
i n the
the
overdrive.
circuit
dynamic v - i p a t h
s e m i c o n d u c t o r , the
concentrations
increases
considerably
figure
the h i g h e r
of m e r i t
the
is
doping
merit,
level
e n e r g y gap,
mass o f t h e
gaps, d i e l e c t r i c
of doping,
the
electron.
constants,
the
dielectric
figure
reduced
of m e r i t
constant,
Semiconductors w i t h
and
and
small
masses have the
is
the
energy
best
merit.
Though the p r e c e d i n g
diode,
the
case).
d e p e n d e n t on t h e
f i g u r e s of
Ignoring
t i m e i s d e p e n d e n t on
large series
time and
(voltage bias
reduced
time.
c u r r e n t o v e r d r i v e ; the
2.
the
rise
i t can be
materials.
a n a l y s i s i s b a s e d on
the
e x t e n d e d w i t h m i n o r m o d i f i c a t i o n s to
Ge
tunnel
other
28
4.
STATIC AND DYNAMIC CHARACTERISTICS OP SERIES-CONNECTED
TUNNEL DIODES
When a number o f t u n n e l d i o d e s a r e c o n n e c t e d
i n series
w i t h a common s o u r c e , a m u l t i s t a b l e v o l t a g e - c u r r e n t c o m p o s i t e
characteristic
i s obtained.
characteristic
will
The form
and c o m p l e x i t y o f t h e
(29)
depend on t h e i n d i v i d u a l
elements.
(2)
Attention w i l l
which r e s u l t s
resistance
certain
in 2
n
elements,
rules
As
of
be c o n f i n e d t o t h e most i n t e r e s t i n g
such a composite
elements
states generated using n negative
provided the i n d i v i d u a l
d e r i v e d i n the f o l l o w i n g
a first
relationship
stable
parameters
Experimental
i n s e r i e s , were f o u n d
obey
sections.
d e v i c e was u n d e r t a k e n
that the s t a t i c
tunnel diodes
elements
step, a study of the s t a t i c
must s a t i s f y .
combination,
characteristic
to e s t a b l i s h the
of the i n d i v i d u a l
curves, obtained using
c o n s i s t e n t w i t h the e s t a b l i s h e d
relationships.
Because o f the i m p o s s i b i l i t y
p h y s i c a l measurements o f t h e c i r c u i t
h i g h f r e q u e n c y , and t h e d i f f i c u l t y
approach,
carried
of
4.1
a computer
out.
Static
device
of a purely a n a l y t i c a l
e q u a t i o n s was
l e d to a b e t t e r
understanding
operation,
Characteristic
Simple
c u r r e n t and v o l t a g e s a t
s o l u t i o n of the c i r c u i t
The computer s o l u t i o n
the c i r c u i t
o f making a c c u r a t e
graphical
o f Two T u n n e l D i o d e s
c o n s t r u c t i o n s based
characteristic-curve
enable
i n Series.
on t h e i n d i v i d u a l
one t o p r e d i c t
the nature of
29
the
composite
characteristics.
Figure
of
a typical
discussion,
voltage"
or
4-1
tunnel
"high v o l t a g e "
most
i s not
the
For
the
i s defined
v<V
The
negative
c o u n t e d as
characteristic
purpose
range
state.
p
of t h e
as
a
following
"0"
or
i s defined
as
resistance
curve
"low
a
"1"
region
a state since i t i s unstable
in
cases,
Multistable
the
diode.
linearized
range 0 < V \ V p
the
s t a t e , and
Vp <^ v <^
shows t h e
tunnel
ranges
and
diodes
are
( c u r r e n t ) the
composite
diodes
composite
are
have p r o g r e s s i v e l y o v e r l a p p i n g
constrained
t o have the
same a t e v e r y
consider
i n s e r i e s and
the
To
case
obtained
quantity
illustrate
of two
d r i v e n f r o m a low
when
multivalued
multivalued
i n s t a n t of time.
c h a r a c t e r i s t i c s , we
connected
characteristics
the
tunnel
impedance
source.
Let
Ip
Figure
>
I
^1
>
*V2
...(4-2)
V
>
V
...(4-3)
p 2
4 - 2 ( a ) shows the
characteristics
the
piecewise
the
fact
(in this
linear
t h a t a t any
diodes
while
across
each.
the
p
...(4-1)
i
p i
linearized
individual
Figure V
= 2 V ^).
p 2
tunnel
characteristic,
point
current
the
terminal voltage
of t h e
sum
superimposed
r e v e a l s which p o r t i o n s
was
is identical
i s the
diode
Figure
p
composite
Inspection
characteristics
2
o f the
4-2(b),
obtained
using
i n both
voltages
individual
overlap
i n current.
Each
30
4*
Figure
4-1.
(a).
Figure
4-2(a)
(b)
Linearized Characteristic
Tunnel Diode.
Curve
of a
(b)
Tunnel Diodes I n d i v i d u a l C h a r a c t e r i s t i c s
Composite C h a r a c t e r i s t i c .
part
o f an i n d i v i d u a l c h a r a c t e r i s t i c
of t h e o t h e r ,
a linear
having
current values
segment i n t h e c o m p o s i t e
combines w i t h
i n common w i t h
be a s s i g n e d
binary
digits
significant
the
digit
a binary notation.
to a.state,
digit) will
to the l e f t
to the second t u n n e l
the
sequence w i l l
t o t h e extreme
Later
with
to Figure
characteristic,
00
circuit
a p a i r of
(the l e s s
tunnel
d i o d e D-^, and
digit) will
refer
as more d i o d e s a r e added,
the l e a s t
significant b i t
a short
d i s c u s s i o n of the
4—2(b),
i t s stable
s t a t e s , and t h e o r d e r i n
l e a d to a b e t t e r u n d e r s t a n d i n g of
State:
both diodes are i n t h e i r
the a p p l i e d voltage
across
the diodes
o f Ip^«
This
i s increased,
increase
i s the f i r s t
Segment AW i s g i v e n
01
states
behaviour.
Initially
As
stable
right.
which they are generated, w i l l
the
to the r i g h t
( t h e more s i g n i f i c a n t
2
i t , t o form
When a s s i g n i n g
r e f e r t o the f i r s t
be p r e s e r v e d
Referring
composite
the d i g i t
diode D «
part
characteristic.
In t h e f o l l o w i n g d i s c u s s i o n , the v a r i o u s
will
every
a
until
low v o l t a g e
the current
the current
and v o l t a g e
reaches the value
peak o f t h e combined
00 b i n a r y
state.
characteristic.
representation,
State:
As
forcing
decrease
voltage
the input v o l t a g e
the voltage
across
(since I p > Ip^)*
2
state while
increases,
to increase
Hence, D
and t h a t
decreases
across
D
2
to
r e m a i n s i n i t s low
D^ s w i t c h e s t o i t s h i g h v o l t a g e
segments BW, CW, DW a r e t r a c e d .
CW and DW i s 0 1 .
2
the current
The b i n a r y
s t a t e and
representation of
32
10 S t a t e :
If
voltage
the v o l t a g e i s i n c r e a s e d f u r t h e r ,
i n both diodes increase u n t i l I p
peak o f t h e
increasing
through
composite
t h e v o l t a g e must r e s u l t
the d i o d e s ,
increases but that
reached.
characteristic).
The
Along
of i d e n t i c a l
However, o v e r t h i s
at
"f".
" i "
1
" f " to
" i
1
n
D
2
"f" is
f o r an
"fghii'".
, : having
the
c u r r e n t a c r o s s the d i o d e s a t a l l p o i n t s .
p a t h the v o l t a g e i s always l e s s
It i s clear
point,
the v o l t a g e a c r o s s
decreases u n t i l point
p a t h from
(second
i n current
v o l t a g e i n c r e a s e v i a the v i r t u a l p a t h
T h i s p a t h i s the u n i q u e
property
Beyond t h i s
system then switches to p o i n t
infinitesimal
c u r r e n t and
i s reached
2
i n a decrease
segment DX
a c r o s s D^
the
that
i n the
than i t i s
s w i t c h i n g p r o c e s s the
excess
(2)
v o l t a g e must be
t a k e n up by
a s t o r a g e element
the t u n n e l diode
capacitance).
decreases
low v a l u e o f Iy2$ d i o d e D
to the
high-voltage
voltage
unless
voltage
During t h i s
s t a t e w h i l e d i o d e D^
s t a t e b e c a u s e D^
i t i s near
generates
r e p r e s e n t a t i o n of
cannot
case
p r o c e s s , the c u r r e n t
ends up
2
in its
i s s w i t c h e d b a c k t o i t s low-
s u p p o r t the
i t s zero v o l t a g e l e v e l .
segments AT
(in this
low v a l u e o f
Iy
2
Further increase i n
and AZ w i t h a b i n a r y
10.
11 S t a t e :
As
peak o f the
the c u r r e n t reaches
composite
a g a i n the v a l u e
characteristic
a p p l i e d v o l t a g e beyond t h i s ,
i s brought
tunnel diode D
2
CZ
representation
and
11.
DZ
are t r a c e d .
