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Transcript
REFLECTANCE MEASUREMENT AND MODELLING OF
HIGH REFLECTIVITY DISTRIBUTED BRAGG REFLECTOR
(DBR) STACKS
BY
GIK HONG YEAP
(200305075)
MSc ADVANCED MATERIALS AND MANUFACTURING
UNIVERSITY OF HULL
ON
22 AUGUST 2005
THESIS SUBMITTED TO
DEPARTMENT OF ENGINEERING
UNIVERSITY OF HULL FOR MSc DEGREE AWARD
SUPERVISOR
PROFESSOR S.K. HAYWOOD
Reflectance Measurement and Modelling of High Reflectivity Distributed Bragg Reflector Stacks
ACKNOWLEDGEMENT
First of all, I would like to thank my thesis supervisor, Professor Stephanie Haywood,
for guiding and helping me in the research project and the thesis. Professor Haywood
has been given me a lot of guidance as well as very useful ideas to work on the thesis.
The interest, enthusiasm and patient that she presented while helping me in the thesis
had motivated me to complete my thesis. It would be a very difficult task for me to
complete the thesis without her guidance and help.
Next, I would like to thank Dr. K.T. Lai who given me a lot of information
regarding the Fourier Transform Infrared (FT-IR) Spectroscopy and shows me the basic
operation of the FT-IR Spectrometer. I also would like to thank Mr. Alvin Lim for
helping me a lot in the Matlab programming, which plays an important role in this
thesis.
Last but not least, I want to thank the staff in University of Montpellier II for
their permission to use the lab facilities and particularly Mr. Jean Baptiste Rodriguez for
assisting me in doing the reflectivity measurement of the sample.
I am also would like to take this opportunity to thank my wife for being very
tolerating and supportive throughout my university years. Finally, I would like to thank
my parents for bringing me to this world and for their support and encouragement.
The guidance, helps, understanding, support and encouragement from all the
above mentioned people have provided me the courage and strength to embark on this
thesis. Thank you once again.
Reflectance Measurement and Modelling of High Reflectivity Distributed Bragg Reflector Stacks
ABSTRACT
Distributed Bragg Reflector (DBR) is becoming an important component in
optoelectronic devices particularly the quarter-wave semiconductor DBR because of its
ease of monolithic integration, hence eliminating the requirement for post-growth
processing and simplifying the design. The nature of the interfaces present in these
materials, grown by sequential process such as Metal Organic Vapour Phase Epitaxy
(MOVPE) or Molecular Beam Epitaxy (MBE) plays a very important role in optical
elements. The DBR mirror structure is incorporated into many optoelectronic devices
such as, light emitting diodes, lasers, optical switches, photodetectors and other devices.
This thesis is about the reflectance measurement of high reflectivity DBR stacks
by using Fourier Transform Infrared (FT-IR) spectroscopy. First, the basic theory of
optical multilayer is reviewed and then the principle operation of FT-IR spectroscopy is
studied. Followed by method used to simulate a DBR stack using a technical computing
language, Matlab, is presented. The simulation results of several properties of DBR are
presented for the investigation purposes as well as for comparison with the measured
spectrum of the sample. This thesis concluded the factors that influence the accurate
measurement of high reflectivity DBR stacks and suggestion for possible future work.
Reflectance Measurement and Modelling of High Reflectivity Distributed Bragg Reflector Stacks
TABLE OF CONTENT
Acknowledgement
i
Abstract
ii
Table of Content
iii
List of Figures
vi
List of Tables
viii
Chapter 1
Chapter 2
Chapter 3
Introduction
1
1.0 Background of The Thesis
1
1.1 Aims and Objectives
2
1.2 Project Explanation and Method
3
1.3 Literature Review or Review Prior To Knowledge
3
1.4 Overview of The Thesis
4
Distributed Bragg Reflector (DBR)
5
2.0 Introduction
5
2.1 Principle Operation of DBRs
6
2.2 1.60 µm DBRs
8
2.3 Material System of Investigating Sample
9
2.4 Growth Methods
10
2.4.1 Metal Organic Vapour Phase Epitaxy (MOVPE)
11
2.4.2 Molecular Beam Epitaxy (MBE)
14
2.4.3 Sample Growth
16
Fourier Transform Infrared (FT-IR) Spectrometers
17
3.0 Introduction
17
3.1 FT-IR Components
19
3.1.1 Radiation Source
19
3.1.2 Interferometer
20
3.1.3 Detector
21
3.1.4 Sample Compartment
21
Reflectance Measurement and Modelling of High Reflectivity Distributed Bragg Reflector Stacks
3.2 Optical Path Different (OPD) and Zero Path Different
22
(ZPD)
Chapter 4
3.3 Interferogram
23
3.4 Fourier Transform
26
3.5 Advantages of FT-IR
28
3.6 Specular Reflectance Spectroscopy
29
Training Program and Sample Measurement in University
31
of Montpellier 2
Chapter 5
Chapter 6
4.0 Introduction
31
4.1 External Specular Reflectance Spectroscopy
33
4.2 Variable Angle Reflection Accessory
34
4.3 Experimental Sample Reflectivity Measurement
35
4.4 Experimental Results
38
Matlab Simulation Program
40
5.0 Introduction
40
5.1 Reflectivity Simulation Program
41
Results and Discussions
45
6.0 Experimental Measurements Results Against Modelled
45
Result
6.1 Measurement Setup in Own Lab
51
6.2 DBR Properties Examination by Matlab Simulation
52
Program
Chapter 7
6.2.1 Varying Number of Periods
52
6.2.2 Varying Refractive Index Different
53
6.2.3 Varying First Layer Material
54
6.2.4 Varying Angle of Incidence
56
Absorption in InGaAs Layer
59
7.0 Introduction
59
7.1 Band Edge Absorption
59
7.2 Free-Carrier Absorption
60
Reflectance Measurement and Modelling of High Reflectivity Distributed Bragg Reflector Stacks
7.3 Effects of Doping on Absorption
60
7.4 Simulation Program With Absorption Taken Into Account
61
Conclusion and Future Work
67
8.0 Conclusion
67
8.1 Future Work
68
References
69
Appendix 1
Sample Growth Sheet and Reflectivity Measurement
75
Appendix 2
Quotations
77
Appendix 3
Basic Reflectivity Simulation Program
80
Appendix 4
Simulation Program For Plotting Reflectivity vs Angle of
83
Chapter 8
Incidence
Appendix 5
Simulation Program With Loop To Calculate The
87
Individual Matrix Elements For InGaAs Layer
Appendix 6
Simulation Program That Take Effects of Absorption Into
91
Account
Appendix 7
Matlab Quick Reference
95
Reflectance Measurement and Modelling of High Reflectivity Distributed Bragg Reflector Stacks
LIST OF FIGURES
Page
Chapter 2
Figure 2-1
A Typical DBR With Alternating High and Low Refractive 5
Indices
Figure 2-2
The Maximum and Minimum Reflection in DBR Stacks
7
Figure 2-3
Normalized Reflectivity Spectra for Low and High Doped 10
Samples of In0.53Ga0.47As/InP and Modeled Spectrum
Figure 2-4
Metalorganic Vapour Phase Epitaxy (MOVPE) Reactor
11
Figure 2-5
MOVPE Process
12
Figure 2-6
MOVPE Precursor Kept in Bubbler
13
Figure 2-7
MBE Growth System
14
Figure 2-8
Schematic Of The Growth Chamber
15
Figure 2-9
Example of Molecular Beam Epitaxy Facility
15
Figure 3-1
FT-IR Spectrometer
19
Figure 3-2
Michelson Interferometer
20
Figure 3-3
Schematic Representation of Waves and Their Phase, Input, 23
Chapter 3
Output, and the Interferometer. (a) OPD=0 case. (b) λ/4 OPD
case. (c) λ/2 OPD case. (d) 3λ/4 OPD case. (e) 1λ OPD case.
Figure 3-4
Interferogram of Radiation From The source
23
Figure 3-5
The Beam Path of a Two Wavelength Source
24
Figure 3-6
Interferogram Consisting of Three Modulated Cosine Wave
24
Figure 3-7
A Typical Interferogram Produced With a Broadband IR 25
Source
Figure 3-8
The Process of Collecting An Infrared Spectrum in an FT-IR 27
Spectrometer
Figure 3-9
Chapter 4
Specular Reflectance
29
Reflectance Measurement and Modelling of High Reflectivity Distributed Bragg Reflector Stacks
Figure 4-1
Photograph Taken in CEM2 Lab
31
Figure 4-2
Multiple Reflections Techniques
32
Figure 4-3
External Reflection Spectroscopy or Specular Reflectance
33
Figure 4-4
Variable Specular Reflection Accessory
34
Figure 4-5
12º Absolute Reflectance Sampling Stage
35
Figure 4-6
Schematic Diagram of the FT-IR Spectrometer
35
Figure 4-7
Variable Specular Reflection Accessory and 12º Absolute 36
Reflectance Sampling Stage in the FT-IR Sample Compartment
Figure 4-8
The Direction of The beam
37
Figure 4-9
Absorption or Transmission of Ambient Water and Carbon 37
Dioxide
Figure 4-10
The ‘V-W’ Mode Double Reflection Technique
38
Figure 4-11
Sample Absolute Reflectance Measurement Spectrum
39
Step by Step Calculation in the Matlab Program
41
Figure 6-1
Measured Reflectance Spectrum Against Wavelength
45
Figure 6-2
The Dimension of The Biggest Piece of Sample
46
Figure 6-3
Simple Reflection Theory
47
Figure 6-4
Transmission vs Wavelength Plot at Different Points Across 48
Chapter 5
Figure 5-1
Chapter 6
The Sample
Figure 6-5
High Resolution Transmission Measurement Spectra
49
Figure 6-6
Reflectivity Spectra Derived From Transmission Measurement
50
Figure 6-7
Reflectivity
vs
Wavelength
for
Model
and
Measured 50
Reflectivity from Transmission Measurement
Figure 6-8
Bruker Optic’s Absolute Reflectance Accessory
51
Figure 6-9
Varies Number of Periods
52
Figure 6-10
Varies Refractive Indices Different
54
Figure 6-11
Reflectivity of DBR with Different Layer Sequence
55
Figure 6-12
Varies Angle of Incidence
56
Figure 6-13
Reflectivity vs Angle of Incidence at 1600 nm
57
Reflectance Measurement and Modelling of High Reflectivity Distributed Bragg Reflector Stacks
Chapter 7
Figure 7-1
Absorption Spectra Of Undoped and n-doped 1 µm 61
In0.47Ga0.53As Layers on InP
Figure 7-2
n+ik Reflectivity Spectra
62
Figure 7-3
Dielectric Constant, ε1 Spectrum of InGaAs
65
Figure 7-4
Reflectivity Spectra With and Without Absorption Taken Into 66
Account
LIST OF TABLES
Page
Chapter 6
Table 6-1
Reflectivity Value and Center Wavelength for Different Angle
57
of Incidence
Chapter 7
Table 7-1
EF and hw Value for Different Doping Level
65
Reflectance Measurement and Modelling of High Reflectivity Distributed Bragg Reflector Stacks
CHAPTER 1
INTRODUCTION
1.0 BACKGROUND OF THE THESIS
One of the most active material research fields since a few decades ago is the
development of semiconductor multilayer systems. [SANYAL, M.K. et al, 1998] DBR
is becoming an important component in optoelectronic devices particularly the quarterwave semiconductor DBR because of its ease of monolithic integration [DEPPE, D.G.
et al, 1990] hence eliminating the requirement for post-growth processing and
simplifying the design [GUY, P. et al, 1994]. The nature of the interfaces present in
these materials, grown by sequential process such as Metal Organic Vapour Phase
Epitaxy (MOVPE) or Molecular Beam Epitaxy (MBE) plays a very important role in
optical elements. [SANYAL, M.K. et al, 1998]
The DBR mirror structure is incorporated into many optoelectronic devices such
as, light emitting diodes, lasers, optical switches, photodetectors and other devices
[MASON, N.J. et al, 1996; DEPPE, D.G. et al, 1990] to improve the extraction
efficiency of the emitters, [GESSMANN, Th. and SCHUBERT, E.F., 2004] to achieve
highly sensitive detectors at the operation wavelength enhance the quantum efficiency,
[MANSOOR, F. et al, 1995] to offer wavelength selectivity with additional advantage
of enhanced absorption in the active region, etc. [MANSOOR, F, HAYWOOD, S.K and
GREY, R., 1995]
It is instructive to determine the DBR reflectivity as a function of wavelength
and angle of incidence in order to evaluate the potential enhancement provided by the
DBR. [GESSMANN, Th. and SCHUBERT, E.F., 2004] The reflectance measurement
can be performed by using Fourier Transform Infrared (FT-IR) Reflectance
Spectroscopy. The terms reflectivity and reflectance are often used interchangeably
although there is a difference between them. Reflectance is a measurement of the
amount of light that is reflected by a sample at a given film thickness while reflectivity
-1-
Reflectance Measurement and Modelling of High Reflectivity Distributed Bragg Reflector Stacks
is the maximum amount of light that can be reflected by the sample and can not be
increased by increase film thickness or smoothness. Reflectance measurements are
typically divided into two categories, which are internal reflectance measurement by
using an attenuated total reflectance (ATR) element in contact with the sample and
external reflectance measurements are made using an infrared beam reflected directly
from the sample surface. [Specac, 2001]
FTIR Spectrometers were developed in 1960s for commercial use but it tends to
be used for advanced research only due to some factors, such as the requirement of large
computers to run them and high cost of the instrument components. Rapid computers
and instruments technology advancements have enhanced the capabilities and reduced
the cost of an FT-IR spectrometer. FT-IR spectrometers have become a standard
instrument for organic and inorganic compound identification work in modern
analytical laboratories. [Thermo Nicolet, 2002]
In practice it is very difficult to measure reflectivity exceeding 99% accurately.
There are many factors that influencing the measurement such as dispersion of the
radiation source due to the presence of incidence angle that not normal to the sample
surface, sample surface contamination (i.e. thumbprint due to handling), the influence of
water vapors in the atmosphere because it is a strong IR absorber, quality of the present
sample, etc. The problems to be solved in this project will involve the eliminating these
influences and the application of multiple reflection technique to obtain the accurate
measurement of the very high reflectivity samples.
1.1
AIMS AND OBJECTIVES
The principle aims and objectives of this project are:
1. To measure the reflectance accurately for very high reflectivity sample, i.e.
R>99% DBR stacks by using multiple reflection technique.
2. Investigate the GaInAs material system for Distributed Bragg Reflector (DBR)
stacks.
-2-
Reflectance Measurement and Modelling of High Reflectivity Distributed Bragg Reflector Stacks
3. Understanding of the operation of DBR stacks incorporated in optoelectronics
devices.
4. Understanding of the principle operation of Fourier Transform Infrared (FT-IR)
Spectroscopy.
5. Compare the experimental model with the simulated model from Matlab
programming.
1.2
PROJECT EXPLANATION AND METHOD
Basically this research project consists of the operation of Fourier Transform Infrared
(FT-IR) spectroscopy to measure the reflectance of the samples and simulation of DBR
stacks using a technical computing language, Matlab for the investigation purposes as
well as for comparison with the measured spectrum of the samples. The aim of this
project is to measure the reflectance accurately for very high reflectivity samples that
exceed 99% by using multiple reflection technique to investigate the optical
characteristics and optimum reflectivity of the samples. The transition of the samples is
acquire during the experiment and will be converted into spectrum by means of
mathematical operation called Fourier Transformation. The experimental model will
then be compared with the simulated model to determine the performance of the
samples.
The project will require further understanding of FT-IR technique as well as
transition and properties of the samples.
1.3
LITERATURE REVIEW OR REVIEW PRIOR TO
KNOWLEDGE
The literature review will include the understanding of DBR theories and operation,
principles of FT-IR spectroscopy technique, components and operation as well as the
transitions and optical properties of the samples. Additionally, the principle or effects of
multiple reflection technique also need to be investigated. The background research will
be discussed further in the next few chapters.
-3-
Reflectance Measurement and Modelling of High Reflectivity Distributed Bragg Reflector Stacks
1.4
OVERVIEW OF THE THESIS
After the introduction which include background of the thesis, aims and objectives as
well as project explanation and method that have been outlined in Chapter 1, Chapter 2
will discussed about the Distributed Bragg Reflector (DBR) including the principle of
operation, the applications of the thesis DBR sample wavelength range, material system
of investigating sample and followed by the sample growth techniques.
Chapter 3 covers the introduction of Fourier Transform Infrared (FT-IR)
Spectroscopy including its components and principle of operation. The advantages of
FT-IR spectroscopy also will be outlined. In Chapter 4, the training program and
experimental sample measurement technique and the commercial accessory used in the
measurement will be described.
Chapter 5 will switch from experimental investigation to the Matlab
programming simulation of the DBR sample. The simulation program will be used to
perform the simulations of several properties of DBR for the investigation purposes as
well as for comparison with the measured spectrum of the sample.
Chapter 6 gives the results achieved from the DBR mirror sample measurement
and from the simulation program. A comparison between the experimental
measurement results and modelled results, as well as the discussion also will be made in
this chapter. Chapter 7 will discuss about the absorption in InGaAs layer and the
simulation program that take into account the absorption. Final conclusions,
recommendations and suggestion for future work are presented in Chapter 8.
-4-
Reflectance Measurement and Modelling of High Reflectivity Distributed Bragg Reflector Stacks
CHAPTER 2
DISTRIBUTED BRAGG REFLECTOR
(DBR)
2.0 INTRODUCTION
Distributed Bragg Reflector stacks (DBRs) are periodic structures with a unit cell of two
dielectric layers having different thicknesses, di (i=1, 2) and alternating high and low
refractive indices, ni. Therefore, DBR stacks also can be called the dielectric mirrors
due to the fact that it is made up of a number of two different refractive index materials.
[LIM, H.C., 2002] DBRs can be regarded as one dimensional photonic crystal with a
high reflectivity stop band or photonic gap comprising the non-propagating light states
in the crystal. [GESSMANN, Th. and SCHUBERT, E.F., 2004]
As mentioned before, the DBR mirror structure is incorporated into many
optoelectronic devices to enhance the performance and efficiency of the devices. Figure
2-1 shows a typical DBR with X layers. n0 is the refractive index of incident medium, n1
is the refractive index of the first layer, n2 is the refractive index of layer 2, while ns is
the refractive index of the substrate.
n0
n1
n2
Layer1
Layer2
Layer X
Substrate
ns
Figure 2-1 : A Typical DBR With Alternating High and Low Refractive Indices
-5-
Reflectance Measurement and Modelling of High Reflectivity Distributed Bragg Reflector Stacks
2.1
PRINCIPLE OPERATION OF DBRs
The DBRs work on the principle that light reflected from material of higher refractive
index than the incident medium will undergo a π phase change at alternate interface.
Also, due to the path length, the reflected light from each interface in a multilayer stack
will have an additional phase change in which all the reflected waves are in phase and
undergo constructive interference if the optical thickness of the layer is of quarter-wave.
[MANSOOR, F., 1995; CONWAY, L.J., 1999]
The DBR reflectivity depends on the number of period (1 period = 2 layers), N,
of the stack and the refractive index difference between the materials, (Δn = n1-n2). The
DBR will only be effective over a narrow wavelength range because the refractive
indices are wavelength dependence. [MANSOOR, F., 1996; HAYWOOD, S.K. et al,
1995; BLUM, O. et al, 1994, CONWAY, L.J., 1999] The number of layers used in a
DBR can be odd or even number. Therefore, there are two equations can be use to
determine the reflectivity. For an odd number of layers DBR, the reflectivity, R is given
by [MANSOOR, F., 1996; HAYWOOD, S.K. et al, 1995]:
2N


