Spectroscopic Ellipsometry Charactarization of Single and Multilayer Aluminum Nitride / Indium Nitride Thin Film Systems
SPECTROSCOPIC ELLIPSOMETRY CHARACTARIZATION
OF SINGLE AND MULTILAYER ALUMINUM NITRIDE /
INDIUM NITRIDE THIN FILM SYSTEMS
A dissertation presented to
the faculty of
the College of Arts and Sciences of Ohio University
In partial fulfillment
of the requirements for the degree
Doctor of philosophy
Jebreel M. Khoshman
June 2005
© 2005
Jebreel M. Khoshman
All Rights Reserved
This dissertation entitled
SPECTROSCOPIC ELLIPSOMETRY CHARACTARIZATION OF
SINGLE AND MULTILAYER ALUMINUM NITRIDE / INDIUM
NITRIDE THIN FILM SYSTEMS
by
JEBREEL M. KHOSHMAN
has been approved for
the Department of Physics and Astronomy
and the College of Arts and Sciences by
Martin E. Kordesch
Professor of Physics and Astronomy
Leslie A. Flemming
Dean, College of Arts and Sciences KHOSHMAN, JEBREEL M. Ph.D. June 2005. Physics and Astronomy
Spectroscopic Ellipsometry Characterization of Single and Multilayer Aluminum Nitride
/ Indium Nitride Thin Film Systems (267 pp.)
Director of Dissertation: Martin E. Kordesch
The optical characteristics of single and multilayer III-V nitride thin film systems deposited by RF magnetron sputtering onto different substrates have measured and analyzed. Spectroscopic Ellipsometry measurements of amorphous (Al, Ga, In) N single layers are carried out at two angles of incidence, 70o and 75o, over the wavelength range
300 – 1400 nm. The measured ellipsometric data are fitted to models consisting of air / a-
AlN/ c-Si (111), air / roughness / a-GaN/ c-Si (111), and air / roughness / a-InN/ c-Si
(111). The roughness is modeled using the Bruggeman effective medium approximation,
assuming 50 % a-GaN (or a-InN) and 50 % voids. The optical constants and the
thicknesses of a-AlN are obtained by analysis of the measured ellipsometric spectra
through the Cauchy–Urbach model whereas the optical constants and the thicknesses of
a-(Ga, In) N are determined using the Tauc-Lorentz model. Analysis of the absorption
coefficient of a-AlN (in the range 200 – 1400 nm) and a-GaN show the optical bandgap to be 5.82 ± 0.05 and 3.38 ± 0.05 eV, respectively. While the analysis of the absorption coefficient of a-InN shows the optical bandgap energy to be 1.68 ± 0.071 eV. These
values are confirmed using different optical methods such as spectrophotometry,
photoluminescence and polarized absorptivity.
From the angle dependence of the p-polarized reflectivity we deduce Brewster and principal angles of the single layers. Measurement of the polarized optical properties reveals a high transmissivity (70 % – 95 %) and low absorptivity (< 18 %) for all three thin films in the visible and near infrared regions. X - ray diffraction analysis verifies the amorphous nature of the (Al, Ga, In) N films.
This work also reports the successful growth of sputtered bilayer and multilayer antireflection coatings on quartz using alternating layers of AlN and InN. The designed layers are optimized at 500 nm and 50o – 70o incidence for their performance as a longwave-pass filter and coatings for solar cells with high efficiency by varying the number of periods of AlN / InN multilayer systems. The XRD patterns of these systems reveal a polycrystalline structure with a maximum diffraction peak at 31.2o (InN (002)).
Approved:
Martin E. Kordesch
Professor of Physics and Astronomy
To the only Star in my Sky, To the Candle of my Life, To my Wife “Shereen” for her Patience, Encouragement and Understanding
Acknowledgments
I would like to express my deepest respect and appreciation to my dissertation supervisor
Prof. Martin E. Kordesch, whose expertise, understanding, and patience, added considerably to my graduate experience. A very special thanks goes out to Prof. P. G.
Van Patten for his technical assistance. Next, I wish to thank the other members of my
committee, Prof. David Drabold, and Prof. David Ingram for the assistance they provided
at all levels of this dissertation.
I would also like to thank my family for the support they provided me through my
entire life and in particular, I must acknowledge my wife. She gave me the stability and
unyielding support I needed to continue with my research. Special thanks to my faithful
friend, Archibald Peters, who has been with me everyday since the beginning of my
graduate career. Without whose love, encouragement and editing assistance, I would not
have finished this dissertation.
viii
Table of Contents
Abstract ...... iv
Dedication...... vi
Acknowledgments...... vii
List of Tables ...... xii
List of Figures...... xiv
List of Symbols and Abbreviations ...... xxx
1 Introduction...... 1
1.1 Aim and Organization of the Dissertation...... 6
2 General Background...... 9
2.1 Formation Techniques of Thin Films ...... 9
2.1.1 Principle of Sputtering ...... 10
2.2 Optical Properties of Thin Films ...... 12
2.3 Optical Constants of Thin Films...... 14
2.3.1 Methods for the Determination of the Optical Constants of Thin
Films ...... 15
2.4 Multilayer Stacks of Thin Films...... 26
2.5 Properties of the Group III Nitride Thin Films ...... 31
2.5.1 Aluminum Nitride Thin Films ...... 32
2.5.2 Indium Nitride Thin Films...... 38 ix
2.6 Amorphous Thin Films...... 43
2.6.1 Optical Properties of Amorphous Thin Films...... 46
2.6.2 Some Potential Advantages of Amorphous Thin Solid Films...... 50
3 Fundamental Theory ...... 54
3.1 Classical Electromagnetic Theory of Thin Film Optics...... 54
3.2 Reflection and Transmission Coefficients of Plane Waves at a Single
Interface...... 59
3.2.1 Brewster and Pseudo - Brewster Angles...... 62
3.3 Theory of a Single Thin Film on a Transparent Substrate ...... 62
3.3.1 Reflectance and Transmittance of Dielectric Films...... 63
3.3.2 Reflectance and Transmittance of Absorbing Films...... 66
3.3.3 Theory of Multilayer Thin Films ...... 67
3.4 Determination of the Optical Constants of Thin Films ...... 71
3.5 Spectroscopic Ellipsometry and Optical Constants...... 72
3.5.1 The Amplitude and Phase Changes on Reflection at a Thin Film...... 74
4 Experimental Setup and Techniques ...... 76
4.1 Sample preparation...... 76
4.1.1 a-AlN and a-InN Single Layer Systems...... 76
4.1.2 AlN / InN Bilayer and Multilayer Thin Film Systems ...... 79
4.2 Variable Angle Spectroscopic Ellipsometry (VASE) ...... 88
4.2.1 VASE Theory...... 89 x
4.2.2 Instrumentation Basics...... 91
4.2.3 Development of an Optical Model...... 96
4.2.4 Fit Model to Measured Data ...... 98
4.3 Optical properties (reflection and transmission) measurements ...... 99
4.3.1 Intensity reflectance data ...... 101
4.3.2 Intensity transmittance data ...... 101
5 Optical Characteristics of III-V Nitride Films ...... 103
5.1 Calibration of the Experimental Optical Setup ...... 103
5.2 The Optical Characteristics of Sputtered Amorphous Aluminum Nitride
Films...... 105
5.2.1 The Optical Constants...... 106
5.2.2 Optical Bandgap Energy...... 118
5.2.3 Polarized Optical Reflectivities and Brewster Angle ...... 125
5.2.4 Polarized Optical Transmissivity and Absorptivity...... 129
5.3 The Optical Characteristics of Sputtered Amorphous Indium Nitride
Films ...... 136
5.3.1 The Optical Constants...... 136
5.3.2 Optical Bandgap Energy...... 150
5.3.3 Polarized Optical Reflectivities and Brewster Angle ...... 162
5.3.4 Polarized Optical Transmissivities and Absorptivities...... 165
5.4 The Optical Characteristics of Sputtered Amorphous Gallium Nitride
Films ...... 171 xi
5.5 The Optical Characteristics of Nitride Bilayer and Multilayer Thin Film
Systems...... 179
5.5.1 Bilayer Antireflection Coatings ...... 179
5.5.2 Multilayer Antireflection Coatings...... 189
5.5.3 Applications of Nitride Multilayer Systems: Broadband NIR-AR
Coating...... 215
6 Summary and Conclusions...... 218
Bibliography ...... 221
xii
List of Tables
2.1 Comparison of evaporation and sputtering...... 10
2.2 Some data on AlN film ...... 34
2.3 Some data on InN film...... 39
4.1 Operating conditions of the RF sputtering system and sample thicknesses ...... 77
4.2 Operating conditions of the RF magnetron sputtering system of multilayer
AlN /InN / Quartz thin film system with period equals to 117 nm...... 81
4.3 Operating conditions of the RF magnetron sputtering system of multilayer
AlN / InN / Quartz thin film system with period equals to 538 nm...... 82
5.1 Optical data types measured on the sputtered amorphous AlN thin films ...... 106
5.2 Best-fit model structural parameters for a-AlN thin films sputtered onto c-Si (111)
substrates determined by the VASE measurements in the wavelength range
300-1400 nm. a The parameter is not chosen as a fitting parameter and it has a
fixed value during the fitting procedure while b represents the beginning of our
wavelength range. The 90 % confidence limits are given by (±)...... 116
5.3 Thicknesses of a-AlN samples measured by using different techniques and
their optical constants at two different wavelengths, 350 nm and 1200 nm. The
90 % confidence limits are given by (±) ...... 118
5.4 Best-fit model structural parameters for a-AlN thin films sputtered onto c-Si
(111) substrates determined by the VASE measurements in the wavelength
range 200-1400 nm. a The parameter is not chosen as a fitting parameter and it xiii
has a fixed value during the fitting procedure while b represents the beginning
of our wavelength range. The 90 % confidence limits are given by (±) ...... 122
5.5 Optical constants of a-AlN thin films at bandgap energy. (a) Experimental
results, a-AlN thin films and (b) Literature data, c-AlN thin films...... 125
5.6 Optical data types measured on the sputtered amorphous InN thin films...... 136
5.7 Best-fit model structural parameters for a-InN thin films sputtered onto c-Si
(111) substrates determined by the VASE measurements in the wavelength
range 300-1400 nm. a The parameter is not chosen as a fitting parameter and it
has a fixed value during the fitting procedure. The 90 % confidence limits are
given by (±) ...... 140
5.8 Thicknesses of a-InN samples measured by using different techniques and
their optical constants at two different wavelengths, 500 nm and 1200 nm. The
90 % confidence limits are given by (±) ...... 150
5.9 Summary of methods of determining bandgap energy of sputtered 160 nm
a-InN thin film on two types substrates...... 168
5.10 The s- and p-polarized transmissivities of one and two bilayer systems through
the visible and near infrared regions at two angles of incidence, 50o and 60o ...... 195 xiv
List of Figures
2.1 Sputtering method: The impact of an atom or ion on a surface produces
sputtering from the surface as a result of the momentum transfer from the in-
coming particle...... 11
2.2 Experimental set-up for measuring specular reflectance: using a calibrated mirror,
in reference position (A) and measurement position (B) ...... 17
2.3 Experimental set-up for measuring specular reflectance: using the VW-set-up,
in reference position (C) and measurement position (D) ...... 18
2.4 Transmittance spectrum of crystalline AlN film deposited on silica glass together
with the envelope curves of peaks (TM (λ)) and valleys (Tm (λ))...... 21
2.5 Optical constants of crystalline AlN film deposited on quartz: (a) refractive index
and (b) extinction coefficient ...... 22
2.6 Principles of Spectroscopic Ellipsometry technique...... 25
2.7 Peak normal-incidence reflectance for the Mo/Be system as a function of the
number of bilayers N. The dashed curve is calculated for multilayers with ideal
interfaces, and the continuous star curve results from a modeled interface
roughness the continuous square curve represent measure data ...... 28
2.8 The transmissivity and reflectivity spectra of a 8 bilayer stack of Ag–SiO. Solid
line, experimental results; dashed-dotted line, computation ...... 29
2.9 Reflectance for normal incidence of different layers of glass/cryolite deposited
on a glass a function of the phase thickness (upper scale), or the wavelength (lower
scale). The number of layers is shown as a parameter on the curves...... 30 xv
2.10 (a) Hexagonal close packed structure with four basis atoms, where 1 and 2 are
anions, 3 and 4 are cations (b) The reciprocal lattice...... 32
2.11 UV-visible spectroscopy of the AlN films on Al2O3(0001) Deposited at a TS of (a)
800 °C and (b) 750 °C with laser energy density of 2 J/cm2, and (c) 550 °C and
laser energy density of 12 J/cm2...... 36
2.12 Spectral dependence of (a) the reflectivity of c-AlN film and (b) the imaginary
parts of the dielectric function ε2 of the same film. The solid line indicates
experimental data while the broken line represents the calculated data...... 37
2.13 Optical constants and transmittance of the InN thin film grown on the
ZnO/α-Al2O3 substrate ...... 40
2.14 Analysis of the ellipsometric spectra of InN thin film grown on the Si (100)
substrate. Experimental tan Ψ (dot line), experimental cos ∆ (solid line),
calculated tan Ψ (dash line), and calculated cos ∆ (dash dot line)...... 42
2.15 (a) Refractive index n spectra of InN thin films on Si (100) substrates at different
temperatures. (b) Extinction coefficient κ spectra of the same samples ...... 43
2.16 The continuous random network structure of amorphous silicon dioxide (a-SiO2),
notice that each Si atom (gold sphere) has 4 bonds, and each oxygen atom (red
sphere) has 2 bonds ...... 44
2.17 (a) Side view of the a-SiO2 /c-Si (001) interface structure, the arrows indicate
the locations of the Si-O-Si bridges. (b) Top view of the stripe structure of the
oxygen bridge bonds at the interface...... 45 xvi
2.18 Schematic density of states as a function of energy for a crystalline and an
amorphous semiconductor. ∆Ev and ∆Ec are band tails due to disorder...... 48
2.19 Imaginary part of the dielectric constant ε2 for amorphous Si compared to ε2 of
the crystal ...... 50
2.20 Melting points of III-nitrides and equilibrium N2 pressures over the III-N(s)...... 52
3.1 A schematic diagram of the path of a light beam within the film and substrate.
The r’s and t’s are the Fresnel’s coefficients of reflection and transmission,
respectively...... 64
3.2 A ray diagram of a light beam with the film and substrate. Tf , Rf , and Tb , Rb
are the respective transmittance and reflectance from the front and back sides of
the film ...... 67
3.3 A schematic diagram of a multilayer system of N-absorbing layers deposited on a
substrate of refractive index ns, and surrounded by an air of index no ...... 68
4.1 XRD pattern of (a) amorphous and (b) crystalline InN thin films sputtered on c-Si
(111) substrates...... 78
4.2 Photograph of the rotating holder of the sputtering system from side and front –
back views ...... 80
4.3 The XRD spectra of quartz substrate (uncoated layer) ...... 83
4.4 The XRD results of one bilayer (1π: 117 nm × 1) thin film system of AlN / InN/
Quartz ...... 84
4.5 The XRD results of two bilayers (2π: 117 nm × 2) thin film system of AlN / InN/
Quartz ...... 84 xvii
4.6 The XRD results of three bilayers (3π: 117 nm × 3) thin film system of AlN / InN/
Quartz ...... 85
4.7 The XRD results of four bilayers (4π: 117 nm × 4) thin film system of AlN / InN/
Quartz ...... 85
4.8 The XRD results of five bilayers (5π: 117 nm × 5) thin film system of AlN / InN/
Quartz ...... 86
4.9 The XRD results of one bilayer (1π: 538 nm × 1) thin film system of AlN / InN/
Quartz ...... 86
4.10 The XRD results of two bilayers (2π: 538 nm × 2) thin film system of AlN / InN/
Quartz ...... 87
4.11 The XRD results of three bilayers (3π: 538 nm × 3) thin film system of AlN / InN/
Quartz ...... 87
4.12 The XRD results of multi-bilayer thin film systems of AlN / InN/ Quartz with
different thicknesses of period 538 nm ...... 88
4.13 Schematic of the geometry of an ellipsometry experiment...... 89
4.14 Photograph of the Woollam Co. Variable Angle Spectroscopic Ellipsometry ...... 91
4.15 A schematic diagram of the optical setup: A Rotating Analyzer Variable Angle
Spectroscopic Ellipsometric (RA - VASE) ...... 91
4.16 Sample analysis procedure ...... 97
4.17 Schematic showing the incident, reflected, and transmitted light...... 100
5.1 Calibration fit from a fine mode calibration...... 105 xviii
5.2 Schematic drawing of the optical model structure for the SE analysis of
a-AlN film...... 107
5.3 Ellipsometric data for parameters (a) Ψ and (b) ∆ for 25 nm a-AlN film sputtered
onto c-Si (111) substrate. The dashed and doted lines represent the experimental
data while the solid line is the model fit for angles of incidence, 70o and 75o...... 109
5.4 Ellipsometric data for parameters (a) Ψ and (b) ∆ for 55 nm a-AlN film sputtered
onto c-Si (111) substrate. The dashed and doted lines represent the experimental
data while the solid line is the model fit for angles of incidence, 70o and 75o...... 110
5.5 Ellipsometric data for parameters (a) Ψ and (b) ∆ for 85 nm a-AlN film sputtered
onto c-Si (111) substrate. The dashed and doted lines represent the experimental
data while the solid line is the model fit for angles of incidence, 70o and 75o...... 111
5.6 Ellipsometric data for parameters (a) Ψ and (b) ∆ for 120 nm a-AlN film sputtered
onto c-Si (111) substrate. The dashed and doted lines represent the experimental
data while the solid line is the model fit for angles of incidence, 70o and 75o...... 112
5.7 Index of refraction and the extinction coefficient obtained from the
Cauchy-Urbach model for 25 nm a-AlN thin film sputtered onto c-Si (111)
substrate...... 114
5.8 Index of refraction and the extinction coefficient obtained from the
Cauchy-Urbach model for 50 nm a-AlN thin film sputtered onto c-Si (111)
substrate...... 114 xix
5.9 Index of refraction and the extinction coefficient obtained from the
Cauchy-Urbach model for 85 nm a-AlN thin film sputtered onto c-Si (111)
substrate...... 115
5.10 Index of refraction and the extinction coefficient obtained from the
Cauchy-Urbach model for 120 nm a-AlN thin film sputtered onto c-Si (111)
substrate...... 115
5.11 Index of refraction of a-AlN thin film sputtered onto c-Si (111) substrate as a
function of thickness at different wavelengths ...... 117
5.12 Extinction coefficient of a-AlN thin film sputtered onto c-Si (111) substrate as
a function of wavelength at different thicknesses ...... 117
5.13 Measured and calculated Ψ and ∆ spectra of a-AlN thin film of 50 nm thickness
sputtered onto c-Si (111). The dashed and dotted lines represent the measured
data while the solid line is the model fit for angles of incidence of 70o and 75o....120
5.14 Measured and calculated Ψ and ∆ spectra of a-AlN thin film of 100 nm thickness
sputtered onto c-Si (111). The dashed and dotted lines represent the measured
data while the solid line is the model fit for angles of incidence of 70o and 75o....121
5.15 Optical constants of a-AlN thin film of 50 nm sputtered onto c-Si (111)
substrate obtained from a Cauchy-Urbach model (200 – 1400 nm) ...... 122
5.16 Determination of the energy bandgap. The (E α n) 2 is plotted vs. photon
energy E for a-AlN thin film (50 nm) sputtered onto c-Si (111) substrate ...... 123
5.17 Determination of the energy bandgap. The (E α n) 2 is plotted vs. photon
energy E for a-AlN thin film (85 nm) sputtered onto c-Si (111) substrate ...... 124 xx
5.18 Determination of the energy bandgap. The (E α n) 2 is plotted vs. photon energy E
for a-AlN thin film (100 nm) sputtered onto c-Si (111) substrate ...... 124
5.19 Spectroscopic reflection data for (a) s- polarized reflectance (b) p- polarized
reflectance for the a-AlN thin film of 85 nm thickness sputtered onto a glass
substrate ...... 126
5.20 The polarized reflectances, Rp and Rs, for the p and s polarizations as functions
of the angle of incidence θi (degrees). These curves are obtained assuming that
medium 1 is air and medium 2 is a-AlN film with n = 1.99 + i 0.001 at λ = 400
nm (solid lines) and n = 1.86 + i 1.88 × 10-5 at λ = 1200 nm (dashed lines).
