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Masters Theses Student Theses and Dissertations
1969
Neutron induced changes to the Modified Ebers-Moll transistor Neutron induced changes to the Modified Ebers-Moll transistor
model used in the NET-1 program model used in the NET-1 program
Kenneth Robert Smith
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Recommended Citation Recommended Citation Smith, Kenneth Robert, "Neutron induced changes to the Modified Ebers-Moll transistor model used in the NET-1 program" (1969). Masters Theses. 6996. https://scholarsmine.mst.edu/masters_theses/6996
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NEUTRON INDUCED CHANGES TO THE MODIFIED EBERS-MOLL TRANSISTOR MODEL USED
IN THE NET-1 PROGRAM
BY
KENNETH ROBERT SMITH J 9 Lf f I
A
THESIS
submitted to the faculty of
THE UNIVERSITY OF MISSOURI - ROLLA
in partial fulfillment of the requirements for the
Degree of
MASTER OF SCIENCE IN ELECTRICAL ENGINEERING
Rolla, Missouri
1969
_..,-. I I ./ I o -,
ii
ABSTRACT
The development of theoretical and emiprical relations which predict
the permanent changes to the electrical characteristics of silicon transistors
is described in this paper. Analytical techniques are also developed to obtain
the parameters used in the Modified Ebers-Moll transistor model.
The method employed is to determine the effects on externally measur
able characteristics and to relate these changes to the parameters in the
Modified Ebers-Moll transistor model. Further, all nonlinear characteristics
are related to the independent variables with interpolating polynomials and
degraded characteristics presented. Then degraded characteristics are used
to establish the parameters for and parameter changes to the transistor model
used in the iterative NET-1 digital computer program.
The result is a complete technique to predict the characteristics
V BE (sat)' V BE (forward) ' V C E (sat) ' ICBO' and hFE as functions of neutron
fluence to the Modified Ebers-Moll transistor model. For common emitter
de current gains down to approximately unity, a satisfactory nonlinear model
exists for the neutron degraded characteristics found in silicon transistors.
Typically, the gain characteristic is established to within 4 percent of the
desired characteristic for the transistor model.
iii
ACKNOWLEDGEMENTS
Acknowledgement goes to Mr. W. C. Watson of the GSE Laboratory at
Redstone Arsenal, Alabama, who supported the off-campus portion of this
writing.
Acknowledgement is made to Dr. C. A. Goben whose comments and
research findings served as the basis for much of the nuclear effects predic
tions. A special note of thanks goes to Mr. C. R. Jenkins who gave of his
time to explain the operation of the data acquisition system and computer data
reduction programs developed at the Graduate Center for Materials Research
at the University of Missouri -Rolla.
Acknowledgement also goes to Dr. Harry Miller of Defiance College,
Defiance, Ohio, who assisted in the establishment of the environmental
algorithm and to my graduate advisors, Dr. Robert C. Peirson and
Dr. N. G. Dillman.
iv
TABLE OF CONTENTS Page
LIST OF ILLUSTRATIONS • • • • • • • • • . • • • • • • • • • . • • • • • • . • • . vi LIST OF TABLES ..• ·. . • . . . . • . . • . . . . . . . . . . . . . . . . . . . . . . viii SYMBOLS . . . • • • • • . • • • . • . . • . • . . . . . . . • . . . . . . . . . . . . . . ix ABBREVIA. TIONS • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • .xii INTRODUCTION • . . • • • . • • • • • • • . . . • • • • • • • • • • • • . • . • • . • . 1
I. ENVffiONMENTAL DEFINITION • • • • • • • • • • • • • • • • • 3
II. DC FORWARD CURRENT GAIN AND NEUTRON EFFECTS UPON GAIN • • • • • • • • • • • • • • • • • • • • • • • 9
lli. TECHNIQUES USED TO CALCULATE POST-ffiRADIATED DC FORWARD CURRENT GAIN • • • • • • • • 18
IV. MODIFIED EBERS-MOLL TRANSISTOR MODEL
V. TECHNIQUES USED TO CALCULATE DC PARAMETERS FOR THE MODIFIED EBERS-
• • • • • •
MOLL TRANSISTOR MODEL •••••••••••••••••••
VI. PREDICTION OF V B'E (forward) , V BE (sat) , AND
V CE(sat) CHARACTERISTIC CHANGES ••••••••••••
Vll. GAIN CURVE FITTING FOR THE 2N1711 TRANSISTOR ••
Vlli. GAIN DEGRADATION TO THE 2N2907 TRANSISTOR AND THE MODIFIED EBERS-MOLL TRANSISTOR
35
44
70
82
MODEL . . . . . . . • • . . . . • • . . . • . • . • . • . . . . . . • . . 90
IX. DISCUSSION, CONSLUSIONS, AND RECOMMENDATIONS • • • • • • • • • • • • • • • • • • • • • • • • 106
Appendix A. FORWARD GAIN MODIFICATION AND "BEND AWAY" APPROXIMATION. • • • • • • • • • • • • • • • • • • • • • • • • • • 110
Appendix B. MODIFIED EBERS-MOLL CONVENTIONS AND MODE DEFINITIONS. • • • • • • • • • • • • • • • • • • • • • • • • • • • • • 117
Appendix C. DETERMINATION OF BASE TRANSIT TIME • • • • • • • • • 119
Appendix D. PREDICTION OF V CE(sat) • • • • • • • • • • • • • • • • • • • • 124
Appendix E. K' FROM REACTOR DATA FOR IC CONSTANT...... • 127
v
TABLE OF CONTENTS (Concluded) Page
Appendix F. REMOVAL OF NONLlNEAR DAMAGE AND K' POLYNOMIAL COEFFICIENTS • . . . . . . . . . . . . . . 133
Appendix G. COLLECTOR LEAKAGE RELATIONS AS A FUNCTION OF FLUENCE . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 137
Appendix H. EMPIRICAL ANNEALING REIA TIONS. • • . . . . . . . • . . 142
Appendix I. ENVffiONMENTAL DEF~ITION • • • • • • • • • • • . • . . • 152
Appendix J.. CONSTANTS AND LEAST-SQUARED-ERROR CURVE FITTING PROGRAM • • • • • • . • • • • . • • • • • • . . . • • . . 172
BIBLIOGRAPHY . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 178
VITA ........ ·. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 181
Figures
Il-l.
III-1.
III-2.
lll-3.
111-4.
111-5.
IV-1.
IV-2.
IV-3.
V-1.
V-2.
V-3.
V-4.
Vl-1.
Vl-2.
Vll-1.
VII-2.
vn-3. Vlll-1.
VIII-2.
VIII-3.
VIII-4.
VIIT-5.
VIII-6.
VIIT-7.
LIST OF ILLUSTRATIONS
Base current components controlling hFE .......... .
hFE versus collector current . . . . . • • . . . . . . . .... .
Gain bandwidth product versus emitter current
t d versus collector current • • • • • . . . . . . . . . . ..... .
Composite damage factor versus emitter current
vi
Page
11
19
22
24
density . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
Current versus voltage for pre- and post-irradiation ....
Modified Ebers-Moll model . . . . . . • .•••.•..•.....
Modified Ebers-Moll representation for forward region ...
Typical neutron degraded h.,FE characteristic • . . . . . . .
Typical forward characteristic . . . . . . . . . . . . . . . . . .
Typical curve fitting results for hFE versus Id . . . . . • . .
Typical ~E versus junction voltage for various <I> • •••••
Typical curve fitting results for Vi versus h . . • o • o • o o
FE Forward collector current changes •••............
Saturation prediction ...•...•............. o •••
2N1711 hFE versus fluence characteristic o • o •••••••
Resulting hFE curve fit using polynomial . . . . . . . . . 0 0
Current gain versus internal voltage for 2Nl711 transistor. 0
t d measurement • • • • • • . . . • . • . . . . . . . . . • . . . . 0 •
Comparison of base transit time . . . . . . o • o • • • • • • • •
Determination of base transit time . . • . . . . . . • . . • . . o
Gain· characteristic for various values of fluence
.Degradation of the 2N2907 versus neutron fluence . . . . . . 2N2907 Forward characteristic ••.•••.•.•.••.....
Gain versus junction voltage for various fluences . • . . . .
28
36
41
42
48
56
57
59
71
77
85
86
88
92
93
94
96
97
99
100
Figures
VIII-8.
VIII-9.
B-1.
C-1.
D-1.
E-1.
E-2.
F-1.
G-1.
I-1.
I-2.
I-3.
I-4.
I-5.
I-6.
I-7.
I-8.
I-9.
I-10.
I-11.
LIST OF ILLUSTRATIONS (Concluded)
Logarithm of hFE versus junction voltage ..
2N2907 Forward characteristic for 10 14 RDU
Current and voltage conventions for tran.sis.tor model .
Base transit time
V CE versus IB ...
Reciprocal hFE versus neutron fluence
Damage conversion
Damage removal . .
Intrinsic recombination rate
Damage-distance relations for targets
Peak ground overpressures for 1-KT burst
Range at optimum height (R ) 0
Scaling factor for initial gamma radiation
Initial gamma radiation for 1-KT air burst .
Deviation of gamma dose for weapon variations .....
Fluence for 1-KT burst in air of 0. 9 sea level density
Neutron fluence versus yield for severe damage to a transportation vehicle .................... .
Neutron fluence versus yield for moderate damage to a transportation vehicle .................... .
Gamma dose versus weapon yield for severe damage to a transportation vehicle . . . . . . . . • . . . . . . . . ..
Gamma dose versus weapon yield for moderate damage to a transportation vehicle . . . . . . . . . . . . . . . . . . . ..
vii
Page
102
104
117
121
125
127
130
133
140
155
157
158
159
161
162
163
164
165
166
167
Table
1-1.
V-1.
VI-1.
VII-1.
VII-2.
VII-3.
VIII-I.
VIII-2.
A-1.
B-1.
1-1.
1-2.
LIST OF TABLES
Neutron/ gamma ratio
Error criteria for Program 4-1 .....•..
Options for predictions of degraded characteristics .
Saturation parameters for 2N1711 transistor ........ .
Gain versus junction voltage for 2N1711 transistor •..••
Polynomial coefficients for 2N1711 transistor .•••••••
Gain-bandwidth product versus emitter-current . . . . ..
Forward parameters for 2N2907 transistor. • • • • • • • . •
Typical &lFE versus R C ..................... ·
Modes for model
Vehicle type and damage . . . . .
Values for vehicle/ damage index ..... .
Page
4
49
78
84
87
89
91
105
115
118
153
153
viii
K v
K'
~B
SYMBOLS
Neutron induced emitter-base depletion region current
Emitter area in square centimeters
Area damage constant for IE cp
Volume damage constant for IEcp
Voltage dependent depletion region width
Collector de current
de Emitter current
de Base current
Collector to base voltage
Empirical damage factor
Base transit time
Pre-irradiated value of hFE
Surface recombination generation current
Surface channel current
Recombination generation current in emitter-base depletion region
Bulk diffusion or bulk recombination generation current
Reverse diffusion from the base to the emitter
Base to emitter forward bias voltage
ix
v BE (forward)
V BE (sat)
1CBO
hFE
N(E)
E
cf>
F(E)
K,k
q,Q
T
n
t e
a N
SYMBOLS (Continued)
Base to emitter forward bias voltage
Base to emitter voltage during saturation
Collector to base leakage current
de Forward current gain
Differential energy spectrum
Energy
Fluence in RDU or n/ em 2
Energy dependent damage ratio for neutrons
Boltzmann's Constant
Electronic charge
Temperature in degrees Kelvin
Pi
Emitter current dependent gain-bandwidth product
Emitter-base transition capacitance
Delay time
Change in IRB
Emitter forward junction voltage
Emission constant
Emitter delay
Total delay time
Emitter current density
Active normal beta
Active-inverted beta
Active normal alpha
X
xi
SYMBOLS (Concluded)
0! Active inverted alpha I
1EF Forward emitter current
1CF Forward collector current
v2 Collector-base forward bias
ME Emitter emission constant
M Collector emission constant c
Rc Collector base leakage resistance
RE Emitter-base leakage resistance
REE Emitter bulk resistance
Rcc Collector bulk resistance
RBB Base spreading resistance
RDU
MeV
FBR
WSMR
n/cm2
LOG
LN- 1n
ABBREVIATIONS
Radiation damage unit
Million electron volts of kinetic energy
Fast Burst Reactor
White Sands Missile Range
Neutron density in neutrons per square centimeter
Common logarithm (10)
Natural logarithm (e)
xii
1
INTRODUCTION
In the past decade much research has been conducted studying the
effects of nuclear radiation on the solid state materials used in the manufacture
of transistors and diodes. These effects are being related to the device
characteristics in a continuing effort, and the understanding of device behavior
is enhanced through these efforts.
Aside from the research and study of specially made devices, there is a need
to predict, within reasonable accuracies, the effects upon the electrical character
istics of the commerically produced devices. The characteristics presented in this
writing are for fast neutrons ( E > 10 KeV) at times greater than 104 seconds after
exposure. The time factor allows the fast, or beta, anneal time to stabilize.
Not only are the predicted characteristics necessary, but for radiation
damage short of destruction, there is a need for a nonlinear analysis capability
using the degraded characteristics. This is primarily a result of a degradation
of the de forward current gain ( hFE) . The immediate need for reasonably
accurate circuit analysis is met by the techniques developed in this thesis.
The characteristics for the irradiated and nonirradiated transistors
are approximated for nonlinear circuit simulation through the relations and
equations for the Modified Ebers-Moll transistor model used in the computa
tionally iterative computer program called NET-1. The NET-1 program, with
its transistor model, was chosen, as it is a mature working program with
iterative capability to solve circuit equations when the de forward current
gain is made a function of the internal emitter-base junction voltage. NET-1
gives reasonably good approximations to device behavior when the normal
manufacturing variations are considered from a design viewpoint.
2
A subtopic of this writing is a definition of the radiation environment
surrounding a nuclear weapon detonation and the assumption made to implement
a mathematical description of silicon transistors subjected to the radiation.
The radiation from a nuclear weapon provides the basis of environ
mental definition; therefore, the means of comparison of damage from
simulated environments becomes necessary. This results because almost all
experimental research is being conducted in nuclear reactors. This correlation is
important in this writing not only because of the necessity to relate back to a
nuclear weapon but because the technique of predicting de current gain degrada
tion is established using data from reactors with different spectra. The dif
ferences are established in Section I along with the assumptions made on the
importance of the types of radiations for the weapon environment.
The primary topic is the degraded electrical characteristics and the
resulting modifications to the Modified Ebers-Moll transistor model.
Section II is presented to establish a knowledge of transistor behavior from an
electrical characteristic viewpoint to allow for the approach to device model
ing to be understood from a theoretical viewpoint as well as the terminal
characteristic approach necessary for computer simulation of circuit
relations.
3
I. ENVffiONMENTAL DEFINITION
In order to predict permanent and semi-permanent electrical charac-
teristic changes to silicon transistors exposed to the radiations from a nuclear
weapon, it is necessary to describe the radiations and the effects of each upon
the electrical characteristics through changes in the physical properties of the
constituent materials.
Neutron, gamma, and thermal radiations are produced in a nuclear
weapon detonation. For the distances established in Appendix I as being of
practical interest, the thermal radiations are of little concern and will be
neglected. Neutron and gamma radiations are considered relative to the dis-
placement and ionization damage caused to the silicon materials in the
transistors.
As a first approximation, the permanent and semi-permanent displace-
ment damage is considered to be caused only by the bombarding neutrons. It
10 is reported by Larin that gamma radiation induced displacements are negligible
compared to neutron induced displacements if the ratio of neutrons per unit
area to roentgens per unit area is greater than 107• This assumes fast
neutrons or neutrons with energies above 0.1 MeV of kinetic energy.
The assumption that the fast neutrons are responsible for the displace-
ment damage resulting from a nuclear weapon detonation is supported by the
4
data in Table I-1. The ratios presented are for electronic equipment in a mili-
tary transportation vehicle that is placed at a distance from the detonation so
as to sustain either moderate or severe mechanical damage. The ratios are
given at this distance as the neutron dose received here is sufficient to
cause significant degradation to the de forward current gain. The data and
information concerning damage, vehicle type, neutron dose, and gamma dose
are given in Appendix I.
Table I-1. Neutron/ Gamma Ratio
Neutron to Gamma Ratio ( n/ em 2) I roentgen
Yield (KT) Moderate Mechanical Damage Severe Mechanical Damage Transportation Vehicle Transportation Vehicle
Ground Burst Ground Burst
1 4 * 10 10 5 * 109
10 1 * 109 5 * 109
100 3 * 109 5 * 108
Air Burst Air Burst
1 2 * 109 3* 10q
10 3 * 108 1* 108
100 1 * 108 1 * 108
For the nuclear weapon environment, surface damage will not be con-
sidered whether it originates from gamma radiation effects to the surface or
from neutron ionized gases in the transistor cans. Although the gamma
radiation is relatively small, there is still the possibility of damage to the
passivated surfaces of the transistors. This results in surface damage
frequently termed nonlinear damage. A technique of surface damage recogni
tion and removal is given in Appendix F.
5
In a fission reaction, such as that existing in a nuclear reactor or a
fission type weapon, there is an initial unique distribution of neutron kinetic
energies. The differential energy spectrum is approximated by Watt's relation
and is termed the Watt's fission spectrum. The data presented in Appendix I
are given with the assumption that the differential energy spectrum is well
approximated by the Watt's relation. In a nuclear reactor the Watt's spectrum
is modified by shielding or moderated by water that slows the neutrons. As
the displacement damage to silicon is a function of the kinetic energy of the
bombarding neutron, the difference between the Watt's spectrum and the dif
ferential spectrum of the reactor used to simulate the environment becomes
important.
To correlate the damage between different spectra requires that a
basis be established for the experimentation and a conversion of the Watt's
spectrum to this basis be made. The basis is established by making the
damage per neutron a function of energy with normalization to unity at 1 MeV.
The arbitrary unit, called radiation damage unit ( RDU) , is then established as
the neutron density at a particular location in the WSMR Fast Burst Reactor.
An important point is that this reactor is identical in structure to the Sandia
Pulsed Reactor.
6
The energy dependence of damage has been computed for elastic
scattering of neutrons and is supported by carrier removal data. Data for
lifetime damage have not appeared, but the function for carrier removal is
assumed to be applicable.
Using the damage rate assumption in the preceeding paragraph, the
relation for converting any differential energy spectrum to the equivalent Fast
Burst Reactor spectrum is given by:
00
4> FBR (RDU) = { F (E)*</> x (E)*dE (I-1)
where F (E) is given in RDU per neutron as a function of energy (E). This
relation is given quite accurately for the 0- to 5-MeV range, but the values
reported for 14 MeV range from. 2 to 3. In the case in which a fusion weapon
is expected, it is recommended that a value of about 2. 5 be used.
As it is intended that the nuclear weapon be used as the reporting basis,
it is necessary to evaluate a multiplicative factor between neutrons per
centimeter squared and RDU for the Watt's fission spectrum. This is most
easily accomplished by performing the integration in equation I-2 where N (E) 00
jN (E)* dE = R (I-2) 0
is the differential energy spectrum of the Watt's spectrum. Normalization of
equation I-2 is done simply by dividing by R. Performing the integration of
this normalized spectrum using equation I-1 results in equation 1-3 as follows:
<I> 00
FBR _ f F (E)* N(E) *dE rpx 0 R
(1-3)
Evaluation of equation I-3 for the Watt's spectrum results in a value of 1/0.83
which indicates that the Watt's fission spectrum is about 17 percent more
damaging than the FBR spectrum for the same r.umber of neutrons per unit
area. A value of cpFBR/cpx greater than unity then indicates that more FBR
neutrons are required to cause the same damage as the spectrum under
consideration.
7
To convert the output of the computer program in Appendix I to RDU,
for the purpose of using the composite damage factor, requires that the output
be multiplied by 1. 17 as indicated in equation 1-4:
RDU = 1. 17* (Y neutrons/ em 2) (1-4)
The displacement damage to silicon transistors is not confined to
changes in carrier lifetime in the neutral base region. There is a neutron
induced base current component which is indicated as resulting from recom-
bination in the emitter-base depletion region.
It can be stated that in general the relative damage rate per neutron
will not be the same in the depletion region as in the neutral base. This means
that F(E) used for carrier removal rate (and assumed to hold for carrier
lifetime changes) cannot be generally assumed to hold for the neutron induced
component in the depletion region. The solution for the technique established
for this writing lies in the fact that the reactor used to establish RDU is
identical in structure to the Sandia Pulsed Reactor that was used to establish
the area dependent coefficients for the neutron induced emitter-base depletion
region current component. It can be safely assumed that the spectra of the
two reactors are near enough identical to equate neutrons/ cm2 to RDU for the
prediction equations.
8
In conclusion, the values of neutrons/ cm2 calculated by the relations
and computer program in Appendix I need only be multiplied by 1.17 and used
in the transistor damage program to calculate degradation for the nuclear
weapon neutron dose. The carrier removal data used to establish F(E) is suf
ficiently correct as is borne out in the continuity of the base current charac
teristic for the 2N2907 transistor analyzed in Section VIII.
II. DC FORWARD CURRENT GAIN AND NEUTRON EFFECTS UPON GAIN
This section presents the techniques used to predict a post-irradiated
value of de forward current gain.
Observation of neutron degraded electrical characteristics shows that
9
de forward current gain is the most sensitive characteristic to neutron induced
changes while lesser changes are observed for V BE (forward), V BE (sat) ,
V CE (sat), and ICBO at a particular collector current.
Presently there are two approaches being used to predict changes in de
forward current gain. The first is based upon the assumption that base recom-
bination is responsible for the base current component that dominates hFE and
that the recombination rate increases linearly with neutron fluence. The
second approach is based upon the forward gain change resulting from an
induced base current component as a result of recombination in the emitter-
base depletion region. Both approaches are used for the regions of dominance
of hFE for the base current components.
The technique of gain prediction using the base recombination rate
increase is limited to a very small collector current region and is complicated
by emission crowding. As a practical means of predicting changes, the
damage to the recombination rate is made a function of emitter current
density, base transit time, and neutron fluence. This results in an empirical
relation that becomes useful over a larger current density range. This
empirical relation is used in this writing for emitter current densities
between 0. 1 amp/ em 2 and 1000 amp/ em 2•
10
It is noted that at the lower current densities the composite damage
factor must predict a base current increase equal to the increase using the
induced component in the depletion region so as to make the base current a
continuous characteristic. Further, the slope of the base current character
istic (when plotted on a log scale versus V BE) must approach the slope of the
depletion region dominating base current component. These two relations
prove to be most useful as will be seen in the section on the 2N2907 modelling.
The neutron induced base current component introduced by recombination
in the emitter-base depletion region dominates gain in the region where the
emitter current density is less than 0. 1 amp/cm2•
There are two relations appearing for the depletion region component.
The first is an emitter area dependent relation and is used as the correlation
between the reactor spectra and is readily accomplished. The second relation is
more nearly related to actual physical conditions by making the component a
function of depletion region volume. This relation is not emphasized here
because of the capacitance measurements needed to establish the width of the
depletion region between the base and emitter regions.
For the purpose of discussion of the base current component dominating
de forward current gain, attention is called to Figure Il-l. In this figure ten
decades of collector current are shown only for discussion and should not be
used for quantitative purposes.
100
Ill 10 u.. J:
1
0.1
-10 -9 -8 -7 -6 -5 LOGic
... -3 -2 -1
..-----'!G 1s ~ ID'
~ PRE·IRRADIATED "\ ( r A ..._
~ POST·IRRADIATED ,/\ j '-----v---/ --------~v ~ S v S , ,
1Et 1s 1F!B D
NEUTRON INDUCED CURREI'tT
ANALYSIS METHOD 2
:11 0.1 amp/em
COMPOSITE DAMAGE FACTOR
Figure Il-l. Base Current Components Controlling hFE 1-' 1-'
12
To determine the practicality of application of the two methods, it is
necessary to study the definition of hFE gain and to define the regions where the
given base current components are dominant along with the qualitative effects of
bombarding neutrons. From Figure Il-l the definition of the reciprocal of de
forward current gain may be observed. This is given by:
where:
= 1B = 1ES + 1EM + ~G + 1RB + 1D' + 1Ecj> + 1CBO
Ic Ic (11-1)
IES is the surface recombination generation current in the emitter
base depletion region. This component is of surface perimeter
origin and therefore does not cause a deviation from the ideal
characteristic. It is reported by Goben that there are no neutron
effects of significance upon this component. This current component
is proportional to exp (q*V/n*K*T) where n is approximately 1. 5
in value.
