SIM UNIVERSITY
SCHOOL OF SCIENCE AND TECHNOLOGY
MODELING OF SEMICONDUCTOR
PHOTODETECTORS
STUDENT
SUPERVISOR
PROJECT CODE
: LIM RENJUN LOUIS (Z0706712)
: YEE MUN CHUN MARCUS
: JAN2010/ENG499/2010
A project report submitted to SIM University
In partial fulfilment of the requirements for the degree of
Bachelor of Engineering (or Bachelor of Electronics)
November 2010
ABSTRACT
This project involves MODELING OF SEMICONDUCTOR PHOTODETECTORS with
Multiplication measurements on GaAs, InP, InGaAs, GaInP, p+-i-n+s with –region
thicknesses, with investigation of applicability of the local ionization theory. Local
expressions for multiplication can predict the measured values surprisingly well in p+-i-n+s
with thicknesses, as thin as 0.2 m before the effect of dead-space, where carriers have
insufficient energy to ionize, and causes significant errors. Only a very simple correction to
the local expressions is needed to predict the multiplication accurately where the field
varies rapidly in abrupt one-sided p+-n junctions. A local ionization coefficient to be
increasingly unrepresentative of the position dependent values in the device as is reduced
below 1 um. The success of the local model in predicting multiplication is therefore
attributed to the dead-space information already being contained within the experimentally
determined values of local coefficients. This suggested that these should therefore be
thought of as effective coefficients, which, despite the presence of dead-space effects, can
be, still be used with the existing local theory for efficiently quantifying multiplication and
breakdown voltages.
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ACKNOWLEDGEMENT
The author would sincerely express his thanks to the following people for their
continued help, support and guidance throughout this project. The work presented in this
report would not been possible without them:

DR YEE MUN CHUN MARCUS, author’s project for all his immensely useful
suggestions, ideas, and time. His encouragement, guidance and advice were
invaluable in keeping the project on track. I also wish to thank Dr Marcus for
constantly giving me encouragement, which have lead me in building positive
attitude towards learning this new technological research project and also
understanding.

The author’s friends for the sharing of their knowledge and guidance for the
improvement for this project throughout the duration of this project.

