E Lectron Bombarded S Ilico N Avalanche Phododiodes by Enseyin

c»(î>^

Electron Bombarded Silicon Avalanche Phododiodes

Enseyin Selçuk Varol

A Thesis submitted to the Faculty of Mathematical and

Physical Sciences at the University of Surrey for the

Degree of Doctor of Philosophy

December 1978 P ro Q u e st N um ber: 27750215

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Summary

The primary aim of the work described in this thesis was to examine theoretically the interaction of an electron beam with commercially available : silicon devices ,; originally intended as avalanche photodiodes s and investigate experimentally the possible application of these photodiodes as electron-bombarded semiconductor targets.

Theoretically the expressions for static and time-dependent current gains in silicon avalanche photodiodes under electron bombardment were derived from solutions of the diffusion equation and boundary conditions. These expressions have been presented also for estimating frequency lim itations of these avalanche photodiode targets. The limiting factor of the high frequency response is shown to be limited by the n contact layer thickness. The current wave forms induced in the external circuit due to in itial bombarding electrons have been derived and computed.

Using capacitance-voltage curves measured at several frequencies, the characteristics of a number of silicon avalanche photodiodes, such as diode structure, doping profile, electric field profile, series resistance, depletion layer width, avalanche zone, ionization rates for electrons and holes, have been calculated.

To perform the experimental investigations an electron bombardment apparatus was set up for bombarding the photodiode. A key component of this apparatus is a deflection system which deflects the electron - IV -

beam to and fro infront of the photodiode aperture at a very hi^i

■speed. .

Experimental measurements have been obtained for the electron

bombardment conductivity (EEC) of avalanche silicon photodiodes under

controlled avalanching conditions using this apparatus. It was shown

that the EEC gain and the avalanche multiplication of the diode can

be cascaded. The current wave forms produced by in itia l bombarding

electrons in the photodiodes were viewed on a sampling oscilloscope

which has a rise-time of 28 picoseconds. - V -

Acknowledgement s

It is a very great pleasure to acknowledge the constant and unstinting advice, encouragement and assistance given "by my supervisor.

Dr. K. W. H. Foulds throughout this project, and also the assistance given from time to time hy the other members of the microwave group.

Also thanks are due to Dr. T. Hinton for in itial guidance with computer programming and also to Mrs. V. Hinton for some sample preparation.

I should also like to thank the many people and organisations outside the University of Surrey who have helped considerably in this study. .1 should particularly, like to thank Dr. D.J. Gibbons (EMI Ltd.) who provided some development avalanche photodiodes. Helpful discussions and some development avalanche photodiodes were also provided by

Dr. Berchtold, Dr. Micheal and Mr. J.S. Suri from AEG-Telefunken, to whom I give my grateful thanks.

For financial support I am indebted to the Ministry of Education o f Turkey.

Finally I would like to thank to my wife, Annemarie, for her continual support and tolerance during the past four years of research and for careful:typing of this thesis. — • V I — '

Table of Contents

Chapter 1 Introduction 1

1.1. Historical Background 8

C hapter 2 Theory 12

2.1. Energy for Electron-Hole Pair

Creation in Silicon 14

2.2 Electron Energy Dissipation in Silicon 15

2.3. Analysis of the Behaviour of a Reverse

Biased P.N Junction Under Electron

Bombardment 19 2.k, Diffusion of .Holes in the n"*” contact

la y e r 21

2.4.1. Solution of the Diffusion Equation for

the Steady State Case 24

2.4.2. Solution of the Diffusion Equation for

Targets under Intensity Modulated

Electron Beam Bombardment 36

2.4.3. Solution of the Diffusion Euqation for

Targets Under Pulsed Electron Beam

Bombardment 43

2.5* Carrier Currents Resulting from

Electron-Bombardment 55

2.5*1. Contribution due to Holes Created in

the p Layer 56 - v i l -

: Essë. 2 . 5 . 2 . Contribution due to Electrons Created

in the p Layer f 58

2.5*3. Contribution due to Holes Diffusing

from the n**" Layer 59

Chapter 3 Characteristics of Silicon Avalanche

Photodiodes 62

3 . 1 . Charge Multiplication Mechanism 63

3 . 2 . Io n iz a tio n R ates 63

3 . 3 . M u ltip lic a tio n 65

3*4. P-N Junction Avalanche Photodiode

With Guard Ring 7 O

3*5* Depletion-Layer Capacitance 78

3*6. Sample Characteristics 8 l

3*7* E rro rs from th e C-V P lo ts 96

Chapter 4 Experimental Arrangement 97

4.1. Electron Bombardment Apparatus 98

4.2. Deflection %rstem 102

4 . 3 * Specimen Holder no

4.4. Signal Detection II 3

Chapter 5 Experimental Results and Discussions II 5

5*1* Experimental Procedure, Results and

Discussion for EBC and Avalanche Gain 117

5*2. Current Wave Forms produced by i44

Electron Bombardment - V l l l -

Page

C hapter 6 C onclusion l6k

References 167

Appendix I a. Computer Program to Calculate

M ultiplication Factors for Hole

In je c tio n 175

h . Computer Program to Calculate

M ultiplication Factors for Electron

In je c tio n 176

Appendix II Computer Program to Plot Current

Waveforms due to Electron Injection 1 7 7 - 1 -

CHAPTER 1

Introduction

Interesting devices may be designed making use of the interaction

of an electron beam with a solid-state junction. These devices are

called electron-bombarded semiconductor (EBS) devices, which are

essentially semiconductor diodes in the same envelopes as modulated

electron beams and appearing on the market as radio-frequency amplifiers

and modulators, switching elements and signal-processing devices.

The idea is not new - an EBS device, after all, is just a diode illuminated

by a high-energy electron beam instead of light. These diodes are

called electron-bombarded semiconductor diodes when illuminated by

high energy electron beam and called photodiodes when illuminated by

light. The diodes employed in our measurements were fabricated to be

used as photodiodes. Therefore for convenience throughout this thesis

they w ill be referred as photodiodes although used as an electron-

bombarded semiconductor diode.

The conductivity in such diodes, may be altered by bombardment

with high energy electrons and this process is referred as electron bombardment conductivity (EBC). The basic phenomenon underlying this

effect is the creation of electron—hole pairs in the semiconductor in

a manner similar to the generation of such pairs by light quanta in photoconductivity. Because of the usually high energy of the bombarding electron in EBC, (for example several thousand electron volts or higher). — 2 -

a single electron may generate hundreds of electron-hole pairs before

its energy is dissipated in the material. In the case of photoconductivity,

by comparison, the energy of the incident light quanta (for example two

to three electron volts) is usually of the same order of magnitude as

the band gap of the material, so that no more than one electron-hole

pair is created per absorbed photon.

Although it is frequently found that a material with a high

photoconductive quantum gain also shows a strong EBC effect, it should

be kept in mind that the phenomena of EBC and photoconductivity differ

in specific aspects. In photoconductivity the free-electron carriers,

for example, may be generated selectively from the valance band to the

conduction band or from impurity levels to the conduction band,

depending on the wave length of the incident radiation. In EBC, where

the high-velocity bombarding electrons transfer energy to the lattice

atoms by collision, such a selective effect cannot be exercised. In

addition depending on the electron-beam or photon energy the depth of

the penetration of electrons or li^ t may be markedly different. The depth,

of the region of generation of electron-hole pairs with an electron beam

can be varied at w ill by means of the accelerating potential applied

to the electron source, '

The phenomenon of carrier production has been used^drirst as the basis

for, semiconductor radiation detectors that permit the energies of hi^ily

energetic particles to be determined accurately From the investigation

of such detectors it has been found that when a sufficiently energetic electron penetrates a semiconductor, it creates a number of electron-hole pairs that is proportional to the initial energy of the electron - 3 -

In other -words, bombardment of a semiconductor by an electron of

incident energy K results in the creation of an average number of

carrier pairs given by

°m - ° Ë (1-1) p a ir wliere is the energy required to create one electron-hole pair.

From the electron bombardment data of silicon E . is experimentally

( 2 ) found to be constant and equal to 3.5 - 3.6 eV . These values fo) are predicted on theoretical results by Shockley . This shows us

that bombardment of silicon with 15 keV-electrons results a current

amplification in the conductor of approximately 4000 or more. Here

gain is defined as, the ratio of increase in diode current to bombarding

current. Should a higher diode gain be desired, it can be obtained, by raising the beam potential. However, a second mechanism exists which, for a properly prepared diode, allows the gain to be increased without increase in beam potential. This is the mechanism of internal avalanche multiplication which occurs at relatively high diode bias voltage This suggestion was made by L.G. Wolfgang et. al. and reference to it has again been made in a paper on microwave / ^ V amplifiers by Silzars et. al. . Avalanche multiplication has the effect on increasing the normal diode gain by a factor M so that the total gain in silicon becomes:-

(1 -2 )

The factor M is a rapidly increasing function of bias voltage, ^here M - 4 -

can be of the order of 100 in a suitable diode. Therefore a gain

of the order 10^ or more is possible. It is thus seen that the

electron bombarded conductivity and avalanche effect together is a

very high-gain phenomenon.

Another important question raised is related to the potential

of such a device to provide high-gain with a very fast transient response. Experimental measurements of electron and hole velocities

in the drift region of a silicon detector have indicated that the velocities tend to saturate towards a fixed value as shown in Figure (l.l)

and Figure (1.2) The results obtained by C.B. Norris and

J.F, Gibbons for the saturation drift velocity of electrons in silicon yields a value of 10^ cm/sec and the field at which the velocity saturates is about 30 kV/cm. For holes the saturation velocity is

8 X 10® cm/sec and the field is 100 kV/cm

The minimum response time.of a photodiode illuminated by h i^ - energy electrons is therefore equal to the depletion width (typically

10 microns) divided by the saturation velocity. This is equal, to

10 X 104/iQ 7 - iglO sec, which indicates that the conductivity induced by the electron bombardment and avalanche effect is not only a high- gain phenomenon, but is also very fast. This fast response,time, and the current gain mentioned in equation (1.2) can only be achived, when an avalanche photodiode has been* fabricated in such a way that all carrier pair creation occured in the fully depleted region of the device. Usually the most common structure employed on avalanche + + photodiodes are the n - p - p guard ring design which is based on - 5 -

(9 ) the model of Biard and Shamfield . Therefore carriers are not only generated in the high-field region, hut as well in the n front contact layer. It is thus apparent that the transport of carriers by diffusion throu^ a low-fi eld n region is involved. Since diffusion is a slow process, the response time of the avalanche photodiode would increase and the current gain would decrease due to bulk and surface recombination. For these reasons, the main goal, of this study is to investigate theoretically and experimentally the properties of silicon avalanche photodiodes under electron bombardment and predict the gains due to electron-bombarded conductivity (EBC) under several electron beam potentials and the avalanche multiplication M for different diode bias voltages, frequency response and risetimes that are obtainable, and show under what conditions the optimum performance may be obtained.

In Chapter 2 of the present work, we calculate theoretically the expressions for static and time-dependent current gain in silicon avalanche photodiode targets under electron bombardment, from the solution of the diffusion equation for different in itial and boundary conditions. These expressions are useful for estimating the effect of the n front contact layer on the current gain, frequency response and current pulse shapes observed in the external circuit. Current waveforms for different avalanche multiplication factors are also derived and plotted.

In Chapter 3 the characteristics of avalanche photodiodes are reviewed. From the capacitance-voltage (C-V) measurements, the density and the electric field profiles in the depletion regions for the 3 • Q) •H k u •HU t Û «Mo § •H •P in Ü tn 8 k eg w eg00 1 1 § o •HH OQ i — G CO m •H CO § 5t= I ■P s d «Po

i •HÜ O H ÜJ g

CM m m

I ' I"I I I % li in

S § o I •HtH to •HCQ C Î •H *o 03 m

silicon avalanche pTaotodiodes used as targets in our experiments

are calculated.

A description of the electron homhardment apparatus used to bombard the semiconductor samples is given in Chapter 4. Also the

deflection systems to deflect the beam to and fro in front of the

sample effective area are shown.

In Chapter 5 we give and discuss the experimental results obtained, and in Chapter 6 conclusion and recommendations for future investigations are presented.

1.1, Historical Background

A very good: review on EBS devices is given in a paper by

Silzars et. al. According to their historical survey, the earliest recorded experiments of conductivity changes in a semiconductor due to electron irradiation appear in a paper by Kronig published in 1924. The first systematic work investigating conductivity changes due to electron bombardment of semiconductors was started in the late

194 o *s. at Philips Laboratories, Hew York. Hittner reported, in a (12) V . paper published in 1948, work on bombarding polycrystalline selenium photoconductors with electrons up to 2 keV in energy. Bittner recently deposited a manuscript in the Archives of the American

Institute of Physics describing his and his associates work. During the period that work was being pursued at Philips, a number of patents

(l4) issued to Bittner and his associates. The Philips work can be thus considered to be the pioneering work on the electron - 9)-

■bonbardment of semiconductors#

Almost at the same time when the work was carried out at Philips,

experiments were being conducted at the Bell Telephone Laboratories

on the effect of energetic electron interacting with insulators and

semiconductors,. No papers were written by the Bell Group on their f IT ^ EBS-related work; however, one patent resulted .

In 1950 work was initiated by a group at the University of London under the direction of Ehrenberg, apparently as a result of the Philips work. In a paper published in early 1951» results of bombarding both (1 8 ) selenium and copper oxide at beam energies up to 90 keV were reported

The work of this group continued for several years although it does not appear that any device related work was pursued.

Starting in the early 1950*s Moore of the RCA Princeton . (20 Laboratoiy worked on the EBS-related phenomena. One paper was published (21) (22) and two patents on devices were granted to the RCA workers ’

The work was bombarding point contact diodes biased in both the forward and reverse directions, in the low-field region. It was also reported that some high-field effects were seen. It was noted that bombardment caused a significant difference in current flow in a germanium point contact diode.

It appears that the first outside funded EBS effort was conducted at the Nuclear Corporation of America for the U.S. Air Force Rome Air

Development Center, in the later 1950*s. The work was concerned with the development of a practical EBS modulator device, sim ilar to - 10 -

present day devices.

The first work to he published describing detailed work on EBS devices was by Brown and his associates, covering work done at the IBM

Yorktown Laboratories during the early IpGo's. Brown in I 963 reported achieving 300-V output across a 9^-0 load with a rise time of 4ns. He also suggested a design for a deflected beam EBS ançlifier which would have a 300-MHz bandwidth.

In approximately I 967 Norris and Gibbons at Stanford University, supported by the Joint Services Committee and later by the Army Evans

Laboratory, initiated work which resulted in a renascence of EBS effort in the United States. He developed a simple but very useful theory, describing the design of many interesting device configurations, and demonstrated the feasibility of many of these configurations (24),(25)^

This work provided a strong foundation for the renewed EBS interest by a number of organizations and especially for the work started in

1968 by the Watkins-Johnson Conqoany (26),(27)^ ^ number of other

Laboratories were actively investigating EBS devices apparently basing their work on what was being demonstrated at Stanford (28),( 2 9)^

One of these efforts was the studying of the application of the EBS principle to some of the future m ilitary electronics problems by the

Army Evans Laboratory, which ended in 1971 when a funding crisis existed in U.S. m ilitary supported research and development.

The electron bombardment df semiconductors was not used only to develop new electronic devices, but also used as a tool to study the carrier transport properties (drift velocity, trapping, etc) of - 11

semiconductor materials • This new method is known as the time-of- (30) fli^ t technique, first introduced hy Spear to measure carrier m obilities in Se, and has since been sue ces s fully us ed to measure high-field properties of some other materials. Using this technique, (3 1 ) Ruch and Kino have measured the electron velocity, diffusion, and trapping coefficient of electrons in GaAs as a function of the electric field. Several other authors (?)»(32),(33) j^ave employed the technique to measure the transport properties of electrons and holes in silicon and CdTe The data in Figure (l.l) and Figure (1.2) is measured using the tim e-of-flight technique by C.B. Norris and J. F. Gibbons. - 12 -

CHAPTER 2

THEORY

We begin this chapter by giving a general picture of how electrons interact with silicon and the first two sections look for various phenomena such as the average rate at which electron-hole pairs are created, electron penetration and the energy dissipation profiles, in the light of the understanding obtained from well established theoretical and empirical relations. When single crystal silicon is bombarded by high energy electrons, electron-hole pairs are created at the average rate of one pair for every 3.6 electron volts of kinetic energy initially in the bombarding electron. This well established value w ill be used throughout the theoretical and experimental work in this thesis.