I is a surface channel component and is presently considered EM
negligible in surface passivated silicon transistors. There are no
reported neutron effects at RDU < 10 15 n/ cm2•
~G is the recombination current in the emitter-base depletion region
and does not modify the ideal characteristic. There are no
reported neutron effects by neutron bombardment. This com
ponent is proportional to exp (q*V/2*K*T) and is apparently
negligible above 0. 3 to 0. 35 volt emitter to base forward bias.
~B is the bulk diffusion or bulk recombination generation current.
This component represents that portion of the normal diffusion
current that does not reach the collector. This component
contributes to a .. deviation from the ideal characteristic and is
affected by neutron bombardment. The current is proportional
to exp (q*V/K*T).
In' is the reverse diffusion current from the base to the emitter.
13
This term exhibits a proportionality of exp (q*V/K*T), but the
multiplying value is two to three orders of magnitude smaller
than the normal diffusion current therefore making it negligible.
This is done by making the doping level in the emitter much
higher than that in the base.
is the component of base current induced by neutron bombardment. 8
This component is reported by Goben to originate in the bulk emitter-
base transition region. This component modifies the ideal diode
characteristic and, because of its magnitude, will dominate
current gain from the low currents through the medium currents.
This component is analyzed by a study of emitter efficiency and
exhibits a proportionality of exp (q*V/nK*T) with n ~ 1. 5.
ICBO is the total leakage current at the collector base junction and is
affected by neutron bombardment. In silicon transistors this
component is assumed negligible, but after irradiation it may
become important in the very low collector current region.
The only two components considered relative to neutron changes are
IRB and IE<P" The changes to IES' IEM' and IRG are assumed negligible.
The change to I is discussed in Appendix G, but is not presently included CBO ·
as only the V = 0 characteristic is considered. CB
For current densities above 0. 1 amp/ em 2, the gain changes are of bulk
or displacement damage origin and are primarily the result of the neutron
induced changes in the recombination rates in the base region. The changes
affect the base recombination current, but the emitter-base translation region
10 recombination current is reported by Larin to not be affected. At higher currents the
diffusion component dominates, but the effects upon this component are not
predictable. However, the composite damage factor will include any effects,
as it is established by measurement of characteristics.
It has been determined that IRB is affected linearly with neutron
fluence and that the relation
1 = + t *K'*¢
b
holds quite well for the normal forward currents encountered where the
following definitions hold:
h = de forward current gain for active normal mode. FE
hFE = pre-irradiated de current gain. 0
~ = average base transit time. Time it takes a carrier to cross
the base region.
K' = empirical damage factor.
¢ = neutron dose in RDU.
14
This reciprocal gain relation is empirical, and the only justification for using
~is that damage correlates better when~ and K' (as a function of emitter
current density) are used in this relation. This relation also allows for
determination of the presence of gamma induced surface damage. This
technique will be presented in the reduction of the nuclear reactor data in
Appendix F.
To reduce the empiricism of the reciprocal gain relation, it would be
necessary to correlate IRG to the emitter transition region volume and con
centration distribution, but presently it is improbable that higher accuracies
15
can be attained by such a correlation, as these functions are not accurately
measurable. From this point, the empirical relation will be utilized and any
variations noted and qualitatively explained.
Therefore~ to calculate an irradiated value of de forward current gain
for emitter current densities above 0. 1 amp/ em 2, it is necessary only to have
numerical values for the original gain ( hFE 0
) , the average base transit time
( ~), and the composite damage factor (K').
The second method for gain prediction is used in the low to medium
collector current range. For Figure II-1 this applies to the first four to five
decades of collector current. In this region it has been observed that gamma
radiation may cause permanent damage. This damage is termed surface or
nonlinear damage and its effects saturate. In the nuclear weapon environment
this component has been assumed negligible (Section I).
The predominant effect upon de forward gain in the low to medium
current region is reported to result from a recombination current in the
emitter-base depletion region. The prediction of this current component,
which adds directly to the base current, requires a knowledge of the recombi-
nation in the depletion or transition region. This then implies that a volume
dependence exists; however, an emitter area dependence has been established
8 by Goben that gives the induced component within a factor of two for all
measured cases. This relation is given by:
where:
K 1 is the area damage constant and has values of approximately
3. 3 Io- 22 to 6. 6 I0- 22 (amp/ cm 2)/ (nvt). A table of typical
values for several transistors is given in Appendix J.
AE is the emitter area in square centimeters.
cf> is the neutron dose in nvt or RDU (Section I).
n is approximately 1. 5. Typical values are given in Appendix J.
Reasonable results are attained by use of the area dependent function;
however, a volume dependent relation has appeared in the literature, and its
8 validity appears certain. The volume relation is given by Goben to be
where:
I = K *X (V )*A >:C cp•:<exp( qV / nKT) Ecp V m E
KV is the volume damage constant for the neutron induced depletion
region recombination current in units of (amp/ cm3)/ (nvt)
(nvt * RDU for this case). Appendix J gives typical values.
X (V)is the voltage dependent transition region width in em. m
AE is the emitter area in em 2•
cf> is the neutron dose in nvt.
n is the constant to modify the slope of the logarithm of IE <P.
The only unique problem for using the volume dependent function
instead of the area dependent function is the determination of the depletion
region width (xm (V)). The width can be attained by a capacitance
16
measurement across the junction. The technique will not be discussed here,
as the area dependent relation will be used for the degradation analysis.
The two techniques for determining IE cp have been established for the
low to medium C()llector current region and the composite damage relation
established for the medium to high currents. By use of the area dependency
and the composite damage factor it is possible to establish a complete
technique for gain prediction. This will be the topic of Section III.
17
18
III. TECHNIQUES USED TO CALCULATE
POST-IRRADIATED DC FORWARD CURRENT GAIN
In order to effect a gain prediction with use of the relations from
Section II, it is necessary to establish the appropriate model for the base
current components in the respectively dominated regions. As the normal
operation of the transistor is above an emitter current density or 0. 1 amp/cm2,
this region will be discussed first.
In order to establish a post-irradiated current gain characteristic for
a transistor bombarded by fast neutrons, there are 4 parameters necessary.
For current densities above 0. 1 amp/ em 2, these are ( 1) original de current
gain (hFE )• (2) base transit time ( \), (3) a value for the composite 0
damage factor (K'), and (4) the neutron dose in RDU.
To realize a numerical calculation, it is necessary to have an
independent variable that can be related to the four dependent variables
previously listed. This variable must be an externally measurable parameter
of which de current gain is a function. As de current gain is usually measured
as a function of collector current (Figure III-1), and the V BE versus LOG10 (Ic)
characteristic change to neutron fluence is negligible relative to base current
characteristic change, the collector current is chosen as the independent
variable upou which to base the post-irradiation de forward gain calculation.
I ' \
~ t'.. ~ "'
~g o
· 0
0 0
.... N
-CIO
\
0 0 0 ..... . 0 ..... 0 • 0
..... 0 0 • 0 ..... 0 0 0 . 0 ..... 0 g 0 . 0
0
19
..., §3 ~
s u ~
0 ..., C
) <U
--~
0 .!.
u u
fll :::s fll ~
<U >
rz:l r:z;..
..c::
....-! I ~ ~
~
<U
~ ...... r:z;..
20
Appropriately, the relations used to relate dependent variables to the
collector current will be presented.
The base transit time is made a function of collector current by the
following two relations :
(III-1)
-2,-:C 17"-,-:c f-: -( I_E_)_ - _K~_T (-~:) (III-2)
A discussion of definitions and a qualitative evaluation of equation III-2
is made in Appendix C.
A point to note is that while the tb relation is valid theoretically, in
practice, difficulty results if manufacturer's data are being used. The
difficulty manifests itself in a nonconstant value of ~ being calculated for
various values of collector current. This difficulty is the result of ft and td
not being specified for the same transistor, but instead being minimum,
nominal, or maximum values. Equation III-2 should be used only when values
of t and f are taken for the same transistor. d t
When manufacturer's data are used, the technique described in
Appendix Cis recommended. It is permissible to put CTE from this technique
into equation III-2, but substitution is unnecessary, as tb is found directly by
the graphic technique in Appendix C.
The term cp , the neutron dose in RDU, represents no difficulty as it is
necessary only to specify the dose for which the degraded characteristic is
desired. Experience has shown that initial numerical analysis is best
21
accomplished using cp in decades from about 10 10 RDU to 1015 RDU. These
six decades will generally cover the range of practical interest.
The term hFE , the pre-irradiated de forward current gain, is made 0
a function of the l.ogarithm of the collector current for purposes of curve
fitting. In the computer program that performs the calculations, individual
values of collector current and corresponding values of h are read directly FE
0
for the temperature in question.
It might be expected that a relation between h E , I , and temperature F C 0
would be appropriate so that the analysis could be performed at any tern-
perature. This presently is not possible, since above 35°C, annealing of
damage occurs. It then would be necessary to relate damage to temperature,
thus making the analysis quite complex.
The term K', the composite damage factor, has been established as a
function of emitter current density. It is theoretically a function of several
other variables, but for the purpose of prediction on a nominal basis, it is
made a function of emitter current density. This relation is empirical but can
be measured for an individual transistor with quite accurate gain prediction
resulting. Gain bandwidth product versus emitter current is shown in
Figure III-2.
For conditions other than 35 o C, passive irradiation, and times greater
than 104 seconds after irradiation, K' assumes functional dependencies upon
time, current, temperature and applied voltage. The data that are presentedby
experimenters is related to the empirical relation K' which makes any analytical
-400
300
-u
!. 200 .. -
100
'
TYPlCAL OF 2N2907 AT 27°C
/ ~
25 50
IE (ma)
7)
Figure 111-2. Gain Bandwidth Product Versus Emitter Current
110
~ ~
23
relation empirical, also. This approach offers little in the way of understanding the
problem and will not be presented here. Appendix H establishes the empirical
relations so that a quantative feel can be realized, but the relations are not
programmed in the degradation program.
The term AE, the emitter area, is needed for the purpose of
calculating the emitter current density. The physical area is measured either
by use of a measuring microscope or through use of photomicrograph
techniques.
As it is not necessary to retain the exact theoretical relations for a
numerical solution, empirical relations are established for h , td, f , and FE t
0
K'. These relations are in the form of interpolating polynomials of degree n.
The coefficients are established using the least-squared-error criterion for a
Taylor series expansion about zero. As the dependent variables hFE , K', 0
and sometimes td are more linear when plotted on a log scale, the exponents
of the dependent variables are fitted with the interpolating polynomial. The
polynomial then functions as an exponent in evaluation of a numerical value for
the dependent variable. This is equivalent to fitting the function with a
logarithmic expansion.
Figure 111-3 shows td versus collector current, and Figure
III-4 shows the composite damage factor versus emitter current density.
Now that the linear interpolating polynomial techniques are established,
the dependent and independent variables are shown below as the respective
interpolating polynomial:
24
\ ' 1\ ~
\ ~ \
8 Q
) ~ ~
=
0 ~
0 .... t,) Q
) ..... ..... 0
Q
0 - a
Ill E
-
=
_u
Ill ~
Q)
>
..,."0
. tv:) I ~ ~
) ~
Q) ~
So -~
--~
•
v Q
/ .... Q
---~
d
6
5
4 -'i' 0 .... .. ::.:
3
2
0 0.1
" \ \.
~ "" ~
10
~ ..........
1'----100
IE/AE (amp/cm2)
1000
Figure III-4. Composite Damage Factor Versus Emitter Current Density 1\j
c.n
c) Let P = ~G 10 (I C) LOG (hFE) = H1 +
d) Let E = 1 + LOG 10 (IE/ AE)
. • . +
. . . +
. . . +
A I n n+ 1 E
As it is desired to determine the new gain characteristic using the
increased base current component approach, it is necessary to develop a
26
relation that gives the base current increase for the region where the composite
damage function is used. At a given value of collector current, the change
in base current for a change in the de current gain is given by:
where I is the recombination generation current in the base region. RB
Establishing a common denominator,
A I = I * ( 1 - h jh ) ,.,. (1/h ) RB C FE FE FE 0
and
(IV-1)
27
but the simplified gain relation is given by:
1/h = 1/h I FE FE (IV-2)
0
Substituting equation IV-2 into IV-1 gives:
which simplifies to equation IV-3.
(IV-3)
Thus it can be seen that for current densities above 0. 1 amp/ cm2 the
base current increase is proportional to the collector current multiplied by
neutron fluence and a relatively constant value t >:CK'. b
In order to extend the gain prediction below an emitter current density
of about 0. 1 amp/ em 2, it is necessary to establish an expression for the base
current increase, at a particular collector current, as a function of neutron
fluence. In the computer solutions, either the area or volume dependence of
the induced component as established in Section II may be used, but for dis-
cussion here the area relation will be used. This relation is given by:
I = K 1 *A >:C ¢ *' exp ( qv 1/ n>:< K':c T) E¢ E
where n ~ 1. 5.
To illustrate the relative dominance of the two components, Figure III-5
is referenced. This figure gives the base current profiles for the nonirradiated
case and the case where the transistor has been neutron radiated to about 10 14
nvt. It is noted that the neutron induced component IE¢ will dominate the lower
28
0 I I I I I
-1
-2
-3
-4
-5
-6
<.:» -7 0 -1
-8
-9
-10
PRE· AND POST-IRRADIATED 'c CURRENTS AS A FUNCTION /I OF FORWARD VOLTAGE ,§ / 8 post
~ 'a -~~ ~ v pre ~
'j )
I ~ /-.. 1.0
!/ v / v
/; /""'1.5
J>+:-Y,f/ ~ ~/~
,~- .(~2.0 ~ /.
/ /j v /
/ v v
-11 I' I
I -12
, I
-13 ,I
I I
-14 /
0 0.1 0.2 0.3 0.4 0.6 0.7 o.s 0.9 1.0
Figure III-5. Current Versus Voltage for Pre-and Post-Irradiation
decades while the component having a slope nearly the same as the collector
current will dominate for the high current, high dose case.
By use of the previously established relations for the increase in the
base current, it is possible to set up an order of solution for current
densities above and below 0. 1 amp/ em 2•
follows:
For current densities above 0. 1 amp/cm2, the order of solution is as
(1) hFE 0
(5) t e
(7) \
( 10) K'
= P2 (Ic)
=
= ( t d * IB 1) I ( 2 * J v BB )
(K* T* CTE) I ( q* IE)
= P3 (IE)
= 1; (2 * rr * ft)
= t - t t e
= IE/AE
= p 4 WG (niE)
29
30
For current densities below 0. 1 amp/ em 2, the order of solution is as
follows:
( 1) hFE = p1 [LOG (Ic) J 0
(2) IB = 1c/hFE 0
(3) AlB = K 1 * A * r:/J * exp ( q * V BE j (n * K * T)) E
(4) h = FE ( r:p) IC /(IB +AlB)
The previous 16 equations can be used for hand calculation if desired;
however, the computer codes are presented at the end of this section for
performing the analysis on a digital computer. The codes are written in
Fortran IV, but no special characteristic of Fortran IV was used. To use
this program in any other version of Fortran, only the READ, WRITE, and
FORMAT statements need be changed. Several variable names contain five
characters and these may be changed with little effort. The functions ALOG10
and EXP may have to be renamed for other versions of Fortran.
These definitions apply
EXP(arg) arg = e
ALOG10 (arg) == log 10 (arg) .
The computer codes presented have been used many times and are
apparently without error. Discontinuity of characteristics will result between
the current density regions, but this is no fault of the program. This problem
will be discussed more in the se?tion on the 2N2907.
31
This completes the prediction of a post-irradiated de forward gain
characteristic. The use of this characteristic will be made more apparent in
the next section, in which modelling is discussed.
PROGRAM 1
c PROGRAM TO CALCULATE A V4LUE OF DC FORWARD CURRENT GAIN FOR A SiliCON IRANSISIOR SOBJECIED 10 NEUtRON BOMBAROMENr
PROGRAM WRITTEN IN FORTRAN IV
32
c c c c t ~
COLLEt lOR CURRENt "Jr~TJifi'"RE~IRRA-olATED RELATTmr·-yrr· VBE. EMISSION CROWDING 1 BENDAWAY 1 APPROXIMATED BY RBB IN MODEL.
c c c c c t c c c t c c c t c c c
LEAKAGE DATA PRESENTLY NOT USED. Z=O CAUSES THE STATEMENT NUMBERS AllOIIEO FOR LEAKAGE EQOAIIONS 10 RE SKIPPED.
DEFINITIONS Of SYMBOLS AND CONSTANTS
XK TMP
iKt XN XM A•S 0 B•S CEF
-ROLIZMANN 1 S CONStANt •TEMPERATURE IN DEGREES KELVIN •COULOMB IC CHARGE •AREA DEPENDENT DAMAGE CONSTANT TO EMITTER-B~~-LDEPLET~=O;:;..-::N __ REG IliN =SLOPE OF NEUTRON INDUCED ANOMOLOUS BASE CURRENT CONPONENT =COLLECTOR CURRENT EMISSION CONSTANT =GAIN POLYNOMIAL COEFFICIENTS =INIEGER TO SHIFf IHEL:O~lTF-cc--ro-zERO -- ----- --=COMPOSITE DAMAGE POLYNOMIAL COEFFICIENTS =EXTRAPOLATED COLLECTOR CURRENT INTERCEPT ON LOG CC AXIS AT EMITTER-BASE VOLTAGE EQUAL ZERO
DEFINITIONS OF VARIABLES c c C HFEO =ORIGINAL DC FORWARD CURRENT GAIN AT VC8•0. C HFEOL *ro-G"llF~I-\iA"nr- -----C HFE =FORWARD GAIN IN GENERAL C HFEL =LOG OF DEGRADED CURRENT GAIN t ttl EtOG OF COlLECtOR CORRENI C CE •EMITTER CURRENT IN AMPS C CC •COLLECTOR CURRENT IN AMPS
C CEL =LOG OF EMITTER CURRENT C DCE •DEGRADED EMITTER CURRENT T DCEl = lOG OF DEGRaucn--""EMT~-ct.JRR'El\IT --------- ----- ------- -- ------ ---C CB •BASE CURRENT IN AMPS C CBL =LOG OF BASE CURRENT C DGCB •DEGRADED BASE CURRENT CC OCBL *LOG OF OEGRAOEO BASE CURRENI
DELCB •BASE CURRENT INCREASE IN AMPS C AE •PHYSICAL AREA OF EMITTER IN SQUARE CENTIMETERS
+------ti~~~~Diril!!---..:o-i~HI~IHfH~i'-'TN~~=~A ~d~~ I!~~ MB I!~~ I ~~~~6m:1tr- --- ----- ---- -C ECD •EMITTER CURRENT DENSITY C CDKN =COMPOSITE DAMAGE CONSTANT NORMALIZED C CDK =COMPOSITE DAMAGE FACTOR C t *NEUTRON DOSE IN ROO -----------·---------- --· C R =INTEGER USED TO SHIFT LOG(HFEOJ TO ZERO TO AVOID NEGATIVE C NUMBERS.NUMBER ADDED TO MAKE LOG(HFEO) EQUAL ZERO AT LOWEST
-t---------V:..:A::..:L=-:U::..:E=--0=-c~-COll_ECTOR_CURR_~~~--C_(),.,~!_~fi~~Q _____ . ___ _
1 2 3 4
c
READ CONSTANTS USED IN PROGRAM READ(1 9 100) XK,TMP~Q, XK1 9 XN READ(1,100) XM,O~CtF,AE,TB REAOCI,tooJ At,A ,l3,A4,A5 READI1 9 100) Bl,B2,B3,B4 1 B5
--.---'C~---...;;D~A~T~A;;...-T.;-'O~.;:C.;;A:.-..l~Co.;:U~l~A~T:..-;E~l:;_E~A~K~A~G~E;.-...----...:.~-~---~-~-~--~---~ READil,lOOJAC,RIO,XRRG,XNA,XNO 6 7 8 9
10
11 12 13 14 5 16 17
18 19
c c
c c
c
READ(1,100)XNC,XNB,VCB,R,XCCL READ( 1,102) I TYPE REA0(1 102) ICODE WRITE(~,300)ITYPE,ICoDE WR ITE{3, 700)
READ INITIAL VALUE OF COLLECTOR CURRENT,NEt.J"T~O"" OOS_E,ANO FL~G READ I 1 I 40 lJ z R E AD ( 1, 40 lJ Y
30 READf1,801)CC,HFEO IFfCCJ 3,3,4
4 CONTINUE HFEOL=ALOGlOlHFEOJ CCL=ALOGlOICC) CALCULA liON OF I HE PRE IRRAOI A I ED VALUE OF---sASF-CURRENT CB=CC/HFEO CBL=ALOG10fCBJ CALCULATION OF EMITTER CURRENT
21 CEL=ALOG10fCE) CALCULATION Of E-MITTER CURRENT DENSITY c
..:;2;-:;2r------r,EC~Dr-==Cr-r.E=-/"'A'T-E~__,.,r-'l,-,---.,.,,-------~ -----~------23 IFCECD=.IJ 20,21,21
33
c c c
21 CALCULATION OF ADDITIONAL BASE CURRENT COMPONENT USING COMPOSITE DAMAGE FUNCTION
24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40. 41
42 43
c
21 CON I INOE ECDE=1.+ALOG10fECDJ D=ECDE CDKN=Bl+B2*D+B3*fD**2.J+B4*fD**3.J+B5*fD**4.J CO~=CO~N/(10.**6.) DELCB=CC*TB*CDK*Y DGCB=CB+DELCB DCBL=ALOG10CDGCB) AFE-CC/OGCB HFEL=ALOG10CHFEJ OCE=CC+DGCB DCEL=ALOG101DCE) WRIIEC3,210JCC,CE1C8 WRITEC3,500JCCL~CtL,CBL WRITE(3,211JHFEu,TB,Y WRITEC3,50ltHFEOL ::lf~t~:~A~I8E~l~gggl~~~et WRITE(3,214)CDK,DELCB,Z
41t ItS . 46 41
c GO To 30 20 CONTINUE
IFIZ )40~50r40 40 CONIINO
GO TO 50
34
48 49
c CALCULATION OF ANOMOLOUS BASE CURRENT COMPONENT FOR ECD L.T.(.l) 50 CONTINUE
VBE=CXM*XK*IMP/Uf*lALUGIOCCCI ALUGlOCCEFit VBE=2.303*VBE .
c
61 WRIIEC3r2Ilt HF 0-.1 -.Y 62 WRITE13r50l)HFEOL 63 WRITE13r212f DCErDGCBrHFE 64 WRITE{3,502JDCELrDC8LrHFEL 65 WRIIEC3r213J ECO,OELCB,Z 66 WRITE{3I400J 67 GO TO 3U
68 69 70 71 72 73 74 75
c
WR If E { 3-.301 )81,82 83,84, 85 · ------------WRITEC3r30l)AC~RI~tXKRGrXNA,XND WRITEC3,301JXN~rXNBrVCB,R,XCCL
·wRITEC3-.600JICODE
77 100 FORMATC5El4.8) 78 101 FORMATC3El4.8) 79 102 FORMATf14) 80 210 FORMATt'OCC -'El4.6r' CE 'El4.-o;-·--~=~ET4-;or-------81 211 FORMAT{ 1 0HFE0= 1 El4.6r' TB = 1 El4.6r 1 DOSE='El4.6) 82 212 FORMATC'ODCE = 1 El4.6r' DGCB = 1 E14.6r 1 HFE =1 El4.6) 83 213 FORMATC 1 0ECD = 1 El4.6r' DELCB= 1 El4.6r' Z =1 El4.6J 84 214 FORMAIC 1 0COK -'El4.6r' OELCB-•Et4.6r' Z - 1 E14.6J 85 300 FORMATC 1 1TRANSISTOR TYPE=2N'14r 1 SAMPLE CODE= 1 14) 86 301 FORMAT(5El4.6) 87 400 FORMAT(//J 88 401 FORMAT(El4.8) 89 500 FORMATC 1 0CCL =•El4.6r' CEL ='El4.6r' CBL ='E14.6J 90 501 FORMAT( 1 0HFEOL='El4.6J 91 502 FORMATI'00CEL= 1 El4.6e 1 DCRL ='El4.6r' HFEL= 1 El4.6) 92 600 FORMA1i 1 0SAMPCE NUMBER = '14) ------93 700 FORMAT(//) 94 801 FORMATC2El4.8) 95 ---c--_..fi._:.~._,~"-'Ao--tF~O'"R:t-10~E11!!1G~RD--A:IH't0ik-A-'~'-T-t-.I-nottdNr---t:~P>t:!R190tr.G~Rt-i!AHMt-- ------------ ----------
IV. MODIFIED EBERS-MOLL TRANSISTOR MODEL
This section presents the Modified Ebers-Moll transistor model, the
relations governing its operation, and the limitations on its numerical
accuracy.