CZ
and DZ
forth.
third
For
any
remains i n the
h i g h v o l t a g e s t a t e w h i l e t h e v o l t a g e a c r o s s D^
segments BZ,
Ip-^? the
i n c r e a s e s and
have t h e b i n a r y
33
The
above d i s c u s s i o n i s v a l i d
applied voltage.
I f the p r o c e s s
coming b a c k i s " n m l k j i c c b a " .
this
case,
showing an
i s reversed,
The
1
only f o r increasing
the path
b i n a r y s t a t e 01
irreversible
effect
traced i n
i s missing
e x h i b i t e d by
in
the
characteristic.
4.2
Static
Characteristics
The
results
of the p r e c e d i n g
to n elements of the
4 r r l j 4 T 2 , 4 T 3 to
of n Tunnel
type
Diodes.
s e c t i o n c a n be
d i s c u s s e d by
extended
generalizing conditions
become:
X
Pn
^(n-l)"--
>
> P1
...(4-4)
I
th
This
to
c o n d i t i o n s t a t e s t h a t the n
i t s "1"
are
state u n t i l
in their
"1"
pass through
Vn
switches
t o i t s "1"
order diodes.(which
t o have e q u a l v o l t a g e
=
^
...(4-5)
zero
ranges the
1
s t a t e i t o b v i o u s l y must
Then c o n d i t i o n 4r-5 e n s u r e s
to s w i t c h back to the
Fm
switch
diodes
n
i t s valley.
V
(n-l) tunnel
Y( -i,)"" '<*Y1
<
diode
cause a l l l o w e r
4-4)
a l l the p r e c e d i n g
cannot
state.
X
When a t u n n e l
tunnel diode
V
p i
^
must be
state.
i n the
For
=
2
to
"1"
the b i n a r y
requirement
m
that i t w i l l
s t a t e from
regions
is:
n
...(4-6)
34
Also
for a l l binary
s t a t e s t o be
distinct*
n-1
V
V1 <
V
F„
-
V
p
"
m
^
•"
( 4
"
7 )
m=l
The
above c o n d i t i o n
positive
regions
case r e s u l t i n g
nsure the
existence
d e r i v e d from n tunnel
of 2
distinguishable
diodes.
In a s p e c i a l
f r o m r e l a x a t i o n of c o n d i t i o n s 4-6
stable p o s i t i v e regions
can be
case a r i s e s f r o m t h e use
having
of diodes
f o u r c o n d i t i o n s , e i t h e r s i n g l y or t o g e t h e r ,
and
i t i s possible
regions
n
t r a c e r ^ ^ was
u s e d t o o b t a i n the
of s e r i e s - c o n n e c t e d
incorporated
m i n i m i z e the
altering
Results.
A curve
characteristics
By
and
2 .
Experimental
features
characteristics.
c e r t a i n numbers^of p o s i t i v e r e s i s t a n c e
(n+l)
this
c o n d i t i o n s 4-4
the
4.3
identical voltage
satisfying
and
between
4-7,(n+l)
generated using n devices;
4-5
to achieve
and
i n the
curve
possible oscillations
tunnel
static
diodes.
The
following
t r a c e r were e s s e n t i a l t o
i n the n e g a t i v e
resistance
region:
1.
a low
s e r i e s r e s i s t a n c e sweep
2.
a low
inductance
Referring
o f GaAs, Ge,
c o n d i t i o n 4-6
T*7
and
to Table
I n A s , and
diode
2.1,
mount.
i t i s seen t h a t a
InSb d i o d e s
approximately.
test
circuit,
can be
made t o
However, a t the
combination
satisfy
time o f
this
These c o n d i t i o n s a r e s i m i l a r t o the ones o b t a i n e d by R a b i n o v i c i
Renton(2) f
n current c o n t r o l l e d negative resistance devices.
o
r
35
writing,
t h e InAs and InSb d i o d e s were n o t y e t c o m m e r c i a l l y
available
The
and t h e i r
parallel
Ge d i o d e
c h a r a c t e r i s t i c v - i c u r v e had t o be s i m u l a t e d .
combination
o f a Ge t u n n e l d i o d e and an o r d i n a r y
s i m u l a t e s t h e InAs d i o d e w i t h a f o r w a r d v o l t a g e
VT, = 0.275v.
The p a r a l l e l
F
a Silicon
backward diode
combination
o f a Ge t u n n e l d i o d e and
(*)
s i m u l a t e s t h e InSb d i o d e w i t h a f o r w a r d
v o l t a g e V = 0.15v.
C
GCLAS
I N 653
T.D.
©
Gt
6c
r.D.
B. ' r.z> B,b, I
<k
Ce
Si
IN 294: H5-I0o\wi^l tfo-lto\iN2<tft
©
®
T.Z>
( b ) , ( c ) , (d)
©•
F i g u r e 4.3(a) E x p e r i m e n t a l C h a r a c t e r i s t i c s of the
Negative R e s i s t a n c e Elements
(b) Two E l e m e n t s Composite C h a r a c t e r i s t i c
(c) Three E l e m e n t s C o m p o s i t e C h a r a c t e r i s t i c
(d) F o u r E l e m e n t s Composite C h a r a c t e r i s t i c
(2$ ma/dry v e r t i c a l ; 0.25 v/div h o r i z o n t a l ) .
The b a c k w a r d d i o d e i s "a t u n n e l d i o d e , w i t h a r e l a t i v e l y ' l o w
i m p u r i t y c o n c e n t r a t i o n . I t i s used w i t h r e v e r s e a p p l i e d v o l t a g e
and e x h i b i t s , o v e r t h a t r a n g e , c h a r a c t e r i s t i c s s i m i l a r t o t h o s e
of a Zener r e f e r e n c e diode b u t w i t h a lower v o l t a g e drop.
The
v o l t a g e d r o p I s 0.28v f o r S i and 0.08v f o r Ge backward d i o d e s .
Figure
istics
(satisfying
negative
these
characteristics.
regions;
these
c o u l d be
consisting
4.4
respectively.
8,
obtained
by
the
parallel
A 32
( c ) , and
and
16
state
of
four
obtain
( d ) , show
the
stable positive
using
two,
composite
a d d i t i o n of a f i f t h
c o m b i n a t i o n of a Ge
This
4-8)
diodes used to
4-3(b),
4,
and
three,
character-
element
tunnel
diode
and
c o m b i n a t i o n would have a f o r w a r d
of 0.08v.
study
of t h e
necessary conditions
for n tunnel
static
characteristics established
f o r the
diodes.
realization
Tire se
at
c o n s i d e r a t i o n ; ' ^
considerations
frequencies,
and
the
the
I t i s the
purpose
determine
the
formulate
o f the
exact
the
the
stable
h o l d when
not
e f f e c t s of
device
conditions
become n e c e s s a r y b u t
p a r a m e t e r s on
(2)
n
speeds w e l l below the
s t r a y elements a s s o c i a t e d w i t h
(1)
2
voltage
switching
device.
However, a t h i g h
operation.
of
conditions
1
current v a r i a t i o n s occur
s p e e d of the
and
the
4-7,
Dynamic C h a r a c t e r i s t i c s .
states
and
4-6,
individual character-
c h a r a c t e r i s t i c s were o b t a i n e d
of t h e
The
the
Figures
backward d i o d e .
voltage
4-5,
c h a r a c t e r i s t i c s having
four devices
istic
a Ge
conditions
separate
r e s i s t a n c e e l e m e n t s , and
composite
and
4 - 3 ( a ) shows the
must be
derived
sufficient
storage
taken i n t o
from
static
f o r proper
following investigation to:
i n f l u e n c e of the
circuit
response,
some e m p i r i c a l
ensure c o r r e c t o p e r a t i o n
c o n d i t i o n s which
and
fast
will
response.
3 7
Most o f
the
i n v e s t i g a t i o n was
s o l u t i o n of
the
circuit
investigated
was
c i r c u i t '\.
equations.
and
the
c a r r i e d out
O n l y a two
by
computer
tunnel
results generalized
diode
wherever
possible.
4.5
Two
Tunnel Diode M u l t i s t a t e
The
and.its
circuit
equivalent
alterations,
a two-bit
o f a Ge
static
investigated
circuit
this circuit
could
and
c h a r a c t e r i s t i c s of
4-4(b).
represent
converter
a GaAs
( 1 N 6 5 3 )
the
a constant
current
s o u r c e I Q J where Ip^ yIQ
R
ideal voltage
i s c h o s e n t o be
larger
of
Appendix
the
two
III).
(a)
ensures
t o be
negative
v^
and
v
2
resistance
are
the
and
second diode r e s p e c t i v e l y
and
i i s the
the
behaviour
instantaneous
of the
o f the
voltage
the
biased
The
line
from
triggering
resistance
diodes
the
(see
stable
operating
and
regions
of
the
drops across
(neglecting
are:
satisfy
The
composite
unstable.
current,
circuit
consists
that:
accessible,
are
It
magnitude of
enough f o r a l l the
characteristic
If
load
adder,
in series.
yIy^.