Section 2.3. describes the analysis of a reversed biased diode. Under the reverse bias a depletion layer is established in a relatively weakly doped region bounded by highly doped end regions of this n"*" - p - p ^ structure.

In section 2.4.1. an approximate linear theory for the current gain of the reverse biased p-n junction EBS diode is given using ~ the dif fusion .equation^ derived ih section 2.4. An earlier theory of static current gain in p-n junctions, employed a simplified constant rate of incident electron-energy dissipation per unit length up to /

— 13 •“

the average range of electron penetration smaller than the thickness

+ . of the n front contact layer. This section describes a more

general analysis of static current gains for energy dissipation profiles mentioned in section 2.2. The theoretical expressions^present the

effects of the n layer thickness,surface recombination and bulk recombination on the current gain. We see from the theoretical results that the dimensionless parameter K = SL/D is an important quantity in the calculation of the targets efficiency.

In section 2.4.2. approximate expressions and calculations are presented for estimating frequency lim itations of p-n junction

EBS targets, under intensity modulated electron beam bombardment.

Calculations indicate that the depth of the junction should decrease with increasing frequency for preservation of constant dynamic gain.

Time-dependent current gains are calculated in section 2.4.3. for different boundary conditions in the n"** layer using the time- dependent diffusion equation. From the graphs plotted it is possible to estimate the time required to empty the carriers from this region.

It takes carriers on the average a time t = vd^/8D to diffuse through an n^ layer of . thickness d. Carrier I diffusion then can ' lim it the rise time of the photodiode.

In section 2.5. the carrier currents resulting from electron bombardment are investigated using Ramo-Shockley principle * The contribution of electrons and holes are separately calculated under avalanching conditions. In our model we ignore the effects of — l 4 —

diffusion in regions of high electric fields, which, leads to

; the discontinuity of. carrier distribution shown in Figure 2.13

and 2.14. Also we assume that thé pairs are created instantaneously.

2.1. Energy for Electron-Hole Pair Creation in Silicon

To calculate the gain in a semiconductor for specified conditions

of electron bombardment, it is necessary to know the average rate at . (o ) which electron-hole pairs are created. The theory given by Shockley

predicts that the electron-hole pairs are created at the average

rate of one pair for every 3.5 - 3.6 eV of initial kinetic energy

in the bombarding electron and the value is independent of the in itial

electron energy. According to this theory the energy loss to the

material by the bombarding electrons is due to two scattering processes,

each having two constants:-

i) Generation of highest energy or Raman phonons, energy

E^ and mean free path L^.

ii) Electron-hole pair production, threshold carrier energy

E. and mean free path L.. 1 ■ 1 . - . ; ...

E^ is determined from neutron scattering data and for silicon E^ = 0 063 eV (3 5 )' The other constants E^ and = r are chosen

to fit data on quantum yield for photons using the formula

' Q = 3 - 2 exp (Eg + 2E^ - hv) / 2rE^ (2 .1 ) - 1 5 -

For silicon this gives E. - 1.1 eV which is equal to the energy hand gap E and r = 17.5* S

The model considered hy Shockley then predicts an energy per pair for ionization hy high-energy particles of about

2.2E.+rE^ (2 .2 )

This gives, a value of 3.5 - 3.6 eV for silicon. In contrast to some of the experimental results CVavilov obtain E . = 4.2 ± 0.6 eV. p a ir ’ ( 23 ) while Brown finds that - 4.7 eV), this value is in good agreement with other measured values, (Koch et. al. obtain

Epair ” 3'53 ± 0.07 eV, Guldberg et. al. and Norris find a value of 3.6 eV). Therefore we use the value 3.6 eV in our work.

2.2. Electron Energy Dissipation in Silicon

To determine the number of electron-hole pairs created at various locations in the material, one has to know the energy dissipated by the electrons at these points. As the electrons move away from their point of origin they lose energy to the material as they go and as well change direction. They slow down continuously until their in itital kinetic energy has been completely exhausted. The energy dissipated and the range of electron penetration R, all depend upon the incident beam energy E^ and the material under bombardment.

If (3.6 eV) is the mean energy required for an energetic electron to create one electron-hole pair, and the electron beam of — i6 —

current density i.^ suffers a mean energy loss of dE in penetrating a layer of material dx thick, the number of electron-hole pairs created per second per unit volume in that layer is

V-) = f (2.3) p a ir

The average rate of energy loss dE/dX is known as the stopping power of the material. The basic theoretical methods used to calculate the stopping power have been given in considerable detail by Lewis

Spencer who combined the results of energy loss theory of Bethe

The mathematical details of the theory involves computer solution of the transport equation which is basically a continuity equation. The other approach to a numerical solution to this continuity equation is the Monte Carlo simulation technique, which is probably the most flexible and convenient one. The use of this technique in electron scattering problems are given by Bishop We will not consider the mathematical details here. The energy distribution function calculated by this theory can provide an analytic expression for electron-hole pair generation function using equation (2.3). This electron-hole pair generation function can be approximated by a third order polynominal which can be prepresented as

■ Go(x) - E E .. (a + P § + Y + <5 §3-) (2 .4 ) p a ir - IT -

where

a - 0 .6 Y = - 12.4

ê . = 6 .2 1 6 = 5 .6 9

The gain parameter \/^pair total number of secondary

electrons generated hy each incident electron that penetrates up to a distance R into the semiconductor. From the experimental results

of Billington and Ehrenberg and Makhov the electron penetration R into silicon can be shown to take the form

E ,= 0.025 ly 1-65 ( 2 . 5 )

where is the voltage of bombarding electrons in keV and R is in microns.

This electron penetration depth is of course conditional, and we define the range R as the distance after which 99% o f th e impinging electrons are absorbed. The residual 1% of the initital electrons still have their initial energy and w ill thus create 1^ > of the total number of electron-hole pairs generated. The range just defined is the maximum range and valid for electron accelerating voltages up to 50 keV.

Figure (2.1) shows the electron—hole pair creation profile vs penetration plotted using equation (2.4) for several electron bombardment voltages in silicon. ir\ en o\

CO o

en Ll I Ll -JO O cr % CL VO

Z Z ^ 00 3 cr — — I— I— ( / ) o< UJ LU _ l CC LU O

Csl en

CO o 00 O

noiJivaao aiva - 19 -

2.3. Analysis of the Behavior of a Reverse Biased P-N Junction

Under Electron'Bombardment.

The diode model analysed is shown in Figure 2.2. A reverse bias is applied to establish a depletion layer in a relatively weakly

doped region bounded by highly doped end regions of this n - p - p

structure. The depletion layer can extend throughout the entire p region upon application of the reverse bias. The front n ’*’ la y e r o f the diode is irradiated by an electron beam with different acceleration v o lta g e s .

The solid curves in Figure 2.2 indicate the shapes of the

electron-hole pair creation profile and we assume that the pairs are

created essentially instantaneously. The dashed curve is the electric

field profile established under reverse bias.

The carrier transport in the highly doped n**" region occurs by

diffusion and in the depleted layer carriers w ill be rapidly separated by the strong electric field, with the result that holes drift in the positive X direction and the electrons in the negative x direction.

The analysis assumes that in the n layer only hole transport contribute to the current under the reverse bias.

The carrier transport in the depletion region is by the drift mechanism due to the strong electric field. The motion of both electrons and holes causes an induced current in the external circuit according to Shockley-Ramo theorem. ^ 20 -

IT\

cn 0) I §

r 4 ITS r § OJ a -S o

M ■p (H o

ir\ 03 0 •H 1 Ü CQ

OJ OJ 0) rH bO •H

•Hnd -P O CJ

+ '■ , In order to investigate the influence of the n layer on the performance of EBS devices, we w ill first consider the diffusion process in this layer. This will then help us to calculate the effect of this layer on the gain, frequency response and the response time of the diode,

2,h. Diffusion of Holes in the N*** Contact Layer

Let us consider a thickness dx and a unit cross-section of + . the n layer of the diode as shown in Figure 2.3

conduction hand

N(x ) E c X + dx donors

n type of silicon

EV valance hand

Figure 2.3 Schematic illustration of the

continuity equation. - 22 -

The volume of the region is dx, the hole concentration in it

is p(x,t), and the total number of holes is p(x,t) dx. A change in the number of holes in the region under study in time dt may be written as

' {p(x,t + dt) - p(x,t)} dx = 1^ dxdt (2.6)

The change occurs: as a result of the following three processes

Firstly, in time dt the bombarding electrons generates

Ggdxdt holes.

Secondly, the change in the number of carriers as a result of bulk recombination will be -(p/s) dxdt, where ç is the life-tim e of holes in n"^ region. The carriers life-tim e may be defined as the mean time spent by excess electrons and holes in the conduction and valance band. They actually depend on the care taken in the crystal growth and do not usually exceed a few hundred microseconds. But times can be changed by electron bombardment, since electron irradiation introduces energy levels in the band gap. In electron bombarded n-type silicon, the b^AViour of the life-tim e as a function of bombardment is given by Wertheim He found the life-time decreased to 50 nanoseconds.

Thirdly, N (x,t) dt holes flow into the region across the boundary defined by the plane x, in time dt. N (x,t) is the hole flux in (cm^ sec^), into the region in the direction of increasing x. - 23 -

In this same time, N (x + dx,t) dt holes flow out across the

boundary defined by the plane x + dx. The change in the number of

holes in the given volume as a result of the difference between the

incoming and outgoing flux w ill be equal to

{N(x) - N(x + dx)} dt = dx dt (2.7)

The total change in the number of holes in the given volume

taking all three processes into account will be

|£ dx dt = (Gq - 1^ - £) dx dt (2.8)

The hole flux can be expressed as N = the current density

being expressed in the case of only diffusion as

H (2 -9 )

where D is the diffusion coefficient and e the electronic charge

consequently, equation (2.8) can be written as

It " “ f + G q (x.t) ( 2 . 1 0 )

This equation is known as the continuity equation, which is the — 2h —

fundamental equation used in determining the expression for static

and time-dependent current gain in the n layer due to the creation

of holes under various types of electron bombardment. This continuity

equation now w ill be solved for the following cases:-

a) For the steady state where the diode active area is

under constant electron bombardment, so that the excess carrier

concentration in the region does not change.

b) The intensity of the incident electron beam is modulated

sinusoidally,

c) Short electron pulses creating excess carriers in the diode.

2.U.I. Solution of the Diffusion Equation for the Steady State

Case

In the highly doped field-free n'*' region of the diode, for the steady state case, 3p/9t = 0, the hole density p per unit length IS obtained from the following diffusion equation:-

Q 2 D “ "f + Ggtx) = 0. , for x

(2 . 1 1 )

D 1^^ - “. = 0 , for x>E - 2 5 -

We assume that the rate of pair production G q (x ) given by equation

(2.4) is changing with distance from the diode surface to a depth

R and becomes zero.

The solution of the differential equation (2.11) for region

xR for the static case are:-

{a+e|-+Y |2'+'5|3-+66 + 2y ^ , , X

(2 .12)

( ^ ) - ( ^ ) ) P jl = Ce + Fe , x>E

Where L = /D ç" is the diffusion length and A, B, C, F are

constants which can be calculated from the following boundary conditions

a) at the vacuum-diode interface (x - O)

where S is the surface recombination velocity.

In the case of a free surface which has no electrode attached, the total current flowing through this surface must be equal to zero. 26 -

However, an equal number of electrons and holes may reach this surface

from the bulk of the semiconductor and recombine there. The number

of holes recombining per unit area per second Sp, is equal to the

number of holes arriving at the semiconductor surface per unit area

per second. The proportionality factor S is the surface recombination

velocity in cm sec^.

The surface recombination velocity S depends on the manner

in which the semiconductor surface has been prepared. Treatment

of the surface with special etchants can reduce the value of S.

However, after a time, as a result of surface oxidation and gas absorption,

the surface recombination velocity increases and reaches a steady

v a lu e .

b) at the junction (x = d)

p = 0

This boundary condition indicates that as soon as the holes arrive to the junction which is at a distance d from the vacuum interface, the strong electric field sweeps the holes into the depletion region.

c) at a depth R

p and -|^ are continuous. - 27 -

The rate of pair production created hy the electron beam becomes zero at a depth R, and the number of carriers crossing the

surface at x = R must be equal to conserve the charge neutrality

at that point. Also the diffusion currents would be equal, since the diffusion coefficient is assumed to have the same value throu^out the n^ region.

There are now two cases to be considered for the solution of the diffusion equation. These solutions would be for ranges of electron penetration smaller and greater than the thickness of n^ l a y e r .

CASE I R < d

When the range of electron penetration is smaller than the thickness of the n layer, the following boundary conditions are used to calculate the constants A, B, 0 and F:-

1) D = S P j a t X = 0

2) P j j = 0 a t X = d

(2.13)

3) Pj = p^^ at X = R — 28 —■

The number of holes collected at the junction per unit time

for each incident electron accelerated to a potential is the

gain of the diode, and the amount of carriers crossing the boundary

X = d i s

X = d

From the expression for the density p^^(x), the expression for

3Pjj/9x at X = d is derived and the gain G is calculated as

r - ^ ,L ' {(Y-KZ) sin h R/L + (KY-Z) cosh E/L - X} . ®pair ® K sinh d/L + cosh d/L U.J.4;

where

X = K (a + - I (e +

Z = I (g + 2y + 36 + ^ ) -'29 -

If all the pairs produced, hy the beam vere collected then

we /would expect a maximum diode gain of

°m- “ Ë p a ir

From equation (2.14) we see that the maximum diode gain is

reduced by a factor. The loss of gain of the diode, as compared

to the theoretical maximum gain was assumed to be due to two processes:

surface recombination and bulk recombination.

For a simplified pair creation profile the gain can be obtained

by setting a = 1,$ = y = 6=0. In this case it is assumed that

the rate of pair production created by the electron beam, is constant with distance from the diode surface to the depth R and then

suddenly becomes zero as shown in Figure (2.4).

n la y e r

p a ir

x=R x=d

Figure 2.4 Simplified pair creation profile, - 30 -

This creation profile is used hy Brown to calculate the

gain of the diode. When a simplified model is considered the equation

( 2 . i 4) reduces to the expression (3) in reference (23) as.

r - r L {sinh B/L + K (cosh R/L - l)} , . m R K sinh d/L + cosh d/L U.l>;

Expanding sinh and cosh terms equation (2.14) can he written as : -

... V--

where - 31 -

When the diffusion length is greater than n*** layer thickness

and the penetration depth, (d < 0.3L), the approximate diode gain

over which these expressions will be correct to within is given

(2 .1 7 )

* ^1- (^1 I ^ sa /f - ^2 I?)

CASE I I . E > d

When.the range of electron penetration is greater than the depth of the n layer, the electron-hole pairs are created not only in the n layer, hut as well in the p drift region. Therefore the total gain is the sum of the gain in both layers. The gain in the

^ l^yGr is calculated from the solution of the diffusion equation using the following boundary conditions

9p.