2




n


1
n
1

 1 − 




  n sn 0   n 2  
R = 
2N 
2
 
 n  
1
 1 +  n1 

 
  n sn 
0  n2 
 

2
(1)
In contrast, for even number of layers, the reflectivity, R is given by:
R =
  n  n  2 N 
 1− s  1  
  n 0  n 2  
    2 N 
 1+ n s  n1  


  n 0  n 2  
2
(2)
-6-
Reflectance Measurement and Modelling of High Reflectivity Distributed Bragg Reflector Stacks
where n1 and n2 are the refractive indices of the first and second layers in the stack
while n0 and ns are the refractive indices of the incident medium and substrate, and N is
the number of periods. When the layer optical thickness is quarter wavelength, t = λ/4n
where λ is the Bragg wavelength and n is the refractive index of the material,
constructive interference of reflected beam will occur because the maximum reflected
intensity occurred at quarter wavelength or at specific thickness as shown in Figure 2-2.
[MANSOOR, F., 1996; LIM, H.C., 2002]
Figure 2-2 : The Maximum and Minimum Reflection in DBR Stacks
Source : LIM. H.C., 2002
The reflection properties DBRs also depend on the polarization of the incident
lightwave. According to Brewster’s law, the reflection of light polarized parallel to the
plane of incidence has a minimum at the incidence angle
tan øB
n1
n2
(3)
This is particularly important for DBRs where the overall reflectivity significantly
decreases at øB. DBRs with improved wide-angle reflectivity can be achieved, e.g.,
using a periodically stacked layers with thickness gradients or random thickness
distributions. Much research was devoted to DBRs with a complete photonic band gap
represented by a certain frequency range where all incoming photons regardless of their
momentum vector, ћk are reflected, where ћ is Planck’s constant while k is a extinction
coefficient. [GESSMANN, Th. and SCHUBERT, E.F., 2004]
-7-
Reflectance Measurement and Modelling of High Reflectivity Distributed Bragg Reflector Stacks
2.2 1.60µm DBRs
The DBR sample that used in this thesis is designed to work at 1.60µm. The 1.60µm
mirror stacks are important for devices such as lasers and asymmetric Fabry Perot
modulators. This wavelength also has medical, military and environmental applications
as emitters or detectors. At shorter wavelengths, the large refractive index differences
between the available DBR stack materials allow fewer periods to be used. For highly
reflecting DBRs, a large refractive index difference between the two alternate layers in
the stack is required. [HAYWOOD, S.K. et al, 1994; MASON, N.J. et al, 1996]
In general, the use of the quarter wavelength semiconductor DBR has been
studied primarily on the AlxGa1-xAs crystal system because of their relatively large
refractive index difference along with the good lattice match between AlAs and GaAs
[DEPPE, D.G. et al, 1989; TAI, K. et al, 1989] contribute to the high performance and
quality with which these mirror structures can be epitaxially grown. For optoelectronic
devices and integrated optoelectronics, however, the InP/InxGa1-xAsyP1-y and InxGa1xAlyAs1-y
lattice-matched crystal system is also technologically important. [DEPPE,
D.G. et al, 1989; LU, T.C. et al, 2003] To date, this has been the material system of
choice for high-speed optical devices operating in the 1.3 and 1.55µm wavelength range
for optical communicating applications. [DEPPE, D.G. et al, 1989]
In the longer wavelength regime the interest is in using InP substrates. The
alloys available for the fabrication of DBR stacks which are not absorbing and are
lattice matched to the substrate include InGaAsP, AlGaInAs and AlInAs. However, the
refractive index difference between these materials is small, and so a large number of
periods are required to obtain a high reflectivity. However, a large number of periods
can lead to growth and processing problems [GUY, P., WOODBRIDGE, K. and
HOPKINSON, M., 1993; CHOA, F.S. et al, 1991] and high mirror resistivities. The
largest refractive index difference is obtained using InGaAs and AlInAs. [GUY, P.,
WOODBRIDGE, K. and HOPKINSON, M., 1993; IBBOTSON, L., 1997]
There are a number of DBRs with peak reflectivities around 1.55µm have been
demonstrated. Using GaxIn1-xAsyP1-y/InP, F.S. Choa et al had demonstrated 79%, 95%
and 100% reflectivity using 10, 20 and 45 periods respectively. Tai et al show over 92%
-8-
Reflectance Measurement and Modelling of High Reflectivity Distributed Bragg Reflector Stacks
using 20 periods, whereas Y.K. Imajo et al obtain 98% with 30 periods. In the
AlxGayIn1-x-yAs/InP system, A.J. Moseley et al achieve 95% peak reflectivity in a 20
period stack.
2.3 MATERIAL SYSTEM OF INVESTIGATING SAMPLE
The high reflectivity of the sample not only relies on the number of periods and the
refractive indices different between materials but also the reproducibility of the layers.
The reproducibility is determined by the control of the lattice match condition which
varies primarily with the In-Ga ratio, the material band-gap wavelength which varies
primarily with the As-P ratio and the layer thickness which is determined by the
material growth rate. [CHOA, F.S. et al, 1991] The reproducibility can be examined by
looking at the top shape of the Bragg band or commonly called stop band. If the flat
region on the stop band has a width of more than 400 Å or 40 nm, this indicates that the
optical quality of the DBR is excellent. The width of the stop band is linearly
proportional to the index difference. Any fluctuation in the index, which equivalent to
the material composition or thickness of each layer will reduce the width of the flat
region. [CHOA, F.S. et al, 1991; TAI, K. et al, 1989]
The DBR used for this research project is 36 periods n-doped indium gallium
arsenic (InGaAs) with In0.53Ga0.47As composition and designed to operate at 1.6 µm.
During the past few decades, considerable research has been devoted to the material
properties, growth techniques and the device behaviour of this ternary material. It is a
good example of semiconductor material where the research and scientific interest are
driven by significant technology needs and opportunities. [BHATTACHARYA, P.,
1993] Different compositions of InGaAs have different lattice parameters as well as
different energy gaps. The most appropriate substrate material proves to be InP. GaInAs
of composition 47% Ga and 53% In, which usually designated In0.53Ga0.47As is latticematched to InP substrate. [IBBOTSON, L., 1997] This composition has an energy gap
of 0.75 eV, corresponding to a cut-off wavelength of 1.65 µm, which is taken as good
enough for 1.3 µm and 1.60 µm devices.
-9-
Reflectance Measurement and Modelling of High Reflectivity Distributed Bragg Reflector Stacks
InGaAs is absorbing when lattice matched to InP, however, the addition of
aluminium or phosphorous will increase the bandgap but reduces the refractive index
difference. Another approach is to decrease the indium content of the ternaries. The
bandgap of the InGaAs is increased, while maintaining a large refractive index
difference between the alloys. [GUY, P., WOODBRIDGE, K. and HOPKINSON, M.,
1993]
Figure 2-3 below shows reflectivity spectra of the low and high doped samples
of In0.53Ga0.47As/InP which will be the sample of this research project that presented by
S.K.Haywood et al. They had shown that the highly n-doped In0.53Ga0.47As/InP DBR
stacks can be used to provide high reflectivity with only a few periods. The effects of
doping level on sample reflectivity will be discussed in more detail later in the thesis.
Figure 2-3 : Normalized Reflectivity Spectra for Low and High Doped Samples of
In0.53Ga0.47As/InP and Modeled Spectrum
Source : HAYWOOD, S.K. et al, 1994
2.4
GROWTH METHODS
The DBRs structures based on InGaAs have been successfully grown by several
epitaxial growth techniques such as Metal Organic Vapour Phase Epitaxy (MOVPE),
Molecular Beam Epitaxy (MBE) and Chemical Epitaxy (CBE). In this section, the
overview of these epitaxial growth techniques will be presented.
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Reflectance Measurement and Modelling of High Reflectivity Distributed Bragg Reflector Stacks
2.4.1
Metal Organic Vapour Phase Epitaxy (MOVPE)
MOVPE is similar to conventional Vapour Phase Epitaxy (VPE) in which the reactant
materials are transported in vapour form to the heated substrate where the epitaxial
growth takes place. The main difference is that instead of using metallic chlorides as the
source material, gallium chloride (GaCl3) or indium chloride (InCl3) for example,
MOVPE uses metalorganic molecules or precursors. Figure 2-4 below shows a typical
MOVPE reactor. In this example, the substrate sits flat on the horizontal graphite slab
inside a quartz tube. Outside the tube and surrounding the graphite is a metal coil
connected to a multikilowatt radio frequency generator. The graphite is heated to around
700 to 800 ºC by infrared lamps.
Figure 2-4 : Metalorganic Vapour Phase Epitaxy (MOVPE) Reactor
Source : RIZZI, A., 2004
Hydrides such as pure phosphine (PH3) and pure arsine (AsH3) gases are used as
the group V sources of the growth while group III sources can be trimethylindium
(TMIn) and trimethylgallium (TMGa). The dopants can be tetraethyltin(TESn),
hydrogen sulphide (H2S), silane (SiH4) or disilane (Si2H6) for n-type dopants and
biscyclopentadienyl magnesium (Cp2Mg) or dimethylzinc (DMZn) or diethylZinc
(DEZn) for p-type dopants. Hydrogen is used as carrier gas. By controlling the ratio of
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Reflectance Measurement and Modelling of High Reflectivity Distributed Bragg Reflector Stacks
constituent gases within the reactor, virtually any composition of structure can be grown.
The reactor is designed in such a way that the thicknesses of the epitaxial layers can be
precisely controlled. [WOOD, D., 1994; DEPPE, D.G. et al, 1990; WILLIAM, M.D.,
2000; MODAK, P. et al. 2000] Figure 2-5 shows the schematic diagram of MOVPE
process.
Figure 2-5 : MOVPE Process
Source : RIZZI, A, 2004
There are many variations on the design of the reactor chamber. For example, in
some existing commercial MOVPE systems, the wafers sit on a horizontal platter and
rotate either slowly or at high speed to achieve uniform growth across the wafer. Other
systems use a barrel-type susceptor inside a large bell jar, similar to VPE and silicon
epitaxy reactors. The method for heating the substrates can be RF induction, resistance
heaters, or infrared lamps. Whatever the configuration, the conceptual nature of the
growth process remains essentially the same.
The metal organic sources under normal room temperature conditions are either
high purity liquids or crystalline solids and are contained in small stainless steel
cylinders measuring about eight inches long by two inches in diameter. Because they
are pyrophoric, these materials are never exposed to air and require careful handing.
The cylinders are equipped with an inlet port connected to a dip tube and an exit port.
Hydrogen gas flowing through the dip tube and up through the metal organic liquid or
solid becomes saturated with metal organic vapours. This type of container is
commonly called a “bubbler”, referring to the action of the hydrogen bubbling through
the liquid.
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Reflectance Measurement and Modelling of High Reflectivity Distributed Bragg Reflector Stacks
The mixture of hydrogen and vapour flows out of the cylinder and to the reactor
chamber. The exact amount of metal organic vapour transported to the reactor is
controlled by the temperature of the bubbler, which determines the vapour pressure of
the metal organic material, and by the flow of hydrogen. The temperature of the
bubblers is controlled by immersion in a fluid bath in which the temperature is regulated
within ± 0.1 ºC or better. Special regulators called mass flow controllers precisely meter
the flow of hydrogen to each bubbler. [R.M. Fletcher et al. 1993; YOUNG, S.J., 2003]
Figure 2-6 below show the MOVPE precursor kept in “bubbler”.
Figure 2-6 : MOVPE Precursor Kept in Bubbler
Source : RIZZI, A., 2004
At the entrance to the reactor chamber, the reactant gasses are mixed. These
gasses consist of phosphine (PH3), a mixture of hydrogen and the metalorganic vapours,
dopant gasses, and additional hydrogen added as a diluent. As the gasses pass over the
hot substrate, decomposition of the phosphine, metalorganics, and dopant sources
occurs. If all the conditions are correct, proper crystal growth takes place in an orderly
atomic layer-by-layer process. Hydrogen, unreacted phosphine and metalorganics, and
reaction by-products such as methane are then drawn out of the reactor and through the
vacuum pump for treatment as toxic exhaust waste. [YOUNG, S.J., 2003; SUNDGREN,
P., 2005]
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Reflectance Measurement and Modelling of High Reflectivity Distributed Bragg Reflector Stacks
The growth of III-V epitaxial materials is typically complex, and the successful
production of high quality films is dependent on many factors. However, the MOVPE
growth technique has a sufficiently high growth rate that a reasonable growth time can
be maintained for required epitaxial thickness, while very good control over layer
uniformity and alloy composition can also be achieved. [RIZZI, A., 2004; DEPPE, D.G.
et al, 1990; WOOD, D., 1994]
2.4.2
Molecular Beam Epitaxy (MBE)
MBE established as the technique for growing thin layers of semiconductor material,
specially containing three or four elements in the 1960s. But by the late 1980s, it was
challenged by MOVPE as the best growth technique for this purpose. [WOOD, D., 1994]
However, it is very attractive for many applications due to it versatility. [STREETMAN,
B.G. and BANERJEE, S., 2000] The MBE system shown in Figure 2-7 and the simpler
schematic of the growth chamber is shown in Figure 2-8.
Figure 2-7 : MBE Growth System
Source : WOOD, D. 1994
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Reflectance Measurement and Modelling of High Reflectivity Distributed Bragg Reflector Stacks
Figure 2-8 : Schematic Of The Growth Chamber
Source : WOOD, D., 1994
In the growth chamber, the sample is held at a high temperature, which depends
on the substrate material and epitaxial layer, in an ultra high vacumm (UHV) at very
low background pressure of 10-10-10-11 torr while molecular or atomic beams of the
constituents impinge upon its surface. The UHV in the growth chamber keeps the
background contamination level low and allow in-situ diagnostic techniques such as
reflection high energy electron diffraction (RHEED) and Auger technique to be used.
Therefore, MBE requires a rather sophisticated setup such as clean room facility.
[STREETMAN, B.G. and BANERJEE, S., 2000; HAYWOOD, S.K. 2004; STRADING,
R.A. and KLIPSTEIN, P.C., 1990] Figure 2-9 shows an example of MBE facility.
Figure 2-9 : Example of Molecular Beam Epitaxy Facility
Source : STREETMAN, B.G. and BANERJEE, S. 2000
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Reflectance Measurement and Modelling of High Reflectivity Distributed Bragg Reflector Stacks
The heated Knudsen cells (K-cell) provide stable atomic or molecular beams
from it crucible which made from pyrolytic Boron Nitride (BN) or graphite, and the
rotation of the substrate improve the deposition uniformity. The chemical composition
and the doping level of the epilayer can be varied via the temperature of K-cells or the
aperture of the shutter that allow rapid changing of beams and abrupt the interfaces and
monolayer growth. The low temperature reduces the arrival rate of unwanted species
and provides heat dissipation for both the K-cells and the substrate heater.
[HAYWOOD, S.K. 2004; STRADING, R.A. and KLIPSTEIN, P.C., 1990]
The growth rate of MBE is typically about 1 µm/hour which equal to a
monolayer within shutter operating time of less than 1 second. MBE is capable in
growing very thin layer at very high quality. [HAYWOOD, S.K. 2004; WOOD, D.,
1994]
2.4.3
Sample Growth
The sample used in this investigation is 36-period n-type In0.53Ga0.47As/InP on InP
substrate with designed operation wavelength at 1.6 µm. This sample was grown by
MBE in EPSRC III-V Central Facility, University of Sheffield in 1997 upon the request
of Professor S.K. Haywood. The growth rates are 1.990 Å/s for the ternary and 2.347
Å/s for the binary at substrate temperature between 475-495ºC range by using silicon as
the dopant with 4 x 1018 cm-3 doping level both InGaAs and InP layers.
The growth was carried out on a 3000Å thick semi-insulating (100) InP
substrate with 2º misorientation. The thicknesses of the layers are 1107Å and 1222 Å
for the ternary and the binary, respectively, corresponding to the quarter wavelength
pathlength and the total thickness of the DBR was 8.6844 µm. These details can be
found in Appendix 1 but the details of the growth process will not be discussed here as
this is not a newly grown sample.
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Reflectance Measurement and Modelling of High Reflectivity Distributed Bragg Reflector Stacks
CHAPTER 3
FOURIER TRANSFORM INFRARED
(FT-IR) SPECTROMETERS
3.0 INTRODUCTION