The n values for a-AlN thin film are taken from Fig.5.8...... 128
5.21 Spectroscopic transmission and absorption data for s- polarized transmittance Ts
(the upper curves) and absorptance As (the lower curves) for the a-AlN thin film
of 85 nm thickness sputtered onto a glass substrate...... 130
5.22 Spectroscopic transmission and absorption data for p- polarized transmittance
Tp (the upper curves) and absorptance Ap (the lower curves) for the a-AlN thin
film of 85 nm thickness sputtered onto a glass substrate ...... 131
5.23 Spectroscopic reflection data at 60o for (a) s- polarized reflectivity Rs and (b)
p- polarized reflectivity Rp for the a-AlN thin films sputtered onto glass
substrates ...... 133
5.24 Spectroscopic transmission data at 60o for (a) s- polarized transmittance and (b)
p- polarized transmittance for the a-AlN thin films sputtered onto glass
substrates ...... 134 xxi
5.25 Spectroscopic absorption data at 60o for (a) s- polarized absorptivity As and (b)
p- polarized absorptivity Ap for the a-AlN thin films sputtered onto glass
substrates ...... 135
5.26 Schematic drawing of the optical model structure for the SE analysis of
a-InN...... 139
5.27 Ellipsometric data for parameters (a) Ψ (λ) and (b) ∆ (λ) of a-InN thin film
(sample no.1). The dashed and doted lines represent the experimental data
while the solid line is the model fit for angles of incidence, 70o and 75o...... 141
5.28 Ellipsometric data for parameters (a) Ψ (λ) and (b) ∆ (λ) of a-InN thin film
(sample no.2). The dashed and doted lines represent the experimental data
while the solid line is the model fit for angles of incidence, 70o and 75o...... 142
5.29 Ellipsometric data for parameters (a) Ψ (λ) and (b) ∆ (λ) of a-InN thin film
(sample no.3). The dashed and doted lines represent the experimental data
while the solid line is the model fit for angles of incidence, 70o and 75o...... 143
5.30 Ellipsometric data for parameters (a) Ψ (λ) and (b) ∆ (λ) of a-InN thin film
(sample no.4). The dashed and doted lines represent the experimental data
while the solid line is the model fit for angles of incidence, 70o and 75o...... 144
5.31 The spectra of the ellipsometric parameter ∆ for a-InN thin films as a function
of wavelength at 70o. The lines (dash, dot, dash dot dot, and short dash dot)
represent the experimental data of the samples 1, 2, 3, and 4, respectively while
the solid line is the model fit ...... 145
xxii
5.32 The relative phase change ∆ for a-InN thin films as a function of film thickness
at 70o and λ = 490 nm...... 146
5.33 Index of refraction and the extinction coefficient obtained from the Tauc-Lorentz
model for 90 nm a-InN thin film sputtered onto c-Si (111) substrate ...... 147
5.34 Index of refraction and the extinction coefficient obtained from the Tauc-Lorentz
model for 160 nm a-InN thin film sputtered onto c-Si (111) substrate ...... 148
5.35 Index of refraction and the extinction coefficient obtained from the Tauc-Lorentz
model for 222 nm a-InN thin film sputtered onto c-Si (111) substrate ...... 148
5.36 Index of refraction and the extinction coefficient obtained from the Tauc-Lorentz
model for 260 nm a-InN thin film sputtered onto c-Si (111) substrate ...... 149
5.37 Optical constants obtained from the Tauc-Lorentz model for a-InN films
sputtered onto c-Si (111) substrates. The lines (solid, dash, and dash dot dot)
represent the optical constants for samples 2, 3, and 4 (Table 5.7),
respectively...... 149
5.38 Bandgap energy for InN films as a function of carrier concentration ...... 153
5.39 Determination of the Tauc optical bandgap energy. Plot of (E α (E) n (E)) 1/2 as a
function of photon energy for 90 nm a-InN thin film sputtered onto
c-Si (111)...... 156
5.40 Determination of the Tauc optical bandgap energy. Plot of (E α (E) n (E)) 1/2 as
a function of photon energy for 160 nm a-InN thin film sputtered onto
c-Si (111)...... 156
xxiii
5.41 Determination of the Tauc optical bandgap energy. Plot of (E α (E) n (E)) 1/2as
a function of photon energy for 220 nm a-InN thin film sputtered onto
c-Si (111)...... 157
5.42 Determination of the Tauc optical bandgap energy. Plot of (E α (E) n (E)) 1/2 as
a function of photon energy for 260 nm a-InN thin film sputtered onto
c-Si (111)...... 157
5.43 Determination of the energy bandgap using (a) Spectroscopic ellipsometry
and (b) Spectrophotometric methods. (Eα(E)n(E)) 1/2 is plotted vs. the photon
energy E for a-InN thin film of 160 nm thickness sputtered onto (a) c-Si (111)
and (b) glass substrates. The absorption coefficient as a function of wavelength
is shown in the inset ...... 159
5.44 Determination of the Cody optical bandgap energy. Plot of (α (E) n (E)/E) 1/2 as
a function of photon energy for 160 nm a-InN thin film sputtered onto
c-Si (111)...... 160
5.45 Why for a-InN thin films the square-root dependence should be used?...... 160
5.46 Photoluminescence (PL) spectra at 20 K of sample no.4 (260 nm)...... 161
5.47 Spectroscopic reflection data for (a) s- polarized intensity reflectance Rs and (b)
p- polarized intensity reflectance Rp for the a-InN thin film of 160 nm thickness
sputtered onto a glass substrate ...... 163
5.48 The polarized reflectances, Rp and Rs, for the p and s polarizations as functions
of the angle of incidence θi (degrees). These curves are obtained assuming that
medium 1 is air and medium 2 is a-InN film with n = 2.26 + i 0.44 at λ = 350 nm xxiv
(solid lines) and n = 2.06 + i 0.00 at λ = 1200 nm (dashed lines). The n values
for a-InN thin film are taken from Fig.5.37 ...... 164
5.49 (a) Spectroscopic transmission data for s- polarized intensity transmittance Ts for
the a-InN thin film of 160 nm thickness sputtered onto a glass substrate ...... 165
5.49 (b) Spectroscopic transmission data for p- polarized intensity transmittance Tp for
the a-InN thin film of 160 nm thickness sputtered onto a glass substrate...... 166
5.50 (a)Sectroscopic absorption data for s- polarized intensity absorptance As for the
a-InN thin film of 160 nm thickness sputtered onto a glass substrate...... 167
5.50 (b)Sectroscopic absorption data for p- polarized intensity absorptance Ap for the
a-InN thin film of 160 nm thickness sputtered onto a glass substrate...... 168
5.51 The p- polarized transmissivity and absorptivity of a-InN thin films as functions of
photon energy at 75o...... 170
5.52 The s- and p- polarized transmissivity of a-InN thin film as functions of its
optical thickness n × h at 75o and λ = 480 nm. The index of refraction value is
taken from Fig.5.37 at the same wavelength...... 170
5.53 Schematic drawing of the optical model for the SE analysis of a-GaN film ...... 172
5.54 Measured and model fit ellipsometric parameters, ∆ (λ) and Ψ (λ), for
a-(Al, Ga, In) N thin films sputtered onto c-Si (111) substrates at 70o ...... 173
5.55 Indices of refraction of a-AlN and a-(Ga, In) N films obtained from the Cauchy-
Urbach and Tauc-Lorentz models, respectively...... 173
5.56 Extinction coefficients of a-AlN and a-(Ga, In) N films obtained from the Cauchy-
Urbach and Tauc-Lorentz models, respectively...... 174 xxv
5.57 Determination of the energy bandgaps for 85 nm a-(Al, Ga) N thin films sputtered
onto c-Si (111) substrates...... 175
5.58 The polarized reflectances for the s and p polarizations as functions of the angle of
incidence θi for a-GaN thin film sputtered onto glass substrate...... 176
5.59 The spectral dependence of the s- polarized transmissivity (the upper curves)
and absorptivity (the lower curves) data at different angles of incident for a-GaN
film of 150 nm thickness sputtered onto glass substrates ...... 178
5.60 The spectral dependence of the p-polarized transmissivity (the upper curves)
and absorptivity (the lower curves) data at different angles of incident for a-GaN
film of 150 nm thickness sputtered onto glass substrates ...... 178
5.61 Phase shifts in an AlN film. The reflected waves are shown separately for clarity.
Reflection from a higher index of refraction produces a change of phase.
Reflection from a lower index of refraction does not cause a change of phase in
the reflected wave...... 181
5.62 Diagram indicating values for indices n1 and n2 (hatched zone) to obtain an
antireflection coating comprising two layers on a substrate with index ns = 1.52
and no = 1 [172]...... 183
5.63 Schematic drawing of quarter / quarter antireflection coating of AlN / InN /
Quartz bilayer system...... 184
5.64 The spectral dependence of the (a) s-polarized reflectivity and (b) p-polarized
reflectivity at 50o, 60o , and 70o of one bilayer system of 53 nm InN film
coated with 64 nm AlN film (the period of the structure d′ = 117 × 1 nm)...... 186 xxvi
5.65 Antireflection coating with two layers of the same optical thickness (one
bilayer system of period d′ = 117 × 1 nm). The coating optimized for 500 nm
at 50o incidence ...... 187
5.66 The spectral dependence of the (a) p-polarized transmissivity and (b)
s-polarized transmissivity at 50o, 60o , and 70o of one bilayer system of 53 nm
InN film coated with 64 nm AlN film (the period of the structure
d′ = 117 × 1 nm) ...... 188
5.67 Schematic drawing of a periodic AlN / InN / Quartz multilayer system with
H, L quarter waves of indices 2.35, 1.94 and design wavelength 500 nm...... 189
5.68 The spectral dependence of the (a) s-polarized reflectivity and (b) p-polarized
reflectivity at 50o, 60o , and 70o of a two bilayer system of 53 nm InN film
coated with 64 nm AlN film (the period of the structure d′ = 117 × 2 nm)...... 191
5.69 Comparisons between the (a) s and (b) p-polarized reflectivities of one bilayer
( d′ = 117 nm) and two bilayer ( d′ = 234 nm) systems of AlN / InN / quartz at
60o incidence, respectively ...... 193
5.70 The spectral dependence of the (a) p-polarized transmissivity and (b) s-polarized
transmissivity at 50o, 60o , and 70o of a two bilayer system of 53 nm
InN film coated with 64 nm AlN film (the period of the structure
d′ = 117 × 2 nm) ...... 194
5.71 The spectral dependence of the (a) s-polarized reflectivity and (b) p-polarized
reflectivity at 50o, 60o , and 70o of a three bilayer system of 53 nm InN film
coated with 64 nm AlN film (the period of the structure d′ = 117 × 3 nm)...... 197 xxvii
5.72 The spectral dependence of the (a) s-polarized reflectivity and (b) p-polarized
reflectivity at 50o, 60o , and 70o of a four bilayer system of 53 nm InN film
coated with 64 nm AlN film (the period of the structure d′ = 117 × 4 nm)...... 198
5.73 The spectral dependence of the (a) s-polarized reflectivity and (b) p-polarized
reflectivity at 50o, 60o , and 70o of a five bilayer system of 53 nm InN film
coated with 64 nm AlN film (the period of the structure d′ = 117 × 5 nm)...... 199
5.74 Comparison of the p-reflectivity spectra of three and five bilayer systems with
one bilayer system over the NIR region at 60o...... 200
q 5.75 The average polarized reflectivity Rave of (quarter-quarter) – wave coatings
of AlN / InN / quartz as function of their total bilayer thickness at λ = 850 nm
and 60o incidence...... 202
5.76 The spectral dependence of the (a) p-polarized transmissivity and (b) s-polarized
transmissivity at 50o, 60o , and 70o of a three bilayer system of 53 nm InN film
coated with 64 nm AlN film (the period of the structure d′ = 117 × 3 nm)...... 203
5.77 The spectral dependence of the (a) p-polarized transmissivity and (b) s-polarized
transmissivity at 50o, 60o , and 70o of a four bilayer system of 53 nm InN film
coated with 64 nm AlN film (the period of the structure d′ = 117 × 4 nm)...... 204
5.78 The spectral dependence of the (a) p-polarized transmissivity and (b) s-polarized
transmissivity at 50o, 60o , and 70o of a five bilayer system of 53 nm InN film
coated with 64 nm AlN film (the period of the structure d′ = 117 × 5 nm)...... 205 xxviii
q 5.79 (a)The average polarized transmissivity Tave of (quarter-quarter) – wave coatings
of AlN / InN / quartz as a function of wavelength at 60o incidence...... 206
q 5.79 (b)The average polarized transmissivity Tave of (quarter-quarter) – wave coatings
of AlN / InN / quartz as a function of wavelength at 70o incidence...... 207
q 5.80 The average polarized transmissivity Tave of (quarter-quarter) – wave coatings
of AlN / InN / quartz as function of their total bilayer thickness at λ = 360 ,
500, and 850 nm and 60o incidence...... 208
5.81 The spectral dependence of the (a) s-polarized reflectivity and (b) p-polarized
reflectivity at 50o, 60o , and 70o of a one bilayer system of 294 nm InN film
coated with 244 nm AlN film (the period of the structure d′ = 538 × 1 nm)...... 210
5.82 The spectral dependence of the (a) s-polarized reflectivity and (b) p-polarized
reflectivity at 50o, 60o , and 70o of a two bilayer system of 294 nm InN film
coated with 244 nm AlN film (the period of the structure d′ = 538 × 2 nm)...... 211
5.83 The spectral dependence of the (a) s-polarized reflectivity and (b) p-polarized
reflectivity at 50o, 60o , and 70o of a three bilayer system of 294 nm InN film
coated with 244 nm AlN film (the period of the structure d′ = 538 × 3 nm)...... 212
5.84 Combined p- reflectance profiles for one, two, and three bilayer systems of
InN / AlN / quartz of identical period of 538 nm at 60o incidence...... 213
5.85 The spectral dependence of the average polarized transmissivity Tave of (a) one
bilayer system and (b) two bilayers system of identical period of 538 nm at 50o,
60o, and 70o incidence ...... 214 xxix
5.86 The spectral dependence of the polarized optical properties of a one bilayer
system of identical period 117 nm and total physical thickness 117 nm at 60o
incidence...... 216
5.87 The spectral dependence of the polarized optical properties of a four bilayers
system of identical period 117 nm and total physical thickness 468 nm at 60o
incidence...... 216
5.88 The spectral dependence of the polarized optical properties of a one bilayer
system of identical period 538 nm and total physical thickness 538 nm at 60o
incidence...... 217 xxx
List of Symbols and Abbreviations
The following table gives those most important symbols used in at least several places in the dissertation.
A Absorptance- the ratio of the absorbed energy to the incident energy
As s-polarized absorptivity.
Ap p-polarized absorptivity.
BBARC Broadband antireflection coating
c The speed of light in vacuum.
d The skin depth.
d′ The period of the multilayer structure. r E The electric vector in the electromagnetic field.
r The amplitude of the electric vector of the incident beam. Ei
r The amplitude of the electric vector of the reflected beam. Er
r The amplitude of the electric vector of the transmitted beam. Et
Eg The bandgap energy
f The variance of the interfacial roughness.
h The physical thickness of the film.
hop The optical thickness of the film.
I The intensity of the wave. A measure of the energy per unit area per
unit time carried by the wave. xxxi
j Subscript used to designate the alternate layers that form the
multilayer structure. ~ K The complex wave vector.
M The elements of the characteristic matrix of a thin layer.
th Mj (hj) The elements of the transfer matrix of the j layer.
M (h) The elements of the transfer matrix of the whole multilayer structure
MSE Mean-square error
N The number of the layers in a multilayer structure.
n~ The complex refractive index.
n The refractive index.
n′ The number of atoms.
nH ,nL The indices of the high-and low-index layer.
p Indicates the plane of polarization in which the electric vector is
parallel to the plane of incidence.
R The reflectance defined as the ratio at a boundary of the reflected
intensity to the incident intensity.
s Indicates the plane of polarization in which the electric vector is
normal to the plane of incidence.
T The transmittance defined as the ratio at a boundary of transmitted
intensity to the incident intensity.
TG Gryogenic temperature xxxii
ts ,t p The ratios of the transmitted amplitudes to the incident amplitudes
of the electric vector for s and p polarizations, respectively.
XRD X-ray diffraction
α The absorption coefficient.
ρ The electrical resistively.
κ The extinction coefficient which is the imaginary part of the
complex refractive index.
ν The frequency of the incident light.
λo Wavelength in vacuum.
λ Wavelength in a medium.
µ The permeability of the medium
ε1 ,ε 2 The dielectric constants.
ε~ The complex dielectric constant.
ε1 (∞) The contribution of the optical transitions at higher energies.
~ δ The complex phase differences of the beam on traversing the
absorbing layer.
θ B The Brewster angle.
θ i The angle of incidence between the boundary normal and the
incident wave.
θ t The angle of transmittance between the boundary normal and
transmitted wave. xxxiii
Ψ The relative amplitude change.
∆ The relative phase change. a-AlN Amorphous aluminum nitride. a-InN Amorphous indium nitride. a-GaN Amorphous gallium nitride.
SR Surface roughness.
RF Radio frequency.
EMR Electromagnetic radiation.
TE Transverse electric.
TM Transverse magnetic.
VASE Variable angle spectroscopic ellipsometry.
ρ (E) Density of electronic state.
PL Photoluminescence. p-IR p-polarized intensity reflectance.
s-IR s-polarized intensity reflectance.
p-IT p-polarized intensity transmittance.
s-IT s-polarized intensity transmittance.
1
CHAPTER ONE
INTRODUCTION
The earliest of what might be called modern thin film optics was the discovery, independently by Robert Boyle and Robert Hooke, of the phenomenon known as
Newton’s rings [1]. Developments became much more rapid in the 1930’s. In 1932
Rouard observed that a very thin metallic film reduced the internal reflectance of a glass plate, although the external reflectance was increased. For nearly the first half of the last century the phenomenal growth of thin film research and development owes much to the stimulus provided by the early utilitarian interest in the application of optical films in mirrors and interferometers [2]. In the decades that followed, the physics of thin films developed greatly and to a large extent remained in the domain of academic interests until developments in vacuum technology systematically progressed.
Thin films are two – dimensional materials “created” by condensation of atoms, molecules, or ions on solid supports defined as substrates. They have unique properties significantly different from those of the same materials in bulk form. This has been attributed to the dimensions, geometry, grain boundaries, lattice defects and voids, non- equilibrium microstructure, and metallurgy which are more predominant in thin films.
The characteristic properties of bulk materials are often related to a unit volume; it is assumed that those properties are volume independent. However, for thin films in which the surface to volume ratio is increased this assumption is not valid. In a film, the two 2 surfaces are so close to each other, inducing a significant influence on the film’s internal physical properties [3]. Such unique properties give thin films considerable significance in industrial and technical applications.
Theoretical and experimental investigations of the optical behavior of a thin film deal primarily with optical reflection, transmission, and absorption, and their relation to the optical constants of the film. As a result of such studies, the reflection, transmission, and ellipsometric properties (Ψ, the relative amplitude change, and ∆, the relative phase change) of thin films have made it possible to determine the optical constants conveniently. Thin film studies have directly or indirectly advanced many new areas of research in various fields of physics, electronics, and chemistry. This is in addition to well established applications such as band pass filters of various kinds, cold mirrors, beam splitters, reflecting coatings, protective and decorative coatings [4-6]. The progress extended most recently towards integrated optics, medical electronics, solar cells, and low-emittance windows [7, 8].
The use of surface coatings to modify the optical properties of materials is well known, and coating technologies have been developed to provide desired results in thousands of applications. One such application is the use of antireflection (AR) coatings to reduce unwanted reflections and enhance total transmitted intensity over a specific wavelength range. If system requirements are such that transmission enhancement is required for only a relatively narrow wavelength interval, single layer coatings may be sufficient. Conversely, if uniformly higher transmission is desired over a wide wavelength interval, or if suppression of reflected energy is significant, multilayer AR 3 coatings must be used. This result is achieved by producing reflections that interfere destructively with those originating from the base material, or substrate. The type of AR coatings used in any particular application will depend on a variety of factors, including the substrate material, the wavelength region, the required performance and the cost.
For many applications, the AR properties achieved by a single layer coating are insufficient, as the bandwidth of the destructive interference is limited. More complex coating arrangements consisting of two or more layers have been developed to address the need for more broadband antireflection (BBAR) coatings, with considerable success.
In the design of the BBAR coating, there are four factors which influence the results: 1) the refractive indices of the materials used; 2) the bandwidth over which the reflection is to be reduced; 3) the over all thickness of the coating, and 4) the number of layers in the design [9]. The bandwidth is usually specified, leaving the materials, total thickness, and the number of layers as variables. The choice of materials is limited by the spectral range of interest and the environmental resistance desired. In general, there are definite lower limits on the total thickness and numbers of layers for an optical design, and it is undesirable to exceed a certain upper range of the total thickness and the minimum number of layers for a given design problem.
The twentieth century has been one of fascinating technological advancement and explorations. Our knowledge has expanded towards the atomic and nuclear scales and our daily lives have benefited deeply from scientific knowledge, giving rise to uncountable innovations: computer, cellular phone, internet, and many more. However, since the beginning of the twenty first century, it has become desirable to make such technological 4 tools smaller, faster, lighter, and cheaper. Thus, we have begun to witness new technological developments in areas such as the growth of novel thin film materials and in the equipments used to investigate their optical, electrical, and chemical properties.
In recent years, the III-V nitride semiconductors, AlN, GaN, InN, and their ternary alloys have attracted much attention in several fields of new technology. This is because of their inherent properties such as high thermal conductivity, thermal and chemical stability, high ionicity, very short bond lengths, low compressibility and high acoustic wave velocity [10]. The III-V nitrides are also very promising materials for their potential use in the fabrication of highly efficient optoelectronic devices (both emitters and detectors) and high power/temperature electronic devices [11-14].
Crystalline, c, and amorphous, a-(Al, Ga, In) N, form a continuous alloy system with optical bandgaps in the range of 6.2, 3.5, and 1.9 eV for c-(Al, Ga, In) N [15], respectively and 5.95, 3.3 eV for a-(Al, Ga) N [16]. These bandgap ranges span a part of the electromagnetic spectrum (from red to violet) not covered by conventional semiconductor technology. Current semiconductor technology covers the infrared (IR) to green range of the electromagnetic spectrum and this is one of the reasons fueling recent interest in these materials and their ternary alloys for optoelectronic device applications.
Specifically, nitrides are suitable for such applications as surface acoustic wave devices
[17, 18], UV detectors [19], UV and visible light emitting diodes (LEDs) [20], and laser diodes (LDs) for digital data read-write applications [21].
Amorphous semiconductors have many present-day applications. One important aspect for the future of these materials and their alloys is that a lack of long range 5 ordering in the atomic network relaxes the k-selection rules for the optical transitions
[22]. This will give rise to a large optical absorption coefficient and most probably a high luminescence efficiency, which can be used in light emitting devices. Amorphous nitrides as large area thin film semiconductors with doping, low cost, and low temperature fabrication capability have not only given a new impetus to the field of optoelectronic devices but they now also form the basis for large area microelectronics [23, 24].