(2 . 1 8 )

2) Pj = 0 at X = d — 32 —

The exact solution for the gain due to the carriers generated in the n layer is given hy the following expression for the measured electron-hole pair creation proflie:-

^ ,L v {(Y» - KZ*) sin h d/L + (KY» - Z») cosh d/L - X} V " ^m ”■ K sinh d/L + cosh d/L

where

K -=

Z* - I (6 + 2y | + 36 | z +

For the simplified pair production profile shown in Figure

(2.5) (a = 1, g = Y = 6 = 0) this expression takes the form

^ ^ {sinh d/L + K (cosh d/L - l)} fn V " % ¥ K sinh d/L + cosh d/L------. —: 33 -

n la y e r p la y e r

p a ir

0 x=d x=R

Figure 2.5 Simplified pair creation profile

Expanding sinh and cosh terms equation ( 2 . 19) can he written as:-

4 * D * Î2- ^ (a^ + |i tp + ------} 1 + Sd/3D , > (2 .2 1 ) 1 + Sd/D ------3U -

where

-2 = It (f - !f - #+#(#)') = & (! + #i + # + #)')

3 = ir (# + ## + K )- # )') ^3 = IT 'f - f f + # f+ # )')

When the diffusion length is greater than the n**" layer thickness and the penetration depth, (d <0.3 L), the approximate solution for the gain in the n"*" layer is given hy,

(2 . 2 2 ) - 35 -

Ihë contrihution to the gain due to electron-hole pairs created in the depletion layer is calculated from the portion of the incident energy which falls within the p region. Therefore,

S = a ^ Gg(x) ax , . (1 - I aj) (2.23)

The total gain is the sum of the gains in hoth layers.

From the solution for ranges of electron penetration smaller and greater than the thickness• of the n layer, one can deduce that the static gain of a p-n junction under electron homibardment is a function of the following parameters:-

l) n**" layer thickness of the diode, d.

2) Penetration depth, R, of the incident electron into the

Semiconductor and to the shape of the electron-hole pair generation p r o f ile

3) To the diffusion length L = which determines the loss due to hulk recombination, and the surface recombination velocity which determines the loss of carriers at the vacuum interface of the diode. . A very important quantity is the dimensionless parameter — 3 6 —

Energy required to create one electron-hole pair, E . p a ir which is a characteristic value of the material under bombardment.

In Figure 2.6 theoretical curves for the diode current gain 6 vs junction depth are presented using different values of surface recombination velocities • The dashed curves represent the gain for the sinçlified electron-hole pair creation function.

2.^.2. Solution of the Diffusion Equation for Targets under Intensity

Modulated Electron Beam Bombardment

We next consider the frequency lim itations of the semiconductor targets under intensity modulated electron beam bombardment, and it w ill be deduced from the following calculation that these frequency lim itations are similar to those results from the decrease of the transport factor in the base of a bipolar transistor at high frequencies.

Let us now assume that the electron-hole pair creation in the semiconductor is not only position dependent, but as veil changing in time with a con$)lex time function f(t) = e^^^, where o) = 2?rf and- f is the frequency of modulation.

The hole density p per unit length for region x < R and

X > R are now obtained from the time-dependent diffusion equation

(2 .1 0 ) as - 37 -

oS OJ 0 <ü S’ 1 i m

"ë

en +3I

•He «

g a •H ê IT\ -§ o OJ ro •H A

VO OJ 0) o bO k o m r—I X nreg GpoTg 38 -

p . = ^ ' ® pair

+ ^ , x

(2 .2 4 )

( ^ ) : ( ^ ) Pjl . - Ce + Fe , x>E

where

^ (1 + i(0Ç )l/2 ’ ^

The only difference between equation (2.24) and equation

(2.12) is the diffusion length. In this case the diffusion length

L is decreased and must be transformed as

L (1 + iü)Ç)^/2

Now there are again two cases to be considered.

CASE I R < d

Using equation (2.16) and making the above mentioned transformation for the diffusion length, for range of electron penetration smaller than the depth of the n layer, the dynamic current gain G t '- 39 -

is given by the following expression:-

Where G is the static gain given by equation (2.17) and.

= 1 /D+Sd/3\Z 1^ ^D+8d/5^ R^>^Da2 + SRb2^2 ^ /Dag + SRbg,, ” ¥ T+Sd ” 12 ^D+Sd ' “ d^ Dai + SRbi' ” 3 a i + SRbi^"^

»■« ë <1 ^Dai igrS> + SRbi‘

The frequency f^ at which the magnitude of the time dependent current gain is A percent smaller than the static gain, is given by the following relation

CASE 2 R > d

Using equation (2.21) and making the above mentioned transformation for the diffusion length, for range of electron penetration greater than the depth of the n"** layer, the dynamic current gain G^, due to , - :4 o -

electron-hole pair creation in the n layer is given/by the

expression:-

'^t = > (2.27)

where G is the static gain given hy equation (2.22)

e' , = 2B' - A' , 0' = IDÇ A'

(y| - yg) (Da^ + Sd-b|) - (Da^ + S d t ^ - (Da' + Sdt.') B' .= ------:------D + 8 d - ^ (a^ - B{)

y, (DaJ + Sdb ' ) - (Dal + S d t') 1/2 ■' = • (#) D + Sd - (aj^ -

_ 1 D + Sd/3 , 1 D + Sd/5 “ 2 D + Sd ’ ^2 24 D + Sd 4 l , —

The frequency f^ at which the magnitude of the time dependent current gain is A percent smaller than the static gain, £ is given hy the following relation

The degrading effect of increasing frequency on the gain is a consequence of the decrease of the effective diffusion length

L/(l + iü)ç)^/^ with increasing frequency. This effect is analogous to the decrease of the transport factor 3 in the hase of a bipolar transistor at high frequencies. To avoid excessive degradation of output with frequency, a safe theoretical lim it for the depth of junction can he calculated hy using the ahove relationship.

Figure (2.7) shows the frequency f^ at which dynamic current gain i s 50 percent smaller than the static gain as a function of junction depth when the diode is homharded with 10 KeV electrons which penetrate 1 micron for the two different surface recombination velocities S. - 1+2 -

Figure 2.7 Frequency at vhich Dynamic Current Gain is 50 percent smaller than Static Gain as a Function of Junction Depth

10 KeV

1 m icron

3 10 - i tn I» Ü

I(U

.2 10

0.1 1 10

Thickness of n+ layer (microns) - 43 -

2.4.3. Solution of the Diffusion Equation for Targets Under Pulsed

Electron Beam Bombardment

. - , r •; + ■ ■ Diffusion in the n low field region is a relatively slow

process compared to the transit times in the h i^ field region.

Now our aim is to find out how the n^ layer empties itself to the

adjacent high field region after creation of electron-hole pairs by

an electron pulse at a time t = 0.

We now may attem pt to fin d a s o lu tio n o f th e tim e dependent diffusion equation (2.10) by letting the desired solution be a

sum of terms each of which has the form X(x).T(t), where X and T

are functions of x and t respectively. The idea is to construct functions of this form which satisfy' the differential equation and the boundary conditions. By superposition of these functions, one then satisfies the initial conditions.

On substituting

p (x,t) = X(x).T(t) (2.29)

in the diffusion equation (2.10) we obtain

1 3T(t) ^ 1 D T "It— + Ç = X 3^ (2 .3 0 )

Left hand side of equation (2.30) is an expression depending on t only - 44 -

while the right hand side depends on x only. They can he equal

only, if either one is equal to the same constant, independent of

X and t which, for convenience we shall choose to he: Thus we obtain the ordinary differential equations

(2 .3 1 )

with the solutions

T = exp - (X^D + ~) t

(2 . 32)

X = A s in (Xx) + B cos (Xx)

Obviously X^>0, if the solution is to remain finite for all values of t. The further restrictions on X would come from the in itial and boundary conditions, so that its values now w ill be restricted to an infinite set of discrete positive values. Therefore the most general solution w ill be represented by an infinite sum over the discrete values of X. Also the arbitrary constants A and B must be determined in such a way as to conform with the in itia l and boundary conditions - 4-5 —

We shall now treat examples which have practical importance

to us. A slab of n layer with a thickness d, having an initital

concentration G q (x ) at a time t = 0 due to the electron bombardment

The concentrations at its faces are being kept at zero for t>0, i.e .,

the surface recombination at both faces is infinite. Therefore in

+ ■ . the n region the initial and boundary conditions are.

p = G q (x ) , for 0

(2 . 32)

p = 0 , fo r X = 0 and x = d, at t>0

We see from equation (2.32) that the boundary conditions are funfilled fo r

B = 0 , X, = (2 .3 3 )

Therefore the most general solution is of the form

p(x,t) = A^ exp (- 2^1^ - |-) sin (^ ) (2.34) n= l

The arbitrary constants A^, now, must be determined in such a way as to conform with the initial conditions, i.e ., for t = 0 we must have - 46 -

G«(x) - Z A sin (^~) , for 0

A solution may be obtained by using Fourier’s theorem,:which • lead s to

= I 0/ sin (SS) ax (2.36)

For the electron-hole pair creation profile G^(x), A^ would be

\ ° ^E E ^ . ) (2 .3 7 ) p a ir where

- 2y + ( (Z&) (2y + 6S | )

- (ct + B ^ + Y ^ + 5 ^ ) ) cos (nir)

The solution of the diffusion equation with these initial and boundary conditions is then given by

2K FKt) = • E - sin (SS) . T(t) (2.38) p a ir n=l - l+T -

where

The amount of carriers crossing the boundary at x = d is

j(t) = i)|| X = d (2 .3 9 )

and from the expression for the density distribution, p (x,t), the expression for 9p/3x at x = d can be derived and J(t) can be calculated as

2RD J ( t ) . = g g -V 2 (-1 )“ . . T (t) (2.1*0) p a ir

The number of carriers lost from the n**" layer in the time interval t = 0 to t = t to the p drift region is the time dependent gain which is given by

2K Gr(t) = E . 0^ ^ n= l -48 —

o r

a “ (-1 )“ = Ë— # 2 + a^ M t) -1) (2.1*1) n= l

In order to understand the physical significance of equation

(2 .4 i ) let us consider the simplified electron-hole pair creation p r o f ile (a=l, g = y = 5 = O) and assume hulk recombination is negligible. When t ->■ «

»<«> - : (ly....) (2 A 2 ) lim t « pair ^ n= l

Since Z (l - cos nir)/n^ = tt^ /4 the gain would be n= l

G(t) = I I (2.1*3) p a ir

When the penetration and the junction depth are equal to each other, it is obvious that the probability of the carriers to diffuse to the surface and to the junction is fifty fifty. Fifty percent of the carriers would be lost at the surface and the other fifty percent would contribute to the current. The term d/R in equation

(2.43) gives the ratio of the carriers created in the n^ layer. - It9 -

For the case of infinite, si^face recombination Yelbcity the solution for the carrier density distribution is shown in

Figure 2.9, from which it can he seen that the carriers initially created at time t = 0 (dashed curve) diffuse and they also recombine.

Figure 2.10 shows thé amount of carriers crossing the junction at

X = d and the amount of carriers collected by the diode. These

Figures indicate that the time necessary to empty the layer is about 2 ns.

S = « 3 n la y e r = 16.3 keV t= 0 d=R=2.6 Microns L=7.1 Microns

t= 0 .1 ns 2

g rH

-P •H

g ft t= 0 .5 ns •H 1 ICO g •H t= 0 .8 ns g Ü t= 2 ns 0 0 d/2 d Distance (Microns)

Figure 2.9 Carrier density distribution at different times for infinite surface recombination - 50 -

X 10

= 16.3 keV R = d = 2.5 M icrons 4000 20 Ç =50 ns

3000

Q c+

2000 10

1000

Figure 2.10 The amount of carriers crossing the junction, J(t)

The amount of carriers collected by the diode, G(t) - 5 1 -

We next consider the problem in vhich there is no flow of particles at X = 0, i.e ., the surface recombination velocity is zero at the surface.

The solution obtained for this example using only the cosine terms in eq u atio n ( 2 . 3 2 ) would be

p(x,t) = Z B , T(t) . cos (t) x) (2.UU) ' n=0 “ “ where

b^ = (2n + 1 ) Tr/2d n

T(t) = exp (-Da^^t - t/ç)

For the electron hole-pair creation profile Gq (x ), would be

®n ) ksn + 1 )tt^ (2.1*5) pair where

n n n n

The solution of the diffusion equation for the particle density with the initial creation Gq (x ) and assuming no loss of carriers due to surface recombination is then given by - 52 -

p ( x ,t ) - jjg _ Z ^2n + 1 )^ • ^ 2d ^ ^ (2.U6)

The amount of carriers crossing the boundary at x = d using eq u atio n ( 2 . 3 9) i s

2E.D J(t) = - Z (-1)“ . . T(t) (2.1*7) V

+ The amount of material lost from the n layer in the time interval t = 0 to t = t to the p drift region is the time dependent gain which is given by

8E.L2 . “ (-1)" I (f) : + 1)w + i*d^ -1) (2.W) n=0

In order to understand the physical significance of equation (2.48) let us again consider the simplified electron-hole pair creation profile

• «> . li. . ^ ^ I <=-‘5)

5mce there is no recombination in the n layer due to bulk and surface recombination all of the carriers would be collected. - 53 -

For the case of surface recombination velocity is zero the

solution for the carrier density distribution is shown in Figure

2.11, from which it can be seen that the carriers initially created

at time t = 0 (dahed curve) diffuse and they also recombine.

Figure 2.12 shows the amount of carriers crossing the junction at

X = d and the amount of carriers collected by the diode. These

Figures indicate that the time necessary to empty the n^ layer is

about 3 ns. Therefore it takes more time to empty the n^ layer for

th is c ase .

8 = 0 3 n la y e r X 107 = 16.3 keV t= 0 d=R=2.5 Microns L=7.1 Microns t= 0 .1 n s

t = l . l ns •H

1 t= 2 .5 ns

•H

0 0 d/2 d

Figure 2.11 Carrier density distribution at different times for zero surface recombination -W-.

G (t)

O . o R

+3 -p CÎ3 h ) LTV 0 a rd o O •H •H ■ "P ü 0 % LfN •O g 0 CM / a CO +3 fd . g ) 0 G +3 - •H U :. M 0 : 0 rH O H O 0 0

üi TO U U 0 0 •H •iH U : u U' cS ( 6 O CM: 0 (H «H O O +3 s ’ O § ■

g ': 5 CM 0 bO ir\ •H

( ^ ) r - 55 -

2.5• Carrier Currents•Resulting from Electron-Bombardment

After the generation of electron-hole pairs hy the electron- bombardment up to a range R in the diode, the holes v ill move to the ri^ t and electrons to the left across the intrinsic drift region, under the action of the applied electric field. The motion of electrons and holes then induces a current at the terminals of the sample. The current can be most simply calculated in circumstances where the electric field is constant so that the carriers traverse the diode depletion region with constant average velocities. The value of this induced current is given by the Ramo-Shockley theorem which states that a charge q. travelling with velocity v between plates separated by a distance W induces a terminal current I given by

I = (2 .5 0 )

Applying this theorem to the holes and electrons generated in the depletion layer we see that the current flows until all the holes reach the p and the electrons to the n^ contact regions.

The electrons w ill disappear in the n**" layer, but in addition the holes created in this layer would diffuse to the junction and enter into the drift region contributing to the induced current at the terminals of the device.

We would now consider the contribution of holes and electrons -5 6

to the current separately.

2.^.1. Contribution due to Holes Created in the p Layer

After the generation of holes with a pair creation profile

G q (x ) in the p layer the holes traverse to the right (positive x-direction) if the applied electric field is in the positive x-direction as shown in Figure (2.13). The whole package of the created holes under the applied electric field moves a distance v^t in time t. Our aim is to find the area under the curves at different times, i.e ., we want to find the number of holes present in this layer after a time t and using Ramo-Shockley theorem to calculate the induced terminal current. In the p layer at t = 0 there a re G q (x ) dx holes. But for t>0 some of the holes leave the region and at t = v^/W there would be no holes left.

The contribution of the moving hole package would then be given by

W-(R-d) — y gq (x ) ta 0 < t <

(2 .5 1 )

< 2- - 57 -

§ *3)

§ •H •ë

•P C •H OCO ■3 A K «H 0 § •H 1

CO I— I CVJ (U g tû ♦H À

a - 58 -

S.5'2* Contribution due to Electrons created in the p Layer

In avalanche diodes which we w ill he dealing with in great detail later in this thesis, the applied electric field can create a very narrow h i^ field region near the junction such that an electron or hole entering this narrow region can absorb enough energy from the field to make an inçact ionizing: collision with a bound electron. For the time being let us only consider any carriers entering in this narrow region would be amplified by a factor M.

