CONDUCTION MECHANISMS IN LOW BREAKDOWN
VOLTAGE SILICON P-N JUNCTIONS
By
Wieslaw A. Lukaszek
A DISSERTATION PRESENTED TO THE GRADUATE COUNCIL OF
THE UNIVERSITY OF FLORIDA
IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE
DEGREE OF DOCTOR OF PHILOSOPHY
UNIVERSITY OF FLORIDA
1974
ACKNOWLEDGMENTS
I am indebted to Dr. Aldert van der Ziel, my most
inspiring teacher, for his generous assistance and
encouragement as my advisor. I also thank Dr. Eugene R.
Chenette for his encouragement and Dr. Karel M. van Vliet
for his friendly advice and many helpful discussions
during the final stages of this work. The friendly help
of Dr. Charles V. Shaffer and Dr. Samuel B. Trickey is
also most warmly appreciated.
The aid of Mr. Don Estreich of Teledyne who supplied
the diode samples and data, and of Mr. Sam Weaver of Texas
Instruments who supplied the low noise JFET's employed
in this study, is gratefully appreciated.
Finally, my deepest thanks go to my parents,
Antoni and Stefania, and my brother, John, whose encourage-
ment and countless sacrifices I shall never be able to repay.
TABLE OF CONTENTS
ACKNOWLEDGMENTS.
ABSTRACT .
CHAPTER I INTRODUCTION .......
CHAPTER II EXPERIMENTAL METHODS ....
CHAPTER III DISCUSSION OF EXPERIMENTAL RESULTS. .
CHAPTER IV DETAILED ANALYSIS OF NOISE DATA .
CHAPTER V CALCULATION OF DC MULTIPLICATION. ..
CHAPTER VI CONCLUSIONS .. . . .......
CHAPTER VII RECOMMENDATIONS FOR FURTHER STUDY . .
APPENDIX A MULTIPLICATION NOISE RESULTING
FROM NO MORE THAN ONE IONIZATION
PER CARRIER TRANSIT ACROSS DIODE
SPACE CHARGE REGION .. . .....
APPENDIX B MULTIPLICATION NOISE RESULTING
FROM NO MORE THAN TWO IONIZATIONS
PER CARRIER TRANSIT ACROSS DIODE
SPACE CHARGE REGION . . . . .
APPENDIX C DC MULTIPLICATION SIMULATION PROGRAM. .
BIBLIOGRAPHY
. . . . . . .154
BIOGRAPHICAL SKETCH ........
. . . . . . . iv
73
87
Abstract of Dissertation Presented to the
Graduate Council of the University of Florida in Partial
Fulfillment of the Requirements for the
Degree of Doctor of Philosophy
CONDUCTION MECHANISMS IN LOW BREAKDOWN
VOLTAGE SILICON P-N JUNCTIONS
By
Wieslaw A. Lukaszek
August, 1974
Chairman: Aldert van der Ziel
Co-Chairman: Eugene R. Chenette
Major Department: Electrical Engineering
White noise spectra of diodes 1N746 to 1N752 have
been used to investigate the reverse conduction mechanisms
in silicon junctions breaking down by a combination of
tunneling and impact ionization multiplication. The reverse
voltages at the onset of multiplication noise indicate
that, regardless of junction field strengths, carrier
multiplication sets in only when the carrier energies
exceed the threshold energies for ionization. Further
examination of noise data yields their values, indicating
that the multiplication process is dominated by the
effects of the threshold energies for ionization. When
they are taken into account, it becomes clear why Tager's
and McIntyre's theories of noise in avalanche diodes,
and the conventional DC multiplication calculations, are
not applicable to low breakdown voltage diodes. Con-
sequently, a new method of performing DC multiplication
calculations as well as a new theory of noise in low break-
down voltage diodes is developed.
CHAPTER I
INTRODUCTION
As the reverse bias across a p-n junction is increased,
the junction eventually begins to pass appreciable current.
Typical voltage-current characteristics for commercially
available diodes are shown in Figure 1. It is evident that
the reverse current characteristics are considerably dif-
ferent for p-n junctions which "break down" at different
voltages.
The type of breakdown typified by Figure la is
characterized by (i) temperature sensitive pre-breakdown
(saturation) current which approximately doubles for each
10C rise in temperature, (ii) a well-defined breakdown
voltage, VB, and (iii) a unique, positive, breakdown
voltage temperature coefficient, B, defined by
VB = VBO[1+B(T-TO)
where VBO is the breakdown voltage at room temperature, T .
This type of breakdown, characteristic of relatively wide
p-n junctions with VB>14 volts, is commonly known as
"avalanche" breakdown since it can be understood in terms
of an impact ionization mechanism resembling an avalanche
(Tyagi, 1968).
Reverse conduction of the type shown in Figure Ic
is characterized by (i) temperature insensitive reverse
current -- temperature change of 2000C changes the current
by a factor of two or three, (ii) lack of well-defined
breakdown voltage, and (iii) a negative temperature
coefficient whose magnitude depends on temperature and
reverse voltage. This type of breakdown, characteristic
of narrow p-n junctions capable of passing considerable
current densities at reverse voltages smaller than about
2.5 volts, can be understood in terms of internal field
emission of valence band electrons, also known as Zener
tunneling (Chynoweth et al., 1960; Tyagi, 1968).
P-n junctions wniich break down between about 2.5 to
14 volts do so as a result of the combined effect of internal
field emission and impact ionization multiplication (Tyagi,
1968). Since most of the measurements which formed the
basis for the understanding of p-n junction breakdown were
DC measurements, this range of breakdown voltages has
never been carefully investigated. The reason for it
was the severe difficulty encountered in distinguishing
the contributions of each of the two mechanisms.
The aim of this study was to investigate the reverse
conduction mechanisms in diodes breaking down in this
range of voltages. The difficulty which previous workers
encountered in distinguishing the tunneling current from
the impact ionization multiplied current was overcome
by using noise as a diagnostic tool, the idea being that
noise would more closely reflect the basic microscopic
conduction mechanisms. This technique was first employed
by Lauritzen (1966) in determining the onset of carrier
multiplication in low breakdown voltage silicon diodes.
Interest in avalanche diodes as photodetectors, micro-
wave signal sources, and amplifiers has already led to
extensive experimental (Haitz, 1965, 1966; Haitz and
Voltmer, 1968; Baertsch, 1966, 1967; Kumo et al., 1968)
and theoretical (Tager, 1965; McIntyre, 1966; Hines, 1966;
Gummel and Blue, 1967) studies of noise in these diodes
at low and high frequencies.
Because noise measurements provide the experimental
foundation of this study, Chapter II is primarily devoted
to the experimental considerations and techniques involved
in collecting the noise data. DC and capacitance measure-
ments are discussed there also.
A general discussion of the data is undertaken in
Chapter III. It is found that Tager's (1965) and McIntyre's
(1966) theories of noise in avalanche diodes are incapable
of explaining the data. The reason for this becomes clear
when the mechanism of the onset of carrier multiplication
is inferred from the noise data. It is then also apparent
that the conventional method of calculating DC multipli-
cation is not applicable to low breakdown voltage diodes.
When a new definition of the DC multiplication
factor, and a new functional form of the expression for
noise are introduced in Chapter IV, the noise data
reveal in detail the influence of the threshold energies
for ionization on the multiplication process. On basis
of the inferences made in this and the preceding chapter,
new theories of noise in low breakdown voltage diodes
are developed and compared with experimental results.
The details of the multiplication process inferred
in Chapters III and IV are incorporated in Chapter V,
which discusses a simulation program implementing a new
method of performing DC multiplication calculations in
low breakdown voltage junctions. The results of the con-
ventional and the new calculations are then compared with
data.
Chapter VI summarizes the highlights.
Chapter VII suggests topics for further study.
CHAPTER II
EXPERIMENTAL METHODS
The p-n junctions investigated in this study are com-
mercially available diodes, types 1N746 to 1N752, manufactured
by alloying small aluminum buttons onto n-type silicon. The
raised appearance of the aluminum buttons, shown in a vertical
cross-section of the diode in Figure 2, is the result of acid
etch used to delineate the junction geometry and eliminate
surface effects around the button periphery.
The samples chosen for study were those which exhibited
very little flicker noise. In final samples, flicker noise
was not noticeable at frequencies greater than 300 Hz. This
selection criterion presumably eliminated from consideration
those diodes in which surface or edge effects might have had
significant influence on the reverse conduction mechanisms
of the junctions. All final samples, whose DC V-I charac-
teristics are shown in Figure 3, came from Continental Device
Corporation, now Teledyne Semiconductor.
In this study the reverse conduction mechanisms were
inferred from two sets of diode terminal measurements: the
DC V-I characteristics and noise. The DC characteristics
provided information on the average behavior of carriers in
the junctions, whereas the noise measurements provided
statistical information on the transient, or microscopic,
details of the conduction process. The noise measurements
thus provided the groundwork details for the conduction
model, which will be discussed in the following chapter.
The noise measurements, which form the experimental
foundation of this study, were performed with the aid of the
noise measurement system shown in Figure 4. The GR 1381
Random Noise Generator coupled to the HP 350-D 600 Q attenuator
supplied a 2 Hz to 50 KHz white noise calibration signal.
The 600 0 resistor following the attenuator provided the
proper impedance match for the attenuator, while the 1 pf
capacitor in series with the 200 K1 resistor provided DC
and impedance isolation from the rest of the circuit. The
impedance isolation was necessary to maintain a constant
impedance level at the attenuator terminals regardless of
the impedance levels existing in the diode bias network,
and to convert the noise calibration network into a high
impedance, current-like source to prevent excessive loading
of the diode. The diode bias network consisted of a large
number of wire wound resistors, Rb. When used to adjust the
diode reverse current, they were chosen so that Rb was more
than ten times greater than the diode dynamic resistance, Rd.
The low noise preamplifier used for its first stage a
preselected sample of a Texas Instruments low noise JFET, the
SFB 8558. The schematic diagram of the preamplifier is given
in Figure 5 and the preamplifier's equivalent noise resistance
referred to the input, Rn, as a function of frequency, is
given in Figure 6. The preamplifier was followed by a
selectable gain amplifier which amplified the noise signal
to a level suitable for processing by the GR 1925-1926
Real Time Spectrum Analyzer. The oscilloscope was used for
visual observation of the noise waveforms.
The GR 1925-1926 Real Time Spectrum Analyzer contains
a bank of 45 third-octave filters, ranging in center frequencies
from 3.15 Hz to 80 KHz. The output of each filter is sampled
for up to 32 seconds and the dB of the RMS voltage of each
filter is computed and displayed on the GR 1926, or is printed
out by the MDS 800 tape printer.
The measurements were performed as follows. First, the
noise calibration signal, provided by the GR 1381 and the
600 Q attenuator, was removed by disconnecting the attenuator
from the circuit and replacing it with a 600 0 resistor
placed in parallel with the 600 2 attenuator termination
resistor. The diode bias resistor, Rb, was then chosen to
obtain the desired value of diode reverse current, and five,
32 second, diode noise data sets were recorded. Next, the
600 0 resistor in parallel with the attenuator termination
resistor was removed, and the attenuator returned to the
circuit. The attenuation level was then adjusted to obtain
a calibration plus diode noise output about 20 dB higher
than the diode noise output alone, and a second set of five,
32 second, readings was recorded. Given the system noise
equivalent input circuit of Figure 7, it is a trivial matter
to determine that the diode noise current spectral density,
Sid, is then given by
S. S
Sig Sva
Sid AdB=- Sir Sib (2.1)
[10 -1] zt12
and
-dBatt
1 2-
ig 200KQ (1
Sv = (20- ) (10 )Svg (2.2)
Sva = 4KTRn (2.3)
1 1
S = 4kT(-l )L (2.4)
Sir 200K 200K) (2.4)
4kT
Sib Rb (2.5)
I tl2 = R2 1 ) (2.6)
Sb 2+W 2 C2R2
1
Rt= 1 1 (2.7)
20Ki)KQ 200Kn R d R b
where S is the spectral density of the noise generator out-
vg
put voltage, Rn is the equivalent noise resistance of the low
noise preamplifier, Rd is the diode dynamic resistance, Rb
is the bias resistance, Ct is the total input capacitance,
dBatt is the attenuator setting, and AdB is the difference
in dB between the calibration plus diode noise level, and
the diode noise level alone. This calculation was carried
out for each filter, and the final results were averaged over
all filters processing a white noise spectrum. Typically,
this resulted in averaging over the 25 filters of center
frequencies greater than 250 Hz.
The system accuracy was verified by measuring the thermal
noise of resistors. When resistors of 200 : to 2 Mn were
inserted in place of a diode, and the above measurement pro-
cedure was repeated, the resistance values predicted from
noise measurements agreed to better than 2% with values
obtained from precision bridge measurements.