In order to meet demands arising from the rapidly growing field of information processing it is very important to understand the fundamental properties of these materials over a wide wavelength range. Accurate knowledge of the refractive indices and absorption coefficients of amorphous semiconductors is indispensable for the design and analysis of various optoelectronic devices. These optical properties reflect essentially the density of states and thus their analysis is one of the most effective tools for understanding the electronic structure of amorphous solids.
In spite of the prospects for amorphous (Al, Ga, In) N thin films in device applications, there is no systematic experimental and theoretical investigation of the optical characteristics over a wide wavelength range, although such studies were performed on crystalline (Al, Ga, In) N thin films using different techniques including
Spectroscopic Ellipsometry (SE) [25 – 35]. The SE technique is based on measuring the change in light polarization upon reflection from a sample surface or interfaces and allows for the characterization of optical and structural thin film properties with extreme accuracy. This technique has recently found much favor in the non-destructive characterization of thin films and bulk materials. It is an excellent technique for 6 investigating the optical response of semiconductors [36] and is very sensitive to surface irregularities such as surface roughness, interdiffusion and interlayer formations in multilayer thin film systems [37]. It derives its sensitivity from the determination of the relative phase change in a beam of reflected polarized light [38].
1.1 Aim and Organization of the Dissertation
The aim of this dissertation is to investigate the optical characteristics of III-V nitrides as single and multilayer thin film systems over a wide wavelength range, i.e. 300-1400 nm which includes some portion of the infrared region. These systems were deposited onto crystalline silicon (111), glass, and quartz substrates using RF reactive magnetron sputtering at a temperature T < 325 K. The main investigatory optical technique used in this dissertation is Spectroscopic Ellipsometry.
This dissertation is divided into six chapters including the introduction (Chapter
One). Chapter two presents a detailed review of deposition techniques of thin films and a qualitative comparison between sputtering and thermal evaporation. In addition, many differing methods for the determination of the complex refractive index variables n and κ are discussed in a comprehensive way. The discussion also covers the advantages and disadvantages in the determination of accurate values of the optical constants. As a main part of this chapter, we discuss the optical properties of amorphous thin films and their potential advantages over crystalline films. Much attention is given to previous studies of the optical properties of group III nitrides in the rest of this chapter. 7
Chapter three covers the basic concepts of classical electromagnetic theory related to the optical properties and optical constants of thin film systems. Two mathematical methods:
Airy’s method, in the case of single layer systems, and the transfer (2×2) matrix method
(in bilayer and multilayer systems) are described in great detail. The mathematical definition of the ellipsometric parameters, Ψ and ∆, and their relationship to the optical constants are also described.
Chapter four gives information about the actual operating conditions of the RF magnetron sputtering system that were used in the preparation of single and multilayer
(Al, In) N samples. The experimental setup and techniques used in the investigation of the optical characteristics of the samples are also discussed. Additionally, the chapter includes X-ray diffraction results used to verify the amorphous and polycrystalline natures of the studied films.
The results from the SE analysis and their discussion are reported in chapter five.
The optical constants and the thicknesses of a-(Al, Ga, In) N were obtained by the analysis of the measured ellipsometric spectra through the Cauchy-Urbach and Tauc-
Lorentz models. The spectral dependence of polarized optical reflectivities and transmissivities of these thin films sputtered onto glass substrates at room temperature using SE are also reported for different angles of incidence. The wavelength range of our measurement was 300 – 1400 nm including some portion of the infrared region. The polarized optical properties of bilayer and multilayer thin film systems of sputtered AlN /
InN onto quartz substrates with two different identical periods are also discussed in chapter five. These systems were primarily built to “create” BBAR coatings that are 8 effective over a fairly broad spectral region and also usable for a wide range of angles of incidence (50o – 70o) in the near infrared wavelength range.
Chapter six contains conclusions drawn from the discussions in chapters four and five. 9
CHAPTER TWO
GENERAL BACKGROUND
2.1 Formation Techniques of Thin Films
There exists a large variety of thin film deposition processes and technologies which originate from purely physical or purely chemical processes. The formation or deposition techniques can be broadly classified under three headings: thermal evaporation, chemical vapor deposition (CVD), and cathodic sputtering. In our research, sputtering was chosen as the technique for growth of group-III nitrides over other well documented methods [1,
2].
Sputtering deposition is the most widely used method for the growth of both thin metal films and insulators onto a substrate, as verified by the many applications of the process such as, tribological coatings, optical coatings, luminescence films, solar cells, microelectronic applications, etc [39]. One of the main reasons for sputtering is the apparent simplicity of getting a large deposition area without the need for substrate motion. Therefore, the film deposited by this method is uniform in thickness and capable of covering areas usually shadowed by other deposition methods. Also, the deposition of alloys and insulators as composite materials are two important benefits of sputtering.
Unlike thermal evaporation, the material to be sputtered does not have to be heated (Fig.
2.1). Moreover, sputtering has additional benefits as a deposition technique when compared with thermal evaporation as illustrated in Table 2.1. 10
Table 2.1 Comparison of evaporation and sputtering
EVAPORATION SPUTTERING Low energy atoms (~ 0.1 eV) Higher energy atoms (1-10 eV) High vacuum path Low vacuum, plasma path • few collisions • many collisions • little gas in film • gas in film • line of sight deposition • less line-of-sight deposition Larger grain size Smaller grain size Fewer grain orientations Many grain orientations Poorer adhesion Better adhesion High deposition rates Low deposition rates Point source: poor uniformity Parallel plate: better uniformity Optical multilayer is easy Optical multilayer is more difficult
2.1.1 Principle of Sputtering
Sputtering is a physical process that can be compared to throwing steel balls at a concrete wall. Upon impact, the ball tears away fragments of the concrete, resulting in fragments which retain the chemical and physical properties of concrete. If the process is continued, surfaces in the vicinity of the impact are covered with a layer of concrete dust. In sputtering, the “steel balls” are ionized atoms. The “wall” is a plate of the material to be sputtered, called a target as shown in Fig. 2.1. The sputtering process takes place in an evacuated chamber. Nitrogen gas is introduced, and then ionized in a chamber which contains the substrate and target (cathode) of the film material to be sputtered. The target is maintained at a negative potential relative to the positively charged nitrogen atom.
When the positive ions with high kinetic energy are incident on the cathode, the 11 subsequent collisions sputter atoms from the material. The process of transferring momentum from impacting ions to surface atoms forms the basis of sputter coating. The nitrogen atom does not become imbedded in the target. It slams into it like a steel ball into the wall and tears off some of the target material. Since the chamber is maintained at low vacuum the liberated material settles on everything in the chamber, mainly the substrates. The plasma is initiated between the cathode and the anode at pressures in the milli Torr (mT) range by the application of a high voltage which can be either direct current (DC) or radio frequency (RF). A strong magnetic field (magnetron) can be used to concentrate the plasma near the target to increase the deposition rate.
Although the coating speed using RF magnetron sputtering is relatively low compared to DC magnetron sputtering, its ability to sputter insulator cathodes has adapted this technique to wide variety of applications [39, 40]. In DC systems, positive charge build-up on the cathode (target) needs 1012 volts to sputter insulators whereas RF magnetron systems avoid charge build up by alternating potential [41].
Substrate
Growing Film
Atom or Ion with Atom or Ion Kinetic Energy
Surface
Figure 2.1 Sputtering method: The impact of an atom or ion on a surface produces sputtering from the surface as a result of the momentum transfer from the in-coming particle. 12
2.2 Optical Properties of Thin Films
The optical properties of thin films are of significant importance, both in basic and applied research. The wide use of thin films in optical devices requires a good knowledge of such properties. The optical behavior of a thin film is a function of its interactions with electromagnetic radiation (EMR). Possible interactive phenomena are reflection, transmission, and absorption of the incident light onto the surface of the thin film. When describing optical phenomena involving the interactions between radiation and matter, an explanation is often facilitated if light is treated in terms of photons (quantum treatment).
On other occasions, a wave treatment (classical treatment) is more appropriate; at one time or another, both approaches are used in this work.
When light proceeds from one medium into another (e.g. from air into a thin film), several things happen. Some of the light radiation will be reflected at the interface between the two media, some may be transmitted through the medium, and some will be absorbed within it if the material of a thin film is absorbing (metal and semimetal). The principle of conservation of energy determines that for any light radiation-matter interaction, the intensity Io of the incident beam on the surface of the thin film at a particular wavelength must equal the sum of the intensities of the reflected, transmitted, and absorbed beams, denoted as IR, IT, and IA, respectively, i.e.,
I o = I R + I T + I A (2.1) 13
An alternate form is
R + T + A = 1 (2.2)
where R, T, and A represent the reflectivity (IR / Io), transmissivity (IT / Io), and absorptivity (IA / Io), respectively. The sum of these macroscopic quantities which are usually known as the optical properties of the thin film must equal unity since the incident radiant flux at one wavelength is distributed totally between reflected, transmitted, and absorbed intensity.
The optical phenomena that occur within thin films involve interactions between the
EMR and atoms, ions, and/or electrons. Two of the most important of these interactions are electronic polarization and electron energy transitions. One component of an electromagnetic wave is simply a rapidly fluctuating electric field. For the ultraviolet and visible regions, the electric field interacts with the electron cloud surrounding each atom within its path in such a way as to induce electronic polarization. Two consequences of this polarization are: (1) some of the radiation energy may be absorbed, and (2) light waves are retarded in velocity as they pass through the medium. The second consequence is manifested as refraction. The optical properties of thin films related to absorption and emission of EMR can be explained in terms of the electronic band structure of the material and the principles relating to outer shell electron transitions. Since electron transitions require high excitation energies, they are most common at short wavelengths - ultraviolet and visible light.
For a complete understanding of the optical behavior of a thin film, a good knowledge of the structure of the film is also necessary. Films are found to exhibit a wide 14 variety of structures, from irregular, amorphous aggregates to monocrystalline layers.
The properties of such layers, and especially the optical and electrical properties, would be expected to depend considerably on the exact form of the film [2].
2.3 Optical Constants of Thin Films
The optical constants of an isotropic material are the index of refraction n (λ) and the extinction coefficient κ (λ). They are the real and imaginary components of the complex index of refraction n. The optical constants of materials have either been completely unknown or determined only in a narrow wavelength range. Accordingly, in recent years there has been an increasing need for an accurate knowledge of the optical constants of thin films over a wide wavelength range. Knowledge of the wavelength-dependent complex refractive index of thin films is very important, both from a fundamental and a technological viewpoint. It yields fundamental information on the optical energy bandgap
(for semiconductors and insulators) and plasma frequencies. Moreover, the refractive index is necessary for the design of optical components and optical coatings such as interference filters.
For an isotropic, homogeneous and plane-parallel thin film, the real and imaginary parts of the complex refractive index at each wavelength completely determine the optical properties of the film. Therefore, two independent measurements are necessary at each wavelength in order to solve for the unknowns n (λ) and κ (λ). If the above mentioned requirements are not met (for inhomogeneous films [42] or rough films [43]), 15 it is still possible to obtain the optical properties of the “non-ideal” thin films, though with somewhat more difficult.
2.3.1 Methods for the Determination of the Optical Constants of Thin Films
The optical constants of thin films can be measured at a given wavelength by a large variety of available methods including spectrophotometry, interferometry, critical angle, transmittance data (envelope), and Ellipsometry. A review of these methods is described in several textbooks on optics [1, 2, 40, and 42] and numerous publications [29, 30, 44,
45]. However, a comparison of several techniques and methods used for the determination of the optical constants showed considerable discrepancies among the same samples. This explains how delicate such methods could be [29].
Of all the methods available to obtain the complex index of refraction of a material, the one most accessible to non-specialists is the photometric method (RT method). In the RT method, the film is illuminated by unpolarized radiation, and the intensities of reflected and transmitted beams are measured as a function of wavelength
[45]. The photometric measurements of reflectance R and transmittance T at normal incidence are recognized as the most suited for determining the optical constants n and κ of a thin film material over a wide wavelength range.
For transparent films the calculation of the optical constants is straight forward. In the case of an absorbing film on a transparent substrate, the general expressions for the reflectivity and transmissivity at normal or near normal incidence are given by [46] 16
2 ~~ 2 ~r + ~r e~ 2 n t t ~e 2 R = 1 2 and T = s 1 2 , (2.3) ~~ ~ 2 ~~ ~ 2 1 + r1r2 e no 1 + r1r2 e
~ ~ respectively. Here, e = exp(i2π nh / λ0 ); h is the film thickness; λ is the wavelength; no is the refractive index of the ambient; ns is the substrate refractive index; ñ is the complex refractive index of film; r1 , t1 and r2 , t2 are the respective Fresnel’s reflection and transmission coefficients for the first (vacuum-layer) and second (layer-substrate) interfaces. These are used to calculate the optical constants, employing the inversion method [2, 47-49]. Moreover, the determination of the optical constants consists of solving the following system of equations [43]:
Tth (n,κ , h, λ ) − Texp (λ ) = 0 (2.4)
(2.5) Rth (n,κ , h, λ ) − Rexp (λ ) = 0 To solve this system of equations, it is necessary to know the complex refractive index of the substrate and film thickness h. These equations are easily solved numerically by successive approximations of n and κ , which minimize the quality function [46]:
2 2 Q = (Tth − Texp ) + (Rth − Rexp ) (2.6)
The iterative solution is stopped when a defined low Q value is achieved.
Despite the variety of numerical techniques that exists for the extraction of information about the optical constants of layers and substrates from the RT method, there are a few practical problems with the RT method. Usually either transmittance or reflectance is measured because conventional instruments can only make one measurement at a time and switching from one type of measurement to the other requires that the instrument be reconfigured. Also, special accessories need to be purchased to 17 adapt a transmittance spectrophotometer for reflectance measurements. Evidently, with such a system it is impossible to ensure that exactly the same part of the film is measured in both transmittance and reflectance; if the film is not perfectly homogeneous (in composition or in thickness), which is inevitable, the overall accuracy of the method is further decreased. Therefore, conventionally only a single spectrum is analyzed.
Secondly, it is very difficult to obtain sufficiently accurate absolute specular reflectance data. In photometric measurements, there are two common methods for measuring specular reflectance. Using the simplest experimental set-up, one measures the light attenuation upon a single reflectance at the sample surface, which is then compared to the reflectance of a calibrated mirror (see figures 2.2 (A) and (B)). This method obviously necessitates such a calibrated mirror, which is both expensive and has a limited shelf life: its reflectance is only guaranteed for a few months due to ageing effects.
Figure 2.2 Experimental set-up for measuring specular reflectance: using a calibrated mirror, in reference position (A) and measurement position (B).
A more elegant way for measuring absolute reflectance is the VW-set-up, introduced by
Strong [50] and shown schematically in figures 2.3 (C) and (D). Here, a moveable mirror is first placed in a ‘reference’ position, and the corresponding combined reflectance of the two fixed mirrors and the moveable mirror is acquired. Then, the moveable mirror is 18 switched to its ‘measurement’ position and the film to be investigated is mounted. Now
(Fig. 2.3 (D)), the only optical difference with the reference measurement (Fig. 2.3 (C)) is a double reflectance off the sample under test. Thus, division of the measurement and reference values exactly yields the square of the reflectance needed, irrespective of the reflectance of the mirrors used. The only obvious disadvantage of the VW-method is that the two reflectances on the sample occur at different spots. Therefore, the sample should be perfectly homogeneous.
C D
Figure 2.3 Experimental set-up for measuring specular reflectance: using the VW-set-up, in reference position (C) and measurement position (D).
Another problem with the RT method is that there is no easy way to measure normal incidence reflectance. In the usual methods (described above), the angle of incidence is
7° [51]. For maximum accuracy, this angle should be taken into account in the subsequent calculations. It also makes the measurements sensitive to polarization effects
(see Chap.4).
Finally, the solutions (n, k) based on a numerical inversion are not unique [52] for a given measured (R, T). In addition, for any film thickness value determined in the RT 19 method, the Fresnel’s equations give multiple n (λ) and κ (λ) solutions due to the periodic nature of the e expression. Therefore, it is necessary to apply a physical criterion in order to identify the physical solutions. Besides, in some regions of the spectrum the solutions may be missed. This has been currently associated with experimental inaccuracies of R and T or film inhomogeneities.
The general problems with RT methods, referred to above, have been minimized or solved either by using measurements of R and T at oblique incidence [47], or by other measurements such as the transmittance T, and the reflectances, Rf and Rb , (from the film and reverse sides of the coated transparent substrate) [45].
Another method that depends only on the transmission measurements was used to investigate the values of n and κ. This method allows an accurate determination of κ, since T= exp (- 4πhκ/λ), and it requires that the thickness of the film must be such that the effects of multiple reflections are suppressed. For metal films, a lower limit of ~ 300 to 400 Å is imposed on the film thickness. Under this condition the normal transmittance of the film of index ñ and thickness h is given by [52]:
16n (n2 + κ 2 )2.exp[ −4πhκ / λ] T = s (n > n ) 2 2 2 2 s (2.7) [(n + no ) + κ ][(ns + n) + κ ]
According to this relation, the intercept as well as the slope of the log T vs. h straight–line plot should give κ [2].
Neglecting interference and multiple reflections, and considering the reflection at the film-substrate interface (n > ns, κs = 0), the reflectivity and transmissivity are related 20 by T = (1− R )2 exp( −4πκh / λ ). This relation together with the normal reflectance R, given by: (n − n )2 + κ 2 R = s s (2.8) 2 2 (n + ns ) + κ s offers the most convenient method for determining the optical constants from the R and T data on the same film. This method has the advantage that there is a small effect of the experimental error on the determination of the optical constants. In general, the transmittance measurements are more reliable than reflectance measurements because they depend less critically on the optical alignments and surface inhomogeneities.
An envelope method was introduced by Manifacier et al. [53] as a clever way to interpret optical transmission spectra for (n, κ) extraction. In the envelope method the transmittance spectrum is obtained after light traverses the film at normal incidence. The method is applicable to any transmission spectrum showing appreciable interference fringes. From the transmission spectrum, envelopes around the maxima and minima are constructed. Then, the envelopes around the maxima and minima are considered as continuous spectra vs wavelength, TM (λ) and Tm (λ), respectively as shown in Fig. 2.4. 21
Figure 2.4 Transmittance spectrum of crystalline AlN film deposited on silica glass together with the envelope curves of peaks (TM (λ)) and valleys (Tm (λ)) [29].
The refractive index n and absorption coefficient α at each wavelength, and film thickness h can be obtained with TM ,Tm, and ns using the following equations [29, 54]:
2 2 2 1 / 2 (2.9) n = (N + N − no ns ) with 2 2 no + ns TM − Tm N = + 2nons (2.10) 2 TM Tm and λ λ h = 1 2 2[n(λ1)λ2 − n(λ2 )λ1] (2.11)
4πκ α = (2.12) λ
22
1 (n +1)(n + n )( T − T ) α = − ln s M m (2.13) h (n −1)(n − ns )( TM + Tm ) where TM and Tm are the transmission maximum and minimum, respectively, on the envelope at a certain wavelength (see Fig.2.4), λ1 and λ2 are the wavelengths of the two adjacent maxima or two adjacent minima of transmittance, and n (λ1), n (λ2) are the corresponding refractive indices at these two wavelengths. This technique proved successful when used to determine the optical constants of a weakly absorbing thick film,
2 2 2 2 κ << (n-no) and κ << (n-ns) [55] and well-oscillating appearance, deposited on a non– absorbing substrate. As an example the optical constants of crystalline AlN thick film, deposited on quartz substrates by reactive radio frequency magnetron sputtering, were determined from the transmittance measurements using envelope curves, shown in
Figs.2.5 (a) and (b). κ Refractive index,n Extinction coefficient, (a) (b) 0.0 0.1 0.2 0.1 0.0 2 2.4 2.8 3.2 3.6 3.2 2.8 2.4 2
200 400 600 800 200 400 600 800 Wavelength (nm) Wavelength (nm)
Figure 2.5 Optical constants of crystalline AlN film deposited on quartz: (a) refractive index and (b) extinction coefficient [29].
23
On the other hand, the envelope method suffers a few pitfalls: Firstly, there is no ‘right’ way to construct the envelopes between interference extremes. Usually, they are constructed using parabolic interpolation, but this is in fact an arbitrary choice.
Secondly, the method cannot handle local absorption features if the film is not very thick. In addition, the accuracy of the method decreases with decreasing film thickness, since at lower film thickness, the interference extremes are spaced further apart and interpolation between these extremes becomes more difficult.
Also, the envelopes should ideally be constructed from the tangent points touching the transmission curve, not from the interference extremes: it is easy to see that, especially in a region where the transmission is changing fast, connecting the extremes will yield TM (λ) and Tm (λ) curves that are actually too close to each other. Furthermore, the method fails if the absorption in the film is so high that interference fringes are not visible and TM (λ) and Tm (λ) curves coincide.
One more method that has recently found much favor in the non-destructive optical characterization of thin films and bulk materials is the Spectroscopic Ellipsometry
(SE) technique. The SE is an excellent technique for the determination of optical properties of materials, as well as the analysis of single-layer and multilayer systems
[42]. Automated ellipsometers make it possible to determine simultaneously the complex refractive index and layer thickness. Hence the SE technique is superior to other optical methods. Also, it is very sensitive to optical path changes (physical thickness or refractive index changes) and surface irregularities such as surface roughness, interdiffusion and interlayer formations in multilayer thin film systems [37, 56]. It has 24 found a widespread use in the microelectronics industry [57] for the analysis of monolayer thick oxide layers on semiconductor wafers.
Ellipsometry measures the state of polarization of light reflected or transmitted from a material medium. When plane polarized light is reflected from an optically smooth material, it becomes, in general, elliptically polarized (Fig. 2.6). The nature of this elliptically polarized light carries information about the optical properties of the material. When change in the polarization state is measured as function of the wavelength
λ (or energy / frequency) of the electromagnetic radiation, the measurement is said to be spectroscopic. The response of materials to electromagnetic radiation i.e., the extent to which it is reflected, transmitted or absorbed by the material also depends on the angle at which the electromagnetic wave is incident. The measurement of the state of polarization as a function of λ and incidence angle is called variable angle spectroscopic ellipsometry
(VASE).
