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- Name: STUDY OF OPTICAL PROPERTIES OF LEAD SELENIDE (PbSe) USING KRAMERS-KRONIG RELATION
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Rapid advances in a semiconductor manufacturing and associated technologies have increased the need for the study of how to make them more efficient which lead to the study of the optical properties of Lead Selenide a group IV-VI semiconductor. The optical properties of Lead selenide in the range of 1.0-6.0eV were obtained using Maxwell’s equation of electromagnetism and Kramers-Kronig relations.
In this energy range the crystal of PbSe is observed to have distinct behavior at energies 1.00, 1.53, 2.05, 2.84, 3.10, 4.42, 4.95 and 5.74eV indicating high absorption, refection and decay of the photon. At higher energy values 5.72-6.00eV the decay and absorption loss rate is reduced.
Thus as a result of these optical properties PbSe can be used for variety of devices.
Thus the result for these optical properties can be used in a wide range of optoelectronic devices, such devices include optical pyrometer and spectrometer
TABLE OF CONTENTS
Title page i
Table of Content v-viii
List of figures x
List of tables xi
CHAPTER ONE – INTRODUCTION
1.1 INTRODUCTION TO SEMICONDUCTOR 1
1.1.2 N-type and P-type semiconductor 2
1.1.3 Intrinsic and Extrinsic semiconductor 2
188.8.131.52 Intrinsic semiconductor 2
184.108.40.206 Extrinsic semiconductor 3
1.2 GROUP IV-VI SEMICONDUCTOR 4
1.3 LEAD SELENIDE 4
1.3.1 Introduction and Application 4
1.3.2 Physical and Structure Properties 5
- Applications of Lead Selenide. 6
1.3.4 Theory of operation 7
1.4 Kramers- Kronig Relationship 7
1.5 Objective of The Study 8
CHAPTER TWO – LITERATURE REVIEW
2.1 Introduction to Optical Theorem 9
2.2 Theory of the Optical Properties 9
2.2.1 Optical Conductivity 10
2.2.2 Relation of the Complex Dielectric Function to Observables 14
2.3 Derivation of Kramers-Kronig Relations 16
2.4 Physical Interpretation of Kramers Kronig Relation 17
CHAPTER THREE – METHODOLOGY
3.1 Index of Parameters 20
3.2 Formulas 21
3.3 Calculations 22
CHAPTER FOUR – RESULTS AND DISCUSSIONS
4.1 RESULTS 26
4.11 The Reflectance Spectrum of Lead Selenide in the Photon Energy Range 1.00-6.00eV 26
4.1.2 The Reflection Coefficient of Lead Selenide in Photon Energy of Range 1.00-6.00eV 27
4.1.3 The Refractive Index of Lead Selenide in Photon Energy of Range 1.00-6.00eV 28
4.1.4 The Extinction Coefficient of Lead Selenide in Photon Energy of Range 1.00-6.00eV 29
4.1.5 The Real Part of the Complex Dielectric Constant of Lead Selenide in Photon Energy of Range 1.00-6.00eV 30
4.1.6 The Imaginary Part of the Complex Dielectric Constant of Lead Selenide in Photon Energy of Range 1.00-6.00eV 31
4.1.7 The Transmittance of Lead Selenide in Photon Energy of Range 1.00-6.00eV 32
4.1.8 The Absorption Coefficient of Lead Selenide in Photon Energy of Range 1.00-6.00eV 33
4.1.9 The Real Part of the Optical Conductivity Of Lead Selenide in Photon Energy Of Range 1.00-6.00eV 34
4.1.10 The Imaginary Part Of The Optical Conductivity Of Lead Selenide in Photon Energy Of Range 1.00-6.00eV 35
4.2 INTERPRETATION AND DISCUSSION
4.2.1 The Reflectance Spectrum of Lead Selenide in the Photon
Energy Range 1.00-6.00eV 36
4.2.2 The Reflection Coefficient of Lead Selenide in Photon
Energy Range 1.00-6.00eV 36
4.2.3 The Refractive Index of Lead Selenide in Photon
Energy Range 1.00-6.00eV 36
4.2.4 The Extinction Coefficient of Lead Selenide in Photon
Energy Range 1.00-6.00eV 36
4.2.5 The Real Part of the Complex Dielectric Constant Of Lead
Selenide in Photon Energy Range 1.00-6.00eV 36
4.2.6 The Imaginary Part of The Complex Dielectric Constant of Lead selenide In Photon Energy of Range 1.00-6.00eV 37
4.2.6 The Imaginary Part of the Complex Dielectric Constant of Lead Selenide In Photon Energy Of Range 1.00-6.00eV 38
4.2.8 The Absorption Coefficient of Lead Selenide in Photon Energy Range 1.00-6.00eV 39
4.2.9 The Real Part of the Optical Conductivity of Lead Selenide in Photon Energy of Range 1.00-6.00eV 40
4.2.10 The Imaginary Part of The Optical Conductivity of Lead Selenide in Photon Energy Range 1.00-6.00eV 41
CHAPTER FIVE – SUMMARY AND CONCLUSION
1.1 INTRODUCTION TO SEMICONDUCTOR
A semiconductor is a solid material that has an electrical conductivity that is intermediate between those of conductors and of an insulator. Semiconductors are important materials, because of their unique electronic properties, they are the materials of choice for modern electronic devices. The tremendous importance of semiconductor in the present-day electronics stems in part of from the fact that their electrical properties are sensitive to small concentration of impurities. Besides their unique properties for electronics applications, semiconductors also have many other important properties that are very useful for photonic device applications. Many semiconductors are also used for acousto-optic devices and nonlinear optical devices. In such applications, which are based solely on the dielectric properties of semiconductors, semiconductors are nothing but another group of dielectric optical materials.