Since the low noise preamplifier employed a JFET for
the first stage, it sensed the voltage, rather than the current,
at its input terminals. Consequently, the diode noise current
was converted to a noise voltage by the diode dynamic impedance
before the noise signal was amplified. The diode dynamic
impedance, therefore, had to be determined, and was measured
using the diode bias network, capacitively coupled to a Wayne
Kerr B601 Radio Frequency Bridge utilizing a Wayne Kerr
SR 268 Source & Detector, as shown in Figure 8. The 100 KHz
signal applied by the bridge to the diode was adjusted until
further signal level reduction produced no difference in the
measured RC values. This typically occurred for applied
signal levels of less than 40 mV RMS. The parallel equivalent
RC values of the diode and its bias network were thus deter-
mined to an accuracy of 1%.
The capacitance values determined above were also
employed in the C-V plots used in estimating the junction
doping profiles and doping densities. The diode areas were
obtained from junction photographs taken after the aluininum
button was etched away in phosphoric acid. The error in
determining the junction area in this manner was estimated
to be about 15%.
A typical junction surface is shown in Figure 9. Although
the surface is full of ridges, it is assumed that the junc-
tions are very nearly planar because they are extremely thin.
On the scale of the several hundred angstrom thin space
charge regions (SCR), the ridge contours should appear as
gentle undulations in the junction topology.
The DC V-I measurements were made using the test setup
shown in Figure 10. It consisted of the diode bias network,
a Keithley Model 615 Digital Electrometer operating as an
ammeter, and a Fluke Model 801B Differential Voltmeter. The
accuracy of the DC measurements was 0.5% for currents greater
-7 -7 -9
than 107 amperes, and 2% for currents between 107 and 10
amperes. Voltage measurements were accurate to 10.05%.
All measurements were taken at room temperature, 2220C.
CHAPTER III
DISCUSSION OF EXPERIMENTAL RESULTS
In order to make the forthcoming discussions of noise
and DC V-I measurements meaningful, it is necessary to
know the junction properties and parameters. Some of them
can be inferred from the ly vs. Vr plots for the four
C
most important diodes, the 1N749 to 1N752, shown in Figures
11 to 14.
Since the 2 vs. Vr plots do not yield straight lines
over the entire range of reverse voltages, it appears that
1
the junctions are not abrupt. Unfortunately, vs. V
C r
plots do not yield straight lines, either. Consequently,
it seems that the junction profiles are composite of
linearly graded and abrupt. If it is assumed, however,
that to a first order approximation the diodes can be
treated as step junctions (provided that the doping densities
are evaluated from -- vs. Vr slopes obtained for large
C r
reverse biases, where the doping densities are most likely
to be representative of the bulk doping densities) the
results of Table I are obtained.
The calculated doping densities on the n-side of
the junctions are seen to lie within the range of doping
densities predicted from the starting resistivities of
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the n-type wafers. This seems to indicate that for reverse
voltages greater than about 1.5 to 2 volLs, the doping
densities at the junction edges attain the bulk doping
densities, and the junctions should begin to assume step
junction characteristics. The changes in the 7 vs. Vr
C
slopes for smaller reverse biases are caused by the extreme
thinness of the diode SCR's: as the reverse biases
decrease, the SCR's narrow and enter the diode metallurgical
transition regions where the doping densities are no longer
constant.
However, demonstrating that the doping densities at
junction edges reach bulk doping densities does not con-
clusively prove that for Vr greater than 1.5 to 2 volts
the junctions may be approximated by step junctions.
Another piece of evidence which strengthens this contention,
though, will be obtained when the DC V-I characteristics
are examined. This discussion will be most meaningful
after the noise measurements are examined.
The typical noise data obtained for diodes 1N746
to 1N752 are shown in Figures 15 to 21. The diode noise
current spectral density, Sid, has been normalized with
respect to the noise current spectral density of a saturated
thermionic diode conducting a DC current equal to the p-n
junction reverse current. The noise ratio, NR, is, therefore,
given by
NR = i- (3.1)
r
where Ir is the p-n junction reverse current.
Since the noise current of a saturated thermionic
diode arises from the reception at the anode of all
individually, and randomly emitted, thermally excited
cathode electrons, a noise ratio of unity indicates that
the p-n diode reverse current is also due to collection
of randomly and singly generated carriers, or carrier
pairs. This is precisely what happens when the p-n
junction reverse current arises from internal field
emission in which thermally excited valence band electrons
tunnel to the conduction band. The noise measurements
thus furnish a foolproof method of determining the range
of reverse voltages for which the diode reverse conduction
arises from a single step tunneling process. (A few
samples exhibiting multi-step tunneling were also observed.
They are characterized by a noise ratio less than unity
(van der Ziel, personal communication), as shown in Figures
21a and 21b. These samples, however, exhibited substantially
1
greater noise than the remaining samples and, therefore,
were excluded from further study.)
An independent method of verifying that over the
range of reverse voltages for which the noise ratio is
unity the reverse current arises from single step tunneling
is obtained from the laws of internal field emission.
Chynoweth et al. (1960) and Tyagi (1968) found that for
a phonon-assisted tunneling process the diode reverse
current is given by
y 3/2
yE- (--) 4 2m
I = AVrEe Y (3.2)
r r 3qh
3ash
where Ir is the diode reverse current, Vr is the diode
reverse voltage, A is a constant for a given temperature,
E is the electric field, E is the band gap energy, m
is the effective mass of tunneling electrons, q is the
electronic charge, T=-, where h is the Planck constant,
2-'
yE 3/2
and n and n are constants. The factor e represents
the phonon-assisted tunneling probability, En accounts
for image fields or Coulombic forces corrections included
in the theory, and V takes into account, in an empirical
r
way, the effect of electron tunneling from the conduction
band back to the valence band so that for zero reverse
bias the net current, Ir, is also zero. For abrupt junctions
Tyagi claims n=l, n=1.5.
YE 3/2
E no
Compared to e AV En is a slow varying
function of the reverse voltage so that the reverse current
may be written as
s
Ir = Ie E (3.3)
where Io and s are considered constant provided Vr 0,
and E is an "effective" electric field such that
_s w s
e T= / e dx (3.4)
o
where the integration extends over the entire width of
the diode space charge region, w. For an abrupt junction
the main contribution to the integral comes from electric
field values close to the maximum electric field value, E .
m
Thus, little error is made in assuming that the "effective"
electric field is given by Em. Consequently, for an abrupt
junction
s
Ir = I0e r (3.5)
where s is a constant and 1 is the built-in potential
of the junction. Thus, if a junction is indeed rtur-.L
and the reverse current is due to tunneling, log i
should give a straight line.
Inspection of Figures 22 to 28, which show i:-i i
1
vs. for diodes 1N746 to 1N752, reveals trne .' -.
Vr
the range of reverse voltages for which the noi-:. rtci: is uir.
log I vs. is, indeed, a straight li'-. 71- L..
vrr
V-I data and the noise data are, therefore, cor.I r.r.i:
and the contention that the Iuncr,i.: .: I a,. t ,: f r r. *.cird
:Cc r.* i',:ti,:.., be c,:.i. .J-r.i rupr. uL'r.ber s t r.r r--r, r, .
If the randomly emitted tunneling electrons undergo
multiplication due to impact ionization, the randomness of
the multiplication process generates additional noise,
and the noise ratio becomes greater than unity. Comparison
of Figures 15 to 21 with Figures 22 to 28 reveals that
over the range of voltages for which the noise ratio
increases from unity, log Ir increasingly departs from
the straight line it followed at lower reverse voltages.
This suggests that the failure of the reverse current
to obey the tunneling relation is due to impact ionization
multiplication of the tunneling current.
If this contention is indeed correct, then it
should be possible to verify it quantitatively with the
help of noise theories of Tager (1965) or McIntyre (1966)
relating the spectral density of noise current generated
in impact ionization multiplication to the terminal DC
current flowing through the diode. According to Tager,
the low frequency spectral density of noise current
produced in an impact ionization multiplication process
in which the electrons and holes have the same ionization
coefficients, is given by
Si = 2qIt(M )3 (3.6)
where It is the current initiating the multiplication
process, which in this case is the tunneling current,
and M is the DC current multiplication factor, defined by
I
M = (3.7)
It
where Ir is the terminal reverse current flowing through
the diode. Thus, if it is assumed that the expression
for tunneling current remains valid in the presence of
impact ionization multiplication (which, in light of the
current densities encountered, is a very reasonable
assumption) then M is simply the ratio of the measured
value of reverse current to the extrapolated value of
the tunneling current.
Thus, if
Sid 2qltM (NR)
log2- = log[ 2---t ] = log[M (NR)] (3.8)
is plotted vs. log M as done in Figures 29 to 32 for
diodes 1N749 to 1N752, a straight line with a slope of
three should be obtained. It is clear that, with the
exception of the 1N752 diode for which the agreement
between experimental results and theory is reasonable,
considerable discrepancies exist. Moreover, the discrep-
Sid
ancies are serious because the measured values of 2Si
2qIt
are larger than the theoretical one, which represents
the upper limit attainable in impact ionization multipli-
cation for which electrons and holes have the same ioniza-
tion coefficient. In addition, at low values of multipli-
Si
cation the rate of increase of 2 also exceeds the
tqI
theoretical, and, therefore,the upper limit value.
At this point it might be argued that these discrep-
ancies arise from the unequal ionization rates of electrons
and holes in silicon. However, if the latest ionization
rate data of van Overstraeten and de Man (1970), shown in
Figure 33,are extrapolated to the field strengths encountered
in diodes used in this study, the extrapolated electron
ionization rates are not substantially greater than the
extrapolated hole ionization rates.
For the sake of completeness, however, the noise
data was also analyzed in terms of McIntyre's (1966) theory
of noise current spectral density for impact ionization
multiplication in diodes for which the electron and hole
ionization rates are not equal. According to McIntyre,
if the carriers initiating the multiplication are electrons,
the low frequency noise current spectral density is given by
Si = 2qI (M )3[l-(l-k) (M-- )2] (3.9)
M
where It is the electron current initiating the multipli-
cation, which in this case is the tunneling current, k=
is the ratio of the hole ionization rate, 8, to the electron
ionization rate, a, and M is the DC current multiplication
I
factor, M = Since 8 and a depend strongly on the
It
electric field, this equation is strictly cor-ect only if
an appropriate "average" value of k is used (MlcIntyre, 1973).
However, for the purposes of this discussion this is not
an important consideration.
S.
According to McIntyre's (1966) theory, 1 is given by
tqI
S.
S-= (M 3[1l- (-k) (M-1)2] (3.10)
2qt M
which is plotted for several values of k in Figure 34.
Comparison of Figure 34 with the experimental values of
..r tija'r.-: r :' l ir, i : rrfar a *:I'-- r a ift- r-rc
Lt ein chr r. .:L r.J L- F r. *:.l. tl c .r : Lr.r d -ri, i
cL t.: r..: i.nr i z a c i :n r ate r r.,L r. an t i a l 1 I l arir 4 r.z.ar,
tr-.e e ei c r.r :n :nir :r.i: rr a- In iiirr. .: L r e re -a :ured
ar,.j c r~r ,p.:1- 'J 'luai .: ri.:,n.rz s :n r, r e .t f tilujre 3
ri r-. :. i :.: r. r r.: A r rr. i : .cr
i L:ur.t-I d r :i i r r:, r r. .r- .r a ,ri rn c- .r
',- u-r: "3 ., ua:- *r, th-- n 'Cr I-*r,5 I ai.ula i. .:r,- ,f
C.": irul r. ip l r. che riaua r. : r:ttrin,.J ir :ii. te h tr-
rultr F i lica i :.i r r.i -u,. J : f.r r tr. ,,:,r,-z~r : alue :-
t '.' r e : c Tr ,. i g l r. t l r ,n .: i ra .-: = :,.u ldJ
.:I e a ., ich i. ". :C, r. _rr ,ad i tr. : I th ti r..
.p .:t r, n t l i r -._r F jr : I r-,- F t i
.li, f f c u.: ir : r _-,e _, ri.-_ i r f t :.: t_ r :r ) r :r' i : ,i .:t r r :i.
Sid
magnitude and functional dependence of 2 on M or
2qIt
require an unreasonable ratio of ionization rates to
possibly achieve an acceptable agreement of theory and
experiment; they fail to correctly predict the magnitude
of the experimentally observed DC multiplication; and
they completely fail to predict the existence of a reverse
voltage threshold for the onset of impact ionization
multiplication. Any theory of reverse breakdown transition
from tunneling to impact ionization multiplication must be
able to eliminate all of these discrepancies, while
remaining consistent with all other observations of break-
down phenomena in silicon p-n junctions.
One of these observations, made by many workers
investigating carrier multiplication in semiconductors,
is the existence of an energy threshold for pair production
(Shockley, 1961; Moll and van Overstraeten, 1963): the
agent attempting to generate a carrier pair must be capable
of supplying at least a certain minimum amount of energy
if it is to succeed in producing a hole-electron pair.