35
The Ebers-Moll transistor model was developed from the diffusion
equations and approximated the de forward current gain and collector current
characteristics only in the region where the base recombination current dom
inated current gain. This region is typically less than a decade of collector
current as the high current characteristic is modified by emission crowding
and the lower decades are dominated by other components of base current as
discussed in Section II.
To approximate the effect of current crowding and the gain decrease
in the region where the surface recombination-generation current in the
emitter-base depletion region began to dominate the base current, additions
or modifications were made, and the Modified Ebers-Moll transistor resulted.
Before discussion of the particular characteristic of concern, it is
appropriate to present the entire Modified Ebers-Moll model as used in the
NET-1 digital computer program. Reference is made to Figure IV-1 and
Figure IV-2 for model schematic and the relations for its numerical
calculations.
36
c
PNP
+
v2 l Rc
RBB
B ~
•a ai 1CF
vl RE
+
l E
Figure IV -1. Modified Ebers-Moll Model
37
The l.Vbdified Ebers-Moll model is the full Ebers-Moll model with the
following additions:
The terms ME and Me (emitter and collector emission constants
respectively) are included to account for departure from the ideal exp (q* v/K*T)
relation for the forward biased junction currents. Respectively, the relations
are now exp(q*v 1/ME*K>:CT) and exp (q>i<v2/Mc>'.cK>i<T)
The terms CTE and CTC (emitter and collector transition capacitances
respectively) are included in analysis regardless of bias state of junction to
partially account for f decrease and t increase for the forward biased t s
junction.
The transition capacitances are made functions of junction voltages by
the following relations:
and the diffusion capacitances are made a function of the junction currents
obeying the following relations:
cde = q [ 1ef +
and
As it is not the intent here to establish the ac parameters, the reader
is referred to the reference in }lrogram 2-1, 2 at the end of Section V for
definition and explanation of the capacitance relations.
38
The terms REE' RCC' and RBB' representing the bulk material
resistances, are included and are constants throughout the analysis.
The terms RC andRE are included across the collector base junction,
respectively, to account for junction leakages.
It should be noted that R C can be used to make current gain ~ N) appear
to be a function of the collector to emitter voltage in the active normal mode
andRE used for (3 I in the active inverted mode. These lowered values of
RC and/ or RE cannot be used when a junction is reverse biased as there
results an erroneously high leakage current (Appendix A).
The terms (3 Nand (3 I' active normal and active inverted de current
gains, respectively, are included as 3rd order Taylor Series expansions with
the independent variable being the junction voltages v 1 and v 2 , respectively.
v 1 is the forward emitter base junction voltage (de).
v 2 is the forward collector base junction voltage (de).
The voltages shown in Figure IV-I are for a PNP transistor as are
the equations in Figure IV-2. For an NPN transistor, the polarities of v 1 and
v 2 are reversed.
For convenience in numerical analysis, the NET-I program holds the
values of (3 and (3 at the values of B >:< A 1 and B ,:, B 1, respectively, when-N I N I
ever the junction voltage is negative. This is analogous to saying that either
39
v 1 or v 2 is entered into the gain polynomial with a zero value if it is calculated
as negative in the circuit equations.
The additions or "modifications" enhanced the capability of the model
significantly by providing for approximations to emission crowding and non-
constant de current gain.
However, the additions imposed several limitations upon the use of the
NET-I program. The model does not give the proper gain change as a function
of temperature. This is equivalent to saying that the gain polynomial does not
predict the correct base current characteristic. A suggested solution to this
is to be found in Section IX on recommendations. This problem is a result of
the fact that the various components of base current exhibit differing exponential
dependencies. This will create no particular problem for this writing as the
characteristics will be established at 27° C.
The emission crowding region is not properly modelled by the gain
polynomial as the collector current is made to bend away from an ideal com-
ponent of emitter current when in actuality the emitter current exhibits "bend
away" also. This is resolved by fitting the current gain to an idealized emitter
current through the collector current and hFE and then using the resistances REE and
R to approximate "bend away" for all three de currents into the transistor. BB
This is the subject of the next section and will not be pursued here.
Although the ''bend away" region is only approximated by RBB' the only
significant error introduced is in the value of V BE at a particular value of base
current. It can very easily be argued that the temperature rise in the junctions
at the currents where emission crowding occurs changes V BE significantly,
40
and a good circuit design overcomes both the temperature problem and the
V BE changes by use of the silicon transistor as a current controlled device,
not a voltage controlled device. This is accomplished by inserting enough
series resistance into the base lead to make the V changes upon the base . BE
current negligible. This is termed "swamping."
At the lower end of the collector current region, it is observed that
errors in V BE - IC characteristic occur below a de forward current gain of
approximately unity. From Figure IV -3, it is observed that the emitter
current must approach equality to the base current at gains less than unity.
In the Modified Ebers-Moll transistor model, the collector current is made to
bend away from the ideal emitter current, thus preserving the current gain
relations but introducing errors into V BE versus IC. In this region the
temperature argument is replaced by the argument that circuits are not
normally designed to operate in this region.
The argument just presented does not provide justification for not
modelling the physical case as with the advent of faster digital computers;
more sophisticated solutions are eminent. It is expected that temperature
feedback from power dissipations will eventually result in the temperature
being dynamic. In this case the base region in particular will have to be
modelled on a component basis (Section IX).
With the possibility of more accurate modelling somewhat in the future,
the present problem of circuit analysis must have a satisfactory solution. This
solution is found in the Modified Ebers-Moll transistor model for the
C) 0 ....1
0
-4
-11
-12
-13 UNITY GAIN
-14
-15 0 0.1 0.4 0.6 0.7 0.8 0.9
Figure IV-2. Modified Ebers-Moll Representation for Forward Region
4]
1.0
I 100
w u..
90
80
70 )
~ 60
..c 50
4
.3 I
2 I
I
I
i I 0-
~o12 RDU' TYPICAL OF 2N2907 T = 300°K
71 I I
// ?o13 RDU
II I / lj /
7/ lj / // /
/-, ~ 7 l------1'0 14 RDU _,_. ~ --? ~__..,... ~ ~ -
0.00001 0.0001 0,001 0.01 0.1 10 100
'c (ma)
Figure IV-3. Typical Neutron Degraded hFE Characteristic ~ l\:)
pre-irradiation and post-irradiation case usihg the NET-1 digital computer
program to effect the numerical analysis.
43
In order to provide incentive for using the Modified Ebers-Moll model,
it is necessary only to observe Figure IV- 3. Over the "useful" range of
collector currents, the forward de current gain hFE is seen to vary con
siderably. In the post-irradiated case, it is observed that the percent gain
reduction is greater at the lower collector currents. This then makes the
ratio of maximum to minimum current gain even greater; therefore, the analysis
problem becomes even greater.
In view of the previous arguments, hFE must be considered the most
important design and analysis characteristic. For this reason, the develop
ment of a nonlinear current gain relation for the Modified Ebers-Moll model
will be made. This is the topic of Section V that follows.
V. TECHNIQUES USED TO CALCULATE DC
PARAMETERS FOR THE MODIFIED EBERS
MOLL TRANSISTOR MODEL
44
There are two regions of operation of the transistor for which the pre
and post-irradiated characteristics are predicted. This section presents the
techniques used to model the Modified Ebers-Moll transistor model parameters
to the characteristics for the active normal and saturation regions of operation.
The establishment of the de transistor parameters upon saturation data
is presented first. The development of the model parameters OJl this basis
assumes the operation in saturation to be most important and the forward gain
characteristic to be least important.
The changes in the saturation voltages V BE (sat) and V CE (sat)
presently are not entirely predictable; however, the approximations as given
in Section VI are worthy of consideration and are used as a basis for modelling
here.
The de constants REE' Rcc' RBB' Me' ME, Ics' and IES are
calculated from saturation data using the equations developed by Sokal. 16 The
equations are modified to utilize four separate values of current gain, (3 N'
insit~ad or an average over the region of interes.t. These points are, for
convenience, evaluated from the gain versus LOGIC l)Olynomial applicable
at the neutron dose for which the model is desired. If the dose is other than
45
zero, the saturation data may have to be modified before entry into the
program. Section VI will establish the necessary changes, techniques for
changing, and a table of options depending upon the changes necessary for a
particular application.
In order to avoid the necessity for the reader to look up the reference,
the definitions of the de constants and parameters are given here.
RBB - base spreading, bulk, and contact resistance.
REE - emitter bulk and contact resistance.
RCC - collector bulk and contact resistance.
RE
Rc
IE
Ic
IB
IES
Ics
0! N
Me
ME
- emitter-base junction ohmic leakage resistance.
- collector-base junction ohmic leakage resistance.
- emitter de current in amps.
- collector de current in amps.
- base de current in amps.
- emitter-base diode saturation current.
- collector-base diode saturation current.
- common base normal current gain.
- common base inverted de current gain.
- common emitter normal de current gain.
- common "emitter" inverted current gain.
- collector-base diode emission constant.
- emitter-base diode emission constant.
M and M are factors to align the observed currents to the C E
theoretical exponential relations.
The inverted characteristics are achieved by interchanging the
collector and emitter terminals and forward biasing the base-collector
junction. The inverted characteristic is a carryover from the days of alloy
transistors that would perform in the inverted mode. The silicon transistors
today have highly doped emitters and relatively low doped collectors. The
collector will not "emit" efficiently and the resulting current gain is very
small with the highest values in the vicinity of 5.
The program written to calculate the constants and parameters from
46
the saturation voltages [ V BE (sat)] , V CE (sat) and the corresponding collector
current, normal gain and forcedgainarepresentas Program 2. This program
is included here so that modelling can be done without running the degradation
program. For a determination of the required input data, the reference at the
beginning of the codes is given.
To avoid possible confusion, the following definitions are given. This
will allow writing program for other versions of FORTRAN.
Definitions for Program 2-1
LOG = natural logarithm
EIS = IES = 1ES
CIS = ICS = 1cs
XME = ME = Me
XMC = MC = Me
CI = IC = Ic
It should be noted that the equations in the reference have several
mistakes and omissions. The codes presented here are believed to be free
from mistakes, and consistent results have been obtained.
IE
Ic
= _s_
+ RE
v2
v2 = +
Rc
Equations Governing PNP Transistor
( qv,jNJE Kf IES e
) ( qv2 jMC KT -1 -ai e
I CS - 1)
=
Ics
1 - Ql 0! N l
v - v +l ':<a I ':' R + v1 c E E EE c cc
( qv2 /Nlc Kf ) e - 1 - 0! l
N ES l - 0!
N
I C iS negative
IE iS positive
0! I
( qv 1jME KT ) e - 1
Upon attainment of the de coostants important in saturation, the
forward de current gain is modelled. to the internal junction voltage by
47
establishment of v 1 for a set of values of collector current and de forward gain
hFE" This is accomplished by prol?;t'amming the equations given in Figure V-1.
For the case where the leakage and O!I ::tre zero, the solution need not be
iterative, but the general case is considered so that if this program is used by
individuals not completely familiar with the model, results should point toward
the errors.
The details of the program will not be mentioned here, but it is
important that several parameters be defined for :Program 4-1 written in
FORTRAN.
0
1
2
-3
-4
-5
-6
u -7 0 ..J
-8
-9
-10
-11
-12
-13
-14
-15
48
I I I 'J/'v. I VCB = 0 ~~/ 'c (pre)
t = 300°K ~
~ J 7. 'c (post)
f:V/V '/ // 18 (pre)
j vv -j
v v I 1/
A ~ /
A ~
,..t.<J/ v ~~ 1-~~~
~~/ ~ ~ A...~..,. A
~/
j ~
lA' ·~ 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0
VBE
Figure V-1. Typical Forward Characteristic
49
The inverted alpha value (a r) which is called ALI in the program must
be put in equal to the value put into NET-1 for the constant value of the inverted
polynomial, i.e. , ALI = [{31 j (!31 + 1) J = [BI * {3I / (BI * {3I + 1) J· This is done as the NET-1 program sets v 2 = 0 for purposes of
calculating a I when v 2 is calculated negative in the circuit being analyzed. It
is not necessary, however, to remove the exp ( q * \1'2 /M C * K * T) as it is
included in the analysis. For a negative v2 its maximum value is Ics/(1- aN* a~
and as ai is very small at v2 = 0, the denominator will stay very close to unity.
Care, therefore, should be exercised to put ALI in as near to the actual value
at v 2 = 0.
The four error inputs (ERR1, ERR2, ERR3, and ERR4) are defined in
Table V-1. The value for ERR1 and ERR2 may be lowered, but at 0. 005 the
Table V-1. Error Criteria for Program 4-1
ERR1 = ERR2 = 0. 01 (collector current)
ERR3 = 0. 0001 for all cases
ERR4 = 0. 0
solution would not converge but continued to try to satisfy the error check.
This is probably a result of the extremely small change in v1 necessary to
effect a 0. 5 percent change in the exponential relation for the current. No
divergent solutions have been noted as a result of a very highly damped iteration
technique using the logarithm of the ratio of the desired-to-calculated value
as the feedback for the next solution.
It is noted that a compoJ?.ent of collector current can be added as a
function of v2 through RC. This component will add directly to the base current
50
and can be used to change the gain as a function of the collector to emitter
voltage. The program presented will correctly model the equations where it
is assumed that a collector to emitter voltage of 1 volt exists and all other
solutions will be approximations. The gain relation must be established with
RC modelling and then RC lowered to give the increased base and collector
currents to approximate terminal gain at some other V cE· The iteration
program is presented as Program 4-1 where the following definitions pertain:
vc = collector voltage (negative)
X = 0.0
Q = coulomb charge (1. 602 * lo-19 )
T = degrees Kelvin (273 + °C)
0 = -5.
ZK = 1.0
XK = Boltzmann's Constant (1. 38* lQ-23)
XER = 5. * 10-6
VE = 0.0
XICS = 1cs
XIES = 1ES
XME = ME
XMC = Me
cc = collector current
TYPE = transistor 2N number
DOSE = RDU
51
The establishment of the Ebers-Moll parameters for the active-normal
or forward operation requires a somewhat different but simpler approach.
For the forward region, there are several conditions existing that are
important in establishing the correct electrical relations to the equations
governing the operation of the model. First, v2 is either zero or negative for
the forward region. As the NET-1 computer program holds v2 equal to zero
for the forward operation, and {31 is assumed to have zero value for this
condition, a 1 is zero for the forward region. Second, the equations governing
the currents are written for the ideal characteristic and do not include the
"bendaway" characteristic due to emission crowding.
When establishing ME and IES, data points must be taken at V BE low
enough so that v 1 = VBE' where V1 is the emitter-base voltage for the ideal
component. If the ideal collector current characteristic is established by
extrapolating the linear region of a LOG IC versus V BE characteristic, values
can be taken in the high current region.
The "bendaway" characteristic of the three de currents is effected by
the relations established in Appendix A. There is an advantage to using only
H to approximate "bend away, " as the assumption in Section VI relative to BB
v predictions will be satisfied. The "bendaway" then will be BE (forward)
modelled with no change in RBB or REE"
To establish the Ma:Ufied Ebers-Moll parameters for the neutron
bomba 1·ded case, it is assumed that the LOG IC versus V BE ideal character
istic is independent of neutron fluence. Upon this assumption, the following
relation is valid:
52
(hFE + 1) I hFE. (V-1)
It is very convenient to make IC equal to zero by measuring I and BO C
hFE with a collector-base voltage equal to zero. This results in the relation:
(V-2)
If the measurements are made at V CB * 0, equation V-1 must be used
with ICBObeing approximated by v2/Rc where:
The difference between v2 and the voltage collector tobase (v cB) is
only the drop across RCC. The inclusion of this equation makes the solution
iterative; therefore, equation V-1 is recommended when h is measured with FE
leakage included in the collector current measurement and ICBO removed
from collector current by data obtained from reverse biased junction leakage
measurements. This removes the necessity to determine RC C for the forward
characteristic. The leakage current is then approximated by v 2/Rc = ICBO
with v 2 ~ V If the bulk collector resistance is available, or is taken from CB
the saturation characteristics, the following relation holds:
(V-4)
Then:
VCB - I >:c R c cc
(V-5)
53
For silicon transistors the term I ':< R may be considered. c cc negligible by use of RCC = 0.
For the forward region operation, the preceeding technique will
preserve the current gain relation, but the V versus LOG I . characteristic BE C
(as shown in Figure V -1) is poorly approximated below an h of approx-FE
imately one. As the Modified Ebers-Moll model relations will not fit the
characteristic below unity gain, the emitter forward characteristic must be
linearized. This requires that ME and IES be established from the collector
current characteristic above unity gain. For this region and the ideal
characteristic, the following equations represent good approximations as in
this region, where the LOG (hFE) is a fairly straight line. This is true when
a single base current component is dominating the gain relation.
The collector current relation is given by:
( V-6)
where V = v1 by extending the ideal collector current component. BE
For the emitter current, the relation is given by:
(V-7)
Upon substituting IE = IC (hFE + 1) /hFE into equation V -7 and
equating equation V-6 to equation V-7, there results this relation:
~\: {LOG LOG (rcc]·
(V-8)
This equation, to be solved in linear form, requires two sets of I and h e FE data points. This results in these relations:
ME LOG [rei (hFEi + 1) J LOG (rEs) = XM LOG (re1) - LOG (ree);
ME LOG [re2. (hFE2 + 1) J - LOG (rEs) = XM LOG (re2) - LOG (ree) .
54
To solve this system of equations by determinants, it is easier if the following
definitions are made with x and y being the dependent variables.
Let
A = LOG [ret (hFEt + 1)/hFEi]
B = XM [LOG (rei) LOG(ree) J e = LOG [re2 (hFE2 + 1)/hFE2]
D = XM [LoG (re2) - LOG(ree) J X = ME
y = ME* LOG (rEs) .
Making these substitutions, there results the following linear set of equations:
A* x + ( -1) >!c y = B
e * X + ( -1) >'.c y = D.
Solving this set of linear equations by determinants, the following solution
results:
B -1, D -1 B-D
X = =
I~ -1 A-e -1
I~ y =
I~
B
D -1
-1
B>:•C-A*D = A- C
Then the desired values for the emission constant and saturation current are
given by:
B-D A-C
( ) B>:•C-A*D LOG IES = B - D
There is no need to resubstitute th.e original definitions as these
55
relations can be solved directly in the digital computer. The program to solve
these equations appears as Program 3 and could be made a part of the degrada-
tion program.
The values for IES and ME should be calculated using IC 1 , IC2, hFE1 ,
and hFE2
as data sets at the minimum and maximum values of collector current
expected. Care must be exercised for the degraded characteristic to be sure
that the minimum collector current data points are not input with hFE below or
in the immediate vicinity of unity.
Upon exercising the gain degradation program, curves typical to the
solid lines of Figure V -2 result for the neutron doses shown. Inputing data from
these curves into the iteration program ( Program-4) results in curves typical
of those in Figure V-3. Now that an empirical relation exists (as shown for
doses of o, 1011 , 1012 , 1013 , and 1014 nvt) for BN versus Vt. the only remaining
task is to fit these curves to the 3rd order gain-voltage polynomial. The
curves in Figure V-3 were fitted to a Taylor Series expansion about v 1 = 0
Ill II.
·..c
J~r-------------~--------------~--------------~------------~--~
ao1 1 I n I -- I I ,.,
," ,"
,/ 6011-~~-+-_J_j
40 I I h' lA/- I I I
./ . ';I'/
. "'/; ;
20 I I I ::::;;;;oooo""'·::;::--F "i I
0~--------------~------------~--------------~---------------L--~ 0.01 0.1 1 10 100
'c (ma)
Figure V-2. Typical Curve Fitting Results for hFE Versus IC en 0')
z ell.
sor---~-----j----i-----}---~-----+----JL __ _j
0
60r-------~--------r-------~--------~------~~~----~------~----~
1012 RDU
40~-------r--------r--------r--------~----~~--------r-------,_----~
1013 RDU
20~-------r--------r--------r--~~~r-~----4-------~--------;-----_,
1014 RDU
I 1015 RDU
0~2
0 o.~t--~--!-~~~~~===1~~--0.8 0.9 0.3 0.4 o.s 0.6 OJ
V 1 (volts)
Figure V-3. Typical hFE Versus Junction Voltage for Various cp c.n -.:J
58
using a least-squared-error criterion. The program written for the IBM 1620
digital computer is presented as Program 6-1 in Appendix J ( 6-2 in FORTRAN
IV).
Section VIII will present the gain plotted from a NET-1 exercise for
the 2N1 711. Here it will suffice to say that curves typical of the dashed curves
in Figure VI-1 were obtained with typically less than 3 percent error. This
verifies the essential correctness of the technique.
A salient point to note is that {3N is definitely a function of v 1 , and for
proper analysis it should be made a function of v 1 through use of the gain-
voltage polynomial. Further, this technique does not assume that VBE = v1
as the drops in the bulk resistances are included and subtracted from V BE
Although there are cases where ~ * RBB is negligible, its effects in the
forward region should be included, as ~B will modify the forward character-
istic at neutron doses where gain is degraded appreciably enough to cause
significant increase in the base current. The iteration program does not use
the value of ~B calculated by Program 3; therefore, it is possible to model a
portion of the bendaway region in the forward case when it is not necessary to
preserve the V ) characteristic. BE (sat
Reference to Figure V-4 emphasizes the values of VBE assigned.
These correspond to ann= 1 component of current at 300°K. The problem
that exists for fitting the third order Taylor series polynomial to these values
is that the polynomial may assume shapes as shown in Figure V-4. Curve A
represents a good fit in the region of 0. 5 to 1. 0 volts but can give appreciable
errors below 0. 5. Curve Band C give unreal negative values which cause the
/ I o.s· V 1 (volts)
/ I / I
/ I a,/ I
I I I I I I I lc I
I I
DESIRED
----ATTAINED
Figure V-4. Typical Curve Fitting Results for V1 Versus hFE
59
1.0
iteration to work improperly and sometimes cause divergence or oscillation.
NET-1 did not allow a negative A1 or B1 coefficient for this reason. To
overcome the problem of curves like B and C, the NET-1 program was
modified to accept the negative A1 or B1; then the value of the polynomial
replaced by unity whenever the value was calculated less than 1. This point
was chosen based upon the errors that are encountered below unity gain and
could be changed to 0. 1, 0. 01, or 0. 001 if desired.
Curve A presents a very erroneous gain below 0. 5 volts, but in all
but 1 of some 60 cases tried, convergence to the proper value was attained.
60
A technique of using false points below 0. 5 represents the best
technique for insuring convergence, but gives curves similar to D when enough
points are used below 0. 5 to insure no negative {3N. This introduces errors
in the gain values above 0. 5 volts.
The best answer to the individual using NET-1 is to use the approach
that gives the best results for the regions desired. It is seldom necessary to
fit gain characteristic over more than several decades of current. Section VII
will present the results obtained using only the desired gain points and limiting
the polynomial value to unity.
To complete the forward region modelling, it is necessary to derive
the relations needed to approximate the bendaway characteristic. From
Appendix A the relation for the difference between the measured voltage and the
ideal junction voltage v 1 is given by:
A VBE = 1B [RBB + (?N + 1 )REEJ.
Evaluation of t:l. VBE requires two values of V and two values for the junction BE .
voltage. The two junction voltages are given by:
v11 = 2. 303 (XMc >:c K* T/ q) [ LOG10 (Ic1) LOG10 (Ics) J , and
v12 = 2. 303 (xMC * K*' T/ q) [ LOG10 (Ic2) - LOG10 (Ics) J . The difference between the actual characteristic and the ideal
characteristic is given by:
v11 , and
Dividing by IB1 and IB2, respectively, and substituting the RBB and REE
relation gives:
t:l. VBE1/~1 = RBB + (hFE1 + 1) REE' and
t:l. VBE2/~2 = RBB + (hFE2 + 1) REE .