The
resistances
points
the
source.
negative
choice
binary
d i o d e s are
l a r g e r t h a n the
R i s small
(b)
The
slightly
This
diode
i n d i v i d u a l diodes
in section 4 . 2 .
i s an
a full
tunnel
derived
g
W i t h minor
or a c o u n t e r .
conditions
source v
4-4(a)
i s shown i n F i g u r e
i n Figure
analog-to-digital
( 1 N 2 9 4 1 )
Circuit.
series
t h e n the
the
first
resistance),
equations
describing
38
~L
C
M l
+
(X
ft
A
r
i
^
O
2
(a)
- -
|
.
i
I
rh
r
|
1
( ^ J * ^
(b)
4-4.
Two
L
dv
k
k dt
ck
where
/ft
Gads
£
Figure
f t
|4,
c
=
T u n n e l Diode: M u l t i s t a t e
f t
=
E
-
R
i
"
(i + I ) - f
Q
(i
+ I ) - f
Q
L = total
l "
V
V
Circuit.
.,.(4-8)
2
k
(v )
k = 1, 2
...(4-9)
f c
(v )
k = 1, 2
...(4-10)
k
k
inductance
i n the
circuit,
E = V. + v ,
b
s'
= voltage
(C^
across
i s assumed
frequencies
ck
capacitive
the b l o c k i n g
a short
circuit
of i n t e r e s t )
current,
and
capacitor
a t the
C^
39
f (v^),
^2^ 2^
v
1
respectively.
mental
a
r
S i
e
v
e
by
n
equations AII-3
These e q u a t i o n s
v - i characteristics
closely
AII-4
approximate
o f t h e d i o d e s , t h e d-c
w h i c h a r e g i v e n i n T a b l e 4.1.
behaviour
and
C o m p a r i s o n of t h e
the
differences.
Consequently
6£
parameters
circuit
w i t h C, h e l d c o n s t a n t r a t h e r t h a n v a r i a b l e
k
only t r i v i a l
experi-
indicated
most of the
computation
was c a r r i e d out w i t h C, c o n s t a n t .
I t s h o u l d be n o t e d t h a t
e q u a t i o n s 4-8 and 4-9 a r e i n v a r i a n t t o t h e s u b s t i t u t i o n s g i v e n
in
equations
3-8
and
3-9.
h
Diode
Ge
D
1
V
P
V
ma
ma
V
4.56
1.16
0.06
V
V
V
|r|
P
~ V1 ~
. ma
l
T
ohms
V
.0.3
6V
0.48
D
V2
30
1N2941
Diode
I
0.82
4.86
2
0.10
0.34
0.6
1.0
40
GaAs 1N653
T a b l e 4.1
4.6
Switching
The
1.35v, cause
11
Two
Diodes
Behaviour.
Three
the diodes
states
Figures 4-5(a),
t o s w i t c h from
respectively.
4 - 6 ( a ) and
under i n v e s t i g a t i o n
study,
the
under
ns
i n p u t v o l t a g e s of amplitudes
dynamic v - i c h a r a c t e r i s t i c s
initial
of the
o f m o d i f i e d s t e p v o l t a g e s w i t h 0.5
investigated.
and
Parameters
tunnel diodes' switching behaviour,
application
10,
D-C
The
are a l s o
listed
inductance
L was
The
diodes
rise
time,
was
0.45,
0.90
and
state
superimposed
o f t h e two
4-7(a).
the 00
are
parameters
the
t o the
static
and
shown i n
of the
i n these F i g u r e s .
assumed z e r o .
01,
This
In
circuit
this
simplifies
01
C
±
c
^Current
2
STATE
= 10 pf
= 10
pf
L = 0
Voltage
B = 50-nv . = 0.45v (0.5ns
r i s e time)
= 2.41 ma
(ma)
(v)
Current.. (
o.l
\
Voltage
0.8
(v)
(b)
(a)
F i g u r e 4-5(a)
(b)
Dynamic v - i C h a r a c t e r i s t i c s o f t h e M u l t i s t a t e
C i r c u i t (00—01)
V o l t a g e and C u r r e n t Waveforms ( 0 0 — 0 1 ) .
41
the
solution,
behaviour
effects
in
c u t s down on c o m p u t a t i o n time
of the c i r c u i t
of inductance
relatively
and makes t h e
easier to analyse.
on t h e s w i t c h i n g t r a n s i e n t w i l l
The
be t r e a t e d
a later section.
01 S t a t e :
Initially
A trigger
both
current
and v o l t a g e
reaches
i t s steady
Both diodes
Unbalance
an e x t e n t
diodes
value, a transient
increase.
ceases
2
constants
line.
causes the
^ becomes z e r o
F u r t h e r decrease
causing diode
state while
D
2
increase.
to such
and t h e
in i ,
t o s w i t c h back t o i t s
D^ assumes i t s h i g h v o l t a g e
adjustment t o the q u i e s c e n t
A plot
load
respect to voltage
to increase.
occurs.
s t a t e and t h e
D^ c a n cause t h e c u r r e n t i t o d e c r e a s e
capacitance
low v o l t a g e
When t h e s t e p
r e l a x a t i o n process
high voltage
D^ w i t h
state.
of the s i g n a l , the
to f o l l o w the t r a n s i e n t
D^ t o r a c e d i o d e
across D
rapidly.
time
that the c a p a c i t i v e current i
discharges
is
across both
low v o l t a g e
i s a p p l i e d to the c i r c u i t
i n the inherent switching-time
E v e n t u a l l y , diode
final
the r i s e
s w i t c h towards t h e i r
f a s t e r diode
The
During
current decreases
voltage
are i n t h e i r
s i g n a l w i t h 0.45v a m p l i t u d e
(see F i g u r e 4 . 5 ( a ) ) .
series
diodes
counterpart.
condition relaxes very
o f t h e t r a n s i e n t v o l t a g e s and c u r r e n t v e r s u s
shown i n F i g u r e
time
4-5(b).
10 S t a t e :
A trigger
circuit.
"0"
s i g n a l w i t h 0.9v a m p l i t u d e
When t h e s t e a d y
state while
diode
D
s t a t e i s reached,
i s a p p l i e d to the
diode
i s i n i t s "1" s t a t e .
D^ i s i n i t s
The s w i t c h i n g
Current
(ma)
Voltage
(v)
'ma)
Voltage
(a)
Figure
4-6(a)
(b)
(v)
(b)
Dynamic v - i C h a r a c t e r i s t i c s o f t h e M u l t i s t a t e
V o l t a g e and C u r r e n t Waveforms
(00—10).
Circuit
(00—i
4^
43
behaviour
differs
radically
f r o m the p r e c e d i n g
step reaches
i t s steady v a l u e ,
high voltage
state
until
zero
is
the v o l t a g e
across
while
constraints
Therefore
low
diode
portion
U
s
e
d-c
i s due
and
t o what may
to
the
of t h e
then
be
in i
diodes|.
relaxes
termed
d u r i n g which the
to
a
The
dynamic
t r a v e r s e s i t s v a l l e y , makes up
the
largest
State:
voltage
than
In the
state.
steady
Quiescent
i n the p r e v i o u s
two
cases.
function
o f time f o r t h i s
i s obvious
considered
the
as
case
t o a l l the
Effect
diodes
and
the
considered
on
more
the
high
rapidly
Figure
voltages
4-7(b)
as
the
a
t r a n s i e n t waveforms,
precursor
pulse
can
speed of o p e r a t i o n of the
possibilities
of C a p a c i t a n c e
in their
the
case.
d u r a t i o n o f the
here,
up
reached
c u r r e n t and
particular
a measure o f t h e
end
i s a p p l i e d to
F i g u r e 4 - 7 ( a ) and
above d i s c u s s i o n and
t h a t the
output
amplitude
c o n d i t i o n s are
dynamic b e h a v i o u r
From the
1.35v
s t a t e , both
show the
4.7
effect
when
of
circuit.
For
occurs
(see F i g u r e 4 . 6 ( b ) ) .
r e l a x a t i o n process
A trigger voltage with
it
decreases
f u r t h e r decrease
characteristics
f duration
0
o p e r a t i n g p o i n t of
portion
Any
state, giving rise
p i
current
increasing; this
i s m o m e n t a r i l y t u r n e d on,
voltage
of the
r e l a x e s towards i t s
The
i s charged; t h i s
the
When the
c a p a c i t i v e c u r r e n t i ^ becomes
stops
state.
e s t a b l i s h e d by
"precursor"
11
where t h e
i n i t s high voltage
discharges
its
(see F i g u r e 4-6(a))«
a p o i n t i s reached
and
each diode
case.
steady
can be
Transient
states
achieved
circuit.
corresponding
in T
Behaviour.
2
= 8
be
ns.
(a)
Figure
4-7(a)
(b)
Dynamic v - i C h a r a c t e r i s t i c s o f t h e M u l t i s t a t e
V o l t a g e and C u r r e n t Waveforms (00 — l l ) .