After the generation of electrons with a pair creation profile

Gq(x) in the p layer, the electrons traverse to the left (negative

X direction) under the applied electric field which is in the p o s itiv e X direction as shown in Figure (2.14). The electrons entering the narrow avalanche zone would now produce electron- hole pairs by impact ionization and the holes produced by avalanche multiplication would travel to the ri^ t in the depletion region and contribute as well to the current at the terminals of the diode.

Therefore there are two contributions to the terminal current due to electrons initially created by electron bombardment.

i) Using Ramo-Shockley theorem the in itial package of

electrons moving a distance v^t, where v^ is the

velocity of electrons, contribute to the terminal current

as - 59 -

R R-d ^2 = - r ; ^ 0 < t < ( 2 . 5 2 ) V t+ d e

ii) The holes created due to the electrons entering into

the avalanche zone and multiplied by a factor

contribute to the terminal current as

h . R-d G q (x ) dx “ e - l f / 0 < t < d

qv, R R-d I 3 =^“e— / °oW ^ < t < ^ (2 . 5 3 ) d

R M / „ G^(x) dx a+VgCt- — ) h

2 . 5 . 3 . Contribution due to Holes Diffusing from the n Layer

We have already calculated the amount of carriers crossing

the boundary at x = d from the n layer into the p drift region.

The mount of carriers is given by equation (2.4o) f o r 8 = 0 and by equation (2.4?) for S = «. Each carrier entering the drift region - 6o -

OJ - 61 -

has to cross through the narrov avalanche zone and get multiplied

by a factor M. The Contribution of holes from the n^ layer for

diffusion times greater and smaller than the drift times can be

w r itte n as

W J(t - dx for W_ % W f t > 0 h Th

(2.54)

W J ( t -) dx fo r t <

In calculating these current wave forms we assume that the

pairs were created essentially instantaneously with their saturation velocities in the targets under bombardment. Also we ignored the

effects of diffusion in regions of high electric fields which resulted in the discontinuities (sharp edges) in Figure 2.13 and Figure 2.14.

The graphs computed in Chapter 5 for the current wave forms, uses the equations 2.51 to 2.54. In the computation the effects of the t , , n layer is ignored since the thickness of this layer is chosen to be only 0.3 microns. This corresponds also to the thickness of one of the commercially available photodiodes used in our work. 6s —

CHAPTER 3

CHARACTERISTICS OF SILICON AVALMCHE PHOTODIODES

To use avalanche photodiodes to best advantage as EBS targets,

it is important that the user has a working knowledge of their

operating mechanism. Therefore in section 3.1. and 3.2. we first

review the basic avalanche mechanism and discuss some sinple ideas

on the theory of ionization rates without considering the mathematical

details of the best theory given by Baraff

In section 3.3. multiplication factors and breakdown conditions

are given for pure electron injection and for pure hole injection into the avalanche zone. The expression for the multiplication initiated by a mixture of holes and electrons is also given. The e 3q>ressions for pure electron and for pure hole injection are used in Chapter 5 to compute theoretically the multiplication factors and breakdown voltages in order to compare with the experimental results. Two different silicon avalanche photodiode structures are discussed in section 3.4. Their electric field and concentration profiles are shown and in section 3.5* the equivalent circuit of the photodiodes is illustrated.

In section 3.6. sample characteristics of two commercially available photodiodes (electric field, concentration profile, depletion layer width at different bias voltages, ionization rates) are calculated from the C—V measurements for the junction capacitance. - 63 -

These measurements are made using a Boonton capacitance meter and as well using slotted line techniques at different frequencies. In personal communications with the manufacturers these results and their measurements were compared;and accepted to he more accurate than their own,

3.1. Charge M ultiplication Mechanism

We start hy discussing some basic knowlege on the operating mechanism of avalanche photodiodes.

The basic physical mechanism upon which charge m ultiplication depends is that of intact ionization. If the electric field in the depletion region of a photodiode is sufficiently high an electron or a hole can gain energy from the field at a rate faster than they lose to acoustical or optical phonons. Electrons and holes then experience a steady acceleration to higher energies. But this acceleration does not go indefinitely and they give up a considerable fraction of their kinetic energy to a valance electron, thereby raising the valance electron to the conduction band and leaving a positive hole in the valance band. In this case, the original electron and hole have created an electron-hole pair. These additional carriers, in turn, can gain enough energy from the field to cause further impact ionization, until an avalanche of carriers has been created. Below the diode breakdown voltage the total number created is finite; above, it can approach infinity.

3.2. Ionization Bates

An important parameter in charge multiplication is the ionization — 64 —

rate.. This is the probability that an electron or hole w ill have

an ionization collision with an atom, in a unit distance of travel,

to create one electron-hole pair. These coefficients are strong

functions of the electric field. Since the electric field in the

depletion region of a photodiode is a function of position, the

ionization coefficients w ill be as well functions of position^ and

are large enou^ to merit consideration only where the electric field

is highest.

The probabilities that a given electron or hole w ill have an

ionizing collision within a distance dx are (E(x)) dx and

(E(x )) dx, respectively, where and are the ionization

rates. The field dependence of ionization rates are calculated

theoretically by Baraff whose theory involves computer solutions

of the Boltzmann collision equation. Experiment ally the ionization

rates of electrons and holes in silicon have been measured by the use

of photomultiplication measurements on p-n junctions by Lee et. al.

These measurements fit to the theoretical calculations of Baraff.

The results can be fitted over wide range of field to the following ex p ressio n

xXg , = A exp (-b/E(x)) (3.1)

wherej for silicon. - 65 -

electrons holes

A (cm^) = 3.8 X lOG 2.25 x 10^

h (V/cm) = 1.75 X 106 3.26 x 10^

Figure 3.1 shows the experimental results of ionization

rates for silicon.

3.3. Multiplication

Impact ionization phenomenon leads to m ultiplication in

avalanche photodiodes. The region of carrier multiplication in

these photodiodes is restricted to a narrow avalanche zone with width w close to the junction. If I^^ holes flow per second

into the avalanche zone at x = 0, the hole current I^ will increase with distance through.the zone and reaches a value M^I^^ at x = w, where is the multiplication factor of holes defined as I^(w)/I^Q,

Similarly, the electron current I^ w ill increase from

X = w to X= 0 and the total current I would he the sum of hole

and electron currents.

The increase of hole current at W would be equal to the

number of electron-hole pairs added in a distance dx, to the initial

stream of holes. We can therefore write

° “h ^ ^ * V e ^ (3 -2 ) - 66 -

'§ 0 « 10 § •H ■g. N •H 1 10

2 3 4 5 6 T 8 9 10

Electric Field (xlO^ V/cm)

Figure 3.1 Measured Ionization Coefficient for Avalanche

Multiplication vs Electric Field for Silicon

(After Lee et. al.) (50) — 67 -

o r

vhere I — is the total current and is independent of p o s itio n .

From the solution of equation (3.3) vith the boundary condition

that I = Ij^(w) = one can obtain a relationship between the multiplication factor and the ionization rates as

1 W X 1 (o^ - a^) dx'} dx . ' (3.4)

If the avalanche process is initiated by electrons instead of holes from the plane x = W, then the electron multiplication factor M e would be

1 V W = / a exp {- / (a - a,) dx'} dx (3.5) e 0 X ® ^

Instead of just considering that a pure electron or hole current initiates the multiplication, let us consider the. case using a mixture of holes, and electrons. In this situation the accompanying differential equation takes the form ^^l) — 68 “

I f we th en d e fin e

(3 .7 )

“ = Ch + le) ' ( V leo)

and use. the boundary conditions that I^(0) = 1^^ , I^(w) = 1^^,

the solution of equation (3.6) is

W X ( l - k) S exp {- / (o^ - a^) dx*)} dx 1 _ — = 2 2.______M W 1 - k + k exp {- / (a, - a ) dx) 0

(3 .8 )

w w w k exp {- / (a, - a ) dx*} { / a exp / (a. - a ) dx*} dx + 0 0 ® X n e ______W 1 - k + k exp {- / (oL - a^) dx} 0 ^ ®

When a pure hole or electron current (k = 0, 1, respectively) is used to initiate the multiplication, equation (3.6) yield the ' ■ . -6 9 -

equation (3.4) and equation (3.5).

Expressions (3.4, (3.5) 6.8) are often called ionization integrals. The avalanche breakdown voltage is defined as the voltage where and M approaches infinity. Hence the breakdown condition is given hy the ionization integral

W X f OL exp {- / (a. - a ) dx*} dx = 1 (3.9) 0 0 0 . 0

If the avalanche process is initiated by electrons instead of holes, the breakdown condition is given by the ionization integral

W W f a exp {- f (a - ou ) dx*}. dx = 1 (3.10) 0 X G a

The ionization integrals approach unity together. This means that the condition of breakdown is unique and it does not matter if the avalanche is started by holes or electrons or both. To calculate these integrals with equation (3.1), it is necessary to use a computer. A simpler expression was proposed by M iller as

(3 .1 1 ) - TO -

where Vg, are the applied, and breakdown voltages, respectively.

The exponent n is know as the M iller’s exponent.

This emprical relation, which satisfies the boundary conditions

(M = 1, Vg = 0 and M - Vg - Vg^) describes the variation

of the multiplication coefficient; however, it has the disadvantage

that n is not constant and varies in particular with substrate

doping and also with the penetration depth of the incident electrons.

Measurement of the multiplication factor is a useful method

of obtaining the ionization rates.

3.4. P-N Junction Avalanche Photodiode with Guard Ring

One of the most common avalanche photodiode structure, often called the "guard ring" structure, is based on the model of Biard and Shaunfield The guard ring is required to ensure that the field does not rise at the edges of the junction area and cause breakdown.

A device of that sort may consist of a semiconductor structure shown schematically in Figure 3.2. A slab of nearly intrinsic semiconductor is bounded on one face by a relatively thin layer of very heavily doped semiconductor of one conductivity type and on the other face by a relatively thick layer of very heavily doped semiconductor of the opposite conductivity type. Figure 3.2 shows an n* - w - p* structure but n* - y - p"**, p* _ % - n’*', and p* _ y - n* are equivalent alternatives. Ohmic contacts to the heavily doped- - 7 1 -

§ •H «§ g & A tH § o o •H •HH »H° H a w •H O) ra K

ü 0)

■P

« nd I 44 o § •H -PO (U CQ CQ CO O ?4 o

CM rô 0 bO •Hk - 72 -

regions serve as means of applying dc reverse "bias to this strnctnre.

When a reverse bias is applied a depletion layer is established in the TT region of the diode. The electric field distribution in this layer is given by

E(x) ------(W - X ) ( 3 . 1 2 ) ^s

The electric field is shown in Figure 3.3.a and is not constant but falls linearly with increasing distance from the junction, the slope being proportional to the constant impurity concentration

Ng shown in Figure 3.3.b.

W is the front edge of the depletion layer and given by

2 e . V_ W = (-TET) (3.13) £

For this type photodiodes the front edge-of"the depletion layer may not spread out to occupy the entire ir-type volume. The generated carriers in the undepleted volume are then collected by diffusion and multiplied as the carriers diffusing from the n^ region of the diode. They are collected slowly and effect the response and efficiency of the photodiode. (a)

Area / E(x) dx

x=W D istance

s •HÎ IO § o

D istance

Figure 3.3 (a) Electric field distribution,

(b) Doping profile — —

A second common structure is the reach-through avalanche ( 53) photodiode -which were first reported by Ruegg . It has several advantages over n* - n - p* structure, since it combines, to the best extent possible, high speed, high gain and low noise. This

is achieved by separating the depletion region into a fully depleted drift region and a narrow avalanche zone in which the carriers are multiplied. A typical structure of this type is shown in Figure 3.4.

The two diffusions, p-type (boron) and n^-type (phosphorus), that create the avalanche region are carried out in sequence to form n"^ p - IT - p structure. When a reverse bias voltage is applied, the depletion region of the diode just reaches-through to the low concentration tt region when the peak electric field at the junction is approximately 10^ less than that required to cause avalanche breakdown.

The applied voltage in excess of the reach-through voltage,

is now dropped across the total intrinsic width causing the depletion layer to increase rapidly out to the p"*" contact. The fields in the multiplication region and hence the multiplication factor increase relatively slowly with increasing bias voltage above reach-through.

In practice a multiple layer structure with discontinuities between layers is difficult to fabricate. Impurity diffusion during fabrication often leads to finite impurity gradients in the structure.

In this case the impurity concentration in the p region of the - 75 -

-S o •H Q

O§ •HiH •po •HCQ d o o (U < T j O •H T) O +3 O f : A (U ü Ü d rH d CC î> a < 5h A % > A f •H U 0 ê W < ! •p 0 O d u 0 4^ A CQ

n 0 k 1 • H À - t 6 -

+ + n — p — TT — p structure is not uniform and the layer can he assumed

linearly graded from the junction to the tt layer as shown in Figure

3 .5 .h .

Knowing the breakdown field at the junction when the diode is

biased into breakdown, the field profile through this type of

structure as shown in Figure 3.5.a is given by the equations

qN, X « E(x) = E ^ 0.< X < b '^'S t"s

(3 .1 4 )

where

b = the width of the p region

1^2 . = the impurity concentration in the p region

Ng = the impurity concentration in the tt re g io n

a = the impurity gradient in the p region

= the maximum field which occurs at % x = 0 - 1 1 -

+ n P TT P f

o •H Area = U •ë

x=W distance

ü 0 •H

1 o § o

d ista n c e

Figure 3.5 (a) Electric field distribution

(b) Doping profile — T8 —

3.5. Depletion-Layer Capacitsince.

It is well known that the voltage dependence of the depletion

layer capacitance in p-n junctions can he used to determine semiconductor

doping profiles (5^)^ Doping profiles can then he used to

determine the electric field profiles. Briefly, the most commonly

used technique consists of forming a diode of known area. A, and measuring the depletion layer capacitance, at some suitable RF

frequency, as a function of dc reverse bias. The doping profile

is then obtained from

HU).= (§) (3.15)

where N is the doping level in a region Ax, C is the measured capacitance at voltage Vg. To find the position one can use the e q u a tio n .

Ax _ ^s dC AV

Most commercial capacitance meters and profiling instruments measure the capacitance of the diode under test by applying a constant RF voltage, and monitoring the imaginary component of the resulting RF current (i.e., the component 90° out of phase with the drive voltage). The imaginary component of the RF current is then directly proportional to the depletion-layer capacitance. !T9"”

In order to characterise avalanche photodiodes more fully

it is desirable to measure the diode*s. impedance over a wide range

of frequencies. ^ this means a more accurate representation of

the equivalent circuit can be built up. For the equivalent circuit

therefore the series resistance of the bulk material and of

the ohmic contacts must also be measured ..as well as the capacitance

of the space charge layer. The series resistance R^ is a parasitic element in avalanche photodiodes that can degrade the

device performance. The equivalent circuit of an avalanche photodiode is shown in Figure 3.6.

One of the important basic measuring instruments that can be

used for measuring the diode*s impedance is the slotted line. With

it, the standing-wave pattern of the electric field in a coaxial transmission line of known characteristic impedance can be accurately determined. From the knowledge of the standing-wave pattern,

characteristics of the photodiode connected to the load end of the

slotted line can be obtained. The degree of mismatch between the photodiode and the transmission line can be calculated from the ratio

of the amplitude of the maximum of the wave to the amplitude of the minimum of the wave, (i.e., the voltage standing-wave ratio VSWR).

The photodiode impedance can be calculated from the VSWR and the position of a minimum point on the line with respect to the photodiode.