Finally, author’s biggest thank you goes to his parents for their unlimited
understanding, guidance and support.
Words can never describe how grateful the author is to those mentioned above.
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TABLE OF CONTENTS
PAGE
ABSTRACT
i
ACKNOWLEDGEMENT
ii
LISTS OF FIGURES
iii - iv
CHAPTER 1
INTRODUCTION
1.1
PROJECT BACKGROUND
1
1.2
PROJECT OBJECTIVE
2
1.3
OVERALL PROJECT OBJECTIVE
2
1.4
PROPOSED APPROACH
2
1.5
SKILLS REVIEW
3
CHAPTER 2
LITERATURE REVIEW
2.1
Semiconductors
4
2.2
Semiconductors materials
4
2.3
III-V semiconductor materials
5
2.3.1
Intrinsic semiconductors
5
2.3.2
Extrinsic semiconductors
6
2.4
Band gap
7
2.5
Gallium arsenide (GaAs)
9
2.5.1
9
GaAs advantages and disadvantages
2.6
Indium phosphide (InP)
10
2.7
Indium gallium arsenide (InGaAs)
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2.8
Gallium indium Phosphide (GaInP)
12
2.9
Merits and Limitations of Local Impact Ionization Theory
13
2.10
Physics of Impact Ionization
2.10.1
2.10.2
2.10.3
2.10.4
2.10.5
2.10.6
Introduction
Impact Ionization Gain Mechanism
Ionization Threshold Energy
Ionization Coefficients and Gain Equations
Impact Ionization Coefficient Measurement
Impact Ionization Response
15
16
17
20
23
25
CHAPTER 3
SIMULATION AND DISCUSSION
3.1
Impact Ionisation Coefficients and Multiplication of
electrons and holes for GaAs
29
3.2
Impact Ionisation Coefficients and Multiplication of
electrons and holes for InGaAs
31
3.3
Impact Ionisation Coefficients and Multiplication of
electrons and holes for InP
34
3.4
Impact Ionisation Coefficients and Multiplication of
electrons and holes for GaInP
36
3.5
Discussion
38
PROBLEMS ENCOUNTERED AND SOLUTIONS
42
CHAPTER 4
4.2
CHAPTER 5
CONCLUSION
5.1
FUTURE WORK
44
5.2
SUMMARY
46
5.3
REFLECTION
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REFERENCES
48
APPENDIX A -
Gantt chart FOR PROJECT PLANNING
50
APPENDIX B -
PROGRAM DESIGN FOR
Gallium arsenide (GaAs)
51
APPENDIX C -
PROGRAM DESIGN FOR
Indium phosphide (InP)
52
APPENDIX D -
PROGRAM DESIGN FOR GaInP
53
APPENDIX E -
PROGRAM DESIGN FOR
Indium gallium arsenide (InGaAs)
54
APPENDIX F-
PROGRAM DESIGN FOR
Multiplication of electrons and holes function
55
APPENDIX G-
PROGRAM DESIGN FOR
Voltage breakdown versus thickness
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LIST OF FIGURES
Figure 1: impact ionization
Figure 2: Semiconductor band structure
Figure 3: Gallium arsenide.
Figure 4: Indium phosphide.
Figure 5: p   n   n  diode layer structure
Figure 6: Energy Band diagram of a reverse biased PIN structure where impact ionization
process occurs as the electrons travels through the high electric field region.
Figure 7: Energy band gap versus lattice constant for III-V compound alloy system
Figure 8: Silicon energy band structure. Notice that due to the large separation between the
Г and X valleys, the inter valley transition shown here is only possible when a high electric
field is applied.
Figure 9: Schematic view of a semiconductor used to calculate current gain. The electric
field direction, current flow, and boundary conditions are also shown.
Figure 10: Multiplication gain M versus αL for pure electron injection. Various α/β value
are used to demonstrate its effect on avalanche breakdown curve
Figure 11: (a) Schematic view of a p-i-n diode used for measuring the electron ionization
coefficients. High energy light illumination is used to ensure that photocurrent is created
very close to the surface. (b) Hole ionization coefficient measurement using the same setup
but illuminating the diode from the n+ side.
Figure 12: Representation of impact ionization process. (a) Only electron initiated impact
ionization.
(b) Both carriers initiated impact ionization.
Figure 13: Calculated bandwidth versus gain in a P-I-N photodiode for various values of
β/α. Plot indicates that the bandwidth will not be limited by the gain as long as M< α/
β[49].
Figure 14: Frequency Response of a P-I-N photodiode with 1m thick multiplication layer.
Notice that parameter k is define as β/α
Figure 15: Measured M e for a w  0.48m device as a function of temperature
Figure 16: Impact Ionisation Coefficients for GaAs versus inverses electric field
Figure 17: simulated of multiplication w  0.48m device as a function of temperature from
20K to 500K
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Figure 18: Measured (symbols) and calculated (lines) multiplication characteristics of 1.3
and 1.9 um thick InGaAs p-i-n diode (filled symbols) from 20–400 K [18]
Figure 19: Impact Ionisation Coefficients for InGaAs versus inverses electric field
Figure 20: Simulated of multiplication characteristics of 1.3 and 1.9 um thick
InGaAs p-i-n diode from 20–400 K
Figure 21: Measured Me (symbols) Calculated Me (solid lines) using bulk ionization
coefficients for InP
Figure 22: Impact Ionisation Coefficients for InP versus inverses electric field
Figure 23: p   n   n  diode layer structure and the measured results of
Mn and Mp as a function of reverse bias
Figure 24: p   n   n  diode layer structure and the measured results of
Mn and Mp as a function of reverse bias
Figure 25: Impact Ionisation Coefficients for GaInP versus inverses electric field
Figure 26: Simulated of multiplication characteristics of 0.24 um thick for GaInP
p-i-n diode from 300 K
Figure 27: Simulated of voltage breakdown versus thickness for GaAs
Figure 28: Simulated of voltage breakdown versus thickness for GaInP
Figure 29: Simulated of voltage breakdown versus thickness for InP
Figure 30: Simulated of voltage breakdown versus thickness for InGaAs
Figure 31: Silicon Avalanche Photodiode
Figure 32: Schottky Barrier diode
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CHAPTER 1
INTRODUCTION
1.1 Project Background
Photo detectors are used primarily as an optical receiver to convert light into electricity.
The principle that applies to photo detectors is the photoelectric effect, which is the effect
on a circuit due to light. Max Planck In 1900 discovered that energy is radiated in small
discrete units called quanta; he also discovered a universal constant of nature which is
known as the Planck’s constant. Planck’s discoveries lead to a new form of physics known
as quantum mechanics and the photoelectric effect E = hv which is Planck constant
multiplied by the frequency of radiation. The photo electric effect is the effect of light on a
surface of metal in a vacuum, the result is electrons being ejected from the surface this
explains the principle theory of light energy that allows photo detectors to operate. Photo
detectors are commonly used as safety devices in homes in the form of a smoke detector,
also in conjunction with other optical devices to form security systems.
A photo detector operates by converting light signals that hit the junction to a voltage or
current. The junction uses an illumination window with an anti-reflect coating to absorb the
light photons. The result of the absorption of photons is the creation of electron-hole pairs
in the depletion region. Examples of photo detectors are photodiodes and phototransistors.
Other optical devices similar to photo detectors are solar cells which also absorb light and
turn it into energy. A similar but different optical device is the LED, which is basically the
inverse of a photodiode, instead of converting light to a voltage, or current, it converts a
voltage or current to light.
Impact Ionization is the process in a material by which one energetic charge carrier can lose
energy by the creation of other charge carriers. For example, in semiconductors, an electron
(or hole) with enough kinetic energy can knock a bound electron out of its bound state (in
the valence band) and promote it to a state in the conduction band, creating an electron-hole
pair.
Figure 1: impact ionization
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1.2 Project objective
Objective of this project is to develop a modelling of semiconductor photo detector.
In order to do so, an assembly language is being use to write a code to calculate the impact
ionization. So as to derive the gain equation for impact ionization process, we need to
define impact ionization coefficient & breakdown voltage.
The secondary aim is to improve the equation to calculate avalanche photodiode.
1.3 Overall objective
Overall objective is able to understand the physics of semiconductor photo detectors as well
as able to use an assembly language to write a program able to calculate the impact
ionizations coefficients with ease.
1.4 Proposed approach
Proposed approaches are to research and study impact ionization theory and understand the
concept of it. Learning the process and the principle to detect and amplify very low light
signals and still used as the most sensitive detectors.
Choosing an assembly language suitable for writing a code for this equation is taken into
consideration. Most important factor is to study and understand with is the correct equation
to implement in which able to use the equation using an assembly language.
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1.5 Skills review
Skills
Methods / Sources
Articles and journals from the internet
Exploring of ideas and
gathering information
Reference books and technical papers from library
Brainstorming of ideas
Literature review
Project tutor
Assessing and evaluating
project progress
Targets and goals
Management of project
Data review
Developing of codes
Software development
Codes testing and debugging
Final codes developments
Final assembly and testing
Final assembly of software
Evaluation and test out
Report writing skills
Presentation
Oral presentation skills
Collecting and organizing of data and diagrams
Computer literacy
Microsoft office and other applications proficiency
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CHAPTER 2
LITERATURE REVIEW
2.1
Semiconductors
Semiconductors have bulk resistivity in the range of 10-4 ohm-cm (heavily doped) to 103
ohm-cm (undoped, or intrinsic). That's seven orders of magnitude! Semiconducting
elements include silicon and germanium, it is no coincidence they are both from group 4 of
the period table. Semiconducting compounds include gallium arsenide, indium phosphide,
and gallium nitride, from groups 3/5 or 2/6 of the period table.
Semiconductor materials have electrons in their outer shell. When bonded together in a
crystal lattice, atoms share electrons such that they each have eight electrons in the outer
shell.
Electrons are somewhat loosely bound so they can become carriers in the presence of an
electric field.
2.2
Semicondictors materials
Semiconductors materials are materials, which are insulators at absolute zero temperature
but which conduct electricity at room temperature. Property of a semiconductor material
that can be doped with impurities that alter its electronic properties in a controllable way
Because of their application in devices like transistors and lasers, the search for new
semiconductor materials and the improvement of existing materials is an important field of
study in materials science.
Most commonly used semiconductor materials are crystalline inorganic solids. These
materials are classified according to the periodic table groups of their constituent atoms.
Semiconductor materials are differing by their properties. Compound semiconductors have
advantages and disadvantages in comparison with silicon. For example gallium arsenide
has six times higher electron mobility than silicon, which allows faster operation; wider
band gap, which allows operation of power devices at higher temperatures, and gives lower
thermal noise to low power devices at room temperature; its direct band gap gives it more
favourable optoelectronic properties than the indirect band gap of silicon; it can be alloyed
to ternary and quaternary compositions, with adjustable band gap width, allowing light
emission at chosen wavelengths, and allowing e.g. matching to wavelengths with lowest
losses in optical fibres. GaAs can be also grown in a semi insulating form, which is suitable
as a lattice-matching insulating substrate for GaAs devices. Conversely, silicon is robust,
cheap, and easy to process. On the other hand, GaAs is brittle, expensive, and just growing
an oxide layer cannot create insulation layers; GaAs is therefore used only where silicon is
not sufficient.
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Some materials are tuneable by alloying multiple compound semiconductors, e.g., in band
gap or lattice constant. The result is ternary, quaternary, or even quinary compositions.
Ternary compositions allow adjusting the band gap within the range of the involved binary
compounds; however, in case of combination of direct and indirect band gap materials there
is a ratio where indirect band gap prevails, limiting the range usable for optoelectronics;
e.g. AlGaAs LEDs are limited to 660 nm by this. Lattice constants of the compounds also
tend to be different, and the lattice mismatch against the substrate, dependent on the mixing
ratio, causes defects in amounts dependent on the mismatch magnitude; this influences the
ratio of achievable radioactive / non radioactive recombination and determines the
luminous efficiency of the device. Quaternary and higher compositions allow adjusting
simultaneously the band gap and the lattice constant, allowing increasing radiant efficiency
at wider range of wavelengths; for example AlGaInP is used for LEDs . Materials
transparent to the generated wavelength of light are advantageous, as this allows more
efficient extraction of photons from the bulk of the material. That is, in such transparent
materials, light production is not limited to just the surface. Index of refraction is also
composition-dependent and influences the extraction efficiency of photons from the
material.
2.3
III-V semiconductor materials
Three-five materials refer to compound semiconductors made from one element from
Group III on the periodic chart (arsenic in the case of GaAs) and one from Group V
(gallium in the case of GaAs). Other three-five (or III-V in Roman numerals)
semiconductors include indium phosphide and gallium nitride.
2.3.1 Intrinsic semiconductors
A semiconductor in which the concentration of charge carriers is characteristic of the
material itself rather than of the content of impurities and structural defects of the crystal is
called an intrinsic semiconductor
Electrons in the conduction band and holes in the valence band are created by thermal
excitation of electrons from the valence to the conduction band. Thus an intrinsic
semiconductor has equal concentrations of electrons and holes. The carrier concentration,
and hence the conductivity, is very sensitive to temperature and depends strongly on the
energy gap. The energy gap ranges from a fraction of 1 eV to several electron volts. A
material must have a large energy gap to be an insulator.
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2.3.2 Extrinsic semiconductors
An ordered bonding of the individual atoms to form the crystal structure forms typical
semiconductor crystals such as germanium and silicon. The bonding is attributed to the
valence electrons, which pair up with valence electrons of adjacent atoms to form so-called
shared pair or covalent bonds. These materials are all of the type; that is, each atom
contains four valence electrons, all of which are used in forming the crystal bonds.
Atoms having a valence of +3 or +5 can be added to a pure or intrinsic semiconductor
material with the result that the +3 atoms will give rise to an unsatisfied bond with one of
the valence electrons of the semiconductor atoms, and +5 atoms will result in an extra or
free electron that is not required in the bond structure. Electrically, the +3 impurities add
holes and the +5 impurities add electrons. They are called acceptor and donor impurities,
respectively. Typical valence +3 impurities used are boron, aluminium, indium, and
gallium. Valence +5 impurities used are arsenic, antimony, and phosphorus.
Semiconductor material “doped” or “poisoned” by valence +3 acceptor impurities is termed
whereas material doped by valence +5 donor material is termed n-type. The names are
derived from the fact that the holes introduced are considered to carry positive charges and
the electrons negative charges. The number of electrons in the energy bands of the crystal is
increased by the presence of donor impurities and decreased by the presence of acceptor
impurities.
At sufficiently high temperatures, the intrinsic carrier concentration becomes so large that
the effect of a fixed amount of impurity atoms in the crystal is comparatively small and the
semiconductor becomes intrinsic. When the carrier concentration is predominantly
determined by the impurity content, the conduction of the material is said to be extrinsic.
Physical defects in the crystal structure may have similar effects as donor or acceptor
impurities. They can also give rise to extrinsic conductivity.
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2.4
Band gap
Energy gap or band gap is an energy range in a solid where no electron states can exist. In
graphs of the electronic band structure of solids, the band gap generally refers to the energy
difference (in electron volts) between the top of the valence band and the bottom of the
conduction band in insulators and semiconductors. This is equivalent to the energy required
to free an outer shell electron from its orbit about the nucleus to become a mobile charge
carrier, able to move freely within the solid material. In conductors, the two bands often
overlap, so they may not have a band gap.
In semiconductors and insulators, electrons are confined to a number of bands of energy,
and forbidden from other regions. The term "band gap" refers to the energy difference
between the top of the valence band and the bottom of the conduction band. Electrons are
able to jump from one band to another. However, in order for an electron to jump from a
valence band to a conduction band, it requires a specific minimum amount of energy for the
transition. The required energy differs with different materials. Electrons can gain enough
energy to jump to the conduction band by absorbing either a phonon (heat) or a photon
(light).
A semiconductor is a material with a small but nonzero band gap, which behaves as an
insulator at absolute zero but allows thermal excitation of electrons into its conduction band
at temperatures, which are below its melting point. In contrast, a material with a large band
gap is an insulator. In conductors, the valence and conduction bands may overlap, so they
may not have a band gap.
The conductivity of intrinsic semiconductors is strongly dependent on the band gap. The
only available carriers for conduction are the electrons, which have enough thermal energy
to be excited across the band gap.
Band gap engineering is the process of controlling or altering the band gap of a material by
controlling the composition of certain semiconductor alloys, such as GaAlAs, InGaAs, and
InAlAs. It is also possible to construct layered materials with alternating compositions by
techniques like molecular beam epitaxy. These methods are exploited in the design of
heterojunction bipolar transistors (HBTs), laser diodes and solar cells.
The distinction between semiconductors and insulators is a matter of convention. One
approach is to think of semiconductors as a type of insulator with a narrow band gap.
Insulators with a larger band gap, usually greater than 3 eV,[citation needed] are not
considered semiconductors and generally do not exhibit semiconductive behaviour under
practical conditions. Electron mobility also plays a role in determining a material's informal
classification.
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The band gap energy of semiconductors tends to decrease with increasing temperature.
When temperature increases, the amplitude of atomic vibrations increase, leading too larger
inters atomic spacing. The interaction between the lattice phonons and the free electrons
and holes will also affect the band gap to a smaller extent. The relationship between band
gap energy and temperature can be described by Vishnu’s empirical expression,
E g T   E g (0) 
T 2
T 
, where Eg(0), α and β are material constants.
In a regular semiconductor crystal, the band gap is fixed owing to continuous energy states.
In a quantum dot crystal, the band gap is size dependent and can be altered to produce a
range of energies between the valence band and conduction band. It is also known as
quantum confinement effect.
Band gaps also depend on pressure. Band gaps can be either direct or indirect, depending
on the electronic band structure.
Figure 2: Semiconductor band structure.
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2.5
Gallium arsenide (GaAs)
A compound of the elements gallium and arsenic it is an III/V semiconductor, and is used
in the manufacture of devices such as microwave frequency integrated circuits, monolithic
microwave integrated circuits, infrared light-emitting diodes, laser diodes, solar cells, and
optical windows.
Figure 3: Gallium arsenide.
2.5.1 GaAs advantages and disadvantages
Advantage
GaAs is a direct band gap semiconductor, while Si is an indirect band gap semiconductor.
Direct bandgap semiconductors can be used to make semiconductor lasers and light
emitting diodes (LED's).
Form high-quality hetero junctions with a large number of alloys, such as InGaAs, AlGaAs,
GaAsP, and GaAsSb. These heterojunctions can be used to construct quantum wells, which
are used for semiconductor lasers. Also it has a peak saturation velocity of 2x107 cm/sec
(twice that of Si) and an electron mobility of 8000 to 9000 cm 2 /Vsec (10-20 times that of
Si). These properties suggest that GaAs devices could switch faster than their Si
counterparts. GaAs has a larger band gap than Si, so it is more immune to the radiation
effects present in space and military applications.
Disadvantage
No high-quality insulators (counterparts to silicon dioxide) grow on GaAs. The
insulator/GaAs interfaces have high levels of defects called "traps" that make GaAs
MOSFET's impossible. This is probably the main reason that GaAs will never supplant Si
as the semiconductor-of-choice for integrated circuits. GaAs transistors are usually either
MESFET's or HEMT's. These transistors have some advantages over MOSFET's but their
disadvantages outweigh the advantages. It is extremely brittle. For this reason, GaAs wafers
are typically smaller than three inches in diameter, as opposed to silicon wafers, which
commonly have diameters of eight or more inches. Since the area on a wafer is proportional
to the square of its diameter, Si has a huge cost advantage over GaAs. The hole mobility in
GaAs is 15-20 times smaller than the electron mobility, which makes CMOS-like circuits
unattractive
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2.6
Indium phosphide (InP)
A binary semiconductor composed of indium and phosphorus. It has a face-centered cubic
("zincblende") crystal structure, identical to that of GaAs and most of the III-V
semiconductors.
Figure 4: Indium phosphide.
InP is used in high-power and high frequency electronics because of its superior electron
velocity with respect to the more common semiconductors silicon and gallium arsenide. It
also has a direct bandgap, making it useful for optoelectronics devices like laser diodes. InP
is also used as a substrate for epitaxial indium gallium arsenide based opto-electronic
devices. Indium phosphide also has one of the longest-lived optical phonons of any
compound with the zincblende crystal structure.
Semiconducting material used as a doping agent in production of microchips. Originally
investigated by DARPA and NASA, this technology is now being integrated by
corporations. Carriers move through it faster than any semi conducting material known to
man, also at lower power consumption. It is also one in a small number of photonic
semiconductors. According to Aaron Bond, the chief technology officer at T-Networks,
among one of the properties of indium phosphate is its unique ability to generate, modulate,
amplify, and receive light at Telco wavelengths (1.55 - 1.3 microns). This wavelength is
used by the telecommunications industry in its single-mode optical fibres. This is to be the
next step in the computing industry between the transistor, and the carbon Nan tubes.
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2.7
Indium gallium arsenide (InGaAs)
Indium gallium arsenide (InGaAs) is a semiconductor composed of indium, gallium and
arsenic. It is used in high-power and high-frequency electronics because of its superior
electron velocity with respect to the more common semiconductors silicon and gallium
arsenide. InGaAs band gap also makes it the detector material of choice in optical fiber
communication at 1300 and 1550 nm. Gallium indium arsenide (GaInAs) is an alternative
name for InGaAs.
The indium content determines the two-dimensional charge carrier density.
Properties
Energy gap versus gallium composition for InGaAs
The optical and mechanical properties of InGaAs can be varied by changing the ratio of In
and Ga, InxGa1-xAs. The InGaAs device is normally grown on an indium phosphide (InP)
substrate. In order to match the lattice constant of InP and avoid mechanical strain,
In0.53Ga0.47As, this composition has a cut-off wavelength of 1.68 μm.
By increasing the ratio of In further compared to Ga it is possible to extend the cut-off
wavelength up to about 2.6 µm. In that case special measures have to be taken to avoid
mechanical strain from differences in lattice constants. GaAs is lattice mismatched to Ge by
0.08%. With the addition of 1.5% in to the alloy, InGaAs, becomes perfectly latticed
matched to Ge. The complete elimination of film stress reduces the defect densities of the
epi InGaAs layer compared to straight GaAs.
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2.8
Gallium indium Phosphide (GaInP)
GaInP, a wide band gap semiconductor lattice matched to GaAs, is of interest for a variety
of device applications such as heterojunction bipolar transistors (HBTs) And heterojunction
field-effect transistors (HFETs) The advantages of GaInP over GaAlAs for GaAs-based
HBT applications include its large energy band gap (1.9 eV), lower conduction band offset,
reduced deep-level concentration, and easier selective etching. GaInP-emitter HBTs with
high current gain and good microwave performance have been reported. The use of GaInP
in the collector region of HBTs is also attracting attention because of its high breakdown
voltage potential. For the collector material of high-voltage microwave HBTs, an important
figure of merit is the product of breakdown electric field and carrier saturation velocity
Which is roughly proportional to the product of breakdown voltage and current gain cutoff
frequency attainable in the transistor GaInP has a potentially higher breakdown field
saturation velocity product than other materials lattice matched to GaAs. To design
structures for high-voltage or high-power applications, an accurate knowledge of impact
ionization coefficients in GaInP is necessary for calculating breakdown characteristics of
junctions. In the work reported here, photocurrent multiplication was used to measure the
electron and hole ionization coefficients in ~100 GaInP by illuminating p   n   n  diode
structures from either side with above band gap radiation.6 The results show that GaInP has
significantly lower values of a and b than those of GaAs or InP, a promising indication for
high-voltage applications.
Figure 5: p   n   n  diode layer structure
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2.9
Merits and Limitations of Local Impact Ionization Theory
Accurate determination of the electron and hole ionization coefficients,  and 
respectively, is important, since these are used to determine avalanche multiplication
characteristics and breakdown. Conventionally  and  are assumed to depend only on the
local electric field, and the mean multiplication due to an electron-hole pair generated at
position is given by
M  x0  
 W