Figure 2.6 gives a schematic representation of this technique. It shows how the incident electric field can be resolved into two mutually perpendicular components rp and rs (the complex Fresnel’s reflection coefficients). The complex quantity rp / rs is split into the amplitude ratio (tan Ψ) and phase difference ∆. Thus, ellipsometry measures these parameters, from which the optical properties of materials are extracted. 25
r Ep
Incoming light wave Plane of incidence r Es Out coming r light wave Ep
r Es Ambient
Film 3
Film 2
Film 1
Substrate
Figure 2.6 Principles of Spectroscopic Ellipsometry technique (Reflected light = combination of five beams).
The data from an SE measurement is not useful by itself, but must be compared with an appropriate optical model in order to extract the useful information [58]. The optical model describes the interaction of light with the sample and it is constructed on the basis of the sample structure. In the process of this modeling, the number of layers must be selected and the optical functions of each layer must be determined. If the optical functions of the layer or the substrate are well known, then tabulated data can be used for the optical functions of materials while for materials with unknown optical functions, a parameterizable model must be employed (for more details see Chaps. 4 and 5). Careful modeling of the investigated layer system for the analysis of ellipsometric measurements gives good results. 26
2.4 Multilayer Stacks of Thin Films
A multilayer thin film is a finite combination of single layers having different optical constants and film thicknesses. Such films may be produced by different techniques, including vacuum evaporation and RF magnetron sputtering, and their thicknesses may be controlled with a very high accuracy. Multilayer thin films are successfully and widely used in antireflection coatings, very high reflectivity broadband mirrors, integrated optical lenses, microelectronics, and other branches of science and technology (even arts). The calculation of optical properties of a multilayer as a function of wavelength of incident light for any arbitrary angle of incidence has been an important problem in recent years.
The computation, design, and investigation of the optical properties of bilayer and multilayer systems are usually confronted with many serious problems. The design of multilayer stacks is today motivated by two goals. One is directed towards the design of optical coatings, filters and beam splitters which under appropriate conditions have enhanced or reduced reflectivity [59, 60]. The other relates to the study of the properties of supperlattices (which represent a distinct condensed state). Within the filter art, stacking is not restricted and only dictated by the required filter characteristic. But for superlattices stacking is restricted since by definition such lattices must strictly be repetitive arrays of identical periods. For example, a Fabry-Perot filter may use
LHLHLHHLHLH stacking, but a superlattice stacking must of necessity be of the
ABABAB....AB type. Thus, to settle a point of nomenclature that is not always 27 sufficiently appreciated in the literature, both examples are multilayer stacks, but only the latter is a superlattice.
A further difference between the two kinds of layer structures is that the individual layers in the filter stack, in general, exceed several hundred angstroms in thickness. But superlattice layer thicknesses should measure below 100 Å.
Dielectric or conductor multilayers suffer from three defects. The first, which is more of a complication than a fault, is that there is a variable change in phase associated with reflection. The second is that the high reflectance is obtained over a limited range of wavelengths [59]. The third, which is an ultimate serious, is that the interface imperfections i.e., interfacial roughness and diffuseness, have the effect of reducing the specular reflectance of an interface either by scattering light into non–specular directions in the case of interfacial roughness or by increasing the transmittance of the interface in the case of interfacial diffuseness. The relation between the specular reflectance R and the variance of the interfacial ( f ) is given by [61]
2 2 (2.14) R = Rο exp( − 4πf / d ′ )
where Ro is the reflectance of a multilayer mirror with perfectly smooth layers; d′ is the
period of the multilayer structure , d′ = (n′a)1 +(n′a)2 +.....+(n′a)i +....; i is the layer number; n′is the number of atoms, and a is the atomic spacing.
A recent study carried out by Gaponov et al. [61] studied the effects of interfacial roughness on the reflectance of Mo – Si multilayer mirrors for the wavelength range 125
– 450 Å. The multilayer structures were fabricated by electron-beam sputtering. It was 28 found that the number of periods increases with increasing interfacial roughness. Their study also showed that the reflectance R of such a system decreases as the interfacial roughness increases.
Skulina et al. carried out a study of the peak reflectance, R, versus the number of bilayers, N, of molybdenum/beryllium multilayer mirrors for normal incidence in the extreme ultraviolet region, and under various deposition conditions [59]. They found that, the measured reflectances do not increase as rapidly with the number of bilayers as the numerical model suggested (see Fig.2.7). This could possibly be due to the increasing value of interfacial roughness with increasing thickness. Also, they concluded that, more experiments were clearly necessary to optimize the deposition conditions to minimize interfacial roughness and optimize peak reflectance.
100
80
60
40 Reflectance (%) Reflectance Measured Fit Ideal 20
0 0 20406080100 Number of Bilayers
Figure 2.7 Peak normal-incidence reflectance for the Mo/Be system as a function of the number of bilayers N. The dashed curve is calculated for multilayers with ideal interfaces, and the continuous star curve results from a modeled interface roughness the continuous square curve represent measure data [59].
29
Madjid et al. [62] have studied the optical properties of Ag – SiO multilayer superlattice stacks. The optical properties of these stacks were studied by theoretical and experimental means. There was a discrepancy between the theoretical and experimental results. This discrepancy was ascribed to the interfacial layers and random variation in the layer thickness. As the number of periods was increased the discrepancy increased.
Fig.2.8 shows the reflectivity and transmissivity spectra of a 8 bilayer stack of Ag – SiO. Reflectivity Transmissivity
Wavelength × 10-4 (Ǻ)
Figure 2.8 The transmissivity and reflectivity spectra of a 8 bilayer stack of Ag – SiO. Solid line, experimental results; dashed-dotted line, computation [62].
A high reflectance can be obtained from a stack of quarter-wave dielectric layers of alternate high–and low–index. This is because the beams reflected from all the interfaces in the assembly are of equal phase when they reach the front surface where they combine constructively. Macleod [63] studied the reflectance R for normal incidence of alternating 30 quarter–wave layers of high–index (nH = 1.52) and low–index (nL = 1.38) dielectric materials on a transparent substrate (ns = 1.52).
Figure 2.9 shows a family of reflectance curves for different odd quarter–wave layers of glass/cryolite. It is noted that the high–reflection central zone is limited in extent. On the either side of the plateau, the reflectance falls abruptly to a low oscillatory value. The author found that the addition of extra layers did not affect the width of the zone of high reflectance, but increased the reflectance within it and the number of oscillations outside. Also, it was found that all curves were symmetric about a certain wavelength (λs= 460 nm) called the design wavelength.
Figure 2.9 Reflectance for normal incidence of different layers of glass/cryolite deposited on a glass as a function of the phase thickness (upper scale), or the wavelength (lower scale). The number of layers is shown as a parameter on the curves [63].
31
2.5 Properties of the Group III Nitride Thin Films
Group III nitrides are semiconductor materials that are expected to play a revolutionary role in modern optoelectronics and high power / temperature electronic devices. In the field of electronic devices, however, use of group III nitride devices are limited to high temperature applications because of their low carrier mobilities. Development of the technological base for growth and processing of these materials demands detailed knowledge of their physical properties. Much effort was expended in the growth and characterization of gallium nitride (GaN), aluminum nitride (AlN), and indium nitride
(InN) in the 1970’s, but significant difficulties were encountered in obtaining high- quality materials. GaN and InN had high n-type background carrier concentrations resulting from native defects commonly thought to be nitrogen vacancies.
Also no suitable substrate material with a reasonably closed lattice-match was available
[64]. Despite these difficulties and the lack of modern crystal growth and characterization techniques, much progress was made in understanding the physical properties of these materials.
In recent years effort has been made to examine various characteristics and parameters of these materials. The hexagonal wurtzite structure (Fig.2.10) of GaN, AlN, and InN form a continuous alloy system (InGaN, InAlN, and AlGaN) whose direct bandgaps range from 1.9 to 6.2 eV [65, 66]. Thus, the group III nitrides could potentially be fabricated into optical devices which are active at wavelengths in the whole of the visible region and extending well out into the ultraviolet (UV) range. Also, they form a complete series of ternary alloys which, in principle, makes available any band gap 32 within this range and the fact that they also generate efficient luminescence has been the main driving force for their recent technological development [67]. Moreover, like most wide bandgap semiconductors, the nitrides are expected to exhibit superior radiation hardness compared to conventional semiconductor materials such as Si and GaAs, making them attractive for space applications [13].
Figure 2.10 (a) Hexagonal close packed structure with four basis atoms, where 1 and 2 are anions, 3 and 4 are cations (b) The reciprocal lattice [68].
2.5.1 Aluminum Nitride Thin Films
Aluminum nitride (AlN) has received increasing attention from the material research community due to its unique properties such as high thermal conductivity, electrical isolation, chemical stability, and a wide band gap energy (~ 6.2 eV) [69 - 71]. It also has a reasonable thermal expansion coefficient match to Si and GaAs, and is resistant to high temperature, making AlN films promising materials not only for applications in 33 optoelectronics and microelectronic devices [13, 72], but also for surface passivation of semiconductors and insulators, and in surface acoustic wave device applications [73].
Table 2.2 shows some data on AlN film. One major area where these properties can find application is in the construction of high-power, high-temperature devices which can operate in harsh conditions. Another major interest in this insulator stems from its ability to form alloys with the group III nitrides such as AlGaN [11] and AlGaInN [82], allowing the fabrication of AlGaN/GaN based electronic and optical device applications.
For these applications, it is very important to study and understand the optical characteristics of AlN film over a wide range of wavelength. For example, knowledge of the dispersion of the refractive index of AlN, AlGaN and AlGaInN is crucial in the design of heterostructure lasers, wave guiding devices and optical filters using these compounds [11, 83]. Furthermore, accurate knowledge of the optical characteristics is essential for better understanding of the microstructure or electronic band structure of a material which is, in general, significant for the understanding of the bonding and forces at the atomic level [84]. 34
Table 2.2 Some data on AlN film.
Property Value References Bandgap ~ 6.2 eV [74, 75] Density 3.3 g/cm3 [70] Refractive index n 1.98 - 2.15 [80] Melting point > 2000o C [70 ] Drift velocity 1.7 × 107 cm/s [70] Resistivity ρ 1011 - 1013 Ω cm [81] Lattice constant a 3.11 Å [10, 68] Lattice constant c 4.98 Å [10, 68] Thermal conductivity 2.0 W/cm.K [77] Theoretical thermal conductivity 3.20 W/cm.K [76] -6 -1 -6 -1 Thermal expansion coefficients αa & αc 4.2 × 10 K & 5.3 ×10 K [78] Relative dielectric constant ε at long r 8.3 -11.5 [70,79] wavelength range
Several experimental and theoretical studies were performed on crystalline AlN films.
Solanki et al. [12] performed calculations of the anisotropic frequency dependent dielectric function of c-AlN film, using the linear muffin-tin orbital method with the atomic sphere approximation. Their calculations show that the anisotropy is very small confirming the fact that the band structure is almost isotropic.
X. Tang et al. [80] used an optical waveguide technique to study the dispersion properties of aluminum nitride in the (200-900) nm wavelength range. The refractive indices of AlN on sapphire substrate at four wavelengths, 632.8, 531.1, 514.5, and 488.0 nm and thicknesses of the waveguide material were determined. The refractive index of c-AlN films was found to be 1.98 - 2.25. The deposition of aluminum nitride on fused quartz, by dc diode reactive sputtering has been studied by in situ soft X-ray emission 35 spectroscopy [26]. The refractive index and the optical band gap for AlN thin films were determined by optical transmission and reflectivity measurements (RT method). The calculated refractive index in the visible range was found to be approximately 2.5 whereas the energy gap is 5.9 eV.
Vispute et al. [85] have grown high quality epitaxial AlN layers on sapphire
(0001) substrates by pulsed laser ablation. The optical transmission spectra of the AlN films deposited under various processing conditions was studied in the 150 - 400 nm wavelength range as shown in Fig. 2.11. The spectrum was highly transparent for films deposited at higher substrate temperature and it indicated considerable improvement in terms of transmission and absorption edge over films deposited at lower substrate temperature and energy density ED (curves b and c). Also, the fundamental absorption edge for the epitaxial AlN film was found to be 6.3 eV. The lower transmission and broad absorption edge of the film deposited at high ED were attributed to nitrogen deficiency, high density of grain boundaries and Al content in the film, all of which introduce defect states in the film. 36
Figure 2.11 UV-visible spectroscopy of the AlN films on Al2O3(0001) deposited at a TS of (a) 800 °C and (b) 750 °C with laser energy density of 2 J/cm2, and (c) 550 °C and laser energy density of 12 J/cm2 [85].
Guo et al. [86] measured the reflectance spectra of AlN on (0001) α-Al2O3 substrates in the photon energy range from 6 to 120 eV by synchrotron radiation. Also, the imaginary part of the dielectric constant, ε2, for AlN was determined by using the Kramers-Kroning analysis. Each of the transitions observed in the spectral dependence of the reflectance
(Fig.2.12 (a)) and the imaginary part of the dielectric function (Fig.2.12 (b)) were interpreted in a conventional way as the interband transitions from the valance to conduction bands.
37
(a) (b)
Figure 2.12 Spectral dependence of (a) the reflectivity of c-AlN film and (b) the imaginary parts of the dielectric function ε2 of the same film. The solid line indicates experimental data while the broken line represents the calculated data [86].
Few studies have been performed on AlN films by Spectroscopic Ellipsometry (SE). The dispersion of the pseudo dielectric function, < ε >, using SE was reported by Edwards et al. [27]. Jones et al. [28] combined spectroscopic ellipsometry with UV/VIS and VUV spectroscopy to directly determine the optical constants of AlN in the 6 - 44 eV range.
The index of refraction for single-crystal AlN was calculated by using the Kramers-
Kronig analysis. The optical functions of AlN thin films deposited on the quartz substrates by the reactive radio frequency magnetron sputtering were determined from the transmittance and reflectance spectra in the range 190 - 820 nm [29]. For analyses an inverse synthesis method was used and SE analysis was performed to confirm the accuracy of the inverse synthesis method. Refractive indices of AlN films were in the range of 1.95 - 2.05 at 633 nm and 2.26 - 2.38 at 250 nm. The extinction coefficients were small (< 5x10-4) and nearly constant at low energy (< 2 eV), but exhibited various dispersion features at 2.2 - 3.5 eV. 38
Recently, spectroscopic ellipsometry was also used to characterize AlN thick films deposited on SiO2/Si substrates by Joo et al. [30]. Refractive index and thickness of each layer were extracted from SE data analysis employing a multi-layer model and effective medium approximation. The refractive indices of the AlN films were in the range of 1.98
- 2.15 while the extinction coefficients, κ, were in the range of 10-3 to 0.1. It is known that the refractive index, n, of AlN at 632.8 nm varies depending on its structure order:
1.8 - 1.9 for amorphous films, 1.9 - 2.1 for polycrystalline films, and 2.1 -2.2 for epitaxial films [70, 87].
2.5.2 Indium Nitride Thin Films
Indium nitride (InN) with a hexagonal wurtzite crystal structure (Fig.2.10) has a direct band gap of approximation 1.9 eV [88]. It is recently gaining interest for its numerous potential applications in visible light optoelectronic devices, optical coatings, and various types of sensors. In addition, it has received much attention with regards to the properties of its alloys with GaN and AlN, permitting the fabrication of InGaN and InGaAlN based shortest-wavelength semiconductor laser diodes [89 - 91].
InN is a potential material for low cost solar cells with high efficiency. Yamamoto et al. [92] proposed InN for a top cell material of a two-junction tandem solar cell. The combination of band gap, 1.9 eV for InN and 1.1 eV for Si, is around the optimum to obtain a conversion efficiency of over 30%. The InN/Si system as a solar cell material has the advantage that it contains no poisonous elements such as arsenic and needs no toxic gases such as phosphine in the fabrication process. Qian et al. [93] recently showed 39 that InN as a good plasma filter material for the widely used GaSb and GaInAsSb photovoltaic cells in thermophotovoltic system. Table 2.3 shows some data on InN film.
Table 2.3 Some data on InN film.
Property Value Reference Bandgap 1.9 eV [88] Density 6.81g/cm3 [94] Melting point 1373 K [97 ] Resistivity ρ 1 – 10-3 Ω cm [98 ] Drift velocity 4.2 × 107 cm/s [35, 99] Lattice constant a 3.53 Å [96] Lattice constant c 5.69 Å [96] Thermal conductivity 0.45 Wcm-1 K-1 [95] -6 -1 -6 -1 Thermal expansion coefficients αa & αc 3.8 × 10 K & 2.9 ×10 K [96]
Relative dielectric constant εr at long 8.4 [70] wavelength range Refractive index n at the bandgap 2.71 ± 0.08 [35]
A precise knowledge of the optical characteristics over a wide wavelength range is particularly important in view of the use of InN films in the above mentioned applications. The study of the optical characteristics of InN is difficult because it lacks a suitable substrate material and is often grown with a high native defect concentration
[13]. Moreover, due to its rather poor thermal stability, InN cannot be grown at the high temperatures presently being used for the growth of high quality AlN and GaN crystals
[70]. The low dissociation temperature of InN makes it difficult to produce thin films.
Despite the problems inherent to InN growth, some work has been reported using a radio frequency (RF) reactive magnetron sputtering technique [31, 35, 64, and 100]. 40
Experimental studies have mostly been limited to photoluminescence and photo- reflectance measurements for the investigation of the position of the fundamental band gap; while work on the calculations of the optical properties of InN film in the wide spectral range has been rare. The optical constants of InN films grown on ZnO/α-Al2O3 by RF reactive magnetron sputtering were obtained by transmittance spectroscopy based on Kramers-Kronig analysis in the photon range 1.0 – 2.8 eV [100]. As shown in the
Fig.2.13, the extinction coefficient κ above the photon energy 1.5 eV is not zero indicateing that the film is not transparent in this region. Also, in the photon energy range
1.0 – 2.5 eV the refractive index is around 1.8. This low refractive index was caused by a lowered film density because of the many pits in the mosaic structure.
Figure 2.13 Optical constants and transmittance of the InN thin film grown on the ZnO/α-Al2O3 substrate [100].
41
The optical constants for the wavelength range of 400 – 850 nm and the band gap energy of InN films were determined from the measurements of transmittance and reflectance spectra (RT method) by Shamrell et al. [31]. The refractive index of the InN films varied from approximately 2.5 to 2.7 over 400 to 850 nm, with the maximum occurring at about
580 nm. The extinction coefficient varied smoothly from 0.05 to 0.5 while the direct bandgap was 1.92 eV.
The reflectance spectra in the photon energy range 2 to 130 eV were measured on
InN single crystals by using synchrotron radiation and the optical constants (n, κ) were determined on the basis of the Kramers-Kroning analysis [101]. This study presented the first experimental data on the optical constants on InN above the band gap energy (in the photon energy above 2.0 eV). The optical constants of InN thin films on (111) GaAs substrates grown by sputtering under different growth conditions have been investigated both theoretically and experimentally in the range 0.2-3.0 eV by Yang et al. [35].
Djurisic et al. [32] have proposed a theoretical model for fitting the optical dielectric functions, ε1 and ε2, of hexagonal InN in the range 2-10 eV. Foley and Tansley
[33] used the pseudopotential method to calculate the dielectric functions, ε, of InN films, while Christensen and Gorczyca [10] and Solanki et al. [34] used the muffintin orbital method. However, optical constants calculated in such a manner agree with experiment only in terms of the position of peaks in the absorption spectrum, while the shape of the dispersion curve is completely different (theoretical calculations produce sharper and stronger peaks than those observed in the experimental data). 42
In recent times, spectroscopic ellipsometry was used to characterize InN thin films deposited on Si (100) substrates at different temperatures by RF magnetron sputtering
[64]. The ellipsometric spectra of the InN films were measured as a function of photon energy (see Fig.2.14). The optical constants for the wavelength range of 410 – 1100 nm
(1.2 – 3 eV) were obtained by analyzing the measured ellipsometric spectra through the
Forouhi – Bloomer model. The refractive index of the films was in the range of 1.8 – 2.7, with the maximum occurring at about 660 nm while the extinction coefficient varied from 0.0 – 0.9 as shown in Figs.2.15 (a) and (b). The absorption edge was found around
1.85 – 1.90 for the samples under study. Moreover, this study showed that the thickness of InN film increases when the substrate temperature increases. This indicated that the deposition rate of InN film was related to the substrate temperature.
Figure 2.14 Analysis of the ellipsometric spectra of InN thin film grown on the Si (100) substrate. Experimental tan Ψ (dot line), experimental cos ∆ (solid line), calculated tan Ψ (dash line) and calculated cos ∆ (dash dot line) [64].
43
Thus far, there is no organized experimental and theoretical investigation of optical characteristics in crystalline or even amorphous (Al, In) N thin films over a wide range from the free carrier absorption region to the above band gap region.
Figure 2.15 (a) Refractive index n spectra of InN thin films on Si (100) substrates at different temperatures. (b) Extinction coefficient κ spectra of the same samples [64].
2.6 Amorphous Thin Films
Amorphous solids have been used as so-called glasses with various modifications for over 2000 years. The earliest glass objects were beads, which have been found in Egypt, and glass was also actually made in early Mesopotamia. However, scientific investigations on amorphous solids only started in this century, particularly, about seventy years ago. Amorphous solids have been intensely studied in modern times because of their technological importance and fundamental interest. These materials, for example, find potential application in fiber optic waveguides for communication 44 networks, computer-memory elements, solar cells, and transformer cores [23]. Because of the complexity of these systems, theoretical studies have relied mainly on numerical simulations [102].
The simplest way to describe the structure of amorphous solids like a-Si and a-
SiO2 consists of considering that the short-range order is preserved whereas the extended atomic network is random [102 - 104], as shown in Fig. 2.16. The Figure shows that the structure of vitreous SiO2 can be described by a continuous random network in which the
SiO4 tetrahedra joined at the corners are connected with each other to form a glassy solid.