The wave len^h, X, of the exciting wave can be measured by obtaining the distance between minima, since they are spaced by half wave lengths. — 80 —

cç

o o

o «CQ E § O %

■ p ü •H0 1 g s I VD PO

The photodiode impedance, Z , can he determined if the

impedance, Z^, at any point along the slotted line is known. The

expressions relating the impedance are;- (see Figure 3.7)

z - jZg tan 9 "x = ^0 ^ - jZ t^e (3.17)

where

= 360 Y degrees

3.6. Sample Characteristics

Photodiodes used in our experiments were obtained from EMI

Ltd. and AEG-Telefunken.

EMI photodiodes were prepared by oxide masking techniques in which surface breakdown was avoided by the guard-ring method.

The active area sensitive to radiation was prepared by diffusion of phosphorus from a phosphorus oxychloride source into a p-type silicon epitaxial layer grown on a p-type silicon substrate. This, resulted in the diode structure shown in Figure 3.2, having a p-n junction approximately 2.5 microns below the surface. The resistivity of the epitaxial layer was 6 - 1 0 cm.

Reach-through n — p — tt — p structure as shown in Figure

3.4 was used to fabricate AEG-Telefunken photodiodes. The correct p - 82 -

doping was achieved hy implanting boron atoms and performing a

drive-in diffusion into a ir-type silicon epitaxial layer grown on

a p-type substrate. Finally the n layer wsis introduced resulting

in a p-n junction approximately 0 .3 microns below the surface (59)^ .

The diode chips were mounted either to TO l 8 cans or in microwave 8-4 packages to reduce various parasitic elements. Figure

3 .8 and 3.9 show the diagrams of the diode mountings.

Capacitance measurements versus applied reverse-bias voltage were made first using a Boonton-Electronics capacitance meter which has a capacitance range of 0 .0 1 - 3000 pf and a test signal of

1 MHz on EMI and AEG-Telefunken silicon avalanche photodiodes.

The Type 8 %4-LBA slotted line is used to measure the photodiodes impedances for several frequencies at different bias voltages. The standing—wave pattern was measured on a 50—ohm coaxial transmission line. The results obtained have been expressed as an equivalent capacitance and resistance in series. The plots of reactance versus frequency with bias voltage as an independent parameter for the EMI photodiode is given in Figure 3.10. The series resistance of this photodiode was found to be practically constant at 6 0 . Small variations .(< 0.5 0 ) were observed, but these were considered to be beyond the system resolution.

The curves of the junction capacitance, versus bias voltage for these diodes are given in Figure 3.11 and Figure 3.12. A plot of 1/0% versus voltage resulted a straight line for the EMI photodiode. — 83 —

"K ê m 'ë O 'ë o üi nd ft +3 'CJ O <ü g) Æ -P l '^ 'd S 5 K S 5 § O *H *H fd 0 •P d0 Xi u +3 •H o I 0 k c 0 ft s 0 § ü •H CQ ft CQ 0 CD ft 1 +5 H § ft ■ p '0 •d bO § Td I 0 g 1 •H i ü fd o5 ' ë I o ft o ft -g +3 O > g d

oo 0 â •H k

î 10 - 84 - diode removable ■protective cap

diode h o ld e r - diode lead main body - polythene d ie le c tr ic

epoxy ^e sin connector pin

SMA connector

F ig u re 3 .8 Cross-section of the mounting of avalanche photodiode in TO I 8 can.

copper contact rin g screv cap spot weld A lm in a

Au w ire Sif package photodiode chip

copper base

indium solder 50 0 N ty p e connector c e n tre p in

F ig u re 3 .9 Cross-section of the mounting of avalanche photodiode in package. ^ 85 -

•§ O O irv ■a a o I H

& I a

O (U O Ü irv § -po S Pi 3 m (U

O •îf O o lf\ - 8 6 -

The slope gave the impurity concentration of the substrate as

2 X IQlS cm^ and the intercept at 1/C^ = 0 gave the built-in potential as 0 .2 v o l t s .

The depth of the depletion layer has been calculated for the

EMI photodiode using equation 3.13 and plotted in Figure 3.13.

Even at the breakdown voltage, which is 192.3 volts, the lightly doped p region cannot be fully depleted and only extends up to

11 microns, leaving an undepleted p-type epitaxial material of

5 m icrons.

Knowing the variation of the depletion layer width with the dc bias and the impurity concentration, one can then plot the electric field distribution inside the p region using equation

3.12. The electric field distribution is shown in Figure 3.1%. The maximum field at the junction is found to be approximately 350 kV/cm at a reverse bias voltage of 190 volts. At this voltage the ionization rates for both electrons and holes versus depth has been calculated and plotted in Figure 3.15• The electrons are more ionizing particles then holes, and the ionization rate of the electrons is approximately

10 times higher than the ionization rate of the holes.

The capacitance voltage characteristic for the AEG-Telefunken photodiode showed an expected reach-through behaviour and indicated that the depletion region penetrates into the tt region between

15 V and about 35 V. From the constant capacitance after complete penetration of the tt region, the width of this region is calculated to be 9 "5-microns. , . ■ — 8^ —

^ (cmZ/pZ)

O . rH VO O co

ê O O ‘•d -d- 0 H 1 O M lO OJ

£ S CO s o\ o is > g 0) OOO s s •H rQ > d <ü o ü VD m

•p§ •H o O ai » d ro OJ o E •H k

o -d- co o CVJ rH 3

(j5) soTTBQ.TOBd'BO uoxq.otinC t b Q-Oj; - 8 8 -

Area

20.

10-

■o

-p

0.1

1 10 100 200

Bias voltage (Volts)

F ig u re 3.1 2 Capacitance vs "bias voltage for AEG-Telefunken photodiode - 89 -

o ON

S o •H CO H

rHI § t • CÜ I CO CQ S CO g ; o o 3" S ° § co co cd m •H ,Q

KI 00 rH 00 g0 •H k

OJ o ON OO VD 00 OJ rH O

( suojDTH)^ ‘qî^pxA J0^'i-uoiq.9id0ci if\

O 03 H IO S

ë •g 5 5? r-i I § •H •P(U 6 A oo ir\

CJ in O

X ( nio/A) PT9TJ - 91 -

CQ PO I •â to I ? A > o o\ p H "ë

I tno q.Ttm jcsd uoTq.'BZTUoi

M ir\

CQ fl cn O u o s 00

o o 8 O

mo q.Ttm. jsd scj.'bjc tio tq .'b z t u o i - 92 -

The impurity concentration profile of the p layer and in the intrinsic ir layer has heen calculated for AEG-Telefunken photodiode, using equation 3.15 and plotted in Figure 3.16. The maximum concentration in p layer was found to he about 1 x lO^G cm^.

The acceptor density in the ir layer was found to he 5 x 10^^ cm^.

The thickness of the p layer, h, is 2.5 microns and the impurity gradient, a, in this layer is 3 x 10^9 cm^. The total depletion layer extends to 12 microns.

The electric field profile is given hy equation 3.14 and shown in Figure 3.17. The maximum electric field at the junction has heen calculated using the expression

\ > (* - &) + (w _ b)Z _ assl (* _ S S S

which gives a maximum ' field of 3.9 x 10^ V/cm at the junction.

At ah out 90 volts the diode is fully depleted.

Figure 3.18 shows the ionization rates for electrons and holes a t 170 v o lts . -/ 93 -

•§ o 0 1 g

u o «H

IS A VO 'S m 1 •H ft - 9 ^

m § u Ü S ir\

CO IÎÎ § t

ir\

in O CVJ iH O

(nro/A) PTSTJ aTJ^asia ■ - 95 -

CO ? > o o o o o o o o o o o ir \ 00

"ë % irto q.tim jred sq.'BH noTq.'BZTnoi

A § CQ

co 0) : OJ § •H ■g N ‘O f l M

00 H to 00 § ü ■â S, •H M lf\ I A

o o O o o o o 8 O o

LA 00

Tno q-TUfl jad uoTq.'BZTUoi - 96 -

3.T» Errors from the C-V Plots

Great care was taken in measuring the junction capacitance of silicon avalanche photodiodes. Any error in these measurements would ofcourse effect the calculations of the electric field and impurity concentration profile inside the depletion region. Firstly, a calibration check on the capacitance meter was performed using standard capacitances. The error associated with measuring a 1 pF standard capacitance did not exceed 0.5 %. Secondly, the effects of the package, the coacial sample mounting and the coaxial cable connecting the sample mounting to the capacitance meter were eliminated by using differential measurement method. The system capacitance was first measured with an empty 8-4 package and a capacitance nearly equal to this parasitic value was connected to the DIFF terminal of the meter to bring the zero setting within the range of the zero control and then the zero adjustment has been made. The 8-4 package with the photodiode chip inside, was then replaced into the system and the capacitance at different bias voltages were plotted using an X-Y recorder. DC bias voltage was applied to the photodiode via the rear-panel bias terminals. These measurements were made on several photodiodes and as well using several length of coaxial cable.

Capacitance measurements were complicated by the presence of the guard ring whose capacitance is an unwanted addition to the capacitance of the active device. Most of the errors in C—V measurements would therefore be due to the uncertainty.' of the area diffused under the guard ring. - 97 -

CHAPTER : k

EXPERIMENTAL ARRANGEMENT

An electron bombardment apparatus was built to investigate the interaction of an electron beam with suitable EBS targets, with the capability of injecting electrons into these targets with a minimum duration of approximately 30 picoseconds and with an energy up to 50 keV.

A description of this apparatus and its components is given in this chapter. The main components are an electron source complete with its focusing coils and a pair of deflection plates perpendicular to each other. The electron beam passes throu^ the plates and is deflected to and fro infront of the target area.

The duration of the electrons on samples effective area is controlled by a sine-wave applied to the plates as presented in section 4.2.

We also introduce in section 4.2. a microwave cavity deflection system used only in our auxiliary measurements.

The EBS diodes are mounted in a coaxial circuit in order to achieve fast rise time and the details of the sample holder and the detection system are described in section 4.3. and in section

4.4. — 9 8 —

■u:..:

4.1. Electron Bombardment Apparatus

A photograph of the electron bombardment apparatus is shown

in Figure 4.1 and a block diagram in Figure 4.2.

The hesirt of the apparatus was an electron gun from a

Metropolitan Vickers EM3 electron microscope, complete with the

focusing coils. The electron gun was of the three-electrode type

with variable bias voltage and mechanical adjustments for centring.

The electron source was a V-shaped hairpin of tungsten wire, surrounded

by a cylindrical metal shield which had a small aperture opposite

the point of the V. The shield is often referred as a Wehnelt cylinder,

after the discoverer of the beneficial effect of its presence. When

a negative potential was applied to the shield the system becomes

a three-electrode gun. The bias potential required was of the order

of 100 volts which was applied by using batteries as shown in

Figure 4.2. The bias control thus gave a useful variation of the beam intensity. The filament was about 0.1 mm in diameter and was

heated directly by a current driven by a 12 volt battery. Typical

filament current under operating conditions were between 1.5 and

2 amps. It was necessary to set the filament exactly on the axis

to within 0.1 mm, and at the correct height to within 1 mm. For this - 99 -

i4>4^ I'. « ^ &Ü&&

¥- > 4%" n S'

>' / - '^p

, ‘4*-i

w x s e m r ^

Figure 4.1 The Electron Bombardment Apparatus - 100 -

IMQ / \ y \ e HT 12 V, 2 K 100 V 0.025 y f : A c y : Ampenneter 777777

Filam ent

G rid

Anode 777777

M agnetic Lens O s c illa to r

• Adjustable 1 Deflection Stub P la te s Po^er D ig ita l D iv id er Ammeter Electron Beam Faraday Cup

rH Photodiode

Power supply & D.C. metering

Coupling C ap acito r Oscilloscope

T rig g er

X-Y recorde] Sampler

FIGURE 4 .2 SCHEt.lATIC DIAGRAM OF THE ELECTRON BOMBARDMENT APPARATUS - 101 -

purpose a small jig was used as in Figure 4.3. Increasing the

distance of the filament tip from the shield aperture has precisely

the same effect as increasing the negative bias preventing the

anode field from reaching to the filament.

The emitted electrons from the filament were attracted through the shield aperture towards the anode by the very strong electric

field existing between these electrodes. The anode was a metal

cylinder with an aperture in the face exposed to the cathode shield.

It was convenient to maintain the anode and the apparatus body at earth and the cathode at high negative potential.

Figure 4.3 Jig for Centring and Adjusting the Filament Height - 102 -

The E.H.T. supply consisted of a Hursant Electronics

Power Supply Model H 437 with an external filter comprising of

0.025 high voltage capacitor and 3 resistor to suppress the

25 kHz ripple generated by the supply. The output voltage was continuously variable from about -5 to -50 keV.

The beam emerging from the anode was focused on to the photodiode effective area by the condenser lens, which was placed about mid-way between cathode and the photodiode. The illuminating system was fitted with tilt adjustments to allow the emitting tip of the filament to be brought exactly on to the axis. The top of the photodiode mounting was coated with a phosphor to enable the electron beam to be aligned. Once the beam was striking the photodiode the final adjustment was carried out by observing the oscilloscope trace or the ammeter.

The specimen chamber consisted of an evacuated (lO^ torr) glass cylinder about 30 cm diameter and coated internally with a conducting aluminium layer which was kept at earth potential. All the vacuum joints were of the rubber gasket type, and a Pirani gauge was fitted as well as a discharge tube for vacuum measurements.

The oil diffusion pump and a shaft driven backing pump, connected to the system with a rubber tube to reduce vibration, gave a h i^ pumping speed.

4.2. Deflecting System

One of the techniques for creating short electron pulses is - 103-^

to deflect the beam to and fro in front of the photodiode effective

area. ' The deflection system used for low frequency modulation of the electron beam consisted of two parallel plates 5 cm long eind

0.3 cm apart to which a sinusodial voltage was supplied by an

Airmec Oscillator type 304. This high power oscillator has a

frequency range of 25 kHz to 100 MHz. Over 100 MHz up to 600 MHz a Philco Sierro Electronic high power source model 470A-500 was used.

An adjustable stub connected to the oscillator output enabled to supply the maximum power to the plates.

It is important for us to know the time that the electron beam illuminates the diode effective area and the time in which it returns on top of the diode area. To calculate these times we have to know the velocity of the electron beam on the plane of the diode area and the amount of deflection. Therefore let us assume the electric field on the plates is E = E^ sin (2Trft), a sinusodial field of maximum value E^ and frequency f , and calculate the deflection and the velocity. From Newton*s Second Law, F= ma we obtain the relation

If the initial velocity of the electron is v (v = 5 .9 7 x 10^ cm/sec, E^ is the accelerating voltage of electrons in volts) the time during which the electron is in the field between the plates is

1/v (see Figure 4.4). - lO it -

Electron Beam

D e fle ctin g P la te s

_L U

Screen

F ig u re k,k Fundamental Dimensions of the Deflecting System and

the Path of the Electron Beam

Assuming the field is uniform between the plates we can replace E^ by V^/d, where is the maximum potential. If t^ is the time at which the électron enters the alternating electric field, the component of velocity perpendicular to the plates is —<105 —

ëv_; , . - ° mâ“ sin (2ïïft) dt (4.2) h

which becomes, upon integration and trigonometric manipulation

eV Vy = sin (^) (4.3)

The deflection D on the fluorescent screen on the photodiode plane depends on the slope of the electron beam as it leaves the field.

V D = 2L tan a where tan a = —^ V

T herefore we can w rite

2eV L „ ® sin (^) (4 . 4 )

Since the phase angle between the deflection and the voltage is of no interest, the deflection produced by a high—fre(%uency potential is

2eV L ^ ~ mnvfd (2ïïft) sin (-^) (4.5) - 106 -

Thus, at higb frequencies a sinusodial potential on the

plates w ill result in a deflection which is decreased in amplitude,

whereas its wave-form remains "^hanged. Thin amplitude distortion!

is shown in Figure 4*5 for plates of 5 cm length and an electron

velocity corresponding to an acceleration potential of 20 keV,

(v = 8.4 X 10^ cm/sec). The deflection is reduced to zero at 8 x 10®

cycles/sec independent of the applied voltage to the plates. The

departure from low-frequency response is noticahle at about 1 0 ®

c y c le /s e c .