Accurate knowledge of the infrared (IR) optical properties of materials is crucial for
understanding the physics of materials such as the phonon and electric structures and for
control of materials processing. There are a variety of spectrometers and accessories for
measuring spectral optical properties of materials in infrared region. [ZHANG, A.M.,
HANSSEN, L.M. and DALTA, R.U., 1995] Recently, FT-IR spectrometers have
replaced conventional dispersive instruments for most applications due to their superior
speed and sensitivity. They have been applied to many areas that are very difficult or
nearly impossible to analyse by dispersive instruments and have greatly extended the
capabilities of infrared spectroscopy. The situations or the areas where the FT-IR
spectrometer is preferred over dispersive instrument are:
-
Working in the infrared.
-
High spectral resolution is needed.
-
High spectral accuracy is needed.
-
Working with weak signals.
-
Quickly acquire of spectra with high signal to noise (S/N) ratio is needed.
FT-IR possess strong theoretical reasons that enable them to excel in the above
categories and the potential advantage is depends strongly on the instrument’s design
and the particulars of sample measurement. [Oriel Instruments, No Date; Thermo
Nicolet, 2002]
First commercially available FT-IR spectroscopy was manufactured by the
Digilab subsidiary of Block Engineering in Cambridge, Massachusett during the late
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Reflectance Measurement and Modelling of High Reflectivity Distributed Bragg Reflector Stacks
1960s. Since then, many others companies such as Nicolet Instruments, Bruker, PerkinElmer Corporation, etc have begun to manufacture and selling FT-IR in United State of
America (USA) and FT-IRs have moved from the research lab, to the quality control lab,
factory floor for online analysis and into industrial and academic lab. [SMITH, B.C.,
1996]
FT-IR is a fast analytical technique and provides very interesting qualitative and
quantitative information [KHANMOHAMMADI, M. and KARGOSHA, K., 2005]
which acquires broadband near IR to far IR spectrum [Oriel Instrument, No Date]. FTIR still holds some secrets for the researchers or chemists who trained to work with
conventional dispersive instrument (i.e.grating monochromator or spectrograph)
although it has been routing used in research and application laboratories and for
process control. [HERRES, W. and GRONHOLZ, J., 1987; Oriel Instrument, No Date]
Instead of viewing each component frequency sequentially, as in a dispersive IR
spectrometer, an FT-IR spectrometer collects and examines all wavelengths
simultaneously, known as multiplex or Felgett advantage. [Oriel Instrument, No Date]
The generation of the spectrum also is not straightforward or in other word, can not be
obtained by control the setting of appropriate knobs to control the slit widths, scanning
speed, etc. [HERRES, W. and GRONHOLZ, J., 1987] FT-IR is a method of obtaining
infrared spectra by first collecting an interferogram of a sample signal by using an
interferometer and involves a mathematical manipulations of Fourier Transform on the
interferogram to obtain the spectrum which may also involves phase correction and
apodization. [Oriel Instrument, No Date; HERRES, W. and GRONHOLZ, J., 1987]
This spectrum generation process may introduce a barrier to understanding of the FT-IR
technique.
However, there are clear advantages of FT-IR spectrometer compared to grating
spectrometers and therefore moderately and low priced FT-IR instrument are now
entering even routine labs. The most important component apart from the optics is the
dedicated computer which directly determines the accuracy of the spectrum by the
quality of its software. Hence, it is recommended that the user must be familiar with the
principles of FT-IR including the operation, data collection and data manipulation.
[HERRES, W. and GRONHOLZ, J., 1987]
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Reflectance Measurement and Modelling of High Reflectivity Distributed Bragg Reflector Stacks
3.1
FT-IR COMPONENTS
In an FT system, there are three basic spectrometer components: radiation source,
interferometer, and detector. A simplified optical layout of a typical FT-IR spectrometer
is illustrated in Figure 3-1.
M1
M2
L
Source
Beam
Splitter
Sample
X
Detector
Figure 3-1 : FT-IR Spectrometer
Source : Based on Oriel Instruments; W.Herres and J. Gronholz; Thermo Nicolet; C.P. Sherman Hsu
3.1.1 Radiation Source
The purpose of the source is to provide radiant energy in the IR region of the
electromagnetic spectrum. The radiation sources used for both dispersive and Fourier
transform spectrometers are the same types. Generally, there are two types of infrared
sources. The simplest one is called an air cooled source due to it temperature is
maintained by air currents in the spectrometer. Typically, air cooled source run at 1100
to 1400 K temperature range. Air cooled source may not provide enough IR intensity
for some applications but it main advantages are they are inexpensive and very
convenient because they not require special cooling.
However, the source in FTIR instruments is more often water-cooled to provide
better power and stability. The disadvantages of water cooled sources is that they
required cooling water, this added expense and may not be available in certain locations
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Reflectance Measurement and Modelling of High Reflectivity Distributed Bragg Reflector Stacks
and that it cost more than air cooled sources. [SHERMAN Hsu, C.P., 1997; SMITH,
B.C., 1996]
3.1.2
Interferometer
The FT-IR interferometer is commonly Michelson interferometer that collects a
spectrum as shown in Figure 3-2 below. The Michelson interferometer was invented by
1907 Nobel Prize in Physics winner, Albert Abraham Michelson in 1880. [SMITH, B.C.,
1996]
M1
M2
L
Source
Beam
Splitter
To sample
and detector
X
Figure 3-2 : Michelson Interferometer
Source: Based on Oriel Instruments; W.Herres and J. Gronholz; Thermo Nicolet; C.P. Sherman Hsu
The Michelson interferometer consists of three active components which are
beam splitter, a fixed mirror or stationary mirror and a moving mirror which can be
moved very precisely back and forth. [Oriel Instrument, No Date; Thermo Nicolet, 2002;
HERRES, W. and GRONHOLZ, J., 1987] The two mirrors are perpendicular to each
other [SHERMAN Hsu, C.P., 1997]. The beam splitter is a semi-reflecting device and is
made of deposited germanium thin film on flat potassium bromine, KBr substrate that
transmits half of the radiation striking it and reflects the other half. [Oriel Instruments,
No Date; SHERMAN Hsu, C.P., 1997]
As radiation emitted by a source, it is collimated and directed to the beam
splitter, and strikes the beam splitter. At the beam splitter, the beam separates into two.
One beam is reflected off the beam splitter, travels to the fixed mirror, M1 and the other
beam travels to the moving mirror, M2. Both beams are reflected back to the beam
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Reflectance Measurement and Modelling of High Reflectivity Distributed Bragg Reflector Stacks
splitter. The reflected beam travels through a distance L and is reflected back to hit the
beam splitter again after a total path length of 2L. Although same thing happen to the
transmitted beam but the total path length of the transmitted beam is according to 2*(L
= x) because M2 is not fixed at the same position L but moving back and forth around L
by a distance x precisely at a constant velocity. [Oriel Instruments, No Date; HERRES,
W. and GRONHOLZ, J., 1987; KOSTERS, P., 2000]
After reflected by both M1 and M2 mirrors, the two halves of the beam
recombine on the beam splitter and exhibit a optical path different of 2*x. This means
the partial beams are spatially coherent and create an interference pattern when they
recombine since some of the wavelengths recombine constructively and some
destructively. [Thermo Nicolet, 2002; HERRES, W. and GRONHOLZ, J., 1987]
3.1.3
Detector
There are two most popular detectors for a FTIR spectrometer, which are deuterated
triglycine sulfate (DTGS) and mercury cadmium telluride (MCT). The disadvantage of
many detectors used in conventional dispersive instruments like thermocouple and
thermistor is the response times are too slow for the rapid scan times of the
interferometer that need to be ≤1 seconds. The DTGS detector is a pyroelectric detector
that delivers rapid responses because it measures the changes in temperature rather than
the value of temperature and operates at room temperature and it is the most commonly
used detector material in the mid-IR.
In contrast, the MCT detector is a photon or quantum detector that depends on
the quantum nature of radiation and also exhibits very fast responses but must be
maintained at liquid nitrogen temperature (77 K) to be effective. In general, the MCT
detector is faster and more sensitive than the DTGS detector but it is cost 5 to 10 times
higher than DTGS detector. [SHERMAN Hsu, C.P., 1997; SMITH, B.C., 1996]
3.1.4
Sample Compartments
Generally, there are two distinct types of sample compartment, namely purgible and
non-purgible compartment. The purgible compartment maximized the amount of IR
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Reflectance Measurement and Modelling of High Reflectivity Distributed Bragg Reflector Stacks
energy that made it to the sample and to the detector because it has holes that let the IR
beam in and out in the side but it required purge gas to keep the KBr beamsplitter from
fogging and to prevent CO2 and H2O peaks from appearing in spectra.
The non-purgible or often known as sealed compartment, the KBr IR transparent
windows are installed over the holes in the sample compartment. The advantages are
that they don’t need to be purged and no water vapor can reach the hygroscopic KBr
beamsplitter. Although there may be some residual CO2 and H2O presence in the
sample compartment and lead to some residual CO2 and H2O peaks in the spectra but
the bands are usually not strong enough to be a problem. The disadvantage is that the
window will partially block the IR beam and cause as much as 20% IR energy lost,
affecting the signal-to-noise ratio. Another disadvantage is that the window and the
desiccant pack need replaced or regenerated regularly. [SMITH, B.C., 1996]
3.2
OPTICAL PATH DIFFERENT (OPD) AND ZERO PATH
DIFFERENT (ZPD)
The detector response for a single-frequency component from the source is considered
for an easier explanation. The differences in optical paths between the two split beams
are created by varying the relative position of moving mirror, M2 to the fixed mirror,
M1. When the M1 and M2 are at the same distance from the beam splitter, the two
beams are totally in phase with each other; they interfere constructively and lead to a
maximum in the detector response. The condition is known as zero path difference or
ZPD and this is the natural reference point for the FT-IR.
When M2 travels in either direction by the distance λ/4 around L, the optical
path is changed by 2*(λ/4) or λ/2. This is known as optical path difference (OPD). In
this position, the two beams are 180º out of phase with each other, thus interfere
destructively. As M2 continue to move by the distance λ/4 each time, it will create an
interval constructive and destructive interference between the two beams as shown in
the Figure 3-3. [SHERMAN Hsu, C.P., 1997; Oriel Instruments, No Date; SMITH, B.C.,
1996]
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Reflectance Measurement and Modelling of High Reflectivity Distributed Bragg Reflector Stacks
Figure 3-3 : Schematic Representation of Waves and Their Phase, Input, Output, and
the Interferometer. (a) OPD=0 case. (b) λ/4 OPD case. (c) λ/2 OPD case. (d) 3λ/4 OPD
case. (e) 1λ OPD case.
Source : Oriel Instruments
3.3 INTERFEROGRAM
The interferogram is the name of the interference signal acquired and recorded by an
FT-IR spectrometer. When M2 is moved at a constant velocity, the intensity of radiation
reaching the detector varies in a sinusoidal manner to produce the interferogram output
as shown in Figure 3-4.
Figure 3-4 : Interferogram of Radiation From The source
Source : SHERMAN Hsu, C.P., 1997
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Reflectance Measurement and Modelling of High Reflectivity Distributed Bragg Reflector Stacks
Interferogram is actually a time domain spectrum and records the detector
response changes versus time within the mirror scan. Extension of the same process to
two or more frequencies results in a more complex interferogram, which is the
summation of the individual modulated waves. Figure 3-5 shows the beam path of a two
wavelength source while Figure 3-6 shows the interferogram for three component
frequencies. The greatest amplitude occurs at the point of zero path difference (ZPD).
[SHERMAN Hsu, C.P., 1997; Oriel Instruments, No Date]
Figure 3-5 : The Beam Path of a Two Wavelength Source
Source : Oriel Instruments
Figure 3-6 : Interferogram Consisting of Three Modulated Cosine Waves
Source : SHERMAN Hsu, C.P., 1997
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Reflectance Measurement and Modelling of High Reflectivity Distributed Bragg Reflector Stacks
The examples above just are some simple and symmetric interferogram. The
interferogram produced with a broadband IR source displays extensive interference
patterns which is a complex summation of superimposed sinusoidal wave where each
wave corresponding to a single frequency. If this IR beam is focused in a sample
compartment with the help of some mirrors and directed through the sample, the
amplitudes of a set of waves are reduced by absorption if the frequency of this set of
waves is the same as one of the characteristic frequencies of the sample [KOSTERS, P.,
2000; SHERMAN Hsu, C.P., 1997] as shown in Figure 3-7.
Figure 3-7: A Typical Interferogram Produced With a Broadband IR Source
Source : Oriel Instruments
Because of the effect of interference, the intensity of the beam as measured with
the detector depends on the difference in path length in the two arms of the
interferometer. The OPD, called the retardation, δ is twice the path difference between
the two arms. Hence, the intensity for a specific wave number, υ, at the detector can be
described as:
I(δ) = ½ Iº(υ){1 + cos 2πυδ}
(4)
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Reflectance Measurement and Modelling of High Reflectivity Distributed Bragg Reflector Stacks
Where, Iº(υ) is the light intensity of the source. Generally, the DC component in the
signal is subtracted and only the varying part of the measured intensity is of the interest
for FT-IR spectroscopy and therefore the interferogram has a function as follow:
I(δ) = B(υ) cos 2πυδ
(5)
where B(υ) gives the single beam spectral intensity or spectrum which is the intensity of
the source at wave number, υ as modified by the system that is studied and by
instrumental characteristics. [KOSTERS, P., 2000; HERRES, W. and GRONHOLZ, J.,
1987]
The interferogram contains information over the entire IR region to which the
detector is responsive. The detector signal is sampled at small, precise intervals during
the mirror scan. The sampling rate is controlled by an internal, independent reference, a
monochromatic beam from a helium neon (HeNe) laser focused on a separate detector
and sometime the interference pattern of the HeNe laser is included in the interferogram.
[HERRES, W. and GRONHOLZ, J., 1987; SHERMAN Hsu, C.P., 1997]
3.4
FOURIER TRANSFORM
Once an interferogram is collected, a mathematical operation known as Fourier
Transform is performed to convert the interferogram into a final IR spectrum which is
the familiar frequency domain spectrum showing intensity versus frequency, i.e.
emission, absorption, transmission, etc. This also explains how the term Fourier
Transform Infrared spectrometry is created. This method is discovered by J.W. Cooley
and J.W. Tukey in 1965, followed by an explosive growth of computational power at
affordable prices has been the driving force behind the market penetration of FT-IR
instruments. [Oriel Instruments, No Date; HERRES, W. and GRONHOLZ, J., 1987;
SHERMAN Hsu, C.P., 1997]
There are a number of steps involved in calculating the spectrum. Instrument
imperfections and basic scan limitations need to be accommodated by performing phase
correction and apodization steps. These electronics and optical imperfections can cause
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Reflectance Measurement and Modelling of High Reflectivity Distributed Bragg Reflector Stacks
erroneous readings due to different time or phase delays of various spectral components.
Apodization is used to correct for spectral leakage, artificial creation of spectral features
due to the truncation of the scan at its limits. [Oriel Instruments, No Date; HERRES, W.
and GRONHOLZ, J., 1987]
A reference or background of a single beam without a sample is also collected
when performing the measurement and the sample single beam is ratio up to the