The latest values of threshold energies for impact ioniza-
tion in silicon, as determined in ionization rate measure-
ments and first reported by Moll and van Overstraeten (1963),
are 1.80.1 eV for electrons and 2.40.1 eV for holes.
Since the noise data of Figures 15 to 21 indicate
that carrier multiplication, as judged from the departure
of the noise ratio from unity, always sets in at a well-
defined value of reverse voltage that is almost independent
of the diode breakdown voltage (if the breakdown voltage
is defined as that value of reverse bias which gives rise
to some constant value of reverse current density), the
author contends that the onset of multiplication is governed
by the magnitude of the threshold energy for ionization.
To substantiate this contention quantitatively, the
following multiplication threshold model, described in
terms of Figure 35, which shows the electric field and
energy band diagrams for a reverse biased abrupt p-n
junction, is proposed. Since the tunneling probability
is strongly dependent on the electric field, most of the
tunneling current originates at x=0. At x=x1 the tunneling
electrons become free carriers, absorbing 1.12Er eV, where
Er corresponds to the emission or absorption of one phonon
whose energy may be 17.9 meV (TA), 43.7 meV (LA), 53.2 meV
(LO), or 58.5 meV (TO), (Logan, 1969). Continuing towards
x2 the electrons acquire energy from the electric field,
and at x2 those which have made the fewest number of colli-
sions with the lattice are sufficiently energetic to
participate in ionizing collisions. The onset of impact
ionization multiplication, therefore, occurs when x2=x .
Denoting the reverse voltage at which this occurs by Vrth'
the abrupt junction approximation yields
xn (1.12Er)+(1.80.1)
x +x V +- (3.11)
n p rth
where
x NA-ND
n -3 .A
x +x N (3.12)
n p A
Consequently,
NA
Vrth = [(1.12Er)+(1.8+0.1)] (NA- )- (3.13)
Table II compares the reverse threshold voltages, Vrth'
computed using the doping densities of Table I and
E =0.06 eV, with the reverse threshold voltages obtained
from Figures 18 to 21. In light of the approximations
made and the experimental uncertainties involved in
determining the doping densities, they are in good agreement.
The holes produced in ionizing collisions between
x2 and xn travel in opposite direction, but they, too,
ionize only after they have acquired at least the necessary
threshold energy for ionization. The electrons produced
in hole ionizing collisions then travel towards xn, and
after acquiring at least 1.80.1 eV participate in more
ionizing collisions. This process repeats itself, but
since the number of ionizing collisions decreases with
each successive set of ionizations, the process converges
to a finite value of DC multiplication.
24
TABLE II
REVERSE VOLTAGES AT THE ONSET OF
IMPACT IONIZATION MULTIPLICATION:
COMPARISON OF DATA WITH THEORY
DIODE TYPE EXPERIMENTAL REVERSE COMPUTED REVERSE
THRESHOLD VOLTAGE THRESHOLD VOLTAGE
1N749 2.650.1 V 2.99 3.45 V
1N750 2.70.15 V 2.39 2.78 V
1N751 2.70.15 V 2.31 2.71 V
1N752 2.70.15 V 2.27 2.65 V
In light of this model it is not difficult to see
why the existing theories could not explain the experi-
mental data: none of the theories take the influence
of the threshold energies for ionization explicitly into
account. The DC multiplication, which will be discussed
in more detail in Chapter V, has conventionally been
calculated by integrating the ionization rates a(x) and
B(x) over the junction SCR in accordance with the solution
of the differential equation
dJ (x) dJ (x)
-- d-- a(x)Jn(x)+6(x)Jp(x) (3.14)
modeling the process (Moll and van Overstraeten, 1963;
Lee et al., 1964; van Overstraeten and de Man, 1970).
J (x) and J (x) denote, respectively, the electron and
hole current densities as a function of position in the
junction SCR. This equation does not explicitly consider
the effect of threshold energies on the multiplication
process because it claims that the differential increase
in current densities at point x is proportional to the
current densities at point x, which does not take into
account a "distance delay" over which the carriers are
incapable of ionizing because they are acquiring the
necessary threshold energy. For diodes which break down
at several hundred volts of reverse bias this "distance
delay" is very small in comparison to the width of the
junction SCR, and the approximation involved in neglecting
it is very good. For diodes breaking down at several
volts of reverse bias, on the other hand, the "distance
delay" could constitute the entire width of the diode SCR.
The approximation of neglecting it in such cases and
employing equation 3.14 is totally unacceptable it leads
to calculation of finite DC multiplication when no multi-
plication is experimentally observed.
Application of conventional calculations of DC multi-
plication to very low breakdown voltage diodes thus leads
to the conclusion that multiplication must be taking place
in these junctions because the electric fields in them
are very strong. The author, on the other hand, contends
that if the electric fields in these junctions are very
strong, multiplication will occur, but only if the
carrier energies exceed the threshold energies for ioniza-
tion. This is precisely why no impact ionization multi-
plication occurs in tunnel diodes, in spite of the
extremely strong electric fields existing in these junctions.
Because McIntyre's theory of noise in avalanche
diodes employs equation 3.14 in its derivation, it is
now clear why his theory is not capable of predicting
the multiplication noise in diodes employed in this study.
Moreover, Tager's theory is also unacceptable because
it does not incorporate in its derivation an upper limit
on the number of ionizing collisions a carrier can undergo
LI
in its transit across the diode SCR. Such a limit must
be imposed if threshold energies for ionization are
taken into account.
Not explicitly incorporating the influence of the
threshold energies for ionization in the existing theories
of noise and DC multiplication, therefore, renders them
inapplicable to the diodes employed in this study. The
next two chapters will attempt to resolve these difficulties.
CHAPTER IV
DETAILED ANALYSIS OF NOISE DATA
Figures 29 to 32 indicate that if the DC multiplication
factor, M is defined conventionally by
I
M r (4.1)
It
it is impossible to obtain a unique slope which would define
the exponent, x, in Tager's (1965) expression for the low
frequency noise current spectral density
Si = 2qIt(M ) (4.2)
The author suspected that this difficulty arose because
the DC multiplication factor was not appropriately defined.
This can be seen most clearly when M = 1, which is per-
fectly acceptable mathematically, but is physically meaning-
less, for a multiplication factor of unity means that there
is no carrier multiplication. Thus, it was the author's
contention that the DC multiplication factor should be
defined by
I It+MIt or (4.3)
I
M = 1 (4.4)
t
This definition reduces to the conventional one for large
values of M, but is more appealing physically for it allows
M to go to zero when there is no carrier multiplication.
However, if M is defined by M = (Ir/It)-1, the expression
for the noise current spectral density has to be reformulated,
for consistency, to
Si = 2qIt[l+f(M)] (4.5)
where f(M) goes to zero when M goes to zero.
The results of plotting f(M) vs. M for diodes 1N749
to 1N752 are shown in Figures 36 to 39. These graphs
contain several interesting features. First, even though
all data points do not lie on one straight line, it is
possible to fit most of them along two or three straight
lines, the slopes of which increase monotonically with
increasing reverse voltages. Second, the departures of
data points from lines of lower slopes to lines of higher
slopes occur for all diodes at roughly the same (although
somewhat increasing) value of reverse voltage. The values
of reverse voltages at which these transitions set in, as
defined, whenever possible, by the intersection of the two
slopes, are:
Diode type Onset of Ist Onset of 2nd
transition transition
1N749 3.95 V 4.35 V
1N750 4.1 V 4.65 V
1N751 4.4 V 4.95 V
1N752 4.7 V 5.1 V
Finally, even though the slope of any line tangent to the
data points does not exceed its maximum allowed value of
three, as did the tangents to the data points of Figures
36 to 39, it is now possible to draw tangent lines whose
slopes approach unity -- which appears to contradict the
established results that, according to the old definitions,
the values of the slopes should never be less than two.
All of these observations can be understood in terms
of the conduction model incorporating the carrier threshold
energy for ionization, as proposed in the preceding chapter.
Since the amount of noise produced in impact ionization
multiplication increases as the maximum number of ionizing
collisions per carrier transit across the SCR increases,
and eventually approaches the upper limit value given by
Tager's (1965) expression for the case of unlimited number
of ionizing collisions on a carrier transit across the SCR,
the breaks in the straight lines in Figures 36 to 39 and
the monotonic increase in slopes are indicative of transi-
tions from a lower to a higher maximum number of ionizing
collisions that a carrier can undergo in a transit across
the diode SCR. In particular, if the built-in voltage of
the junctions is taken into account, and an electron threshold
energy of 1.80.1 eV is assumed, as given by Moll and van
Overstraeten (1963), then it becomes clear that the onset
of the first break in the slopes of Figures 36 to 39 is
due to a transition from a maximum of two to a maximum of
three ionizing collisions per electron transit across
the diode SCR. This is easily verified by dividing
1.80.1 eV into the total diode potential, the sum of
the built-in potential listed in Table 1, and the potential
at the onset of the first slope transition, listed above.
The results are:
Diode type Total potential divided by 1.80.1 eV
1N749 2.92 2.62
1N750 3.01 2.69
1N751 3.19 2.85
1N752 3.36 3.01
When the procedure is reversed and the total potential
at the onset of the first slope transition is divided
by three, the following electron threshold energies for
ionization are obtained:
Diode type Electron threshold energy for ionization
1N749 1.66 eV
ln750 1.7 eV
1N751 1.8 eV
1N752 1.9 eV
This is in excellent agreement with Moll and van Over-
straeten's (1963) value of 1.80.1 eV. The apparent increase
in the threshold energy for ionization with increasing
diode breakdown voltage is also reasonable. Both Moll and
van Overstraeten's (1963) and these values include the
"intrinsic" threshold energy as well as energy lost in
phonon collisions. Since the rate of energy acquisition with
distance is smaller in higher voltage breakdown diodes (the
electric fields are weaker), but the rate of energy losses
is the same (because it depends on mean free path between
phonon collisions, which, presumably, is not strongly
dependent on field strength), it can be expected that an
increasingly larger number of phonon collisions would occur
between successive ionizations, resulting in increasingly
higher observed threshold energies in higher breakdown
voltage diodes.
An analogous analysis can be made for the second slope
transition in Figures 36 to 39. Since the threshold energy
for ionization for holes is, according to Moll and van
Overstraeten (1963), greater than the threshold energy for
ionization for electrons, the second slope transition may
be suspected to arise from a transition from a maximum of
two to a maximum of three ionizing collisions per hole
transit across the diode SCR. When the total potential
at the onset of the second slope transition is divided
by three, the following values of threshold energy for
ionization for holes are obtained:
Diode type Hole threshold energy for ionization
1N749 1.79 eV
1N750 1.89 eV
1N751 1.99 eV
1N752 2.04 eV
These values are considerably smaller then the 2.40.1 eV
obtained by Moll and van Overstraeten (1963). Thus it may
be suspected that the second slope transition is due to a
transition from a maximum of one to a maximum of two ionizing
collisions per hole transit across the SCR. This, however,
results in hole threshold energies that are substantially
higher than those given by Moll and van Overstraeten (1963):
Diode type Hole threshold energy for ionization
(assuming transition from a maximum of
one to a maximum of two ionizations per
transit)
1N749 2.69 eV
1N750 2.84 eV
1N751 2.99 eV
1N752 3.06 eV
Moreover, if the hole threshold energies obtained under this
assumption are used in the simulation of DC multiplication,
a serious difficulty arises which does not occur when the
hole threshold energy of 1.90.15 eV is used. As a con-
sequence, it is suggested that the value of 2.40.1 eV is
probably too high.
An immediate application of this interpretation of
noise data can be made in conjunction with the conduction
1
model proposed in the preceding chapter and the -- vs. V
C
data to compute the doping densities on the n and p sides
of the junctions. With the help of Figure 35, the analysis
proceeds as follows. At the onset of impact ionization multi-
plication
V +V NA-ND
9 A D (4.6)
rth A
where V = E /q, Ve E /q and E is the bandgap energy
and E is the electron threshold energy for ionization.
1
Moreover, from -- vs. V data
C
(NA-ND)ND AV
A D D 2 -()p (4.7)
NA Eq A-
C2
Therefore,
PV
N = (4.8)
D Vg+V
g e
Moreover,
N 2
NA NDP or (4.9)
PV2
NA = h (4.10)
S V (V +V )-(V +V )
rth g e g e
AV
Since r and, therefore, P, can be determined from the 1 vs V
A- C
C
data, and Vrth and V can be determined from the noise data,
rth e
ND and NA can be calculated. A comparison of ND and NA
obtained in this manner with the values of ND and NA given
in Table I is shown in Table III. The agreement is quite
good, which lends further credence to the proposed conduction
model and the subsequent interpretation of noise data.