(J)
Circuit
(00—ll)
4^
F o r L = 0,
the
s w i t c h i n g times
figures of merit
of b o t h
diodes:
the
overall
switching times.
s m a l l e r the
t h a t the
and
the
capacitance
ratio;
o p e r a t i o n o f the
:C
the
are
2
v o l t a g e waveforms f o r c a p a c i t a n c e
The
waveforms f o r the
being
three
a s t e p i n each
s m a l l e r the
the
switching
F i g u r e 4-8
ratios
s t a t e s are
figures
of merit
However, i t was
affects
circuit.
dependent ori the
time
shows the
o f 1:2,
1:1
found
transient
and
1:0.75.
shown; the v o l t a g e
input
case.
A c o m p a r i s o n o f t h e waveforms shows t h a t f o r a
capacitance
and
the
quite
o f 1:2,
d u r a t i o n and
large.
reduced
ratio
ratio
The
g e n e r a l response
peak a m p l i t u d e
ratio
peak amplitude
corresponds
the
to equal
C | r | time
F u r t h e r i n c r e a s e i n the
does n o t
improve t h e
when the
an
end
on
response
operation
capacitance
ratio
input corresponding
up
i n the
and
then
settles
as
01
may
r e s e t to the
state,
low
voltage
for fast
c h o s e n as
s m a l l as p o s s i b l e , and
s h o u l d have t h e
larger
particular
C^:C
two
2
operation.
An
shown) i s o b t a i n e d
causes the
In t h i s
case,
circuit
to
i s switched
state while
t h e use
p a r a m e t e r w h i c h can be
response.
a
partially
diode
P
2
region.
above d i s c u s s i o n s u g g e s t s
an a d d i t i o n a l
ratio
are
and
f o r the
cause f a l s e
tunnel diode
designing
gap
capacitance
This
i s i n c r e a s e d t o 1:0.5.
to the
state;
and
constants
(waveforms n o t
i n i t s valley voltage
The
ratio
10
a b e t t e r response
precursor pulse.
diodes.
example o f f a l s e
of t h e p r e c u r s o r p u l s e
1:0.75 o f f e r s
f o r the
i s s l o w e d down
The
diode
capacitance
adjusted i n
capacitances
the diode
capacitance.
of the
with
the
should
be
s m a l l e r band
Furthermore,
the
F i g u r e 4-8»
E f f e c t of C a p a c i t a n c e
(R = 50-n.)
on T r a n s i e n t
Behaviour
47
capacitance
fast
ratio
should
fall
r e s p o n s e and
proper
operation.
(equal
4.8
time
constants)
i n the
range
The
seems t o o f f e r
valley
great
£)Ly,
fixed
capacitances,
c u r r e n t s o f the
extent
the
C-^ and
the
the
(5yl
diodes
=
d u r a t i o n o f the
l a r g e r the
t h e r e f o r e the
the
v - i characteristic
a n a l y s i s was
first,
but
r e p l a c e d by
with
the
ma
previously
8 ns.?
Figure
shows the
4-9
with(5l
v
o f the
= 0.32
The
diodes
larger
circuit
the
determines to
1^'
pulse
Figure
on
the
u s e d i n the
characteristic
This reduction
d u r a t i o n t o 19 ns
4-8(b) f o r the
01
and
11
duration
(5yl
of
to
the
was
in(5ly
( i t was
case ^ = 0 , 2
change i n (^ly d i d n o t
time f o r t h e
larger
capacitance
identical
dynamic v - i c h a r a c t e r i s t i c
The
the
a
preceding
o f 0,656 ma. i . e . ,
ma.
pulse
the
= 10 p f ) .
f o r the
affect
s t a t e s by
the
10
state
values
any
amount.
d i f f e r e n c e between the
i s hot
operation
see
ma.
switching
appreciable
a new
t o 0,32
precursor
o f 6yl
effect
a valley current
r e d u c e d f r o m 0,82
increased
show t h e
the
r e l a x a t i o n t r a n s i e n t towards
T^>
diode
~ \2^
precursor
To
Ge
T u n n e l Diodes on
capacitive current discharging
faster
for
response.
d i f f e r e n c e between
(^y^
a stable state.
the
( C ^ : ^ ) <1.9
r a t i o (C-^:C^) = 1:0.75
optimum
E f f e c t o f the D-C P a r a m e t e r s o f the
Switching Behaviour,
For
1<
critical
as
f a r as
peak c u r r e n t s
switching
time
of the
or
two
proper
i s c o n c e r n e d , however," i t i s w e l l w o r t h m e n t i o n i n g t h a t
I /I
p
ratio
v
response.
f o r each i n d i v i d u a l
diode,
the
faster
the
the
48
Current
(ma)
STATE
_
r
i
C
Voltage
Figure
4-9.
(R = 50 a).
o f Vp , V y , V
p
of Inductance
investigation.
two v a l u e s
Figure
on t h e s w i t c h i n g
voltages
on T r a n s i e n t
m
a
behaviour
were assumed t o be
of inductance
4-10
Behaviour.
were c o n s i d e r e d
shows t h e o u t p u t
o f i n d u e t a n c e ? 1 0 and 30 nh.
resistance
load l i n e s
multistate
circuit,
switching
= 0.32
semiconductor m a t e r i a l .
Only small values
for
pf
v
f o r a given
Effect
10
10
f
(v)
were n o t i n v e s t i g a t e d s i n c e t h e s e
4.9
=
n
Dynamic v - i C h a r a c t e r i s t i c f o r 6l
The e f f e c t s
fixed
2
n
voltage
i n this
waveforms
W i t h t h e low
and l a r g e i n p u t v o l t a g e s e n c o u n t e r e d i n t h e
t h e main e f f e c t s
behaviour are:
of inductance
on t h e
Figure
4-10.
c =14.4(1
1
v (0.5ns
g
-
E f f e c t of Inductance
576)"V;
c
2
on T r a n s i e n t
= 14.4(1 -
r . t . ) = 0 . 4 5 , 0.90, 1.35v
respectively
Behaviour
R = 50^- ,
for 01,10,11
states.
50
1.
The
inductance
i n t r o d u c e s a s m a l l d e l a y i n the
o u t p u t v o l t a g e waveform.
2.
The
higher value
overshoot
form.
i n the
of i n d u c t a n c e
initial
This overshoot
manner t o w a r d s the
F i g u r e s 4-10(a),
3.
The
higher value
relaxation
state.
pulse
rise
steady
state.
of inductance
T h i s i s shown i n
causes
transient
and
i n c r e a s e s the
speed
a
faster
towards i t s s t a b l e
o f o p e r a t i o n (see
4-10 (b)).
c h o i c e of the
transient
inductance
inductive
relatively
t o p o u t p u t waveforms, and
and
load
line
can be made t o c o n t r i b u t e
to speed
up
the
operation.
Effect
The
10 ns
oscillatory
4-10(b).
r e s i s t a n c e , the
4.10
an
of the v o l t a g e wave-
d e c a y s i n a damped
of t h e d i o d e
Hence, by p r o p e r
circuit
causes
T h i s s h o r t e n s t h e d u r a t i o n of the p r e c u r s o r
Figure
flat
(30 nh)
rise
of I n p u t P u l s e R i s e Time on T r a n s i e n t
behaviour
time was
of the
circuit
investigated.
Behaviour.
f o r i n p u t s t e p s w i t h 3,
The
following
5,
observations
were made?
1.
The
dynamic v - i c h a r a c t e r i s t i c s
quite
similar
4-7(a) except
rise
to those
;
characteristics
closer
are
shown i n F i g u r e s 4 - 5 ( a ) ,
f o r the f a c t
t i m e , the
i n a l l cases
that
the
slower
the
4-6(a),
pulse
i s t h e dynamic p a t h t o t h e
of the . d i o d e s , i . e . ,
the
d-c
s m a l l e r the
51
capacitive
2,
The
d u r a t i o n o f the p r e c u r s o r
extent
3,
currents,
independent
Over t h e
positive
instantaneous
input pulse
4,
Over the
switch,
latter
4,11
Effect
important
d-c
4,2
obvious
The
t a b l e and
Similarily,
capacitive current;
rise
affect
the
Operation,
coefficients
and
GaAs t u n n e l
the
switching
both
i n switching
o p e r a t i o n of t h e
speed
forward—voltage
the
of
the
diodes.
speed.
it
It
effect
circuit
tunnel diodes
2
the
stable
critical.
coefficients
appreciable
does
coefficients.
not
extent.
for application in a
i t i s a d v i s a b l e to p i c k diodes
same v a l l e y - c u r r e n t t e m p e r a t u r e
a
temperature
i s not v e r y
t o any
was
(T ).