background single beam to produce a transmittance spectrum. This transmittance
spectrum can be converted to absorbance by taking the negative log10 of the data points.
The x-axis of the FT-IR spectrum is typically represented the wave numbers in cm-1.
This unit is a product of the Fourier transform algorithm operating on the interferogram
and is the reciprocal of the actual wavelength of light measured in centimeters at a point
in the infrared spectrum. [Thermo Nicolet, 2002] The Figure 3-8 below shows the
process of collecting an infrared spectrum in an FT-IR spectrometer.
Figure 3-8 : The Process of Collecting An Infrared Spectrum in an FT-IR Spectrometer
Source : Thermo Nicolet, 2002
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Reflectance Measurement and Modelling of High Reflectivity Distributed Bragg Reflector Stacks
3.5
ADVANTAGES OF FT-IR
FT-IR instruments have distinct advantages over conventional IR spectrometers such as
dispersive spectrometer:
•
FT-IR spectrometers have a built-in wave number calibration of high precision
that practically about 0.01 cm-1 due to the use of a helium neon laser as the
internal reference. This is known as the Connes advantage. This eliminates the
need for external calibrations.
•
FT-IR spectrometers have better speed and sensitivity known as Multiplex or
Felgett advantage. A complete spectrum can be obtained during a single scan of
the moving mirror, while the detector observes all frequencies simultaneously.
An FTIR instrument can achieve the same signal-to-noise (S/N) ratio of a
dispersive spectrometer in a fraction of the time ≤1 sec versus 10 to 15 min.
The S/N ratio is proportional to the square root of the total number of
measurements. Because multiple spectra can be readily collected in 1 min or less,
sensitivity can be greatly improved by increasing S/N through coaddition of
many repeated scans.
•
Increased optical throughput which known as Jaquinot advantage because FT-IR
instruments do not require slits due to dispersion or filtering is not needed.
Instead, a circular optical aperture is commonly used in FTIR systems. The
beam area of an FT instrument is usually 75 to 100 times larger than the slit
width of a dispersive spectrometer. Thus, more radiation energy is made
available. This constitutes a major advantage for many samples or sampling
techniques that are energy-limited.
•
Elimination of stray light and emission contributions. The interferometer in
FTIR modulates all the frequencies. The unmodulated stray light and sample
emissions (if any) are not detected.
•
Modern FTIR spectrometers are usually equipped with a powerful,
computerized data system. It can perform a wide variety of data processing tasks
such as Fourier transformation, interactive spectral subtraction, baseline
correction, smoothing, integration, and library searching.
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Reflectance Measurement and Modelling of High Reflectivity Distributed Bragg Reflector Stacks
FTIR spectrometers are the preferred choice for samples that are energy-limited
or when increased sensitivity is desired although the spectrum of many samples can be
satisfactorily run on either FTIR or dispersive instruments. A wide range of sampling
accessories is available to take advantage of the capabilities of FTIR instruments. [Oriel
Instruments, No Date; HERRES, W. and GRONHOLZ, J., 1987; SHERMAN Hsu, C.P.,
1997]
3.6
SPECULAR REFLECTANCE SPECTROSCOPY
Reflectance spectroscopy allows the samples to be analyzed without modification.
Generally, reflectance measurement can be divided into two categories, which are
internal reflectance measurement and external reflectance measurement. The internal
reflectance measurements using an attenuated total reflectance (ATR) element to make
the measurement in contact with the sample. In contrast, the external reflectance
measurements are made using an infrared beam that reflected directly from the sample
surface. There are also two types of external reflectance, which are specular reflectance
and diffuse reflectance.
For this research project, the specular external reflectance measurement is of
interest in making the reflectance measurement of the high reflectivity DBR stacks. If
we define a line that drawn perpendicularly to the surface of the sample as surface
normal, the specular reflectance occurs when the angle of incidence equals to angle of
reflectance as shown in Figure 3-9.
Surface Normal
θ1 θ2
Sample
Figure 3-9 : Specular Reflectance
Source : SMITH, B.C., 1996
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Reflectance Measurement and Modelling of High Reflectivity Distributed Bragg Reflector Stacks
where θ1 is angle of incidence while θ2 is angle of reflectance. The specular reflectance
spectra are obtained with accessories that can be mounted or slide into the sample
compartment of a FT-IR spectrometer. The details of the specular reflectance
measurement will be discussed in more detail in the next chapter.
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Reflectance Measurement and Modelling of High Reflectivity Distributed Bragg Reflector Stacks
CHAPTER 4
TRAINING PROGRAM AND SAMPLE MEASUREMENT
IN
UNIVERSITY OF MONTPELLIER 2
4.0 INTRODUCTION
Training program and sample measurement in University of Montpellier 2 is one of the
activities in project planning schedule. The program was took place at Centre
d’Electronique et de Micro-Optoelectronique de Montpellier 2 (CEM2) from June 13th
to June 15th. Figure 4-1 below shows the photograph taken in the CEM2 lab. From left,
Mr. Jinyu Li, Mr. Jean Baptiste Rodriguez, me and Professor Stephanie Haywood.
Figure 4-1 : Photograph Taken in CEM2 Lab
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Reflectance Measurement and Modelling of High Reflectivity Distributed Bragg Reflector Stacks
The main objective of the program is to observe the reflectivity measurement
attachment or accessories that available in CEM2 as well as to do some absolute
measurement of reflectivity for our samples in order to setup a similar system in our
own lab. Unfortunately, the attachment does not work in the same way as what initially
proposed. The proposed multiple reflections are as shown in Figure 4-2 below.
R1
P2
R2
P0
R1
P4
P0
R2
Figure 4-2 : Multiple Reflections Techniques
For the above multiple reflections techniques, the reflectivity can be calculated
as follow:
P2 = P0R1R2
(6)
P4 = P0R12R22
(7)
For the systems above, either the reflectivity R1 or R2 has to be known in order to obtain
the reflectance of the sample DBR. The attachment available in CEM2 is ideal for
external specular reflection and the detail of how this attachment work will discuss later.
The attachment needs to be redesigned in order to perform the multiple reflection
measurement technique as proposed.
However, some reflectivity measurements of the samples have been done by
using the attachment as well as the transmission measurements of the samples. Some
photographs of the FT-IR and the accessories that available in CEM2 were also taken
for further reference.
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Reflectance Measurement and Modelling of High Reflectivity Distributed Bragg Reflector Stacks
4.1 EXTERNAL SPECULAR REFLECTION SPECTROSCOPY
In the external specular reflection spectroscopy, also known as specular reflectance,
light is reflected from a smooth or mirror-like sample to record its spectrum as shown in
the Figure 4-3. [Specac, 2001; Harrick Scientific, 2005]
Figure 4-3 : External Reflection Spectroscopy or Specular Reflectance
Source : Specac, 2001
This technique was analyzed theoretically by Francis and Elison in 1959 as a
spectroscopy technique for films on mirror surfaces followed by some other
applications in the middle of 1960s and this technique found much wider use in the
1970s when accessories became more readily available. [Harrick Scientific, 2005]
External reflection spectroscopy is a non-contact, non-destructive technique and
particularly useful for film thickness, refractive index measurements as well as
recording spectrum of thin films on metal substrate. However, the external reflection
spectrum may look different from transmission spectrum in many ways. For example,
bands may be shifted to higher wavenumbers, the spectrum may follow the dispersion
in the refractive index, and spectral contrast may not depend linearly on sample
thickness. [Specac, 2001; Harrick Scientific, 2005]
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Reflectance Measurement and Modelling of High Reflectivity Distributed Bragg Reflector Stacks
4.2 VARIABLE ANGLE REFLECTION ACCESSORY
The attachment that being use in CEM2 for their FT-IR spectroscopy is Variable
Specular Reflection Accessory produced by Harrick Scientific as shown in Figure 4-4.
Figure 4-4 : Variable Specular Reflection Accessory
This attachment is the industry standard for variable angle specular reflectance
studies and there are various sample stages that readily available for incorporate with
the attachment for absolute reflectance measurements or convenient horizontal
reflectance measurements at a 12º incident angle. For the absolute reflectance
measurements, the sample stage that in used was 12º absolute reflectance sampling
stage, which particularly well suited for examining highly reflective samples, features a
12º incident angle and retains the polarization of the incident beam. Figure 4-5 below
shows the 12º absolute reflectance sampling stage.
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Reflectance Measurement and Modelling of High Reflectivity Distributed Bragg Reflector Stacks
Figure 4-5 : 12º Absolute Reflectance Sampling Stage
4.3 EXPERIMENTAL SAMPLE REFLECTIVITY MEASUREMENT
The sample measurement is done by using Thermo Nicolet Nexus 870 E.S.P FT-IR.
The Figure 4-6 below shows the schematic diagram of the FT-IR. From the diagram,
one can see clearly the movement of the beam from the source through the beam splitter,
then through the sample compartment and finally being detected by the detector.
Figure 4-6 : Schematic Diagram of the FT-IR Spectrometer
Source : SHERMAN HSU, C.P., 1997
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Reflectance Measurement and Modelling of High Reflectivity Distributed Bragg Reflector Stacks
In order to do the measurement, the Variable Specular Reflection Accessory and
the 12º absolute reflectance sampling stage have to be located in the sample
compartment as shown in Figure 4-7. The measurement is performed by using the
software supplied, called Omnic and the setting is as follow:
Ø Resolution = 32 cm-1
Ø Gain = 8
Ø Speed = 0.6329 kHz
Ø Wavelength = 12500 – 2100 nm
Ø Aparture = 1
Ø Source = White light
Ø Detector = Internal (DTGS)
Ø Beam Splitter = CaF2
Ø Final format = % Reflectivity
Figure 4-7 : Variable Specular Reflection Accessory and 12º Absolute Reflectance
Sampling Stage in the FT-IR Sample Compartment
The optical setup in the sample compartment is illustrated in Figure 4-8. Before
the sample is being placed on the sampling stage, the beam from the source is focused at
the sampling stage reference mirror to obtain maximum signal of interferogram by
adjusting the parabolic mirrors of the reflection accessory. After the sample being
placed on the sampling stage, a background spectrum is obtained by collecting an
interferogram with the sampling stage in the V mode where the beam is directed by
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Reflectance Measurement and Modelling of High Reflectivity Distributed Bragg Reflector Stacks
reflection accessory mirrors to the sample stage mirror, then from sample stage mirror
to the detector.
Figure 4-8 : The Direction of The beam
Source : Harrick Scientific Products Online Catalogue, 2005
This background spectrum actually is a response curve of the spectrometer
which take into account the performance of the source, interferometer and detector as
well as any ambient water and carbon dioxide that present in the optical bench in a
specific wavelength and concentration as shown in Figure 4-9.
Figure 4-9 : Absorption or Transmission of Ambient Water and Carbon Dioxide
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Reflectance Measurement and Modelling of High Reflectivity Distributed Bragg Reflector Stacks
In order to collect the reflectivity spectrum of the sample, the sample stage is
rotated to the ‘W’ mode. Once again, the parabolic mirrors is adjusted to focus the beam
to the sample and to obtain maximum signal. In ‘W’ mode, the beam from the source is
directed to the sample first, then reflected by the sample to the sample stage and the
sample stage mirror reflects the beam back to the sample and finally the beam is
directed to the FT-IR spectrometer by reflection accessory mirrors. The alignment and
optical path length for the sample and reference spectra is maintained in this
configuration. It is important to note that the measured quantity is the ratio of the
sample spectrum to the background spectrum and is the square of the reflectance, i.e. R2.
Figure 4-10 below shows the ‘V-W’ mode double reflection technique.
Figure 4-10 : The ‘V-W’ Mode Double Reflection Technique
Source : Harrick Scientific Products, 2005.
4.4 EXPERIMENTAL RESULTS
The reflectance spectrum of the sample is obtained and displayed by the software called,
Omnic. The measurement data also can be extract and save as csv extension file in text
file format. With this csv file, the measured R2 value can be manipulated into single
reflection spectrum by take the square root and plot it against wavelength as shown in
Figure 4-11 but the details of the experimental results will be discussed further in
Chapter 6 later. However, the wavelength cannot be obtained straight away because the
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Reflectance Measurement and Modelling of High Reflectivity Distributed Bragg Reflector Stacks
R2 spectrum is plotted against wavenumber in the Omnic software. In order to obtain the
wavelength value, we need to take the reciprocal of the wavenumber.
Measured Reflectance vs Wavelength
100
90
Reflectance (%)
80
70
60
50
40
30
20
10
0
1400
1480
1551
1629
1715
Wavelength (nm)
Figure 4-11 : Sample Absolute Reflectance Measurement Spectrum
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1811
Reflectance Measurement and Modelling of High Reflectivity Distributed Bragg Reflector Stacks
CHAPTER 5
MATLAB
SIMULATION PROGRAM
5.0 INTRODUCTION
It is relatively simple to calculate the reflectance and transmittance of a monolayer or
double layer system by using Maxwell’s equation. However, when the number of layers
increases or in a multilayer system, the size of the equations grows and their calculation
becomes more complicated. In order to cope with this problem, the matrix method can
be applied due to a few advantages. In particular in the application to periodic structures
like DBRs, the calculation can be done with a simple processing by a computer. For
homogeneous and isotropic multilayer systems, a 2x2 matrix method is adequate.
[STUMPF, W., 2001]
In order to create the simulation program, Matlab program language would be
the best option because it is a mathematical language that capable of handling all of the
function needed to model the DBR mirrors and allow for very advanced matrix and
vector function calculation. Another advantage of Matlab language is that the basics of
the language is easy to learn and has a nice set of Graphical User Interface (GUI)
command and option.
The Matlab program used to implement the characteristic matrix method of
calculating the reflectivity and simulation of the spectrum will be explained in this
chapter.
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Reflectance Measurement and Modelling of High Reflectivity Distributed Bragg Reflector Stacks
5.1
REFLECTIVITY SIMULATION PROGRAM
The simulation programs used to simulate the reflectivity and several properties of DBR
in this thesis are amended from the thesis from LIM, H.C., 2002, CONWAY, L.J., 1999
and BILBY, R, 2000. This simulation program was written based on the transmission
matrix calculations from BORN, W. and WOLF, E., therefore the theory and equations
or formulas that used in this program will not be outlined in this thesis. Generally, these
programs are base on the steps shown in Figure 5-1 below.
Figure 5-1 : Step by Step Calculation in the Matlab Program
Source : CONWAY, L.J., 1999
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Reflectance Measurement and Modelling of High Reflectivity Distributed Bragg Reflector Stacks
The steps or the stages of the calculation are as follow:
Step 1 : User input parameters
Ø Number of layers of the DBR
Ø Refractive index of layer1, layer 2, incident medium and substrate
Ø Angle of incidence, θ
Ø Bragg wavelength
Ø Lowest wavelength in range
Ø Highest wavelength in range
Ø Step of wavelength in range
Step 2 : Convert appropriate parameters into nanometer (nm) = (x 10-9) and the angle of
incidence, θ from degree to radian:
θ i = θx
π
(8)
180
Step 3 : Calculate all the necessary angle of incidence at each interface using Snell’s
Law:
 ni
For 1st layer, θ1 = arcsin
 n1 sin θ i