Z u
M 1
0 I
a o4
Co
m
if (N ( tn C
co r 0 co
(N (N <( s 0
X
rE
El
S II
HZ
O
0 E
0
F:4
u 0
M
HH
Z H
H H
H
o
OH
OZH
H
* en n in
H --H -- H
r~ -^ r~~ r-
2 2 S Z
r-1 r-1 -1 i-
S'an mn 0 e
H E- 0 w H r0
HElo Z
HO N N rH H- I
H l00
O H rH
O 0
C c rCl r N
0
H 00
0 40 0 o 0
E C
U > (N in
H co en
3> en N rH '
H I U
Co H I I I Co
SH oo (N r en o
Co (N (N Hl Hl '
*H
41
.t
0
0
0
41
c,
0 0
0 q
*'
43
0
-H
O e-
-Sb
Moreover, if the proposed conduction model is truly
valid, then a suitable derivation incorporating the essential
features of the model should yield expressions for the mag-
nitude of the noise current spectral density which should be
in good agreement with experimentally observed results.
From the point of view of noise as a statistical fluctuation
phenomenon, the most important feature of the proposed con-
duction model is the limit it imposes on the maximum number
of ionizing collisions a carrier can experience in one
transit across the diode SCR. Consequently, any derivation
of the noise current spectral density must incorporate this
constraint. This complicates the problem considerably for
it requires that the multiplication process be examined
on basis of individual, successive transits of carriers
across the diode SCR, rather than on monitoring the end
product of the multiplication process at the device terminals,
as was done by Tager (1965) and McIntyre (1966).
To obtain some insight into how the analysis of multiple
ionizing collisions might be carried out, the author began
by considering the limiting case of a maximum of one ionizing
collision per carrier transit across the diode SCR. This
is more than just an exercise. According to the author's
interpretation of Figures 36 to 38, about half of the
data points correspond to multiplication resulting from
no more than two ionizing collisions per carrier transit
across the SCR. Moreover, even when up to two ionizing
collisions are energetically possible, the probability of
the second collision will be considerably smaller then
the probability of the first collision because the space
charge regions of the diodes examined in this study are
very narrow, whereas the mean free path between ionizing
collisions, as quoted by previous workers (Moll and van
Overstraeten, 1963; Lee et al., 1964) is relatively
large. Thus the approximation of no more than one ionizing
collision per carrier transit across the diode SCR should
yield reasonable results, especially for the case of the
diode with the narrowest SCR, the 1N749.
The analysis of the noise current spectral density
is best carried out by referring to Figure 35. The tunneling
electrons, originating at x=0 at the rate of n0 per second,
are accelerated from x=x1 to x=x2, and undergo ionizing
collisions from x=x2 to x=xn, producing n, electron-hole
pairs. The n tunneling electrons and the nl ionization
produced electrons are swept to the right and out of the
SCR, The nl holes travel to the left and, after
acquiring the necessary threshold energy, undergo ionizing
collisions, producing n2 electron-hole pairs. The (nl+n2)
holes are then swept to the left and out of the SCR, while
the n2 electrons travel to the right and, after acquiring
the necessary threshold energy, ionize, producing n3 electron-
hole pairs. This process continues, so that the total
number of carriers, N, collected at the device terminals
becomes
N = n. (4.11)
i=0
Letting nl=aln0, n2=b2nl=b2aln0, n3=a3n2=a 3b2an0, etc.,
where the denotes averages and ai and bi represent the
probability of ionization for electrons and holes, respectively,
on the i-th transit across the diode SCR,
i
N = n0 E H aj oddbj even ;a01, b01 (4.12)
0i=o j=0
This converges to a finite number because a.
since no more than one ionizing collision per carrier transit
across the SCR is allowed. A reasonable simplification
which can be made at this point is that a.=a and b.=b. This
means that the probability of ionization differs for holes and
electrons, but remains the same on all transits. In this case,
N = n0(l+a+ab+a25+ab+ ...)
= n ( +a ) (4.13)
1-ab
and the DC multiplication factor, M = 1, becomes
n0
a(l+b)
M = a (4.14)
The low frequency spectrum of the current fluctuations
resulting from the multiplication process can be obtained from
SN = 2var N
(4.15)
where SN denotes the magnitude of the spectrum of N, the
number of events occurring during a unit time interval
(van der Ziel, 1970). Since the diode current, I is
given by Ir=qN,
SI = 2q2var N (4.16)
r
where
var N = N2 -2
2
= (n )2 ( n)
i=0 i=0
= i Z (nnj-n.nj ) (4.17)
i=0 j=0 1 ] 1 3
In order to evaluate Sr it, therefore, remains to evaluate
nin This is accomplished by representing ni by
0 b. ; i even
n = a b a b r (4.18)
1 r=ir 2r3r4r.
air; i odd
where ajr and bmr are either 1 or 0 (depending on whether or
not an electron or hole ionizes on the j-th or m-th transit
across the SCR) and ensemble averaging over the products
n.n.. This analysis is discussed in detail in Appendix A.
The resulting expression for var N is given by
-- 2-
var N = n l+3a+ab+a b} (4.19)
S(1-ab)2
I-I
and, therefore,
2- 1+3a+3ab+a b2
Sr = 2q n0o (1-a,)2
*r (-ab)
= 2qIt l+3a+3a+2 (4.20)
(1-ab)
If, for the moment, it is assumed that the probability
of ionization is the same for holes as it is for electrons,
a=b, then
M (4.21)
1-a
and
(l+a)
Sr = 2qIt (1l) (4.22)
r (1-a)
Eliminating a,
S = 2qIt(1+3M+2M2) (4.23)
r
which is, indeed, in the previously postulated form
S = 2qIt[l+f(M)] .(4.24)
r
This result provides a more formal justification of the
previously proposed redefinitions of the DC multiplication
factor and the functional form of Si, and lends greater
credence to the interpretation of noise data based on the
new definitions.
When equation 4.20 is rewritten in the form
Sr = 2qIt[l+f(a,b)] (4.25)
r
and f(a,b) is plotted as a function of M for various values
of k=-, Figure 40 is obtained, which contains the somewhat
a
surprising result that f(a,b) is rather weakly dependent on k.
This result is exploited in the derivation of the magnitude
of the noise current spectral density for the case of up
to two ionizations per carrier transit across the diode SCR.
Moreover, since
SI = 2qlt[l+f(a,b)] = 2qlt(1+3M+2M2) (4.26)
r
over a fairly wide range of k= for low values of M, it
a
demonstrates that the slope of f(M) vs. M can, indeed,
approach unity, as it does for the data of Figure 36.
To compare the results of this analysis with experiment,
Figure 41 shows, along with f(M)=3M+2M2 plotted versus M,
the experimental data for the 1N749 diode. Although the
theoretical curve shows the same functional dependence on
M as do the experimental data, it is about a factor of
three too low. However, in view of the fact that the
derivation of the theory is strictly statistical, taking
no account of any junction parameters, this result was
most encouraging.
At this point the author suspected that the discrepancy
between theory and data resulted from dismissing the possibility
of the second and third ionizations, and proceeded to
extend the theory to more than one ionization per carrier
transit across the SCR. Although this is easily done in
principle, it becomes extremely tedious algebraically
because the analysis must take into account the various
branching possibilities, and the correlation between the
branches. In fact, the extension of the theory to the
case of up to two ionizations per carrier transit across
the SCR is a composite of an infinite number of the up-to-
one-ionization analyses. In order to restrict the algebraic
details to manageable size, the author chose to analyze
the case of up to two ionizations per carrier transit under
the additional restriction that the probability of the
second ionization is much smaller than the probability of
the first ionization, and, therefore, that terms containing
-2
a2, where a2 is the probability of the second ionization,
could be neglected. It was felt that this would be a
reasonable approximation for the noise data of Figures 36
to 39 below the first slope transition.
The analysis of this case under the additional approxi-
mation that the electron and hole ionization probabilities
are the same (as justified by DC calculations of Chapter V
and by the relative insensitivity of the results of Figure 40
to k=b)is given in Appendix B. The algebraically tedious
a 2
results, evaluated for 2 = 0.3, which is the highest ratio
al
of the probability of the second ionization to the probability
of the first ionization that the author felt could be assumed
under the approximation that a2<
with the results of the up to one ionization per transit
theory, evaluated for k--=l. Although the two curves inter-
a
sect giving rise to a slope transition, in accordance with
the previous interpretation of the slope transitions in
the experimental data, the results are somewhat disappointing
in that f(al,a2) is not sufficiently larger to give sub-
stantially better agreement between theory and the noise
data of Figures 36 to 39.
A careful examination of the various assumptions and
approximations made in the theoretical analysis revealed
that the source of the final discrepancy between the theoret-
ical results and the experimental data lay in the simplifying
approximation that the carrier ionization probability is the
same on all transits. A typical plot of the number of
carriers generated on a given transit per one tunneling
electron, obtained from DC calculations for the 1N752 diode
operating at reverse bias of 5.2 volts, and shown in
Figure 43, demonstrates that the ionization probability
for electrons on the first transit across the diode SCR
is considerably smaller than the ionization probability
for carriers traversing the SCR on succeeding transits.
This may be deduced by comparing the ratio of consecutive
terms for transit numbers greater than about five which,
in the case of Figure 42 is 0.88, with the value of the
first term,0.516. If the ionization probability were
identical for all transits, the value of the first term
would have been 0.88. In retrospect, it is easy to see
why the ionization probability on the first transit is
less than on succeeding transits. The electrons under-
going ionizations on the first transit come from tunneling,
which, in terms of Figure 35, means that they do not become
free carriers until x=xl. Consequently, they ionize over
a much smaller portion of the SCR than do the succeeding
holes or electrons originating as free carriers closer
to the edges of the SCR.
To determine what effect the smaller ionization
probability on the first transit would have on the noise
theory, the derivation of the up-to-one-ionization-per:
SCR-transit theory was repeated for the case of a=b, -=r
a
where al is the ionization probability on the first
transit. The result, derived in detail in Appendix A,
is given by
Si = 2qlt(1+3M+2M2/r) (4.27)
clearly indicating that theoretically predicted noise
output will increase for r
This analysis was also repeated for the up to two
ionizations per SCR transit theory. Indeed, this is the
derivation that appears in Appendix B. Because the results
are algebraically tedious, they are evaluated numerically
once the values of all, al and a2 have been determined.
Here, all denotes the probability of the first ionization
on the first transit, and al and a2 are the first and
second ionization probabilities on succeeding transits.
For the range of voltages where the up to two ionizations
theory is valid, the probability of second ionization on
the first transit is zero. The details of evaluating
all' a, and a2 from the DC multiplication program are
discussed in Appendix C.
The final results of the up to two ionizations per
SCR transit theory for the case of all
with the experimental data for the 1N749 to 1N751 diodes
in Figures 44 to 46. In view of the many approximations
made in the derivation of the theory, the agreement is
quite good, even for those values of M where the probability
of a third ionization is no longer zero. In the case of
the 1N751 diode, where at low values of M the theory predicts
a larger noise output than appears to have been observed,
the discrepancy is probably due to incorrect values of M,
which at low values, can be subject to substantial error.
The analysis of the noise data was terminated at this
point for it was felt that even though an up-to-three-
ionizations-per-SCR-transit theory would undoubtedly
yield better agreement between theory and experiment, it
would probably offer no new insights into the details of
the multiplication mechanism, and, therefore, in view of
the algebraic tedium required for its derivation, would
not be worthwhile.
CHAPTER V
CALCULATION OF DC MULTIPLICATION
As has already been mentioned in Chapter III, the
calculation of DC multiplication in avalanche diodes has
conventionally been done by integrating the electron and
hole ionization rates, a and B, over the junction SCR,
in accordance with the solution of the differential equation
dJ (x) dJ (x)
---d (x)Jn(x)+3(x)J (x) (5.1)
modeling the process. Since this equation incorporates
the effect of the threshold energies for ionization
implicitly in the ionization rates and, therefore, is
not capable of modeling those segments of the carrier
trajectories over which multiplication does not occur
because the carriers do not have sufficient energy to
ionize, it is not applicable to calculation of DC
multiplication in low breakdown voltage diodes in which
these "inactive" segments of carrier trajectories con-
stitute a very sizeable portion of the diode SCR.
These "inactive" segments could be taken into account
in the above equation by incorporating "distance delays,"
L1(x) and L2(x), so that
(5.2)
dJ (x) dJ (x)
~- x x = a(x)Jn[x-L (x)]+S(x)J[x+L2(x)] .
dx dx 1 pt2
However, if Ll(x) and L (x) are taken so that
x
V = -qf E(x) dx (5.3)
e x-L (x)
and
x+L2(x)
Vh = -qf E(x) dx (5.4)
x
where Ve and Vh represent, respectively, the threshold
potentials for ionization of electrons and holes, then
the magnitudes of J and J must be increased from their
n p
actual values at xL(x) since, in general, there will be
some carriers with non-zero energies which will not need
to travel the full distance L(x) to acquire the necessary
threshold energy for ionization. On the other hand, if J
n
and J are left unaltered, then L1(x) and L2(x) need to be
evaluated as suitable averages over all carrier energies.