exp      dx 
 x0

W
 x

1    exp      dx  dx
0
 0

As described by Stillman and Wolffe [7] where W is the total depletion width. The electric
field exists between x=0 and x=W causing electrons to move from left to right. This
expression is also traditionally used to derive the values of,  and  from photo
multiplication measurements performed with carrier injection from the depletion region
edges. [7]-[8]
For electron multiplication, M e x o =0 and for pure hole multiplication M h , x o =W.
However, carriers entering the high field region with energy much less than the ionization
threshold must traverse a dead space, distance d e , for electrons or d h for holes, before they
acquire sufficient energy to impact ionize. With estimation this dead space distance is given
by equating it to the ballistic distance a carrier requires to reach the ionization threshold
energy, Eth i.e. d  Eth / qF .
For a device where electrons are injected at x  0 , these corrections usually disallow
electron ionization in a dead-space region from 0  x  d e and hole ionization in the region
from W  d h  x  W . Okuto and Crowell [9] presented an approximate expression relating
multiplication to the ionization coefficients while accounting for the reduced multiplication
by these regions. Bulmanet al[10]. simplified their expression to interpret the measured
multiplication results from p -n-n junctions by assuming no electron-initiated ionization
occurs within a distance d e from their injection point.
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Hole dead-space effects in the region from W  d h to were ignored since the electric field
there was small so its contribution to the multiplication was assumed negligible. In recent
years, several groups have suggested that the effect of dead-space is to reduce the mean
value of multiplication below the prediction of a local model [11]-[14]. To account for
dead-space regions within the local framework, Di Carlo [15] and Lugli [16] and Wilson
indicated that they should be included in the electron and hole current multiplication
equations, which are then solved numerically. They concluded that the simple
modifications to one of the type implemented by Bulman et al.[10] do not fully correct for
dead-space and lead to an overestimation of the high multiplication values and thus to an
underestimation the breakdown voltage. However, the validity of their comparisons is
unclear since coefficients, which enter the theories of, describe carriers, which have already
travelled the dead space and so are generally different to those in the conventional local
theory described or used by Bulman et al.[10]
As the size of devices continues to shrink leading to higher electric fields, impact ionization
will become increasingly important in device design. In submicron devices it would be
expected that nonlocal aspects of carrier transport would have to be considered. However,
practically all published ionization data to date have been in the form of local coefficients.
Moreover, the complexity of the alternative methods of analysis make it difficult to
interpret multiplication measurements especially where structures are investigated where
the field varies rapidly, as argued in G. E. Bulman, V. M. Robbins, and G. E. Stillman[10],
It is therefore important to understand both when nonlocal effects become important and
their effect on the multiplication characteristics and breakdown voltage. From this
understanding, the limitations on the applicability of the local model can be identified
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2.10
Physics of Impact Ionization
2.10.1 Introduction
The impact ionization process in semiconductors has been observed and studied since 1950.
In the past several decades, there has been more widespread interest in its applications due
to the rapid progress in optical communication systems. Avalanche photo detectors (APDs)
use the impact ionization principle to detect and amplify very low light signals and they are
still used as the most sensitive detector for most systems. It is found that to have a highgain, low-noise APD, the ionization rate for electrons and holes needs to be very different.
Efforts to find the best material to perform impact ionization reveal that silicon has the
largest ionization rate difference between electrons and holes among all semiconductors.
Due to its cut-off wavelength around 1 μm, however, silicon has rarely been employed for
today’s optical communication systems that use 1.55 μm wavelengths light. Suitable
materials using combinations of binary and ternary/quaternary III-V semiconductors such
as InGaAs on InP substrates have provided solutions for longer wavelength detection.
It is very important to note that most III-V semiconductors have similar ionization rate for
electrons and holes. Therefore, they create much more noise than silicon for equivalent gain
regions at equivalent gains. Because the SIM can operate with arbitrary photodiodes,
silicon of course is the ideal material to enhance detector sensitivity and amplification for
minute signals which cannot be obtained by other materials. In this chapter, we start with a
brief discussion of the gain mechanism for impact ionization. Threshold energy and impact
ionization gain equations will then be derived. Methods used for measuring the impact
ionization coefficient will also be discussed. At the end, the noise associated with the
impact ionization process will be addressed.
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2.10.2 Impact Ionization Gain Mechanism
The impact ionization gain mechanism can be demonstrated by using a thermally generated
electron, or an electron created by an absorbed photon, travelling inside a semiconductor
where a depletion region is formed. Figure 6 illustrates the impact ionization process in a
reverse biased PIN. As shown in the figure, electrons can gain sufficient kinetic energy
while travelling in a high electric field. If the electric field is high enough, this high-energy
electron may initiate the electron-electron scattering so that an electron in the valence band
can be excited to the conduction band. As a result of this, another electron-hole pair is
produced by promoting an electron from the valence band into the conduction band. Due to
the strong electric field, the subsequent electron and hole will continue to collide with the
lattice and create more electron-hole pairs. Therefore, numerous carriers are generated and
the result is a multiplied current output. This phenomenon is sometimes referred to as the
avalanche breakdown.
Figure 6: Energy Band diagram of a reverse biased PIN structure where impact
ionization process occurs as the electrons travels through the high electric field region.
The energy required to initiate impact ionization depends on the band gap of the material.
The reason for this can be found in Fig. 6 where an electron transition from valence band to
conduction band is necessary for carrier multiplication. In low band gap semiconductors,
such as InAs, an electric field of 104 V/cm is required. For wide band gap materials, such
as GaP, the field required is greater than 105 V/cm.
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The energy band gap diagram and lattice constant for various III-V compounds
semiconductors is shown in Fig. 7[2] addition to these, the band gap of silicon is 1.12 eV
which is not shown in the plot.
Figure 7: Energy band gap versus lattice constant for III-V compound alloy
system[2]
2.10.3 Ionization Threshold Energy
The minimum energy required to excite an electron from the valence to the conduction
band is equal to the band gap energy of the semiconductor. Impact ionization, however, is
more than just freeing an electron from the valence band. In order to decide the ionization
threshold energy, various methods including parabolic, non parabolic, realistic, and non
local pseudo potential band structure have been utilized. It is found that the threshold
energy differs for most semiconductors. It is a function of the band structure, effective mass
ratio between electron and hole, density of state, phonon interaction, and spin-orbit splitting
energy.
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The best way to estimate the ionization threshold energy involves a simple two parabolic
band model. In this model, we consider one conduction band with effective mass me and
one valence band with mass m h . As shown in Fig. 6, prior to the collision, the electron
travelling from the left-hand side has a kinetic energy of 1/2 me vi2 and a momentum
of m e vi , where vi is the initial velocity of the electron. After collision, three carriers exist: a
new electron-hole pair plus the original electron. Electrons continue travelling to the right
and the hole to the left.
Assuming that the collision is elastic, the conservation of energy and momentum must be
satisfied these two assumptions are summarized as
1
1
1
2
2
2
me v t  E g   me v e  2   m h v h 
2
2
2