(9)
 n1 

For 2nd layer, θ 2 = arcsin
n
sin
θ
1 
 2

n2
For substrate layer, θ sub = arcsin
 n sub sin θ 2
(10)



(11)
Step 4 : Calculate the optical thickness of each layer:
For 1st layer, d1 =
For 2nd layer, d 2 =
λB
(12)
4n1
λB
(13)
4n2
Step 5 : Calculate the optical admittance for each layer in s-polarization:
For incidence medium, µi = niYcosθi
(14)
For 1st layer, µ1 = n1Ycosθ1
(15)
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Reflectance Measurement and Modelling of High Reflectivity Distributed Bragg Reflector Stacks
For 2nd layer, µ2 = n2Ycosθ2
(16)
For substrate, µsub = nsubYcosθsub
(17)
Step 6 : Loop from lowest wavelength in range to highest wavelength in range with step
wavelength in range in order to calculate reflectivity as a function of wavelength.
Step 7 : Calculate the phase change at each layer
For 1st layer, δ 1 =
(2πn1d1 cos θ1 )
For 2nd layer, δ 2 =
λ
(2πn2 d 2 cos θ 2 )
λ
(18)
(19)
Step 8 : Finding the characteristic matrix for layer 1 and layer 2
 cos δ 1
For 1st layer, M 1 = 
iµ1 sin δ 1
 cos δ 2
For 2nd layer, M 2 = 
iµ 2 sin δ 2
i sin δ 1 / µ1 
cos δ 1 
i sin δ 2 / µ 2 
cos δ 2 
Hence: cm = M1 x M2
(20)
(21)
(22)
Step 9 : Layer control
Ø For even number of layers, the characteristic matrix, M = (M1M2)N/2, where
N is the number of layers.
Ø For odd number of layers, the characteristic matrix, M = (M1M2)N/2 x M1,
where N is the number of layers.
M
M =  11
 M 21
M 12 
M 22 
(23)
Step 10 : Calculate the elements of the characteristic matrix, B and C.
Ø B = M11 + (M12 x µsub)
(24)
Ø C = M21 + (M22 x µsub)
(25)
Step 11 : Now we can calculate the reflectivity value at different wavelength. The
reflectivity, R is calculated as follow:
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Reflectance Measurement and Modelling of High Reflectivity Distributed Bragg Reflector Stacks
 µ B−C 

ρ =  0
 µ0 B + C 
(26)
R = ρρ * , where ρ * denoted the conjugate of ρ .
(27)
Step 12 : Finally, the results or the array of the reflectivity for the range of input
wavelength range will be printed and store in a file then plotted as a function
of wavelength.
The above steps are the steps for the basic program for the simulation of the
DBR mirrors. The coding and comments of the program can be found in Appendix 3.
The Matlab program for the investigation of the various DBR mirror properties can be
modified from this program and the results from the simulations will be extended to the
next chapter to include more specific properties.
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Reflectance Measurement and Modelling of High Reflectivity Distributed Bragg Reflector Stacks
CHAPTER 6
RESULTS AND DISCUSSIONS
6.0
EXPERIMENTAL
MEASUREMENT
RESULTS
AGAINST
MODELLED RESULT
The reflectance of the In0.53Ga0.47As/InP DBR is shown in Figure 6.1. As mentioned
before in Chapter 4, the initial measurement is in R2 value, in order to obtain the
reflectance value of the sample, the square root of the initial measurement value need to
be taken. After the square root has been taken, it shows that the peak reflectivity has
gone up from ~90% to ~96%. This reveals that multiple reflection technique can be
used to measure the high reflectivity DBR more accurately compared to single
reflection technique.
Reflectance (%)
Measured Reflectance vs Wavelength
100
90
80
70
60
50
40
30
20
10
0
R Square Value
R Value
Modelled
1400 1450 1500 1550 1600 1650 1700 1750 1800 1850
Wavelength (nm)
Figure 6-1 : Measured Reflectance Spectrum Against Wavelength
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Reflectance Measurement and Modelling of High Reflectivity Distributed Bragg Reflector Stacks
From the figure, the center wavelength of the DBR is shown to be 1.55 µm with
peak reflectivity value of 96% but the DBR was designed for 1.6 µm wavelength
operation with theoretical peak reflectivity value of 99.96%. This may due to some
inhomogeneities of layer thickness towards the edge of the mirror which shifted away
from the optimum quarter-wave values resulting from a small drift in the growth
temperature from bottom to top of the DBR and alloy composition fluctuations resulting
from flux variations during the growth. The decrease in reflectivity may also due to the
increased absorption at the lower wavelength.
The flat region of the reflection band or also known as stop band has a width of
more than 100 nm and this indicates that the optical quality of the DBR is excellent
although the modelled spectrum has a stop band of about 150 nm. The narrowing of the
stop band may due to the fluctuation in the thickness that deteriorates the optical quality
of mirrors by introducing asymmetry into the stop band. The fluctuation of refractive
index of each layer and the change in the refractive index with doping also will reduce
the width of the stop band. This would also correspond to the reduced peak reflectivity.
In practice, the ‘V-W’ mode double light reflection on two different points of
the DBR about 10 mm apart and beam size of about 5 mm requires high sample
homogeneity in order to obtain the accurate measurement. Therefore, the size of the
sample also had a significant effect on the reflectivity measurement. However, the DBR
samples used had been cut into few pieces for some other measurements done
previously and some of the samples had broken into much more smaller pieces. The
approximated dimension of the biggest piece of sample is shown in Figure 6-2. With
this dimension, some lost of light source may be expected at the point marked a cross in
the figure and this reduces the intensity of the light as well as the peak reflectivity.
15 mm
17 mm
3 mm
18 mm
Figure 6-2 : The Dimension of The Biggest Piece of Sample
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Reflectance Measurement and Modelling of High Reflectivity Distributed Bragg Reflector Stacks
Considering the simple theory of the reflectivity where a light source is directed
to DBR with the angle of incidence, θ and the light being reflected by the DBR with the
angle of reflection equal to θ as shown in Figure 6-3, the peak reflectivity of the DBR
will reduced if the intensity of the reflected beam is reduced due to lost because the
reflectivity is given by:
Reflectivity, R =
P
P0
(28)
R
θ
P0
P
Figure 6-3 : Simple Reflection Theory
With reference to the sample growth sheet in Appendix 1, there is an evidence
of a drop of about 15% of the peak reflectivity towards the edge of the mirror. In
addition to the lack of homogeneity, band to band absorption also may lead to a peak
reflectivity value underestimated of about 1%. Another factor that may affect the
reduced in peak reflectivity is that the alignment of the sample slide to the sampling
stage is done manually and not mechanized; this may lead to uncertainty of human error.
The higher peak reflectivity of the transfer matrix model may also imply that
any free carrier absorption in the semiconductor materials is negligible and is not taken
into account. However, the absorption at the design wavelength may be negligible but
over the wavelength range of the spectrum in Figure 6-1 this may not be the case and a
lot of discrepancies between modelled and measured reflectance spectra arise because
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Reflectance Measurement and Modelling of High Reflectivity Distributed Bragg Reflector Stacks
of this problem.The investigation on the effect of free carrier absorption will be
discussed further later in this thesis. Nevertheless, reasonable layer thickness variation
and residual absorption are not able to account for the 3.96% reduced peak reflectivity.
The measured reflectivity is also likely to be affected by the quality of the present
sample. This sample is an eight years old sample; therefore it may have some defects
such as scratches and contamination. The reflectivity spectra reveals accurate
information about DBR properties, therefore the reflectivity measurement should be
carried out first in order to prevent any unwanted defect or contamination of the mirror.
In order to investigate regarding any defect or non-homogeneity on the sample,
transmission measurement was carried out at different points on the sample. The result
is shown in Figure 6-4.
Transmission vs Wavelength
30.00
Transmission (%)
25.00
Point 1
Point 2
20.00
Point 3
15.00
10.00
5.00
0.00
1200
-5.00
1300
1400
1500
1600
1700
1800
1900
2000
Wavelength (nm)
Figure 6-4 : Transmission vs Wavelength Plot at Different Points Across The Sample
The figure reveals that there is a slight variation of the sample thickness towards
the edge of the DBR and as one can see, the transmission spectra consists of a large,
central minima which is centred at the design wavelength, in this case λ =1600 nm. This
indicates that the operation wavelength of the DBR is 1600 nm instead of 1550 nm that
shown in the reflectivity measurement spectra. This also reveals that the reflectivity
measurement may not being done at the center but near the edge of the DBR. As the
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Reflectance Measurement and Modelling of High Reflectivity Distributed Bragg Reflector Stacks
measurement point shift towards the edge, the central minima of the spectrum shifted to
the lower wavelength, indicating the DBR is thinner towards the edge. Theoretically,
the reflectivity spectra are able to be obtained from the transmission spectra by using the
formula as follow by ignoring any absorption in the InGaAs layer:
Reflectivity (R) = 1 – Transmission (T)
(29)
For this purpose, a high resolution transmission measurement was carried out at the
point that believed to be the centre of the DBR and the transmission spectra is as shown
in Figure 6-5 while the reflectivity spectra derived from the transmission measurement
is shown in Figure 6-6.
Transmission (%)
High Resolution Transmission Measurement
35.00
30.00
25.00
20.00
15.00
10.00
5.00
0.00
1200 1300 1400 1500 1600 1700 1800 1900 2000 2100 2200
Wavelength (nm)
Figure 6-5 : High Resolution Transmission Measurement Spectra
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Reflectance Measurement and Modelling of High Reflectivity Distributed Bragg Reflector Stacks
Reflectivity Spectrum From Transmission
Measurement
Reflecivity (%)
100.00
90.00
80.00
70.00
1450
1500
1550
1600
1650
1700
1750
1800
Wavelength (nm)
Figure 6-6 : Reflectivity Spectra Derived From Transmission Measurement
The reflectivity spectrum show the peak reflectivity of over 99.9% for 100 nm
wavelength range centered at 1600 nm and this reveals that the reflectivity of the sample
is in good agreement with the model as shown in Figure 6-7 although the width of the
stop band is narrower due to the factors that mentioned earlier.
Reflectivity vs Wavelength
Modelled
120
Reflectivity from
Transmission
Measurement
Reflectivity (%)
100
80
60
40
20
0
1400
1450
1500
1550
1600
1650
1700
1750
1800
1850
Wavelength (nm)
Figure 6-7 : Reflectivity vs Wavelength for Model and Measured Reflectivity from
Transmission Measurement.
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Reflectance Measurement and Modelling of High Reflectivity Distributed Bragg Reflector Stacks
6.1
MEASUREMENT SETUP IN OWN LAB
In order to setup the reflectivity measurement in the lab, a few quotations had been
obtained and these quotations can be found in Appendix 2. The quote from Harrick
Scientific Products is for the Variable Angle Specular Reflection Accessory and the 12º
Absolute Reflectance Stage, same as the one used in CEM2. The quote from Dr. Paul
Turner is for the Bruker Optic’s A519-A absolute reflectance accessory which also
utilize 12º V-W measurement mode but required even larger sample size compared to
Harrick Scientific Products’ Variable Angle Specular Reflection Accessory as shown in
Figure 6-8.
Figure 6-8 : Bruker Optic’s Absolute Reflectance Accessory
Source : www.brukeroptics.com
However, the cost, delivery time delay and time needed for re-design and
fabrication of the sampling stage had become the limitation on the setup to meet the
thesis deadline. Although Dr. Paul Turner also recommended a relative reflectance unit
from Pike Technologies that requires a calibrated gold mirror in order to obtain the
absolute reflectance value and it was claimed to be simpler to setup and lower cost
compare to the V-W absolute reflectance accessories but the cost of the calibrated gold
mirror also need to be taken into account. Although the reflectance of gold mirror is
assume to be unity but in fact it is not hence it is not a good long term solution for us.
Therefore, we decided to use only the results obtained from CEM2 and move on to
perform the modeling of the DBR properties using Matlab programming.
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Reflectance Measurement and Modelling of High Reflectivity Distributed Bragg Reflector Stacks
6.2
DBR
PROPERTIES
EXAMINATION
BY
Matlab
SIMULATION PROGRAM
The basic Matlab simulation given in Appendix 3 can be modified in order to
simulation the results of several properties of the DBR. The DBR will be examined by
altering a few parameters, such as the number of layers or periods, angle of incidence,
refractive indices different, sequence of the layer and observing how the reflectivity of
the DBR is affected by these parameters. The refractive indices used in this modelling
were taken from HAYWOOD, S.K. et al, 1994 and DEPPE, D.G. et al. 1990.
6.2.1 Varying Number of Periods
The effect of the number of periods on reflectivity will be demonstrated in this section.
This can be done by modify the basic Matlab program in Appendix 3 by fixed all the
user input parameters but allowing for a variable number of periods with ni = n(air) = 1;
n1 = n(InGaAs) = 3.6; n2 = n(InP) = 3.17 and ns = n(InP) = 3.17. The results of
reflectivity against wavelength with varied number of periods produced from the
simulation are shown in Figure 6-9.
Reflectivity vs Wavelength
1
N = 36
0.9
N = 25
0.8
N = 15
N=5
Reflectivity (%)
0.7
0.6
0.5
0.4
0.3
0.2
0.1
1400
1450
1500
1550
1650
1600
Wavelength (nm)
1700
Figure 6-9 : Varies Number of Periods
- 52 -
1750
1800
1850
Reflectance Measurement and Modelling of High Reflectivity Distributed Bragg Reflector Stacks
From the figure, it is clear that reflectivity is highly affected by the number of
periods where increasing number of periods will increases the reflectivity of the DBR
but the reflectivity curve saturates at large values of number of periods. Any further
increases in the number periods will not increases the reflectivity further and the control
of growth condition become very critical with the increasing number of periods.
6.2.2 Varying Refractive Index Difference
This section will demonstrate the effect of the refractive index difference on reflectivity
and the bandwidth of the stop band. Now, if we consider the limits of the graph at
number of periods, N = 0 and N bigger than 20, where the curve begins to saturate as it
approaches 1.0 or 100% reflectivity. The Equation (2) can be rewritten in a simpler term
as follow,
1 − ab 2 N 
R=
2N 
1 + ab 
2
(30)
n 
n 
where a =  s  ; b =  1 
 n2 
 n0 
From Figure 6-8, it can be observed that the graph saturates at large values of N. This is
because ab 2 N become >>1 as N increases, hence [1- ab 2 N ] → −ab 2 N and if N is large
2
enough ab
2N
1 − ab 2 N 
→ 1 when N is large enough, and the curve
→ ∞ . Eventually, 
2N 
1 + ab 
will saturates. At the other extreme where N = 0, b 2 N = 1 and equation can then be
written as:
1 − a 
R=
1 + a 
2
(31)
and hence the reflectance is solely dependent on the refractive indices of the input and
exit media. If n0 = 1 (air as entry medium) and ns =3.17, then, at N = 0, the reflectance
is ~27% if we calculate the reflectivity from Equation (30). If the entry medium is not
air but a semiconductor, say InP, where n0 =3.17 at 1.6 µm, then the reflectance is ~0%.
This difference is due to the large ∆n in the first case, ∆n =2.17 as compared to the
second case, ∆n =0. The reflectance will be greater for a larger value of ∆n . The
refractive index difference also determines the bandwidth of the stop band as mentioned
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Reflectance Measurement and Modelling of High Reflectivity Distributed Bragg Reflector Stacks
before which increases with increasing ∆n and approaches a constant for high values of
N as shown in Figure 6-8 and Figure 6-10.
Reflectivity vs Wavelength
1
Δn = 0.43 (nInGaAs=3.6)
0.9
Δn = 0.33 (nInGaAs=3.5)
Δn = 0.28 (nInGaAs=3.45)
0.8
Δn = 0.13 (nInGaAs=3.3)
nsubstrate = 3.17
Reflectivity (%)
0.7
0.6
0.5
0.4
0.3
0.2
0.1
1450
1500
1550
1650
1600
Wavelength (nm)
1700
1750
1800
Figure 6-10 : Varies Refractive Indices Different
6.2.3 Varying First Layer Material
In this section, the effect of the order of the layers in the periods on the overall DBR
reflectivity will be examined. Figure 6-11 below shows two reflectivity curves that
employ different order of the layers. The figure shows that the reflectivity for
InGaAs/InP order, which is InGaAs is the first is higher than the InP/InGaAs order.
This result indicates that if the first layer of the DBR is a higher index layer, the
reflectivity obtained will be higher. Therefore, the order of the layers in the stack has a
significant effect on the peak reflectivity.
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Reflectance Measurement and Modelling of High Reflectivity Distributed Bragg Reflector Stacks
Reflectivity
1
1st layer = InGaAs
1st layer = InP
0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
1450
1550
1500
1600
1650
1750
1700
1800
Reflectivity
1
0.9998
0.9996
0.9994
0.9992
0.999
0.9988
0.9986
0.9984
1599.7
1599.8
1600
1599.9
1600.1
1600.2
Figure 6-11 : Reflectivity of DBR with Different Layer Sequence
- 55 -
1600.3
Reflectance Measurement and Modelling of High Reflectivity Distributed Bragg Reflector Stacks
6.2.4 Varying Angle of Incidence
Figure 6-12 below shows the reflectivity against wavelength plot for vary angle of
incidence value.
Reflectivity vs Wavelength
1
0º
12º
0.9
60º
30º
0.8
45º
Reflectivity (%)
0.7
0.6
0.5
0.4
0.3
0.2
0.1
1450
1500
1550
1600
1650
Wavelength (nm)
1700
1750
1800
Figure 6-12 : Varies Angle of Incidence
It is an interesting feature to examine the variation of angle of incidence and
how it affects the reflectivity. From Figure 6-12, it can be observed that the stop band
shifted to a lower wavelength with increasing angle of incidence. This is due to the
increases in penetration depth with increasing angle of incidence. The same
investigation had been carried out by SHEN, J.L. et al for GaAs/AlAs quarter-wave
stacks with InGaAs/InGaAsP cover layer. The result from this investigation agreed with
their result.
For further investigation of whether or not the variation angle of incidence will
increases the reflectivity, the reflectivity value and center wavelength for different angle
of incidence is obtained and is shown in Table 6-1.
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Reflectance Measurement and Modelling of High Reflectivity Distributed Bragg Reflector Stacks
Angle of Incidence
Center Wavelength
(nm)
1600
1595
1590
1570
1550
1530
0º
12º
20º
40º
60º
80º
Reflectivity (%)
99.9867
99.9874
99.9886
99.9926
99.9964
99.9990
Table 6-1 : Reflectivity Value and Center Wavelength for Different Angle of Incidence
It can be seen from Table 6-1 that as the angle of incident approaches 90º, the
reflectivity tends toward 100% reflection as expected. This shows that overall, with
increased angle of incidence, there is an increase in reflectivity for s-polarization but the
center wavelength will shift towards lower wavelength instead of at Bragg wavelength
of interest.
As mentioned earlier, it is also instructive to determine the DBR reflectivity as a
function of angle of incidence. The idea to determine the reflectivity as a function of
angle of incidence arise after referred to CHUNNILALL, C.J. et al, 2002 work on FTIR measurement. For this purpose, a Matlab simulation program as in Appendix 4 by
adds in a loop for angle of incidence calculation. Figure 6-13 shows the DBR
reflectivity as a function of angle of incidence at Bragg wavelength (1600 nm).
Reflectivity vs Angle of Incidence at 1600 nm
100
99.98
Reflectivity (%)
99.96
99.94
99.92
99.9
99.88
99.86
0
10
20
30
40
50
Angle (degree)
60
70
80
90
Figure 6-13: Reflectivity vs Angle of Incidence at 1600 nm
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Reflectance Measurement and Modelling of High Reflectivity Distributed Bragg Reflector Stacks
From the figure one can see that the DBR has high reflectivity over the range
from 0º to 65º. After the angle of 65º, the reflectivity drop dramatically up to the angle
of incidence of ~85º but the DBR reflectivity increase again for grazing angle incidence,
which is not in the interest of this thesis.
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Reflectance Measurement and Modelling of High Reflectivity Distributed Bragg Reflector Stacks
CHAPTER 7
ABSORPTION IN InGaAs LAYER
7.0 INTRODUCTION
As mentioned earlier, considerable research has been devoted to the material properties,
growth techniques and the device behaviour of InGaAs ternary material during the past
few decades. It is a good example of semiconductor material where the research and
scientific interest are driven by significant technology needs and opportunities.
Different compositions of InGaAs have different lattice parameters as well as different
energy gaps. The most appropriate substrate material proves to be InP. However,
InGaAs is absorbing when lattice matched to InP. Therefore, in order to reduce the
absorption of InGaAs at certain wavelength, the content of indium in the ternary need to
be reduced as mentioned by GUY, P., WOODBRIDGE, K. and HOPKINSON, M.
The calculated indium content in the ternary material to reduce the absorption is
only correct at the wavelength of interest only. However, DBR is operated over a
wavelengths range between the wavelength of interest and the absorption is depends on
wavelength. Therefore, nevertheless the InGaAs layer absorption is exists in the DBR.
There are a few different mechanisms of absorption, such as photon assisted absorption,
band edge absorption, direct absorption, free carrier absorption, etc but the most
common absorption in DBR are band edge absorption and free-carrier absorption.
7.1
BAND EDGE ABSORPTION
Band edge absorption is an absorption process that involves the charge carriers electrons
and holes which attract each other and they can bind to a lattice site or move as a free
exciton through the DBR structure. Although the band edge absorption is weak but the
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Reflectance Measurement and Modelling of High Reflectivity Distributed Bragg Reflector Stacks
increased optical path length due to increasing angle of incidence or variation in DBR
optical thicknesses across the DBR will increases the probability of band edge
absorption. [STUMPF, W., 2001]
7.2
FREE-CARRIER ABSORPTION
Free-carrier absorption occurs when the free charge carriers does not generate electronhole pairs. There is a distinction between inter-band absorption and intra-band
absorption for free-carrier absorption. Free-carrier absorption is temperature and doping
dependence. According to LI, A.Z. et al., the free-carrier absorption can be obtained
from the permittivity of the semiconductors as follow:
α=
ωε 2
(32)
nr c
where ε2 is the imaginary part of the permittivity, nr is the effective refractivity and c is
the speed of light in free space.
7.3
EFFECTS OF DOPING ON ABSORPTION
There are many publications had demonstrated that doping of semiconductors will shift
the absorption to a lower wavelength and enable the DBR to be used in the wavelength
of interest. This displacement of the absorption edge is known as the Burstein-Moss (BM) shift. When the mobility of a carrier with respect to its small effective mass is
significantly greater than it’s opposite at high doping level, the (B-M) effect becomes
larger due to the multi body effect of interactions between the charge carriers and
doping atom which constrict and reduce the band gap. [STUMPF, W., 2005;
HAYWOOD, S.K. et al., 1995; BENNETT, B.R., SOREF, R.A. and DEL ALAMO,
J.A., 1990]
According to literature [DEPPE, D.G. et al., 1990], either n-type or p-type
doping, has a significant effect on DBR reflectivity, where, the near band edge
absorption will significantly reduce with n-type doping due to (B-M) shift, while the p- 60 -
Reflectance Measurement and Modelling of High Reflectivity Distributed Bragg Reflector Stacks
type doping tends to work the other way round due to the impurity bands which form
around the acceptor levels in the crystal. The author also stated that for the n-type DBR
in their study, the band-to-band absorption decreases when the electron concentration is
increased but the free-carrier absorption increases and the free-carrier absorption may
dominate the band-to-band absorption for wavelength of ~1650 nm and limit reflectivity
at high enough free-carrier or electron concentrations (larger than 1018 cm-3).
The effect of n-type doping on absorption also had been demonstrated by
HAYWOOD, S.K. et al., 1995 on 1 µm thick layers of In0.47Ga0.53As with different
doping levels on InP substrate in a series of experiments. The absorption spectra are
shown in Figure 7-1.
Figure 7-1 : Absorption Spectra Of Undoped and n-doped 1 µm In0.47Ga0.53As Layers
on InP
Source : HAYWOOD, S.K. et al., 1995
7.4
SIMULATION PROGRAM WITH ABSORPTION TAKEN INTO
ACCOUNT
The complex refractive index for absorbing materials is n* = n + ik instead of n, where
k is the extinction coefficient that is wavelength dependence. However, most of the
publications in literature ignore the effect of free-carrier absorption in their studies
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Reflectance Measurement and Modelling of High Reflectivity Distributed Bragg Reflector Stacks
because the k value is much less than the n value and the effect of k on the reflectance is
negligibly small in near-IR region. But in this thesis, the effect of the absorption on the
reflecivity is included as an investigation purpose. Initially, the refractive index for
InGaAs layer in basic reflectivity simulation program was replaced by n + ik to observe
the changes of the reflectivity as a try and error approach by using the equation from
DEPPE, D.G. et al. to obtain k value as follow:
α 
k =
λ B
 4π 
(33)
where α is free-carrier absorption and λB is bragg wavelength.
The α value of 50 cm-1 was used for the investigation purpose but the result from
simulation is not credible as shown in Figure 7-2. The reflectivity curve seems to go
over 100% reflectivity where this is not possible because free-carrier absorption will
only reduce the reflectivity but not increase it. This may due to both the refractive index
and extinction coefficient are wavelength dependence and it is not as simple as
replacing the refractive index value by n + ik.
Reflectivity vs Wavelength
1
0.9
0.8
0.7
Reflectivity
0.6
0.5
0.4
0.3
0.2
0.1
0
1500
1550
1600
Wavelength (nm)
1650
Figure 7-2 : n+ik Reflectivity Spectra
- 62 -
1700
1750
Reflectance Measurement and Modelling of High Reflectivity Distributed Bragg Reflector Stacks
Due to the first approach is not succeeded, the investigation was move on to
another approach by trying to modify the program to calculate the reflectivity of every
interface in the DBR one by one instead of calculating it at one go as in the basic
program. The idea is that the transmitted light from each interface will deducted by the
InGaAs layer absorption. However, this approaches also not giving much success
because there is an error while running the program due to the elements of the argument
is not match as shown below:
??? In an assignment A(I) = B, the number of elements in B and
I must be the same.
Error in ==> I:\testabsorptionmatrix.m
On line 69 ==> Mnew(zz) = Ma;
This may due to the Matlab program not recognize the loop to calculate the
matrix elements for InGaAs when the loop is inserted as shown in Appendix 5 with the
loop being written in red. In order to solve this problem, further familiarization of
Matlab programming language is needed and this is time consuming. With the whole
project has be done in such a short period, it is not possible to expertise in this
programming language although the basic of the language is easy to learn.
Another approach had been tried which based on the dielectric constants
function which has strong connection with optical spectra such as the refractive index,
extinction coefficient and absorption coefficient. The below equations that gave the
relation between the refractive index, extinction coefficient and absorption coefficient
as a function of dielectric constant were taken from ADACHI, S., 1989 in Journal of
Applied Physics.
n* = n + ik = ε1/2
[
]
[
]
 ε 2 + ε 2 1/ 2 + ε 
2
1 
n= 1