An experimental check was made using the mathematics involved

in the above derivation^ on the performance of the electron bombardment

apparatus. The deflection was measured experimentally on a flourescent

screen at a distance 25 cm from the deflection plates, as a

function of the beam accelerating potential and the frequency. It

is interesting to note that the experimental results verify the mathematically calculated results very closely. At a beam voltage

of 40 keV and a plate potential of 100 MHz a deflection of 2.5 cm

could be obtained. Therefore we can write

D = 2.5 sin (2.n.lO® t) and

Vy = 2 . 5 x 2 x 7t x 10® cos (2 . tt .10® t ) - 107 -

m o rH a ü i f \ il A +3 W C 0) iH 0) -P (à rH A

00 m 0) S - p ü rH 0) A m Ü m O Q) •H rH +5 ü (U & rH > <ü O fd M CQ 0) bO G •H t>» •p O c3 c IH

<ü d o o o o o o bO o co vo c\] •H A

noTq.'BjqTTBO 'O'p jo qusojaj — 108 —

Therefore the ■velocity of the electron beam on the diode vould be

V = 1.5T X 10^ cm/sec. The time that the beam takes to transit the diode depends on the diode effective area. If the diameter of the diode active region is 0 .5 mm this time would be t^ = 5 x 10^ /

1.57 X 10^ = 31 psec.

For jaicrowave modulation of the electron beam another deflection system must be used. One such system is the microwave lecher wire deflection system it consists of resonant lecher lines X /2 long enclosed in a circular wave-guide. End plates can be fitted to reduce loss from the system due to radiation. The electric microwave standing-wave pattern has an antinode half way along the lecher line and the beam passes between these lines at this point.

The alternating electric field here produces a transverse velocity on the beam and the beam passes the diode area twice per microwave c y c le .

The amplitude D of the sinusodial deflection of the electron beam on an observation plane at a distance L from such a deflection system turns out to be

D =

where is the power in watts fed onto the deflector, E^ is the beam voltage in volts and A ,U are two constants defined by m m — 109 —

iÇ-.-

4 s^ ir-^f^ ff.

" A

g i=A%l

• T-*“V

Figure 4.6 Microwave Cavity Deflection System 110 -

the geometry of the system.as

4 A m + d,

where d^ is the spacing of the two wires, d^ is the diameter of

the wire and A is the free space wavelength of the applied signal.

Figure 4.6 shows the deflecting system used. The lecher

lin e s are 7 cm long, the distance between the wires is 4 mm and

the diameter of the wires is 0.5 mm. The cavity formed has a cut

off frequency at 1.92 GHz. Input power to this system was supplied

by General Radio 900-2000 MHz unit oscillator type 1218-B. The

system produced a measured deflection sensitivity at the diode of

0.04 mm/mW input power.

4.3. Specimen Holder

The method of mounting the photodiode on the base plate of

the specimen chamber is shown in Figure 4.7.a. and in Figure 4.7.b.

The diode was in a 50 ohm coaxial connector. The specimen holder was a 0 .5 cm thick brass disc and pressed under vacuum to the

stainless-steal base plate by an 0-ring. All the electric connections - Ill -

Figure 4.7.a. Specimen Holder (Side View) 112 -

Figure 4.7.1 Specimen Holder (Top View) - li s

to the hase plate and to the specimen holder were made hy vacuum t i g h t 50 ohm connectors. A combined shutter, Faraday cap and a fluorescent screen with a ruler was mounted onto a stud which passed throu^ a hole in the specimen holder. The stud could he operated from outside the specimen chamber hy means of a Wilson sead.

This system facilitated the visual positioning and focusing of the electron beam, prevented accidental over-exposure of the photodiode while setting up, and measured the beam deflection.

The Faraday cup measures the electron beam current and was made of a small copper cylinder closed at one end and the opposite end of the cylinder was closed by a thin copper foil having a small hole in the centre. The foil and the Faraday cup are isolated in such a way that they do not make metallic contact. This foil was changed when different diodes were tested, since the hole in the foil must be the same area of the photodiode active areas under test.

4.4. Signal Detection

The generated current waveforms from the photodiode were viewed on a Hewlett-Packard l4l A sampling oscilloscope which has a rise-time of 28 psec. The traces were recorded using a

Bryans X-Y plotter model 2900 A4.

Photodiodes were mounted in a 50 coaxial line for proper matching to the rest of the circuit and a bias T circuit was used to provide the dc biasing for the photodiode and ac coupling of the detected current waves to the oscilloscope. The use of a - I l k - '

resistance in the dc supply path vas found to he much better than the use of an inductor since it was observed that the latter

introduced a lot of distortion on the falling edge of the pulse.

This fact was checked using an ordinary T junction (R = O) and

comparing the output pulse waveshapes. The disadvantage of using no resistance on the dc supply path was the loss of considerable

amount of power. The high resistance also protected the photodiode.

The details of the coaxial circuit in which the EBS diode is mounted is shown in Figure k. 8 *

Photodiode

—> to sam pling o s c illo sc i

G eneral General Radio Radio Coupling Capacitor T -Ju n ctio n

Stabilized DC Power Supply

Figure 4.8 Coaxial Circuit in which the Photodiode is mounted. ■ - ‘ . ' f “ ■

CHAPTER 5

EXPERIMENTAL RESULTS ARP DISCUSSIONS

In this chapter we present our experimental results and

discuss these results under the li^ t of understanding the theory

and assumptions given in chapter 2 and chapter 3 .

In section 5.1. we report the experimental results of the

simultaneous application of the EEC and avalanche m ultiplication of

silicon p-n junction targets. From these measurements it is concluded

that the EEC and the controlled avalanche effect can occur together

in the EES targets and these phenomena ' can he cascaded as predicted.

To compare the theory and experimental results for EEC gain,

theoretical curves for electron acceleration potentials similar to

potentials used in our experiments are given in Figure 5.2. a, h, c, d,

e, f. These theoretical curves-are based on the assumption that the.

ideal gain is E^/3.6, and with the use of the experimental curve given

in Figure 5*1 &re useful in estimating the surface recombination velocity.

We next give the experimental results on the variation of multiplication coefficients, for two different avalanche photodiodes having different structures, with the reverse bias up to breakdown, for primary electron energies in the range 7 - 30 keV. Figure 5.4 and

Figure 5*7 show the dépendance of the multiplication coefficients on the penetration depth of the electron beam, since the penetration depth decides pure hole or a mixture of electron and hole injection - 116 - into the avalanche zone to initiate the multiplcation process.

Consistency of the observed curves with the theoretically excepted ones in Figure 5.4 suggests that the electron bombardment of a p-n junction can be used as an alternative excitation source to light obtaining the multiplication data.

Since the calculation of Miller exponent can be a short cut to calculate the multiplication factors, general Miller exponents n are calculated for two diodes from the experimental measurements.

In section 5*2. we give the experimentally observed and theoretically computed pulse shapes • for EBS targets under electron bombardment. The experimental pulse shapes given in Figure 5-9- and 5» 10 made us decide to make a model as given in chapter 2. To develop the mathematics of this model some assumptions are made.

The primary assumptions in this analysis are that

i) the carrier distribution is generated instantaneously with the carriers having their saturated velocities and that is why the computed current wave forms given in Figure 5-10 a,b. Figure 5»11 a,b and Figure

5 .1 2 a,b do not start from zero at t = 0

ii) the rise time is independent of the multiplication process which in fact is not in accordance with small signal theory of avalanche photodiodes and perhaps indicates the simplicity of the model used

iii) diffusion effects in high electric field are ignored in the m odel.

The important points to look at in these computed curves are the shapes - 117 -

of the current wave forms and one must notice that these shapes are

changing with multiplication factor, electron penetration depth and

the difference of velocities of the carriers generated.

5 . 1 . Experimental Procedure, Results and Discussion for EBC and

Avalanche Gain

After the system had been evacuated to a pressure of about lü^ torr, the electron gun filament was switched on and the E.H.T. was set to, say 10 keV. The bombarding voltage was measured directly using the meter supplied with the Hursant electronics power supply.

The beam intensity was controlled with the bias control of the

Wehnelt cylinder. The current in the focusing coil was adjusted

to focus the beam to a fine spot and the tilt adjustments were used

to bring the spot to a mark on the shutter placed above the specimen holder, This precaution was necessary to avoid over-exposure of

the photodiodes while setting up the apparatus. Wien properly

focused and positioned the spot was vertically just above the photodiodes * active area.

In all experiments currents were measured with the electron beam scanning one line rather than the beam at one spot. Parallel

deflection plates were used to convert the electron bombardment

apparatus to the scanning mode. This was done to reduce the effects

of local heating and possible damage at higher electron beam voltages

to the photodiode surface.

Determination of the photodiode gain requires the measurement - 118 -

of electron beam current and the photodiode current. These currents were measured by a Keithley type I 50 B picoammeter. The beam current i^ was kept in the range of 10^ and 10^^ A to reduce contamination, which causes loss of excitation electrons, particularly at low energies. The photodiodes were first reversed biased well under the breakdown voltage so that the increase of the photodiode current was due only to electron bombardment. The resulting induced currents were above the diode dark reverse saturation current.

Knowing the diode currents i^ and i^ under electron irradiation and without irradiation respectively, we may deduce the current resulting from the external excitation

and hence the EBC gain can be calculated using

^EBC " i^ (5 *2 )

The avalanche multiplication effect on the photodiode gain was measured increasing the reverse bias above the breakdown voltages.

When the reverse bias was increased the photodiode current i^ and ig became voltage dependent and the multiplication factor M(V) at the voltage V was calculated using - 119 --

i (V) i.( V ) - i (V) = i W = l '-- L ^5.3)

Therefore the overall photodiode gain can he written as

°T ° °EBC • (5.W

The experimental results are shown in table 1 for the EMI photodiode. These measurements for each different incident beam energy were taken at about one hour intervals. The penetration depth of primary electrons has been calculated for each bombarding voltage and is included in the table. This distance is defined as the range after which 99 1» of the impinging electrons are absorbed.

The residual 1 % electrons creates , 1 ^ of the total number of electron-hole pairs generated. This neglected fraction is well within the measuring accuracy assumed for this work.

To reduce the effects of local heating the currents were measured with the electron beam in the scanning mode. Since we know that a number of factors limit the rise-time of the observed signal, it was necessary to test for such factors by varying the line scan speed and determining whether the form of the signal variation changed.

When a voltage with frequency in the kHz range was applied to the deflecting plates the amplitude of the signal was independent of the scan speed. . i O ^ Oj oi OJ cn m CO tn m m. O o o o o O O o o O o O rH rH rH rH rH rH H rH rH rH rH rH X X X X XXX X XXX X C\I VO -V o\ rH o A «H to •p

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F igure 5 .1 shows the variation of the EBC gain with incident

beam energy for the EMI photodiode. The photodiode has an n"*" layer

thickness of 2.5 micron. The reverse bias used to measure the EBC

gain was of the order of 60 volts at which the depletion layer extended

up to 6 micron. This reverse bias was well below the breakdown voltage

of the photodiode and no avalanche m ultiplication of generated carriers

was taking place. Therefore the gain measured was due only to the

increase in the electron bombardment conductivity. Primary beam

e n erg ies 7 to 25 keV were used to create electron-hole pairs in + the n layer and the depletion region of the photodiode, since the

theoretical calculation for the EBC gain given in chapter 2 does

not include the gain from the undepleted material. Theoretical

curves of EBC gain versus junction depth are presented in Figure

5 .2 a, b, c, d, e and f using different values of surface recombination velocity as a parameter and for beam voltages used in the experiments.

Also the experimental points for the EBC gains are shown on these

figures. The experimental results and the theoretical curves

suggest that a surface recombination velocity of 7 x 10^ cm /sec exists at the photodiode surface. For the calculation of the surface recombination velocity the energy required to create one electron- hole pair is taken to be 3 .6 electron volts. - 127 -

6000—

5000—

liOOO—

3000_.

o § 2000----

1000__

10 20

Ey(keV)

Figure 5*1 Charge collected per electron penetration into

the EMI photodiode versus primary beam energy

veil before breakdown - 128 -

1500.

G ‘cô Ü 1000 —

g • experimental poi:

500 —

0 1 2 3 4 5 6 T 8

n layer thickness (microns)

Figure 5«2.a. EBC Gain vs Junction Depth taken at a BEAM /\

V oltage o f 7 keV • — 129 —

3000

2900—

2000__

1500 •S ë experimentaî\S = 2 x 10^ cm/sec o p o in t S

1000—

500----

0 1 2 3 h ^ 6 T 8

n layer thickness (microns)

Figure 5«2.h. EBC Gain vs Junction Depth taken at a Beam

V oltage o f 10 keV - 130 -

4000

3000----

c o 2000__ o • experimentais^int^ “ 1 3 s = 2x10^ cm/sec

1000 —

n'*’ thickness (microns)

Figure 5.2,c. EBC Gain vs Junction Depth taken at a Beam

voltage of 13 keV 131 -

kooo

3000

e experimental point

C3 2000 1 .3 I cm/ sec §

1000

n layer thickness (microns)

Figure 5«2.d. EBC Gain vs Junction Depth taken at a Beam

Voltage of 15 KeV - 132 -

6000

5000 — ■

• eicperiiiientar^oint 4000__

.3 3000 — S = 2x10^ cm/sec o §

2000 —

1 0 0 0 __

G 1 3 4 6 8 10

layer thickness (microns)

Figure 5*2.e* EBC Gain vs Junction Depth taken at a Beam

v o lta g e o f 20 keV - 133 -

8000

7000 —

:rimental po:

5000 —

4000 — k = 1 .3 s S = 2x10** cm/sec

3000

2000 —

1000 —

n layer thickness (microns)

Figure 5*2.f. EBC Gain vs «Junction Depth taken at a Beam Voltage of

25 keV - 134 -

As the reverse bias was increased over 60 volts, avalanche multiplication of the'carriers occured. The experimental results

on the variation of the multiplication coefficient with the reverse bias up to breakdown, for primary electron energies in the range

7 “ 25 keV are shown in Figure 5•3» A plot of the reciprocal values

of the multiplication coefficient versus the applied bias voltage has been adopted. If we consider the two extreme curves of Figure

5 . 3 , a remarkable difference appears. This is due to the fact that electrons and holes have different avalanche multiplication characteristics. At the primary beam energies 7 to 15 keV all the electron-hole pairs are created within the 2.5 micron diffused

layer near the surface. This results in a pure hole injection into the multiplication region.

At a beam voltage of 20 keV about 88 percent of the generated electron-hole pairs are created within the n'*’ layer and 12 percent in the depletion region. Also for the beam voltage of 25 keV 64 percent of the electron-hole pairs are created within the n^ layer and 36 percent in the depletion region. Therefore for these cases the carrier multiplication process was initiated by a mixture of electrons and holes.

The two extreme curves of Figure 5«3 then correspond to pure hole current injection (7, 10, 13, 15 keV) into the avalanche zone from the junction and to a mixture of electron and hole current injection for 20, 25 keV. The increase of the primary electron beam energy shifts the multiplication curve towards larger multiplication s

•H

CO •H

CO 0\

0\ -P

ir\ co ■VD

tf\ OJ

o o\ vo o

H | S —136 —

coefficients» since an increase of causes an increase of the

injection electrons, which are the more ionizing carriers in silicon.

This property I S also shown hy the dashed theoretical curves

in Figure 5»3 which have heen computed using ionization rates based on m ultiplication measurements performed by Lee et. al. using carrier injection by light. The depletion region was divided

into small intervals and the ionization integrals given in equation

(3.U) and equation (3.5) were computed using a computer by Simpson*s rule. The computer program is given in appendix I.