2.1
And
me vt  me vi  mh vh
2.2
Where E g is the energy band gap of the semiconductor, ve is the electron velocity, and vh
is the hole velocity after the collision. Note that when me  mh and ve  vh  vi , (2.1) and
(2.2) can be derive that the required initial electron energy for ionization process as
Ei 
1
2
m e v i  1 .5 E g
2
2.3
This is the well-known 3/2-band-gap rule for ionization threshold energy in semiconductor
[1].
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It is found that for wide and indirect band gap materials such as silicon, the calculations for
the conduction band in the high energy regime become more complicated. Lots of research
has been devoted to this area. An overriding principle is that the ionization process must
always satisfy the energy and momentum conservation. It is found that for silicon, the
ionization threshold energy is 3.6 eV for electrons and 5.0 for holes. The reason why they
are both greater than 1.5 E g ( E g = 1.12 eV) is because of the indirect band gap nature of
Si which requires extra energy for electrons to transit from the Г valley to the X valley.
This type of transition is called inter valley scattering. The Si energy band diagram using a
non local pseudo potential calculation is shown in Fig 8 [3] to illustrate the process.
Figure 8: Silicon energy band structure. Notice that due to the large separation
between the Г and X valleys, the inter valley transition shown here is only possible
when a high electric field is applied.
There is no simple explanation for why electrons ionize much more readily than holes in
silicon. One major reason for this is because the minimum energy of the second conduction
band in the X valley located very close to the main conduction band effectively increases
the total density of states for the conduction band. The energy difference between these two
bands is calculated to be only ~ 0.1 eV which allows electrons to transfer between these
two with little resistance.
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2.10.4 Ionization Coefficients and Gain Equations
To derive the gain equations for impact ionization process, we need to first define the
impact ionization coefficient. For most semiconductor materials, the impact ionization
process is asymmetric to some degree for electrons and holes. For example, the ionization
rate for electrons is about 5 times greater than holes in Germanium. For silicon, however,
electrons can ionize 50 times to 1000 times more readily than holes depending on the
electric field amplitude. The probability for initiating impact ionization is quantified as the
impact ionization coefficient. It is defined as the reciprocal of the average distance travelled
by an electron or hole to produce an electron-hole pair. Therefore, its unit is cm 1 . For
electrons, the coefficient is denoted as α. For holes, it is denoted as β. Materials with very
different value for α and β, such as in silicon can create less impact ionization noise
because only one type of carrier is dominant during the ionization process. With these
definitions, we are ready to derive the gain equations in the following paragraph.
Figure 9 shows a schematic geometry for a semiconductor region. We will use this simple
structure to explore the gain equations. In this figure, the current density for holes and
electrons is denoted as j p and j n . All current flows in the same direction as the electric
field. The electron current increases with increasing x while hole current decreases with
increasing x. Under dc conditions, the total current J is the sum of the electron current and
hole current, j  j n ( x)  j p ( x) = constant.
Figure 9: Schematic view of a semiconductor used to calculate current gain. The
electric field direction, current flow, and boundary conditions are also shown.
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Prior to the gain derivation, several assumptions need to be made. First, the length of the
semiconductor is long (L >1 μm) such that non-local theories [4] are not considered.
Secondly, the current density is low to avoid the space charge effect which may screen the
electric field and lower the gain. A differential equation can be used to describe the
multiplication process in terms of ionization rates α and β.
If β is much smaller than α, (2.4) can be rewritten as
Mn 
Mp 
Mn 
1

1
 
exp  x   
1
 
    exp L   
   exp L   
2.4

    exp L   
   exp L   
2.5
L
0
1


exp  x   
exp( L )

1  exp( L )

L
0
2.6
A positive feedback factor (β/α) in (2.6) shows the effect of ionization coefficients on
multiplication gain. As β gets bigger, but still much smaller than α, M n can reaches
breakdown in a shorter distance.
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If we consider an extreme case, β = 0, (2.6) can be further simplified to
L

M n  exp   dx   exp L 
0

2.7
From (2.7) we observe that when β = 0, there is no avalanche breakdown because M n just
continues to increase exponentially with L . A plot is given in Fig. 10 using (2.6) to
demonstrate the gain versus L for various value of (α/β):
Figure 10 - Multiplication gain M versus αL for pure electron injection. Various α/β
value are used to demonstrate its effect on avalanche breakdown curve
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2.10.5 Impact Ionization Coefficient Measurement
The discussion above reveals the relation between carrier multiplication gain and ionization
coefficients. In order to find out the ionization coefficients for electron and hole,
experimental measurement is necessary because there is no good model for calculating
these values. We know that α and β are strong function of electric field in the multiplication
region. Electric field, however, also depends on the bias voltage, doping profile, and the
device geometry. In this section, we will explain how to measure the ionization coefficients
accurately for different device structures including p-i-n diodes.
For p-i-n Diode is the best one to measure α and β because the electric field is very close to
constant due to the lightly doped intrinsic layer as shown in Fig. 11.
Figure 11: (a) Schematic view of a p-i-n diode used for measuring the electron
ionization coefficients. High energy light illumination is used to ensure that
photocurrent is created very close to the surface. (b) Hole ionization coefficient
measurement using the same setup but illuminating the diode from the n+ side.
In measurement, a very high energy light is used to illuminate the diode. Due to the high
energy of the photons, electron-hole pairs can be created very close to the surface to obtain
the pure electron or hole injection condition. Electron ionization is measured by shining
light from the p+ side. Photon-excited electrons are injected into the intrinsic layer while
holes are swept to the left. Hole ionization is measured by illuminating light from the n+
side, injecting holes into the intrinsic layer.
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The electron gain M n can be calculated by dividing the total current measured for a given
electric field by the photocurrent created by the light. The photocurrent can be precisely
measured when the diode is operated without avalanche gain. The hole gain M p can be
obtained through the same process. Once M n and M p are known, we can use equation
(2.4) and (2.5) to derive formulas related α and β to M n and M p . They are calculated to be