Solving by determinants results in the following relations for ~B and
Thenecessaryrelations to determine RBB and REE are included in
Program 3 and require only two corresponding sets of values for V BE' IC,
hFE' and~·
61
PROGRAM 2-1
C CALCULATION OF TRANSISTOR PARAMETERS FOR NET-1 C REFERENCE SOKAL, SIERAKOWSKI, SIROTA ELECTRONIC DESIGN 13 C JUNE 21,1967 PAGE 60-65
21 READ ltNT Y=NT IFIY-O.Ol199t10Q,199
199 PUNCH 2tNT PUNCH 11 PUNCH 3
22 READ 4 ,vBE1tVBE2tVBE3,VBE4 23 READ 4, CI1,CI2,CI3,CI4 24 READ 4tVCE1,VCE2,VCE3,VCE4 25 READ 4•BN1•BN2•BN3•BN4 30 READ 31•BF,BF4,BI
PUNCH 5 PUNCH 4•VBE1,VBE2,VBE3tVBE4 PUNCH 3 PUNCH 6 PUNCH 4,CiltCI2,CI3tCI4 PUNCH 3 PUNCH 7 PUNCH 4•VCEl,VCE2,VCE3,VCE4 PUNCH 3 PUNCH 32 PUNCH 4,BNltBN2,BN3tBN4 PUNCH 3 PUNCH 33 PUNCH 31tBFtBF4,BI PUNCH 3 PUNCH 11
62
C VT=K*T/Q WHERE K=BOLTZMAN CONSTANT T=KELVIN TEMPERATURE=273+C C Q=ELECTRON CHARGE VT=0.0257V AT 25 DEGREES C
VT=Oe0257 51 D=VT*CIICI2-Clll/CCI3-CI1ll*LOGICI3/Clll-LOGCCI2/CI1) l 52 XME=C CVBE3-VBEll*C CCI2-CI1l/CCI3-Clll )-CVBE2-VBElll/D
PUNCH S,XME PUNCH 3
53 RX=CIVBE3-VBE1l-XME*VT*LOGCCI3/ClllJ/ICI3-CI1l Y=ICVBE2-VBEll-CVCE2-VCE1l l
54 XMC=CCIVBE3-VBE1l-CVCE3-VCElll*CICI2-Clll/CCI3-CI1ll-Yl/D PUNCH 9,XMC PUNCH 3 RY=CC CVBE3-VBE1l-CVCE3-VCE1ll-XMC*VT*LOGCCI3/Cl1))/CCI3-Cl1l
55 D2=2e718**< CVBE2-VCE2-CI2*RYl/(XMC*VTJl-1e0 56 CIS=CCI2*1l.O/CBN2+1.0l l*( CBN2/BFI-l•Ol l/02
PUNCH 12tCIS PUNCH 3
57 D3=CI3*C1e0/CBN3+1.0ll*CBN3/BF-le0l+CIS 58 Z=XMC*VT*LOGCCCI4*(1.0/CBN4+1.0ll*IBN4/BF4-l.Ol+CIS)/D3l 59 C5=CCVCE4-VCE3l+Z-CCI4-CI3l*CRX-RYll/XME*VT 60 D4=1eO-ICI3/CI4l*CBF4/BFl*C2·718**C5l 62 D5=2.71B**CCVBE2-CI2*RXl/(XME*VTll-1•0 63 EIS=CCI2*C I 1e0/CBI+1.0l 1+11.0/BFl l l/D5
PUNCH 14tEIS PUNCH 3
64 D6=CI4*CC1e0/IBI+1.0ll+1.0/BF4l+EIS 65 Z2=XME*VT*LOGCCCI3*1l.O/IBI+l.Ol+1•0/BFl+EISl/D6l 66 RX4=CZ2-CVBE3-VBE4l+ICI3/CI4l*RXl/CI4 67 REE=CRX-CBF4/BFl*RX4l/Cl.O-CBF4/BFll
PUNCH 15tREE PUNCH 3
68 RBB=BF*RX-CBF+1.0l*REE PUNCH 16,RBB PUNCH 3
69 RCC=IRBB/BFl-RY PUNCH 17,RCC PUNCH 3 PUNCH 11 GO TO 21
100 STOP 1 FORMATCI4l 2 FORMATC19H TRANSISTOR NUMBER I4l 3 FORMATC//l 4 FORMATC4E14.6)
63
5 FORMATC52H VBE1,VBE2,VBE3tVBE4, BASE-EMITTER VOLTAGES IN VOLTS) 6 FORMATC51H CI1,CI2,CI3,CI4t MEASURED COLLECTOR CURRENTS IN MAl 7 FORMAT155H VCE1,VCE2•VCE3tVCE4 COLLECTOR-EMITTER VOLTAGES IN VOLT! 8 FORMAT150H EMISSION CONSTANT FOR EMUTTER-BASE DIODE •••••• ME=El0e3l 9 FORMATC50H EMISSION CONSTANT FOR COLLECTOR-BASE DIODE •••• MC=El0.3l
10 FORMAT13El4e6l 11 FORMATC//l 12 FORMATI50H COLLECTOR-BASE SATURATION CURRENT UN MA •••••• ICS=E12.4l 14 FORMATI50H EMITTER-BASE SATURATION CURRENT UN MA •••••••• IES=E12.4l 15 FORMATC50H EMITTER OHMIC SERUES RESISTANCE IN K OHM ••••• REE=E12.5l 16 FORMATI50H BASE OHMIC·SERIES RESISTANCE INK OHM •••••••• RBB=E12.5l 17 FORMATI50H COLLECTOR OHMIC SERIES RESISTANCE IN K OHM ••• RCC=E12e5l ~1 FORMATI3E14.6l 32 FORMATI21H NORMAL CURRENT GAINS. 33 FORMATI39H FORCED CURRENT GAINS AND INVERTED GAIN!
END
PROGRAM 2-2
r. C:ALCtJ[ATlON DF TRANSISTOR SA.fURATHH-J PAPA"1FTFRS FOR NET-1 C REFERENCE Sf1KAL,SJERAKOWSKI~STROTA ELECTRONIC DESIGN 13 c II IN E ? 1 ' 1 ° 6 7 p A c;. E 6fl- 6 5
1 21 RFA0(1,l) lTVPF,IfOOE 2 XVZ=fTVPE . . 3 IF(XVZ-C.Cl199,100,1~9 ~ }99 WPJTE(3,21ITYPF,ICODE
5 22 ~EAD(},4) V~F1,VRE2 9 VAF3 9V~E4 6 23 REAn(l,4) CIJ!CI2,CI3 1 CI4 7 24 Rf.A0(1,4) VCE ,VCE2,V~E3,VCF4 8 ?5 REAO(l,4) BNJ,BN?,BN3,BN4 o 30 ~EA0(1,31JPF,RF4,RI,T
10 WRTTE(3,5JVRFl,VRE2,VRE3,VRE4 11 WRITF(1,~lCil,CT2,CJ3,r.I4
..12____ laiR I IF ( 3-, 7) VC E ] , V( F 2 , VC E 3, VC F 4 13 WRITE(3,3?JBN1,RN2,BN1,8N4 14 WRJTE(3,33JRF,RF4,RI,T 15 XK=l.3RE-23 1 6 o- 1. 6D2 E 1 9 17 VT=XK*T/Q 18 51 D= VT* ( ( ( C I 2-C I 1 J If C B-C I 1)) * I\Lf1G f C I3/C J 1) -A LOG ( C J 2/C I 1) ) 19 52 XME=((VAE3-VBEl)*lfCI2-CI1)/(CI1-Cil)}-fVBE2-VBElJ)/0
64
20 WR II E (.3, 81XME -----------~-----------~-~-------- _____ _ 21 53 RX=({VAE3-VBE1J-XME*VT*ALOG(CI3/Cil))/(CJ3-CJ1) 22 Y= ( (VBF.2-VAE1 J-(VCE2-VCE1) 1 23 54 XMC=(({VRE3-VREl)-(VCE3-VCEl)J*ffCI7.-Cil)/(CI3-CI1JJ-Y)/0 24 WRITEI3,9)XMC 25 RV=(((VBE3-VRF1J-(VCE3-VCE1JJ-X"1C*VT*ALOGCCI3/Cil)l/CCI3-Cilt 26 55 D2=EXP((V~E2-VCE2-CI2*RY)/(XMC*VT) J-1.0 27 56 CIS=CCI2*(l,O/CBN2+l.Oll*Cf~N?/RFl-l.OJJ/D2 ~ WR(Tf(3.121CIS -zq 57 n3=CI3*(1,0/(BN3+l.O)l*(BN?/~F-l.O)+CIS 30 58 Z=XMC*VT*ALOGCCCI4*fl.O/(BN4+l.OJI*IA~4/RF4-l.Ot+CISt/03) Jl 59 C~=(fVCE4-VCE3J+Z-fCI4-CI3J*(~X-RVJJ/XME*VT 32 60 04:1.0-ICI3/CT4l*fRE4/REJ*EXPIC5t 33 62 05=E~Pf(VRF.2-CI2*RX)/(X~E*VTJI-l.O 34 63 EIS=(CI7.*((].0/(8J+l.OJI+(l.O/RFJJ./05 35 WRITEf3,14JEIS -36 64 06-CI4*((1-0lfBI+l.Oll+l.OLB£4J+FIS ---~---37 65 Z2=XME*VT*Al0G((CI?*(1.0/(RI+l.O)+l.O/RFJ+EJSJ/06t 38 66 RX4=(Z2-(VAE3-VRE4J+(CJ3/CI4l*P.XJ/Cl4 39 67 REE=CRX-fBF4/BFl*RX4)/(t.0-1~F4/qFJJ 40 WRJTEf3,151BEE 41 68 RBB=BF*RX-(Rf 1.0J*REE 42 WRITf(3,16JRAR 43 69 ~CC=(RRB/BFJ-RY ~- WRITEl3t171RCC 45 GO TO 2 · 46 100 STOP 47 1 FOBMAT(I4,2X,J4) 48 2 FOB~AIJtlTRANSISIDR TYPE-?N'I4,2X, 1 SAMPIE CODE-•IAJ 49 4 1 FORMATf4F14.8) 50 31 FORMAT(4~14.8) 51 5 FORMAT( 1 0VBE1= 1 E12.4t 1
.5.2._ ----~---~- __ fL EQR.M.AT.l •.CCI L = 'F 1 2. 4 I I 53 7 FORMAT( 1 0VCE1= 1 El2.4,' ~~ ~~· F~~MftT( 1 (R~l ~i~f2.4~i sr, ~~ Fn~~ATC•~nF = 1 ~12.4,'
VBE2= 1 El?..4,• V8E3='El2.4,' CI 1 = 1 E12.4_.__~~~-C.L3. ='__ill,-4,• VCE2-'E12.4,' VCE3= 1 El?..4, 1
R~? ='~1?.4,' ~N~ ='E12.4t' RF4 = 1 ~1Z.4, 1 qy ='El2.4,•
VRE4= 1 E12.4) £14 _ =' El.2.4L VCE4= 1 E12.4) -BN 4 :' E 12. 4) TMP =1 E12.4)
~6 B EOPMAT('rX~F ='Fl?.41 ~5~7~----~~9~E~n~R~~~A~T~. ~(~'~(~X~M~C~~=~·~t~l~2~,~4rrt------------------------------------------------5A 12 FnP;Att•rXIC5 =•FI2.4 SQ 14 FnRMATC•0XIFS ='Fl2.4t 60 15 FOR~AT('OqE~ ~'E12.4J _6_L6 _ _ 16 FOPMAIL!J)R~g ___ =:'El2.4) ____ _ 6-z- 17 FOR'UIT' rPrr = •TP e4 J 6~ F~O
65
PROGRAM 3
c·· PRO~RAM-T!'l ESTAR(JSH ME;.IES;Vl, C XK =BOLTZMANN'S CONSTANt C I :IfMPFRAI!!RE TN DEGREE KFI VTN C Q =COULOMBJC CHARGE C XMC =SLOPE CONSTANT OF COLLfCTOR CURRENT CHARACTER 1ST IC NOT = TO MC
1 2 3 4 5 6 '1 8 9
10 u
22. 23
~~ 2.6 l'T
C OEfiNlTION OF VARIARLES C XME -FMITTER-BASF OIO~E EMISSION CONSTANT C XIES=EMITTER-BASF 010f}f SATURATTON CIJRP!:NT C XIESl=LOGARITHM OF XJES C Vl =EMITTER-BASE DIODE FORWARD BtASED JUNCTION VOLTAGE C CC( :TNTERSECTJON Of fXTRApO! ATfO CO! I ECTOR C!!RRfNT AT VBF&O.
A•ALOG O(CCl*CHFEl+l. /HFElJ B=XMO*fALOGlOCCCli-ALOGlOCCCCJJ C=ALOG10fCC2*(HFE2+l.J/HFE2J O:o:XMC*(A!OGJOICC21-AI OGlOICCCII XME•fB-DJ/(A-CJ XIESL•(l./XMEI*(B*C-A*DI/(A-CJ XlES=lO.**XIESL XMCI=A~DGlOlXMCI
CCCL=ALOGlO(CCCJ WRITEf3,300el z·ooo oo zoo 1 1 =1 160
•~tabt1188~ 1 s6~~~~003 2003.CE•CC*(HFF+ .1/HfE CCL•ALOGlO CCI
::
"" E-O CRL•ALOGlOCCBJ .
31 Vl=2.303*(XME*XK*T/Ol*(ALOG10(CEJ-XIESLt 32 Hff!:AIOGlOfHEfl
42 WR TF.(3,3005JCCl,HFEl,CC2!HFE2 43 WRITFf3,30061CClL,HFElL,CC2L,HFE2L
-t·~~~~,2~o~on5r;C~I~IHf~l~:~5691~x~Kr,~r~,o~,ux .. ~~c-,~c7t~cr-------------------------------
51 52 53 54 55 56
~~ 59 60 61 62 63 64 65 66 67 68 69 70 11 72
~~ 75 76 77 78
~~~2=~f~i:tl: Vll=2.303*CXMC*XK*T/QJ*CALOG10lCC1J-CCCLJ Kii=f~3R~*!X~~*XK*T/Q)$(ALOG1Q{CC2J-CCCLJ
WRITE 3,1210JDY8EI,DY~E2 · W~ITE(3,1200JR88,REE,Vll,Vl2 GO TO 1
ttA8 ~~~~Att4F14.AI . . .
66
3006 FORMAit'~CCll =1 EI2.4, 1 HFE1L=~EI2.4, 1 CC2l 1 EI2.4, 1 HFE2l-'El2.41 .3005 FORMAT('vCCl = 1 El2.4t 1 HFEI ='El2.4t 1 CC2 ='El2.4, 1 HFE2 = 1 E12.4J 3004 FORM~TC 1 CXHC- = 1 El2.4t 1 XMCL ='El2.4t 1 CCC =1 El2.4, 1 CCCL =1 El2.4J
"3002 FORMAT(5El2 4). . 3003 FORMAtt•otY~E= '14,8x,•SAMPlE NO- 1 14,8x,•OOSE =1 El2.4J 3020 FORMAT(//) . 3010 FORMATC'OHFE = 1 El2.4, 1 VI = 'El2.4J ~8bl ~R;~tfl:s~~rl: :~t~:4:: ~~~= :~~~=t~· esc= •et2.4J 3000 FORMATC'OCC = 1 El2.4t 1 CE = 1 El2.4t 1 CB = 'El2.4) 3008 FORMAH 1 1SOLUTION 1 1 5000 FORM AT.( 12 J
ENIJ
PROGRAM 4-1
C PROGRAM TO FIND BETA VS V1 C ITERATION PROGRAM TO FORCE A CURRENT GAIN CURVE INTO THE C EBERSS MOLL TRANSISTOR MODEl FOR A PNP TRANSISTOR C INITIAL CONDITIONS C DEFINITIONS C BI1 INVERTED CURRENT GAIN C SECC SATURATION EXPONENTIAL COLlECTOR CURRENT C SECE SATURATION EXPONENTIAL EMITTED CURRENT C CE EMITTER CURRENT C CC COLLECTOR CURRENT C XLCE LEAKAGE CURRENT IN EMITTER JUNCTION C XLCC LEAKAGE CURRENT IN COLLECTOR JUNCTION C ALN ALPHA NORMAL C ALI ALPHA INVERTED
1 READ 501, TYPE PUNCH 600 PUNCH 60l,TYPE PUNCH 800 READ 500,REE,RCCtRBBtVCtX READ 102, Q, RE• XIES, XME READ 102, T, RC, XICS, XMC READ 500, o,zK,XK,XER,VE
2 READ 502,CC,HFEO,DOSE PUNCH 503 PUNCH 504,DOSE PUNCH 800 IFICCl 1000, 1000, 3
3 CONTINUE READ 501, ALI READ 102tERR1,ERR2,ERR3,ERR4 B=HFEO BN=HFEO CE=I!B+l.l/Bl*CC CC=-CC ALN=B/!B+l.Ol
4 D=l.-ALN*ALI EXP=LOGICE*D/XIESl V=<EXP*XME*XK*Tl/Q 53=1.0
7 V11=V 51=.001 S2=.001
C ITERATE FOR V1 TO SATISFY EMITTER CURRENT 31 V2l=VC-VE+CE*REE-CC*RCC+V11
EXP1=Q*V11/(XME*XK*Tl
67
EXP2=Q*V21/!XMC*XK*Tl CE1=V11/RE+!XIES*!2e71828**EXP1-1el-ALI*XICS*!2.71828**EXP2-l•l)/0
32 IFIABSICE-CE1l-ERR1l 26,26,34 34 R1=ABS!CE1/CEl
IF!R1-1.)35,35,36 36 S1=S1/10.
GO TO 7 35 V11=V11-S1*e4343*LOG!R1l-e0001
GO TO 31 C ITERATE FOR V2 TO SATISFY COLLECTOR CURRENT
26 PRINT 1011,V11 V12=V
11 V22=VC-VE+CE*REE-CC*RCC+V12 EXP1=Q*V12/!XME*XK*Tl EXP2=Q*V22/IXMC*XK*Tl CC2=V22/RC+!XICS*I2e7l828**EXP2-l·l-ALN*XIES*I2e7l828**EXPl-lell/D
12 IFIABSICC-CC2l-ERR2l 5lt5ltl4 14 R2=ABSICC2/CCl
IFIR2-lel15tl5t16 16 S2=S2/lO.
GO TO 26 15 V12=V12-S2*.4343*LOG!R2!-.000l
GO TO ll 51 PRINT 1011tVl2
IF!ABSIV11-Vl2l-ERR3l100t100t75 PRINT lOlOtALI
75 IF<SENSE SWITCH 2!100,76 76 ALI=ALI+ABSCV1l-V12l*S3
PRINT 1010tALI GO TO 4
100 BI=ALI/(1.0-ALil CONTINUE PUNCH l004tS1 PUNCH 1005tS2 PUNCH 1006tS3 PUNCH 1007tBN PUNCH 1009tALN PUNCH 1008tBI PUNCH lOlOtALl PUNCH 1002tCEl PUNCH 1003tCC2 PUNCH lOlltVll PUNCH 1001tV2l PUNCH 800 IF<SENSE SWITCH 2! 76t2000
2000 GO TO 2 1000 PUNCH 703;REEtRCCtRBBtVCtX
PUNCH 702tQ,REtXIEStXME PUNCH 702tTtRCtXICStXMC PUNCH 702t0tZKtXKtXER STOP
502 FORMATI3El4.8) 503 FORMATI13H NEUTRON DOSE! 504 FORMATIE14.8) 702 FORMAT14El4e6l 703 FORMAT<5El4e6l
1001 FORMAT14H V2=E20.12l lOll FORMAT<4H Vl=E20.12l 1002 FORMAT!4H CE=E20.l2l 1003 FORMAT14H CC=E20.l2l 1004 FORMAT!4H Sl=E20.l2l 1005 FORMAT<4H S2=E20.l2l 1006 FORMATI4H S3=E20el2l 1007 FORMATI4H BN=E20.l2l 1008 FOR~AT!4H BI=E20.l2l 1009 FORMAT!4H AN=E20.12l 1010 FORMAT14H AI=E20.12l
102 FORMAT14El4e8l 500 FORMATI5El4.8l 800 FORMATII!l 601 FORMAT1El4e6l 600 FORMATI8H TYPE 2Nl 501 FORMAT1cl4e8l
END
68
69
PROGRAM 4-2
r PP!!i~Ci\~:l-Tri (.)f.Tfr.IMf"·ll- ii!'.t;-vr·P5!JC::, Vl IJSINr. St\TIJQ,~TJ'~'\J C PJ\I~fiMJ:"TFPS 1\Nn ~EWTf1f\Jt<;, JTrQ,'\ll'l''-!
~ E 1\1') ( 1, Q4 It TYPF., I U''l[ ;nrJc;_r: ____ _ ? RE~nlt,lOC\DFF,~rc,~c,vc,xrF~ 3 R F A I"' ( 1 , 1 C 2 l X~: , T , 8 , X ME , A I
___ 4.. __________ J!.l:U_:;: . .Q.LU!.fiE >'.<..XJL>tli _____ ... _____________________ -- -------- _. __ .... _ ...... _ ~ ~PJT[(7,na) 6 WRTT[(1 ,<:"•3) ITYP;::;, JCOO~,rlflSF 7 60 R~An(ltl(JJCC,HFE
~8~---..,...,,--~IJE'E-'l~C:,-JC~'~-~r~,,._7-L+C~,-!6:.:-,l~~,....,........,.-=-=,..,.'"="'=".,...,...,..~-~~_,~:'""!'""--------·--·-·--· __ .. ____ _ q 61 Cl=CC*Il,+(r~t:C/Rf")-(Rr:r::tiH:l*ll.+l,/HFEil
10 A~=HFE/Il.+HFFI 11 C2=XIES*(AN/( 1.-AN*I'll))
..J.2. __ . _____ -ll.=..t.L.LLXKl~~.l...l.>l< I A I DG I C' I C 2 I -_M_O_G.Hl.U:.L~ LL _________ _ 13 E=Cl-C? 14 F=V/PC-C2*'XPCV*XK1l 15 IF!AP.SIE-FI-.C·Cl*CC l40,ttO.l2 l 6 l Z K= J 17 l ~=C2*EXPIX~l*Vl-IV/RCl+Cl-C2 18 3=C?*XI"l*f:XP(XKl*Vl-l 1./RC) 19 Z:V-IA/R) 2Q._ _________ K.::.K.± 1 __ 21 IFIJ\~SIV-71-.C0Gn~tl5,!5,1" 22 10 V=7 2':\ IFCK-1(()],1,?5 24 ]5 V:Z 25 4G WPITEI~,ll)V,~FE,~C,A~ 26 Gn rn AC 7.1 25 WPJTf(~,~r! 28. _ ~0 Tf.L 6L _ __ 29 70 ~~ITFI3,n()PFF 9 WCC,DC,XIES 30 ~PTTF(3,0l)X~E,AT,VC 31 WR IT r ( l, 9? l T, '), XI< , X K 1
? p 3:>. 9Cl FOR~~IIT('lS'lliJTfOI\ql 3 4 q 3 p-, r. ..., A r 1 • 1 Tv P r = ? ~J 1 I 4 , .:: x, 1 s 11 -~ PL E N n. = • r r; , "i x , • F L t J f NC E = • r: 1 2. 4 1 35 90 FnPM/IT('CRF> ='~-'17,4 9 ' 1f"C ='El?. 1io' RC ='flZ.4,' ffS = 1 1:12.4)
.3.6. ______ ...... .9.l .. FO.P.~.t\T!. 1L.'!1E .. =:'ELZ.!f_,_,_ ,\1 .. ='E1.2 ... 4.t' .IJ.C_ .... ='£.12.4) , 3 7 q 2 F n R ~ " T ( ' r T I•' p = ' [ 1 2 • 4 ' ' c L !'A q = ' "' 1. 7 • 1t ' ' R n 1. T = ' f 1 2 • 4 ' ' E X p = I E 1 ? • 4 r 3A Q4 FnA~AT(I4,T4,Fl4.P) ~ Q 1 00 F r"l P. '.1 AT ( <; ~ 1 4. F ) 40 ]02 FDPMAT(~fl4 °) --------------·----·-- ·-- .. .... - ..
70
VI. PREDICTION OF V V BE (forward)' BE (sat)'
AND V CE (sat) CHARACTERISTIC CHANGES
The characteristic changes to V (f d), V , and V BE orwar BE(sat) CE(sat)
as functions of neutron fluence are necessary for the prediction of the
parameters for the Modified Ebers-Moll transistor model. This section
presents the techniques used for predicting these changes and the relations
used to establish the model parameters for the forward region and the
saturation region. Included are a table of options, the changes necessary,
and the computer program used to effect a suitable model. Forward predictions
are shown in Figure VI-1.
For the forward characteristic, at currents low enough for emission
crowding to be negligible, the LOG IC versus VBE characteristic is observed
to change only slightly with neutron fluence. The bendaway characteristic,
however, occurs at a lower value of collector current because of the increase
in base current. The assumption is made that the emission crowding character-
istic maintains the same relation to the base current for the pre- and post-
irradiated case. For this reason, it is recommended that ~B be used solely
to approximate the bendaway characteristic. This insures that the VBE due to
emission crowding maintains an invariant relation to IB, with the exception
71
_u -7 g ._J
0 I /IDEAL
b.V "' 1/ VPRE
b.V 2 ~ I VI /
POST
~v/; v/ I ~/
;0 / ) 7
/. ~J' /. v 1/
! '//
i~ / /
;/'/ h l
;'f uf
I ~
t'
-1
-2
-3
-4
-5
-6
-8
-9
-10
-11
-12
-13
-14
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0
V BE( volts)
Figure VI-1. Forward Collector Current Changes
72
of the V BE change in the emitter-base diode emission constant and saturation
current.