Since neither correction is easy to make at the outset, and
since the modified differential equation would still need
to be solved numerically, the author chose to abandon this
approach in favor of a numerical simulation of the mul-
tiplication process.
The simulation, whose program listing is given in
Appendix C is accomplished as follows. An abrupt junction
doping profile is assumed, and the electric field triangle
is cut into a large number of variable width segments such
that the integral of the electric field is the same over
each segment. (In the results shown in Figures 47 to 50,
x+Ax
dv = -qf E(x) dx = 0.01V, so that the electric field
x
triangle was cut into several hundred segments.) The
integral of the ionization rates, which represents the
probability of an ionization, and the fraction of the
tunneling current originating in a given interval are
then evaluated approximately for each interval by
x k+Ax
dPn(k) = x a dx .[a(xk+tAxk)+a(xk)]Axk (5.5)
xk+Axk
dPh(k) = I B(x) dx = [B(xk+AXk)+6(xk)]Axk (5.)
Xk
and
xk+Axk
s
x e dx
dIt (k) = k
w s
I e-t dx
0
s s
[e E(xk+Xk) + e Exk)Axk
s' s (5.7)
,1 L-FTx Ax) r ETxk
[e k+ + e- ]Axk
2k
where dPn(k) and dPh(k) represent, respectively, the
probability of ionization for electrons and holes in
the k-th interval, and dIt(k) represents the normalized
tunneling current originating in the k-th interval. The
multiplication'process is then simulated by following the
motion of electrons and holes back and forth across the
diode SCR, and allowing them to participate in ionizations
after they acquire their respective threshold energies,
which can be kept track of very easily by counting off
the appropriate number of intervals. Thus, the tunneling
current originating in the k-th interval becomes a free
V
electron current in the (k+m)-th interval, where m= ,
which may then ionize starting in the (k+m+nl)-th interval,
V
where n1= and where V and Ve denote, respectively, the
bandgap potential and the electron threshold potential for
ionization. That fraction of the tunneling current which
ionizes in a particular interval, and the ionization produced
electrons originating in that interval, are then not allowed
to ionize again until they have traveled at least another
n1 intervals. The tunneling electrons are traced in this
manner across the entire SCR, while the holes produced in
ionizations occurring in a given interval are traced in the
opposite direction, not being allowed to ionize until they
V
have traveled at least n2= intervals, where Vh is the hole
threshold potential for ionization. The fraction of the
hole current ionizing in a particular interval, along with the
ionization produced holes originating in that interval,
are then not allowed to ionize again until they have
traversed at least another n2 intervals. In this manner
the hole current is traced across the entire SCR.
This procedure is alternately repeated for electrons
and holes, and the contribution to the terminal current
due to ionizations on each carrier transit across the SCR
is recorded. When the contribution on the p-th transit
is less than 0.005 of the contribution on the first transit,
the simulation is stopped and the contributions to the
terminal current from all transits are added. The same
procedure is followed for the multiplication process initiated
by the holes left in the valence band when the electrons tunnel to
the conduction band. The total multiplication is then the
sum of the contributions from the tunneling electron
initiated multiplication and the typically much smaller
hole initiated multiplication.
The simulation thus circumvents both of the previously
discussed difficulties which would be encountered in the
differential equation approach to the computation of DC
multiplication. It incorporates unambiguously the threshold
energies for ionization and, by keeping track of the past
history of the carriers, automatically computes the fraction
of the total current crossing a given interval which is
sufficiently energetic to ionize in that interval. This is
equivalent to automatically adjusting the magnitudes of
Jn [-L (x)] and J [x+L2(x)] to account for the fact that
not all carriers need to travel the full distance L(x) in
order to acquire the necessary threshold energy for ionization.
Unfortunately, the simulation creates a different
problem which, too, is difficult to resolve at the outset.
Namely, it assumes that when an ionizing collision occurs
the primary carrier and the secondary carriers are scattered
in a way which, on the average, does not impart any excess
energy to either the primary or the secondary carriers, so
that all carriers will be accelerated again essentially
from rest. Since the scattering produced by an ionizing
collision is probably not isotropic, this assumption may
not seem very reasonable. However, in the actual calcula-
tions the non-isotropic effects should, at least partially,
cancel, for if the scattering favors the direction of the
ionizing carrier, then the carriers of the same polarity
as the ionizing carriers will not need to accelerate the
full distance L(x) to acquire the threshold energy, whereas
the carriers of opposite polarity will need to accelerate
farther than the distance L(x) to acquire their threshold
energy. Since the simulation is performed by alternately
tracing the motion of electrons and holes across the SCR,
these effects should alternately cancel to some extent.
However, without knowing the exact details of the ionization
and scattering mechanism, it is difficult to estimate how
much cancelling actually takes place.
Although the simulation program was written especially
for computation of DC multiplication occurring in low break-
down voltage diodes, where it is necessary to take the
threshold energies for ionization explicitly into account,
it may also be used to perform DC multiplication calculations
in the conventional manner simply by letting the threshold
energies for ionization, and the bandgap energy, approach
zero. In actual calculations the bandgap and threshold
potentials were set at 0.01 V. The ionization rates
employed were those of van Overstraeten and de Man (1970).
From the results obtained for diodes 1N750 to 1N752, shown
in Figures 47 to 49 under the label of "conventional
calculations," it is apparent that the conventional method
of performing DC multiplication calculations is inadequate
for it predicts much greater multiplication than is experi-
mentally observed, even to the point of predicting finite
multiplication when no multiplication actually occurs, as
evidenced by the noise data of Figures 15 to 21.
Although, as shown in Figures 47 to 49 under the
label of "conventional calculations, including tunneling,"
the agreement between the experimental and the calculated
results improves somewhat when, on the first transit, the
bandgap energy for tunneling is taken into account, the
calculated values of DC multiplication are still considerably
larger than what is experimentally observed, and finite
values of multiplication are still calculated when no
multiplication actually occurs.
When the threshold energies for ionization are
finally taken into account, the calculated values of
DC multiplication do go to zero when no multiplication
is observed in the noise data, but the predicted magnitudes
of the DC multiplication are smaller than those experimentally
observed. This, however, is understandable for the integral
of the ionization rates over a given distance represents,
by definition, the probability of ionization over that
distance. Therefore, eliminating large portions of the
diode SCR from integration by explicitly taking the threshold
energies for ionization into account, but employing conven-
tional ionization rates derived from integration over the
entire SCR, reduces the calculated ionization probabilities
and, therefore, the calculated DC multiplication factor.
It, therefore, seems appropriate to increase the ionization
rates to compensate for this shortening of the ionization
rates' integration region. Unfortunately, this is not a
trivial matter since the functional dependence of the
ionization rates on the electric field implicitly reflects,
along with other parameters, the influence of the threshold
energies for ionization on the ionization rates, and,
therefore, should probably be modified when the threshold
energies for ionization are explicitly taken into account.
Since there exists no theoretical groundwork to indicate
how this should be accomplished, the author chose not to
modify the functional dependence of the ionization rates,
but merely introduce scaling constants cl and c2 so that
the adjusted ionization rates a and B are given by
a =cle and B =c2 where a and B are the "reference",
conventional ionization rates given by van Overstraeten
and de Man (1970). The constants cl and c2 are adjusted
to give the best possible agreement between the calculated
and the experimentally obtained values of DC multiplication.
Since the adjusted ionization coefficients would be
useless if cl and c2 were determined individually for each
diode, the author chose to adjust cl and c2 on diode 1N752,
whose DC calculations are most sensitive to the values of
cl and c2 because they need to attain the largest values
of DC multiplication without diverging. Once this was
done, calculationswere performed on the remaining diodes,
using their respective values of the threshold energies
for ionization, as determined in Chapter IV. The results
of these calculations are shown in Figures 47 to 49 under
the label of "calculated". The discrepancy between the
calculated and the experimental values of DC multiplication
occurring for low values of M is attributed to the tri-
angle approximation for the junction electric field,
which gets progressively worse with decreasing reverse
voltage since the diode SCR narrows and gets progressively
closer to the changing doping densities of the junction
metallurgical transition region. In that region the
triangle approximation predicts stronger electric fields
than are actually present. This, in turn, leads to larger
calculated values of ionization probabilities and, hence,
larger than observed values of DC multiplication. In
spite of this discrepancy, the DC calculations performed
by explicitly taking the threshold energies for ioniza-
tion into account and employing the adjusted ionization
rates are in considerably better agreement with the
experimental data than are the results obtained by con-
ventional calculations.
The values obtained for cl and c2 are quite reasonable
also: c1=3, c2=5. In fact, it was on the basis of the
magnitude of c2 that the author dismissed as unlikely the
2.850.2 eV value for the hole threshold energy for ioniza-
tion, for when this value was employed in the DC calcula-
tions it was not possible to obtain reasonable agreement
between the calculated and the experimental values of DC
multiplication for values of c2 as large as twenty.
It was also observed that when the best agreement
between the theoretical and experimental results was
obtained, the electron and hole ionization contributions
to the DC multiplication decreased monotonically on
successive transits, as shown in Figure 43. Since the
ratio of successive terms remained essentially constant
regardless of which terms were picked (with the exception
of the beginning few) this implies that the ionization
probabilities for electrons and holes were essentially
the same. This information is used in Appendix B to
simplify the derivation of the expressions for the magnitude
of the low frequency spectral density of noise current
produced in a multiplication process resulting from no
more than two ionizations per carrier transit across the
diode SCR.
CHAPTER VI
CONCLUSIONS
The principal result of this study is a detailed under-
standing of the mechanism of reverse conduction transition
from tunneling to impact ionization multiplication in low
breakdown voltage silicon p-n junctions. It is concluded
that the transition occurs when the energy of the tunneling
electrons exceeds the electron threshold energy for ionization.
If the DC multiplication factor is redefined by
M = (Ir/It)-1, and the functional form of the expression for
the magnitude of the noise current spectral density is changed
to Si = 2qIt[l+f(M)], then the electron and hole threshold
energies may be obtained from slope transitions in the plots
of log f(M) vs. log M. The electron threshold energy for
ionization determined in this manner is found to be in excel-
lent agreement with the value of 1.80.1 eV found by Moll
and van Overstraeten (1963). However, the hole threshold
energy of 1.90.15 eV disagrees with their value of 2.40.1 eV.
Moreover, it is found that the threshold energy for ionization
is not constant, but increases slightly with increasing junc-
tion width.
The noise data may also be used to estimate the average
minimum distance between ionizing collisions. This is
accomplished by obtaining the junction width at the reverse
voltage corresponding to the onset of the transition from
no more than two to no more than three ionizing collisions
per electron or hole transit across the diode SCR, and dividing
that distance by three. The results are:
Diode type Average min. distance Average min. distance
for electron ionizing for hole ionizing
collisions collisions
1N749 192 A 199 A
1N750 213 A 224 A
IN751 223 A 234 A
1N752 242 A 250 A
When the threshold energies for ionization, the total
potential at the onset of carrier multiplication, and the
value of (NA-ND)ND/NA are interpreted in terms of the transi-
tion model (Figure 35), the values of NA and ND can be
determined. The doping densities on both sides of a step
junction may thus be evaluated from diode noise and C-V measure-
ments. This technique could be used in some cases for the
evaluation of impurity solubilities in silicon.
Conversely, when the junction doping profile and the
threshold energies for ionization are known, the transition
model may be used to predict the reverse potential for the
onset of impact ionization multiplication. This furnishes
a method for distinguishing the tunneling section from the
multiplication section of the diode reverse V-I characteristics.
Besides determining the onset of carrier multiplication,
the threshold energies for ionization dominate the mechanism
of the multiplication process. Because they allow no more
than two or three ionizations per carrier transit across
the diode SCR, theyrender Tager's (1965) and McIntyre's (1966)
theories of noise in avalanche diodes inapplicable to the
diodes of this study. This is amply demonstrated by the
severe discrepancies between their theoretical results and
the experimental data, and by the considerable improvement
in agreement between the data and the theoretical results
obtained when an upper limit of two ionizations per carrier
transit across the SCR is explicitly imposed. Indeed, for
low multiplication values, a noise theory based on the explicit
constraint of no more than one ionization per carrier transit
across the diode SCR gives considerably better agreement with
the experimental data than do the theories of Tager (1965)
or McIntyre (1966).
Moreover, unless the effect of threshold energies for
ionization is explicitly taken into account, a finite value
of DC multiplication is calculated when no multiplication
is experimentally observed. This discrepancy does not arise
when a simulation program incorporating the threshold energies
for ionization is used for calculating the DC multiplication.