s w i t c h i n g time and
d i f f e r e n c e i n peak-voltage
circuit,
the
time,
Circuit
seriously affect
d i f f e r e n c e i n the
the
on
d i f f e r e n c e i n the v a l l e y - c u r r e n t temperature
Hence, i n c h o o s i n g
multistate
the
the d i s c u s s i o n i n s e c t i o n 4.8
s t a t e s ; however, t h i s
the
the
speed depends m a i n l y
temperature
change of + 15$
coefficients will
affect
and
on the
the
the
diodes
t h a t a change of + 25°C i n t e m p e r a t u r e p r o d u c e s
corresponding
steady
constant
gives
the
r e s i s t a n c e r e g i o n s , the
switching
depends on
t h a t the
coefficients will
estimated
resistance regions,
p a r a m e t e r s f o r the Ge
Prom t h i s
is
the
o f Temperature
Table
time,
time,
negative
C | r | time
rise
i s to a l a r g e
dynamic o p e r a t i n g p o i n t f o l l o w s
rise
and
of the
pulse
having
the
52
Germanium
Symbol
Coefficient
Gallium Arsenide
(1N2941)
(1N653)
Peak P o i n t v o l t a g e
Temp. C o e f f . * *
AV /AT
-60
(xV/°C
-120 [J,V/°C
V a l l e y Point v o l t a g e
Temp. C o e f f . * *
AV /AT
-1
mv/°C
-1
mv/°C
Forward P o i n t v o l t a g e
Temp. C o e f f . * *
AV /AT
-1
mv/°C
-1.5
mv/°C
V a l l e y P o i n t current
Temp. C o e f f . *
I AT
AT = T - T
p
y
F
0.75%/°C
0.6/o/°C
y
R
* Measured
T = operating
* * Manufacturer's Data
T
R
temperature
= room temperature
Table 4 . 2 Temperature C o e f f i c i e n t s of the D-C Parameters
for the Ge and GaAs Tunnel Diodes
4.12
Extension to Three Tunnel Diodes i n
If
Series.
three tunnel diodes are connected
s t a b l e states r e s u l t
(see
section 4 . 3 ) .
in series,
Table 4.3 gives
eight
the
e i g h t s t a t e s of the composite device i n terms of the i n d i v i d u a l
diode's
states.
53
State
process
3
D
2
1
0
0
0
2
0
0
1
3
0
1
0
4
0
1
1
5
1
0
0
6
1
0
1
7
1
1
0
8
1
1
1
T a b l e 4.3
It
D
S t a b l e S t a t e s f o r Three Tunnel
i s seen
that f o r every
state
i n v o l v e s o n l y two t u n n e l d i o d e s
preceding
analysis
c a n be u s e d
Diodes
except
5, t h e s w i t c h i n g
and t h e r e f o r e t h e
t o o b t a i n an e s t i m a t e
switching time.
For state
to
and f o r c e s t h e two p r e c e d i n g d i o d e s
i t s "1" s t a t e
back to t h e i r
in
this
diodes
of
case
"0" s t a t e .
5, t h e t h i r d
switches
to switch
i s complicated
and t h e d u r a t i o n o f t h e r e l a x a t i o n t r a n s i e n t s f o r
D^ and D^ w i l l
Experimental
F i g u r e 4-11
circuit
of the
tunnel diode
The s w i t c h i n g b e h a v i o u r
determine
o p e r a t i o n of the c i r c u i t
4.13
Device
to a c e r t a i n extent
(see s e c t i o n
the
speed
4.13).
Results.
shows t h e e x p e r i m e n t a l
response
of the
o f F i g u r e 4-4 t o t h e a p p l i c a t i o n o f p u l s e s o f
0.45, 0.90 and 1.35v.
The p u l s e s were o b t a i n e d from
amplitudes
a Tektronix
54
111 p u l s e g e n e r a t o r and have a r i s e
of
1 n s , and a d u r a t i o n o f 20 n s .
on a T e k t r o n i x 581 o s c i l l o s c o p e
parameters
of the experimental
R = 50 ohms) were a p p r o x i m a t e l y
time
The r e s p o n s e
(rise
time
circuit
No a t t e m p t
experimental
account
time
The e x p e r i m e n t a l
(0^=20
The
pf, Cy
= 18 p f ,
2
of the c i r c u i t
(waveforms o f F i g u r e
results
i n the
(taking
into
o f t h e o s c i l l o s c o p e ) show r e l a t i v e l y
agreement w i t h the computer s o l u t i o n s .
p r e c u r s o r p u l s e shown i s a p p r o x i m a t e l y
the p o s s i b l e
3.3 n s ) .
pulse r e p e t i t i o n rate
time
observed
was made t o e v a l u a t e t h e . i n d u c t a n c e
circuit.
the r i s e
was
t h e same as t h o s e
a n a l y s e d on t h e computer i n s e c t i o n 4.9
4-10).
o f 0.5 n s , a f a l l
good
The d u r a t i o n o f t h e
12.5 n s ; t h i s
would
limit
t o 80 Mc/s.
•4-
01 S t a t e
10 S t a t e
11 S t a t e
A
1
m t
X
IS.
I
Diode
F i g u r e 4-11.
D.
Diode
D,
E x p e r i m e n t a l V o l t a g e Waveforms
f o r Two T u n n e l D i o d e
Circuit
(0.5 v/diY v e r t i c a l ; 10 ns/diiv h o r i z o n t a l )
55
001
11
111
r
nif
i
010
HI'
1111
3
A
111
: N
T*
|
I'
Diode
Figure
4-12,
D3
Diode
D.
Diode
D,
E x p e r i m e n t a l V o l t a g e Waveforms f o r a Three
T u n n e l D i o d e C i r c u i t (0.5 v/div v e r t i c a l 5 lOns/clvV
horizontal)
56
F i g u r e 4-12 shows t h e r e s p o n s e
of the eight
state
circuit
(see s e c t i o n s 4.3 and 4.12) t o p u l s e s o f magnitude 0.2, 0.4, 0.6,
0.8,
and
1.0, 1.2, and 1.4v.
the b i a s
c u r r e n t I Q = 4 ma.
diodes to r e l a x to t h e i r
The
time
c i r c u i t was s l o w e d
stable
line
r e s i s t a n c e was 25 ohms
The time
stateswas
down i n t h i s
o f t h e germanium d i o d e
tunnel diode
4.14
The l o a d
necessary f o r the
a p p r o x i m a t e l y 20 n s .
case b e c a u s e
(Q5-100) u s e d
of the r e c o v e r y
t o shunt
the f i r s t
t o o b t a i n a low f o r w a r d v o l t a g e .
Summary.
1.
The speed
of operation of the c i r c u i t
i s l i m i t e d by
the d u r a t i o n o f t h e p r e c u r s o r p u l s e .
2.
The s w i t c h i n g t i m e s
the f i g u r e s
3.
i n t h e range
d i o d e s and on
the l a t t e r
1^(0-^ S C 2 X I .9 f o r b e s t
should
response.
The d u r a t i o n o f t h e p r e c u r s o r p u l s e i s i n v e r s e l y
proportional
to
(
5
l
y
« The
thus a f f e c t s
the e f f e c t
it
A proper
dependence
the s w i t c h i n g time.
of temperature
temperature
To r e d u c e
on t h e s w i t c h i n g t i m e ,
speed
valley-
coefficients.
choice o f s e r i e s
contribute
to
temperature
i s a d v i s a b l e to p i c k diodes having equal
current
4.
a r e dependent on
of merit of the i n d i v i d u a l
of the capacitances C ^ >
the r a t i o
be
of the c i r c u i t
relatively
i n d u c t a n c e c a n be made t o
flat-top
o u t p u t waveforms and
up t h e c i r c u i t o p e r a t i o n .
57
5.
The
preceding
APPLICATIONS
m u l t i s t a t e composite d e v i c e s
chapter
c a n be u s e d t o f u l f i l l
discussed
i n the
a v a r i e t y of d i g i t a l
(2)
functions.
Some o f t h e p o s s i b l e a p p l i c a t i o n s
discussed b r i e f l y p o i n t i n g
5.1
devices.
Full
Binary
The
driven
Adder.^ ^'^ ^
2
circuit
from v o l t a g e
signals
3 5
i s shown i n F i g u r e
sources.
two o u t p u t s :
Sum
and C a r r y
receives
representing
input
corresponding
The
circuit i s
variable
inputs
There a r e e i g h t
to the f o u r output
One n o t e s from T a b l e
"ON"
on i t s
numbers t o be added, and t h e Sum and
i n d i c a t e the r e s u l t a n t outputs.
possibilities
"1" o r "0"
s i g n a l s a r e 1.
5.1, where X,T,Z d e n o t e t h e t h r e e
the b i n a r y
inputs
a 0,0; 0,1; 1,0; o r 1,1 a c c o r d i n g t o
a r i t h m e t i c f u n c t i o n p e r f o r m e d by t h i s
shown i n T a b l e
biased
and has t h r e e
i n p u t c h a n n e l s and y i e l d s
whether none, one, two, o r a l l t h r e e
Carry
4-13
The c i r c u i t
on e a c h o f i t s t h r e e
detailed
be
o u t t h e a d v a n t a g e s and d i s a d v a n t a g e s
7
of these
will
states.
5.1 t h a t i f any o f t h e i n p u t s a r e
( l ) o r "OFF" ( 0 ) a v a r i e t y o f u s e f u l l o g i c a l
functions
(35)
can
be p e r f o r m e d .
If
of T a b l e
is
input
5.1
assigned
Z i s so a d j u s t e d
shows t h a t :
Sum
as t o be always
i s assigned
to e i t h e r X or T but not both.