The relations to predict a neutron fluence dependent emission constant
and saturation current are derived in Section V. These relations are to be
used for predictions of the forward region model only and are coded for the
digital computer to make numerical analysis possible with a minimum of
effort.
H the collector current characteristic does not follow an ideal
characteristic, another method of predicting VBE changes is reported to give
satisfactory results. The technique is presented here for predictions above
emitter current densities above 0.1 amp/cm2 only, and is analogous to the
technique of Section VI, except that this technique will predict a new collector
current characteristic instead of a new emitter current characteristic.
Further, this technique will predict changes in the slope and intercept of the
LOGIC characteristic which have been assumed invariant in Section VI.
In order to calculate post-irradiated values for V BE as a function of
collector current, it is necessary to establish a parameter that remains
constant and is a part of the current gain relation. This is obviously the base
current, and, therefore, it is necessary only to establish a .6. V BE as
measured between the ideal characteristic and the measured characteristic at
a V CE high enough to insure no saturation. The value of V BE is then shifted
from the pre-irradiated value of collector to the post-irradiated value
according to the following equation:
73
(VI-1)
It should be noted that K' is calculated by use of I and hFE to C (pre)
0
determine emitter current density. Further, the quantities in equation VI-1
are available in the degradation program for current densities above 0. 1
amp/cm2•
This technique allows for the collector current to be recalculated for
a given VBE" If it is desired to use the original values of I and to have new c V BE values, it is necessary only to shift the value of the difference between
the ideal curve and the actual curve for several points. A new curve can be
constructed and new values of V ) read for the original values of BE{post
collector currents. This has benefit if all other characteristics are specified
as functions of collector or emitter current, and this usually is the case for
The V characteristic is observed to change negligibly BE (saturation)
at low to medium currents, but significant changes are observed for high
collector currents. Presently there is no technique available for prediction,
as the change cannot be predicted from forward gain changes. This results
from the fact that M and M play a part in the saturation voltages at E C
particular values of collector current. If only the value of ME is changed using
the technique described for V changes, it cannot be expected that saturation BE
voltages would correspond. For the Ebers-Moll model it is possible that by
using the gain degradation technique, a V BC characteristic for the inverted
mode could be developed resulting in a new value of M . However, for the c silicon transistors of today that possess very low values of {3 , it is more
I
appropriate to determine Me and a 1 to provide the best approximation to
saturation characteristics. For this reason the inverted gain is not an input
into the modelling program.
The program used here to determine the de parameters only requires
74
saturation characteristics as V BE is not considered as important as V BE (sat) .
The reason for this is that V BE is a very sensitive characteristic to tempera
ture and most circuit design techniques place a fairly large resistance in
series with the base lead so that V BE changes do not significantly affect the
base current and the resulting eollector current.
In summary, VBE(sat) values must be entered from data as no
prediction techniques are available. When entering changed values, the value
of gain must also be inputed for 1:he corresponding values of collector current.
V CE(sat) CHARACTERISTICS.
The collector to emitter saturation voltage, V CE (sat) is observed to
increase with increasing neutron dose for a particular value of base current.
The saturation voltage is made up of the 5 components as given by the
following relation:
V == VBE - VCB + IE* REE + I * R + I * R , CE (sat) C BBL C CC
where:
IC >:< RBB = voltage drop in the longitudinal direction in the base L
region and is usually very small.
75
voltage drop in the emitter bulk material and, in today' s
transistors with highly doped emitters, is very small.
IC ,:< RCC = voltage drop in the collector bulk material and is the
significant component of the V . CE (sat)
Presently the changes in the collector body resistance are not
predictable to any degree of accuracy. As radiation reduces the minority
carrier lifetime of the carriers in the collector region material, the con-
ductivity will be reduced if the length of the collector is large compared to the
diffusion length.
A second effect occurs when the exposure is large enough to cause
carrier removal. This causes an even greater increase in resistance.
In summary, the collector bulk resistance change with radiation is
not accurately predictable presently, as there is a complex interaction of
current crowding, diffusion length reduction, and carrier removal.
As most of the saturation voltage occurs for the collector body
resistance if the diffusion length is short, it is noted that for epitaxial
transistors that have a minimized collector length, large junction areas, and
low resistance collectors, it is possible to accurately predict V CE (sat)
changes. This is done by the following method:
Measure V at forced gains from 1 to BN. This should be CE (sat)
done at the desired operating temperature with a low duty pulsed de instrument.
76
It is also necessary to do this at a constant collector current. The desired
operating current is used (Figure VI-2) .
P is the point where the transistor comes out of saturation. From the
gain change the point P will move to a higher base current. This indicates
that more base current is needed to drive transistor into saturation. For the
type transistor previously described, P moves to the right to P'. P' then
defines the point where transistor comes out of saturation and is given by:
This equation requires that b' IE' and AE be known so that K' can be found
from the mean data.
IB (pre) - current at P
IB (post) - current at P'
I is current at which data are taken. c The problem that exists for modelling V CE(sat) with the Modified Ebers-Moll
model is that V ) must be specified at four values of collector current CE(sat
at particular values of forced current gain.
Using the previous technique then, it will be necessary to measure
V . characteristics at constant collector currents corresponding to the CE (sat)
currents chosen from the V BE (sat) versus IC characteristics. The curve of
V versus I will be used as a basis for calculation as VBE ( ) changes BE (sat) C sat
negligibly with radiation. Once the VBE(sat) versus IC curve is established,
the values of IC and B F will be chosen as the bases. Then for those values
of I chosen v characteristics will be run to attain I versus VC E ( C ' CE (sat) B sat).
1
I I
ACTIVE
,_.._.. _____ _ ;t---- ------=:.:.:
SATURATION
10 100 18 (ana)
Figure VI-2. Saturation Prediction
IC = CONSTANT
1000
These data then will be presented so that the previous equation can be used
to shift the V CE (sat) curves right along the ~ axis. Once this is done, the
curves resulting will be entered at particular values of forced gain to attain
V CE (sat) values for the computer program that calculates the de parameters
for the transistor model.
The following table (Table VI-1) summarizes the necessary data
changes and the effects and inclusions for the Modified Ebers-Moll model.
77
In concluding this section, it is worthwhile to make several observations
relative to the differences between saturation region modelling and forward
region modelling.
Table VI-1. Options for Predictions of Degraded Characteristics
78
Characteristic Relation for Prediction
VBE
v BE (sat)
V BE versus IC
No data necessary for calculation of saturation parameters. If this characteristic is relatively unimportant, run iteration program with degraded gain values only.
For forward parameters, the prediction is based upon bendaway being invariant relative to IB. No change in RBB or collector current
characteristic need be made as the gain change will model the emission bendaway of all three current components. The forward changes of the emitter current will be modelled by the program that predicts forward characteristics.
Accuracy is lost below unity gain for the Modified Ebers Moll model.
VBE(sat) versus IC
No suitable prediction technique has appeared; therefore, actual data are necessary to model this characteristic through use of the saturation program. If used with the original values of saturation parameters, it will result in saturation occurring at a higher base current if the degraded gain is used. This will result in a V BE increase due to the
increased base current necessary to cause saturation to occur.
This characteristic is of no importance or consideration for the forward operation and has no effects upon the forward region determination of ME and IES ·
Characteristic
v CE (sat)
Relation for Prediction
V CE (sat) versus JB
Prediction for gain effects are calculated in the iteration program using the saturation parameters and, therefore, no inclusion is necessary.
If RCC is to be included, it is necessary to
increase the saturation data for V CE (sat) by
RCC ,.,. IC for use in the saturation program.
79
There is no provision in the forward program as V CE (sat) is not of any importance.
In the saturation program, the inclusion of de degraded gain is made to account for the increased base drive current required to drive transistor into saturation. If this parameter is changed only in the iteration program, v ( ) will shift along the IB axis as
CE sat desired and V ( t) will change in the BE sa bendaway region because of the increased drop across RBB as the edge of saturation
is reached. V BE (sat) will also increase by
the component aJB >:c REE because of the
slight increase in emitter current. At values of h approaching unity this component may
FE become significant, but this condition will be seldom found in practice, as at this large degradation, most circuits will not come close to saturation; B F is usually chosen above 5.
80
Characteristic Relation for Prediction
For the forward program, h is instrumental FE
in determinining M and I Its inclusion E ES
is mandatory or no gain change would occur. It is the only varying parameter input into the forward program.
The value of RCC as calculated in the saturation program is sufficient
for the forward region, as RCC has little effect upon the forward characteristic
unless the load resistor in the collector happens to be of comparable magnitude.
The value of RBB from the saturation calculation should not be used to
approximate the forward region bendaway. Use the technique presented in
Appendix A.
REE is of little concern as its value is typically very small for today's
transistors having highly doped emitters.
V BE (forward), as established when using the saturation parameters,
should not be used for quantitative purposes, since ME and IES for the
saturation characteristics are not necessarily anywhere near ME and IES
essential for accurate forward region modelling.
The term h for the saturation characteristic should be used only to FE
satisfy numerically the saturation relations and to establish the base current
where the transistor saturates. The V BE (forward) characteristic will be in
gross error if saturation values of ME and IES are used in the forward region,
unless M and 1 happen to be calculated the same for both regions. E ES
The {3 polynomial must be used only at the temperature for which it N
was established since the gain will not change with temperature even though
the collector and base currents vary. The base current varies only by the
polynomial relation.
81
82
VII. GAIN CURVE FITTING FOR THE 2Nl711 TRANSISTOR
It is the purpose of this section to present data to show the results and
limitations of the technique used to establish the 3rd order gain polynomial
using the parameters established for saturation values of V , V BE (sat) CE (sat)'
It should be noted that ME and IES are determined for saturation, and,
therefore, the forward voltage, VBE' will not be accurately modelled even
though the current gain is accurately modelled for the forward region. The
model parameters given are, therefore, recommended only for the forward
region when considerable resistance is placed in series with the base lead.
The bendaway region will not be accurately modelled for the saturation
value of ~B. In the case where bendaway modelling is desired and accurate
V is essential, the forward region modelling program must be BE (forward)
used. The techniques and an example using the 2N2907 appear in the next
section.
It is recommended that the model established here be used for a
saturated model. The gain polynomial is required to establish currents and
voltages where the transistor enters the saturated mode.
The de current gain degradation is established using data from an
irradiation experiment perforlll:ed at the Diamond Ordnance Reactor Facility
(DORF).
The data presented in Figure VII-1 give percent of f3 remaining after N
passive irradiation at 27°C. It is assumed that measurements were taken at
times greater than 105 seconds after irradiation and that negligible annealing
resulted as a result of the measurement currents.
83
The results of an exercise using the gain polynomial coefficients as an
input to the NET-1 computer program are shown in Figure VII-2. It can be
seen that there are no gross errors in the relations used to establish the
coefficients. In fact, the 3rd order representation is shown to give more than
reasonable approximations to the nonlinear gain characteristic.
It is noted that none of the gain polynomials had a negative A1 coefficient.
This prevented data to verify that the modification to the NET-1 program
provided a valid analysis; however, the validity has been verified by use of the
coefficients for the 2N1132 transistor.
The remainder of this section deals with the implementation of the
technique used to establish the gain polynomial coefficients by indicating the
order of solution and the necessary inputs.
To implement the iteration calculation that establishes descrete values
of f3 and the corresponding v1 requires that the values of the five resistances, N
the emitter and collector saturation currents, and the emitter and collector
emission constants be established from saturation data. These are attained
by implementation of the saturation program given in Section V or by taking
values from the NET-1 library. The values presented here are taken from
the library, as it is the purpose here to verify the technique and to explain the
method.
84
In addition to the nine constants indicated in the previous paragraph, it
is necessary to have values of the de current gain ( hFE) and the corresponding
collector current. A value for {31 is also required and is taken as 0. 0 for the
2N1711. These values are tabulated in Table VII-1.
Table VII-1. Saturation Parameters for 2N1711 Transistor
hFE re (rna) RBB = 5. >:c 10-3 KQ
35 0.1 Ree = 1. * 10-3 KQ
52 1 REE = o. 5>'r: 1o-3 Kn
75 . 10 Re = 6 :::c 107 Kn
99 100 RE = 1>'r: 104 Kn
40 500 Me = 2.5
BI = 0 ME = 2.69
Type NPN Si 1ES = 5* 10-16 ma
1es = 1 •:< 10-15 rna
Upon input of the eleven constants and corresponding values of collector
current and de current gain, the values shown in Table VII-2 are attained as
output of the iteration program where the following important definitions hold:
( 1) BN = {3N ( 5) BI = {3!
( 2) AN = aN (6) ALI= a I
( 3) eE = IE ( 7) V1 = vi
( 4) ec = re ( 8) V2 = v2
Ill loL.
.z: t-:z Ill u 1:11: w 0..
100
90 f- 100 ma
\\\ ft\G ~~ 80r DORF REACTOR TEST DATA
lma
2H1711 T = 300oK
70
60
50
40
30
20
tor--------------------+------------------------------~------~~~--~~~---, E> 0.01 MeV
2 * 1011 1012 1013 7*1013
FAST NEUTRON FLUENCE (n/cm2)
Figure VII-1. 2Nl711 hFE Versus Fluence Characteristic 00 01
1001 I I I ~
Q REACTOR DATA POINTS 0 RESULTS FROM NET·l V ORIGINAL de GAIN
•r-------------+---------------+---------------1-~~------~~~--~
.. .. 3:
2N1711 T = 300°K
M~-------------+----------------~--~~--------~~~------------~--~
.....-:r I ~ 1 ~ I :::?"1 el I ~ 1013 n/cm2
I 1014 n/cm2
0 ~
0.01 0.1 1 -- • ""
'c (ma)
Figure VII-2. Resulting ~lyE Curve Fit Using Polynomial C1J ~
87
The data from Table VII-2 are plotted in Figure VII-3 for the o. 1-ma
to 200-ma collector current region. These data represent the data necessary
to establish the 3rd order gain polynomial by fitting to a linear interpolating
polynomial. The polynomiaL must be an expansion about the origin, and is
established here by using a least-squared-error criterion. The program used
appears in Appendix J.
Table VII-2. Gain Versus Junction Voltage for 2N1711 Transistor
rc (rna) 0 1012 n/ cm2 1013 n/cm 2 1014 n/ cm 2
hFE V1 hFE V1 h Vt h Vt FE FE
0.1 35 0.693 15 0.693 2 0. 716
1 52 0.849 33.5 0.849 11 0. 854 1 0.897
10 75 1. 01 56.6 1. 01 23.3 1. 01 3 1. 03
100 99 1. 17 84.2 1. 17 41. 6 1. 17 5.5 1. 18
150 6 1. 21
200 43.5 1. 22
300 1. 25 80 1. 25 40 1. 25
500 40 1. 28 1. 28
The coefficients attained are presented in Table VII-3 and represent
gain polynomial coefficients only for the temperature, gain, and saturation
characteristics used in the iteration program. These should not be used unless
the previously mentioned conditions are met, and these coefficients should
never be used if forward voltage V is required to significant accuracy. BE
V for this model is only a numerical abstraction to satisfy the equations BE .
during saturation.
120 r 1 , l I I I
2
0 -
' vc / 110 50
"/k: /V "!a
I _J/. /~ ~ v I /I .100
I ~-~ 1/1 V 100 ~-.~.,...--" / ~v
0.1 ~ -
D I ~.1 ~ :.- • 1 ma 1_0 100 -.... 1150
100
80
~N 60
4
0.4 0.5 0.6 0.7 0.8 0.9 1.0 1.1 L2
v1 (volts)
Figure VII-3. Current Gain Versus Internal Voltage for 2Nl711 Transistor 00 00
89
Table VII-3. Polynomial Coefficients for 2N 1711 Transistor
2N1711
Dose B A1 A2 A3 A4 N
0 1 202.984 -695.284 851.009 -284.773
1012 1 92.4365 -377.453 483.047 -142.557
1013 1 69.2323 -273.140 312.014 - 84.9216
1014 1 305.484 -904.075 878.812 -277.874
B = 0 I
This concludes the gain curve fitting as it has been shown that a very
accurate gain relation can be modelled into the Modified Ebers-Moll model by
the techniques presented in this writing. The next section presents the
modelling of the 2N2907 where the technique of forward region modelling will
be shown by example.
VIII. GAIN DEGRADATION TO THE 2N2907 TRANSISTOR
AND THE MODIFIED EBERS-MOLL TRANSISTOR MODEL
This section presents the 2N2907 transistor as an example of the
techniques used for predicting the neutron induced changes to the forward
region characteristic and the resulting Modified Ebers-Moll representation of
these characteristics. Also presentedaretherecommended modification of
the 3rd degree de current gain polynomial and the reasons that make it
imperative that this modification be made.
90
The prediction of a post-irradiated current gain characteristic is made
using the composite damage factor for the 0. 1-ma to 200-ma collector current
range and the emitter area dependent depletion region induced base current
component for collector currents below 0. 1 rna.
To perform the analysis with use of the computer program given in
Section III as Program 1, there are several characteristics that are necessary.
The next several paragraphs discuss these characteristics and the origin.
To implement the analysis for collector currents above about 0. 05 rna
and less than 200 rna, it is necessary to have the base transit time. The two
techniques established in Appendix C were used, and the results of both will
be given here as the results serve as an example of the inaccuracy that may
result when switching delay data are used.
Initially the base transit time was determined using the circuit
arrangement in Figure VIII-1 and the•equations and definitions of Appendix C.
From oscilloscope traces using a rise time plug-in, the value of t d
is taken to be 10 nsec. This value with the corresponding voltages and
currents is entered into the equation in Appendix C along with manufacturers'
data for gain-bandwidth product, ft (~E) . The results are plotted in Figure
VIII-2 as the dashed curve. As the value of base transit time did not remain
constant, the technique is deemed unsatisfactory for this case. See Appendix
C for a discussion of this topic.
The base transit time was then established by use of the data from
Table VIII-1 and the intersection of the extrapolated linear region of a total
time versus reciprocal of emitter current characteristic. The resultant base
transit time is taken from Figure VIII-3 as 0. 43 nanoseconds. This value is
used for the degradation analysis.
Table VIII-1. Gain-Bandwidth Product Versus Emitter-Current
IE 1 rna 10 rna 40 rna 100 rna 200 rna
ft 150 me 280 me 350 me 290 me 245 me
\ 1. 06 nsec 0. 56 nsec 0. 45 nsec. 0. 55 nsec 0. 65 nsec
To implement a degradation analysis using Program 1, it is necessary
to have values of original gain hFE and the corresponding collector current 0
I . As the computer program used to reduce the data from the Automatic c
91
Data Acquisition System puts out values of hFE and IC, these data are directly 0
read into the degradation program. This eliminates the need to establish a
30 vdc
200Q
OUTPUT
1KQ
INPUT
SQ
TIME---0~--------~---------------------------------
-16
100 PERCENT
t v
10 PERCENT
I I
-----1---1 I
INPUT
OUTPUT
0~--------._--~-----------------------------TIME--
Figure VIII-1. t d Measurement
92
-u • "' c -
0.7
0.6
o.s
._A 0.4
0.3
0.2
o.
~
I
I
USING MEASURED DELAY td
~ ).._
USING RECIPROCAL CURRENT VERSUS tt ~ ~
10
'c (ma)
Figure VIII-2. Comparison of Base Transit Time
/ v
100
/
(,0 c.v
94
0
\ • -
v' ~ ~ CD
a ..... E-t +
>
..... l:l.l
a J-4
\ E-t CD
-l:l.l
-d
~ o\= .. A
..
\ \
., • 0
I IX!
a .!
- 0 IU
5
<:::: -
..... ~ ..... E
CD t) t:l
C":)
I
-l=l > CD
J-4
\ So
..... ~
.....
\ 0~
o.o
N
o
1i~ - •
IJ 0
N
0 0 •
• 0
-
95
gain polynomial as enough points are available to give a well defined
characteristic. There should be no need for intermediate points.
The remaining inputs necessary for the degradation program (Program
1) are found from the following sources in this writing:
( 1) Coefficients for the composite damage polynomial . . . . . .
( 2) Numerical values for the constants used in the
. Appendix F
analysis . . . . . . . . . . . . Appendix J
( 3)
( 4)
Physical area of the emitter in cm2
CCC and XMC.
( 5) Area dependent damage constant for induced current IE cf> and the
Appendix K
Figure VIII-6
emission constant XN . . . . Appendix J
Upon exercising the degradation program (Program 1) with the
indicated inputs, the degraded current gain is obtained and is plotted in
Figure VIII-4 as a function of LOG10 IC.
As the current gain is supported by irradiation test data only at 1 rna
and 10 rna, the comparison of results is limited. However, for the limited
data, the comparison of measured to predicted gain is consistent and relatively
accurate.
As it is the intent to establish the parameters for the Modified Ebers-
Moll model and the NET-1 inputs, the data for the pre- and post-irradiated
characteristic are presented in Figure VIII-5.
140----~~----~----~----~----~----~----~----~----~----~----~-----
120 I
6 REACTOR DATA FOR 1 AND 10 ma
2N2907 T = 300°K 100~--~~----~----~----~----T-----.-----~----4-----~~~-r~--~~--~
w· u..
..c
80~---4-----+-----+----~----~----+-----~----~--~~---+-----r;r--~
60~--~-----+----~----~----4-----~----~--~-A~-+--~-+~---rT---~
40~--~~----~----~----~----~----~----~~~+---~~----+-----+---__,
w~---+----~----4-----~--~----~--~q_~~+---~,---~--,_~--~
-11 -8 -7 -6 ~5 -4
LOGIC
Figure VIII-4. Gain Characteristic for Various Values of Fluence (,0
~
w
""' .z: 1-:z w u 1:11: w D.
2N2907 300°K
0~----------------~------------------~L-----------------~ 1ot2 1o13 tot4 to15
fRDU
Figure VIII-5. Degradation of the 2N2907 Versus Neutron Fluence co --J
The establishment of the model parameters is best accomplished by
making a plot of the logarithm of the current components as a function of the
emitter to base forward voltage V BE. This characteristic is presented in
Figure VIII-6 for the 2N2907 silicon transistor for a temperature of 27 o C and
a collector-to-base voltage of less than 0. 005 volts.
98
The forward region parameters are established by the following method.
First, the collector current logarithm is linearized and extrapolated to V BE= o.
This establishes the parameters CCC and XMC for Program 4 (page 69).
Program 4 also requires that two sets of corresponding values for
collector current and current gain be established. It is recommended that
these points be taken in the vicinity of the expected operating point if this point
is known. In the case where the transistor is degraded by neutron bombard-
ment, these values should be taken at the maximum value of gain and at a
value of gain approximately equal to 10. This prevents the large errors
encountered in the vicinity of unity gain. It should be realized that when the
transistor is degraded to gains of less than 1, a poor approximation to the
forward voltage VB E wi 11 be obtained.
Upon exercising Program 4, the values of IES and ME are attained as
are the values of h , I , Vto and the logarithms of these variables. From FE C
these outputs any desired characteristic can be established.
As the input to NET-1 for a gain relation is a 3rd degree Taylor series
expansion about v 1 = 0, it is appropriate to make this plot and then plot the
corresponding values given by a least-squared-error curve fit of the data
from Program 4. This appears as Figure VIII-7. It can readily be seen by
99
2902 '~ v 1
...., .;
~ Q . .....
2 J
I ~AVBE ~
3 BE2 v J
-5 1/ -/
I / v
~v .I v h v
) ~ -i/
WI~
~ I ~ -~CCL
·8
·1 0
~11
·13
0 0.1 o.~ 0.3 0.4 o.s 0.6 0.7 0.8 0.9 1.0
V BE( volts)
Figttt~ VIII-6. 2N2907 Forward Characteristic
w II.
.I:
12oi--T-----r---,---r-----,---..,.-------0
2N2907 300~
100~-------+--------+--------+--------~-------;--------~----~
sor---~-----t-----r----~----}---~-hL_~ __ JLJ
60r--------r-------~~------l--------t--------t-----~~~--_i 40~------~-------;--------~-------+--------r--.hr--~~----~------~
201 I I I I ://1 / I 1--. I
0 0.1 0.2 0.3 0.4 o.s 0.,6 0.7 0.8
v1 (volts)
Figure VIII-7. Gain Versus Junction Voltage for Various Fluences ..... 0 0
101
this figure that a 3rd degree polynomial gives a rather poor approximation
over the rangeof collector currents. It is recommended that this relation be
used over a limited range of collector currents and that the modification
indicated in Secti<:>n IX be adopted as this allows the best approximation to the
actual current gain characteristic.