To calculate the ionization integrals by a computer is

somewhat cumbersome and a sim ilar expression was proposed by M iller given by equation (3.11). According to Miller*s Law a plot of

I log (l - 1/M)| versus |log (V/Vg^)| would be a straight line with

slope n. In Figure $.4 a plot of |log (l - 1/M)| versus |log (V/V^^) has been adopted. The curve corresponding to pure hole injection, i.e ., for multiplication coefficient measured using beam energies

7, 10, 13, 15 keV is nearly rectilinear and therefore can be characterized by a definite M iller * s exponent (n - 12). But the other curves corresponding to beam energies of 20 keV and 25 keV and the computed curve for pure electron injection are not straight lines.

To each value of V/V^^ an exponent n must be associated. The values of the exponent n obtained are shown on Figure 5.5. The value of exponent n decreases with the increase of bias voltage.

Therefore it would be interesting to find a law linking the exponent n to the voltage applied. - 137 -

4» A Jl 138 1. 0,

Br 192.3 volts

n 0 .5

,1/2

0.90 0.95 1.00

Figure 5.5 Variation of the Exponent n with the Voltage for pure Electron Injection

Since a plot of n versus CV/Vg^) give nearly a straight line, the linking law would be

( V ^ n = a (:— 1 + C (5 .5 ) ^Br where a is the slope and C is the intercept at (v/V^^) = 0.

For pure electron injection we can then write

V 1 / 2 . = - 4.6316 (^p^) + 4.9584 (5 . 6 ) - 139 -

Table 2 shows the experimental results for EBC and avalanche

m ultiplication for AEG-Telefunken photodiode. The m ultiplication

versus reverse bias is plotted in Figure 5*^ for electron

bombarding voltages of T keV and 25 keV. The kink in the ;

multiplication curve represents a correctly diffused and micro plasma

free photodiode. The reach through voltage is approximately

45 volts. The voltage in excess of 45 volts is dropped across

the total intrinsic width of the tt region. The field in the

multiplication region and hence the multiplication factor increase

relatively slowly with increasing bias voltage above reach-through.

Figure 5 .T is a plot of |log (l - 1/M)| versus |log (V/V^^)

The Miller exponents at different bias voltages are calculated and

included in the plot. For avalanche multiplication initiated by

electrons produced by 25 keV beam energy, n^ can be represented

by the equation

- - 6.357 ( ÿ ^ ) ^ +0*571 ( 5 . 7 ) Br

where 7^^ is equal to 175*3 v o lts . A •H & m en on on on on on ^ ^ ^ ^ o o o o o o o o o o . H r—I H H r—I H S H i—i rH H rH S O X X X X X X X X X t— f—I o\ O on rH vo OO lA o VO on m o \ vo -d- o vo on vo ïï I CVJ on on on ir\ co

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S VO I o

g U 0— vo vo VO VO vo VO VO lA lA lO lO lO lO lO lO lO lO lO lO lO lO O rH rH rH rH rH rH rH rH rH H rH rH X X X X XX X X X XXX vo co 0\ vo co OJ co o \ on on CM « lA O O o CVJ -=r O co lA rH lA CM -V vo 0 \ rH rH rH CVJ CM *« 0 \ rH CM O D) rH d •H f » m S o O O O O P o o o o o O lA »d rH on lA VO t-* o \ rH on lA VO VO O rH rH rH H rH A

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lA - 1 4 2 -

!'■ -

100.

70 - Figure 5,6 Avalanche multiplication vs hias voltage

60 - for ABG-Telefunken photodiode 50 _

ItO _

• » 25 keV

30 -

7 keV

10 — 9 - 8 - 7 - 6-

2 -

BIAS VOLTAGE (volts)

100 150 - 143 -

00 m

cu cv ro

H | g O

• o

CO VO 00 VO - i4 4 -

5*2. Current vavefoms produced by electron bombardment

These measurements were ohtained from observation of current

"Waveforms produced by an in itia l electron beam which oscillated back

and fro infront of the sample. The duration that the beam was on

the sample effective area should be varied by adjusting the frequency

of the voltage applied to the deflection system. The minimum time

was 32 picoseconds.

Data was collected graphically using the X-Y recorder in

conjunction with the sampling oscilloscope. The output pulse caused

by the moving charge sheet in the photodiodes was recorded for a

range of input dc voltage. A typical set of data as it was recorded

following bombardment is presented in Figure (5.8) for EMI photodiode

and in Figure (5.9) and Figure (5.10) for AEG photodiode.

It was noted that the electrical pulses in Figure (5.8) for

EMI photodiode show a distinct kink, which may be explained by the

release of the charge beyond the depletion layer and from the front n^

contact layer. The rise time of this photodiode is about 800 picoseconds

This rise time is effected by the holes entering into the depletion

region from the n^ layer by diffusion, and also by the triangular

electric field established in the depletion layer. If the electrons

and holes had moved with their saturation velocities, v , in the drift s ' region one might have expected a rise time of 266 picoseconds. To

find an exact solution for the transport of carriers with the triangular electric field one has to solve the following continuity eq u atio n - 1%5 -

i t - + GgCx) (5.7)

This equation contains both the diffusion and the drift terms.

'When the electric field is high near the junction the drift term is dominant and when the electric field is low at the end of the depletion layer, then the diffusion process takes over. Every carrier produced . and multiplied in the photodiode has to pass through this diffusion dominated region. Since diffusion takes much longer time than drifting the rise time of the photodiode would be mainly characterized by the diffusion time.

For the AEG-Telefunken reach-through avalanche photodiode the n^ contact layer is very thin so that one can neglect the effect of this layer. The rise time and the fall time of this photodiode when it had not reached-through would be limited again by carrier diffusion. To compare the experimental and theoretical results the computer plot of the current pulse shapes are also given for different primary electron energies using different multiplication factors. Figure (5.10.a) shows a Set of current pulse shapes under

25 keV electron bombardment for electron drift velocity of 10^ cm/sec and hole drift velocity of 8 x 10^ cm/sec. Figure ( 5..1 0 .b ) shows a set of current pulse shapes with the same in itial electron bombardment voltage for electron velocity of 1 0 ^ cm/sec and hole drift velocity 1 0 ^ cm /sec.

Figure C5.11.a) also gives a plot of family of curves for - 146

primary electron energy of 35 keV -with, unequal electron and hole

drift velocities and Figure (5*ll«h) is for equal drift velocities.

+ For electron beam penetrating up to the p contact, the

current pulse shapes are as shown in Figure (5.12.a) with unequal

electron and hole drift velocities and with equal drift velocities in Figure (5.12.b).

The computer program plotting these family of curves is given in Appendix II.

From the.experimental and the theoretical curves one“can see that the shapes of the electrical pulses changes with electron bombardment energies and the voltage applied to the photodiode.

Although experimentally observed and theoretically computed current wave forms are similar in shape for the same multiplication factors and penetration depthstheir time scales are different. This may be due to the assumptions made in the theoretical model or due to the effects of the external circuit used to detect these current wave forms. - li(7 -

00 S «H (D m § CQA (U . K ■P g 5h OJ S CQ 0 § -P -PO rH

(Aui) 0pnq.Txdurv — lU 8 —

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o I 'g - o r—I0) rü (D O rHd *H nd o G CD\ •HPi +3 O O È“ -a -O

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~~CHAPTER 6~~

~~CONCLUSIONS~~

It has been shown with theoretical and experimental results that the excitation of electron-hole pairs by an electron beam makes it possible to investigate the parameters which characterize the performance of silicon avalanche photodiodes. These parameters are those which can be controlled by the careful manufacture of the diodes. The expressions for static and time-dependent current gains, which were derived from solutions of the diffusion equation and boundary conditions using the mathematical model suggested for analysis, are useful for estimating the effects of these diode parameters on efficiency and frequency lim itations.

Frequency limitations of these diodes are similar to those resulting from the decrease of the transport factor in the base of a bipolar transistor at high frequencies. Calculations indicate that the depth of the junction should decrease with increasing frequency for preservation of constant dynamic gain.

Experimental measurements which have been obtained for the electron bombardment conductivity under controlled avalanching conditions have given good agreement between measured and calculated gains for several photodiodes. It was shown that the EBC gain and the avalanche multiplication of these diodes can be cascaded and - 165 -

~~an overall current multiplication of the order 10^ was demonstrated.~~

It has been concluded that for the achievement of high gain and efficiency the depth of the junction should be decreased to 0 .3 microns. Then an array of such junctions can be made individually conducting by a low-current electron beam deflected to the desired junction, one can control the flow of relatively high currents in the corresponding external circuit.

Therefore one of the future aims could be to determine the response of silicon avalanche photodiode arrays to electrons, and to asses their suitability for use in electron imaging devices and instruments.

The present results also indicate the possibility of obtaining meaningful m ultiplication data by electron excitation in the same way as previously done by light. In the case of electron excitation the electron energy plays the role of the wavelength of light, as it determines the depth of the excitation and hence the type of injection. The electron energies which were available by electron bombardment apparatus allow excitation depths in silicon from 0 .5 microns to approximately 15 microns. This electron excitation depth is characterized by a definite range, whereas this does not occur with excitation by light, which has an exponential generation function. This property of electron excitation could give a better selectivity in exciting deep structures. The much lower spot size attainable with electrons in comparison with light could prove — i6 6 — particiilariÿl' useful for observing the uniformity of multiplication at high resolution or for localized multiplication studies on microplasma or other defects. - 167 -

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~~(July 1954) - 175 -~~

~~Appendix I~~

~~a) multiplication for pure hole injection~~

~~This program computes the m ultiplication factor for pure hole injection into the avalanche zone from the positive x direction.~~

~~nrvitr I FT -, I CT Q CÛ25 let S=ll. 8+8. 854E-14 8 8 :8 FOR V=140 TO 195 8TE!=' . 2 0040 LET 5 0050 LET V=5E-09 0055 -LET R6=0~~

0055 cnp x=o TO |.i-20+Y E“E-' Y 0.070 let E= ( 0+N/S ) N- ( X+ ( Y/2 ) / ) 0080 LET 91=22500000=^' = E: :P-5250000/E) 0090 LET R2=:8O0OOO=^=\EXP(-i75OO00/EX' 0100 LET 93=93+(A2-R1 ."*+8 0110 LET R5=95+(R1=+=Y:' + \EXP(R3) ) f.:.-! Tm r;r-v’T 0125 LET Ml=l/'::1-A5) O l50 C'OINT V.= Ml.' H5 0150 NEXT 0170 END _ . . ■ • ' b) multiplication for pure electron injection

~~This program computes the multiplication factor for pure~~

~~electron injection into the avalanche zone from the negative x~~

~~d ire c tio n .~~

0015 LET N=2E-^15 0020 LbT 0=1. bE—18 0025 LET E=il. 8*8. 85:]E-14 0030 FOR V=140 TO 185 S~E P . 2 0040 I.rj 1.1= r v;+T.' -1+ > t- = 0050 LET Y=5E-08 0055 LET R6=0 0060 LET .93=0 0065 FOR X=W-1000*Y TO V STEP '-Y) 0070 LET E=(Q+N/5 0080 LET 91=22500000* r ' C ' -• •- F p r« n p cr 0080 i nr T H ■-•'=T m m m m m T: rtrv‘C*i' - 1 -T •• 0100 :+:U 0110 LET A6=A6+ 0150 PRINT V.. Ml.. R6 0160 NEXT V 0170 END

~~■ The difference between m ultiplication integrals for pure~~

~~hole and electron injection resides in the fact that the field has been turned around. - 177 -~~

~~Appendix II~~

~~This program computes the current wave forms produced by electron injection into an avalanche silicon photodiode. -178 --~~

-710 RiM NAME. -"PROG^VAFÜOL"- BY DIPL. INC. H. 1 YRROL 0020 PRINT " .'I'll- 21O0+-?- QPPPMF r+x+xirirHxjxkf+xtx+xK^z^xk kixM+Hx^: K+^x+x+x+xf:*:*: 0022- DIM RTC10L Y$C33.. Gf!:33 0024 LJLT pl=jLRL2l)*4 0020 OM ESC -n-'EN GC‘SUB.0508 ' 00-10 '-''RirTr ■ ■ 0 0 2 0 G0 0 !.’L 0 0 ! ^0 1-Ü- 4P PfjCMp -i PPM 0042 000UE' 00-10

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~~OOlt! pxlMT . U ' 12 .IF '"'S-':"-' " 1 ! {EM p r/N T "PRESS PPOF }•:.P-Y Tft n pap FRcrcr,,.~~

M ■ : n r •: • T k rr >• m . ■ i r r ,-j «; . . r- _ .%M ; r , ' j : 2 0 ' ■“"•■' ' . k.-l | ;l r-Cjj..! y K|pi r ' ,0';:4T FR'Ii . lJ s :-“ V" THEN GOTO 035A 0:4-6 : - T A1"G=! 23

0 :5 0 r ' NIC.!.. .ToiN oosuo oooo 0 :6 0 II HIT: .!._= "L" 'H:2N GOSUB 2100 0: 70 I'" PTC: : : "7" H'SH COSUO 2200 OSTi! 1" A.T'J. i.jT."S" THEN G0SU2 2300 7 :2 0 If- p-iNS'. I .:--"A" '"MS!' '-‘ppJO 3H00 0-2'! !' ;:'T 1. '"'-'■ 1-:" in- . 0400 I" I'i>;r>| ;/M':;! M - U 41! '' 1 p ’■ r •• 1 • " r. " ’! HEN GCrSR P 3 0 0 0 0430 •'■■ !:-[;i ;] ■' " C " THEN G'jSUB 6000 ! : 2 i!“ ci^Li.- ■;. ] -■■■-■ "M" THEN GC-SUi: 6100 0-136 IF 'TT'Ol. I 1 — up II THEM GO .. J ~ "0" THEM SUP 7000 0450 .! A - :"!E[ J "H" THEN GOSUB 7 0 00 U’-S O I; :-7 1 1 1 J •■ "E" THEf.: GO TO "iOO.Cj •-!jr. 1 CM, . '■! :~-0 THE”! GOT'O •Zi ~j Z<‘Zi _H70 OLHO. 012 0 :0 0 REM i- ! ‘•;:J;-^::|.-.-+ -.--r- .c-.j y Kj- '■I*-! ESC.. PAUSE PLOT"! NO 1 STOP rr,i. J - . ! 0:10 •100! M ,k i - 179 - L^CL-:0:0 8Oî80 RET!URN L ! ci 1 iO'IjCi REM USER DEFINED CONSTANTS 10C2 ■ ET rp 000 8 3 ' 1002 LET 00003 LET 1006 LET V l=10000000 10’00 LET V:L~-8OO0U0O .1010 LET 1. 6E--1?^:'"'1/W>.+ ( 35 ■1 r-i -i LET Û2 1. 6É-19+V2/W :• + '335 1014 LET X0=0 LL LET 00.T.0 LET:X:8::(;i/Y2)-KR-D)XY1 LET Y. - 155 LET X 8=X ?/5 1018 LET 1020 LET R0=4 ■ 1022 lE T A8=16 1024 LET H8=4 1025 LET Y 8=Y ?/4 1026 RETURN 108? REM 1100 LET M-:'A 1102 LET T-4" 1104 LET X1=P '.'1106 1 PT I'l 1100 LET X3=D-KY1+T> 1110 LET X4::NfD-’(V2*") 1112 LET ;::5=D-f-vi+(T-':w/v2).' 1113 LET G1=0 6+Xl. 'P) + (3. 105+( 1144 LET G2='L 6+X2/P) + (3. 105+( 1116 LET 03-'::. 6+X3/R 3. 1 0 5 * 0 1118 LET 0 4 = 0 6+X4/R::' + ':3. 105+( 1120 LET 05=0 6+X5/R 3. 105+'0 1122 LET 11= (0 1 -0 2 :'+ 0 2 1124 LET 12=': 01-03::'*01 1126 LET 13.- ( 0 3 -0 2 ::' +M+Q2 LET 14=': 04 -0 2 :'+ 0 2 1130 LET I5=(Gl-02:'+M+02 1132 LET I6=(Gl-05::'+M+02 1134 LET 51 = '. 11+12+13.:'+1000000 1136 LET 52=': 13+14:-'+1000000 1138 LET S3=( 14+15::' ''4000000 1140 LET 54-I6+1000000 1142 IF T>=0 THEN IF ""OrW-'IP-D) )/Y2 "HEN LET V=S1 1144 IF T:: (W-(R-D)::'/Y2 THEN IF T{=':R-D)/Vi 1 HEN LET Y=S2 1146 IF T;>':R-D>/V1 t h e n i f I'O W /V C ■THEN L.ET Y=S3 1142 IF TMF-V2 THEN IF T{=U'!.'V2) + (R -D )/Yi THEN LET Y=S4 ■188? RETURr! 2000 F;EM ■' ■ >";n •[■' I F fT ft'J " MNIT'* 2010 PRINT 202G PRINT "PLEASE EN'rEP TH E MINIMUM iWiD MAXIMUM VALUE OF X" 2022 PRINT "CURRENT VHLUES XU-^ XO.; "X 8=’U X?. 207 0 INPUT XO, X8 204 0 PRINT 2000 PRINT "-'LEPSE ENT ET: THE MINIMUM AND MAXIMUM VALUE OF Y" 2052 P RINT •'T ORRENT VALUES Y0=’'.! V0j " V^=" : Y; 2060 IMPOT ’. 0, Y? 2070 "ÔOSUE "04 0 ' .