Therefore, the structure of the amorphous solids is characterized as an irregular arrangement of atoms in contrast to crystalline solids whose structure has a periodic array of atoms. This fundamental difference is evident at a glance in Fig. 2.17 (a).
Figure 2.16 The continuous random network structure of amorphous silicon dioxide (a-SiO2), notice that each Si atom (gold sphere) has 4 bonds, and each oxygen atom (red sphere) has 2 bonds [104]. 45
c-Si (001) a-SiO2
(a) (b)
Figure 2.17 (a) Side view of the a-SiO2 /c-Si (001) interface structure, the arrows indicate the locations of the Si-O-Si bridges. (b) Top view of the stripe structure of the oxygen bridge bonds at the interface [104]
In brief, amorphous semiconductors are noncrystalline and lack long range periodic ordering of their constituent atoms, but they do have a local order on the atomic scale.
This short range order is directly responsible for observable semiconductor properties such as optical absorption edges and activated electrical conductivities [105]. Moreover, the amorphous solids, which are solid materials exhibiting an amorphous structure, can be produced in thin film form by evaporation, sputtering and chemical vapor deposition techniques (see Chap. 2, Sec. 1), under the appropriate conditions. A low substrate temperature promotes an amorphous structure. The optical properties of such films can be considerably different from those of bulk material, and also depend strongly on preparation conditions. 46
2.6.1 Optical Properties of Amorphous Thin Films
The scientific and technological interest in amorphous semiconductors stems mainly from their optical and electrical properties. Investigations of the electrical properties of amorphous semiconductors rely heavily on the procedures and analyses used in studies of crystalline semiconductors like Si and the III-V compounds. However, exploration of the electrical properties of amorphous materials encounters many difficulties [106] while the optical properties are easier to interpret.
The optical properties such as the dielectric functions and absorption coefficient reflect essentially the density of states, more or less modified by the transition probabilities between the states. Therefore, the analysis of optical spectra is one of the most effective tools for understanding the electronic structure of amorphous solids. Pierce et al. [107] carried out photoemission measurements from and optical studies of amorphous Si, a-Si, to study the electronic structure of the material. They measured the energy distribution curves of a-Si by using a conventional retarding-field energy analyzer and a higher-resolution screened-emitter analyzer. When the energy distribution curves from crystalline silicon, c-Si, are examined, one finds variations in the position and strength of the structure as a result of the conservation of wave vector k in the crystal.
However, in amorphous materials no such variations had been found in the energy distribution curves. This is not unexpected because the structure of amorphous solids is characterized as an irregular arrangement of atoms, in contrast with crystalline solids whose structure has a periodic array of atoms (see Figs. 2.16 and 2.17), and it leaves the crystalline momentum ћk undefined. Thus the simplification of Bloch theorem and the 47 concept of wave vector k as a good quantum number in crystalline materials are no longer valid in amorphous materials.
The absence of k-conservation in amorphous semiconductors does not imply that the density of electronic states DOS, ρ (E), is not a meaningful concept [108]. In fact, techniques such as photoemission showed [107 - 109] that, just as for crystals, the electronic states fall into bands separated by energy gaps. A schematic illustration of the
DOS, ρ (E), for an amorphous semiconductor compared to that of a crystal is shown in
Fig.2.18. As shown in the figure the electronic states of amorphous semiconductors have different features than that of crystalline semiconductors: (1) the absence of sharp band edges and tailing of the DOS into the gap (quasi-gap); (2) The states in the gap are localized in energy regions where the DOS is much smaller than in the valence and conduction bands. The localization of states in energy regions may be associated with a particular defect or cluster of defects. Also, charge carriers (electrons) in these states have very low mobility, about zero, at low temperatures [110].
48
Crystalline ρ (E)
Valence band Gap Conduction band
E
Amorphous ρ (E)
Mobility gap Valence band Conduction band
E Extended Localized Extended states states states
Ev Ec ∆Ev ∆Ec
Figure 2.18 Schematic density of states as a function of energy for a crystalline and an amorphous semiconductor. ∆Ev and ∆Ec are band tails due to disorder.
The electronic states in the body of the valence and conduction bands are probably extended throughout the material. These states will have much higher mobility than the localized gap states, although their mobilities will be very much smaller than the mobilities of free carriers in crystalline semiconductors [105, 110]. The optical transition between these states of relatively high mobility and the states of low mobility is called the mobility edge. The mobility edges define a mobility gap. To a first approximation the optical transitions in amorphous semiconductors are described by the non-direct transition model [107] which ignores k as a quantum number and requires only conservation of energy. 49
In general, the optical spectra of amorphous materials differ mainly in three aspects from those of crystalline materials:
1. Since in a disordered structure there is no requirement for momentum
conservation, the fine structure of the crystalline spectra disappears.
2. The imaginary part of the dielectric function ε2 (ω), which depicts absorption
processes, peaks at lower energy.
3. The peak height of ε2 (ω) is decreased and depends on the details on film
preparation.
A typical example of the difference in optical properties between crystalline and amorphous solids is shown in Fig. 2.19. It can be seen that the imaginary part of the dielectric constant ε2 for a-Si is situated approximately in the same energy range as that of c-Si, but it lacks the sharp structure characteristic of crystals. Also the ε2 of a-Si shows more strength than the crystal at lower energies and peaks at 3.5 eV whereas the center of the strength of the crystalline ε2 curve is at approximately 4 eV. This is easily understood: with the loss of the long-range order, the k–vector ceases to be a good quantum number and is not at all conserved during optical transitions. Moreover, Ewald et al. [111] found
that the height and peak position of the ε2 and reflectance spectra of a-Si depend on the preparation parameters. In films evaporated slowly the maximum value ε2max = 25 is found at 3.7 eV while ε2max = 20 at 3.5 eV in the fast evaporation case. They also found that the deposition temperature has little effect on the peak position of ε2 (ω) but has a considerable effect on the peak height.
50
Figure 2.19 Imaginary part of the dielectric constant ε2 for amorphous Si compared to ε2 of the crystal [107].
2.6.2 Some Potential Advantages of Amorphous Thin Solid Films
Amorphous semiconductors have many present-day applications. One important aspect for the future of amorphous semiconductors and their alloys is that a lack of long range ordering in the atomic network relaxes the k-selection rules for the optical transitions.
This gives rise to a large optical absorption coefficient and a high luminescence efficiency, which can be used in light emitting devices (one of the often-named
“technologies of the future” in optoelectronics – an industry which generated revenues of over $20 billion in 2002 in North America alone [112]).
When an amorphous solid can be used in place of a crystalline one in an application calling for large area sheets or thin films, it is generally advantageous to do 51 this and thereby avoid the problems associated with polycrstallinity or the expense of preparing large signal crystals. Thus it would, for example, be too expensive to fabricate large windows out of crystalline SiO2 (quartz) while it is extremely reasonable to do so using SiO2 – based silicate glasses.
Technically, it is easier to prepare amorphous films than crystalline films because
of fewer requirements on the amorphous film quality. This may help to lower the cost on
equipment and production atmosphere. The growth of the crystalline nitride films such as silicon nitride and group III nitrides requires high temperature (500 – 1400) oC [113 -
115], while amorphous nitrides can be prepared at low temperature, even at room temperature. In addition, the growth of c-InN is the most difficult among the III-nitrides because the equilibrium vapor pressure of nitrogen over the c-InN is several orders higher than c-AlN and c-GaN as shown in the Fig.2.20. Because of the low InN dissociation temperature and high equilibrium N2 vapor pressure over the InN film [116, 117]; the preparation of InN film requires a low growth temperature. The low temperature processes are less expensive due to reduced energy demands during manufacturing and
less stringent substrate requirements.
Of course, besides the above general advantages of amorphous nitride over crystalline thin films, it is also true that amorphous nitride thin film growth is substrate independent. Most crystalline nitride film growth has very strict requirement on substrate.
The highest quality substrate materials, such as sapphire and silicon carbide are
commonly used [118, 119]. These materials can be expensive and limited in size. 52
Figure 2.20 Melting points of III-nitrides and equilibrium N2 pressures over the III-N(s) [116].
Additionally, the lattice mismatch to these substrates is quite large and even the highest
quality substrate materials contain a high concentration of lattice mismatch related
structure defects. Silicon substrates have been successfully used for nitride growth [94,
95], because they have the required hexagonal surface symmetry. The low cost, large size
and availability of this substrate makes it an attractive alternative to sapphire and silicon
carbide despite the well known difficulties in this material system. These difficulties are
caused by a lattice mismatch of 17% and by thermal expansion coefficient 53 incompatibility [95]. Hence, for amorphous nitride films, there is no such limitation on a substrate.
In summary, the amorphous nitride as a large area thin film semiconductor with doping, low cost, and low temperature fabrication capability has not only given a new impetus to the field of optoelectronic devices such as solar cells, flat panel displays, printer heads, but it now also forms the basis for large area microelectronics.
In order to meet the demands arising from the rapidly growing field of information processing it is very important to understand the fundamental properties such as optical characteristics of these materials over a wide wavelength range. This is the major motivation of this research. 54
CHAPTER THREE
FUNDAMENTAL THEROY
The optical characteristics of a material can be interpreted as the interaction of light with matter. Our primary interest in this chapter is to present results that are essential for the ellipsometric measurement on film-coated substrates. In this chapter the basic concepts of classical electromagnetic theory related to the optical properties and optical constants of thin films are briefly presented in Section 3.1. The simple case of planner interface between two homogeneous optically isotropic media and progress to the system that involves one or more thin films (bilayer and multilayer thin film systems) are discussed in Sections 3.2 and 3.3. Section 3.4 summarizes a simple and direct method for the determination of the optical constants of very thin films. Finally, in Section 3.5 we address an important definition of the spectroscopic ellipsometry parameters Ψ and ∆. In addition, the expressions for the amplitude Ψ and phase changes ∆ on reflection for transparent and absorbing thin films are presented.
3.1 Classical Electromagnetic Theory of Thin Film Optics
Thin film optics may be developed more or less completely through several independent approaches [2, 120]. The most straightforward approach is derived from classical electromagnetic theory where the problem of the determination of the reflectance R, transmittance T, and absorptance A of a single film reduces basically to the solution of a 55 boundary value problem that involves the solution of Maxwell’s equations [121]. This leads to the determination of the steady state amplitudes of the electric and magnetic field vectors at the interfaces of a single film on a substrate [48, 122].
The present discussion is planned to rely on the findings of this method. The discussion is based on the assumptions that the medium is homogeneous and isotropic with a dielectric constant ε, permeability µ, conductivity σ, and thickness h.
After handling Maxwell’s equations, worked side by side with the appropriate material equations [121], it can be shown that Maxwell’s damped wave equation of the electric field of an electromagnetic wave in a conducting medium is given by [48]
4πµσ ∂ Ε εµ ∂ 2 Ε ∇ 2 Ε − ( ) − ( ) = 0 (3.1) c 2 ∂ t c 2 ∂ t 2 where the second term implies that the wave suffers progressive attenuation through the medium; c is the speed of light in vacuum. The simplest solution to equation (3.1) is a plane time harmonic wave [123]:
→ → ~ (3.2) i(K .r − ωt) E = E0 e → Here, E0 is the amplitude of the electric vector of the incident wave, ω is the angular
~ frequency of the incident light, and Κ is the complex propagation vector. If this solution
is substituted in equation (3.1), we obtain
~ ω 2 µ 4 πσ Κ 2 = ( ε + i ) (3.3) c 2 ω We may write
~ ~ ~ ω 2 µ Κ 2 = Κ .Κ = ε~ (3.4) c 2 56 with ε~ representing the complex dielectric constant, defined as 4πσ ε~ = ε + iε = (ε + i ) (3.5) 1 2 ω the frequency dependent conductivity σ describes the response of the medium to the
driving field. We may also introduce a complex refractive index ñ in the form
n~ = n + i κ (3.6)
where n is the real part of the refractive index which defines the phase velocity (vp) of light in a material, vp = c / n ; c is the speed of light in the material and κ is the imaginary
part called the extinction coefficient, (also called the attenuation index); this index
describes the attenuation of a light wave in matter. Hence, the optical behavior of a
material is generally utilized to determine its n and κ, known as the optical constants. The
optical constants define how light interacts with a material.
We can thus suggest the general relation
~ ω ω K = n~ = (n + iκ ) (3.7) c c The quantities n and κ can be determined in terms of the material constants ε, µ and σ by squaring equ. (3.7) and equating the real and imaginary parts to those in equ. (3.3). This gives the following relations:
n 2 − κ 2 ε = (3.8) µ
2nκω nκν σ = = (3.9) 4πµ µ 57
Where σ is the static dc electrical conductivity [120] and ν is the frequency of the incident light. Upon solving equations (3.8) and (3.9) simultaneously, we obtain the following expressions for the optical constants n and κ
µ 1 (3.10) n = ([ε 2 + 4(σ /ν )2 ] 2 + ε ) 2 and
µ 1 κ = ([ε 2 + 4(σ /ν )2 ] 2 − ε ) (3.11) 2 ~ Therefore, it follows from the definition of Κ and by using equations (3.8) and (3.9) that the real and imaginary parts of the dielectric functions ε~ (equ. (3.5)) can be expressed as:
2 2 n − κ 2nκ 4πσ (3.12) ε = ε2 = = 1 µ µ ω
Hence, the optical constants in terms of the dielectric functions ε1 and ε2 take the form:
µ 1 n = [(ε 2 + ε 2 ) 2 + ε ] (3.13) 2 1 2 1 and µ 1 κ = [(ε 2 + ε 2 ) 2 − ε ] (3.14) 2 1 2 1 where ε1 and ε2 are given by equation (3.12). The solution of the wave equation (3.2) in the conducting medium for a wave propagating in the z- direction takes the form:
~ − (ωκ z ) iω [( nz ) − t ] (3.15) r r i ( K z − ω t ) r c c E = E o e = E o e e
This form represents a wave traveling in the z-direction with a velocity c/n that is
r 2 attenuated by exp ( −ω κ z / c ). The attenuation of the intensity ( I ∝ E ) is thus given by exp ( − 2ω κ z / c ). We can define an absorption coefficient α by the inverse of the 58 distance along the direction of propagation in which the intensity of a wave falls 1/e times its original value [1]. From Beer-Lambert law, α is expressed in terms of the intensity of the incident light Io and the intensity of transmitted light Ih after passing through a thickness h [124] as
−α h I h = I o e (3.16)
Hence, from equations (3.15) and (3.16) we can show that the absorption coefficient α is given by [125]
2 ωκ 4 πκ (3.17) α = = c λ o
λo being the wavelength in vacuum. This helps to define another physical quantity called the skin depth d as the distance at which the intensity of the incident light drops to 1/e of its value at the surface [48]. In terms of the absorption coefficient the skin depth takes the form
1 λ d = = o (3.18) α 4 πκ
This quantity is usually a very small fraction of the wavelength. This shows why Ag, Au and Cu are the best conductors and it also shows why they are of a high opacity. 59
3.2 Reflection and Transmission Coefficients of Plane Waves at a
Single Interface.
The derivation of the reflection and transmission coefficients of electromagnetic radiations at a plane interface between two homogeneous isotropic media is a first step towards the derivation of the optical properties of a material. Since these coefficients are dealt with extensively in common standard text books on optics [122, 126], they will be overlooked here except for the final formula.
When a monochromatic plane wave is incident at an angle θi at the n1/n2 interface separating media of refractive indices n1 and n2, it will be divided between a reflected wave at an angle θi in the same medium and a transmitted wave at an angle θt in the adjacent medium. By applying the boundary conditions at a surface of discontinuity which requires that the tangential components of the electric and magnetic induction vectors for a plane wave in oblique incidence are continuous [121], the ratios of reflected to incident (r1) and transmitted to incident (t1) amplitudes assume the following relations
[120, 127]
⎡ Er ⎤ n1 cos θi − n2 cos θt (3.19) r = ⎢ ⎥ = 1s E n cos θ + n cos θ ⎣ i ⎦TE 1 i 2 t
⎡ Er ⎤ n2 cosθi − n1 cosθt r = ⎢ ⎥ = (3.20) 1p E n cosθ + n cosθ ⎣ i ⎦TM 2 i 1 t
⎡ Et ⎤ 2n1 cosθi t = ⎢ ⎥ = (3.21) 1s E n cosθ + n cosθ ⎣ i ⎦TE 1 i 2 t
60
⎡ Et ⎤ 2n1 cosθi t = ⎢ ⎥ = (3.22) 1p E n cosθ + n cosθ ⎣ i ⎦TM 2 i 1 t where n1 and n2 are the refractive indices of the two media; θi and θt are the angles of incidence and transmittance, respectively. These ratios are called the Fresnel’s equations for plane-polarized components parallel (p) and perpendicular (s) to the plane of incidence (see
Fig.2.6). The Fresnel’s equations (3.19) – (3.22) describe the action of the sample on the electric field components (p and s) to the plane of incidence. The p-reflectance Rp is defined as the intensity reflectance (intensity of reflected beam divided by the intensity of the incident beam) measured with the incident beam linearly polarized in the p-plane, and an analogous definition holds for the s- reflectance Rs. The p- and s- reflectances may be calculated from the Fresnel reflection coefficients as follows
2 ∗ R p = r1p = r1p .r1p (3. 23)
2 ∗ (3. 24) Rs = r1s = r1s .r1s
Similarly, the p- and s- transmittance (defined as the ratio of transmitted to incident energy) may be calculated from the Fresnel transmission coefficients as follows
n2 cosθ2 2 Tp = t1p (3. 25) n1cosθ1
n2 cosθ2 2 Ts = t1s (3. 26) n1 cosθ1
Generally, when unpolarized radiation is incident upon a film at non-normal incidence, both the reflected and transmitted light becomes partially plane polarized. If the incident radiation is elliptically polarized, the degree of elliptical polarization of both the reflected and transmitted radiation is altered. This is because not only is the reflectance and 61 transmittance different in the two planes, but also because the phase shift on reflectance is different for the two planes. If the radiation which is obliquely incident on a film is initially unpolarized then the polarizing effect of the film can be neglected and the average reflectance and average transmittance is the mean of the two polarizations (i.e.
Rave = (Rp + Rs) / 2, and Tave = (Tp + Ts) / 2).
For normal incidence the Fresnel’s equations (3.19-22) are simplified since the distinction between planes of polarization vanishes. In this case, Fresnel’s equations reduce to
n1 − n2 n2 − n1 (3. 27) r1s = r1 p = n1 + n2 n1 + n2
2n1 (3.28) t1s =t1p = n1 +n2
The reflectivity R and transmissivity T for normal incidence at n1/n2 interface are given by:
2 2 2 (3.29) R=Rs = Rp = r1s = r1p =[](n1 −n2)/(n1 +n2) and n 2 n 2 4n n T = T = T = 2 t = 2 t = 1 2 s P 1s 1p 2 (3.30) n1 n1 (n1 + n2 ) respectively.
All the previous values can either be real as in the case of dielectrics or complex for absorbing media. 62
3.2.1 Brewster and Pseudo - Brewster Angles
If both media are dielectric in nature (i.e., κ1 = κ2 = 0) then the p-polarized reflectance,
Rp, becomes zero at a specific angle of incidence θB:
−1 n2 (3.31) θB = tan ( ) n1
This can be readily proved by using the Snell’s law [122] and equation (3.20) (r1p = 0 at
θi = θB). The significance of this is that when unpolarized light makes a reflection with this specific angle of incidence, the light polarized parallel to the plane of incidence is never reflected and thus it is all transmitted [120]. The resulting reflected light is polarized perpendicular to the plane of incidence. This specific angle is called the
Brewster angle or the polarizing angle [120].
When the reflecting surface is not a dielectric material (i.e., κ ≠ 0), the situation becomes more complicated. The Fresnel’s reflection coefficients (equations (3.19) and
(3.20)) are complex numbers, so the polarized reflectance does not go to zero at any angle of incidence. However, the angle for which Rp is minimum is called the principal
(or pseudo- Brewster) angle of incidence.
3.3 Theory of a Single Thin Film on a Transparent Substrate
There are several methods available for the determination of the optical properties R, T, and A of single or multi thin film systems. Many of these entail rather heavy computations and are now greatly assisted by the availability of electronic computers [62,
127]. Such methods are the Airy’s summation method, Rouards’ method, the boundary conditions method, transfer matrix method … etc [2, 48, and 120]. 63
In this work the discussion will be concerned with the most important methods
(summation and transfer matrix method). Since these methods have become well established, again the detailed treatment of any of them will be overlooked here except for the final forms and expressions.
3.3.1 Reflectance and Transmittance of Dielectric Films
For a parallel–sided, homogeneous isotropic film of refractive index n between media of indices no and ns (Fig.3.1), the amplitudes of the reflected and transmitted waves off and through a thin film are readily given by Airy’s method [129] as follows:
Combining with the Fresnel coefficients, the reflected amplitude is given by
t t ' r e−2iδ (3.32) r = r + t t ' r e− 2iδ − t t ' r r 2e− 4iδ + .... = r + 1 1 2 1 1 1 2 1 1 1 2 1 − 2iδ 1 + r1r2e The transmitted amplitude is given by t t e−iδ t = t t e−iδ − t t r′r e−3iδ + t t r 2r 2e−5iδ + .... = 1 2 (3.33) 1 2 1 2 1 2 1 2 1 2 −2iδ 1 + r1r2e From the conservation of energy we can obtain
2 ' 1− r1 (3.34) t1 = t1 So equations (3.32) and (3.33) become
r p , s + r p , s e −2iδ r = 1 2 p , s p , s p , s − 2iδ (3.35) 1 + r1 r2 e and t p , s t p , s e − iδ t = 1 2 p , s p , s p , s − 2 iδ (3.36) 1 + r1 r2 e 64
Equations (3.35) and (3.36) are generally valid. For a non-normal incident each takes two possible forms, depending on the state of polarization of the incident light (using the appropriate Fresnel coefficients for each kind of polarization). In the above equations r1, t1 and r2, t2 are the respective Fresnel’s reflection and transmission coefficients for the first and second boundaries (Fig.3.1). These coefficients are given by the Fresnels’ equations (3.19) – (3. 22) and δ is the phase change of the beam that occurs upon traversing in the film; this phase shift is given by [48]:
2π δ = nh cos θ t (3.37) λ0
Here, λo is the wavelength of the incident light in vacuum, and h is the film thickness.