As the gain characteristic is considered of highest importance, it is
appropriate that it be modelled more accurately than is given by the 3rd degree
polynomial of f3N versus v1 . An observation of Figure VIII-8 suggests a much
better approach to the problem of gain curve fitting. It is noted that for the
pre- and post-irradiated case, the logarithm of hFE versus v1 is very nearly
a straight line and can be represented quite accurately by a 3rd degree
polynomial.
To effect a modification to the NET-1 program, it is quite simple to let
the 3rd degree polynomial be used to represent the logarithm of hFE. To
effect this modification requires first that a negative A1 be allowed by removing
the error check on the input data, and second, the following relation be
established in the NET-1 program.
f3 = 10BNE N
With this accomplished, the gain can be accurately represented over the entire
useful range of operating currents. Further, the problem of nonconvergence
of solutions due to a negative value of gain being calculated by the polynomial
w LL
..c C) 0 .J
21 I I I I I :.;?""- f=-....-: 10 1~ RDU I 10 13 RDU
2N2907 T = 300°K
I
ol I 7/1'~ 17/ I I I I I
· 1 1 I 7f I I I I I I
•21 7/ I I I I I I I I
0.1 0.2 0.3 0.4 o.s 0.6 0.7 0.8
vl (volts)
Figure VIII-8. Logarithm of hFE Versus Junction Voltage ~
0 1.\:)
103
is completely eliminated as the current gain cannot be calculated negative using
the exponential relation.
Thus far only the ideal characteristic has been modelled. To model
the bendaway ch3:racteristic as seen in Figure VIII-6 requires that the
equations in Appendix A be used to approximate the emission crowding charac
teristic in the high current region. This can be done using two sets of data
points and Program 4. However, the dependence only upon base current will
not be maintained unless REE = 0. For the post-irradiated case, then, only
one value for .6. V BE is necessary for input into the equation given in Appendix
A. The accuracy of the approximation will be decreased in this manner, but
the degraded bendaway characteristic is not theoretically established at the
present.
Upon completion of the degradation analysis, the results for a neutron
· fluence of 1014 RDU are chosen for presentation in Figure VIII-9. Here the
pre-irradiated characteristic is given for both the ideal case and with the
actual and predicted bendaway characteristic.
The complete set of parameters for several values of fluence is given
in Table VIII-2. These are to be taken as the parameters for the character
istic in Figure VIII-9 and are not intended to be used for all 2N2907's. The
characteristics and parameters vary considerably from one commercial
specimen to the next, and the analyst must develop the parameters for the
characteristic most likely to fit the particular case in question.
-10
-11
-12
-13
0
2902 2N2907 300°K
0.1 0.2 0.3 0.4 o.s 0.6 0.7 o.s 0.9
VBE(volts)
Figure VIII-9. 2N2907 Forward Characteristic for 1014 RDU
104
1.0
Flue nee
0
1012
1013
1014
Flue nee
1ot2
1ot3
1014
Flue nee
1ot2
1ot3
1014
XMC CCC M 1ES RBB E
1 2. 5 * 10-14 0.529
1 2. 5 * 10-14 1. 007 3. 09 * 10-14
1 2. 5>!< 1o-14 1. 012 3. 59 >:c 10-14
1 2. 5* 10-14 1. 054 11. 23 * 10-14
At A2 A3
98.47 -916.8 2332
50.35 -446.2 1065.6
7.807 -60.93 118.0
At' A2' A3'
-2.3568 7.8811 3.1207
-2.6762 8.5633 0.9373
-3.4135 8.0689 1. 8603
Coefficient Definition
At, A2, A3, A4,
At I' A2'' A3'' A4''
v t versus hFE
Vt versus LOG 10 hFE
R Rcc EE
0.322
A4
-1500.6
-597.36
-28.566
A4'
-7.9679
-5.8631
-5.8430
Table VIII-2. Forward Parameters for 2N2907 Transistor
105
DISCUSSION
IX. DISCUSSION, CONCLUSIONS, AND
RECOMMENDATIONS
The techniques presented in this writing are adequate for prediction
106
of neutron degraded electrical characteristics of silicon transistors when the
limitations of the NET-1 computer program and the Modified Ebers- Moll model
are considered.
The accuracy of the numerical approximation to the electrical
characteristics is limited relative to emission crowding bendaway, relative to
gains approaching and below unity, and relative to region where leakage can
significantly affect the current gain. Accuracy will also be limited where the
collector current characteristic deviates substantially from the exponential
relations.
Variations of the collector current emission constant and saturation
current with neutron fluence are not included. It may be desirable to include
this change in the degradation program when precise prediction is being done
using several actual data points. However, for prediction based upon nominal
values of the constants and characteristics, the overall accuracy will not be
enhanced by changing the collector current characteristic, as noticeable
changes do not occur at fluences of practical interest.
107
CONCLUSIONS
The current gain polynomial used in the NET-1 computer program
limits the range for which the model is useful, as the polynomial cannot give a
good approximation over the entire range of collector currents. It is imperative
that the gain polynomial be modified to represent the logarithm of de current
gain.
The emission crowding bendaway characteristic is most appropriately
modelled for the general case using REE = 0. This assures that the bendaway
is a nonvarying function of the base current as is assumed for V E (f d) B orwar
prediction. In the case where only the pre-irradiated characteristic is to be
considered, a better approximation is made using REE and RBB and the
relations derived in Appendix A. (Section VIII gives an example and a solution
for R and R . ) BB EE
The characteristic change resulting from temperature changes cannot
be readily modelled by varying the temperature input to the NET-I computer
program as the current gain characteristic is invariant with respect to tern-
perature. This indicates that the base region modelling is lacking in its
relation to the actual physical case.
Collector multiplication was not considered, since the Modified Ebers-
Moll model has no accurate means for inclusion. The technique suggested for
using R proved to be useless for the ranges of collector currents encountered, c
and, therefore, it is not considered for application.
As the composite damage factor will predict a commendable value of
hFE for collector-to-base voltages other than zero, it is appropriate that
current gains be used at the desired V CB and the gain polynomial be used to
establish the ideal base current characteristic. Emission crowding then is
modelled by use of ~Band REE so as not to modify the gain characteristic.
108
The base transit time should be established using the extrapolated total
time delay technique, because the technique using the delay time and switching
characteristic predicts a value of emitter transition capacitance that may be in
error. This results in a value of b that does not remain constant, as the
emitter current varies throughout the solution.
RECOMMENDATIONS
It is recommended that an interim change of the NET-1 de gain
polynomial be made so that the polynomial represents the logarithm of the
current gain with the independent variable remaining the internal junction
voltage ( v 1) • Subsequently the base current should be modelled as the sum
of exponentials representing the various base current components. This will
allow a temperature input to modify the current gain as well as the junction
voltage for a particular value of collector current.
It is recommended that a means of modelling collector multiplication
be developed and a suitable approximation made as a function of the ideal
collector-to-base voltage (v2).
It is recommended that the emission crowding bendaway be modelled
more accurately by making ~B a function of ~ through the junction voltage
109
v 1· This would make the bendaway a function of ~ and, therefore, could be
preserved for the irradiated case. This could be accomplished by a derivation
of the transverse voltage under the emitter as a function of a generalized
geometry, and the effect upon emission across the emitter. A reasonable
approximation could then be made through a base current dependent resistance
so as to avoid a time-consuming calculation in a network analysis problem.
It is recommended that a suitable set of relations be established for
leakage from the relations in Appendix G and incorporated into Program 1.
APPENDIX A. FORWARD GAIN MODIFICATION AND "BEND AWAY" APPROXIMATION
110
In Section IV it was indicated that RC could be used to make the de
forward current gain appear to be a function of the collector to emitter voltage
during active-normal operation. It was also suggested that RBB and REE
be used for the forward region bendaway approximation. This appendix
presents the derivation first for bendaway then the derivation for the gain-V CE
function.
In the active-normal, or forward region operation, the base to emitter
voltage deviation from the ideal characteristic is defined as:
As this increase occurs for all three currents, it is appropriate to
establish relations in terms of each component.
In general,
but
Substitut).ng then gives the relation in terms of emitter current. This relation
is given by:
Ll.VBE = 1E [REE+RBB/ ( hFE + l )] '
substituting the relation that,
Ill
there results the relation in terms of collector current. This relation is given
by:
Making the substitution in the general equation that:
I = E
the relation in terms of base current is:
These relations all assume that I = I + I , where only positive E C B
quantities are used in the equations.
It can be seen that the values calculated for saturated conditions will
introduce bendaway in the forward region. If the bendaway is of considerable
importance in the forward region, it should be modelled to the desired current
by one of the equations given here. Note that there exists only an approxima-
tion. Note further that this increase will not affect the gain versus collector
current relation; therefore, gain in bendaway must be modelled by the 3rd
degree polynomial using the ideal collector current and emitter currents.
Second, the V gain modification derivation is presented. CE
The current gain in the absence of leakage is given by:
where IC is the ideal collector current without leakage. As the collector 0
112
current is made up of the ideal component and the leakage component -v2/Rc where v 2 assumes negative values for forward operation. This means then that
(3N and hFE cannot be considered equal when -v2JRc is of relative importance.
Then the de forward current gain is given by:
However, the ideal component is given by:
substituting this then results in the relation that:
For convenience, then,
and defining,
but as v 2 is negative during forward operation, it can be seen that gain will
increase with an increase in the V CE" For discussion, then,
for the current convention of all currents into the transistor •
v =-I *R +vt-vz+Ic*Rcc' CE E EE
113
where IE will be negative or out of the transistor and v 2 is negative for the
forward case making V CE positive, as v 2 is larger than the other components
when V CE = 1 volt is used for the initial gain.
Solving for v 2 ,
Substituting into ~FE relation,
V CE + IE*REE- v 1 - Ic".<RCC
Ah.FE = R >!<I C B
Solving for RC,
V + I *R - v 1 - I *R CE E EE C CC
R = C ~hFE~B
Then,
VCE R = +
C Ah.FE*IB
(eN+ 1)*REE
~FE ~ *I FE B
{3 *R N CC
Ah.FE
1 [V CE Vt J R = - + (!3 + 1)*R - {3 >!•R - - , C ~hFE IB N EE N CC IB
solving the ideal relation for v 1 in the forward direction.
I = IES [exp (q*v yME':•K*T) - 1 J E 1 - a >:C a
N I
IE ( 1-aN>'.cai) =
IES
114
which is given approximately by:
then,
Solving for v 1,
ME*K*T. v 1 = (O. 4343 ) q [ LOG 10IE + LOG 10 ( 1-aN *ai)- LOG 10IEs] , (A-1)
and
R = 1 [veE+ (/3 +l)"'.cR - f3 *R - vt J C Ah I N EE N CC I
FE B B (A-2)
As the gain is normally found as a function of IC, the additional relation
is needed that
where use of {3N indicates hFE is measured at V CB = 0.
In order to establish the effects of RC upon gain, the following example
is presented.
Writing the relation for v 1 in terms of collector current, there results:
M *K*T v 1 = O. :a43*q LOG 10(Ic/aN) + LOG 10 ( l-aNai)-: LOG 10(IES).
115
but
Substituting then gives All in terms of collector current FE •
1 [V CE *.BN v t*.B J All = - + R >!< (.8 + 1 ) - .8 *R - N
FE RC IC EE N N CC I . 0 c
. 0
Using collector current as the basis of calculation and the gain curve for
the 2N2907, a typical variation will be generated (Table A-1).
Table A-1. Typical .6..hFE versus RC
VCE Ic hFE IB REE Rcc Rc Vt Ah.FE
10v 0. 001 rna 4 0.00025 0.5 2 106 0.47 3.83
10v 0. 01 rna 10 0. 001 rna 0.5 2 106 0.53 9.5
10v 0. 1 rna 25 0.004 0.5 2 106 0.59 2.2
10v 1. rna 60 0.017 0.5 2 106 0.65 0.56
10v 10 rna 115 0.087 0.5 2 106 0. 71 0. 11
10v 100 rna 150 0.670 0.5 2 106 0.77 0.014
It can be readily seen that the approximation can be used over a very
limited range of collector currents. Very roughly speaking, the gain increase
is inversely proportional to I with the increase being typically 90 percent of the B
1 VCE term ----- The equation for .6..h should be used when exact numerical
RC IB . FE
values are desired with the approximation for v 1, given by the logarithm
expression.
116
If desired, equations A-1 and A-2 can be pro~ammed using 1he !l Ic(13N+ lL N
versus v 1 polynomial and relation that IE = 13 to give the AhFE for any N
RC. This is recommended, as the approximation is very limited in current
range and effects at other collector currents must be closely observed or
significant errors may be overlooked.
APPENDIX B. MODIFIED EBERS-MOLL CONVENTIONS AND MODE DEFINITIONS
Given here are the current voltage conventions used by the NET-1
digital computer program for the modified Ebers-Moll transistor model
(Figure B-1, Table B-1).
c c
NPN •cJ 1cc PNP
•cJ 1cc
+ f
•• •c v2 •a
•c
••• • •• •• v, RE
-!
•e f RI!E •e f 1 EE
• •
Figure B-1. . Current and Voltage Conventions for Transistor Model
117
+ f v2
vl
+l
Active Normal
v 1 positive
v 2 negative
Active Inverted
v2 positive
v 1 negative
Saturated
v 1 positive
v 2 positive
Cut Off
v 1 negative
v 2 negative
Table B-1. Modes for Model
Emitter base junction forward biased
and collector base junction reverse
biased.
Emitter base junction reverse biased
and collector base junction forward
biased. Emitter acts as ncollector";
collector acts as "emitter. "
Both junctions forward biased.
Both junctions reverse biased
118
119
APPENDIX C. DETERMJNATION OF BASE TRANSIT TIME
Two techniques for determining the base transit time have been used in
this writing resulting in a choice of method depending upon the particular appli-
cation. The explanation relative to this will follow the derivations for each
technique.
where:
The base transit time 1, is given by:
tt = ~ + .,. c + .,. e ' (C-1)
tt is the total time delay as a function of emitter current.
~is the base transit time or the average time required for a minority
carrier to cross the neutral base region.
-r is the time constant of the emitter transition capacitance and the e
dynamic emitter resistance, r . e
.,. is the time constant of the collector transition capacitance and the c
collector rest stance, r sc •
These time delays simply add, resulting in equation C-2:
t = t. + C *r + C *r . t -b TE e TC sc (C-2)
For small emitter currents the dynamic emitter resistance is given by:
re = dVBE/diE ,
120
but
Differentiating with respect to IE:
Substituting
r = (K*T/ q) (1/I ) e E
into equation C-2 gives equation C-3:
(C-3)
where 'T is small at the low emitter current and is assumed negligible here. c
The emitter delay can be determined by a time delay measurement made
by driving the transistor from cutoff to saturation with the test circuit configura-
tion shown in Figure VIII-I. The resulting collector current waveforms also
appear in Figure VIII-I.
The time delay td as shown in Figure VIII-I is the time in seconds it
takes the collector current to rise from zero to 10 percent of its final value and
is given by equation C-4:
The total timet is related to the gain-bandwidth product by: t
(C-4)
121
Solving equation C-4 for CTE and substituting the re~mlt into equation
C-3 along with \(IE) relation gives equation C-5:
( t >!<I ) ) d Bl K':'T ~ = lf2':'7r'!•f ( ) - _,_ (- (1/I \
tIE 2-·-~ q E) BB
(C-5)
This relation appeared in the original degradation program as a means
for calculating ~. However, it was replaced by the following technique to
attain a constant value of ~ in the FORTRAN IV version.
The delay td is made a function of collector and/ or emitter current, but
it is observed that the quantity (t >!<I )/ (2':' ~) remains very nearly con-d Bl BB
stant in the low emitter current region, resulting in Figure C-1.
-., --
Figure C-1. Base Transit Time
122
Use of this graphical representation for equation C-5 leads to the second
technique for determining ~. By extrapolating the low emitter current linear
region characteristic to infinite emitter current, the intercept is taken as b.
Below emitter currents corresponding to point A in Figure C-1, tt = ~ + T e.
In practice, it is relatively easy to make a rough plot of Figure C-1 and then to
least squared error curve fit the data from A to B using y = mx + b, where
y = \ = 1/[2*7T*ft(IE )]
b= b m = CTE (K*T/q)
x = 1/IE .
The graphical technique using the curve fit has several advantages.
First, a simple plot verifies the low currents assumptions, and, second, a
quick approximation to ~ is found.
The approach using the switching delay has several drawbacks. If
manufacturer's data are being used, it will assuredly result that the value of td
nominally presented does not correspond to the nominal value presented for ft.
In the original version of the degradation program, this resulted in a varying~
in the region where it should have remained constant. This technique was
replaced in the FORTRAN IV degradation program.
In general, then, the method represented by equation C-5 may be used
when f and t are measured for the same transistor, but it is recommended that t d
the graphical extrapolation be used as it requires fewer measurements.
123
A comment is in order relative to the measurement of the gain-band
width product from which~ is determined. The gain-bandwidth measurement
method is subject to error at fluences where hFE is significantly degraded. :For
an explanation and proper derivation, attention is called to the reference by
Manlief. 12
124
APPENDIX D. PREDICTION OF VCE(sat)
To determine a new V CE (sat) characteristic as a function of IB, 10 ,
and rJ, it is necessary to have a pre-irradiation characteristic of V CE (sat)
versus IB for the values of collector current that are input into the saturation
modelling program. This normally requires four values of collector current,
but the same value can be used at the two values of forced gain taken in the high
current region thus reducing the number of values to three.
Presently the effects upon the saturation characteristics are not
theoretically predictable, but several observations have been made that
warrant attention. It has been observed that the saturation characteristic
shown in Figure D-1 does not vary in shape, but as a result of a gain reduction,
shifts along the I axis as given approximately by equation D-1: B
AI = I (post) - I (pre) = t. *K'* cJ> • B B B o
(D-1)
In addition to this component, the characteristic will shift up the voltage
axis by the increased voltage drop across the collector bulk material as given
by equation D-2:
AV CE (sat) = IC *ARc ( cJ>) • (D-2)
125
ACTIVE IC = CONSTANT
~
·~ k~.;;;::::---~ E-1~ _'\. ___ tJ._I B=--_ _.,..1 ~ lctJ. Rcc I ~ -.p•
p 1---- _:-:_::::----=l.:~~--=-- ________ ... SATURATION
1 10 100 1000
I 8 (ma)
Figure D-1. V CE versus IB
The composite relation is shown in Figure D-1 where P is the point
representing the edge of saturation for the pre-irradiated case, and P' is the
point representing the edge of saturation for the post-irradiated case.
In using the saturation data modelling program, the post-irradiated
characteristic can be established without using equation D-1, since the shift
along the IB axis results through the change in the gain polynomial. Then
equation D-2 can be used to increase the value of RC to give satisfactory
results.
It must be noted that the values of V CE (sat) must be changed before
input to the saturation data modelling program if the values of f3N 1• f3N 2' f3N 3 '
and {3N4
are to be changed. To input these changes requires that Figure D-1 be
entered at values of IB that correspond to the values of {3F used as inputs.
126
These values are given by equation D-3:
IC t/{3F = IB . (D-3)
Three or four curves similar to Figure D-1 will be needed to obtain
values at all collector currents inputed into the saturation data modelling
program.
127
APPENDIX E. K' FROM REACTOR DATA FOR IC CONSTANT
It is the purpose of this Appendix to present the characteristic of
reciprocal common emitter· de current gain and its relation to neutron dose and
the composite damage constant (K'). From the relation that~l/hFE) = ta, *K'*cp,
tt is noted that for a constant emitter current, the relation is a straight line
with cp as the independent parameter. Superimposed upon the straight line will
be two additional characteristics as can be seen in Figure E-1. At low doses
and low currents, it is observed that surface damage adds to the reciprocal gain
relation. ~-------------------------------------/~------~
IU
-e. ... -<l
tiDU
,::.. ~~,(7'
~\) .. ~ v.,~~
Figure E-1. Reciprocal hFE Versus Neutron Fluence
128
In the nuclear environment definition in Section I, it was shown that the
surface damage from gamma radiation was probably negligible. As the
ionization effects saturate, it is appropriate that the surface effects be
removed from the nuclear reactor data to allow K' to be made a function of
emitter current density. Care must be used in this operation as the surface
damage may be a result of ionization caused by the neutrons bombarding the
gases used in the transistor cans. In this case, the surface effects cannot be
assumed negligible even for a neutron bombardment. However, for the purpose
of determining K' the component must be removed.
At high doses and high emitter current an upward bend in the reciprocal
gain relation is noted as a result of operation in saturation. Presently, the
theory or empirical relations do not provide for gain relations in this region.
Then, in the determination of K', data in this region must be deleted. Care
must be exercised in the subsequent analysis not to extend the gain prediction
into the saturation region even though a value of K' exists empirically.
It can be seen that the slope in the linear region is the minimum slope on
the reciprocal gain characteristic. Then for the linear region, the slope is
given by:
The reciprocal gain relation then is given by equation E-2:
1/hFE ~ 1/hFE o + [a (1/hFE)/ a .p] .p '
but this is equivalent to:
1/h = 1/h + t. *K'* cf> • FE FE b
0
(E-1)
(E-2)
129
Then by equating corresponding components,
and assuming ~ has been found by the technique suggested in Appendix C, K'
can be found by:
minimum . (E-3)
This value must be measured from a characteristic of gain versus
neutron dose at a constant emitter current to establish the emitter current
density dependence of K'. Then knowing the physical area of the emitter, K'
can be determined with the correct dependence upon emitter current density.
As de forward current gain is usually measured as a function of neutron
dose at a constant collector current, it is necessary to establish the relations
to convert the data to the reciprocal relation in terms of emitter current
instead of collector current. Equations E-1, E-2, and E-3 apply to the rela-
tions when emitter current is held constant and must be modified if collector
currents are used.
At any value of collector current where K' is appropriately used,
where h is the gain at the value of IC for which the calculation is being made. FE
This is the degraded value of gain.
Previously much work was done in which the de forward current gain
was presented as a function of neutron dose with the collector current held con-
stant. To use these data, it is necessary to establish a technique to correlate
the collector current to the emitter current.
130
If there is no surface or nonlinear damage (Appendix F gives method of
recognition) a plot of reciprocal current gain (l/hFE) versus neutron dose 4>
will appear as shown in Figure E-2 for a constant collector current (IC).
There is a slight bendaway from the straight line representing the curve for
some higher value of emitter current. To determine the value ~ *K', it is
necessary to measure the slope of the constant emitter current characteristic.
This can be done by following the steps given next.
0 tRDU
Figure E-2. Damage Conversion
Step 1
At some arbitrary point A, take the degraded value of hFE and using IC
for which the characteristic was measured, calculate the emitter current by:
131
Step 2
Using the relation for the de current gain as a function of emitter
current at zero neutron dose, establish the value of reciprocal current gain at
the value of IE recorded in Step 1. This value of reciprocal gain determined
point B.
Step 3
Construct a straight line between points A and B and label with the value
or emitter current determined in Step 1.
step 4
Calculate the product (~*K') by:
>:< ¢must be zero and h must be the nondegraded de current gain. FEB
step 5
Calculate K' by dividing out the base transit time (tB) ·
step 6
Calculate the emitter current density by dividing the emitter current
from Step 1 by the physical area of the emitter·
132
where ECD is the emitter current density.
Step 7
Record values of K', ECD, and LOG 10ECD for curve fitting to the
polynomial relation:
K' = K 1+K2 [LOG 10 (ECD)J + K.[ LOG 10 (ECD)J 2 + ... + Kn [ LOG 10 (ECD)J n-l
This completes the technique for determining K' from the constant
collector current characteristic where no surface or nonlinear damage exists.
Appendix F gives the method of removing the surface damage component of
reciprocal current gain ( 1/hFE) .
APPENDIX F. REMOVAL OF NONLINEAR DAMAGE AND K' POLYNOMIAL COEFFICIENTS
133
In Appendix E it was shown how to determine K' from a gain measure-
ment at a constant collector current in the absence of surface or nonlinear
damage. In the use of shielded core nuclear reactors or in the case where the
bombarding neutrons ionize the gases used to fill the transistor cans, surface
damage may exist. If it is desired to separate this component to leave only the
linear damage, the following technique is suggested for consideration. Figure
F-1 shows the reciprocal gain with and without surface damage.