._:10O F F H •k-V;; V v;.r- .TCy NÎND'lH ^ Pl 'l'I-- PP ItrFT’-, 2-17. "! iJ.'T ' 5 29++,:' '4‘'C'r* c TO".; UC.' " (X.- .V; 0 '! 0".: V8.: " «"Y < " PFX IY' . ' " ' 2140 PRINT ."PLEASE. ENTER rF.NIMUM AND MAXIMUM '..'AL.UP OF X" = 'il'L: "LURÜLNr VALUES U0=".:UÛ:"J8='UU?: 2130 irr*uT uo, u? •

L134. zr U3: LET U8=IGX LLlOi '"Hi NT ;:i7 0 PRINT " F LEASE ENTER MINIMUM AND MAXIMUM '"'SUE OF Y". 2 1 :2 O lIN T "CURRENT . VALUES V0-'UvO="V3=:'UV9.. 2180 INPUT '/g, V? 21C2 i r V 0 (0 THEN LET . v : 2184 IF VÛXYÜ THEN LET V8=Y8 :. ,

213Ô RETURN • LLYÜ REM -4:.k.+.-r: k-K: : i'- ' T i ; k y -:i CCT I,". ;i" T.r T'f* r>itir.'>pMITTpp p 2 2 :0 (RRINT ...... XEEO '"PINT " ''Mil'- RANEE OF VALUES FOR RRR:-!! 'E !-ER R DO YOU ::IS-i USE" "

22^1'? RPIN'- "INPUT I F 'r i N L PrlD FINOL VALUE AND STFr:' siZ E ": _2<2 PRZF^T "CURRENT VALUES .9 0 = "/AO.. "99=".; A3.: "98=".. ASM T.'pt?T ,VPH C O p p 2 LEO prTMRK- 2 TOO RRT i'I ■*•■•*■ y;>::--::4;p.+:;!;;J.> ;>S:-.l-:+:>;:;+::4-..+;;J. | ;l ;.!.;;4;;.,-::.;:+: CrrT Kji PT r 1 '.Tp C 7 Cp •TT'?P T*-, 221.0 PRINT 2220 P';'U;T 'T'i ro ss EPM'E- +RF RLOTTINC ETE":' SIZE H! X ':':U"'1^NT '.T'ONR 222L' LN^l!! / F

~~J' 0 A !"i p C t . - : •■' i';7 :i.S I-y: y.r.i' j-y •::pr T p'i.j i- r^. r r . C T “' C !-.-r- T r.rr 2020 r:.pp;T ""F.F':::SE EN+E- T!-^E ERIE SIZE FOR y:~~

~~2 0 :0 ,UFMJ" ;N' ^0 '-’O . R2~ U! llO O F\C 7--+ I I ■■<■■•■>: I J : 4. I J : ,J - : : k . ;.k ;4. i.i-y: - -j. . t- y - ;, 7.:. . : -I r.p - p: I ,Tp i r . , . C T ! pOyf-:~~

~~7 ' ]_ l ' . ÿ ■: I I .'.I c ' : - :.i~~

~~■■ ■ 1 7: rrpp. - Tf i - 'o.,.' - -'rrr ' vo~~

~~31 I"0 r r . \. '_'o~~

~~" O : r -V I • . • O '. 0 0~~

~~r-- r p v •.■■ • r-i *'o .:.'■':• ,.*o ■—irp '_*■:>~~

~~72^111 nr:r;|r. Otpp~~

~~•; . •■•'. , » IT7 • • -r ; I~~

r., :\ ,r. 3 3 RO REM ’ - 4 :i M r-C'-f ' fir-; Tf - | p; f"; 7 ! ‘ 3 02 f'RTNT- 3338 NUMERU: ‘.•PLUES"; 3-1.0 LET U=U-2+E2 _ 3RP LF" V-E’ ' E-’ • ■ ■ 3222 IF ' .0=:Xx THEN LE+ U U-Ei'

~~3 3'2E OÛSO0 ôÀ:"n~~

r- IF ::PS(Y321E"3':' -ME^i'LE" Y^O - 101 - ■ I ' ■■ ■ ■ I : I " '7 I i_! ' V 0 58 :244 LET V = Y -E 9 /2 1 _4E , L . " V QOS! ME/iT GO sue ■PE"U"' Ô400 REM C401 LET S: 2402 LET T' LET F' C404 IF :08 3406 IF VC 3408 IF NO IF YO: I F ,F8 : 414 3500 PRINT 3502 PRINT 3504 PR I.NT “•Rir.r'- 350c INPUT 3520 PETUP! 4000 - REM '■■■■• 4010 PRINT 4020 PRINT 4020 I NP'J'^ 4040 f=’PINT 4 050 1020 ■ G'Tp ,'C ci ' I - ? ' 4 p - T I j p 500C P::M ••+■■■ 5 0 0 2 . ■ G0SUB 5004 GOSUP

~~" O R A 5020 LET 5030 G OS 5032 GOS' 504 0~~

~~5050 . Gi~~

~~5070 NEX 50'30 NLiLT~~

~~5000 PC ! UP 6000 REM 2010 r - ' C - I t- 6020 PRPRINT 1 NT "SPECIFY YC'UP FUNCTION AS A SUBROUTINE STARTING AT LINE NO. lOGT: }■ I r; 1~~

~~EOE'P r.-'p T f (T 0070 c -n -1 ►,!i- 6072 PRINT 6074 PRINT 6080 STOP 6000 RETURi~~

~~6100 REM + ' : 6104 PRINT 6106 PR IN " !''.RI.N r 6110 PRINT 6112 • PPir.!T 6114 PRINT STOP 6118 GOSUB~~

p Q T I ! C ' < T ; . ■•T* T' * *1 - •• • 1. - •-1. .1. • 1. .1 . . .’f. .« f. «1". .f. «'f. .V. •*• •• .f. 01011 C‘L G 2 U -; ' GOSUB 0700 6210 PRIN+ 6220 PR IN - "PLOTTER X U:SEP COOpf.'I'IATES " PR I NI U.: V, X, Y

'22+0 RET'.'P”.1 ■; T-M r+; :+:;.k . j.- 7000 REM +" W:+:.•f : :+: .4: :fr+: ;4::+; + * * OPTION L I SI 7002 c o s t Ip 0050 • 7004 PRINT 7010 • • ;+: :+: : H :+• :+: ;+. k::+::-k;. PRINT OPTION LIbT ::+:*:+x+::. l"K +: 7020 PRINT 7022 PRINT "TYPE FIRST *_ET""ER OF OPTION REOUIR ED" 7024 PRINT 7030 ___ PRINT "WINDOW SETS WINDOW :N USER COOF(DINATES" 7040 PRINT "LOCATION — — SETS PLOT LOCATION IN PLOTTER UNITS" 7050 PRINT "FUNCTION.---- —- DEFINES FUNCTION AS Rfi FALGORITHM 7060 PRINT "RANGE ------—- SETS RANGE OF PARAMETER A" 7070 PRINT "STEF" ------— ■— SETS .PLOTTING STEP SIZE IN X" ------7080 PRINT "GRID ------SETS SIZE OF GRID" ":'RINT "MODIFY-—— .• *— CHANGE ANY GRAPHICS PART'METER " 7000 PRINT ------DRAWS GRID & AXES" 7100 ------PRINT "TEXT ------ENABLES TEXT TO BE WRITTEN" 7110 . PRINT PLOTS GRAPHS" 7112 PRINT "DIGI"!SE — INPUT COORDINATES " 7120 PRINT "OPTION ------REPEATS OPTION LIS T 7130 "W Pj_p ------PRINT — REPEATS OPTION LIST 7140 PRINT -— lERM!NATES PROGRAM" 7150 RETURN

~~0000 REM +••* ++++ GRAPHICS : +++++ TEKTRONIX GRAF'HIC S SUBROU"'INES ':27/0 .'78 )~~

:1 • ; k :+: -f; : •+; ; j.; y. ;f ; 0002 REM ; ;f : :k :+; :+; :+; :f: :J.; :+; :ÿ ; ;t; :+: ;+. :+ :+• : i ; : ;4: INITIHLIZA TION 0004 ' GOSUB 05*50 0006 :+: :+: :■ •; :::f. REM ++ DEMENSION GRAPHICS STRJMGS. SET CHARACTER :SIZE

~~0008 DIM 0$[06], T$l:80], w !f[0 ]~~

~~0010 REM U':lMAX), Y (MAX), CMARACTER SIZE 4::+::+:.+:+::.»:-k:-f--f::+;:kr:+::+; :4r ;+: ;+: ;4: ;4: ;4r 4• : .•+: r+: ;+: :+: >f : :+• ;+r ; :4::4:~~

0 0 1 2 P E R D IIP.. Y8, EO 0 0 1 4 DATA 1023, 780/ 14 SOIS LET Û l= " 'CjYTFTXi?-'; >++, /012345679O : , <=:>?" 001? LET 01=0$, "0ABCDEFÛHI TKLMNOPORSTUVNXYZC' J t . ' ' 0018 l-ET ül =U$, " .■■”-'l-'7.><100I>7'l02O'110:: 71047L10R 1 0 F "> "

O0c:0 LET L'.?=U$, "Lllc7"Lll0><12GL><1211><122><123:><124><125>612b><127ï " ■= i'-tifi LL’L.RTI UN I NL'Ul'-l, USER l'ilNDul*!, X—oTEF', X V “ GF:ID, R~F!PNGE 0024 PEAT' U0, V0, ÜO, VÔ, T:0, V0, XO, VO, X7, X8, yp. RP. R8. pc- 0026 DATA 100.. 25.. 1000.. 600, G, 0, 10, 10, . 2, 2.. 2 ,1 , 2.. 1 0040 REM :T:*4:4:M::+:>k:+:**4:*:+::+::+:^::+::+::+:4:4::+::+: SCALE FACTORS IN PLOTTER HNTTS/I'PCR HNITP 0042 l e t oO=ABS'.: (UO-U0)/(XO-X0::') 0044 LET “?=RBS'-:':;V?-V0>/''V8-VR-> > 0046 RETURN 0050 REM CLEARS SCREEN X SETS HL PHA M‘'DE

~~0052 PR I N I ‘'<27><12> 0056 RETURN 0060 REM :| PETS APRP’HTC MPPE~~

~~0062 PRINT "<20::..: " 0064 RETURN 0070 REM +'^++**++:+:+::+++1:+::+::+:++:+:.:>::+:+:+::+.:+::+::+::+::+::+:H:M::,. gETS ALPHA MODE .F HOME~~

0072 LET U=0 0074 ■ LET '.'=V8 0076 GOSUB 0150 0078 PRINT "<3i:>": 0070 RETURN .Ji-iOC C-CM C.,jl iC.C.._CUCTJ..ÜCü JZ__ - l83 ' "’-’-u; : riiL., 0084 RETURN . . 0000 REM 4::+:4:+:4:;.:+:+::4:4r*4::+:4::+:4::+:4::+:**:+:4::+:: 4:4:+*:^ N0 CONTINUE AVAILABLE

8094 RETURN 8100 REM MOVE PEN TO %, V 9102 . PRINT "<29>"; 9104 GOSUB 9300 9106 RETURN 9150 REM 4::+::+::.^::+::^::+::4::+::+::+::+::+::+::+::+::+::i:4::+:H-:>N4:4:*4:H<4:MO'"'E PEN TO | I. 9152 PRINT ’'<29>‘U 9154 GOSUB 9350 9156 RETURN 9200 REM +'+'+:+'‘-"-'-"+:+-+:+”’+”++:+”t:+:i:.v;:+:;+:;+:;+;:+;;+:;+; QP|ÿj_.^ VECT'OR TO %' Y 9202 GOSUB 9300 ' 9206 RETURN 9250 REM +*;++X+^--H”+ + -+:H::4:;+::+::|:>hk+;>f::+;:4::+::4;:+::+: VE'" T'+R TO II.'"' 9252 GOSUB 9350 ' 9256 RETURN 9300 REM :+::+::+::+:4:H::k:f:4::+::+::+::+:4::+:4::+::+:Y::::+:4: J:!, V JQ IJ- V CON'vERSlON 9302 LET U=(X-X0)+S9+.5+U0 9304 LET V=(Y-V0>+T9+. 5+V0 9 :5 0 9352 IF U>U8 ‘"HEN LET U=U8 9354 IF U60 THEN LET U=0 9356 IF \C+-'8 THEN LET V=V8 9359 IF V60 THEN LET V=0 9360 LET N9=:NT': V / 3 2 ) - l PRINT 0$[N9 , N93; 936+ LET n 9= V -32 *N 9+97 9366 PRINT OTCM?, N91; 9368 LET N9=IN'P. U .'32:'+ l

~~9330.. PRINT Of[N9 , N 9j; 9372- LET H9=U-32 +N9+65 9374 PRINT Of[W9 H93; 9376 LET N=N+1 IF N=15+INT •:N/’15> th e n GOSUB 9380 RETURN • 9400 REM : :4:4 : ••+:+:f: :+: :-f : :+: :+: :4: :+:~~

9402 PRINT "631> 9404 GOSUB 9 1 5 0 9+06 RETURN 9500 RE M *'■ + :+; ;+• ;+: :+: :4; ;+; p p , J ^ 9504 GOSUB 9100 9506 PRINT "631> T f 9510 RETURN 'ZiC-cr,-:, _'U RE M 9554 GOSUB 9150 9556 PRINT "63l>' T f 9560 RETURN 9600 REM *+ + 9602 GOSUB 9100 9604 PRINT "<3l>' T6 9606 RETURN Q lT cr Ktl'l •++-i-t'+-+. 9652 GOSUB 9150 9654 PRINT "631>";T6 9656 RETURN •t) —jJiiTy :++++ DIGITISE 9701 LET W f= "" 9702 ■ PRINT 9704 PRINT "MO'-'E CURSOR L THEN PRET 9706 INPUT "627X :26:>'L Hf ' 9710 IF LENCMf)C :5 th e n pPINT "610: :'V Q 7 - 1 :*• icr 1 FN, !.l.f'X:::5_TyCjJ_Q g_Tj:L_.'P3.M P - 9716 LET 19=2 9710 GOSUB 9750 9720 LET 0=32+19 972 2 LET 19=3 9724 GOSUB 9750 9726 LET U=U+I9 ■ 9728 LET 19=4 ■ 9730 GOSUB 9750 9732 LET V=32+I9 9734 LET 19=5 9736 GOSUB 9750 9738 LET V =V +I9 9742 LET X=(U-U0 9744. LET Y=(V-V0 9746 RETURN 9750 REM :+ CONVERSION OF STRING TO A NUMERICAL VALUE

~~9754 FOR I 3=1 TO 9756 IF Wf [ 19.. . 9758 LET 19= IS 9760 LET 18=32 9762 NEXT IS 976.4 RETURN QQQQ END~~