In metals highly mobile electrons roam freely within the structure, in a semiconductor the electrons are trapped by the covalent forces between the atoms that form the lattice. The conductivity of semiconductors is therefore dependent on the amount of charge carriers that participate in the conduction. Conduction in semiconductors occurs via charge carriers known as “electrons and holes”. The electrons are negative charge carriers while the holes are positively charged.
1.1.2 N-type and P-type semiconductor
Depending on the type of impurity added there are two types of semiconductors N-type and P-type, in N-type semiconductor the majority carriers are electrons and minority carriers are holes, while in the P-type semiconductor majority carriers are holes and the minority carriers are electrons. The properties of a pure semiconductor can be changed very significantly by adding very small amount of impurities of materials having five electrons in their outer most shell and the process is called “doping”.
N-type doping changes a semiconductor into a material with a surplus of electrons compared to the amount of holes, in that case the semiconductors is called an N-type semiconductor. If the density of electrons is n, density of holes is p. then in an n-type semiconductor, n will be greater than p, the doping atoms are therefore called donors in this case.
P-type doping changes a semiconductor into a material with a surplus of holes compare to the amount of electrons. The semiconductor is called the P-type semiconductor. Similarly in a P-type semiconductor p is greater than n, in this case doping atoms are called acceptors. The process of doping has reduced the number of electrons and increased the number of holes.
1.1.3 Intrinsic and Extrinsic semiconductor
220.127.116.11 Intrinsic semiconductor: Semiconductor in its pure, un-doped form is called intrinsic semiconductor, in intrinsic semiconductor the number of electrons equals the number of holes and thus conductivity is very low as valence electrons are covalently bonded also the free carriers are generated by thermal energy and thus while an electron is created a hole occurs simultaneously. It is called electron-holes pair.
18.104.22.168 Extrinsic semiconductor: These are semiconductors that are doped are said to have higher and better conductivity. Impurities, donors and acceptors are introduced in the pure intrinsic semiconductor, thermal energy frees electrons from donor atoms and holes from acceptor atoms. Donor doping creates an n-type material by donating an extra electrons. On the other hand the acceptor doping creates a p-type material. The density of dopants is normally higher than free carriers in a semiconductor. Thus as a consequence of doping the conductivity of a semiconductor can be changed. Conductivity is directly related to the amount of free carriers in a semiconductor. Therefore decreasing the amount of free carriers will lead to an increase in conductivity.
Semiconductors are defined by their electric conductive behavior. Metal are good conductors because at their Fermi level, there is a large density of energetically available states that each electron can occupy, metal conductivity decreases with temperature increase because thermal vibrations disrupt the free motion of electrons. Insulators by contrast are very poor conductor of electricity because there is a large difference in energies between the conduction and the valence band. The valence band is occupied by electrons which are free from their parent atoms. Semiconductors on the other hand have intermediate level of electrical conductivity when compared to metals and insulators. Their band gap is small enough that a small increase in temperature promotes sufficient number of electrons from the lowest energy level to the conduction band.
1.2 GROUP IV-VI SEMICONDUCTOR
When any member of the group IV elements, is bounded to the group VI element an IV-VI semiconductor is formed, IV-VI semiconductors are used primarily in optoelectronic devices designed for detection and emission of mid-infrared electromagnetic radiation. Technologically important IV-VI semiconductors include binary PbS, PbSe, PbTe and pseudobinary, tenary and quaternary alloys such as PbSnSe, PbSnTe, PbEuTe, and PbSnSeTe. Depending on the alloy composition and temperature their band gaps range from 0 to over 500 meV. they are direct gap materials with conduction and valence band minima at the L-point in k-space . Such properties make IV-VI semiconductor useful for the fabrication of optoelectronic devices covering the 3µm to 30µm spectral range.
1.3 LEAD SELENIDE
1.3.1 INTRODUCTION AND APPLICATION
Lead selenide (PbSe), is a semiconductor material. It forms cubic crystals of the NaCl structure; it has a direct band gap of 0.27 eV at room temperature. It is a grey crystalline solid material. It is used for manufacturing of infrared detectors for thermal imaging, operating at wavelengths between 1.5–5.2 µm. It does not require cooling, but performs better at lower temperatures. The peak sensitivity depends on temperature and varies between 3.7–4.7 µm single Nano-rods and polycrystalline nanotubes of lead selenide have been synthesized via controlled organism membranes. The diameter of the Nano-rods were approximately 45 nm and their length was up to 1100 nm, for nanotubes the diameter was 50 nm and the length up to 2000 nm. Lead selenide Nano crystals embedded into various materials can be used as quantum dots, for example in Nano crystal solar cells.
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