1.1h,. I
1A,1•~------------------~~--------------------_. t tRDU Figure F-1. Damage Removal
134
It is observed that the surface component of reciprocal gain saturates,
and therefore the two characteristics become parallel at doses greater than
that depicted as A-A.
Then in the region AA-BB it is necessary only to use the slope of the
characteristic with surface damage for the slope of the characteristic without
surface damage. The region from¢ = 0 to AA is determined by extrapolating
the relatively linear region of IC to the intercept C and then shifting IC 0 0
characteristic until C coincides with 1/hFE . It should be noted that the upo
turning beyond BB represents operation in saturation and should not be used to
determine K'.
To effect the operation described in a digital computer, the following
steps should be followed:
Step 1
Curve fit the region AA to BB of characteristic IC with a Taylor Series 0
expansion about ¢= 0 using a least squared error criterion. This results in a
polynomial of the form:
2 A *A-n C + A2*¢+ Aa*¢ + · · · + (n+1) '¥ •
Step 2
Replace C by 1/h at the corresponding collector current to give the FE
0
relation for linear damage. The result is as follows:
135
1/h = 1/h + A2*r+.+ A3'>',cr+.2+ +A •:~r+.n FE FE 't' 't' ' ' ' (n-1) 'f" • (Ic0 ) o
This polynomial can then be used to find the gain at any neutron dose, cp, for a
particular collector current.
It is necessary to establish the coefficients for the composite damage
curve polynomial for the mean, minimum, and maximum values for NPN and
PNP silicon transistors. The equation used in the program is as follows:
ECDE = 1 = 0. 4343 LOG (ECD)
CDKN = Bl + B2 (ECDE) + B3 * (ECDE**2) + B4 >'.c (ECDE*•:<3) ...
CDK = CDKN/ (10. ** 6. ) .
For passive radiation, V CE = 5 volts, and T = 35°C, the following data are
fitted to fourth order least-squared-error Taylor expansion about zero.
Minimum (NPN)
0.000001 3.450 B1 +3. 423
1.0 1. 600 B2 -2.564
2.0 0.830 B3 +0. 8651
3.0 0.58 B4 -0. 1316
4.0 0.58 B5 0.007738
0.0 0.0
1 + LOG ( IE/AE) K' Coefficient
Mean (NPN)
0.000001 5.65 Bl +5. 697
1.0 2.67 B2 -4. 488
2.0 1. 85 B3 +1. 789
3.0 0.87 B4 -0. 3701
4.0 0.88 B5 +0. 03154
0.0 0.0
Maximum (NPN) Mean (PNP)
0.000001 9.00 Bl +9. 005
1.0 4.08 B2 -7.606
2.0 2.20 B3 +3. 385
3.0 1. 32 B4 -0. 8149
4.0 0.96 B5 0.0805
0.0 0.0
Although there is not an abundance of data points to verify that mean
(PNP) is nearly the same curve as the maximum (NPN), the data points will
be used here.
136
APPENDIX G. COLLECTOll. LEAKAGE .RELA'l'lONS AS A FUNC'l'ION Oli' FLtrENC:B
137
In the typical silicon transistor of today having a lightly doped collector
region, the collector-base junction leakage is in tlle nanoampere range and fol:'
most practical uses is considered negligible. Bo-w ever, as it is desired to
extend the forward region gain characteristic down to gains approaching unity
for the Modified Ebers-Moll model, it is aptlropriate that a techniqae be pre-
sented to model the leakage as a function of neut:t-on doee received, as leakage
effects upon de current gain hFE become significant in tbe low gain region.
In the absence of surface damage, \\lbether from garPnla radiation or
from gas ionization by neutrons, leakage iS a recombinatioJl process and,
therefore, depends upon the collector-base depletion region volu:rne, the carrter
concentrations, and the intrinsic recombiJlation rate per ca:rrier.
In general, the junction leakage fot the reverse biased collector base
junction is given by equation G-1.
1coo == q* 0c *Xc) [ ni (Rio +- I<rg *<P )]
where
q = 1. 602 * lo- 19 coulomb
A = collector area in em 2 c
X = depletion region width in em c n1 = intrinsic carrier concentratioJl
(Q ... l)
R. = pre-irradiation recombination rate per carrier 10
K = recombination rate neutron damage constant equal to rg
¢ = neutron dose in RDU.
138
To implement a calculation using equation G-1, the following techniques
are suggested:
The term AC is found by measuring the physical area of the collector
metallization either by photo micrograph techniques or by use of a measuring
microscope having a moving bed with micrometer calibration.
The term n. is calculated using the three halves law for intrinsic 1
semiconductors as given by equation G-2:
3 87 ,, 10+ 16, T3/2 ,, -1. 21/2'!<K':<T n = . -·• >;c -;ce , i
where
T = oKelvin = oc + 273
q = 1. 602 '".c 10- 19 coulomb
e = base of natural logarithms.
(G-2)
The determination of X presents a somewhat more difficult problem in c general, but it has been observed that thermal equilibrium concentration values
will give results that agree with actual leakage measurements. This then
implies that:
n = N n D
n = (n.) 2/NA , p 1
where
N = donor concentration D
N A = acceptor conce~tration,
and in the case of an npn transistor,
N = N D c
139
This assumption then allows for the built-in junction potential (IJ! 0 ) to
be calculated by equation G-3.
[N *N J IP = K*T * LN A D
o q (n.) 2 1
(G-3)
The total voltage across the junction is given by V in equation G-4
V = V CB + IP o ' (G-4)
where V CB and IP 0 are positive quantities for a reverse biased junction.
To complete determination of XC requires that the necessary equation
for a stepped or graded junction be used. As the concentration· of the collector
region is small compared to the base region and the base spreads into the
more uniformly doped collector region, the equation for a stepped junction is
used to determine XC. Further as the collector doping is much smaller than
the base doping, equation G-5 will reduce to equation G-6 with N equal to NC:
~ 1/z
X = [ 2V e- (_!_ + __!_ \ c q ND NAL (G-5)
[2V ]%
XC= -t (~)
where
140
(G-6)
V = V CB + 1/Jo
N = doping concentration of collector.
The remaining parameter R. can best be obtained from the manufactur-10
er; however, it is possible to determine this quantity by measuring the reverse
junction leakage at several values of V CB' The value of Rio can be obtained by
using equation G-7 with ¢ = 0:
Rio = 1cBcj( q*Ac *Xc".cni) · (G-7)
This equation will give constant values of R. for mesa transistors, but 10
will not do the same for planar transistors. For planar types, Figure G-1
results and the value of Rio is found as point A by extrapolating data to V CB = 0.
3R. 10
2R. 10
R. 10
oL--------------------------------------vcB --•
Figure G-1. Intrinsic Recombination Rate
To convert this to a leakage correction factor for planar transistor,
divide by R. and label the axis as leakage correction factor. 10
141
Figure G-1 should not be construed to indicate that R. is a function of 10
the collector-ba~e reverse bias voltage as this characteristic is only an
empirically developed technique to determineR. and the leakage correction 10
factor, because for planar transistors, the leakage increases faster with applied
voltage than can be accounted for by the dependence of the collector-base deple-
tion layer width upon applied voltage. It has been observed that the leakage
correction factor is nearly the same for several types of planar transistors and
represents a practical means for attaining leakage predictions.
Having attained the quantities as described, it is now possible to calcu-
late the leakage after neutron bombardment by equation G-1.
The previously described technique is for transistors having low doped
collectors where the stepped junction equations are valid. Further, it has been
assumed that no surface damage has resulted from gamma radiation or neutron
ionization of the gases in the transistor cans.
APPENDIX H. EMPIRICAL ANNEALING RELATIONS
This Appendix is presented so that previous work done relative to
annealing of neutron damage will be included.
142
Thus far the bulk of material presented deals with observations. and
only a few theoretical relations have appeared in the literature. The
predominance of information is given with correlation to the composite damage
function and therefore only gives information for current densities above 0. 1
amp/ em 2• The relations appearing are empirical and are recommended only as
a means to analyze a transistor using test data for a particular device and test
conditions.
The polynomial relations established in this section are intended only to
implement the computer calculations 1n the high current density regions and to
give a relation that can be referenced to for discussion purposes. The
polynomials represent no particular data; therefore. values are set numerically
equal to unity or zero unless test data are read into the program. Test data
must be curve fitted to a Taylor series least-squared error interpolating
polynomial of 4th degree and the coefficients entered as data. This somewhat
reduces the utility of the program but will suffice until theoretical relations
are established for the annealing characteristics.
143
In Section II the composite damage function was established as a function
of emitter current density; however, there are three other important variables
upon which damage is dependent. This section deals with the three variables
and outlines the analytical approach used to incorporate their effects upon the
damage function K'.
The damage function is empirically related to current density where the
following test conditions are specified. K' is given at a temperature of 35°C,
passive (no operating current) operation, and at time greater than 105 seconds.
If other conditions exist for the application being considered, variations from
time, current, and thermal effects must be included to account for the differ
ence in behavior. The variations that are to be incorporated are:
( 1) Transient or time anneal with the thermal and current modifications,
and
(2) Active/passive damage ratio as a function of emitter current with
temperature variations.
Transient (time or beta) anneal has been observed to reduce damage by
as great a factor as 5 in silicon transistors. That is, damage may be 5 times
greater at short times after neutron bombardment than that given by K'. The
decrease or anneal is probably brought about by a decrease in damage resulting
from annihilation of interstitial defects and possible reordering of large defect
clusters. As reordering appears to be a function of how energetic the atoms
are, it is expected that increased temperature and current will result in reduc
tion of damage if temperature is elevated above 35 o C and/ or if current is flow
ing during irradiation. This expectancy is supported by data, but presently
144
conclusions are not drawn on the limited data. As of yet, no theoretical
relations have appeared in the literature; therefore, the inclusion will be made
in an empirical fashion and based upon test data for the particular device in
question.
The annealing factor has been defined by the following relation:
D.l/hFE (t) AF=--.;;..o;;~~
t:&l/hFE(oo)
where t:&l/ hFE (t) is the change in reciprocal gain at time (t) from original
reciprocal gain and D.l/hFE (oo) is the change in reciprocal gain at time
(infinity) from original. This is understood more easily and its relation to K
established, by the following derivation:
D.l/hFE (oo) = b * </>*K' (oo)
D.l /hFE (t) = b * cp*K (t) •
As the base transport time ('~) and change·in neutron dose (D.¢) are held
constant throughout measurement, it is correct to form the following equation:
t:&l/ hFE (t) ~ * C/J*K' (t)
t:&lfhFE (oo) = b * C/J*K' (oo) =
K' (t)
K' ' (oo)
but this is the definition of the annealing factor (AF). Using this fact and the
definition of the changes:
K' (t) AF= K'
(oo)
145
Then, to include the annealing factor, it is necessary to form a time
dependent polynomial to multiply times K'. This is not a polynomial with only
time as the independent variable as there are temperature and current
dependencies invqlved.
It has thus far been observed that for temperatures above 35 o c,
temperature will have small effect upon the transient damage relation. This
allows for the establishment of an annealing factor as a function of time for a
constant current. It is necessary to form a polynomial for each desired cur
rent, as there would be no justification for establishing a polynomial with two
independent variables, since a constant value of current must be entered. It
cannot be assumed that damage for a varying current is related to the damage
at a constant current by a simple polynomial of two independent variables, as
nothing indicates that the variables are really independent.
For temperatures below 35 o C, there exists a complex relationship of
four variables that cannot be described in two-dimensional space while holding
one variable at a constant value. For this reason current and temperature must
be held constant at some value for which the annealing factor is desired. For
the purpose here, the same polynomial will be used as that used for T > 35 .. C
except that the additional requirement is made that the temperature be defined.
As the temperature parameter must always be included for the degradation
program this imposes no problems. Care must be taken to insure that the sub
routine that establishes the polynomial also establishes the temperature for any
further analysis. The current and temperature are also needed so that the
annealing factor can be multiplied by the correct value of damage. This value
146
of damage must first be established by evaluation of the active/passive ratio
which is described next.
The active/passive ratio (A/P) is the factor that reduces the value of
damage at times greater than 105 seconds as a result of current flowing during
radiation and/ or as a result of temperatures above 35 o C. To incorporate a
mean value of A/P for operating currents during irradiation, the factor will be
defined as a function of emitter current density and temperature. Presently,
data indicate that the dependency is a function of current density at high
currents:
K'ocT = K'* (A/P) •
where K'ocT means the value of K' (oo) as a function of operating current and
temperature during irradiation. Then the ratio is given in general by:
A/P = Polynomial (IE/ AE) .
As the current is lowered this polynomial approaches unity until at zero cur-
rent, unity exists by definition. At low currents, or below the current where
theA/Pis practically equal to unity, there is evident an increasing dependency
upon temperature. For the computer program there will be no temperature
effects considered where theA/Pis practically less than unity. For the low
current region or no current region, the function used to multiply times K' is
primarily a temperature correction factor, that is, a function of emitter
current. For this region,
K' = K' * TCF, T
147
where
TCF - temperature correction factor.
TCF will be specified at a particular temperature and as a function of emitter
current. Both factors will be incorporated as polynomial expansions with a
determination as to whether K' or K' is used. It is necessary to deter-OCT T
mine from data where the K' modification changes from temperature dependency
to current dependency. This is accomplished by observing where the A/ P
becomes practically unity. Below this point in emitter current K' will be used T
to determine damage. It is possible to determine analytically the crossover
point between temperature and current dependencies, but it is appropriate for
operator to input information so that understanding of operation and meaning will
be insured.
In order to make use of gain degradation data from reactor tests, it is
necessary to have the data qualified relative to temperature, current, voltages,
etc. Present practice is to radiate passively with neutrons in an approximate
fission spectrum at a temperature of 35°C and to measure de current gain after
105 seconds to give transient annealing effects time to become negligible. The
information is then presented as a function of emitter current density which
necessitates that measurements be qualified relative to emitter current, base
transport time, emitter area and neutron dose. It has been found that reason-
able data will be obtained if gain is measured in decades of collector current
and presented for each decade of neutron dose.
148
The damage factor to be derived from the data is empirical in nature
and is functionally dependent on emitter current density in the literature only
because correlation presently is better using this approach than trying to
separate the various effects on the five base current components. Section II
on gain theory gives a more detailed explanation.
Observation of the minimum, maximum, and nominal values of K' for
many transistors shows more of a variation in the value of K' at high and at low
values of emitter current density. At high current densities it is observed that
diminished accuracy occurs probably as a result of current crowding or opera
tion in saturation. Current crowding results in an increased current density at
the emitter perimeter and therefore a reduction in effective area. Then it is
expected that the geometry of the emitter will have a part in determining the
value of Kat high current densities. If many predictions are necessary in the
high emitter current density region, it may be beneficial to make an attempt at
correlation using an effective area based upon the emitter area, the emitter
perimeter, and the base spreading resistance.
Another effect that causes prediction errors in the high current region is
the effects of saturation. There presently is little in the literature that would
allow for an analysis in this region, but the effects can be seen as an upward
bending in the reciprocal gain versus neutron dose plot resulting in an increase
in the value of K'.
As it appears that damage from neutrons is almost entirely independent
of rate of application, and because the energy dependency is established through
149
an energy versus damage relation for conversion of damage from one spectrum
to an equivalent spectrum, the calculation of K' can be done on an incremental
basis instead of a differential basis; then it is necessary only to establish two
points of gain in the linear region and calculate K' from a reciprocal gain
change and a corresponding change in neutron dose. The derivation is presented
here to clarify questions that have arisen during reading of the literature.
Using the damage relation for two points,
hpE 0
1 + hFE0 *b *K'*«f>1
hFE 0
where these values are taken at the same emitter current so that K and tb and
h are equal in the two equations. Subtracting FE
0
b - h = h [ FE2 FE1 FEo l+h *t.*K'*«f>
FE ~b 2 0
1
establishing a common denominator,
but
and
Substituting
then cancelling hFE and solving forK', 0
h -h FE2 FE 1
K' =
This is the form necessary when reducing raw gain data from a reactor test.
150
This form will be programmed in a subroutine and the development appears in
Appendix E.
To establish the form presented, it is necessary only to establish the
two following definitions:
and
hFE2-hFEt
hFE2*hFEt =
hFEt
hFE/.chFEt
which upon substitution into (1) gives
K' = b
1 1 .6 1/hFE
= ·-
hFEt hFE2 hFE
or change in reciprocal gain is proportional to dose change multiplied by the
darnage constant of the same emitter current. A salient point here that must be
considered is that the equations (1) and (2) are only good for the linear region
of the reciprocal gain versus neutron dose curve. More will be said about
this in the development of the subroutine in Appendix E.
151
152
APPENDIX I. ENVmONMENTAL DEFlNITION
It is the purpose of this Appendix to establish an algorithm from the
various data to determine range, neutron fluence, and gamma dose for a chosen
yield weapon as a function of moderate or severe mechanical damage to a
variety of types of vehicles and equipment.
The method for a given yield weapon is to determine the variables
suggested in the preceding paragraph by polynomial curve fitting of data for
ground bursts and air bursts at the corresponding radii of optimum mechanical
damage.
The following is an algorithm with reference to necessary data and
normalization techniques used to allow for a computer program to be written to
perform the numerical calculation.
ALGORITHM
(1) Establish vehicle type and damage (moderate or severe) from
Table I-1.
(2) Enter Table 1-2 to determine a value for the vehicle/ damage index.
Example:
153
Table 1-1. Vehicle Type and Damage
Type No.
(1) Truck mounted engineering equipment (unprotected)
(2) Earth moving engineering equipment (unprotected)
(3) Transportation vehicles
( 4) Box cars, flat cars, full tank cars, and gondola cars.
(side on orientation)
(5) Locomotives (side on orientation)
(6) Telephone lines (radial)
(7) Telephone lines (transverse)
(8) Average forest stand
(9) Box cars, flat cars, full tank cars, and gondola cars
(10) Locomotives (end on orientation)
( 11) Merchant shipping
Table 1-2. Values for Vehicle/ Damage Index
1M= 5.832 lS = 4.568
2M= 4.071 2S = 3. 100
3M= 5.832 3S = 4.568
4M = 5.832 4S = 5.084
5M = 4.815 58= 3.100
6M = 6.429 6S = 6.815
7M = 7.000 7S = 6.429
8M = 5. 516 8S = 6. 429
154
Enter type number (Table I-1) and letter M for moderate or S for
severe damage from Table I-2.
a. Transportation vehicle
b. Moderate mechanical damage
Table I- i - number 3
Table I-2 - number 3M
Index= 5. 832
P2 = 5. 832 for computer input
(3) Determine yield for which analysis is desired. Use W for yield in
computer program. (Yield in kilotons.)
Applicable equation for Figure I-1
N 1 = LOG 10 (yield) in tons
N = Nl + (M-N1) Z/ 10
V = A>!<N + B
R = Ground Range
V = A [N1 + (M-Nl) (Z/ 10)] + B
R= 10v
R= 10A{[N1+ (M-N1)(Z/10)] +B}
where
A = 0. 5367
B = 1. 0485
z = 2. 56
M = P2
N1 = Pl
}:as I
20MT 5
7 + lOMT I 4 +4 -
'f lhiT r. I
700 Jnn
4
l -
t 5t10CKT "i
Q 70 • ..J :!:. ! 40 w > ~
20~~ > ... I - OKT 0!
11 ! % m 9
7 9 m
4 3 5 115
-10
m
__ a 1 10 s
2
I. z I .j Figure 1-1. Damage-Distance Relations for Targets
8 1 m
t7m
6 17 18 m 5 5 - 6 I 5
4 5
5 m
1 , 3 5 5
2 I m
2 I 5 5 s
1 ~
1-' en en
156
R = 100. 5367[N1 + (M-N1) (0. 256)] + 1. 0485
In the computer program this appears as:
R = 10 { 0. 5367 [P 1 + (P2-P1) (0. 256 >] + 1. 0485}.
(4) Normalize Range (R) toRN using the following equation:
RN = R/300 ~W .
(5) Using Figure 1-2 enter RN and read from curve, the normalized height
(H1N) of burst by using characteristic A-A (dashed curve).
(6) Unnormalize to attain height at which a given yield will have the greatest
damage radius by H10 = H1N ff. (7) Follow constant pressure line back to ground level to find lesser range
that has equivalent damage effects as for the optimum height. This is
termed RG in the computer program and is found by
RG = R >'.c POLY ,
where POLY is a LaGrange polynomial approximating the normalized
relation for the ratio of the ground distances for air and ground bursts.
This ratio is plotted in Figure 1-3.
(8) Calculate slant range (RSO) using the Pythagorean relation
RSO = ~R2 + H102 •
(9) Enter yield into Figure 1-4 to determine the gamma dose scaling factor
as a function of yield using the following equations:
C = LOG 10 (yield in kilotons)
SFL = C * POLY1
SFL SFA = 10 ,
1000
, 900 --.......
800 ~~ r--.
-.. : 0 700
::=. 600 ~ a::
-
; 50 &L. 0
,r-.....
or---... ... % 400 C) -w :c
30 0
20 IQ
~
~
~ /
100 « II /. ··v
~
~ ~
--........
"" ~ r--... ' ~ ~ .........
"" ~i' "' 50 v ~ ~ ~
100 7 ly ~ / / IJ .......
D v / I /
~ ~ I
J
j
10 I I I I v~ REGULAR,REFLECTION RECIO~ ....
'-...... _,..,. ~ NS / 1\
.......
~ ~ ' ~ ~
H 7 MACH REGION A ~ /
~ IJ I ........... ..,...
r\ ~ P" 1----L"" l---..... ,...
vr ~ v v / I I / I
v I I I
J J
I v I I I I
I
J / I I I
Ol 100 200 300 400 500 600 700 800 900 1000 1100 1200 1300 1400 1500
RANGE FROM GROUND ZERO (feet)
0 1 2 3 4 5 RN---....
Figure I-2. Peak Ground Overpressures for 1-KT Burst 1-1 ~:.;,
"'I
~ C) D::
1.0 -I -'~..-_._ I T
: o--,___ -2.__ I 1 ~--0.8 ~
6 0---0 - 1'\ -
0.6
o. .t-
0 2----c=±-+-+-+-----J--~ ~I I ......
200 400 600 1600 800 1000
R (feet) 0
1200
Figure I-3. Range at Optimum Height (R0 )
1400
1-' 01 (X)
3 1000 r---r
700 r-
400 lr--t-
200 '0 t-
2 100
I ~ 70 0 ..... u 4( 4 u. C) z ::i 2 -< u .,
J 1-
)r-
)r--
7~
~
4
0
z 2cz1 1
1 KT
0
2
I I I I I I 71 I ' I
/ /
I I
J v
" / -p
I / / 7
I I
/ 7 /
v I I I /.1 I
4 7 10 20 40 70 100 200 400 700 1 2 KT KT MT
EXPLOSION YIELD
1 2 3
c-Figure l-4. Scaling Factor for Initial Gamma Radiation
I I I -t 106 ~ 7
- 4
fl u
j_j ] j
I -
2
105
7
-I -
1-!-- -4
/ -
I -2
J 104 --7
--
4 -
2
-I I
.1.- 103 4 7 10 20
MT MT
4
IX 0 ..... u -< u.. C) z ..J -< u V)
6
5
4
3
J--1 Cll ~
160
where
SFL = C if C is less than 1
SFL = 1+ 1.116(C-1) + 0.0615(C-1) (C-2) -0.0065(C-1) (C-2) (C-3)
if C is greater than 1.
( 10) Enter range into Figure l-5 to determine gamma dose for a 1 kiloton
weapon then multiply by scaling factor (SFA) to attain actual gamma
dose.
Do this for minimum and maximum ranges (RG and RSO) to attain
minimum and maximum gamma dose.
(11) Apply deviation factor from Figure l-6 to gamma dose to account for
weapon variations as a function of yield.
(12) Enter range into Figure l-7 to determine neutron dose for a 1 kiloton
weapon then multiply by yield in kilotons to attain fluence for desired
yield.
If it is desired to study neutron fluence and gamma dose only for a given
yield and vehicle/ damage index, the computer program at the end of this
Appendix is recommended.
In order to attain a quantitative measure of the problem facing silicon
transistors, several outputs from the computer program appear in Figures l-8
through I-11. These figures present the neutron fluence and gamma dose for a
transportation vehicle at a distance from the detonation where moderate or
severe mechanical damage will be sustained.