"or" i s thus achieved.
Furthermore Carry
"OFF"; t h e t o p
"1" i f and o n l y
The b i n a r y
i s assigned
i f "1"
exclusive
"1" j u s t
Gt
0
Sum
CJI, M
C
-
Figure
5-1.
Full
a
r
r
y
1
Binary
Adder.
Inputs
CARRY
X
Y
z
0
0
0
0
0
1
0
0
1
0
0
1
0
1
0
1
1
0
0
1
0
0
1
1
0
1
0
1
0
1
0
1
1
0
1
1
1
1
1
1
Table
5.1
Truth
t a b l e of F u l l
Binary
Addition
59
in that single
connective
case where
"and"
i s thus
"1" a r e a s s i g n e d
table reveals that
cases
where b o t h X and
"0".
The
and
Y are
"1"
i s assigned
Y receive like
b i n a r y f u n c t i o n " i f and
o t h e r hand, C a r r y
i s assigned
assigned
"0";
the
"0"
"ON";
full
Secondly,
flow
the
a t the
circuit
just
This problem
i s encountered
those
"1"
or
On
or" i s therefore
the
the
adder has
for
noted
no
in
gain
so
fan-out.
unidirectionality
from t h e
X
realized.
f e a t u r e s t o be
i s necessary
isolation
of
i n t h a t case where b o t h
does n o t p r o v i d e
o f i n f o r m a t i o n and
in
o n l y i f " i s achieved.,
Firstly,
output
just
assignments of
"inclusive
a d d e r d e s c r i b e d above.
that amplification
The
the bottom h a l f
t o Sum
T h e r e a r e a number o f u n d e s i r a b l e
the
Y.
realized.
Suppose however t h a t Z i s always
the
t o b o t h X and
i n the
f o l l o w i n g stages.
i n most t u n n e l d i o d e
circuits
and
any
(36)
of
the
the
Sum
existing
output
methods o f c a s c a d i n g
i s above g r o u n d .
a d v a n t a g e s of c i r c u i t
o p e r a t i o n and
circuit
full
c o n v e r t e r by
IQ
from a s i n g l e
triggering
sampled f i r s t h a n d
composite
is
the
circuit.
available
stability,
high
speed
i n two
sampling
The
output
forms.
The
the
of
suited for
(37)
'
b i n a r y adder d e s c r i b e d above becomes a
analog-to-digital
and
Thirdly,
integration.
Analog-to-Digital Converter.
The
used.
a d d e r , however, o f f e r s
compactness w h i e h make i t w e l l
m i c r o m i n i a t u r i z a t i o n and
5.2
The
simplicity,
' can be
v
removing the
source.
The
steady
state current
analog voltage i s
pulses
are f e d to the
o f the
analog-to-digital
voltage
two-bit
across
the
diode
converter
tunnel
diode
60
stack i s a staircase with
voltage
of the lowest
available
The
The
resolution
signal
resolution
characteristic
without
f o r each d i o d e .
i s l i m i t e d by t h e
carefully
using diodes.
that avoids
noted.
controlled v - i
T h i s imposes a s e v e r e
of a p r e c i s e predetermined
device
i n this
diodes.
encoder n e c e s s i t a t e s v e r y
the generation
diodes.
(see s e c t i o n 4 . 3 ) .
f o l l o w i n g u n d e s i r a b l e f e a t u r e s h o u l d be
characteristics
resistance
the i n d i v i d u a l
resistance devices
extension of the converter's
accurate
i s also
o f t h e c o n v e r t e r c a n be e x t e n d e d t o f o u r
by u s i n g f o u r n e g a t i v e
The
on
Binary output
to the forward
however, t i m i n g p r o b l e m s may a r i s e
presently available
An
diode,..
equal
c a n be made t o encode t h e a n a l o g
sampling;
The
bits
order
steps
by s e n s i n g t h e v o l t a g e s a c r o s s
circuit
initial
incremental
restriction
composite
Means o f o b t a i n i n g a n e g a t i v e
this
shortcoming
have been
(2)
d i s c u s s e d by R a b i n o v i c i and R e n t o n
critical,
however, t h e t u n n e l d i o d e
simplicity
was
5.3
found
and h i g h
.
When t h e a c c u r a c y
circuit
speed of o p e r a t i o n .
t o have an i n h e r e n t e n c o d i n g
has t h e advantage o f
A three-bit
time
i s not
encoder
o f 20 n s .
Counter.
Counting
possesses
t o a base n c a n be p e r f o r m e d by any d e v i c e
that
n d i s t i n g u i s h a b l e s t a t e s w h i c h c a n be a t t a i n e d i n a
definite
sequence.
The c o u n t e r
pulses.
The s y s t e m changes s t a t e a f t e r
r e m a i n s i n i t s hew s t a t e u n t i l
i s designed
to r e c e i v e a s e t of
each incoming
the a r r i v a l
of the next
p u l s e , and
pulse.
61
t h
The
n
incident
trigger
the
a reset
p u l s e , i n a c o u n t e r t o t h e base n, i s made t o
circuit
same t i m e d e l i v e r s
The
counter.
proper
amplitude
the next.
triggers
Iy-^^ I Q ^ l p ^ .
enables
When t h i s
c a n be u s e d
as a
a c o n s t a n t d-c c u r r e n t
Application
of pulses of
t h e s t a c k t o s w i t c h from
source.
staircase
the reset c i r c u i t .
removal o f I Q , e i t h e r
one s t a t e t o
After
reaches
the a p p r o p r i a t e
level,
R e s e t t i n g i s a c c o m p l i s h e d by
by s h u n t i n g t h e d i o d e s , o r by o p e n - c i r c u i t i n g
t h e r e s e t p u l s e , I Q f l o w s a g a i n , and t h e
diodes are a l l i n t h e i r
staircase
circuit
stage.
The s t a c k v o l t a g e waveform t a k e s t h e a p p e a r a n c e o f
a staircase.
tunnel
tunnel diode
The d i o d e s a r e b i a s e d from
I Q such t h a t
the
a c a r r y output t o the next
multistate
source
it
w h i c h r e t u r n s t h e c o u n t e r t o z e r o , and a t
upon f u r t h e r
"0" s t a t e ,
application
ready
t o form
of the input p u l s e s .
another
62
6.
The
behaviour
main o b j e c t o f t h e
of a simple
diodes
from the
states
are
in principle,
for
the
this
by
by
fabrication
In the
diode,
on t h e
figure
of m e r i t
o f the
the
multistate
solutions
of t h e
diode
of the
switching time.
computer
the
(l)
and
study,
the
diodes
i s an
static
non-linear
ratio
important
i t is
(see
of a
time
Summary,
a b a c k g r o u n d f o r the
characteristic
o f the
c o n d i t i o n s f o r the
was
The
study
digital
o f the
compatible
The
investigation
existence
o f the
with
of
of
the
the
conclusions
are:
c a p a c i t a n c e s o f the
This r a t i o
of
computer
estimate
most i m p o r t a n t
parameter i n determining
static analysis.
generation
system t o o b t a i n an
results
composite
dynamic b e h a v i o u r
i n v e s t i g a t e d by
presented,
circuit.
range t o e n s u r e the
f r o m the
circuit
dynamic
speed of the
number n,
dynamic b e h a v i o u r
current overdrive
Experimental
The
While
dependence o f i t s s w i t c h i n g
necessary
s o l u t i o n s are
drawn f r o m t h e
diodes.
circuit.
r e v e a l s the
tunnel
stable
n
diodes.
r e q u i r e d number o f s t a b l e s t a t e s .
a two
and
and
f o r any
2
tunnel
semiconductor m a t e r i a l s a v a i l a b l e
i s d i s c u s s e d to provide
A study
device
of s e r i e s - c o n n e c t e d
is valid
of t h i s
tunnel
i n v e s t i g a t e the
dynamic p o i n t s of v i e w .
of t u n n e l
course
i s to
interconnecting n tunnel
the
single
S e c t i o n 3.5)
and
technique
limited
study
combination
static
obtained
at present
CONCLUSION
two
the
tunnel
operation
must l i e w i t h i n a
certain
stable states predicted
63
(2)
its
The d i f f e r e n c e i n t h e d i o d e s '
v a l l e y c u r r e n t s , and
dependence on t e m p e r a t u r e a f f e c t t h e s w i t c h i n g
composite
device.
The
multistate
circuit
performance
of d i g i t a l
logic
analog-to-digital
together
speed o f t h e
has many a p p l i c a t i o n s i n t h e
functions
s u c h as b i n a r y
conversion,and counting.
These a p p l i c a t i o n s ,
w i t h many u s e f u l f e a t u r e s , m a k e t h e m u l t i s t a t e
w e l l worth f u r t h e r i n v e s t i g a t i o n .
1.
inherent
high
2.
simplicity
3.
small
of design,
number o f components,
w h i c h make t h e c i r c u i t
microminiaturization.
circuit
Some o f t h e s e f e a t u r e s a r e :
speed o f o p e r a t i o n ,
s i z e and s m a l l
addition,
well
suited f o r
64
APPENDIX I
Al.