The many details of the transition mechanism thus lead
to the conclusion that the conventional treatment of carrier
multiplication phenomena, based on the equation
61
dJ (x) dJ (x)
d- d= = a(x)Jn(x)+B(x)J (x) (6.1)
is not applicable to carrier multiplication in low break-
down voltage diodes.
CHAPTER VII
RECOMMENDATIONS FOR FURTHER STUDY
The first undertaking in further work on this topic
should be the removal of the triangular electric field
approximation from the DC multiplication simulation program.
The electric field should be obtained from the Poisson
equation and the impurity doping profile. If this correction
does not remove the discrepancy between the experimental and
the calculated values of DC multiplication at low reverse
voltages, the cause of the discrepancy should be investigated.
When good agreement between the experimental and the
calculated values of DC multiplication is obtained, the cal-
culations should be extended to a case of greater practical
importance: the reverse breakdown characteristics of base-
emitter junctions of bipolar transistors. Because of the
very heavy doping densities encountered in base-emitter
junctions, it will probably be necessary to take bandgap
narrowing into account. Moreover, if the reverse DC V-I
characteristics of the base-emitter junctions or of the low
breakdown voltage diodes are to be calculated directly from
impurity doping profiles, it will also be necessary to refine
the theory of tunneling breakdown (Chynoweth et al., 1960;
Tyagi, 1968).
Furthermore, the program should be modified to calculate
carrier multiplication resulting from a single pass across
the space charge region. This would permit calculation of
carrier multiplication in short channel FET's and other
small devices in which one carrier type is somehow removed
so that it does not participate in the multiplication
process.
Since the threshold energies for ionization play an
important role in the simulation program, accurate calcula-
tions of DC multiplication in structures of arbitrary lengths
and arbitrary impurity doping profiles will require accurate
knowledge of the threshold energies under these conditions.
Consequently, it will be necessary to carry out additional
DC and noise studies to establish means of predicting the
threshold energies for ionization in such structures.
The effect of different junction widths and different
electric field profiles could be investigated further with
the aid of the recently developed replacement series for the
1N746 to 1N752 diodes employed in this study. The new series
of diodes, manufactured by alloying aluminum onto the lightly
doped p layer of a p-n diode, break down more abruptly than
their older counterparts. The more abrupt breakdown is
obtained from the resulting p -p-n doping profile (author's
conjecture!), which, for any given reverse voltage, widens
the space charge region and lowers the peak electric field.
The lower peak electric field decreases the tunneling current
and the wider junction sets up a more favorable environment
for impact ionization. Thus, when impact ionization multi-
plication finally sets in, it occurs more abruptly than in
the conventional p -n alloy diodes employed in this study.
Since the new diodes should, for a given value of DC multi-
plication, exhibit more noise than their older counterparts,
this conjecture should not be difficult to verify.
Moreover, it would be possible to fabricate narrower
junctions by employing stronger p-type material, such as
boron doped aluminum (Chynoweth et al., 1960). The electric
field profiles in such p -n diodes would be more asymmetrical,
which should permit impact ionization multiplication at lower
reverse voltages. This should reduce the likelihood of
multiple ionizations per carrier transit across the space
charge region, and make the noise data more amenable to
interpretation in terms of the already developed noise theory.
Due to phonon involvement in the carrier transport and
energy balance mechanisms, the threshold energies for ioniza-
tion should also be determined as function of temperature.
Low temperature measurements might also shed some light on
the mechanism of phonon collision involvement in the ioniza-
tion and multiplication processes. Tunnel diode DC V-I
characteristics measured at 4.20K revealed structure which
reflected the cooperation of different energy phonons in
the tunneling process (Chynoweth et al., 1962; Logan and
Chynoweth, 1963; Logan et al., 1964). Such structure might
also be observed in the DC or noise data of these diodes.
In conjunction with the experimental work, the noise
theory should be extended to the case of a multiple, but
finite number of ionizing collisions per carrier transit
across the diode space charge region. This might be possible
by combining the author's method with the approach employed
by van Vliet in the derivation of avalanche photodetector
noise (van Vliet, 1967). Incorporating the quantum efficiency
and the Boson factor in the primary current would also make
the theory useful for evaluating avalanche photodetector per-
formance.
Finally, the improved multiple ionization noise theory
could be used to extract from the noise data information on
the ionization probabilities. This would be of considerable
help in undertaking a theoretical study of the adjusted
ionization coefficients. Such a study might also lead to
new approaches to the treatment of other aspects of carrier
transport in small devices for which the conventional,
differential equations approach, with its implicit assumption
of unlimited scattering, is not valid.
APPENDIX A
MULTIPLICATION NOISE RESULTING FROM NO MORE THAN ONE
IONIZATION PER CARRIER TRANSIT ACROSS DIODE SCR
The derivation of expressions for the magnitude of the
low frequency spectral density of noise current produced in
impact ionization multiplication resulting from no more than
one ionization per carrier transit across the diode space
charge region proceeds as follows. As indicated in Chapter IV,
the total number of carriers, N, collected at the diode ter-
minals in one second is given by
N= n.
i=0 '
where n0 is the number of electrons tunneling in one second,
and ni, i=l, 2, 3, ... is the number of hole-electron pairs
produced in one second on the i-th carrier transit across the
diode SCR.
If it is assumed that nl=an0, n2= bn=aEn0, n3=an2=a2 n0,
etc., where the denotes averages and a and b represent the
ionization probabilities of electrons and holes, respectively,
then
----------- -2------2-2
N = n0(l+a+ab+a b+a b +...)
and the DC multiplication factor, M = N/no-1 becomes
M = a(l+b)
1-ab
The low frequency spectrum of the current fluctuations
can be obtained from
SN = 2var N
where SN denotes the magnitude of the spectrum of N, the
number of events occurring per unit time interval (van der
Ziel, 1970). Since Ir = qN,
SI = 2q2var N
r
= 2q2 (N N2)
S 2
= 2q2{(( ni)2-( ni) }
i=0 i=0
= 2q2{ Z Z (n.n.-n.n.)}
i=0j=0 3
To evaluate ninj, it is necessary to have explicit expressions
for n. and n.. These may be obtained as follows. If the emission
of tunneling electrons is pictured as a sequence of l's, then
the terms n., j=0, 1, 2, ... can be pictured as
n0: 1 1 1 1 1 1 1 1 ... 1 ... 1
nl: 0 1 1 0 1 0 0 1 ... a ... 1
n2: 0 1 1 0 1 0 0 0 ... alrb2r ... 0
n3: 0 1 0 0 1 0 0 0 ... alrb2ra3r ". 0
n4: 0 1 0 0 0 0 0 0 ... a lrb2ra3rbr 0..
etc.
where 1jk j=l, 2, 3, ..., k=l, 2, 3, ... no, denotes the
occurrence of an ionizing collision on the j-th transit,
initiated by the k-th tunneling electron, and where the O's
denote the absence of ionizing collisions. The transitions
from 1 to 0 occur independently and at random. From the above
tabulation, it is apparent that n. is given by
n0 ajr; j odd
n alrb2ra3r ... or
r=lr2r3r4r bjr; j even
where ajk and bjk, j=l, 2, 3, ... k=l, 2, 3, ... n0 are
1 or 0, for eventhough ajk and bjk are allowed to fluctuate
independently and at random, H akrbk retains the two essential
k=l r r
features of the above table, namely, that (i) if all preceding
entries in the k-th column are 1, then the next entry can be
1 or 0, and (ii) that if any of the preceding entries is 0,
then the next succeeding entry must be 0. The average
nO air no a.
nin- = ( alrb2r .. r )( p alpb2p
r=l ir p= bjp
may now be evaluated by ensemble averaging. Averaging over the
mN subensembles containing n0 = N elements
mN N air N ajp
n.n. = { ( E alr b ... or )( a b ... or )}
3 N m=l r=l b.i p=l p b.
2 ajr
1 N N 2 ir (i+l)r ...or
mN Z I a b ...or
m=l r=l 2
ir (i+l)r ...ajr
or
bjr
1 N N N air ap
+ { Z(a br.. or )(ab2...or)
mN =l r=l p=l bir bjp
r/p
Since all a's and b's are equal, then if both i and j are even
-- N 2 (7-) (1) 2) -2'2
n.n. = N(a2) (b2) (a) )
j>i
+N(N-1)(a) 2 (b)
2-2
Moreover, since a and b are 1 or 0, a =a and b2=b. Therefore,
-N =N() ()(i)( )
n N .= Na b +N(N-l)a b
j>i
Finally, averaging over all subensembles
( 4) b(-) i (1+') i+J,
-2 2 + 2 2
n"in = na b + (n0-nO)a b
j>i
Thus,
iFF -ii. ii. 2 b +) -2
nin-nin = n0a + (ng-nO)a b
j>i
(i+_ ) ("+)
-2- 2
-n0a b
) () 2 -
= n0ga + [(n0-n0)-n0]a b
However, since the tunneling electrons have a Poisson distri-
bution, for which var no = no
(n.n.-n.nj) = na ; i,j even
j>i
The same expression is obtained if j is even and i is odd.
If j is odd and i is odd or even,
_ (3+1) (JJ.-
(n n -n n) = n0
If i j, the above expressions hold provided j is replaced by i.
Consequently,
var N = E Z (ninj-ninj)
i=0 j=0
-5 * __ _
S (n. -ni ) + 2 Z Z (n.n.-n.n.)
i=0 i=0 j=0 3
j>i
-2- -2-2
= n0 (1+a+a+a b+a b +...
2- -2-2
+2n 0(a+ab+a b+a b +
--2- -2-2
+a2 +a22+ ..
S2)2
1+3a+3ab+a2i
= no0 (abl- }
Therefore,
SI = 2q2var N
r
2- 1l+3a+3ab+a b
= 2q no{ L----
= q1{l+3a+3ab+a b j
(1-ab)2
If a=b
S = (l+a)
r (l-a)
and
M a
(1-a)
Therefore,
Sr = 2qIt(l+3M+2M2)
If, however, the probability of ionizing collision on the
first transit, al, is not equal to a, the probability of
ionizing collision on succeeding transits, then
(n n -n n )= --al-(j-1)
(ninj-nin) = n0ala1 ; j
and
2 -2
(n0-n0)=
Consequently,
var N = z i (n.n.-n.n.)
i=0 j=0
(l+a)2+a (l-a)+2a
= n{ (1-a) 2
and
M a
(1-a)
al
Therefore, if = r
a
Sr = 2qt (l+3M+2-)
APPENDIX B
MULTIPLICATION NOISE RESULTING FROM NO MORE THAN TWO
IONIZATIONS PER CARRIER TRANSIT ACROSS DIODE SCR
The derivation of expressions for the magnitude of
the low frequency spectral density of noise current pro-
duced in impact ionization multiplication resulting from
no more than two ionizations per carrier transit across
the diode space charge region proceeds as follows.
It is assumed that carrier multiplication taking place
during one transit across the diode SCR is, in general,
of the form
al ala2 al
1 \ a \ ala2
Sx x
1 2
-xx
/I
a / 1
where "l" denotes the "primary" carrier initiating the
multiplication process, "x" denotes an ionizing collision,
al and a2 denote the probabilities of the first and second
ionizing collision, and the dashed lines indicate carriers
of opposite charge to the "primary" carriers.
If the above diagram is extended to include several
transits across the diode SCR, it takes on the appearance
of a "tree" from which it becomes apparent that the multi-
plication process can be decomposed into an infinite number
of "branches" of the form
S-2- --2-2 -3-2
n nal na nala nala2 nala2
--P x-- 1x -L x -x x... etc.
which is precisely the up to one ionization per carrier
transit across the SCR process for the case of unequal
probabilities for holes and electrons. (It should be
observed, however, that in this analysis, the ionization
probabilities for electrons and holes are assumed to be
the same. The evidence supporting this assumption is dis-
cussed in Chapter V. Even if this assumption were not
strictly valid, however, Figure 40 indicates that the results
obtained for unequal ionization probabilities do not differ
substantially from the result obtained for equal ionization
probabilities in the one ionization per carrier transit case.
And since the up to two ionizations per carrier transit
across the SCR case is a composition of the one ionization
per carrier transit cases, the final result should not be
very sensitive to the difference in ionization probabilities
for holes and electrons.)
To reduce the number of "branches" that need to be
considered, it will also be assumed that a2<
-j-k
branches beginning with allala2, k>2, will be neglected.
This assumption is justified in Appendix C Moreover,
since the probability of ionizing collision is smaller on
the first transit than it is on succeeding transits, the
probability of the first ionizing collision on the first
transit across the diode SCR will be denoted by all.