1  M n  1
L  M n  M p
  Mn
 ln 
 M
  p




2.8
1  M p  1
L  M p  M n
 Mp
 ln 
 M
  n



2.9
And

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2.10.6 Impact Ionization Response
The carrier build-up time in the multiplication process depends on the contribution of
carrier feedback as illustrated in Fig. 12. In the ideal condition, Figure 12(a), where only
one type of carrier is capable of initiating impact ionization process (electrons for β = 0),
the output current pulse increases with the transit time for the initially injected electron.
The output current decreases to zero as all the ionized holes arrive at the negatively biased
electrode. Thus, the current pulse lasts about twice as long as the transit time. Since the
pulse width is independent of the multiplication gain, there is no gain-bandwidth product
limitation when β or α = 0.
Figure 12: Representation of impact ionization process. (a) Only electron initiated
impact ionization.
(b) Both carriers initiated impact ionization.
In the other extreme case when β = α, the build-up of the impact ionization process is
shown in Fig. 12(b). The gain of the process can be very high which may cause the output
current pulse to become very long too. There is a gain-bandwidth product limitation for this
case. The detailed derivation for the gain bandwidth product as a function of β/α was given
by Emmons. It is found that the avalanche multiplication process does not affect the device
bandwidth as long as the dc multiplication gain M is less than α/β. On the other hand, if M
> α/β, the multiplication gain becomes a function of frequency and is expressed as
M ( )  M 0 /( 2 M 0  1 )
2
2
1
2
2.10
Where τ1 is an effective transit time and is approximately τ1 = N(β/α)τ. N is a number
varying slowly from 1/3 to 2 as β/α varying from 1 to 10 3 , and τ is the transit time equal to
L / v s where L is the length of the avalanche gain region and vs is the saturation velocity.
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Simulation result based on (2.10) is shown in Fig. 13. Clearly we see that the 3dB
bandwidth decreases as β/α value increases, and the decreasing rate is greater for higher
multiplication gain.
Figure 13: Calculated bandwidth versus gain in a P-I-N photodiode for various values
of β/α. Plot indicates that the bandwidth will not be limited by the gain as long as M<
α/ β[6].
The gain-bandwidth product for M > α/β can be obtained using (2.10) for high frequencies
and is expressed as
M ( )   
M 0
( 2 M 0 1 )
2
1