Measurement o f T u n n e l
The
tunnel diode
known f o r n e a r l y e v e r y
tions^the
to
Diode
equivalent c i r c u i t
application.
junction capacitance
s e t the upper
attainable
Parameters
limit
and
on u s a b l e
s w i t c h i n g speeds.
For high frequency
lead inductance
frequency,
AI.l
i s described
Bias C i r c u i t
provided
and
If this
region,
then
biasing
point
i s i n the n e g a t i v e
signal
prevent
-A
parameters,
tunnel diode
does n o t
a r r a n g e m e n t can be
measurements.
a-c
tunnel diode
instability
stability
1602
these
to t h e
some means must
desired operating
used.
and
any
conductance
convenient
However, i f the
p r e c a u t i o n s must be
desired operating
the
taken
network
to
obtain
for accurate,steady-state,small-signal
I t c a n n o t be
a c r o s s the
occur
conductance r e g i o n , then
special
necessary
e r r o r s due
the
U.H.F. A d m i t t a n c e
o p e r a t i n g p o i n t i s i n the p o s i t i v e
may be u n s t a b l e , and
the
to determine
Stability
t o b i a s the
point.
Radio
applica-^
here.
To measure the
be
and
be
are b e l i e v e d
A method o f m e a s u r i n g
parameters u s i n g a m o d i f i e d General
Bridge
p a r a m e t e r s need to
overemphasized
tunnel diode
must be
to the n o n - l i n e a r i t y
t h a t the
less
measuring
t h a t 5 mv
of the d i o d e
to
character-
istics.
The
1602-A A d m i t t a n c e B r i d g e
impedance t o t h e unknown,and has
presents
a d-c
path;
a
negligible
therefore,the biasing
65
configuration
equivalent
its
shown i n F i g u r e A l - l a was
c i r c u i t of F i g u r e
negative
Al-lb,
with
r e s i s t a n c e r e g i o n , the
conditions for stable operation
r
j ^
r
<
R,
+
used.
the
From
diode
necessary
and
the
biased
in
sufficient
are:^^
...Al-la
r <|r|
d
and
L<
where | r [ i s the
at
the
|r|
2
absolute
C
...Al-lb
magnitude o f the
p o i n t u n d e r c o n s i d e r a t i o n , and
inductance
i n the
negative
L i s the
resistance
total
series
circuit.
T.D.
Test
under
,—AA/W-
C(v)
Tunnel
R, «
fit
•-r
Diode
Equivalent
Circuit
Bridge Input
Impedance N e g l e c t e d
(a)
F i g u r e AI-1
(b)
(a)
Tunnel Diode Test
(b)
Equivalent
Circuit
Circuit
66
AI.2
Tunnel
The
minimize
D i o d e T e s t Mount
test
stray
mount f o r t h e d i o d e must be
inductance
and
t e s t mount made of r e a d i l y
R^
i s a disc
negligible
shunt
been e s t i m a t e d
f o r the diode
The
disc
as p o s s i b l e , i n
1
the
( i t has
i s of
the
i s l o c a t e d as c l o s e t o
order to minimize
stray
inductance.
Results
complete d i a g r a m of the
test
F i g u r e AI-3b shows t h e b r i d g e e x t e r n a l c o n n e c t i o n s .
The
series
series
inductance, being
constant
for.a
measured o n l y o n c e , u s i n g
r e s i s t a n c e may
be
given
a dummy
measured a t c o n d i t i o n s of
diode.
very
reverse voltage.
t h e measurement o f t h e d i o d e
admittance,the
procedure
as f o l l o w s ;
l)
The
has
and
For
is
mount i n d u c t a n c e
Measurements and
package d e s i g n , need be
large
connectors*
u n d e r t e s t w i t h L = 2L^
r e s i s t o r R,
F i g u r e A I - 3 a shows t h e
The
Radio
i s chosen to s a t i s f y
t h a t t h e b r i d g e and
Experimental
circuit
F i g u r e A i r 2 shows a
a v a i l a b l e General
c a p a c i t a n c e and
same o r d e r as L , ) .
a
AI.3
to
r e s i s t o r which i s e s s e n t i a l l y n o n - i n d u c t i v e ,
i n e q u a l i t i e s AI-1
"the d i o d e
capacitance.
designed
The
connections
bridge
from
been i n t e r c h a n g e d .
generator
end,
and
i s connected
the
generator
Normally,
i t appears
the
as
and
shown i n F i g u r e A I - 3 b .
the
signal
without
signal
from
i s a p p l i e d at
the
attenuation across
unknown impedance*, o n l y a p o r t i o n of the
However, a p p l y i n g the
s i g n a l d e t e c t o r have
signal
the g e n e r a t o r
the
i s detected.
a t the d e t e c t o r
67
874-QNJ
-874-QUP C o n n e c t o r
Connector
«.562 B.
Disc
Tunnel
Resisto
R,
Figure
AI-2
Tunnel Diode
Coaxial
Diode
Mount
h-p 608-D
VHF SIGNAU
GENERATOR
80 Mc/s
Detector
3 50 mv
Input
Conductance
Standard
-»-To B r i d g e ^Unknown—
Coaxial
mount
Generator
Inpu
Susc e p t a n c e
Standard
RADAR
RECEIVER
(a)
Figure
(b)
AI-3(a)
(b)
Diagram o f the Test C i r c u i t
Bridge External Connections
Tuned t o
80 Mc/s
68
terminal
ensures
fed to the r e s t
that this
of the b r i d g e
the t u n n e l diode
normal
i s attenuated
circuits
signal
operation.
A n o t h e r m o d i f i c a t i o n was
a c h i e v e d by r o t a t i n g
multiplier
lever
2)
replacing
through
of the diode
diode.
T h i s admittance
f o r various bias voltages.
value
specific
diode
inductance
of capacitance
n
diode
The
c a n be e x p r e s s e d
pf
n =
as a
with
capacitance
ass
0.42
0.6
o f t h e mount and d i o d e was
from the s h o r t - c i r c u i t e d
i s subtracted
o f a Ge 1N2939 d i o d e
variation
=
1 -
The t o t a l
diode
F i g u r e AI-4 shows
as o b t a i n e d f r o m t h e above measurements.
C(v)
this
Measurements were made a t 80 Mc/s.
F i g u r e AI-5 shows t h e e x p e r i m e n t a l
variation for this
conductance/
i n p l a c e to o b t a i n the
characteristics
f u n c t i o n of b i a s v o l t a g e .
bias
introduced i n
a dummy s h o r t - c i r c u i t e d
true diode values
admittance
millivolts
mount must be measured
taken with the diode
typical
across
90°.
resistor,and with
the a c t u a l
of negative
from the r e a d i n g s
the
appearing
the c o u p l i n g loop a s s o c i a t e d w i t h the
The a d m i t t a n c e
w i t h the d i s c
the s i g n a l
being
i s a t t e n u a t e d by t h e same amount as
the b r i d g e t o a l l o w f o r measurement
was
(40 db) b e f o r e
c a n t h e n be o f t h e o r d e r o f a few
while"" t h e d e t e c t e d
for
signal
reading.
found
t o be 10 nh,
3( m mho)
G-Cmmho)
Y s h o r t C r c u i f -. 51.5
mho
• 10
F i g u r e AI-4
Admittance C h a r a c t e r i s t i c s of a
1N2939 D i o d e as a F u n c t i o n o f V o l t a g e
i
—
10
•
• —
»
4
(v\
/
/
*
>•
"
*-•—
•
c
.
>3.Z
x
F i g u r e . AI-5
Capacitance V a r i a t i o n
of V o l t a g e
Pt
4.1'
as a F u n c t i o n
70
APPENDIX I I
All.
Methods o f A p p r o x i m a t i n g
In the study
is
i
necessary
Tunnel Diode
of the tunnel diode
to represent
the tunnel
= f ( v ) i n terms o f an a n a l y t i c a l
theoretical
quite
expressions
elaborate
observed
characteristic
the
relationship
the
diode
This
found
tunnel
i n their
between t h e v a r i o u s
diode
diode
i s mainly
to approximate
characteristic
expression.