Io
θo t1t2′ r2 r1 no
−t1r2′r2 h θt n t1 t1r2
ns
θs
t1t2 −t1t2r2r1′
Figure 3.1 A schematic diagram of the path of a light beam within the film and substrate. The r’s and t’s are the Fresnel’s coefficients of reflection and transmission, respectively.
Hence, we may state the reflectance R and transmittance T of a dielectric film as [48,
120] 65
2 2 2 R = r = r1 + r2 + 2r1r2 cos 2δ (3.38) 2 2 1 + r1 r2 + 2r1r2 cos 2δ and 2 2 n cos θ 2 n cos θ ⎡ t t ⎤ T = s s t = s s 1 2 (3.39) ⎢ 2 2 ⎥ no cos θ o no cos θ o ⎣⎢1 + r1 r2 + 2r1r2 cos 2δ ⎦⎥ receptively. Here, ns is the refractive index of the substrate. The condition should be such that (R + T = 1), which means that no energy is absorbed by the film.
We notice that for normal incidence,θ o = θ s = 0 , the general expressions for R
(equ.3.38) and T (equ.3.39) become more compact and take the forms [2, 120]
(n2 + n2 )(n2 + n2 ) − 4n n2n + (n2 − n2 )(n2 − n2 ) cos 2δ R = 0 s 0 s 0 s (3.40) 2 2 2 2 2 2 2 2 2 (n0 + n )(n + ns ) + 4n0n ns + (n0 − n )(n − ns ) cos 2δ and 8n n2n T = 0 s 2 2 2 2 2 2 2 2 2 (3.41) (n0 + n )(n + ns ) + 4n0n ns + (n0 − n )(n − ns )cos2δ
respectively. In the above formulae n0 is the refractive index of air, n is the refractive index of the thin film, ns is the refractive index of the substrate, and δ is the phase difference. It is noted from equations (3.40) and (3.41) that the normal reflectance and transmittance do not change when δ is replaced by δ+π. This means that R and T are periodic functions (oscillatory variation) of the optical thickness with period equal to λο
[48]. 66
3.3.2 Reflectance and Transmittance of Absorbing Films
When a light beam (electromagnetic wave) falls on an absorbing film supported by a transparent substrate, part of that beam is transmitted, another part is absorbed within the film and the rest is reflected. The transmitted beam is subject to multiple reflections within the film as shown in Fig.3.2. The equations of propagation of light in a transparent medium (Sec.3.4.1) for the case of an absorbing medium can be obtained by replacing the real refractive index by a complex term. The Fresnel reflection and transmission coefficients become complex (equs.3.19 – 3.22). Replacing n with n~ = n + iκ , we notice that
n sin θ sinθ = 0 0 (3.42) t n + iκ or
~ ~ 2 (3.43) cos θ t = 1 − (n0 sin θ 0 / n )
So θt is also complex and will not represent the angle of transmission (except for the
special case θ 0 = θ s = θ t = 0 , for which the Fresnel reflection coefficients can be easily found). For other than normal incidence [130 - 132], exact expressions for the reflectance and transmittance by using the Airy’s summation method are somewhat complicated [2,
48]. However, we will discuss the absorbing medium case by using the transfer matrix method (Sec. 3.4.3). 67
I0 Rf Tb
n0 Air
~ n Film
n s T f I o R b Substrate
n0 Air
Figure 3.2 A ray diagram of a light beam with the film and substrate. Tf , Rf , and Tb , Rb are the respective transmittance and reflectance from the front and back sides of the film.
3.3.3 Theory of Multilayer Thin Films
When an optical system consists of a stack of thin films (multiple layers) the method of summation of the amplitudes becomes quite difficult to apply for a multilayer structure
(Fig.3.3). Explicit single expressions for the polarized reflectance and polarized
transmittance of multilayer films are cumbersome. As a matter of fact, the most
conventional method in this case is the transfer matrix method derived by Abele’s in
1953. Applying the transfer matrix technique [48, 62, and 128] each primary layer is represented by a 2×2 characteristic matrix M, the element of which contain all pertinent
geometrical and optical parameters. 68
n~ n~ n~ ~ ~ ~ 1 2 3 n j n nS θo N
θs
h1 h2 h3 hj hN h
Figure 3.3 A schematic diagram of a multilayer system of N-absorbing layers deposited on a substrate of refractive index ns, and surrounded by an air of index n0.
For a TE wave (electric vector perpendicular to the plane of incidence, s-polarized (Fig.
2.6)) an absorbing thin film can be characterized by the matrix (Fig.3.3)
⎡m m ⎤ ⎡ ~ i ~⎤ 11 12 cosδ − sinδ (3.44) ⎢ ⎥ ⎢ ~p ⎥ M=⎢ ⎥=⎢ ⎥ ⎢ ⎥ ⎢ ~ ~ ~ ⎥ ⎣m21 m22⎦ ⎣−ipsinδ cosδ ⎦ where
~ ~ ~ ~ ~ ~ (3.45) p = n cos θ t δ = (2π )n h cos θt / λ0 ~ ~ Here, n is the complex refractive index of the layer, θ t is the complex transmitted angle
~ (equ.(3.42)), and δ is the complex phase difference. From the matrix equation (3.44) two expressions can be derived. The square of their moduli gives the s-polarized reflectance
Rs and the s-polarized transmittance Ts as
2 (3.46) ⎡ p o (m11 + p s m12 ) − (m 21 + p s m 22 ) ⎤ R s = ⎢ ⎥ ⎣ p o (m11 + p s m12 ) + (m 21 + p s m 22 ) ⎦ and
2 ⎡ ⎤ p s 2 po (3.47) Ts = ⎢ ⎥ po ⎣ po (m11 + p s m12 ) + (m 21 + p s m 22 ) ⎦ 69 respectively. Here, m11, m12, m21, m22 are the elements of the transfer matrix which are given by equation (3.44) and
(3.48) po = no cos θ o p s = ns cos θ s
The corresponding formulae for a TM wave (electric vector parallel to the plane of incidence, p-polarized (see Fig.2.6)) are immediately obtained from equation (3.44) -
~ (3.47) on replacing the quantities p , Po, and Ps by:
~ ~ ~ q = cos θ / n q = cos θ / n (3.49) q = cos θ t / n o o o s s s In a multilayer structure of N-layers (Fig.3.3) a layer j can be characterized by a transfer matrix ⎡ ⎤ ⎢m j m j ⎥ (3.50) ⎢ 11 12 ⎥ M j (hj ) = ⎢ ⎥ ⎢ j j ⎥ ⎢m21 m22 ⎥ ⎣ ⎦
⎡ 2π ~ ~ i 2π ~ ~ ⎤ (3.51) cos( n jh j cosθ jt ) − ~ sin( n jh j cosθ jt ) ⎢ λ ~ λ ⎥ ⎢ 0 n j cosθ jt 0 ⎥
M (h ) = ⎢ ⎥ j j ⎢ ⎥ ⎢ ~ ~ 2π ~ ~ 2π ~ ~ ⎥ ⎢− in j cosθ jt sin( n jh j cosθ jt ) cos( n jh j cosθ jt ) ⎥ λ λ ⎣⎢ 0 0 ⎦⎥
Here, j is the subscript which denotes the alternate layers that form the multilayer structure; it is an integer in some fixed range1 ≤ j ≤ N . The above result can immediately be extended to the general case of an assembly of N-layers obtained from the product of the individual matrices taken in the correct order, i.e 70
N ⎡M11 M12 ⎤ (3.52) M (Nh) = ∏M j (hj ) = ⎢ ⎥ j=1 ⎣M21 M22⎦
It is noted from equ. (3.51) that we regard the medium as consisting of a very large number of thin films of thicknesses h1, h2, ..., hj , ... hN (see Fig.3.3). If the multilayer structure is composed of N identical periods, each of a thickness h (= h1 + h2) the product of all individual matrices takes the form
N ⎡ 2 ⎤ ⎡Μ11 Μ12 ⎤ M (Nh) = ⎢ ∏M j (hj )⎥ = ⎢ ⎥ (3.53) ⎣⎢ j=1 ⎦⎥ ⎣Μ21 Μ22⎦
After computing the elements of the transfer matrix of the whole structure and substituting into equations (3.46) and (3.47) Rs and Ts of all the layers can be calculated.
Hence, the transfer matrix method is convenient to use when the optical properties of a multilayer system are of concern. An extremely efficient method for calculating the optical properties, R, T, and A, and their first order derivatives with respect to wavelength for thin films and multilayers was implemented through a computational program [62,
128].
71
3.4 Determination of the Optical Constants of Thin Films
The optical behavior of a material is generally utilized to determine its optical constants n and κ. The optical constants and thickness of thin films are determined by a variety of methods (see Chap.2, Sec.4). We will refer to a simple direct method derived by Wolter
[133] for very thin films. In this method the optical constants n and κ and a film thickness h can be simultaneously determined from the measurements of the transmissivity T, and the reflectivities, Rf and Rb, of the film. The resulting expressions are [133]
n λ (1 − R − T ) n λ A (3.54) ε = 2nk = s o f = s o f 2 2πh T 2πh T or
n λ (1 − R − T ) n λ A (3.55) ε = 2nk = o o b = o o b 2 2πh T 2πh T
2 2 ε 1 = n − k 2 2 n + ns λ o = o ± × 2 2π h
(R + R ) A π 2h 2 (3.56) 2n n f b − (n − n )2 − (n )2 + (n − n ) 2 o s o s s 2 o s T T λo
Rf , Af and Rb , Ab , are the reflectivity and absorptivity of the film for light incident from the air, and the substrate sides, respectively as shown in Fig.3.2. It is noted from equ.
(3.54) and (3.55) that these equations are valid for the case where the thickness of the film h deposited on a transparent media is much smaller than λo, and ns Af = n0 Ab . Also, we notice from equ. (3.56) that there are two values for ε1, but only one of those values can have a physical significance. By employing a feed back method we can choose 72 between the two solutions such that the values of n and κ that are obtained yield the experimentally recorded values of R and T. Solving the equations (3.54) and (3.56)for n and κ we obtain:
1 1 ⎡ 2 ⎤ (3.57) n = ⎢ (ε 1 + ε 2 ) + ε 1 ⎥ 2 ⎣ ⎦
1 1 ⎡ 2 ⎤ (3.58) κ = ⎢ (ε 1 + ε 2 ) − ε 1 ⎥ 2 ⎣ ⎦
3.5 Spectroscopic Ellipsometry and Optical Constants
A plane polarized light reflected from an absorbing surface at non-normal incidence assumes elliptical polarization (Fig.2.6). The ellipticity (the ratio of minor to major axis) of the reflected beam is determined by the relative phase change ∆ (defined as the difference between parallel and perpendicular components of the incident wave that occurs upon reflection) and the relative amplitude change Ψ (the ratio of the reflection amplitudes of the p and s components). The p and s components of a reflected beam, in general, have different amplitudes for non-normal incidence (equ.(3.35)). The ratio of the amplitudes of reflectance (equ. (3.35)) of the p and s components can be written in a generalized form [134] as R ρ = p = tan(Ψ) exp(i∆) (3.59) Rs where
Rp tan Ψ = ∆ = δ s − δ p (3.60) Rs 73 which is the fundamental equation of ellipsometry. Ellipsometry is a technique based on measurements of the states of polarization of the incident and reflected light waves, leading to the determination of the ratio ρ (equation (3.59)) of the complex Fresnel’s reflection coefficients [134]. Since the standard ellipsometric angles ∆ and Ψ are related to the optical constants, a determination of the latter is, in principle, possible from the measurements of ∆ and Ψ. This method of measurements forms the basis of the ellipsometry (for more details see Chapter 4).
The explicit relationship between the ellipsometric angles, ∆ and Ψ, and the optical constants ns, κs of a substrate and n of the medium are given by Drude equations
[120] as follows
2 2 2 2 2 2 2 2 tan θ t (cos 2Ψ − sin 2Ψ sin ∆) (3.61) ε1 = ns − κ s = n sin θ t (1 + ) (1 + sin 2Ψ cos ∆)2
2 2 2 n sin θt tan θt sin 2Ψ cos 2Ψ sin ∆ nsκ s = (3.62) (1 + sin 2Ψ cos ∆)2 or 2 2 2 n sin θ t tan θ t sin 4Ψ sin ∆ ε 2 = 2ns κ s = (3.63) (1 + sin 2Ψ cos ∆)2
The general Drude equations (equations (3.61) and (3.62)) as such do not allow direct calculations of the optical constants and thickness from the measured value of ∆ and Ψ.
Computations of the optical constants are conveniently handled by an electronic computer using an appropriate optical model (see Chapters 4 and 5 for more details). 74
3.5.1 The Amplitude and Phase Changes on Reflection at a Thin Film
By separating the complex expression for the reflected amplitude into real and imaginary parts, we may readily obtain the modulus and the phase terms of the reflected beams off a transparent thin film. Thus from equ.(3.35) we have
r + r r 2 + (r + r r 2 ) cos 2δ (r 2 − r 2r )sin 2δ r = 1 1 2 2 1 2 1 − i. 2 1 2 1 2 2 2 2 (3.64) 1+ 2r1r2 cos 2δ1 + r2 r2 1+ 2r1r2 cos 2δ1 + r1 r2
s p The single equation (3.64) represents both r and r . Values of r1 and r2 are inserted appropriate to the plane of polarization under consideration (equs.3.19 – 3.22). The phase changes ∆ on reflection at the thin film are thus given by
r (1− r 2 )sin 2δ tan ∆ = 2 1 1 (3.65) 2 2 2 r1(1+ r2 ) + r1 (1+ r1 )cos 2δ1 where 2π δ = n h cos θ (3.66) 1 λ 1 t
The quantities generally derived from polarimetic measurements are Ψ and ∆ which are given by equations 3.59 and 3.60, respectively. These quantities are readily obtained from equations 3.64 to 3.66. It is convenient to write:
p p p a1 ≡ r1 + r2 cos 2δ1 b1 ≡ r2 sin 2δ1 p p p p a2 ≡ 1+ r1 r2 cos 2δ1 b2 ≡ r1 r2 sin 2δ1 s s s a3 ≡ r1 + r2 cos 2δ1 b3 ≡ r2 sin 2δ1 s s s s a4 ≡ 1+ r1 r2 cos 2δ1 b4 ≡ r1 r2 sin 2δ1
A ≡ a1a2 + b1b2 B ≡ a1b2 − a2b1
A′ ≡ a3a4 + b3b4 B′ ≡ a3b4 − a4b3 75
We then have
(AB′ + A′B)2 + (AA′ − BB′)2 tan Ψ = 2 2 2 2 (3.67) (a2 + b2 )(a3 + b3 ) and AB′ + A′′B tan ∆ = (3.68) AA′ − BB′
The expressions for the amplitude and phase changes on reflection for an absorbing thin film may be obtained from equations (3.64) and (3.65) by substituting the appropriate complex values of the Fresnel’s reflection coefficients (equs. (3.19) – (3.22)). 76
CHAPTER FOUR
EXPERIMENTAL SETUP AND TECHNIQUES
In this work two types of optical measurements were carried out. The first is the measurements of the polarization states, Ψ (λ) and ∆ (λ), for sputtered a-AlN, and a-InN thin films onto c-Si (111) substrates at two angles of incidence, 70o and 75o. The second is the s- and p- polarized transmissivities and reflectivities, Ts,p (λ) and Rs,p (λ), measurements of the same thin films as single layer systems on glass substrates at different angles of incidence. Also, Ts,p (λ) and Rs,p measurements for bilayer and multilayer structures of sputtered AlN/InN onto quartz substrates were performed at three angles of incidence,50o, 60o, and 70o , in the visible to near infrared regions.
4.1 Sample Preparation
4.1.1 a-AlN and a-InN Single Layer Systems
As mentioned in Chap.2.6.2, it is easier to prepare amorphous nitride films than crystalline films. The growth of crystalline nitride films such as silicon nitride and group
III nitrides, (Al, Ga, In) N, requires high temperature (500 – 1400) oC [113, 114], while amorphous nitride can be prepared at low temperature, even at room temperature. Also the film growth is substrate independent whereas that of most crystalline nitride film has very strict requirements on the type of substrate. High quality substrate materials, such as sapphire, silicon, and magnesium oxide, are commonly used. 77
The single layer amorphous (Al, In) N samples for the SE measurements were prepared by a reactive RF magnetron sputtering method (see Chap.2.11). The method has been in use for several years to produce amorphous thin films for lighting applications [135]. An
Al target and In target of 99.999% purity were each separately sputtered in a pure nitrogen atmosphere. Aluminum nitride and indium nitride films were simultaneously deposited independently onto cleaned c-Si (111) and glass substrates. The substrates were clamped to a thick copper block that limited the temperature during deposition to T < 325
K. Typical deposition conditions and sample thicknesses are shown in Table 4.1. In order to obtain a homogeneous thin film, the a-(Al, In) N films were deposited at a rate 0.1 –
0.2 and 0.3 – 0.5 Å/s, respectively. The deposition rate as well as the nominal film thickness was monitored with a vibrating crystal thickness monitor. Using a surface profilometer the film thicknesses were checked. From Table 4.1, it is noticed that the thicknesses measured by a surface profilometer (rows 6 and 8) are concordant with the nominal thicknesses (rows 5 and 7) as measured by a crystal oscillator inside the vacuum chamber during the sputtering process.
Table 4.1 Operating conditions of the RF magnetron sputtering system and sample thicknesses. Base pressure 4-6 x 10-7 Torr Sputtering pressure 5-8 x 10-3 Torr RF power 139 W Average back power 2-10 W Nominal thickness of a-AlN film (nm) 25 50 80 120 Thickness by profilometer (nm) --- 48.6 87.5 122 Nominal Thickness of a-InN film (nm) 90 154 218 255 Thickness by profilometer (nm) 87.3 157.8 226 264.3
78
To verify the amorphous nature of the studied films, X-ray diffraction (XRD) measurements were performed. The diffraction pattern of amorphous and crystalline structures of InN thin films (as an example) grown on c-Si (111) substrates was collected between 20o and 100o as shown in Figs.4. (a) and (b), respectively. Our XRD results for crystalline InN films are concordant with the previously reported results in the literature
[136]. All thin films characterized in this work show XRD patterns similar to the one shown in Fig. 4.1 (a).
106
105 Si (333) Si (111)
104 Si (222) Si
103 Intensity (log-scale,a.u)
102
1 10 ( a )
6 10
5
10 Si (111) Si (333)
104 Si (222)
3 10 (002)InN Intensity (log-scale, a. (log-scale, u) Intensity
102
( b ) 101 20 40 60 80 100 2θ (degrees)
Figure 4.1 XRD patterns of (a) amorphous and (b) crystalline InN thin films sputtered on c-Si (111) substrates. 79
4.1.2 AlN / InN Bilayer and Multilayer Thin Film Systems
In order to prepare bilayer and multilayer thin film systems of nitrides using RF magnetron sputtering, a new mechanically rotated holder (see Fig.4.2) was used. This holder is designed to control the position of a sample relative to the chosen target without opening the vacuum system. Therefore, it protects the sample from any probable contamination. Also, it contains a liquid N2 cold finger which helps to keep the temperature of the substrate low during the sputtering process. The substrate temperature
(Ts) is a significant factor for the quality of nitride films. Higher Ts generally means better crystal structure, whereas low Ts often leads to amorphous structures. Although the sample holder is cooled by the 78 K (-195 oC) liquid nitrogen, the substrate temperature may be higher because of thermal conduction and radiation from the plasma during growth.
Typical deposition conditions for each single layer and its thickness in the multilayer systems with different periods are tabulated in Tables 4.2 and 4.3. In these
* Tables the parameter TG which refers to the Gryogenic temperature is very low less than
32 K. It is noticed also that the total thicknesses measured by profilometer are concordant with the nominal thicknesses as measured by a crystal oscillator inside the vacuum chamber during the sputtering process. Further, the deposition parameters for each single layer and its thickness in a periodic AlN / InN / Quartz multilayer system were constant, leaving the number of bilayers as variable. In order to design of a Broadband
Antireflection Coating (BBARC), the periodic system was carefully built according to the 80 refractive indices of AlN and InN as single layers. Details will be discussed in the following chapter.
Sample stage
The rotating part ( 360o )
Liquid nitrogen cold finger
Figure 4.2 Photograph of the rotating holder of the sputtering system from side and front – back views. 81
Table 4.2 Operating conditions of the RF magnetron sputtering system of multilayer AlN/InN/ Quartz thin film systems with period 117 nm.