According to the conduction model of Figure 35, the prob-
ability of the second ionizing collision on the first transit,
a21, for the range of voltages where the up to two ionizations
per transit across the SCR theory may be considered valid,
is zero. This follows from equation 3.13, with 1.80.1
replaced by 2V The results of this calculation are
Diode type Reverse voltage for which a21>0
1N749 4.89-5.06
1N750 4.85-5.00
1N751 4.90-5.06
1N752 4.95-5.10
The reverse voltages given above are greater than the voltages
corresponding to the first slope transition in Figures 36 to
39, thus confirming the contention that for the range of
voltages where the up to two ionizing collisions per carrier
transit theory is valid, a21=0.
Under the condition stipulated, the "tree" for this
multiplication process is given in Figure 50. The heavy
arrows denote electron ionizations, the light arrows denote
hole ionizations, and the ]ij denote "branch" numbers. To
avoid excessive clutter, only the beginnings of each "branch"
are indicated. Thus, "branch" Il is the branch given by
-3-2
allala2
x
--4-2
allala2
etc.
All branches, therefore, continue vertically downward,
and their continuation is indicated by dashed arrows. The
beginning of each new branch is indicated by a horizontal
translation to the right, followed by the downward arrows
of that branch.
The analysis of this multiplication process proceeds
as follows. Let N be the total number of carriers collected
at the diode terminals as a result of multiplication initiated
by n0 tunneling electrons. Then,
N = Zm.
where mi denotes the number of carriers produced by the i-th
branch. Hence,
var N = E var m.+2ZE (mimj-mi mj)
i 1ij
j>i
mi = E nik
k=0O
where nik is the number of carriers produced in the i-th
branch, on the k-th transit across the SCR. Following the
procedure of Appendix A, nik is, in turn, represented by
n0
nik = i (product of all, all's and a2's)r
r=l
where all, all's and a2's are 1 or 0, and the product function
of all, al's,and a2's reflects the past history of the r-th
primary carrier, in the i-th branch, on the k-th transit
across the diode SCR. Thus the problem of up to two ionizing
collisions per carrier transit across the SCR is reduced
to a sum of an infinite number of up to one ionization per
carrier transit analyses.
To obtain the expression for M, the branch contributions
are summed as follows:
N = m = 10 + mlj + Z m0j
i j=l j=o
+ E Z mOij + i: m. + i: m..
i=2 j=l j=2 i=2 j=l
resulting in
1+al
N = n0[l + all -
1-ala2
a lal a2 (l+aI)
-2 2
(1-ala2) (1-a )2
a llal (l+a2)
(1-ala 2) (1-a )
+ 11 1
S-3- (+2)
allaa2(l+a2)
-- 2
(l-ala2) (1-a) 2
-4-
allala2 (1+a2)
(1-ala2) (1-a 1
a a^ ( 1-a 1)
Hence, M = 1 becomes
n0
ala2 1+a1
M = al [ (1+- ) (--- )
1-al 1-ala2
-2
a(1-a+2ala2) (1+a2)
+ (1-l) 2(--a2
(1-a1) (1-aYa2)
To calculate the results for var N, the procedure of
Appendix A is duplicated to obtain
- -2-
a11(3+5al+al 2-ala2)
var mi = 1 + -
(1-ala2)
and the variances of the branches b, and b2 whose averages
are given by
b +a +a 2- 2 2+a 22+...
bl: 1+al+ala2+al a2+a a +
and
b- a+- -a +2- 1 2-2 ..
b2: al+ala2+al a2+a a2 +
They are, respectively,
2-
l+3al+3ala2+a 2a2
var b_ 2
(1-ala2)
and
al (l+3a2+3ala2+ala2)
var b2 = a 2)2
(1-ala2 2
Observing that the variances of branches whose averages
are represented by
alala 2 (1+al+ala2+a2 )
or
k- --- -2- -2-2
alla1 a2a 1 +ala2+al a2 +a a2+ ...)
are given, respectively, by
j- k
allal a2 var b1
or
-j- k
allal a2 var b2
and repeating the summation performed to obtain N = Em.
i I
2-
all(3+5al+ala2-ala2)
Z var mi = 1 + 2
i 1 (1-ala2)
-2-
all1l12 1+3a1+3ala2+ala2
+ ----
(1-a1) (1-ala2)
-2 ------ 2
a la2(1-a+2aj 2) 1+3a2+3ala2+ala2
(1-a l) (1-ala2)2
It, therefore, remains to compute 2 IE (mm.-mm.).
ij
j>i
Fortunately, it is necessary to compute only seven (mm-mm.)'s,
since the remaining can be obtained by inspection. The pro-
cedure employed in calculating these is illustrated for the
case of (m0mlmll-m0 mll)
(m ml lmlm1) i = E 1(nloinllj-nl0inllj)
i=l j=l 1
where
"n0
nl0i = 1 (1 all a a2 a a2 a )r
r=l
i terms
and
nn0.
"llj = E [1 all al a2 al (a1 a2 a .. .)]s
s=1
j terms
81
Here, denotes identically the same factor in nl0i and
nllj, and the factors bracketed by are to be treated
as one, since they correspond to i=l or j=l. Since this
approach is identical to that employed in Appendix A, it
may be observed there that
(nl0inllj-nl0inllj) =(nlOinllj r=s
2 -2-
since n -n0 =no
2a -- 2- 2-
Moreover, when r=s, a 1=a a =a and =a2. Consequently,
(ml01ll-mlmll) = j I(n"l0ilj11 r=s
i=1 j=1
--3- -(+24i2
Sallaa2 (l+a2) (5+a2-4aa2)
(= n0 l-al a2)2
The other six (mm.-im.im.)'s of importance are
i 3 iJ
_____ l+a
-4- l+a2 2
(mllm12-mllml2) = a 2 -2
1-ala2
_1 a a2 (l+a.) a2(l+al)
'^IT0o-"o-i = -[3+]-- ]
S-(~ +a2) (l-ala2)
13- 2 l2+ a 2)
(ml0m021-mllm021) = a a22( _
11l-a-a (1-ala2)
(mllm020-mllm020) =
(m020m021-m020m"21)
- -3-2 -
alla a2(l+al) (1+a2
= 2 l+a2 +a
Sall ala ( ) (+ l )
l-ala2 1-ala2
4- l+a2 l+a2
(m221-m20m21) = all a2 ( ---) (2+ )
1-ala2 1-ala2
All other (mimj-mimj)'s may be obtained from these by observing
that, for example,
(mimlj-mlimlj) = al (mliml (j-1) -mlml(j1)
i>l, j>3
Generating additional (mim -mi )'s in this manner and adding
the results
- -3-
n0allala2 (l+a2) (5+a2-4ala2)
(1-al) (1-aa)2
iZ (mlimlj,-m imlj )
i=0 j=1 1
j>i
-- -4-
nOallala2 1
+ 0 1( )2
(1-al) 1-ala2
3-22 2)
n,,llala (l+al)(3+a-2aaa)
B: (m10m02j-m10m02j)
j=0) 1-al)(1-al 2)
n0allala2 (l+al) (3-2ala2+a2)
(1-ala2 2
i1 j (m lim02j-lim02j
- -3-2 ) (-
n0allala2(1+a (1I+a2
(-al)(-a 22
-5-2 -
0allala2 1+a2 2
+ 2 l- 2
Snalal(l+a2) (3+a2-2ala2)
i= 10 2 1- -a1) la)2
-2-
(l-al+ala2)
(1-al) (1-ala2)2
- -4- -2-
E 0 --i an0allaa2-a +aa2a ) 1+a 2
E: Z (m m2j -mlim2 ) 2 -2
i=1 j=0 (-al) l1-a a2
- -3- -
--- _Pn a aa (1+a ) (2+a2-1a2)
(m02im02j-m02im02j) n0allala2+a2)
i 02 02 0 (1-al) (1-ala2
j>i 2
- -4-
nallala2 1+a2 2
+ (1-al) 2 1 2
(1-a1) 2lala2
i Z (m r n m
Sj=l 21 2j 2im2j)
j>i
- -4- -
allala2( 1+a2 (2 +a2
(l-ai -a- )a(2+ -aa2
(1-al) l-a a2 1-ala2
- -5- -
+ nallala2 1+a2 2
(1-al) 1-ala2
-2-
H: Cm n0allala2(l+al) (l+a2)
j=0 (m2002j -2002j) (-ala2i2
-4-
n0allala2 +a 2
(1-al) 1-ala2
I:il 2im2j-m2imo2j)
i=1 j=0
- 4-2
(1-al) (l-ala2
- -6-2
+ n0allala2 1+a )
(1-al) 2 l-ala2
85
_______ ______ _2_2 1+a
- -2-2 1+al
J: (m020m03j-m020m03j) = nallala2( -
j=0 l-ala2
-4-2
Snllala2 (1+al) (1+a2)
(1-al) (1-ala2)2
S_ 4-2 -
n 0n aalla a2 (1+al) (1+a2
iK: =l j (m02im03j-m02im03j (1-a) (l-aa1 )2
- -6-2
n0allala2 1+a2 )2
+ ( 2 ia
(1-al) 1-ala2
L (m -a2 (1+al)(+a2)(1-a 1+a 2)
j=0 (1-a1) (1-ala2)
- -5- -2-
: (m2nallala2 (1-aala2
iZ=l j=O(mo2im~ j -m oij (l3alm2
1+a 2
(la_2 )2
l-ala2
-2-
n0allala2 (1+ 1 (l+a 2
N: (m20m03j-m20m03j) (1 ala 2
- -4-
n+ allala2 l+a2 2
(1-al) l-ala2
- -2-
n0allala2(1+a1) (l+a2)
0: E E (m2imj-03j m2im03j ()i 2
i=l1 j=0 l-a
-4-
n"011a1a2 1-a
n0allala2( a2 )2
n_ alal(la-a a ) l+a
P: Z (m20m3i-m20m3j) -
j=o (1-a0) 1-ala
na
n0a 11la a2 (l-a1+aa 2)
Q: (m2im3j -m2im3j)2
i=1 j=0 (1-al
1+a2 2
l-ala2
Finally,
2Z E (m-mm.m) = 2[A+ (B+C+D+E+F+G+H+I)
i j 1-a
j>i
+ ---=--- (J+K+L+M+N+O+P+Q)]
(1-a1)
After all, al, and a2 are calculated using the DC multiplica-
tion program of Appendix C, the above results are evaluated,
and (var N) 1 is plotted vs. M in Figures 44 to 46.
.0
APPENDIX C
DC MULTIPLICATION SIMULATION PROGRAM
The DC multiplication simulation program performs
calculations of carrier multiplication resulting from
impact ionization in p-n step junctions. Its input variables
are:
NSETS : number of diodes whose DC V-I characteristics
will be calculated
NVREV : number of reverse voltages for which the DC
multiplication will be calculated
XNA : NA-ND = net doping density on p-side (cm-3
XND : ND = net doping density on n-side (cm-3)
ALPHAl electron ionization rate constants: adjusted
I B1_
B1 ionization rate is given by a =ALPHAl e E
ALPHA2 hole ionization rate constants: adjusted ioni-
B2 zation rate for E < 4 x 105 V/cm is given by
B2
= ALPHA2 er
ALPHA3 hole ionization rate constants: adjusted ioni-
B3 zation rate for E > 4 x 105 V/cm is given by
B3
2, = ALPHA3 e E
PHI : 4, junction built-in potential
Xk+Axk
DV : dV = -q fk E(x) dx, where Axk is the length
S k
of the k-th interval in the E(x) vs. x diagram
TV : tunneling potential
EV : electron threshold potential for ionization
HV : hole threshold potential for ionization
SLOPE : the tunneling current, It, is given by
SLOPE
It= Ioe Fr
VREV : reverse voltage for which DC multiplication
will be calculated
The different portions for the program perform the
following computations. Statements 29 to 66 divide the p
and n sides of the electric field triangle into equipotential
intervals, DXN(K), DXP(K), and compute the electric field
values, EN(K), EP(K), at the end points of the intervals.
Statements 67 to 95 number these quantities consecutively,
starting on the p side with the first interval and the
first electric field value. Statements 96 to 164 compute
the approximate values of the ionization probabilities
for electrons and holes, dP (k) = XINTN(K) and
dPn(k) = XINTP(K), in each interval, DX(K). The spatial
distribution of the tunneling current, dIt(k) = TI(K), is
computed in statements 173 to 219.
The electron initiated multiplication is calculated
in statements 220 to 241, 251 to 266, and 276 to 287.
TV
Statements 220 to 225 shift the tunneling current by DV
intervals to obtain the free electron current. (In "con-
ventional calculations" TV is set to one DV.) The book-
keeping algorithms which trace the electrons and holes
back and forth across the SCR and compute the DC multi-
plication on each transit are contained in statements 237
to 241 and 262 to 266. (If the available storage is
sufficiently large, the encremental increase in current,
DIE(N) and DIH(N), resulting from carrier multiplication
in each interval can be replaced by DI(K,N), where K is
the transit number and N is the interval number. Combining
I DI(K,N) with the interval number as function of distance
k
then yields the distribution of carrier multiplication as
function of distance across the SCR.)