5
3
2
0
-Ill c Ill D' .. c
7
4
2
o4
7
4
2
~ 10 3 .. -UJ 7 cl)
0 0 4 % 0 1-...( 2 E ...( 0:: ...( 10 2 ~ ~ ...(
" 7
4
2
10
7
4
2
\ \
\
' \ \ \
\~
1\ \ \ \ \ \ \
........ \ \ \ \
\ \ \
\ \ \ \
\ \ ' 1 mile 2 miles 13 ~iii\
I 1 I I
0 1000 2000 3000 4000 5000
SLANT RANGE FROM EXPLOSION (yards)
1
7
4
2
4
4
w 4n 0 0 % 0
161
0
2 ~ 0 ~
1o·3 o:: -3 ~
7 ~ "' ..
2
1 -4
7
.. 2
1o·5 .s 6000
Figure 1-5. Initial Gamma Radiation for 1-KT Air Burst
IU V)
8
i C)
~ 0 .... ~
e u ~ .... ~ i= ~
> IU Q
10
8
6
4
2 ~
I
~
' -6
-1 ' 3
/ ~
/ /
v ~
I
+ DF I
l - DF --~
" "" ~ ............
"'---. 4 -5 . 6 7
WE-LOG (yield in tons)
Figure I-6. Deviation of Gamma Dose for Weapon Variations
8
1-1 0) l'\)
15
14
12
11
N' E ~
011 c 0 .. .. ;)
• c -
1o 15cr--,---\T------,-------,----....., 1- \ 1011 71- -
4~~-t-------t--------~~~~~----~--------~~-------J= 4
2,~-~--~------r-"~--4-----~----~2 10141=1---t-----r-------~---\~~~------~~----~
1
1- \ 1010
7~~\~--~-------r-----\~+-------~------~-7
4r~-r-\r----r--4\~--~~>
163
11
10
2 r----1---t-\ -t----+--~1\\~---------~-2
1o 13·r~-------~+--------+--------+--+----~------~1o9 ~ 9 1- T \ - ~
7 \ -1 ~
~ \ \ = ! 4r-------~--------~--------~----~~------~· ~ \ - ~ 2r-----~\~\--~----~----~,~--~2 ~
1012 ~ - 108 z
::: \ \ :7
\ \ :4
\ \-2 o.~m~ll• 1 m,n. \ 107
1-
7
4
2 1-
8
1 500 1000 1500 2000 2500
SLANT RANGE FROM EXPLOSION (yards)
0 2 4 6 8 10
SRY-
Figure I-7. Fluence for 1-KT Burst in Air of 0. 9 Sea Level Density
-N e ~
VI c
6r-~~--~~----------------------~----------------------------------~------~
~ 4 .. :I • c
("') -0 --w u z w :::» ..J u. z ~ 2
GROUND BURST
AIR BURST
o~I--------------------------------tLo--------------------------------~loto~------YIELo (kilotons)
Figure l-8. Neutron Fluence Versus Yield for severe Damage to a Transportation Vehicle 1-' ~ >!>-
25L_~~~------------~------------------~
20 N-
E ~ -c::
0 .. .. :II 15 • c::
N -0 --w u z ~ 10 _, II.
~ a&: t-::)
I I "'-.. GROUND BURST w z 5
0 t ---TO 100
YIELD (kilot••)
Figure 1-9. Neutron Fluence Versus Yield for Moderate Damage to a Transportation Vehicle 1-l m 01
20~------------------~~------------------~~------------------~
16~----~--------------~--------~~----------+---------------------,
.... ~ c • r12r----------------------r---------------_:~~-+----------------------e ~
C)
w
g aL-----------------------~~------------------------t-~~------------------~ ~
! ~
AIR BURST
4~--------------------~--------~~--------~~--------~~--------j
0 -1 10 100 1000
YIELD (kilotons)
Figure I-10. Gamma Dose Versus Weapon Yield for severe Damage to a Transportation Vehicle ....... 0') 0')
-.. c • ~ c • 0 ..
(")
c --1&1 ., 0 0 ~ ~
~ \)
10~--------------------------------------------------------~----------~
Bl ;;;"' ~K I I
6
"
2~----------------~~~----~~---------------------------+~~-------.
0~1---------------------------llO-----------------=========~lOOb---------_j YIELD (kilotons)
Figure 1-11. Gamma Dose Versus Weapon Yield for Moderate Damage to a Transportation Vehicle '""' ~ -.:a
168
The pertinent point to note relative to Figures 1-8 and 1-9 is that the
threshold of permanent de current gain degradation is of the order of magnitude
of 10 10 fission neutrons/square centimeter for the most susceptible silicon
power transistors.
Another important point is that de current gain degradation from neutron
bombardment is accumulative, almost entirely independent of rate of application.
This observation was made by operation of a reactor similar to the WSMR-FBR
in the pulsed and steady state modes.
The gamma dose output from the computer runs is in roentgens which for
the purpose here is considered numerically equal to the rad (silicon) . This
assumption will be considered valid as it is intended to show that the gamma
dose plays an insignificant part in the analysis of permanent and semipermanent
damage resulting from a nuclear bomb detonation.
Figures I-10 and I-ll present the total gamma dose for the same condi
tions for which neutron dose was established. Looking at the worst case,
there are doses of approximately 2 x 10 14 and 10 13 roentgens, respectively, for
severe and moderate damage.
As the gamma dose is effective in causing surface damage, the
possibility of a sufficient dose to cause damage was investigated. It is reported
in the literature that surface damage is neglected if the ratio of neutron dose to
gamma dose is greater than 107•
169
PROGRAM-5
C REFERENCE THE EFFEC~S OF NUCLEAR WEAPONSt ED. SAMUEL GLASSTONEt C US DOD AND US AECt UF767 U5 1962 IUNCLASSIFIEDl C W=YIELD IN KILOTONS C P2=VEHICLE/DAMAGE CONSTANT FROM TABULATION BELOW C R=RANGE IN FEET FROM GROUND ZERO FOR PARTICULAR VALUE OF P2 c C TABULATION OF VEHICLE/DAMAGE CONSTANTS
p 166t 174 c C SEVERE DAMAGE lSI MEDIUM DAMAGE IMl C 1S=4.568 1M=5.832 TRUCK MOUNTED ENGR EQUIPMENT !UNPROTECTED! C 2S=3e100 2M=4~071 EARTH MOVING ENGR EQUIPMENT !UNPROTECTED! C 3S=4.568 3M=5.832 TRANSPORTATION VEHICLES C 4S=5.084 4M=5e832 BOX CARStFLAT CARStFULL TANK CARS,AND GONDOLAS C CARS (SIDE ON ORIENTATION! C 5S=3e100 5M=4.815 LOCOMOTIVES ISIDE ON ORIENTATION! c
21 READ 1,W,P2 80 PUNCH 15
PUNCH 6tW PUNCH 4 PUNCH 16 PUNCH 6t P2 PUNCH 4
C CALCULATION OF RANGE FOR YIELD AND VEHICLE/DAMAGE CONSTANT c p 175
71 P1=1LOGIWll*•4343 +3. R=10•**(,5367*1P1+1P2-P1l*l•2560ll+1.0485l
100 PUNCH 2 101 PUNCH 3tR 102 PUNCH 4
c C CALCULATION OF HEIGHT AT WHICH A GIVEN YIELD WILL HAVE THE C GREATEST DAMAGE RADIUS c p 137
c c c c c
c c
c c
RN=R/(300•*1W**•3333)) HlN=280e+235e*IRN-1el-70e*(RN-1el*IRN-2el+15e*IRN-1.l*IRN-2•l* 1CRN-3.)-(5e/24•l*IRN-1el*IR~-2.l*IRN-3e)*IRN-4el
H10=H1N*CW**e3333) 103 PUNCH 5 104 PUNCH 6tH10 105 PUNCH 4
CALCULATION OF GROUND RANGE FOR EQUIVALENT DAMAGE AS FOR OPTIMUM HEIGHT
p 137 RG=SLANT RANGE MINIMUM
22 POLY=.91Q-.045*1RN-1e)-.015*1RN-1el*IRN-2e)+.0067*(RN-lel*IRN-2al* 1CRN-3el-e0015*1RN-1el*IRN-2•l*(RN-3el*CRN-4eJ
RG•R*POLY 106 PUNCH 7 107 PUNCH 6tRG 108 PUNCH 4
By PATHAGOREAN THEORM = RSO CALCULATION OF SLANT RANGE RSO=SQRTCR**2e+Hl0**2•l
112 PUNCH 9 113 PUNCH 6tRS0 114 PUNCH 4
CALCULATION OF GAMMA DOSE SCALING FACTOR AS A FUNCTION OF YIELD
170
c p 378 C IN KILOTONS
c
c
24 C=.4343*1LOGIWll IFCC-lel 50t50t51
50 SFL=C GO TO 52
51 SFL=l.+l.l16*CC-lel+e0615*CC-lel*(C-2el-.0065*1C-l.l*IC-2.l*IC-3el 52 SFA=10•**SFL
115 PUNCH 10 116 PUNCH 6tSFA 117 PUNCH 4
C CALCULATION OF GAMMA DOSES FOR MAXIMUM AND MINIMUM RANGEStROENTGENS c p 377
c
c
c
c c c c c c
25 RSl=RG/3000. IFIRSl-1.1 60t60t61
60 RS3=5.*RS1 D5=-0.00042*1RS3-1.l*IRS3-2.l*IRS3-3.l*IRS3-4.) D4=-0.0095*CRS3-1el*CRS3-2.l*CRS3-3.l+D5 GD1=4.127-0e613*<RS3-l.l+0.066*CRS3-1.l*(RS3-2el+D4 GO TO 62
61 D2=1RS1-2•>*lRSl-3•l*IRSl-4.l Dl=I0.009/6•l*lRS1-l•l*IRS1-2el*IRSI-3·>-IOel34/24el*IRSl-l•l*D2 GD1=2.233-le716*1RS1-1.l+C.l07/2el*IRSl-l•l*IRS1-2el+D1
62 GRDO=ClO•**GD1l*SFA 121 PUNCH 12 119 PUNCH 6tGRDO 120 PUNCH 4
RSZ=RS0/3000. IF<RS2-1el 63t63t64
63 RS4=5.*RS2 E5=-0.00042*!RS4-1el*IRS4-2.l*CRS4-3el*IRS4-4el E4=-0.0095*CRS4-1.l*lRS4-2.l*IRS4-3.l+E5 GD2=4.127-0e613*CRS4-1el+0.066*CRS4-1.l*IRS4-2el+E4 GO TO 65
64
65 118 122 123
E2=1RS2-2•l*IRS2-3•l*IRS2-4el E1=10.009/6el*IRS2-1•l*IRS2-2•l*(RS2-3el-l0e1340/24el*IRS2-l.l*E2 GD2=2.233-1•716*1RS2-lel+lel07/2el*IRS2-l•>*<RS2-2el+El GRDM=C10e**GD2l*SFA PUNCH 11 PUNCH 6tGRDM PUNCH 4 IT IS NECESSARY TO APPLY A FACTOR TO THE GAMMA DOSE AS IT IS VARIABLE FROM WEAPON TO WEAPON OF THE SAME YIELD
CALCULATION OF NEUTRON DOSE FOR MINIMUM AND MAXIMUM RANGES IN NEUTRONS PER SQUARE CENTIMETER
p 392 RSS=RG/750. RS6=RSOI750. 05=-t.042/6el*IRS5-l.J*IRS5-2ei*(RS5-3el FE1=13e55-le02l*CRS5-lel+l.l35/2el*IRS5-1•l*IRS5-2el+D5 06=-(.042/6el*lRS6-l·l*IRS6-2el*IRS6-3el FE2=13.55-1e021*1RS6-lel+l.l35/2el*IRS6-1•l*IRS6-2el+D6 ANFl=llO.**FEll*W
124 PUNCH 13 125 PUNCH 6•ANF1 126 PUNCH 4
ANF2=ClO.**FE2l*W 127 PUNCH 14 128 PUNCH 6•ANF2 129 PUNCH 4
GO TO 21 72 STOP
1 FORMAT12E14.6l 2 FORMATC42H RANGE FROM GROUND ZERO FOR OPTI~UM HEIGHT! 3 FORMATCE14e61 4 FORMATC/1) 5 FORMATC25H HEIGHT ABOVE GROUND ZERO! 6 FORMATIE14e6l 7 FORMATC44H RANGE FROM GROUND ZERO FOR GROUND EXPLOSION! 8 FORMATI20H SLANT RANGE MINIMUM! 9 FORMATI20H SLANT RANGE OPTIMUM!
10 FORMATI26H GAMMA DOSE SCALING FACTOR! 11 FORMATI19H GAMMA DOSE MINIMUM! 12 FORMATI19H GAMMA DOSE MAXIMUM! 13 FORMATI21H NEUTRON DOSE MAXIMUM! 14 FORMATI21H NEUTRON DOSE MINIMUM) 15 FORMAT llBH YIELD IN KILOTONS! 16 FORMAT 124H VEHICLE/DAMAGE CONSTANT!
END
171
APPENDIX J. CONSTANTS AND LEAST-SQUARED-ERROR CURVE FITTING PROGRAM
172
This Appendix contains the constants used in the computer programs in
this writing.
XK = K =Boltzmann's Constant= 1. 380*10-23 joule/molecule*°K
Q = Coulombic Charge = 1. 602* 1o- 19 coulomb
TMP = T = Degrees Kelvin = 273 + o C
XK1 = K1 • 3. 3 * 10:-22 for planar-epitaxial (NPN)
= 6. 0 * lo-22 for planar (NPN)
= 6. 4 * lo-22 for mesa (NPN)
= 5. 5 * Io-22 for grown-diffused (NPN)
= 6. 8 * lo-22 for diffused (NPN)
XN= n = 1. 5 for planar epitaxial (NPN)
= 1. 47 for planar (NPN)
= 1. 45 for mesa (NPN)
= 1. 60 for grown diffused (NPN)
= 1. 38 for diffused (NPN)
The two computer programs that follow are polynomial curve fit
programs using the least squared error criterion. Program 6-1 is written in
FORTRAN and Program 6-2 is written in FORTRAN IV. Program 6-2 contains
several statements, beginning with statement number 51, which allow for input
data to be modified by any equations. It is necessary only to put the desired
relations beginning with statement number 51 and ending before statement
number 11.-
173
174
PROGRAM 6-1
C PROGRAM TO DETERMINE POLYNOMIAL FITTING AN ARBITRARY NUMBER OF C EQUALLY SPACED POINTS USING THE LEAST SQUARES CRITERION c C TENTH DEGREE MAX. FOR 200 POINTS MAXe c C SECTION TO DETERMINE POWERS OF DEPENDENT VARIABLE, Y!ll c
c
DIMENSION XC200)t Yl200lt All1t lllt B111h C!lllt PPOl READ 20t M PUNCH 20tM
20 FORMAT CI2l DO 11 I= 1' 2 01 READ lOt X!Ilt Ylll PUNCH 902tXIl)tY!Il
902 FORMATI2Fl5e7l PUNCH 1000
10 FORMAT12E14.8l IFIX(l) l 11• 12t 11
11 CONTINUE STOP
12 NUMB=I-1 MX2=M*2 DO 13 I =1 tMX2 PII)=O.O DO 13 J=1tNUMB XP=I
13 PII>=P!l)+X(Jl**XP 40 DO 41 I=1tMX2 41 PUNCH 903• PCil
PUNCH 1000 903 FORMAT!1F15e7l
C THE X MATRIX IS NOW COMPLETE c c C ATTAINING COEFFICIENTS FOR THE NORMAL EQUATIONS c
c c c
N=M+1 DO 30 I=hN DO 30 J=1tN K=I+J-2 IFIK) 29, 29t 28
28 AlltJ)=PIKl GO TO 30
29 A ( lt 1) =NUMB 30 CONTINUE 50 DO 5 1 I = 1 ' N
DO 51 J=1tN 51 PUNCH 903• AIItJl
PUNCH 1000 Blll=O.O DO 21 J=ltNUMB
21 Bl1l=BI1l+Y!Jl DO 22 1=2tN Bill =o.o DO 22 J=ltNUMB PWR=I-1
22 Blll=Bill +Y(Jl*XIJl**PWR
REDUCTION BY PIVOTAL CONDENSATION
c
NMl= N-1 DO 300 K=ltNMl KPl= K+l L=K . PUNCH 904tNMltKPltL
904 FORMAT<315) PUNCH 1000 DO 400 I=KPltN IFCABSFCACitKll-ABSF(ACLtKlll 400t 400t 401
401 L=I 400 CONTINUE
IFCL-Kl 500, 500t 405 405 DO 410 J=KtN
TEMP=ACK,J) 1000 FORMAT(//)
PUNCH 903tACKtJl ACKtJl=ACLtJl
410 ACL,JJ=TEMP TEMP=BCK) BCK!=BILl BILl=TEMP PUNCH 903tBCLl
C ELIMINATIONt BACK SUBSTITUTION, AND PRINTING OF RESULTS c
500 DO 300 I=KPlt N FACT=ACitKl/ACKtKl AC I.K!=O.O DO 301 J=KPltN
301 ACI,JJ=ACitJl-FACT*ACKtJl 300 BCIJ=BCIJ-FACT*BCKl
CCNl=BCN)/ACNtNl I=NM1
710 IPl=I+l SUM=O~O DO 700 J=IPltN
700 SUM=SUM+ACitJ)*C(J) C C I l = C B ( I l -SUM) I A I I ' I l I=I-1 IF<Il aoo. eoo. 110
800 DO 900 I=1tN 900 PUNCH 901• ltCCil 901 FORMATCI5tFl5.7l
STOP END
175
17~
PROGRAM 6-2
:v FORilllhtt4El4.8) 21 STnP
·-·-~--···-~ -··· .. - ..
2?. 12 NIIMF\=J-1 23 ~X:?=f"l*2 -----------------· ·24 - · -------- nn--n-r=r,-ror-rr--2~ P(JJ=O.O ?h DO 13 J=l,NUMR 27 )(P= I 2A 13 PftJ-P(tJ+XtJ):t¥xP 2 9 4 0 DO 4 1 I = 1 t M X 2
j~ 9~! ~~~~~ffl6~~~l~(llF15.71 ·3-z--------- WR If E I 3 t 1000,--
~ THf X MATRIX IS NOW COMPLETE CFFICIENIS FOR fRE NORMAL EQOAYIONS (. A 1 J A IN l Nu l. U t
33 N=M+ 1 34 00 30 I= 1 t N 35 DO 30 J=1,N
··n--------------- K= I +J-2 37 JF(K) 2J,2J,28 38 28 ACJfJ)=P(K) ~9 GO 0 30
0 209 l(( If! J=NUM8 41 3 CON INUE
c 42 --------lB~C~l~)~=~0~·~0~7~wonr--------------------------------------~- DO 21 J=l,NOMB
R r D\JC T InN ~-y P fv{)-rAL--cfi-No~ N s frr·c-~-(---------- -S~ N~l=N-1
-----5..l-------~Di!::n'-~3~C:-.i.i~·;-JKs..;:;w...l.,_,~::.~.I\I~M..L.J---------------------~? KDl=K+l "~ t=¥
56 '51
--s"R-·
59 60 61 6?.
r
401 4CO
_QD 4 C C __ J_::;_KP L, N. ______________________________ .. ____ _ JF( 1\RS( A( J,K) )-.1\A<;(A( L,K))) 400,400,40i L= J CONTINUf
.\..----;l;-;F:-(;-:l:---K:-:-7} --;::5-:::0:-:::0:-· ,--~~. C::-:0::-. ,-4=-0~5=----------------- --- ········ -·--· c
't05
410 (
no 410 J-=K N -TEJ1P._=_A( K •. JL_ - -. ---t'\(K,J)=A(L,J). A(L,Jl=HMP
··----b..3..-----------~-==~B'-!(~K~)~._ ____________________ _ 64 R(Y)=8Cl) A~ R(ll=TEMP
6h 67 68 69 70 71
7?. 73 74 75
11:> 77
7R. 70 80 81 82
c c r
c
c
c
"iOO
~01 ~00
?10
EL I M I NAI lDN_._ B.A C K __ .SUB.S.TliUILON.t.AND _J>IHJ'll'tl.HG_ . .0~ RE SU L T S
DO 300 J=KPl,N FACT=ft( J 9 K)/ACK,Kl Afi,Kl:O.O on 3C1 J=KPl,N A_ ( ItJl=A( t,J)-FACT*A(K,Jl Rl I J::::EH I )-FACT*EHKl t tN) ;;RTNTTAc·N~-NT ______________ --------- ------- -- ··-- --- -I=NMl IP1=T+l
83 STOP __ __84 _____ ___20_1 EO RMAI.L!JlL. = ' I 2, ' C « 'I l , ' I = 1 E 1. 2 • 4 I
85 END
177
178
BIBLIOGRAPHY
1
CHANG, Y. F. (1967) "The Conduction-Diffusion Theory of Semiconductor
Junctions," Journal of Applied Physics, vol. 38, No. 2, pp. 534-
544, February 1967. 2 CHO TT, J. R. and C. A. GOBEN ( 1967) Annealing Characteristics of Neutron
Irradiated Silicon Transistors, Space Sciences Research Center,
University of Missouri at Rolla (C00-1624-4). 3 CHOW, M. C. and J. L. AZAREWICZ and C. A. GOBEN (1968) Recombination
Statistics for Neutron Bombarded Silicon Transistors, Space
Sciences Research Center, University of Missouri at Rolla
( C00-1624-12) . 4 FRANK, MAX and CARL D. TAULBEE (1968) Handbook for Predicting
Semiconductor Device Performance in Neutron Radiation, the
Bendix Corporation Technical Report No. AFWL-TR-67-54 (Rev). 5
GOBEN, C. A. {1965) A Study of the Neutron-Induced Base Current Component
in Silicon Transistors, Sandia Corporation Reprint SC-R-65-912. 6
GOBEN, C. A. (1964) Neutron Bombardment Reduction of Transistor Current
Gain, Sandia Corporation Monograph SC-R-64, 1373, Physics
TID-4500 (37th Edition).
179 7
GOBEN, C. A. and F. M. SMITS and J. L. WIRTH (1968) Neutron Radiation
8 Damage in Silicon Transistors IEEE GINS NS-15:2, 14-29.
GOBEN, C. A. and F. M. SMITS (1964) Anomalous Base Current Component
in Neutron Irradiated Transistors, Sandia Corporation Reprint
SC-R-64-195. 9
GWYN, C. W. and D. L. SCHARFETTER and J. L. WIRTH (1967) The
Analysis of Radiation Effects in Semiconductor Junction Devices,
Sandia Laboratories, SC-R-67-1158. 10
LARIN, FRANK ( 1966) Prediction of Radiation Effects in Semiconductor
Devices (Notes for U. S. Army Missile Command Seminar) ,
Bendix Corporation. 11 MALMBERG, A. F. and F. L. CORNWELL and F. N. HOFER (1964) NET-1
Network Analysis Program 7090/94 Version, LA 3119, Los Alamos
Scientific Laboratory. 12 MANLIEF, S. K. A Method of Measuring the Minority Carrier Base Transit
Time in a Junction Transistor Exposed to a Neutron Environment,
Sandia Laboratory, Albuquerque, SC-TM-314-63(14).
13 PURDUE, C. H. ( 1966) Computer Programs for Obtaining the Modified Ebers
Moll Diode and Transistor Model Parameters, Sandia Corporation
SC-DR-66-2613. 14 SU, L. S. and G. E. GASSNER and C. A. GOBEN (1968) Radiation and
Annealing Characteristics of Neutron Bombarded Silicon Transistors,
Space Sciences Research Center, University of Missouri at Rolla
( C00-1624-13) .
180
15. SMITH, K. R. ( 1968) Techniques for Determination of Transistor
Characteristics in a Neutron Environment, Part 1, Army Missile
Command Report No. RL-TR-67-6.
16. SOKAL, SIERAKOWSKI, SIROTA (JUNE 21, 1967) Calculations of
Transistor Parameters for NET-1, Electronic Design 13, pp. 60-
65.
181
VITA
The author, Kenneth Robert Smith, was born in Alton, Illinois, on
December 15, 1941. Primary and secondary education was received in the
Roxana school system until JWle 1960. He received the degree of Bachelor of
Science in Electrical Engineering from the University of Missouri-Rolla in
January 1965, and this thesis is submitted for partial fulfillment of the
requirements for the degree of Master of Science in Electrical Engineering at
the University of Missouri-Rolla. Expected date of graduation is 2 August 1969.
The author is employed by the U. S. Army Missile Command as a
civilian engineer, and the work~presented here was done for, and under the
auspices of, the Ground Support Equipment Laboratory of the Research and
Engineering Directorate (Provisional) located at Redstone Arsenal, Huntsville,
Alabama.
The Ground Support Equipment Laboratory, under the direction of
William c. watson, supported this effort to completion both in time and
finances.