2
1
1

1
N (   )
2.11
Equation (2.11) indicates the basic requirements for an impact-ionization based device to
obtain a high gain-bandwidth product. These requirements include a small β/α value and
short intrinsic time. Therefore, a correct choice of material, multiplication layer thickness,
and carrier transport velocity are essential.
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Using (2.10), the calculated frequency response for a P-I-N structure with 1m thick gain
region when operating at M=50 is given in Figure 14.
Figure 14: Frequency Response of a P-I-N photodiode with 1m thick multiplication layer.
Notice that parameter k is define as β/α
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CHAPTER 3
SIMULATION AND DISCUSSION
In Chapter 3, simulation results using Matlab to create a program are indicated in this
chapter. It also explains the methodology used in this Capstone project, simulation study
and results obtained working on materials from III-V compounds.
Using Matlab to simulate the both the measure of α and β since Bulmanet al [10]. found
that β > α, these two sets of parameters for the coefficients since they enable the data to be
more accurately quantified over the wide field range the coefficients are parameterized in
The impact ionization coefficients can be fitted into an exponential form:
 ( E )  Ae exp(  Be / E )
 ( E )  Ah exp(  Bh / E )
 eH  AeH exp[ ( Beh /  ) C
eH
 h  A exp[ ( B /  ) C
For the local calculation the values of α and β were taken from Bulmanet al.[10] for electric
fields, F, This investigation involved measuring both the multiplication and excess noise
characteristics of several samples with overlapping field regions and represents the most
extensive and rigorous to date. By developing a model for the ionization coefficient that
was independent of the multiplication-region width, each carrier and each material, a single
electric-field-dependent model for the ionization coefficient was developed that was
suitable for devices of all thicknesses.
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3.1
Impact Ionisation coefficient and Multiplication of electrons & holes for GaAs
Impact Ionisation Coefficients for GaAs
 ( E )  1.89  10 7 [(exp( (5.75  10 7 ) / E )1.82 ]cm 1
 ( E )  2.21 10 7 [(exp( (6.57  10 7 ) / E )1.75 ]cm 1
Figure 15: Measured M e for a w  0.48m device as a function of temperature [17]
Measurements of M e (M h ) were carried out at temperatures between 20 and 290 K on
p  in  (n  ip  ) diodes of all thicknesses, and also between 290 and 500 K on
the w  0.48m p in  diode. Fig. 15 shows a diode at temperatures between 20 and 500 K.
As expected, increasing the temperature causes the multiplication characteristic to shift to
higher voltages, since higher electric fields are required to offset the increase in carrier
cooling by phonon scattering and maintain multiplication. it also shows that the
temperature dependence of multiplication is not uniform, since the multiplication curves
crowd together at low temperatures. This is probably due to the saturation of the phonon
emission (~n+1) and absorption (~n) scattering rates at low temperatures where n
approaches zero.
Figure 16: Impact Ionisation Coefficients for GaAs versus inverses electric field
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Multiplication of electrons and holes for GaAs
Simulation of calculate Electron and hole multiplication factors for a range of ideal p-i-n s
with from 1 um down to 0.48um for GaAs with temperatures of 20K to 500K
Figure 17: simulated of multiplication w  0.48m device as a function of temperature
from 20K to 500K
Shown in figure 17 increasing temperature causes the multiplication of electrons and holes
to shift to higher voltage therefore higher electric fields are required to offset the increase in
carrier cooling by phonon scattering and maintain multiplication.
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3.2
Impact Ionisation coefficient and Multiplication of electrons & holes for
InGaAs
Figure 18: Measured (symbols) and calculated (lines) multiplication characteristics of
1.3 and 1.9 um thick InGaAs p-i-n diode (filled symbols) from 20–400 K [18]
 eH  AeH exp[ ( Beh /  ) C
eH
 h  A exp[ ( B /  ) C
Impact Ionisation Coefficients for InGaAs (narrowband gap semiconductor) great
importance in the case of an ionization coefficient which increases with temperature that
can result in an unstable positive power dissipation feedback which the voltage breakdown
can be obtained by extrapolating the multiplication curve
For alpha
AeH  7.2597  10 4  (24.204T )  (0.3259T 2 )cm 1
BeH  5.9988  10 5  (3.4763  10 2 T )  (2.4768T 2 )V / cm
C eH  1.783  (7.2548  10  4 T )
For beta
A  6.1026  10 5  (9.6637  10 2 T )  (1.1384T 2 )cm 1
B  1.3394  10 6  (1.0699  10 3 T )  (20.4507T 2 )V / cm
C  1.0910  (2.3505  10  4 T )
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Figure 19: Impact Ionisation Coefficients for InGaAs versus inverses electric field
Fig. 18 shows the temperature dependence of electron multiplication characteristics of the
1.3 and 1.9 um InGaAs p-i-ns from 20–400 K and the temperature dependence of the hole
multiplication for the 3.0 um InGaAs n-i-p from 20–300 K.
The results from all the layers investigated show a very limited increase in photocurrent
initially and then the sudden and clear onset of the avalanche multiplication process
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Multiplication of electrons and holes for InGaAs
Simulation of Electron multiplication factors that depends on temperature with the
thickness of 1.3um and 1.9 um for InGAas p-i-ns from 20K to 400K and hole
multiplication with thickness of 3.0um as n-i-p from 20K-300K.
Figure 20: Simulated of multiplication characteristics of 1.3 and 1.9 um thick
InGaAs p-i-n diode from 20–400 K
The results from figure 20 show a very limited increase in photocurrent initially and then
the sudden and clear onset of the avalanche multiplication process. The avalanche
multiplication of all the p-i-n and n-i-p structures clearly decreases with increasing
temperature, indicating a negative temperature dependence of electron and hole ionization
coefficients respectively
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3.3
Impact Ionisation coefficient and Multiplication of electrons & holes for InP
The electron and hole ionization coefficients can be extracted from the measured
multiplication results if both electrons initiated and hole initiated multiplication
results are available for the same structure When an electron (or hole) initiates the
multiplication process, an electric current is induced by the moving electrons
and holes within the multiplication region
InP has high electron peak velocities resulting from large inter valley separation and
good breakdown properties owed to a relatively low electron impact ionization
coefficient.
Figure 21: Measured Me (symbols) Calculated Me (solid lines) using bulk ionization
coefficients for InP [19]
Compared to the data of GaAs and InP illustrated in Fig.16 and Fig. 22, GaInP has the
lowest values of α and β, signifying a higher breakdown voltage. Moreover, the slopes of
the curves of α and β vs 1/E for GaInP shown in Fig. 20 are steeper than for the others.
Impact Ionisation Coefficients for InP
 ( E )  2.93  10 6 [exp( (2.64  10 6 ) / E ]cm 1
 ( E )  1.62  10 6 [exp( (2.11 10 6 ) / E ]cm 1
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Figure 22: Impact Ionisation Coefficients for InP versus inverses electric field
Multiplication of electrons and holes for InP
Figure 23: Simulated of multiplication characteristics of 0.24 and 2.40 um thick for
InP p-i-n diode from 20–400 K
The results from figure 23 show that the temperature dependence of multiplication is not
uniform, since the multiplication curves with thickness of the materials from 0.24 to
2.40um respectively. Simulation of Electron multiplication factors for a range of ideal p-ins with from 0.24um down to 2.40um for InP with temperatures of 20K to 500K
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3.4
Impact Ionisation coefficient and Multiplication of electrons & holes for
GaInP
As a test of the parameterized coefficients, test shows for all the materials, the predicted
multiplication for an ideal diode, compared with the measured characteristic. It can be seen
that the agreement is good, both at low and high multiplications using matlab for these
simulations for wideband gap semiconductor and narrowband gap semiconductor.
The structures for high-voltage or high-power applications, an accurate knowledge of
impact ionization coefficients in GaInP is necessary for calculating breakdown
characteristics of junctions. In the work reported here, photocurrent multiplication was used
to measure the electron and hole ionization coefficients from either side with above
bandgap radiation. The results show that GaInP has significantly lower values of α and β
than those of GaAs or InP, a promising indication for high-voltage applications.
Figure 24: p   n   n  diode layer structure and the measured results of
Mn and Mp as a function of reverse bias.[20]
Impact Ionisation Coefficients for GaInP(wideband gap semiconductor)
 ( E )  3.85  10 6 [exp( (3.17  10 6 ) / E ]cm 1
 ( E )  1.71 10 6 [exp( (3.19  10 6 ) / E ]cm 1
The electron and hole multiplication factors, Mn and Mp, are defined as the total output
photocurrent divided by the electron or hole current injected at the contacts. Results for Mn
and Mp from respective measurements are shown in Fig. 24.
In these results, slight corrections were made to the injected currents due to slightly
voltage-dependent intrinsic reverse currents. Because the widths of the depletion regions
depend on bias, the number of thermally generated minority carriers that diffuse into the
depleted n  region also varies. This leads to a slight increase of the injected current as the
reverse bias is increased
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Figure 25: Impact Ionisation Coefficients for GaInP versus inverses electric field
Multiplication of electrons and holes for GAInP
Simulation of Electron multiplication factors for a range of ideal p-i-n s with from 0.24um
down to 2.40um for GaInP with temperatures of 300K
Figure 26: Simulated of multiplication characteristics of 0.24 um thick for GaInP
p-i-n diode from 300 K
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3.5
Discussion
Voltage breakdown due to avalanche multiplication, also defined as the reverse bias voltage
where multiplication rate goes to infinity naturally of great practical interest.
The voltage VB is defined as the reverse-bias voltage across the multiplication region at
which the mean gain becomes infinite.
In doing so rather then testing with hands on materials in fab or clean room physically
testing the materials, with this simulation we can create a model and simulate any materials
first with respect to their voltage breakdown versus thickness (width) without wasting
unnecessary time and cost for testing of materials.
Case 1
Figure 27: Simulated of voltage breakdown versus thickness for GaAs
GaAs wide band gap, highly resistive which makes it a very good electrical substrate