2


 ε 2 + ε 2 1/ 2 − ε 
2
1 
k = 1


2


(34)
1/ 2
(35)
1/ 2
(36)
- 63 -
Reflectance Measurement and Modelling of High Reflectivity Distributed Bragg Reflector Stacks
[
]
2
2 1/ 2
− ε 1 
4π  ε 1 + ε 2
α=

2
λ 


1/ 2
(37)
where, ε1 is real and ε2 is imaginary parts of the dielectric function. The real part of the
dielectric constant can be obtained from ADACHI, S., 1989 in Physical Review B or
ADACHI, S., 1982 in Journal of Applied Physics as shown in Figure 7-3 and the
absorption coefficient was obtained from DEPPE, D.G. et al. The Equation (37) is rearranged as follow to obtain the ε2 value:
2

  2α 2 λ2

2

ε2 = 
+
ε
ε
−
1
1

  16π 2



1/ 2
(38)
The simulation program that taken into account the effect of absorption on
reflectivity can be found in Appendix 6. The ε1 can be obtained as shown in Figure 7-2
by taking photon energy of InGaAs equal to 0.75 eV and the ε1 was found to be ~12.75.
For investigation purpose, the optical active energy (OAE) of InGaAs also had been
calculated as follow from CHAKRABORTY, P.K., DATTA, G.C. and GHATAK, K.P.:
 2
 m
OAE, hω =  E g + 2 1 +  c

  mv



 E g E F 



1/ 2
(39)
where Eg is band gap of InGaAs, mc is effective electron mass at conduction band, mv is
effective hole mass in valence band which can be obtained from SARKAR, C.K. et al.
and Landolt-Börnstein, while EF is Fermi energy that can be calculated as follow:
 n
E F = k B T ln
 Nc