Most o f t h e
characteristic
due t o t h e f a c t
The o n l y
that
through
theoretical
( w i t h i n 20fo) t h e
characteristic
are
r e p r e s e n t a t i o n of the
currents flowing
i s not y e t w e l l understood.
expression
Germanium
.
switching c i r c u i t s i t
diode
f o r the tunnel
and u n s u c c e s s f u l
Curves
observed
i s t h e one d e v e l o p e d
by
(9) ( 2 l )
Kane,
'
I t c o n s i s t s of three
expression
one
f o r the t u n n e l i n g c u r r e n t
o f t h e f o r m Kexp
where K i s a c o n s t a n t .
was
found
the
characteristic
approximations.
a polynomial
exponential
with
curve
Two
types
an e x p r e s s i o n
current,
of r e p r e s e n t a t i o n
f o r t h e case
o f GaAs
theoretical
of approximations
approximation,
the c h a r a c t e r i s t i c
case),
diodes.
expression f o r
l e d t o the c o n s i d e r a t i o n of a n a l y t i c a l
approximation.
reverse voltage
by
However, t h i s t y p e
l a c k of a s a t i s f a c t o r y
theoretical
c u r r e n t , and an
(40v) f o r t h e d i f f u s i o n
t o be q u i t e i n a d e q u a t e
The
the
(direct tunneling
t e n t h o f t h e peak c u r r e n t f o r t h e e x c e s s
expression
is
components*
were u s e d ,
and t h e s e c o n d i s a two
Both approximations
the f i r s t
term
are concerned
i n the forward
voltage
region.
r e g i o n , the conduction
current
can be a p p r o x i m a t e d
of the form I =
exp (-0v)
In the
where *J and
0. a r e
71
c onstants.
i
AII.l
Polynomial
The
Approximation
polynomial, approximation
o f l e a s t mean s q u a r e s
polynomial
the
error
specified.
generated
u s i n g t h e IBM 1620 computer.
c o n t a i n s as many terms as n e c e s s a r y
standard
range
of t h e dependent v a r i a b l e
I t s h o u l d be n o t e d
contains a constant
curve
does n o t go t h r o u g h
error
i s negligible
origin
attempt
was o b t a i n e d by t h e method
i s of l i t t l e
i n the i - v plane.
importance
go' t h r o u g h
the
a seventh
actual
characteristic
and c a l c u l a t e d
tunnel diode
ensured
i n most c a s e s .
the o r i g i n
An
resulted
i n a very
Ge t u n n e l d i o d e
(T1925),
calculated
The c u r r e n t d i f f e r e n c e
between
i s normalized with r e s p e c t to the
on t h e lower
p o r t i o n of the graph.
It is
i s a c c u r a t e t o w i t h i n + 5$ o v e r t h e
range.
A similar
4.3$.
curves
t h a t the approximation
whole
near the
and t h e c h a r a c t e r i s t i c
order polynomial.
peak c u r r e n t and p l o t t e d
seen
This
accuracy.
experimental
from
polynomial
term,and t h e r e f o r e t h e a p p r o x i m a t e
F i g u r e A I I - 1 shows, f o r a t y p i c a l
the
(current) w i t h i n the
s i n c e the p o r t i o n of the curve
t o make t h e c u r v e
poor o v e r a l l
i n order to b r i n g
however, t h a t e a c h
the o r i g i n
practical
The
approximation
(1N653).
a maximum e r r o r
Table A I I . l
In t h i s
was o b t a i n e d f o r a t y p i c a l
case
a n i n t h order
polynomial
of + 2$,except a t the o r i g i n w h e r e
gives the polynomials
GaAs
7
i t i s
f o r t h e two d i o d e s .
72
Experimental
... —
v
v
—
Approximation
1
\\
\\
A
V
if'/
llif
II
if
% ^*****»^
o.t
/
Error
4-y.
i
0.3
•
•
•
•
•
•
•
-4X
•
•*
»
•
1
•
•
•
•
•
•
Figure AII-1
1 •
•
T1925 Germanium
Tunnel
Diode
(a) A c t u a l and C a l c u l a t e d C h a r a c t e r i s t i c s
(b) P e r c e n t - E r r o r Between t h e A c t u a l and
Calculated Characteristics
I
Polynomial Approximation
Diode
Ge
f ( v ) . = 0.03281 + 44.6474v - 6 8 3 . 0 1 3 v
GaAs
(lp = 1
f(v)
7211.73V
- 5624.81v
4
5
2
ma)
+ 3744.82v
+ 36259.62v
6
33114.68v .
...AII-1
7
= -0.043518
+ 25.372706v -
+ 676.02962v
3
- 497.832v
+ 490.08273v
+
3
6
207.321040v
- 1086.3072v
7
4
2
+ 899.342610v
- 459.487v
5
8
I60i9966v .
9
...AII-2
Table A I I . l
P o l y n o m i a l Approximations f o r Tunnel
Diodes
73
All.2
Two
Term E x p o n e n t i a l
The
Approximation
s e c o n d method o f a p p r o x i m a t i o n u s e s a two
exponential
(19)»(20)
expression.
term
a p p r o x i m a t i o n i s of
the
form:
f(v)=
+ I
h
= Av
where a, b, A,
tunneling
Figure
B are
current
2-3).
s e t s of c u r r e n t
fitted
region
region.
Table
The
All.2
(IN 653)
the
accuracy
gives
diodes.
by
ratios
can
c h a n g i n g the
diode
It
other
term r e p r e s e n t s
diffusion
current
points
should
i n the
be
points
i n the
diffusion
expressions
The
main a d v a n t a g e of t h i s method
be
two
(see
to
does n o t
represented
constants
f o r a Ge
require
f o r the
B and
b.
investigation
10$.
a GaAs
of
different
ComputerLand
same t y p e o f t u n n e l
This
last
o f the
be
current
(IN2941) and
a
four
tunnel
of t h i s type of a p p r o x i m a t i o n i s +
i t
the
d e t e r m i n e d by m e a s u r i n g
at four p i l o t
points
feature
diode
was
series-connected
circuit.
should
be
noted that
approximations f o r diodes with
ma.
are
values
o f the
f o u n d q u i t e u s e f u l i n the
tunnel
first
the
approximation i s that
Ip/ly
+ B(exp(bv) - l )
The
constants
Two
and
(-av)
s e c o n d i s the
voltage
accurately.
current
the
four
and
exp
constants.
and
The
2
Tables A I I . l
peak c u r r e n t s
and
AH.2
give
normalized
to
the
one
74
Exponential Approximation
Diode
f ( v ) = 0.044v e x p ( - l 6 . 8 v ) + 5.4x10
Ge
(Ip = 1
7
ma)
(exp(l5.4v)-l)
...AII-3
GaAs
f ( v ) = 0.026v. exp(_-10v) + 1. 5 x l O ~ (exp ( 8 . 7 6 v ) - l )
7
...AII-4
Table AII.2
Exponential Approximations
f o r Tunnel
Diodes
75
APPENDIX I I I
AIII.
F a c t o r s I n f l u e n c i n g the Choice
R i n a Multistate Circuit
In choosing
state
circuit,
a value
of the Load L i n e
Resistance
f o r R i n a two t u n n e l d i o d e
multi-
we must compromise between t h e n e c e s s i t y o f making
all
t h e s t a b l e s t a t e s a c c e s s i b l e (i»e., R as s m a l l as p o s s i b l e ) ,
and
the requirement
unstable
(switching load l i n e ;
a p p l i e d to give
provided
L
A stability
test
c a n be
t h e minimum l o a d r e s i s t a n c e f o r s w i t c h i n g ,
known.
fe
static
r e s i s t a n c e r e g i o n s be
R large).
and C ( k :yi>'2) *are
The
the
t h a t the negative
problem of determining
characteristic
w h e t h e r a g i v e n p o i n t on
i s s t a b l e when t h e c o m p o s i t e d e v i c e i s
c o n n e c t e d i n a g i v e n e x t e r n a l c i r c u i t , c a n be answered i n terms
of
the
l i n e a r network a n a l y s i s .
circuit
by a s m a l l
The d e v i c e
signal
linear
itself
i s replaced i n
e q u i v a l e n t c i r c u i t , which i s
valid
f o r s m a l l v a r i a t i o n s o f c u r r e n t and v o l t a g e
point
i n question.
The e q u i v a l e n t c i r c u i t
about t h e
must i n c l u d e a l l s t r a y
(32)
reactive
elements.
The c h a r a c t e r i s t i c
equation
can be d e r i v e d f r o m t h e s e t o f d i f f e r e n t i a l
the
linearized
equation
unstable.
circuit.
have p o s i t i v e
' of the system
equations
representing
I f the r o o t s of the c h a r a c t e r i s t i c
real
p a r t s , the p o i n t i n v e s t i g a t e d i s
I n o t h e r words, t h e t r a n s i e n t f o l l o w i n g any s m a l l
displacement
For
f r o m t h e p o i n t i s a sum o f i n c r e a s i n g e x p o n e n t i a l s .
t h e s e t o f two t u n n e l
r o o t s a r e g i v e n as e i g e n v a l u e s
diodes,
a , a , a
n
the c h a r a c t e r i s t i c
of the matrix:
76
evaluated
a t the p o i n t
of i n t e r e s t .
This
matrix i s obtained,
by
(33)
adapting,., t o o u r s p e c i f i c
stability
o f two i d e n t i c a l
c a s e , Moser's
tunnel
W i t h L and C ^ f i x e d ,
the
negative
resistance
diodes
i n v e s t i g a t i o n of the
i n series.
one c o n d i t i o n f o r i n s t a b i l i t y i n
region
i s that the load
line
resistance
R should
be l a r g e r t h a n t h e magnitude o f t h e l a r g e r o f t h e two
negative
resistances
of the d i o d e s .
o f R = 50 ohms, f o r t h e c i r c u i t
4.5,gave a f a i r
satisfactory
under c o n s i d e r a t i o n
margin of i n s t a b i l i t y
switching.
I t was f o u n d t h a t
a
value
i n section
- sufficient for
77
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