therefore makes it a very good material for ideal material for microwave and millimeter
wave integrated circuits. It can also be operated at higher power levels because they have
higher breakdown voltage.
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ENG499 CAPSTONE PROJECT REPORT
Case 2
Figure 28: Simulated of voltage breakdown versus thickness for GaInP
GaInP wideband gap semiconductor exhibits high breakdown voltage characteristics
indicating that it is a good choice of material for high power applications. This suggests that
the ideal electric field distribution assumption is valid and that edge effects are not
important in this material system for mesa geometry structures. Note that since these
punches through devices have a low carrier concentration in the n  region and high
breakdown voltage, the effects of dead space could be ignored.  And  data were used
to calculate the expected breakdown voltages for p-i-n diode with various thicknesses
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ENG499 CAPSTONE PROJECT REPORT
Case 3
Figure 29: Simulated of voltage breakdown versus thickness for InP
InP wideband gap have an advantage compared to GaAs for many applications when used
in high-field regions of the device profile, can significantly improve device performance.
These applications include high performance power amplifier for cellular phones, ultraefficient ultra-linear power amplifiers ideally suited for digital communication systems and
satellite networks ICs, and highly integrated mixed signal and high-speed fiber-optic
circuits. Wide bandgap semiconductors are associated with a high breakdown voltage. This
is due to a larger electric field required to generate carriers through impact mechanism.
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ENG499 CAPSTONE PROJECT REPORT
Case 4
Figure 30: Simulated of voltage breakdown versus thickness for InGaAs
InGaAs with narrow band gap is also a popular material in infrared detectors and some
short wave infrared cameras. It also has lower multiplication noise than germanium when
used as the active multiplication layer of an avalanche photodiode. Also has a low
breakdown voltage
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ENG499 CAPSTONE PROJECT REPORT
CHAPTER 4
4.2
PROBLEMS ENCOUNTERED AND SOLUTIONS
Problem 1:
Do not have the basic knowledge about semiconductor?
Solution 1:
Research on books and internet to get more information and understand it
Problem 2:
Do not know what is impact ionization?
Solution 2:
Research on books and internet to get more information and understand it and also
approach my project supervisor for guidance
Problem 3:
Do not have the basic knowledge on diode, photodiode?
Solution 3:
Research on books and internet to get more information and understand it
Problem 4:
Do not know how to use what program to do the design and simulation portion?
Solution 4:
Go online to search for the software manual and spent some time on exploring the features
of the software.
Problem 5:
During the design and simulation portion of using matlab, found out need to have a lot
more information to design the simulation.
Solution 5:
My project supervisor requests me by searching some information on the internet.
Problem 6:
Do not have strong programming skills in matlab?
Solution 6:
Research on books and internet to find examples and understand it
Problem 8:
During the designing portion, not able to achieve the desired output out as per required.
Solution 8:
Due to the values of the impact ionization formula written wrongly in matlab code
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ENG499 CAPSTONE PROJECT REPORT
Problem 9:
Spending too much time on the researching for solutions to get it work on the simulation?
Solution 9:
Find experimental papers and research papers to find the correct materials information to
work on the simulations base on them.
Problem 10:
After simulating for 1 material then try to simulate for 4 materials
Solution 10:
1 material is not enough to satisfy the need of modelling the materials therefore need to do
more research for 4 different materials both narrowband gap and wideband gap materials
Problem 11:
Multiplication of electrons and holes with a fixed temperature
Solution 11:
Multiplication of electrons and holes with a fixed temperature does not prove that
temperature of the materials will affect the voltage breakdown of the materials therefore
need to create a function with various temperature to prove.
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ENG499 CAPSTONE PROJECT REPORT
CHAPTER 5
CONCLUSION
5.4
FUTURE WORK
For this 1 year project materials are simulated base on p-i-n diodes therefore very
little studies on Avalanche Photodiode (APD) and Schottky Barrier diodes.
Another project can be done with Avalanche Photodiode (APD). The device
operation works as following: Arriving photons pass through thin n+p junction.
The carriers are absorbed in a π- region. The absorption leads to the generation of
electron-hole pairs in this region. The electric field in the π-region is high enough to
separate the carriers. The electric field across the π-region is not high enough for the
charge carriers to gain enough energy for multiplication to take place.
The electric field, however, in the n+ p- region the electric field is significantly
higher so that the charge carriers (in this case electrons only) are strongly
accelerated and pick up energy
Figure 28 shows a silicon based avalanche photodiodes. It is of interest to mention
that the carrier mobility of holes in silicon is significantly lower than the electron
mobility. Furthermore, the impact ionized holes have to travel all the way from the
n+ p- region to the right p+-region, whereas the electron only have to travel to the
n+- region. The probability of having electron multiplication is much higher than
the probability of having hole multiplication. Therefore, the electrons mainly
contribute to the overall current.
Figure 31: Silicon Avalanche Photodiode
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ENG499 CAPSTONE PROJECT REPORT
Also for Schottky Barrier diodes a thin metal layer replaces either the p- or the n- region of
the diode. Depending on the semiconductor and the metal being involved a barrier is
formed at the interface of the two materials. This barrier leads to a bending of the bands.
Due to the applied voltage the bands can be bended more or less. In the region of the band
bending electron hole pairs can be separated.
Figure 32: Schottky Barrier diode
Lastly, using matlab to simulate the simulation where there is a certain level of limitations.
Many other programs are able to simulate and model materials for realistic usage.
Detail values can be shown with the output for the simulation which can eliminate issue
like values not accurate or running a few matlab programs for different usage. Another
thing is that we can achieve much more stable readings. It would enhance better
understanding of the whole project going on.
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ENG499 CAPSTONE PROJECT REPORT
5.5
SUMMARY
Generally for this project, it enhances me with a lot of knowledge and exposure in
the designing of software simulation. And also creating a chance in allowing me to
have hands on experience on how to manage a project, improving my research skills
and learn more skills.
The main portion of the project is to model semiconductor photodetector.
With minimum knowledge and experience on the designing of software
programming, I faced difficulty in the designing portion, also in the selection of the
program to use and different equations for materials with the necessary
requirements needed for the simulation. In addition, I have no experience in using
semiconductor topic that made me have to start from scratch to learn how to use
matlab to create the simulation.
Pin-diodes with the depletion region extended across the intrinsic or lightly doped
layer and therefore more photo-generated carriers contribute to the photocurrent.
The pin-diode can be realized as a homo-junction or a heterojunction. If the
structure is realized in silicon the device will be usually a homojunction. Under such
conditions all three layers (p-,i- and n-region) have the same optical bandgap.
Depending on the application the thickness and thin dividual layers can be adjusted.
For each of the four materials (InP, In GaAs, GaAs, and GaInP) we were able to
satisfied the exponential model provided in independent of the multiplicationregion width. This enabled us to estimate the electron and hole ionization
coefficients, and, respectively. Since the impact-ionization rate for holes in InP is
greater than that for electrons, the carriers were reversed in the recurrence
equations, as discussed earlier. The optimized sets of width-independent parameters,
and that yielded the best fit in the universal exponential model, for both electrons
and holes.
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ENG499 CAPSTONE PROJECT REPORT
REFLECTION
I personally feel that the project was not very successful even I have completed the
project on-time, but at least I have met the main objectives. I am able to achieve the
simulation and compare to same as the experimental result to prove for all 4
materials. As the model is always assumed as an ideal case with fixed variable
temperature and electric field, where there is no noise distortion which will mean a
change in the simulation.
During the 10 months and the time that I spent in UniSIM, a lot of valuable skills
and knowledge are being picked up along the way; I become more confident in
handling problems of my project and also improved my management skill. Not
forgetting that report writing and oral presentation will greatly enhances in my
future career path, and I can say that it is a very good experience to have in order to
make my life more exciting and meaningful.
47
ENG499 CAPSTONE PROJECT REPORT
REFERENCES
[1]
T. P. Pearsall, “Impact ionization rates for electrons and holes in Ga0.47In
0.53As,” Appl. Phys. Lett. 36, 218-220 (1980)
[2]
S. Wang, R. Sidhu, X. G. Zheng, X. Li, X. Sun, A. L. Holmes, Jr., and J. C.
Campbell, IEEE Photonics Technol. Lett. 13, 1346 (2001).
[3]
R. Poerschke, Shpringer –Verlag, Madelung, O. (ed.), Semiconductor: group
IV elements and III-V compound. Series "Data in science and technology",
Berlin, 164 (1991).
[4]
J. R. Chelikowsky, and M. L. Cohen, Phys. Rev. B14, 556 (1976).
[5]
R. J. McIntyre, “A New Look at Impact Ionization – Part I: A Theory of
Gain, Noise, Breakdown Probability, and Frequency Response,” IEEE
Trans. Electron Devices, vol. 46, 1623—1631 (1999).
[6]
R. B. Emmons, J. Appl. Phys. 38, 3705 (1967).
[7]
H. Ando and H. Kanbe, “Ionization coefficient measurement in GaAs y
using multiplication noise characteristics,” Solid-State Electron., vol. 24
pp. 629–634, 1981.
[8]
O. Konstantinov, Q.Wahab, N. Nordell, and U. Lindefelt, “Ionization rates
and critical fields in 4H silicon carbide,” Appl. Phys. Lett., vol. 71,July 1997
[9]
Y. Okuto and C. R. Crowell, “Ionization coefficients in semiconductors,”
Phys. Rev. B., vol. 10, pp. 4284–4296, Nov. 1973.
[10]
G. E. Bulman, V. M. Robbins, and G. E. Stillman, “The determination of
impact ionization coefficients in (100) gallium aresenide using avalanche