(40)
where kB is Boltzman constant, T is absolute temperature, n is carrier concentration and
Nc is intrinsic carrier concentration.
- 64 -
Reflectance Measurement and Modelling of High Reflectivity Distributed Bragg Reflector Stacks
Figure 7-3 Dielectric Constant, ε1 Spectrum of InGaAs
Source : ADACHI, S., 1989 and ADACHI, S., 1982
A few values of EF and OAE had been calculated as shown in Table 7-1.
Doping Level (cm-3)
EF (eV)
OAE (eV)
1 x 1017
-0.1442
0.41217
1 x 1018
-0.0843
0.57707
4 x 1018
-0.0483
0.6565
6 x 1018
-0.037
0.6787
7 x 1018
-0.0337
0.6876
1 x 1019
-0.0244
0.7043
Table 7-1 : EF and OAE Value for Different Doping Level
With the value of ε1 from Figure 7-2 and the ε2 value calculated in the program,
the reflectivity spectra with absorption being taken into account can be simulated. This
simulation is based on the data from DEPPE, D.G. et al. The DBR is a 25 periods InPIn0.53Ga0.47As with InP on either side of the DBR designed for 1.65 µm operation
wavelength with 7 x 1018 cm-3 and the absorption loss of 20 cm-1. The simulation result
shows the exact results with DEPPE, D.G. et al. result as shown in Figure 7-4. The peak
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Reflectance Measurement and Modelling of High Reflectivity Distributed Bragg Reflector Stacks
reflectivity without taking into account the absorption is 99.31% while the peak
reflectivity is 99.13% with absorption taken into account. This shows that the
simulation program is work.
It is possible to estimate the absorption coefficient for 4 x 1018 cm-3 doping
based on the results from Figure 7-1 as follow if we take the estimate the absorption to
be 1%:
Pabs = 1 – exp (-αd) = 1% , where d = 1 µm = 1 x 10-4 cm
1
Hence, α = 0.99
= 100 cm-1
−4
1x10
(41)
ln
(42)
However, when input this value of α into the program in Appendix 6, the same problem
as in n + i*k program arise.
Therefore, in order to determine the dominant loss mechanism in the DBR, more
analysis is required. However, the limited experimental data exist on the absorption loss
in In0.53Ga0.47As has limit further investigation of the effect of doping on absorption and
reflectivity.
Reflectivity
1
0.9
Without absorption
0.8
With absorption
0.7
Reflectivity (%)
0.6
0.5
0.4
0.3
0.2
0.1
0
1500
1550
1600
1650
1700
Wavelength (nm)
1750
1800
Figure 7-4 : Reflectivity Spectra With and Without Absorption Taken Into Account
- 66 -
1850
Reflectance Measurement and Modelling of High Reflectivity Distributed Bragg Reflector Stacks
CHAPTER 8
CONCLUSION AND FUTURE WORK
8.0 CONCLUSION
The reflectance measurement of high reflectivity In0.53Ga0.47As/InP on InP semiinsulating substrate DBR had been carried out by using specular reflectance technique
in FT-IR spectroscopy. Although there are some disagreement between the
measurement result and the Matlab simulation result for the absolute reflectance
measurement due to the measurement may not carried out at the center of the DBR but
the reflectivity spectra derived from the transmission measurements at different points
of the DBR shows high correlation between the measured and modeled result with
reflectivity over 99.9% and the width of the stop band over 150 nm.
Simulation results of several properties of DBR using a technical computing
language, Matlab, are also presented. The simulation results also produce highly
correlated results with theoretical prediction. The investigated properties of the DBR
using the simulation programs revealed that reflectivity is increased with increasing
number of periods, larger refractive index difference and with the higher refractive
index material as the outer layer of the DBR. The investigation also shows that the
reflectivity will increase with the increasing angle of incidence but the center
wavelength tends to shift to lower wavelength due to increasing penetration depth with
increasing angle of incidence.
Further investigation on the DBR properties by taking the InGaAs layer
absorption into consideration shows high reflectivity is demonstrated even at the
wavelengths corresponding to the band edge when high n-type doping is used in the
DBR. Therefore, it can be concluded that In0.53Ga0.47As/InP DBR can be used to provide
high reflectivity with fewer periods and enables considerably simplified growth
- 67 -
Reflectance Measurement and Modelling of High Reflectivity Distributed Bragg Reflector Stacks
compared to quaternary materials implying a refractive index difference of 0.43.
However, the samples must be carefully selected and the effect of instrument accuracy,
human error and experimental setup must be taken into consideration.
8.1 FUTURE WORK
Although the experimental measurement gave credible results but the results were not
accurate enough to warrant further investigation due to the limited equipment available
in the lab for absolute reflectance measurement. Purchasing or design of reflectance
accessories is a must in order to provide accurate and reliable results on the reflectivity
measurements and DBR properties investigation.
Regarding the Matlab simulation program, although the program that take into
account the effect of absorption is shown to be working for the model outlined by
DEPPE, D.G. et al. but more analysis is required in order to fine tune the program.
There are still a lot of improvements can be done on the program in order to take into
account all the factors that affecting DBR reflectivity and other properties during the
simulation. For example, the current simulation program not taking into account the
effect of internal multiple reflections that occur in the DBR and the interface reflection
at the air/semiconductor interface. It is very useful if we can choose whether or not to
take into account these two effects while doing the simulation.
However, it is a very complicated task to write a simulation program that takes
all the affecting factors into account because it needs very deep understanding of the
programming language as well as the relation between these affecting factors and it is
time consuming.
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Reflectance Measurement and Modelling of High Reflectivity Distributed Bragg Reflector Stacks
REFERENCES
ADACHI, S., 1989. Optical Dispersion Relations for GaP, GaAs, GaSb, InP, InAs, InSb,
AlxGa1-xAs and In1-xGaxAsyP1-y. Journal of Applied Physics, 66(12) pp. 6030-6040.
ADACHI, S., 1989. Optical Properties of In1-xGaxAsyP1-y Alloys. Physical Review B,
39(17) pp. 12612-12621.
ADACHI, S., 1982. Refractive Indices of III-V Compounds: Key Properties of
InGaAsP Relevant to Device Design. Journal of Applied Physics, 53(8) pp. 5863-5869.
BENNETT, B.R., SOREF, R.A. and DEL ALAMO, J.A., 1990. Carrier-Induced
Change in Refractive Index of InP, GaAs and InGaAsP, IEEE Journal of Quantum
Electronic, 26(1) pp. 113-122.
BHATTACHARYA, P., ed., 1993. Properties of Lattice-Matched and Strained Indium
Gallium Arsenide. London: INSPEC, Institution of Electrical Engineers.
BILBY, R., 2000. The Modelling of Bragg Mirrors and Resonant Cavity Structures.
BEng Diss. Hull: University of Hull
BLUM, O. et al, 1994. Molucular Beam Epitaxy Grown AlAsSb/GaAsSb Distributed
Bragg Reflector On InP Substrate Operating Near 1.55 µm. Journal Of Vacumm
Science And Technology B, 12(2) pp. 1122-1124.
BORN, W. and WOLF, E. 1991. Principles of Optics. Oxford: Pergamon.
Bruker Optics online catalouge, available: www.brukeroptics.com
CHAKRABORTY, P.K., DATTA, G.C. and GHATAK, K.P., 2003. The Simple
Analysis of The Burstein-Moss Shift in Degenerate n-type Semiconductors. Physica B,
339(4) pp. 198-203
- 69 -
Reflectance Measurement and Modelling of High Reflectivity Distributed Bragg Reflector Stacks
CHOA, F.S. et al, 1991. High Reflectivity 1.55 µm InP/InGaAsP Bragg Mirror Grown
By Chemical Beam Epitaxy. Applied Physics Letters, 59(22) pp. 2820-2822.
CHUNNILALL, C.J. et al., 2002. FT-IR Measurements – Standards and Accuracy.
Vibrational Spectroscopy, 30(1) pp. 25-29.
CONWAY, L.J., 1999. Modeling of DBR Mirrors for Vertical Cavity Surface Emitting
Lasers. BEng Diss. Australia: University of Queensland.
DEPPE, D.G. et al, 1990. Quarter-wave Bragg Reflector Stack Of InP-In0.53Ga0.47As
For 1.65 µm Wavelength. Applied Physics Letters, 56(4) pp. 315-317.
FLETCHER, R.M. et al. 1993. Hewlett-Packard Journal. High-Efficiency Aluminium
Indium Gallium Phosphode Light-Emitting Diodes – Includes Related Article on
Structure
of
LEDs
–
Technical
[Online]
Available:
http://www.fondarticles.com/p/articles/mi_m0HPJ/is_n4_v44/ai_14190965/pg_1
[Accessed 22 January 2005]
GESSMANN, Th. And SCHUBEERT, E.F., 2004. High-Efficiency AlGaInP LightEmitting Diodes For Solid-State Lighting Applications. Journal Of Applied Physics,
95(5) pp. 2203-2216.
GREY, R. et al, 1996. Growth of GASb on GaAs/ AlAs Mirrors For 1.68 µm Detectors,
Optical Material, 6(1-2) pp.69-74.
GUY, P. et al, 1995. A Comparison Of 1.55 µm Distributed Bragg Reflector Stacks For
Use In Multi Quantum Well Micro Resonator Modulators. Semiconductor Science And
Technology, 10(9) pp. 1283-1286.
GUY, P., WOODBRIGDE, K. and HOPKINSON, M., 1993. High Reflectivity And
Low Resistance 1.55 µm Al0.65In0.35As/Ga0.63In0.37As Strained Quarter Wave Bragg
Reflector Stack. Electronics Letters, 29(22) pp. 1947-1948.
Harrick Scientific Products, 2005. Solutions in Optical Spectroscopy. Online Catalogue.
- 70 -
Reflectance Measurement and Modelling of High Reflectivity Distributed Bragg Reflector Stacks
HAYWOOD, S.K. et al, 1994. Highly Doped 1.55 µm GaxIn1-xAs/InP Distributed
Bragg Reflector Stacks. Electronics Letters, 30(18) pp. 1526-1527.
HAYWOOD, S.K. et al, 1995. A Comparison of 1.55 µm Distributed Bragg Reflector
Stacks for Use in Multi Quantum Well Micro Resonator Modulators. Semiconductor
Science and Technology, 10(9) pp. 1283-1286.
HERRES, W. and GRONHOLZ, J., 1987. Understanding FT-IR Data Processing.
Karlsruhe: Bruker Analytische Meβtechnik GmbH.
IBBOTSON, L., 1997. Introduction To Solid State Devices. London: Arnold.
ISHII, H. et al, 2005. Infrared Spectroscopy of Pentacene Thin Film on SiO2 Surface.
Applied Surface Science, 244(1-4) pp. 607-610.
KHANMOHAMMADI, M. and KARGOSHA, K. 2005. Application of Attenuated
Total Reflectance Fourier Transform Infrared Spectrometry To The Determination Of
Sodium Percarbonate In Washing Powder Detergent. Talanta, 65(3). pp. 824-827.
KOSTERS, P. 2000. FT-IR Spectroscopy of Thin Biological Layers. PhD diss.,
University of Twente.
Landolt-Börnstein, 1986, Numerical Data and Functional Relationships in Science and
Technology, New Series on Semiconductor (Group III), Vol. 22. Berlin: Springer
LI, A.Z. et al., 2001. The Effect of Dispersion of The Refractive Index on The
Performance of Mid-Infrared Quantum Cascade Lasers. Journal of Crystal Growth,
227-228(July) pp. 313-318.
LIM, H.C. 2002. Organic Resonant Cavity Enhanced LEDs. BEng. Diss., Australia:
University of Queensland.
- 71 -
Reflectance Measurement and Modelling of High Reflectivity Distributed Bragg Reflector Stacks
LU, T.C. et al, 2003. InP/InGaAlAs Distributed Bragg Reflectors Grown By LowPressure Metal Organic Chemical Vapor Deposition. Journal Of Crystal Growth,
250(3-4) pp. 305-312.
MANSOOF, F., 1996. Study of Ga-based Resonant Cavity Photodetector. PhD. Diss.,
University College London.
MASON, N.J. et al. 1996, Growth of GaSb on GaAs/AlAs Mirrors for 1.68µm
Detectors. Optical Materials, 6(1-2) pp. 69-74
MODAK, P. et al. 2000. Journal of Crystal Growth. InAlGaP Microcavity LEDs on GeSubstrates, 221(1-4) pp. 668-673
MOSELEY, A.J. et al, 1989. High-Reflectivity AlGaInAs/InP Multilayet Mirrors
Grown By Low-Pressure MOVPE For Application To Long-Wavelength High Contrast
Ratio Multi Quantum Well Modulators. Electronics Letters, 25(25) pp. 1717-1718.
OLIVIER, M. et al, 2001. Multiple Internal Reflection Spectroscopy: A Sensitive NonDestructive Probe For Interfaces and Nanometric Layers. Materials Science in
Semiconductor Processing, 4(1-3) pp. 15-18.
Oriel Instruments, No Date. Introduction To FT-IR Spectroscopy. Product catalogue.
RIZZI, A., 2004. Semiconductor Physics [Online]. Germany: Physikalishes Institut,
University
of
Goettingen.
Available:
http://www.physik4.gwdg.de/rizzi/download/monroy-o3-growth.pdf [Accessed 5 March
2005]
SANYAL, M.K. et al, 1998. Interfacial Profile Of A Bragg Mirror. Applied Surface
Science, 133(1-2) pp. 98-102.
SARKAR, C.K. et al., 1985. Effective Masses and Non-Parabolicity in GaxIn1-xAs.
Journal of Physics C: Solid State Physic, 18(1985) pp. 2667-2676
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SHEN, J.L. et al., 2001. Reflectivity and Photoluminescence Studies in Bragg
Reflectors With Absorbing Layers. Semiconductor Science and Technology, 16(2001)
pp. 548-552
SHERMAN HSU, C.P., 1997. Infrared Spectroscopy. In: SETTLE, F., ed. Handbook of
Instrumental Techniques for Analytical Chemistry. New Jersey: Prentice-Hall. pp. 247283.
SMITH, B.C., 1996. Fundamentals of Fourier Transform Infrared Spectroscopy. Boca
Raton, Florida: CRC Press, Inc.
SOLE, R.D., 1982. The Effect of Multiple Reflections on Surface Optical Spectroscopy.
Surface Science, 123(1-3) pp. 231-238.
Specac, 2001. IR Sampling Solution. Product catalogue.
STRADING, R.A. and KLIPSTEIN, P.C., ed., 1990. Growth and Characterisation of
Semiconductors. Bristol: J W Arrowsmith Ltd.
STREETMAN, B.G. and BANERJEE, S., 2000. Solid State Electronic Devices, 5th
Edition. New Jersey: Prentice Hall International, Inc.
STUMPF, W., 2001. Investigation and Simulation of The Optical Properties of Doped
Silicon. Germany: University of Constance.
SUUNDGREN, P., 2005. Development of 1.3µm GaAs-based Vertical-Cavity SurfaceEmitting Lasers. PhD. Diss. Stockholm: Royal Institute of Technology.
TAI, K. et al, 1989. High Reflectivity AlAs0.52Sb0.48/GaInAs(P) Distributed Bragg
Mirror On InP Substrate For 1.3-1.55 µm Wavelengths. Electronics Letters, 25(17) pp.
1159-1160.
TAI, K. et al, 1987. Chemical Beam Epitaxially Grown InP/InGaAsP Interference
Mirror For Use Near 1.55 µm Wavelength. Applied Physics Letters, 51(11) pp. 826.
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Reflectance Measurement and Modelling of High Reflectivity Distributed Bragg Reflector Stacks
Thermo Nicolet, 2002. FT-IR vs. Dispersive Infrared. Theory of Infrared Spectroscopy
Instrumentation. Madison: Thermo Electon.
WILLIAM, M.D. et al. 2000. Comparison of InGaAs(100) Grown by Chemical Beam
Epitaxy and Metal Organic Chemical Vapor Deposition. Applied Surface Science, 157(3)
pp. 123-128.
WOOD, D., 1994. Optoelectronic Semiconductor Devices. New York: Prentice-Hall.
YOUNG, S.J., 2003. High Brightness Light Emitting Diode Chapter 3-4 [Online].
Taiwan:
National
Changhua
University
of
Education.
Available:
http://www.ncue.edu.tw [Accessed 28 December 2004]
ZHANG, Z.M., HANSSEN, L.M. and DATLA, R.U., 1996. Polarization-Dependent
Angular Reflectance of Silicon and Germanium in The Infrared. Infrared Physics &
Technology, 37(4). pp. 539-546.
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Reflectance Measurement and Modelling of High Reflectivity Distributed Bragg Reflector Stacks
APPENDIX 1
SAMPLE GROWTH SHEET
AND
REFLECTIVITY MEASUREMENT
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Reflectance Measurement and Modelling of High Reflectivity Distributed Bragg Reflector Stacks
X
X
centre C1
X
X
C3
C2
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Reflectance Measurement and Modelling of High Reflectivity Distributed Bragg Reflector Stacks
APPENDIX 2
QUOTATIONS
Quotation
Quotation number:
TO:
University of Hull
45 Cranbrook Avenue
Hull, HU6 7SP
ATTENTI
ON:
PHONE:
FAX:
E-mail:
Quantity
1
1
1
05-230
Please refer to when ordering.
Date:
6/28/2005
Prices quoted are F.O.B. Ossining,
NY
Terms net 30 days, US Dollars
JinYu Li
G. Yeap
Approximate Delivery:
[email protected]
[email protected]
6-8 weeks
QUOTATION VALID FOR 90
DAYS
Description
Price
Amount
Variable Angle Specular Reflection
Accessory for the Varian/Digilab Excalibur.
P/N VR1-DI8
$3,615.00
$3,615.00
12º Absolute Reflectance Stage for the VR1.
P/N VR1-VWA-12.
$1,856.00
$1,856.00
$400.00
$400.00
Small Sample Support for the VR1-VWA-12.
Designed to accommodate samples 25-15mm
in diameter. (Features a 12mm dia. aperture.)
Please reference quotation number when placing order