Statements 242 to 250 and 267 to 275 retrieve and save
the electron multiplication in each interval on the 10-th
and 12-th transits, and the hole multiplication on the Il-th
and 13-th transits. These quantities are used later in the
program to compute the values of ala2, and a3' the first,
second, and third ionization probabilities. (There is
nothing special about these transits. Any other corresponding
set of transits for which the multiplication has achieved
a steady state condition would do as well.)
The multiplication initiated by holes left in the
valence band when electrons tunnel to the conduction band
is calculated in statements 293 to 338. They are essentially
mirror images of the statements handling the electron initiated
multiplication. The total DC multiplication is then the
sum of the electron initiated and the hole initiated multi-
plication. It is denoted by SUM and computed in statement
342. This ends the DC multiplication simulation.
The remainder of the program is devoted to the calcula-
tion of the ionization probabilities, al, a2 a3, and to
the evaluation of the final results of the up to two
ionization per SCR transit noise theory.
The electron ionization probabilities are computed
in statements 345 to 421. The probability of the first
ionization, al, is computed by taking the multiplication
due to holes on the 11-th and 13-th transits and calculating
the electron ionizations on the 12-th and 14-th transits,
allowing no more than one ionization. This is performed
in statements 345 to 356 and 379 to 390. The square root
of the ratio of the first electron ionizations on the 14-th
transit, El, to the first electron ionizations on the 12-th
transit, Dl, is al. If this calculation is repeated starting
with the first electron ionizations on the 12-th and 14-th
transits (the results of the computation just completed),
as done in statements 358 to 367 and 391 to 401, a2 is
obtained. Repeating the calculations once more in state-
ments 368 to 378 and 402 to 412 leads to a3.
The corresponding hole ionization probabilities are
computed in the same manner, but starting with the electron
multiplication on 10-th and 12-th transits. These calcula-
tions are carried out in statements 452 to 519.
Because the two ionizations noise theory does not
distinguish between electron and hole ionization proba-
bilities, it is now necessary to evaluate the "average"
ionization probabilities to be used in the noise theory.
They are simply the square root of the product of the
electron and hole ionization probabilities, as given in
statements 558 to 560.
According to Appendix B, the probability of the second
ionization on the first transit is zero for all reverse
voltages for which the probability of the third ionization
is zero. Hence, all = S(2), the probability of ionization
on the first transit.
Now, that the ionization probabilities all, al and a2
are known, the theoretical expressions for noise resulting
from up to two ionizations per carrier transit across the
SCR may be evaluated. This is performed in statements
561 to 577. This completes the program.
The calculations described above yield the following
ionization probabilities for the 1N751 diode:
Diode reverse voltage all al a2
5.05 0.485 0.909 0.114
5.0 0.474 0.897 0.107
4.95 0.462 0.886 0.0994
4.9 0.451 0.874 0.0926
4.8 0.428 0.851 0.0796
4.7 0.403 0.829 0.0676
Diode reverse voltage all al 2
4.6 0.378 0.807 0.0567
4.5 0.353 0.786 0.0469
4.4 0.327 0.766 0.0381
4.3 0.299 0.746 0.0304
4.2 0.273 0.728 0.0237
4.1 0.245 0.710 0.0180
4.0 0.218 0.693 0.0133
3.9 0.190 0.677 0.0094
3.8 0.163 0.661 0.0064
The range and functional dependence of the above values
are typical of the other diodes employed in this study.
Consequently, the approximation that a2 << al, invoked
in the derivation of the up to two ionizations per SCR
transit noise theory, is justified.
0001 DIMErNlIN XN(600),VN(600),EMAX(C600),DXN(0OO)EN(600,DrP(600),
SP(60)rVP(6OO)EP0(600),t( 900iINITN ( o00), tNTP( 9q0).0X( 900),
VBEPv30)I 910. 3(301),3 IN(0900),A80I900),SSDI0900),
lDIE(1200)s6TH(?,no1,
0002 REAO 13 SFTS,NVREV
003 13 FnRMAT( )
00004 on o00o Nq=SNSE
0005 RF A i, N, l 0 IPMA1Rl,ALPFA2,R2,ALPHIA3,93
000 810 t FRiT(2E .9 i E7')..
0007 REOA 11 PHI,0V,TOV,EV.3VSLOPE
0008 11 FRB"tTF5.FI"' 0,7l.,
0000 REAO 1, 1(EV 2I=l,NVREV)
0010 12 FnP"AT(l6F5 3)
00 1 POINT 1,C(VEAy(1),11.NVPFV
00 1 FORMAT(l1VOEV /,1C(IX,FS.3.2X))
0011 ES'P :.=1.17
0014 X SrP W -Z,
001. l=tbE-19
001* *Il=P TC(2.*EPS3EL*EPSOI/)e((SXNtXNO)/(XNA*WND)))
001 n 000 NV.lVREV
t% .;5ltf*SPT(PI*+VREV(NV))
0019 G2.SLPE/r
0026 EAX=(2.*S99T(PM,*VR9VfV)))/I
0001 IF(F/0y.rE. 1.2Ei) Gn TO 98
0 PIT 97
00 24 STOP
0027 98 XENZ(TXA/(Xm0A+X0 D)*y
0024 VXPp=W-YNN
C CnMPUTE OX AND E FOR 510E
0029 XN(l)sXNN
0030 G( IY V'9
S033 KaVNk/t)V
3 IIFK- 101,99,99
0036 JJ=+1
003 DOXN! 1)=P(DCII)*(1,-9OT(1,-DV/V1(II))
00 7EW(T I)FMlx(II)*(l.-DYN lI)/XH(lI))
o00 V"(JJ)=VN(IT)-DV
0039 XO JJ)NC(IIT)-0rN(II)
000 EM X(JJ)=rNCIII
00 4 100 CONTINUE
00 0N(K+1t )=2.*(VNM-W*DV)/EN(K)
003 GO TO 102
009 11ot CONTINUE
00 5 31.1
0006 DXNJJ3)=22*V0N/FMAXY
C CnMPUTE OX AND F OR P SlOE
00 102 XP()=Xpp
004 VP(1)=VPP
00 9 EMAX(I=)EEAXX
0050 M VPP/OV
0051 IF M;-l) 201,199.199
005? 199 On 00 a =1,I
0o5 DXPcI)=Wpci)*(I.- 7(i,-y-D/VPt(I)))
EP(I)=EMAYi '(I.-o.p(i)IxP(l))
0050 ED"APi(J)1=EF 9I)
0059 200 CONTINUE
0000 Gn TO 202
0061 201 Jl1
0062 OYP(J)=2 *VPP/EMAXX
0i 0 202 CNT0NiiE
0069 DoP(*l )=?.([VPP-M*OV)/EPCMl
0066 203 CONTINUE
C RENIMliEP OX
0067 3-
0069 3 =
0060 I(N D IPJI
8898YO 8?M 1
0071D NJM JAJJJ
007T DMcXNl-nDYN~i)
00310 COiNTIU
C RFgOiPE0 E
00 31 JIs
00832-a
08 o 320 NE IAJO
00P8 ENEr)EPJSN
008o E P! :?o "
1011, 1: 3 k: JO
00 32 +I)EMAXX
00 IF (K C 330,330,322
000 322 JF 3 2
0096 JEErM+K+I
00o0 9 2=3l NE:JE,JEF
0091 E( NE)cEN(JP
0094 JP1J p
009 330 C~:NTNUF
COM 3PUTE A NT XTNTP T ZEGRILO FOR ELE T ALUS
0098 331 On30 l 1, 1
344 KE S TV,,
B!E=(N E)E!(J P)
0100 EsTaEtI+
010 I 8atTEI
3 -K40 E: ESTTS.I Eo TO T41
010 331 0n IA? 0:o1b11
0 00 M RTOStI
010 342 CNTILNUF
C SFT3 INTNIS AN XSNTP'S TO ZERO FOR FIELD VALUES WHICH ARE TOO SMALL
0 13 4 XINT (II)AR
30118 35 I IIKETOP*
F .J-LF. ,T O ) GO TO 345
1 NT Ia It,
23 XINTTPII)O
011 6, 733-1 S.LT.03 GO TO 347
8012 346 TCn1TI UE
C TPIANGLF OPPPnxTMATTON FOR FIRST 0ND LAST NON ZERO XINT.
817 347 SINTNCMIKE1TAB)I(nXES7ARI):ALPHl.tXP C( 4/EI(EftARII)/7
C TBAPECnTO MFTOVn [s3FD Fnh BFPia'ING XI4T07S
0129 7 IFEJT.PJK5T4 .LT. 1 0) GTn 401
1on u00 IoKESTAr.II
XINTNC(J)IDXIJ]*LPMA1I*(EXP.hBIEII)]IEOPC-81/ECJI /2.
01003 K j SPU
0 0 C UTF F ST 1n 24O XXTNTP
2r F( F .T .0) L .E Ol GU T O 3 02
o311 7 INTP(KFSTIA (xKESTA* ALPHA 3*EXP(-B3/, CKETAR)))/2.
01302 XNTP(KEST0P)=(oX(KFSTA)*ALPHA2*EXPC(-. /E(KESTAR) i/2.
C COMPUTE LIST NON 7ERO XINTP
0139 03 IF(F(FESTnp+lI .LF. o ELP GO TO a0
OI XITPCESTOPl) X(l STp P ) ALPHA3EXP(-B3/tEKESTnp)))/2.
0G TO P(
014A 400 X KNTP(K3TOP+I)=OX KRFSTUP+I1*ALPHA2*EXP(-B2/E(KESTOP)))/2,
S USE TRPAP ZOIO APPROSIATION FOR REMAINING XINTPIS W4ILE CONSIDERING FIELD
0113 012 IFFMt TnP KESTAR .LT. 1) GO Tn 501
61isESTOP *I
01450n 50T D=KE TAR,II
0146 JtI-i
7IF(FI) .LE. .E5) CGO TO 451B3
A 450 XINTP(J)D J),'ALH 43(C P(B3 ))EX(3/IFJ))]/2.
S50 451 XTNTP(J)=OX(J)ALPHA2*(EXP(-B2/E(I))+EXP(.62/E(J)))/2.
p19 F T sIy ,a v = ',F5,3)
0150 to FORHT07,'1',7EUrF a lF5.3)I
OIf'T 65i XP^DALP"*l..BltLPHA2.B2,ALPMt3tB3
015 PN1T 5i. PH,6Tv V
l'.F, SSX3IHV = I.F.3)
PPIlnf 18f
0161 P1rT 7. FAX
016 17T FRTT1/,ItX,'PEIK FIELD IN JUNCTION ',EIO.,3X, IVOLTS/CmI)
BRA 32 FmbIT?: IX.'JUIWCTION WIDTH x 'EIO ,l ,' CM ')
C PRINT STATEMENT FnR DYIINTNXINTP,IND E GCES HERE
0165 NTO=TV/DV+I.
0166 NTT.= TO.1
0167 NFO=EV/OV+1.
0168 NETONEO*A
01691 NHOeHV/DVI.
0170 JWOnJ.,IJr*n
0170 Jm0HJJJ-3NH
0172 jMH5Cj-1
C CnMPUTE OISTRTIBITON nF TUNNELING PROBARILITY
C COMPUTE STAT AND STOP POINTS FOR TPAPEZOID METHOD nF APPrOX SUBINTEGOALS
0175 On 502 T=1.II
0176 KESTA =I
0177 BnE=G/E(I)
7AF (RnE LE.50.) Gn TO 503
79 502 CJNTINUF
0180 503 on 50a T=,.IIJ
010 SrE I/E(KFSTOP
DI OIF (IPE LE.O0U. Gn TO 505
0180 50 CoNTIUE
C8S T TUNNELING PPORARILITY TO ZERO FOR SMALL FIELD VALUES
015 505 ITESTAR-
016 I(IIILE.0) GO TO 507
01 7 KKxs+t
01 R Dn 506 T=1,KK
1T1 1$ t-l
5 IF0 F .LE.) GO TO 507
0506 LTINUE
01 3 507 CnNTINUE
0 I9 IIsF5TnP+2
01 9 IF (JO-KESTOP.LE.0 GO TO 509
0196 KK =K+
0107 DO SD K IsIRs
019H TI9II)=0.
020 IF (JJJ-II.LT.0) GO Tn 509
0201 508 NTINUF
0202 509 CnN,TINUE
C TRndldLF APORnXIMATIONM FUR FIRST AND LAST TUNNELING SUBINTEGRAL
8?80 TltK[AA';(0fIT)*F P-/(ETA lP)/)2,
838r1 V STnp*l)-(D( E-T p.1)Ep(-G/KEET)