noise and photocurrent multiplication measurements,” IEEE Trans. Electron
Devices, vol. ED-32, pp. 2454–2466, Nov. 1985.
[11]
M. M. Hayat, B. E. A. Saleh, and M. C. Teich, “Effect of dead space on gain
and noise of double-carrier-multiplication avalanche photodiodes,” IEEE
Trans. Electron Devices, vol. 39, pp. 546–552, Mar. 1992.
[12]
M. M. Hayat, W. L. Sargeant, and B. E. A. Saleh, “Effect of dead space on
gain and noise in Si and GaAs avalanche photodiodes,” IEEE J. Quantum
Electron, vol. 28, pp. 1360–1365, May 1992.
[13]
A. Di Carlo and P. Lugli, “Dead-space effects under near breakdown
conditions in AlGaAs/GaAs HBT's,” IEEE Electron Device Lett., vol. 14,
pp. 103–105, Mar. 1993.
48
ENG499 CAPSTONE PROJECT REPORT
[14]
S. P. Wilson, S. Brand, and R. A. Abram, “Avalanche multiplication
properties of GaAs calculated from spatially transient ionization
coefficients,” Solid-State Electron., vol. 38, pp. 2095–2100, Nov. 1995.
[15]
A. Di Carlo and P. Lugli, “Dead-space effects under near breakdown
conditions in AlGaAs/GaAs HBT's,” IEEE Electron Device Lett., vol. 14,
pp. 103–105, Mar. 1993.
[16]
S. P. Wilson, S. Brand, and R. A. Abram, “Avalanche multiplication
properties of GaAs calculated from spatially transient ionization
coefficients,” Solid-State Electron., vol. 38, pp. 2095–2100, Nov. 1995.
[17]
IEEE TRANSACTIONS ON ELECTRON DEVICES, VOL. 50, NO. 10,
OCTOBER 2003 2027 Temperature Dependence of Impact Ionization
in GaAs C. Groves, R. Ghin, J. P. R. David, and G. J. Rees
[18]
IEEE TRANSACTIONS ON ELECTRON DEVICES, VOL. 50, NO. 10,
OCTOBER 2003 2021 Temperature Dependence of Breakdown and
Avalanche Multiplication in In0:53Ga0:47As Diodes and
Heterojunction Bipolar Transistors M. Yee, W. K. Ng, J.P. R. David,
Senior Member, IEEE, P. A. Houston, C. H. Tan, and A. Krysa
[19]
Shanghai Institute of Technical Physics Theory Study of SAGCM InP
L. Lin, W. J. Wang, N. Li, X. S. Chen and W. Lu Shanghai Institute of
Technical Physics, Chinese Academy of Sciences National Lab for
Infrared Physics,
[20]
Impact ionization coefficients in (100) GaInP S.-L. Fu, T. P. Chin, M. C.
Ho, C. W. Tu, and P. M. Asbeck Department of Electrical and
Computer Engineering, University of California, San Diego, La Jolla,
California 92093
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ENG499 CAPSTONE PROJECT REPORT
APPENDIX A -
Gantt chart FOR PROJECT PLANNING
50
ENG499 CAPSTONE PROJECT REPORT
APPENDIX B -
PROGRAM DESIGN FOR Gallium arsenide (GaAs)
Ionisation coefficient for GaAs
clc;
clear all;
close all;
%CARRIER DIFFUSION LENGTH
a=1.899*10^5%alpha
d=5.750*10^5%alpha
f=1.82%alpha
g=2.215*10^5%beta
h=6.570*10^5%beta
i=1.75%beta
e=1/(1*10^-6)
%e1=1*10^-6
c=a*exp(-(d/e)^f)
b=g*exp(-(h/e)^i)
e=1/(1.5*10^-6)
%e1=1.5*10^-6
c1=a*exp(-(d/e)^f)
b1=g*exp(-(h/e)^i)
e=1/(2*10^-6)
%e1=2*10^-6
c2=a*exp(-(d/e)^f)
b2=g*exp(-(h/e)^i)
e=1/(2.5*10^-6)
%e1=2.5*10^-6
c3=a*exp(-(d/e)^f)
b3=g*exp(-(h/e)^i)
e=1/(3*10^-6)
%e1=3*10^-6
c4=a*exp(-(d/e)^f)
b4=g*exp(-(h/e)^i)
e=1/(3.5*10^-6)
%e1=3.5*10^-6
c5=a*exp(-(d/e)^f)
b5=g*exp(-(h/e)^i)
e=1/(4*10^-6)
%e1=4*10^-6
c6=a*exp(-(d/e)^f)
b6=g*exp(-(h/e)^i)
e=1/(4.5*10^-6)
%e1=4.5*10^-6
c7=a*exp(-(d/e)^f)
b7=g*exp(-(h/e)^i)
% alph=alpha value in/cm
alph=[c c1 c2 c3 c4 c5 c6 c7];
% e=inverse electric field in terms of cm/v
e=[1*10^-6 1.5*10^-6 2*10^-6 2.5*10^-6 3*10^-6 3.5*10^-6 4*10^-6 4.5*10^
6]
bet=[b b1 b2 b3 b4 b5 b6 b7];
figure(1);
semilogy(e,alph,'b');
hold on
grid on
semilogy(e,bet,'g');
hold on
legend('alpha','beta');
xlabel('1/E(10^-6)cm/v');
ylabel('Ionisation Coefficients/cm');
title('Impact ionisation coefficients versus inverse electric field');
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ENG499 CAPSTONE PROJECT REPORT
APPENDIX C -
PROGRAM DESIGN FOR Indium phosphide (InP)
Ionisation coefficient for InP
clc;
clear all;
close all;
%CARRIER DIFFUSION LENGTH
e=1/(1*10^-6)
a=2.93*10^6 %alpha
d=2.64*10^6 %alpha
g=1.62*10^6 %beta
h=2.11*10^6 %beta
%e1=1*10^-6
c=a*exp(-(d/e))
b=g*exp(-(h/e))
e=1/(1.5*10^-6)
%e1=1.5*10^-6
c1=a*exp(-(d/e))
b1=g*exp(-(h/e))
e=1/(2*10^-6)
%e1=2*10^-6
c2=a*exp(-(d/e))
b2=g*exp(-(h/e))
e=1/(2.5*10^-6)
%e1=2.5*10^-6
c3=a*exp(-(d/e))
b3=g*exp(-(h/e))
e=1/(3*10^-6)
%e1=3*10^-6
c4=a*exp(-(d/e))
b4=g*exp(-(h/e))
e=1/(3.5*10^-6)
%e1=3.5*10^-6
c5=a*exp(-(d/e))
b5=g*exp(-(h/e))
e=1/(4*10^-6)
%e1=4*10^-6
c6=a*exp(-(d/e))
b6=g*exp(-(h/e))
e=1/(4.5*10^-6)
%e1=4.5*10^-6
c7=a*exp(-(d/e))
b7=g*exp(-(h/e))
% alph=alpha value in/cm
alph=[c c1 c2 c3 c4 c5 c6 c7];
% e=inverse electric field in terms of cm/v
e=[1*10^-6 1.5*10^-6 2*10^-6 2.5*10^-6 3*10^-6 3.5*10^-6 4*10^-6 4.5*10^-6]
bet=[b b1 b2 b3 b4 b5 b6 b7];
figure(1);
semilogy(e,alph,'b');
hold on
grid on
semilogy(e,bet,'g');
hold on
legend('alpha','beta');
xlabel('1/E(10^-6)cm/v');
ylabel('Ionisation Coefficients/cm');
title('Impact ionisation coefficients versus inverse electric field');
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ENG499 CAPSTONE PROJECT REPORT
APPENDIX D -
PROGRAM DESIGN FOR GaInP
Ionisation coefficient for GaInP
clc;
clear all;
close all;
%CARRIER DIFFUSION LENGTH
e=1/ (1*10^-6)
a=3.85*10^6 %alpha
d=3.71*10^6 %alpha
g=1.71*10^6 %beta
h=3.19*10^6 %beta
%e1=1*10^-6
c=a*exp(-(d/e))
b=g*exp(-(h/e))
e=1/(1.5*10^-6)
%e1=1.5*10^-6
c1=a*exp(-(d/e))
b1=g*exp(-(h/e))
e=1/(2*10^-6)
%e1=2*10^-6
c2=a*exp(-(d/e))
b2=g*exp(-(h/e))
e=1/(2.5*10^-6)
%e1=2.5*10^-6
c3=a*exp(-(d/e))
b3=g*exp(-(h/e))
e=1/(3*10^-6)
%e1=3*10^-6
c4=a*exp(-(d/e))
b4=g*exp(-(h/e))
e=1/(3.5*10^-6)
%e1=3.5*10^-6
c5=a*exp(-(d/e))
b5=g*exp(-(h/e))
e=1/(4*10^-6)
%e1=4*10^-6
c6=a*exp(-(d/e))
b6=g*exp(-(h/e))
e=1/(4.5*10^-6)
%e1=4.5*10^-6
c7=a*exp(-(d/e))
b7=g*exp(-(h/e))
% alph=alpha value in/cm
alph=[c c1 c2 c3 c4 c5 c6 c7];
% e=inverse electric field in terms of cm/v
e=[1*10^-6 1.5*10^-6 2*10^-6 2.5*10^-6 3*10^-6 3.5*10^-6 4*10^-6 4.5*10^-6]
bet=[b b1 b2 b3 b4 b5 b6 b7];
figure(1);
semilogy(e,alph,'b');
hold on
grid on
semilogy(e,bet,'g');
hold on
legend('alpha','beta');
xlabel('1/E(10^-6)cm/v');
ylabel('Ionisation Coefficients/cm');
title('Impact ionisation coefficients versus inverse electric field');
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ENG499 CAPSTONE PROJECT REPORT
APPENDIX E -
PROGRAM DESIGN FOR Indium gallium arsenide (InGaAs)
Ionisation coefficient for InGaAs
clc;
clear all;
close all;
%CARRIER DIFFUSION LENGTH
t=300
a=(7.2597*10^4-(24.204*t)+(0.3259*(t^2)))%alpha
d=5.9988*10^5+(3.4763*10^2*t)+(2.4768*(t^2))%alpha
f=1.1783-(7.2548*10^-4*t)%alpha
g=6.1026*10^5+(9.6637*10^2*t)+(1.1384*(t^2))%beta
h=1.3394*10^6+(1.0699*10^3*t)+(0.4507*(t^2))%beta
i=1.0910-(2.3505*10^-4*t)%beta
e=1/(4*10^-6)
%e1=1*10^-6
c=a*exp(-(d/e)^f)
b=g*exp(-(h/e)^i)
e=1/(5*10^-6)
%e1=1.5*10^-6
c1=a*exp(-(d/e)^f)
b1=g*exp(-(h/e)^i)
e=1/(6*10^-6)
%e1=2*10^-6
c2=a*exp(-(d/e)^f)
b2=g*exp(-(h/e)^i)
e=1/(7*10^-6)
%e1=2.5*10^-6
c3=a*exp(-(d/e)^f)
b3=g*exp(-(h/e)^i)
e=1/(8*10^-6)
%e1=3*10^-6
c4=a*exp(-(d/e)^f)
b4=g*exp(-(h/e)^i)
e=1/(9*10^-6)
%e1=3.5*10^-6
c5=a*exp(-(d/e)^f)
b5=g*exp(-(h/e)^i)
e=1/(10*10^-6)
%e1=4*10^-6
c6=a*exp(-(d/e)^f)
b6=g*exp(-(h/e)^i)
e=1/(11*10^-6)
%e1=4.5*10^-6
c7=a*exp(-(d/e)^f)
b7=g*exp(-(h/e)^i)
% alph=alpha value in/cm
alph=[c c1 c2 c3 c4 c5 c6 c7];
% e=inverse electric field in terms of cm/v
e=[4*10^-6 5*10^-6 6*10^-6 7*10^-6 8*10^-6 9*10^-6 10*10^-6 11*10^-6]
bet=[b b1 b2 b3 b4 b5 b6 b7];
figure(1);
semilogy(e,alph,'b');
hold on
grid on
semilogy(e,bet,'g');
hold on
legend('alpha','beta');
xlabel('1/E(10^-6)cm/v');
ylabel('Ionisation Coefficients/cm');
title('Impact ionisation coefficients versus inverse electric field');
ENG499 CAPSTONE PROJECT REPORT
54
APPENDIX F function
PROGRAM DESIGN FOR Multiplication of electrons and holes
Multiplication of electrons and holes function
w=10*10^-6;
a=(1.899*10^5-(24.204*t)+(0.3259*(t^2)))%alpha
d=5.750*10^5+(3.4763*10^2*t)+(2.4768*(t^2))%alpha
f=1.82-(7.2548*10^-4*t)%alpha
g=2.215*10^5+(9.6637*10^2*t)+(1.1384*(t^2))%beta
h=6.570*10^5+(1.0699*10^3*t)+(0.4507*(t^2))%beta
i=1.75-(2.3505*10^-4*t)%beta
e=1/(1*10^-6)
%e1=1*10^-6
c=a*exp(-(d/e)^f)
b=g*exp(-(h/e)^i)
e=1/(1.5*10^-6)
%e1=1.5*10^-6
c1=a*exp(-(d/e)^f)
b1=g*exp(-(h/e)^i)
e=1/(2*10^-6)
%e1=2*10^-6
c2=a*exp(-(d/e)^f)
b2=g*exp(-(h/e)^i)
e=1/(2.5*10^-6)
%e1=2.5*10^-6
c3=a*exp(-(d/e)^f)
b3=g*exp(-(h/e)^i)
e=1/(3*10^-6)
%e1=3*10^-6
c4=a*exp(-(d/e)^f)
b4=g*exp(-(h/e)^i)
e=1/(3.5*10^-6)
%e1=3.5*10^-6
c5=a*exp(-(d/e)^f)
b5=g*exp(-(h/e)^i)
e=1/(4*10^-6)
%e1=4*10^-6
c6=a*exp(-(d/e)^f)
b6=g*exp(-(h/e)^i)
e=1/(4.5*10^-6)
%e1=4.5*10^-6
c7=a*exp(-(d/e)^f)
b7=g*exp(-(h/e)^i)
e=1/(5*10^-6)
%e1=4.5*10^-6
c8=a*exp(-(d/e)^f)
b8=g*exp(-(h/e)^i)
e=1/(5.5*10^-6)
%e1=4.5*10^-6
c9=a*exp(-(d/e)^f)
b9=g*exp(-(h/e)^i)
e=1/(6*10^-6)
%e1=4.5*10^-6
c10=a*exp(-(d/e)^f)
b10=g*exp(-(h/e)^i)
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ENG499 CAPSTONE PROJECT REPORT
% alph=alpha value in/cm
alph=c;bet=b;
M1=1/(((1+(alph/(alph-bet))*exp(-(alph-bet)*w)-1)))
Mh1=1/(1+(bet/(alph-bet))*(1-(exp(-(alph-bet)*w))))
alph=c1;bet=b1;
M2=1/(((1+(alph/(alph-bet))*exp(-(alph-bet)*w)-1)))
Mh2=1/(1+(bet/(alph-bet))*(1-(exp(-(alph-bet)*w))))
alph=c2;bet=b2;
M3=1/(((1+(alph/(alph-bet))*exp(-(alph-bet)*w)-1)))
Mh3=1/(1+(bet/(alph-bet))*(1-(exp(-(alph-bet)*w))))
alph=c3;bet=b3;
M4=1/(((1+(alph/(alph-bet))*exp(-(alph-bet)*w)-1)))
Mh4=1/(1+(bet/(alph-bet))*(1-(exp(-(alph-bet)*w))))
alph=c4;bet=b4;
M5=1/(((1+(alph/(alph-bet))*exp(-(alph-bet)*w)-1)))
Mh5=1/(1+(bet/(alph-bet))*(1-(exp(-(alph-bet)*w))))
alph=c5;bet=b5;
M6=1/(((1+(alph/(alph-bet))*exp(-(alph-bet)*w)-1)))
Mh6=1/(1+(bet/(alph-bet))*(1-(exp(-(alph-bet)*w))))
alph=c6;bet=b6;
M7=1/(((1+(alph/(alph-bet))*exp(-(alph-bet)*w)-1)))
Mh7=1/(1+(bet/(alph-bet))*(1-(exp(-(alph-bet)*w))))
alph=c7;bet=b7;
M8=1/(((1+(alph/(alph-bet))*exp(-(alph-bet)*w)-1)))
Mh8=1/(1+(bet/(alph-bet))*(1-(exp(-(alph-bet)*w))))
alph=c8;bet=b8;
M9=1/(((1+(alph/(alph-bet))*exp(-(alph-bet)*w)-1)))
Mh9=1/(1+(bet/(alph-bet))*(1-(exp(-(alph-bet)*w))))
alph=c9;bet=b9;
M10=1/(((1+(alph/(alph-bet))*exp(-(alph-bet)*w)-1)))
Mh10=1/(1+(bet/(alph-bet))*(1-exp(-(alph-bet)*w)))
alph=c10;bet=b10;
M11=1/(((1+(alph/(alph-bet))*exp(-(alph-bet)*w)-1)))
Mh11=1/(1+(bet/(alph-bet))*(1-exp(-(alph-bet)*w)))
% e=inverse electric field in terms of cm/v
% bet=beta value in/cm
%Formula for Mh
%Mh=(1+(bet/(alph-bet))*(1-exp(-(alph-bet)*w)))^-1
%Formula for Me
%Me=((1+(alph/(alph-bet))*exp(-(alph-bet)*w)-1))^-1
v=[1 1/1 1/2 1/3 1/4 1/5 1/6 1/7 1/8 1/9 1/10]
%v=[0:0.0001:10];
figure(3)
Mh=[Mh11 Mh10 Mh9 Mh8 Mh7 Mh6 Mh5 Mh4 Mh3 Mh2 Mh1];
Me=[M11 M10 M9 M8 M7 M6 M5 M4 M3 M2 M1];
plot(Me,v,'b-');
grid on;
hold on;
xlabel('V');
ylabel('Me&Mh');
title('Me&Mh versus V Characteristics');
end
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ENG499 CAPSTONE PROJECT REPORT
APPENDIX G-
PROGRAM DESIGN FOR Voltage breakdown versus thickness
clc;
clear all;
close all;
%t=thickness in mm
%vbd=breakdown voltage in volt
t=[0*10^-6 1*10^-6 2*10^-6 3*10^-6 4*10^-6]
%ev= charge of an electron in eV
ev=1.424*10^6;
%N=doping concentration in /cm3
%Formula for breakdown voltage in volt
%Vbd=(60*(E(in eV)/1.1)^3/2)*(N(cm^-3)/10^16)^-(3/4))
for N=1:5
vbd(N)=(37.5*(ev))*((N))
end
figure(4);
plot(t,vbd);
grid on;
ylabel('break down voltage vbd(volt)');
xlabel('thickness t(um)');
title('Plot for Vbd Vs width');
57
ENG499 CAPSTONE PROJECT REPORT