* System TG Sputtering RF Back Deposition Deposition Total thickness by pressure power power rate time Monitor Profilometer (K) (mT) (W) (W) (Ǻ / sec) (min.) (nm) (nm)
1π
1st InN 25 7.5 104 2 0.2 3.40 53 2nd AlN 23 – 25 8 119 5 – 8 0.1 – 0.2 17.5 64 117 109.8
2π
1st InN 25 – 26 8 119 1 – 2 0.3 3.15 53 2nd AlN 23 – 25 9 119 5 – 8 0.1 19.1 64 3rd InN 27 – 28 8 119 3 0.3 3.40 53 4th AlN 26 – 28 8.5 119 5 – 7 0.1 18.51 64 234 230.5
3π
1st InN 29 – 30 8 120 1 0.3 3.0 53 2nd AlN 27 – 28 7 119 6 – 8 0.1 – 0.2 15.25 64 3rd InN 28 8 119 3 0.4 2.30 53 4th AlN 27 – 29 7 119 4 0.1 – 0.2 16.0 64 5th InN 29 8 119 5 0.3 2.40 53 6th AlN 27 – 29 8 119 3 0.1 17.33 64 351 345.0
4π
1st InN 20 – 22 8 120 4 – 6 0.3 3.02 53 2nd AlN 20 – 23 7.5 119 6 0.1 – 0.2 16.28 64 3rd InN 24 – 25 8 119 3 – 4 0.3 3.0 53 4th AlN 24 – 27 8 119 4 – 5 0.1 – 0.2 16.30 64 5th InN 27 – 28 8 119 3 – 5 0.3 3.01 53 6th AlN 26 – 28 7.5 119 4 0.1 – 0.2 17.16 64 7th InN 27 – 28 8 119 1 – 2 0.3 – 0.4 2.47 53 8th AlN 24 – 27 8 119 4 0.1 – 0.2 17.04 64 468 461.3
5π
1st InN 23 – 24 8 119 4 0.3 2.52 53 2nd AlN 24 – 26 8 119 5 – 6 0.1 – 0.2 17.20 64 3rd InN 26 – 27 7.5 119 7 0.3 2.40 53 4th AlN 26 – 27 9 119 4 – 5 0.1 19.2 64 5th InN 27 – 28 8 119 4 0.4 2.38 53 6th AlN 26 – 28 8 119 2 – 3 0.1 17.29 64 7th InN 28 – 29 8 119 6 0.4 2.30 53 8th AlN 26 – 30 8.5 119 5 0.1 21.07 64 9th InN 30 – 31 8 119 6 0.4 2.03 53 10th AlN 28 – 30 8 119 3 0.1 17.35 64 585 581.25 82
Table 4.3 Operating conditions of the RF magnetron sputtering system of multilayer AlN/InN/ Quartz thin film system with period equals to 538 nm.
* System TG Sputtering RF Average Deposition Deposition Total thickness by pressure power back power Rate time Monitor Profilometer (K) (mT) (W) (W) Å/Sec (min.) (nm) (nm)
1π
1st InN 20 8 119 6 0.4 15.0 294 2nd AlN 25 - 27 8 119 5 - 9 0.1 – 0.2 71.0 244 538.0 525
2π
1st InN 21 - 24 8 119 5 - 3 0.4 15.0 294 2nd AlN 24 - 27 8 – 8.5 119 5 - 10 0.1 73.0 244 3rd InN 25 - 30 8 119 5 - 8 0.3 - 0.4 15.5 294 4th AlN 28 - 31 8 119 4 - 6 0.1 73.0 244 1076.0 1049.1
3π
1st InN 19 - 20 8.5 - 9 119 3 - 8 0.3 - 0.4 15.0 294 2nd AlN 20 - 24 8 119 4 - 6 0.1 – 0.2 71.0 244 3rd InN 27 8 119 3 - 4 0.2 - 0.3 17.0 294 4th AlN 27 - 28 8 119 4 - 6 0.1 -0.2 68.0 244 5th InN 30 - 31 7.5 119 7 - 8 0.3 16.0 294 6th AlN 28 - 31 7 119 5 - 6 0.1 – 0.2 65.0 244 1614.0 1586.9 * Gryogenic temperature
X-ray diffraction θ – 2 θ spectra for two different identical periods of AlN / InN / Quartz bilayer and multilayer thin film systems were measured, in sequence to study the structure of these systems. The diffraction patterns of the quartz substrate (uncoated layer) and one bilayer (1π), two (2π), three (3π), four (4π), and five (5π) bilayers of
AlN/InN thin film systems grown onto quartz substrates (Fig.4.3) were collected between
10o and 80o as shown in Figs.4.3 – 4.11, respectively. XRD pattern of the quartz substrate shows diffuse scattering centered at 21.15o while the XRD patterns of multilayer thin film systems show the strongest InN thin film (002) peak at 31.2o. As the number of bilayer or layres increases, the maximum intensity of that peak is also increased as shown in Figs. 83
4.8 and 4.12. This indicates that the degree of crystallinty depends on the total bilayer thickness or deposition time (Tables 4.2 and 4.3) as well as the substrate temperature.
Additional, the number of peaks increases when the total time deposition increases. Thus, the XRD patterns of these systems reveal a polycrystalline structure with a maximum diffraction peak at 31.2o. Further, it is to be noted that the degree of crystallinty of InN film is stronger than that of AlN film which has a diffraction peak at 35.6o with preferred
(002) orientation.
Quartz substrate Intensity (log-scale, a.u) (log-scale, Intensity
20 30 40 50 60 70 80 2 θ (degrees)
Figure 4.3 The XRD spectra of quartz substrate (uncoated layer). 84
One bilayer: 117 nm x 1 InN (002) InN
Intensity (log scal, a.u.) scal, (log Intensity
20 30 40 50 60 70 80 Diffraction angle 2 θ (degrees)
Figure 4.4 The XRD results of one bilayer (1π: 117 nm × 1) thin film system of AlN / InN/ Quartz.
Two bilayers: 117 nm x 2 InN (002) InN
Intensity (log scale, a.u.) (log scale, Intensity
10 20 30 40 50 60 70 80 90 Diffraction angle 2θ (degrees)
Figure 4.5 The XRD results of two bilayers (2π: 117 nm × 2) thin film system of AlN / InN/ Quartz. 85
Three bilayers: 117 nm x 3 InN (002) InN (101) Intensity (log scale, a.u.) scale, (log Intensity InN (102) InN InN (103) InN
10 20 30 40 50 60 70 80 Diffraction angle 2θ (degrees)
Figure 4.6 The XRD results of three bilayers (3π: 117 nm × 3) thin film system of AlN / InN/ Quartz.
Four bilayers: 117 nm x 4 InN (002) InN InN (101)
Intensity (log scale, a.u.) (log scale, Intensity AlN (002) AlN InN (102) InN (103) InN
10 20 30 40 50 60 70 80 90 Diffraction angle 2θ (degrees)
Figure 4.7 The XRD results of four bilayers (4π: 117 nm × 4) thin film system of AlN / InN/ Quartz. 86
Five bilayers: 117 nm x 5 InN (002) InN InN (101) InN
AlN (002) AlN Intensity (log scale, a.u.) (log scale, Intensity InN (102) InN InN (103) InN (004) InN
10 20 30 40 50 60 70 80 90 Diffraction angle 2θ (degrees)
Figure 4.8 The XRD results of five bilayers (5π: 117 nm × 5) thin film system of AlN / InN/ Quartz.
One bilayer: 538 nm x 1
InN (002) InN (101) InN (103) InN Intensity (log-scal, a.u.) Intensity InN (102) InN
10 20 30 40 50 60 70 80 Diffraction angle 2 θ (degrees)
Figure 4.9 The XRD results of one bilayer (1π: 538 nm × 1) thin film system of AlN / InN/ Quartz. 87
Two bilayers: 538 nm x 2
InN (002) InN (101) InN Intensity (log-scale, a.u) (log-scale, Intensity AlN (002) InN (103) InN (102) InN
10 20 30 40 50 60 70 80 Diffraction angle 2 θ (degrees)
Figure 4.10 The XRD results of two bilayers (2π: 538 nm × 2) thin film system of AlN / InN/ Quartz.
Three bilayers: 538 nm x 3 InN (002) InN (101) Intensity (log-scal, a.u.) (log-scal, Intensity InN (103) InN AlN (002) InN (004) InN InN (102) InN
10 20 30 40 50 60 70 80 Diffraction angle 2 θ (degrees)
Figure 4.11 The XRD results of three bilayers (3π: 538 nm × 3) thin film system of AlN / InN/ Quartz. 88
Identical period = 538 nm
Three bilayers InN (002) InN Two bilayers One bilayer
InN (101) InN 1π AlN (002)
2π InN (103) InN (004) InN InN (102) InN
Intensity (log-scal, a.u.) Intensity 3π
0 1020304050607080 Diffraction angle 2 θ (degrees)
Figure 4.12 The XRD results of multi-bilayer thin film systems of AlN / InN/ Quartz with different thicknesses of period 538 nm.
4.2 Variable Angle Spectroscopic Ellipsometry (VASE)
Spectroscopic Ellipsometry (SE) is a sensitive optical technique for determining the properties of surfaces and thin films. It derives its sensitivity from the determination of the relative phase change in a beam of reflected polarized light [38, 134]. Through the analysis of the state of polarization of the light that is reflected from the sample (Fig.
4.13) ellipsometry can yield information about layers that are thinner than the wavelength of the light down to a single atomic layer or less. Depending on what is already know about the sample, the technique can probe a range of properties including the layer thickness, optical constants, morphology, chemical composition, surface and/or interfacial roughness, which affect the optical properties [72]. Moreover, ellipsometry is also becoming more interesting to researchers in other disciplines such as chemistry, biology, and medicine. 89
1. Linearly polarized light 3. Elliptically polarized light v E p-plane v p-plane E
s-plane s-plane θi
2. Reflect off sample
Figure 4.13 Schematic of the geometry of an ellipsometry experiment.
4.2.1 VASE Theory
Spectroscopic ellipsometry is a non-destructive optical technique, which measures the change in polarization state of light reflected from the surface of a sample as shown in
Figure 4.1. It uses the fact that for any angle of incidence greater than 0° and less than
90°, p-polarized light and s-polarized light will be reflected differently from the surface of the sample under study. The s-direction lies perpendicular to the p-direction such that the p- direction, s- direction, and the direction of propagation define a right-handed cartesian coordinate system. The measured ellipsometric parameters (or angles) are expressed as Ψ (λ), the relative amplitude change, and ∆ (λ), the relative phase change.
These parameters are related to the ratio of Fresnel reflection coefficients, rp and rs for light polarized parallel (E-field polarized parallel) and perpendicular (E-field polarized perpendicular) to the plane of incidence, respectively. Therefore, the complex ratio ρ measured by the VASE as a function of wavelength is described by [134, 137 - 139]: 90
(rp− polarized ) ρ = = tan(Ψ) exp(i∆) (4.1) (rs− polarized )
As is obvious from the above equation, the complex reflectance ratio ρ is completely determined by the amplitude (tan Ψ) and a phase (∆) and characterize the differential changes in amplitude and phase. These changes are related to a transformation of the shape and orientation of the ellipse of polarization. Ellipsometry is a self-consistent technique in which one polarization component serves as amplitude and phase reference for the other. Also, because ellipsometry measures the ratio of two values, it can be highly accurate and very reproducible. From equ.(4.1) the ratio is seen to be a complex number, thus it contains “phase” information in delta which makes the measurement very sensitive [134]. The phase information in an ellipsometry measurement provides very high sensitivity to ultra-thin surface films (< 10 nm) [38].
The standard ellipsometric angles, Ψ and ∆, depend on the photon energy E, the sample layer structure, the material dielectric function, ε, and the angle of incidence θ.
These two angles indicate the polarization change of the incident light wave as it is reflected at the different interfaces in the sample. Properties of the sample therefore are obtainable if the properties of both incident and reflected waves are known. In the
Fig.4.13, a linearly polarized input beam is converted to an elliptically polarized reflected beam, thus the name ellipsometry. 91
4.2.2 Instrumentation Basics
A Photograph and a schematic diagram of our main investigatory experimental setup (a rotating analyzer variable angle spectroscopic ellipsometry, 190 - 1700 nm, RA - VASE) are shown in Figs.4.14 and 4.15, respectively.
Figure 4.14 Photograph of the Woollam Co. Variable Angle Spectroscopic Ellipsometry (VASE).
Sample mount Rotating analyzer Polarizer and Photodiode detector Collimating lens
Fiber optics
Electronic Box
35 Light source:
30 a-InN film Variable angle 25 20
15 in degrees
Goniometer base Ψ Monochromator + Xenon lamp 10 Model Fit 5 Exp 70o Exp 75o 0 (a)
200 400 600 800 1000 1200 1400 Wavelength (nm)
Computer
Figure 4.15 A schematic diagram of the optical setup: A rotating analyzer variable angle spectroscopic ellipsometric (RA - VASE). 92
As shown in the Fig.4.15 the optical setup is composed of the following parts:
1. Light Source
Unlike single-wavelength ellipsometers, which can utilize laser sources, spectroscopic ellipsometry (SE) requires a broad spectral output to match the desired range of measurements. To cover wavelengths from the ultraviolet to the infrared, a high pressure
Xenon (Xe) lamp filled with Xenon gas at above atmosphere pressure is utilized. The Xe arc lamp can be used from 190 nm in the deep ultraviolet to over 2 µm in the infrared.
The lamp is focused into the monochromator.
2. Monochromator
A monochromator is used to select the desired operating wavelength.The selected wavelength of light enters the chopper/filter unit.
3. Chopper/ Filter Unit
The beam with the desired wavelength enters this unit, which modulates the beam and prevents the harmonics of the chosen light wavelength from entering the fiber optical cable.
4. Fiber Optical Cable
The fiber optical cable is used to couple the modulated beam from the chopper /filter unit to the collimator on the input unit of the ellipsometry.
5. Input Unit
The input unit conditions the beam before it hits the sample. This unit consists of a collimator which contains a collimating lens to collimate the beam to ensure that the maximum amount of light passes through the optical system, a polarizer stage that 93 transforms unpolarized light beam into a linearly polarized beam (Fig.4.13), and an alignment detector which is used to align the sample to the beam. The alignment detector is a four quadrant silicon photodiode detector with a 1.27 mm hole bored through the center. When the alignment detector is mounted on the ellipsometer, the light beam passing through the hole is reflected from the sample and returns to strike the alignment detector. If the sample is perfectly aligned, the signal from each of the four quadrants will be equal.
6. Sample Stage
The sample stage has two major functions: to hold the sample flat and firm for measurement, and to allow the user to alter the position of the sample during alignment.
A vacuum hole is presented to hold the sample in place during alignment and data acquisition.
7. Output Unit
The output unit conditions the reflected beam and converts the beam into electrical signals. This unit consists of: an iris, which controls the amount of light passing through the analyzer to detector end cap, an analyzer which determines the state of polarization for the light beam (elliptically polarized, Fig.4.13), and a detector which converts the modulated light beam into electrical signals appropriate for analysis by the computer.
8. Base Unit
The base unit brings together the sample stage, input unit, and output unit. It is used to change the angle of input and output units, where the configuration used is called a
Theta-two-Theta stage (θ -2 θ). 94
9. VB-250 VASE Electronic Box
The VB-250 contains all power supplies and drivers for the stepper motors in the standard VASE system. Also, it controls the analyzer motor, and the monochromator chopper blade and filter wheel.
10. Computer
The computer controls the electrical and measurement systems of the ellipsometer. It initiates all pulses to the stepper motors drivers within the VB-250. In addition to collecting and storing data for further analysis.
The VASE performs measurements as a function of wavelength, λ, and angle of incidence, θi .The angle of incidence in this ellipsometer is computer controlled, and is generally in the range of 50o to 80o, depending on the sample. Hence the VASE is a powerful tool for characterizing thin films since the reflected light behaves differently at various wavelengths and angles. Scanning a wide range of wavelengths and angles helps eliminate ambiguities that occur in single wavelength ellipsometers [137 - 139].
Moreover, adding multiple angles to spectroscopic capability (VASE) provides new information because of the different optical path lengths traversed, and it optimizes senility to the unknown parameters [38]. It operates by reflecting linearly polarized light from (or transmitting through) the surface of a material. The light undergoes a polarization change due to the reflection (or transmission) related to the thin film properties (thickness, index of reflection, etc.).
The light reflecting from the top of the thin film (see Figs.2.6 and 4.14) will combine with the other light components (back- surface reflections) before reaching the 95 detector. This introduces constructive and destructive interference (especially for thicker films) [122]. The nature of the interference depends on the film thickness, film optical properties, angle of incidence, and wavelength of light used to probe the sample under study. Therefore, measurements using the VASE technique take complete advantage of how the light interacts with the sample, providing a ‘signature’ of the sample properties
(thickness, optical constants, n and κ, etc.). Moreover, the VASE is an accurate method for nondestructively measuring thin films and multilayer structures (see Fig.2.6). With multilayer structures there are a greater number of reflections (from front and back sides) in the thin film layers. Single wavelength ellipsometers are less likely to succeed when measuring these complicated structures, because there are too many variables (thickness and optical properties of each layer, surface oxidation and roughness, interfacial roughness, and many more) which increase the difference between the generated values by the fit model and the experimental data.
On the other hand, there are several factors, which determine the limits of information about a given sample that may be determined by ellipsomerty. Most of these factors are related to the length scale of the probe used in ellipsomerty, i.e., the wavelength of the incident beam. Eillipsomerty works best for film characterization when the film thickness is not too much smaller or larger than the wavelength of the light used in the measurement. It is relativity difficult to use a probe of 500 nm (or so) wavelength to characterize a 0.5 nm “ultra thin” or 10000 nm “bulk” film, where as film thickness from (say ) 5 nm to 1000 nm are much simpler to characterize in general. Also, roughness variations on the sample surface or at film interface should be less than ~ 10% of the 96 probe wavelength for the ellipsometric analysis to be valid. Larger features can cause non-specular scattering of the incident beam and depolarization of the specularly reflected beam. Finally, the thickness of the film under study should vary by no more than ~ 10% over the width of the spot on the sample surface, otherwise the assumption of the parallel interfaces of the film will not be valid, and the calculated values cannot be expected to match the experimental data.
4.2.3 Development of an Optical Model
The standard ellipsometric parameters, Ψ and ∆, describe a change in the light’s polarization state caused by the sample under investigation. These two values by themselves are not very useful in characterizing a sample. One is really interested in optical characteristics such as film thickness, dielectric functions, ε1 and ε2, and refractive index, n, etc. These optical characteristics are found by using the measured values, Ψ and
∆, in equations [134, 120, and 122] and algorithms [38, 134] to produce a mathematical model that describes the interaction of light with the sample. Figure 4.16 outlines this process. The first step in data analysis is to build a model of the material that is being measured. This model needs to include the order of the layers, their optical constants and their thickness (unknown physical parameters). If these values are not known a “best guess” should be entered. The model should also contain some known parameters, such as the wavelength of the incident light, the incident beam polarization state, and the angle of incident. In order to get a unique optical model which does not contain any strongly correlated parameters one should start with the simplest model and add complexity as needed. 97
Experimental Data Ψ (λ) Measurement Data: Ψ (λ) ∆ (λ) λ
Model Generated Data Air Generated Data Ψ (λ) a-AlN film
SiO2 layer c-Si (111)
Compare Fit Parameters Fit
2.15 0.008
2.10 n 0.007 )
κ κ 0.006 ( Results 2.05 0.005
2.00 0.004
0.003
Index of(n) refractionIndex 1.95
0.002 Extinction coefficient
1.90 0.001
0.000 1.85 200 400 600 800 1000 1200 1400 Wavelength (nm)
Figure 4.16 Sample analysis procedure. 98
4.2.4 Fitting of Model to Measured Data
The next step in data analysis is to vary the unknown physical parameters and generated data from the model by successive iterations until one finds the set of variable parameters which yield a calculated data that closely match the measured optical data. It is then hoped that this set of parameters accurately represents the true physical description of the sample under study. The fitting process is therefore used to adjust the theoretical data to the experimental data (real measure). The data fitting is performed using an iterative fitting algorithm (Levenberg-Marquardt algorithm). It is designed to minimize the value of the mean square error (MSE) by adjusting the sample fitting parameters. WVASE32 uses MSE to describe how close the generated data matches the experimental ones. A smaller MSE implies a better fit. The final stage of any optical experiment is to prove that the best-fit result which is obtained from the modeling process is unique.
In other words, there is one set of values of the model parameters which will best fit the experimental data. A non-unique result indicates that two or more of the parameters in the model are strongly correlated.
Let us say that we have performed a fit to a given set of data, and now we wish to determine if the fit is unique and the parameters in the model are not strongly corrected.
In the case of strongly correlated parameters, the fitting algorithm makes a number of iterations, each of which reduces the mean square error (MSE) only slightly. The algorithm cannot determine which of the correlated parameters to adjust, or how to adjust them. It makes a slight adjustment at each iteration, which only has minimal effect on the
MSE. 99
If the parameter set is uncorrelated and the model is physically accurate for the given sample, the algorithm will usually minimize the MSE in few iterations. In addition, if the parameters change radically with each iteration, but the MSE does not change very much, common sense indicates that if a large change to one of the variables does not significantly change the MSE, there is no well define best-fit value of the parameter. This may be due to correlation, or the fact that the MSE may not be sensitive to the given parameter. Furthermore, because we are dealing with real-world sample conditions, one can take into account the roughness of the surface, interface layer, dioxide layer, and the anisotropy of the layer in order to obtain a good fitting to the experimental data. The fitting procedure in our work includes experimental data of standard VASE: Polarization states (standard ellipsometric parameters, Ψ and ∆).
4.3 Optical properties (reflection and transmission) measurements
An illustration of the transmitted, reflected, and incident beams is shown in Fig.4.17. A beam of light is incident on a sample at some arbitrary angle of incidence, the angle of incidence is defined as the angle between the input beam direction and the direction normal to the sample surface. At the boundary of the medium, part of the light will be reflected at angle θi while the other part will be transmitted through the sample at angle
θr. Snell's law requires that all three beams be in the plane of incidence (shaded green in
Fig. 4.17). The plane of incidence is defined as that plane which contains the input beam, the output beam, and the direction normal to the sample surface. 100
The transmission and reflection measurements acquire the intensity ratios, T and R respectively, over a given range of wavelengths. T and R are defined as the ratio of the light intensity being transmitted or reflected over the incident light intensity on the sample, as shown in equs. (4.2) and (4.3):
I s, p (λ) T s, p (λ) = t (4.2) s, p Ii (λ)
Ir s, p (λ) R s, p (λ) = (4.3) s, p Ii (λ) where Ii refers to the intensity of the beam without the sample present on the sample stage (see Figs. 4.14 and 4.15) at each specified wavelength (the background scan) and Ir
( or It) refers to the intensity of the beam with the sample in place at each specified wavelength. The superscripts s and p refer to the polarization states of the incident beam perpendicular and parallel to the plane of incidence, respectively.
