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

Z 0

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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)