Quote written by
Susan Berets
Harrick Scientific Products, Inc. has earned a reputation for expertise, innovation and quality products.
We will appreciate the privilege of filling your order and we will give immediate and careful attention
to your instructions
Harrick Scientific Products, Inc. • 141 Tompkins Ave., 2nd floor• PO Box 277 • Pleasantville, NY
10570 • 914-747-7202 • Fax 914-747-7209 • www.harricksci.com • e-mail: [email protected]
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Reflectance Measurement and Modelling of High Reflectivity Distributed Bragg Reflector Stacks
Date:
From:
To:
Subject:
Part(s):
Mon, 18 Jul 2005 12:08:18 +0100
[email protected]
[email protected]
Fwd: Reflection units
2 A519A.pdf
application/octet-stream 160.09 KB
3 10Spec_PDS.pdf application/octet-stream 225.99 KB
----- Forwarded message from Paul Turner
<[email protected]>
----Date: Thu, 14 Jul 2005 12:56:15 +0100
From: Paul Turner <[email protected]>
Reply-To: Paul Turner <[email protected]>
Subject: Reflection units
To: [email protected]
Dear Stephanie
I have spoken your student? re a reflection accessory for the IFS
66/S. I think there are several possibilities to consider here,
please see below:
1.A 519-A
Reflection unit for recording absolute values (VW-configuration),
gold-coated mirrors, angle of incidence appr. 12° For spectrometers
with Bruker baseplate in the sample compartment only £3,668.00 + VAT
to include delivery
It is also possible to run this unit in the W mode only, but to get
absolute reflectance values, you would need a calibrated reference
mirror, see below.
This mode is also known as a "relative" relectance accessory
=================================================
Please see the attached product note.
2. One could buy a simpler type of relative only reflection unit eg:
the 10Spec from Pike Technologies, please see attached:
http://www.piketech.com
This is a relative reflectance unit, so you would need a calibrated
gold mirror in order to get the absolute reflectance values (the
A519-A in VW mode does not need a calibrated reference mirror). Such
a calibrated mirror could be procured I believe through the NPL, my
contact there is Barry Scott at [email protected], who could
probably help you. He is our FT-IR user who deals with standards. The
price of the 10Spec p/n 010-1050 is $2150.00 = £1230 + VAT. This can
be ordered through us. In addition you would need an extra sample
compartment baseplate A191/3 to mount the accesory on, price of this
is £76 + VAT. For these items there is a carriage charge of £15 + VAT
I have used the 10Spec myself successfully on the mid IR reflectance
(in order to get the emissivity = 1-R) of various coated glasses from
the glazing industry, but I had a calibration gold mirror as
reference. Optically, it is simpler to set up a relative unit like
the 10Spec. I must say that the feed back we get is that the
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Reflectance Measurement and Modelling of High Reflectivity Distributed Bragg Reflector Stacks
alignment of A519-A is critical to get proper absolute values,
although mechanically the unit is well put together eg the switching
of the mirror between V and W modes is good.
For both units, we would send them for user own installation since
you should be able to set them up yourselves. But if a service visit
is needed for alignment, this would be chargeable extra (£105 per
hour including travel time plus travel costs).
Delivery time of the 10Spec from Pike is usually a matter of a few
weeks. For A519-A, I have to check with Bruker Germany and get back
to you.
If you need a quotation, please let me know.
Best regards
Paul
Dr Paul Turner
FT-IR Applications Scientist
Bruker Optics Limited
Banner Lane
Telephone 024 7685 5200
Coventry
Fax
024 7646
5317
CV4 9GH
email : [email protected]
England
Website :www.brukeroptics.com
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APPENDIX 3
BASIC REFLECTIVITY SIMULATION PROGRAM
%User input parameters
num_layers = input('The Number of Layers in the structure = ');
r_index1 = input('refractive index of layer 1 = ');
r_index2 = input('refractive index of layer 2 = ');
i_index = input('refractive index of the incident medium = ');
s_index = input('refractive index of the substrate = ');
angle_incidence = input('angle of incidence (degrees) = ');
bragg_wave = input('bragg wavelength (nm) = ');
min_wave = input('lowest wavelength in range (nm) = ');
max_wave = input('highest wavelength in range (nm)= ');
step_wave = input('step wavelength in range (nm) = ');
%converting to nm
bragg_wave1 = bragg_wave * 1E-9;
min_wave1 = min_wave * 1E-9;
max_wave1 = max_wave * 1E-9;
step_wave1 = step_wave * 1E-9;
%converting degrees to radiance
theta_incidence = angle_incidence*pi/180;
%calculating incidence angle at each layer (snells laws)
theta1 = asin(i_index/r_index1*sin(theta_incidence));
theta2 = asin(r_index1/r_index2*sin(theta1));
theta_s = asin(r_index2/s_index*sin(theta2));
%calculating layer quater-wavelength optical thickness
d1 = bragg_wave1/(4*r_index1);
d2 = bragg_wave1/(4*r_index2);
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%calculating optical admitance, for s-polarisation (TE is in normal to the plane of
incidence)(assuming in free space Y = 2.6544*E-3)
Y = 2.6544*1E-3;
QA_I = i_index*cos(theta_incidence)*Y;
QA1 = r_index1*cos(theta1)*Y;
QA2 = r_index2*cos(theta2)*Y;
QA_s = s_index*cos(theta_s)*Y;
x=1;
y=1;
Wavelength = [];
Reflectivity = [];
for lambda = min_wave1 :step_wave1 : max_wave1
%calculating delta for n1 and n2
del1 = (2*pi*d1*r_index1*cos(theta1))/lambda;
del2 = (2*pi*d2*r_index2*cos(theta2))/lambda;
%calculating characteristic matrix for n1 and n2
%M1
r1 = [cos(del1), i*(sin(del1)/QA1)];
r2 = [(i*(sin(del1)*QA1)), cos(del1)];
M1 = [r1 ; r2];
%M2
t1 = [cos(del2), i*(sin(del2)/QA2)];
t2 = [(i*(sin(del2)*QA2)), cos(del2)];
M2 = [t1 ; t2];
cm = M1*M2;
if (num_layers/2) == round(num_layers/2)
M = cm^(num_layers/2);
else
M = cm^((num_layers-1)/2)*M1;
end
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%calculate the elements of the characteristic matrix
b = M(1,1) + M(1,2)*QA_s;
c = M(2,1) + M(2,2)*QA_s;
%reflectivity
demin = (QA_I*b) + c;
numin = (QA_I*b) - c;
r = numin/demin;
r1 = conj(r);
Reflect = r*r1;
fprintf('%f, %f\n', Reflect, lambda);
lambda1 = lambda *1E9;
Reflectivity(x)= Reflect;
x= x+1;
Wavelength(y)= lambda1;
y=y+1;
end
plot(Wavelength, Reflectivity);
title('Reflectivity vs Wavelength');
xlabel('Wavelength (nm)');
ylabel('Reflectivity (%)')
grid on;
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Reflectance Measurement and Modelling of High Reflectivity Distributed Bragg Reflector Stacks
APPENDIX 4
SIMULATION PROGRAM FOR PLOTING REFLECTIVITY vs
ANGLE of INCIDENCE
%Prompt user to input data
num_layers = 72;
i_index = 1;
r_index1 = 3.6;
r_index2 = 3.17;
s_index = 3.17;
Bragg_wave1 = 1600;
fprintf('\n\nProgram running, please wait...\n\n');
%setting lowest wavelength, highest wavelength and separation wavelength
Lowest_lambda1 = Bragg_wave1 - 180;
Highest_lambda1 = Bragg_wave1 + 180;
Step_lambda1 = 2;
%converting input wavelength to nano-meter
Bragg_wave = Bragg_wave1*1E-9;
Lowest_lambda = Lowest_lambda1*1E-9;
Highest_lambda = Highest_lambda1*1E-9;
Step_lambda = Step_lambda1*1E-9;
%declaring array
Reflectivity = [];
Wavelength = [];
Angle = [];
%loop for finding Reflectivity, Wavelength and Angle
z = 1;
pp = 1;
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for I = (0*pi)/180 : (2*pi)/180 : (90*pi)/180;
theta_incident = I;
%calculating the refracted angle using the Snell's Law
theta_1 = asin((i_index*sin(theta_incident))/r_index1);
theta_2 = asin((r_index1*sin(theta_1))/r_index2);
theta_s = asin((r_index2*sin(theta_2))/s_index);
%calculating the layer1 and layer2 optical thicknesses
d_layer1 = Bragg_wave/(4*r_index1);
d_layer2 = Bragg_wave/(4*r_index2);
%calculating the optical admittances of the layers
Y = 2.6544*1E-3;
QA_incident = i_index*cos(theta_incident)*Y;
QA_layer1 = r_index1*cos(theta_1)*Y;
QA_layer2 = r_index2*cos(theta_2)*Y;
QA_substrate = s_index*cos(theta_s)*Y;
x = 1;
y = 1;
qq=1;
for lambda = Lowest_lambda : Seperation_lambda : Highest_lambda;
% Caluculating the phase factor "del" for the layers.
del1 = (2*pi*r_index1*d_layer1*cos(theta_1)) / lambda;
del2 = (2*pi*r_index2*d_layer2*cos(theta_2)) / lambda;
% Calculating the characteristic matrix (B C) for layer_1
b1 = [cos(del1), (sin(del1)/QA_layer1)*i];
c1 = [(QA_layer1*sin(del1))*i, cos(del1)];
M1 = [b1; c1];
% Calculating the characteristic matrix (B C) for layer_2
b2 = [cos(del2), (sin(del2)/QA_layer2)*i];
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c2 = [(QA_layer2*sin(del2))*i, cos(del2)];
M2 = [b2; c2];
cm = M1*M2;
% Check if the number of layers to be multiplied is odd or even and
% multiply the matrices accordingly.
if (number_layers) == round(number_layers)
M = cm^(number_layers);
else
M = cm^((number_layers-1)/2)*M1;
end
% Calculate the reflectivity
c = M(2,1) + M(2,2)*QA_substrate;
b = M(1,1) + M(1,2)*QA_substrate;
%reflectivity
demin = (QA_incident*b) + c;
numin = (QA_incident*b) - c;
r = numin/demin;
r1 = conj(r);
Reflect = r*r1;
Reflect1=Reflect;
Lambda = lambda*1E9;
Reflectivity(qq,pp) = Reflect;
x=x+1;
Wavelength(y) = Lambda;
y=y+1;
Angle(z) = I;
z=z+1;
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qq=qq+1;
end
pp=pp+1;
end
plot((0:2:90),Reflectivity(91,:)*100, 'b');
hold on;
title('Reflectivity vs Angle of Incidence');
xlabel('Angle (degree)');
ylabel('Reflectivity (%)');
grid on;
zoom;
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Reflectance Measurement and Modelling of High Reflectivity Distributed Bragg Reflector Stacks
APPENDIX 5
SIMULATION PROGRAM WITH LOOP TO CALCULATE THE
INDIVIDUAL MATRIX ELEMENTS FOR InGaAs LAYER
%Prompt user to input data
num_layers = 72,('The Number of Layers in the structure = ');
r_index1 = 3.6,('refractive index of layer 1 = ');
r_index2 = 3.17,('refractive index of layer 2 = ');
i_index = 1,('refractive index of the incident medium = ');
s_index = 3.17,('refractive index of the substrate = ');
angle_incidence = 12,('angle of incidence (degrees) = ');
bragg_wave = 1600,('bragg wavelength (nm) = ');
min_wave = 1450,('lowest wavelength in range (nm) = ');
max_wave = 1750,('highest wavelength in range (nm)= ');
step_wave = 1,('step wavelength in range (nm) = ');
absorption = input('absorption (cm-1) = ');
%converting to nm
bragg_wave1 = bragg_wave * 1E-19;
min_wave1 = min_wave * 1E-19;
max_wave1 = max_wave * 1E-19;
step_wave1 = step_wave * 1E-19;
absorption1 = absorption * 1E2;
%converting degrees to radiance
theta_incidence = angle_incidence*pi/180;
%calculating incidence angle at each layer (snells laws)
theta1 = asin(r_index1/(r_index1*sin(theta_incidence)));
theta2 = asin(r_index2/(r_index2*sin(theta_incidence)));
theta_s = asin(s_index/(s_index*sin(theta_incidence)));
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%calculating layer quater-wavelength optical thickness
d1 = bragg_wave1/(4*r_index1);
d2 = bragg_wave1/(4*r_index2);
%calculating optical admitance, for s-polarisation (TE is in normal to the plane of
incidence)(assuming in free space Y = 2.6544*E-3)
Y = 2.6544*1E-3;
QA_I = i_index*cos(theta_incidence*Y);
QA1 = r_index1*cos(theta1*Y);
QA2 = r_index2*cos(theta2*Y);
QA_s = s_index*cos(theta_s*Y);
x=1;
y=1;
Wavelength = [];
Reflectivity = [];
for lambda = min_wave1 :step_wave1 : max_wave
%calculating delta for n1 and n2
del1 = (2*pi*d1*cos(theta1))/lambda;
del2 = (2*pi*d2*cos(theta2))/lambda;
%calculating characteristic matrix for n1 and n2
%M1
r1 = [cos(del1), (i*sin(del1/QA1))];
A = -exp(-2*absorption1*d1);
r2 = [(i*sin(del1/QA1)), cos(del1)];
M1 = [r1 ; r2];
%M2
t1 = [cos(del2), (i*sin(del2/QA2))];
t2 = [(i*sin(del2/QA2)), cos(del2)];
M2 = [t1 ; t2];
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Reflectance Measurement and Modelling of High Reflectivity Distributed Bragg Reflector Stacks
for z = 1 : 1 : 72
%Mnew
zz = 1;
A = -exp(-2*z*absorption1*d1);
Mnew = [];
Ma = A * M1;
Mnew(zz) = Ma;
zz = zz + 1;
end
cm = Mnew*M2;
if (num_layers/2) == round(num_layers/2)
M = cm^(num_layers/2);
else
M = cm^((num_layers-1)/2)*M1;
%calculate the elements of the characteristic matrix
b = M(1,1) + M(1,2)*QA_s;
c = M(2,1) + M(2,2)*QA_s;
%reflectivity
demin = (QA_I*b) + c;
numin = (QA_I*b) - c;
r = numin/demin;
r1 = conj(r);
Reflect = r*r1;
fprint('%f, %f, %f\n', Reflect, lambda, phase);
Reflectivity(x)= Reflect;
x= x+1;
Wavelength(y)= lambda;
y=y+1;
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end
plot(Wavelength, Reflectivity);
grid on;
end
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Reflectance Measurement and Modelling of High Reflectivity Distributed Bragg Reflector Stacks
APPENDIX 6
SIMULATION PROGRAM THAT TAKE EFFECTS OF ABSORPTION
INTO ACCOUNT
%Prompt user to input data
num_layers = input ('The Number of Layers in the structure = ');
r_index1 = input ('refractive index of layer 1 = ');
r_index2 = input ('refractive index of layer 2 = ');
i_index = input ('refractive index of the incident medium = ');
s_index = input ('refractive index of the substrate = ');
angle_incidence = input ('angle of incidence (degrees) = ');
bragg_wave = input ('bragg wavelength (nm) = ');
min_wave = input ('lowest wavelength in range (nm) = ');
max_wave = input ('highest wavelength in range (nm)= ');
step_wave = input ('step wavelength in range (nm) = ');
absorption = input ('free carrier concentration (cm-1) = ');
E1 = 12.75, %('dielectric constant1 = ');
%converting to nm
bragg_wave1 = bragg_wave * 1E-9;
min_wave1 = min_wave * 1E-9;
max_wave1 = max_wave * 1E-9;
step_wave1 = step_wave * 1E-9;
absorption1 = absorption*1E2;
%converting degrees to radiance
theta_incidence = angle_incidence*pi/180;
for lambda = min_wave1 :step_wave1 :max_wave1
%calculating E2
E2 = sqrt(((2*(absorption1^2*bragg_wave1^2/16*pi^2)+E1)^2 + E1)^2 - E1^2);
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Reflectance Measurement and Modelling of High Reflectivity Distributed Bragg Reflector Stacks
%calculating complex refractive index
n = (((E1^2 + E2^2)^(1/2) + E1)/2)^(1/2);
k = (((E1^2 + E2^2)^(1/2) - E1)/2)^(1/2);
n1 = n + i*k;
end
%calculating incidence angle at each layer (snells laws)
theta1 = asin(i_index/n1*sin(theta_incidence));
theta2 = asin(n1/r_index2*sin(theta1));
theta_s = asin(r_index2/s_index*sin(theta2));
%calculating layer quater-wavelength optical thickness
d1 = bragg_wave1/(4*n1);
d2 = bragg_wave1/(4*r_index2);
%calculating optical admitance, for s-polarisation (TE is in normal to the plane of
incidence)(assuming in free space Y = 2.6544*E-3)
Y = 2.6544*1E-3;
QA_I = i_index*cos(theta_incidence)*Y;
QA1 = n1*cos(theta1)*Y;
QA2 = r_index2*cos(theta2)*Y;
QA_s = s_index*cos(theta_s)*Y;
x=1;
y=1;
Wavelength = [];
Reflectivity = [];
for lambda = min_wave1 : step_wave1 : max_wave1
%calculating delta for n1 and n2
del1 = (2*pi*d1*n1*cos(theta1))/lambda;
del2 = (2*pi*d2*r_index2*cos(theta2))/lambda;
%calculating characteristic matrix for n1 and n2
%M1
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Reflectance Measurement and Modelling of High Reflectivity Distributed Bragg Reflector Stacks
r1 = [cos(del1), i*(sin(del1)/QA1)];
r2 = [(i*(sin(del1)*QA1)), cos(del1)];
M1 = [r1 ; r2];
%M2
t1 = [cos(del2), i*(sin(del2)/QA2)];
t2 = [(i*(sin(del2)*QA2)), cos(del2)];
M2 = [t1 ; t2];
cm = M1*M2;
if (num_layers/2) == round(num_layers/2)
M = cm^(num_layers/2);
else
M = cm^((num_layers-1)/2)*M1;
end
%calculate the elements of the characteristic matrix
b = M(1,1) + M(1,2)*QA_s;
c = M(2,1) + M(2,2)*QA_s;
%reflectivity
demin = (QA_I*b) + c;
numin = (QA_I*b) - c;
r = numin/demin;
ra = conj(r);
Reflect = r*ra;
fprintf('%f, %f\n', Reflect, lambda);
lambda1 = lambda *1E9;
Reflectivity(x)= Reflect;
x= x+1;
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Reflectance Measurement and Modelling of High Reflectivity Distributed Bragg Reflector Stacks
Wavelength(y)= lambda1;
y=y+1;
end
plot(Wavelength, Reflectivity, 'r');
title('Reflectivity');
grid on;
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Reflectance Measurement and Modelling of High Reflectivity Distributed Bragg Reflector Stacks
APPENDIX 7
MATLAB QUICK REFERENCE
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