I i I r
plane of θ i θ r incidence
θ t I t
Figure 4.17 Schematic showing the incident, reflected, and transmitted light. 101
4.3.1 Intensity reflectance data
The acquisition of intensity reflectance data with the VASE instrument requires some care. Reflectance data may not be acquired at normal incidence, as the finite size of the source and detector arms prevents positioning at angles of incidence less than about 12o.
Therefore, reflectance may only be measured at oblique angles of incidence (i.e, 20o –
80o). At oblique incidence, two types of intensity reflectance data may be acquired,
depending on the polarization state of the incident beam. p-polarized intensity reflectance
(pIR) may be acquired when the input beam is linearly polarized with the beam
polarization in the plane of incidence; s-polarized intensity reflectance (sIR) may be acquired with the incident beam polarized in the s-direction as shown in Fig.4.13.
Both pIR and sIR were undertaken for the amorphous III-V nitride samples on
different substrates in air at room temperature in the wavelength range 300-1400 nm over
a wide range of incidence angles (i.e., 20o – 80o). Dedicated software is used for data
acquisition and analysis. The polarized reflectivities as a function of wavelength of the
incident light, Rs,p (λ), were therefore calculated by using equ.(4.3).
4.3.2 Intensity transmittance data
Intensity transmission is very useful for samples with transparent substrates. Therefore, p-and s-polarized intensity transmittance (pIT and sIT) measurements were undertaken
for the sputtered samples (Tables 4.1 – 4.3) onto glass and quartz substrates. Both pIT and sIT were carried out in air at room temperature in the wavelength range 300 – 1400 nm over a wide range of incident angles (i.e., 10o - 80o). The polarized transmissivity 102 measurements began by taking the intensity of the incident light, Ii, (the background scan)
with no sample present on the sample stage (see Figs.4.14 and 4.15). After Ii is recorded
the sample was mounted and then the intensity of the transmitted light, It, (as shown in
Fig.4.17) was measured in the same wavelength range. The transmittance of the sample
was then calculated by using equation (4.2). The obtained values of Ts,p against wavelength are stored and plotted using Woollam Co.WVASE32 software. 103
CHAPTER FIVE
OPTICAL CHARACTERISTICES OF III-V NITRIDE FILMS
This chapter is basically devoted to study of optical constants and polarized optical properties of III-V nitride thin films as single, bilayer and multilayer systems. These systems, deposited onto different types of substrates, were produced at low temperature using a sputtering technique described in chapter two. The deposition conditions were summarized in Tables 4.1 – 4.3 The optical characteristics such as index of refraction, extinction coefficient, optical bandgap, Brewster and pseudo- Brewster angles, and s- and p-polarized transmissivities of these systems were measured using the Variable Angle
Spectroscopic Ellipsometry (VASE). The VASE technique was described in details in chapter four.
5.1 Calibration of the Experimental Optical Setup
Data acquisition begins with the calibration of the rotating analyzer-variable angle
spectroscopic ellipsometry (RA-VASE) instrument. To calibrate the RA-VASE, the
standard sample (a 20 nm thermal oxide on silicon wafer, see Fig.4.4) must be mounted,
aligned, and calibrated at 500 nm wavelength and 75o angle of incidence. The two
important quantities measured by the VASE are α and β, which are the normalized
Fourier coefficients of the detector signal. They can be represented in terms of the
ellipsometric parameters Ψ and ∆ (see Chap.4.3) for the sample and the (known)
polarizer azimuthal angle as follows [134, 138]: 104
tan2 Ψ − tan2 P α = , (5.1) tan2 Ψ + tan2 P
2 tan Ψ cos∆ tan P β = . (5.2) tan2 Ψ + tan2 P where P is the input polarizer azimuthal angle with respect to the plane of incidence (P =
0o is the plane of incidence). The above equations may be inverted to obtain Ψ and ∆
from the measured α and β and the known P as follows:
1+ α tan Ψ = tan P , (5.3) 1−α
β tan P cos ∆ = . . (5.4) 1−α 2 tan P
These equations form the basis of the ellipsometric measurement with the rotating
analyzer ellipsometer. To summarize the above discussion, the detector signal is
measured as a function of time; the measured signal is Fourier analyzed to obtain the
Fourier coefficients α and β, and finally Ψ and ∆ are calculated from α and β and the known polarizer azimuthal angle.
The measured and calculated Fourier coefficients α and β, and the residual, ζ = 1-
α2- β2, of the standard sample are plotted as a function of polarizer angle P in Fig.5.1.
Note that the best-fit calculated data are indistinguishable from the measured data,
indicating that the chosen model is very good and the ellipsometer is functioning nearly
ideally. 105
Figure 5.1 Calibration fit from a fine mode calibration.
In order to gets a high quality fitting and accurate data, the above calibrating step must be repeated not only when we replace the chosen sample by another one but also when we want to do other optical measurements on the same mounted sample.
5.2 The Optical Characteristics of Sputtered Amorphous Aluminum Nitride
Thin Films
With the purpose of getting a good background about the optical behavior of amorphous
AlN as single layers sputtered on different substrates, five types of measurements were
carried out in air at room temperature in the wavelength range 300 – 1400 nm in steps of
10 nm. These types of measurements were summarized in Table 5.1. 106
Table 5.1 Optical data types measured on the sputtered amorphous AlN thin films. Substrate type Data type Angle of incidence Description c-Si (111) SE 70 o –75 o by 5 o Standard VASE: Ψ and ∆ Glass s-IR 20 o – 80 o by 10 o s- polarized intensity reflectance Glass p-IR 20 o – 80 o by 10 o p- polarized intensity reflectance Glass s-IT 10 o – 80 o by 10 o s- polarized intensity transmittance Glass p-IT 10 o – 80 o by 10 o p- polarized intensity transmittance
5.2.1 The Optical Constants
The set of samples (1 – 4) shown in Table 4.1 were used to characterize amorphous
aluminum nitride, a-AlN, thin films. The model structure used to fit the SE data taken on
all a-AlN thin films is shown in Fig.5.2. The optical constants of the crystalline silicon
substrate and silicon dioxide (SiO2) over layer were taken from the literature and were
not allowed to vary during the fitting process [140]. Since most of a-AlN thin films were
optically transparent in the spectral region of interest (i.e., in the visible and near infrared
regions) [141] the Cauchy – Urbach dispersion model was then applied to model this
region. In general, the Cauchy – Urbach relation is only valid over a region of the
spectrum where there is normal dispersion. As soon as the index of refraction starts to
decrease toward shorter wavelengths it is in a region of “anomalous” dispersion. The
Cauchy – Urbach model cannot describe the optical constants in a region of anomalous dispersion and hence a Kramers – Kroning consistent oscillator model [142] must be used to describe optical constants in such a region of anomalous dispersion. In the Cauchy-
Urbach dispersion model, the refractive index n (λ) and the extinction coefficient κ (λ) as
a function of the wavelength are given by [143], 107
B C n(λ) = A + + (5.5) λ2 λ4
1 1 κ (λ) = α exp β (12400( − )) (5.6) λ γ
, respectively. The six parameters in this dispersion model are A, B, C (index parameters which specify the index of refraction), the extinction coefficient amplitude α, the exponent factor β, and the band edge γ (absorption parameter that specify the shape of the absorption tail). Each of these parameters except for the band edge can be defined as a variable fit parameter in the Cauchy layer, with an Urbach absorption.
Air
a-AlN thin film
SiO2 layer
c-Si (111)
Figure 5.2 Schematic drawing of the optical model structure for the SE analysis of a-AlN films.
To model the a-AlN film, wavelengths greater 600 nm were used and the 3-parameter
Cauchy dispersion formula was then applied to model this region. This is because the extinction coefficient κ is almost zero. The film thickness and Cauchy parameters were determined and the fitting then extended to cover shorter wavelengths below 600 nm with the addition of an Urbach’s tail to account for the absorption tail. The fitting process was done by minimizing the mean-square error (MSE) function (eqn.5.7), which is
Exp Exp appropriately weighted to the estimated experimental errors σ Ψij and σ ∆ ij [144]. 108
Exp Normalization of MSE by the standard deviation of the experimental data σ Ψ ij and
Exp σ ∆ ij reduces the weight of noisy points. A smaller MSE implies a better model fit to the
data.
2 2 ⎡⎛ Mod Exp ⎞ ⎛ Mod Exp ⎞ ⎤ 1 N ⎢ Ψi, j − Ψi, j ⎜ ∆i, j − ∆i, j ⎟ ⎥ MSE = ∑ ⎜ ⎟ + (5.7) 2N − M ⎢⎜ Exp ⎟ ⎜ Exp ⎟ ⎥ i, j=1 ⎢ σψ ,i, j σ ∆,i, j ⎥ ⎣⎝ ⎠ ⎝ ⎠ ⎦
The indices i and j indicate Ψ and ∆ data sets at photon energy Ei and angle of incidence
θi. N is the number of measured Ψ and ∆ pairs while M is the total number of real valued fit parameters. We should note here, however, that due to correlation between parameters, the fitting procedure was not performed for all parameters at the same time (in order to get a unique optical model). We have first achieved the fitting for A, B, C and sample thickness, keeping the extinction coefficient amplitude α at 0, and the exponent factor β at 1.5. The fitting to α and β was then added afterwards. A good fitting with a minimum
MSE for our samples was obtained for the desired wavelength range.
The measured SE data, Ψ and ∆, of (25 – 120 nm) a-AlN thin films sputtered onto c-Si (111) over the spectral range 300 – 1400 nm at two angles of incidence, namely, 70o
(dashed lines) and 75o (dotted lines) are shown in Figs. 5.3 – 5.6. These angles were
chosen to maximize the sensitivity near the Brewster angle of the silicon substrates. The
fitted Ψ and ∆ spectra, simulated with the best-fit Cauchy-Urbach model parameters, are
also shown by solid lines in the figures. The fitting was performed by minimizing the
mean square error, with four parameters fitted: the film thickness and three parameters
from the Cauchy-Urbach model, A, B, and α (C, β, and γ are held fixed during the fitting
procedure). Generally, the simplest optical model which fits the data is the best. As 109 shown in the figures, the best-fit calculated data are indistinguishable from the measured data, indicating that the model is unique over a wide wavelength range.
50
45 Model Fit o Ψ Exp 70 40 o Ψ Exp 75 35
30
25 (degrees) Ψ 20
15
10
5 (a)
200 400 600 800 1000 1200 1400 Wavelength (nm)
140 Model Fit 130 o ∆Exp 70 o 120 ∆Exp 75
110
100
(degrees) 90 ∆ 80
70
60 (b) 50 200 400 600 800 1000 1200 1400 Wavelength (nm)
Figure 5.3 Ellipsometric data for parameters (a) Ψ and (b) ∆ for 25 nm a-AlN thin film sputtered onto c-Si (111) substrate. The dashed and doted lines represent the experimental data while the solid line is the model fit for angles of incidence, 70o and 75o.
110
80 Model Fit 70 o Ψ Exp 70 Exp 75o 60 Ψ
50
40 (degrees) Ψ 30
20
10 (a)
200 400 600 800 1000 1200 1400 Wavelenght (nm)
120
100
80
60 (degrees) ∆ 40 Model Fit Exp 70o 20 ∆ Exp 75o ∆ (b) 0 200 400 600 800 1000 1200 1400 Wavelength (nm)
Figure 5.4 Ellipsometric data for parameters (a) Ψ and (b) ∆ for 55 nm a-AlN thin film sputtered onto c-Si (111) substrate. The dashed and doted lines represent the experimental data while the solid line is the model fit for angles of incidence, 70o and 75o. 111
50 Model Fit Exp 70o 45 Ψ o Ψ Exp 75 40
35
( degrees) 30 Ψ
25
20 (a) 15
200 400 600 800 1000 1200 1400 Wavelength (nm)
180
160 Model Fit Exp 70o ∆ Exp 75o 140 ∆ 120
100
in degrees 80 ∆
60
40
20 (b)
0 200 400 600 800 1000 1200 1400 Wavelenght (nm)
Figure 5.5 Ellipsometric data for parameters (a) Ψ and (b) ∆ for 85 nm a-AlN thin film sputtered onto c-Si (111) substrate. The dashed and doted lines represent the experimental data while the solid line is the model fit for angles of incidence, 70o and 75o. 112
60
55 Model Fit o Ψ Exp 70 50 o Ψ Exp 75 45
40
35 (degrees) Ψ 30
25
20 (a) 15
200 400 600 800 1000 1200 1400 Wavelength (nm)
180 Model Fit 160 Exp 70o ∆ Exp 75o 140 ∆
120
100
(degrees) 80 ∆
60
40
20 (b) 0 200 400 600 800 1000 1200 1400 Wavelength (nm)
Figure 5.6 Ellipsometric data for parameters (a) Ψ and (b) ∆ for 120 nm a-AlN thin film sputtered onto c-Si (111) substrate. The dashed and doted lines represent the experimental data while the solid line is the model fit for angles of incidence, 70o and 75o. 113
It is noticed that the spectral dependence of the relative amplitude change, Ψ (λ), and relative phase change, ∆ (λ), of the samples are also functions of incidence angles.
Moreover, the difference between the spectral dependence of ∆ (λ) at the chosen angles is larger than that in the case of Ψ (λ), especially when the film thickness is less than 55 nm.
This confirms the fact that, the sensitivity of the VASE comes from measuring ∆.
Furthermore, it is to be noted that there are very sharp peaks in the dispersion curves of Ψ
(λ) and ∆ (λ) indicating that the imaginary part of the complex refractive index, n + κ, is very small. Furthermore, the Ψ (λ) and ∆ (λ) figures show that as the thickness of the film increases the number of peaks increases with a constant maximum ∆ value of 177o at λ ~
380 nm. This increase comes from the interference effects.
The optical constants of these thin films exhibit strong dispersion and decrease monotonically with increasing wavelength (see Figs.5.7 – 5.10). The figures show a good example of physically reasonable spectra from the Cauchy-Urbach model for the optical constants of insulator and semiconductor materials with wide bandgap. These results give further proof to the fact that our fitting parameters are not strongly correlated to each other. The best-fit model parameters for the a-AlN samples (Table 4.1) are summarized in Table 5.2. We observe that the values of the same parameter for all samples are very close, which testifies to the validity of the model adopted for a-AlN thin films. The close values for the parameters of the Cauchy-Urbach model are the outcome of the fitting procedure which indicates thickness independence. In general, the thickness dependence may appear in samples well below 10 nm (ultra-thin film) where the film characteristics are dominated by its surface. 114
2.10 0.014 n κ 2.05
0.012 κ
2.00 0.010
1.95 0.008
1.90 0.006 Index of refractionIndex n Extinction coefficientExtinction 1.85 0.004 1.80 0.002 1.75 0.000 200 400 600 800 1000 1200 1400 Wavelength (nm)
Figure 5.7 Index of refraction and the extinction coefficient obtained from the Cauchy-Urbach model for 25 nm a-AlN thin film sputtered onto c-Si (111) substrate.
2.15 n 0.008 κ 2.10 0.007 ) κ 0.006 2.05 0.005 2.00
0.004
Refractive index (n) index Refractive 1.95 0.003 Extenction coefficients ( Extenction 0.002 1.90 0.001
1.85 0.000 200 400 600 800 1000 1200 1400
Wavelength (nm)
Figure 5.8 Index of refraction and the extinction coefficient obtained from the Cauchy-Urbach model for 50 nm a-AlN thin film sputtered onto c-Si (111) substrate. 115
2.15 0.010
n
2.10 κ ) 0.008 κ
2.05 0.006
2.00
0.004 Index of Index refraction (n) 1.95 coefficientExtinction (
0.002 1.90
1.85 0.000 200 400 600 800 1000 1200 1400 Wavelength (nm)
Figure 5.9 Index of refraction and the extinction coefficient obtained from the Cauchy- Urbach model for 85 nm a-AlN thin film sputtered onto c-Si (111) substrate.
2.15
n 0.010 2.10 κ ) κ
0.008 2.05
2.00 0.006
1.95 0.004 Extinction coefficient ( Extinction Index of refractionIndex (n)
1.90 0.002
1.85 0.000 200 400 600 800 1000 1200 1400 Wavelength (nm)
Figure 5.10 Index of refraction and the extinction coefficient obtained from the Cauchy-Urbach model for 120 nm a-AlN thin film sputtered onto c-Si (111) substrate. 116
Table 5.2 Best-fit model structural parameters for a-AlN thin films sputtered onto c-Si (111) substrates determined by the VASE measurements in the wavelength range 300-1400 nm. a The parameter is not chosen as a fitting parameter and it has a fixed value during the fitting procedure while b represents the beginning of our wavelength range. The 90 % confidence limits are given by (±).
Sample no./ 1 2 3 4 Parameter
Film thickness (nm) 25.8 ± 0.07 49.7 ± 0.11 83.58 ± 0.15 117.13 ± 0.29 SiO2 thickness (nm) 2.0 a 2.0 a 2.0 a 2.0 a 1.74 ± 1.03 1.87 ± 5.5 1.85 ± 2.7 1.77 ± 2.8 A × 10-2 × 10-3 × 10-3 × 10-3 3.06 × 10-2 ± 2.3 × 10-2 ± 2.61 × 10-2 ± 9.86 × 10-3 ± B 1.3 × 10-3 7.7 × 10-3 4.8 × 10-4 4.8 × 10-4 C 0 a 0 a 0 a 0 a 1.4 × 10-2 ± 1.1 × 10-2 ± 1.25 × 10-2 ± 1.12 × 10-2 ± α 1.0 × 10-3 8.2 × 10-3 3.6 × 10-3 6.9 × 10-3 β 2.0a 3.8 a 3.8 a 3.8 a γ (nm) 300 b 300 b 300 b 300 b MSE 0.89 0.66 0.79 1.23
Figure 5.11 shows the indices of refraction as a function of film thickness while figure
5.12 shows the extinction coefficients as a function of wavelength. These figures prove that there is no significant change in the optical constants of the thin films as a function of thickness. The thicknesses of the thin films by SE are in substantial agreement with those obtained by the crystal oscillator inside the vacuum chamber (nominal thickness) and profilometer as shown in Table 5.3. Also, the table shows some values of the optical constants of the films at different wavelengths. Refractive indices of a-AlN films in this study were found to be in the range of 1.76 - 2.13. The extinction coefficients were very small (< 0.013) and of the order of 10-5 at high wavelength regions (> 600 nm), indicating
that the films are transparent in this region. 117
2.25 2.20 2.15 2.10 2.05 2.00 1.95 1.90 1.85 1.80 1.75 1.70 1.65 λ = 300 nm 1.60 λ = 350 nm Index of refraction n Index 1.55 λ = 600 nm 1.50 λ = 1000 nm 1.45 1.40 1.35 1.30 1.25 1.20 20 40 60 80 100 120 Film thickness (nm)
Figure 5.11 Index of refraction of a-AlN thin film sputtered onto c-Si (111) substrate as a function of thickness at different wavelengths.
0.016 Sample #1 (25 nm) 0.014 Sample #2 (50 nm) )
κ Sample #3 (85 nm) 0.012 Sample #4 (120 nm)
0.010
0.008
0.006 Extinction coefficient ( coefficient Extinction 0.004
0.002
0.000 200 400 600 800 1000 1200 1400 Wavelength (nm)
Figure 5.12 Extinction coefficient of a-AlN thin film sputtered onto c-Si (111) substrate as a function of wavelength at different thicknesses. 118
Table 5.3 Thicknesses of a-AlN samples measured by using different techniques and their optical constants at two different wavelengths, 350 nm and 1200 nm. The 90 % confidence limits are given by (±).
Nominal Film Film Index of Extinction Sample thickness thickness by thickness by refraction coefficient No. profilometer SE at at (nm) (nm) (nm) 350 1200 nm 350 1200 nm 1 25 25.8 ± 0.07 1.99 1.77 4.5 × 10-3 3.9 × 10-5 2 50 48.6 49.7 ± 0.11 2.07 1.88 2.1 × 10-3 0.9 × 10-5 3 80 87.5 83.6 ± 0.15 2.04 1.86 2.7 × 10-3 1.8 × 10-5 4 120 122 117.2 ± 0.3 2.05 1.87 2.6 × 10-3 1.5 × 10-5
As stated in chapter one, experimental and theoretical data on the optical constants of
amorphous AlN thin films are surprisingly rare. In addition, most of such studies were
done on crystalline AlN films. The refractive indices of c-AlN films deposited on
sapphire substrates at 700o C varied between 2.06 and 2.24 and the extinction coefficients
varied between 0.08 and 0.26 for the wavelength range of 300-800 nm [145].
Theoretically the refractive indices of single-crystal AlN films were found in the range
2.15 - 2.4 at 300 - 820 nm and the extinction coefficients were small (< 0.08) and nearly constant (about zero) at low energy region (3 eV) [28]. Our values for n and κ are comparable to those of crystalline AlN film, the difference attributable to the fact that the loss of long-range order on the electronic structure in the amorphous materials (see
Chapt.2, Sec.6).
5.2.2 Optical Bandgap Energy
To obtain the optical bandgap of a-AlN thin film, we extended our wavelength range to cover a larger portion of ultra violet. Again the optical constants were parameterized 119 using the Cauchy-Urbach model described above. The measured Ψ and ∆ spectra, of 50 and 100 nm a-AlN thin films sputtered onto c-Si (111) substrates over the spectral range
200 – 1400 nm are shown in Figs.5.13 and 5.14. The dashed and dotted lines represent the experimental data for angles of incidence of 70o and 75o, respectively. The fitted Ψ and ∆ spectra, simulated with the best-fit Cauchy-Urbach model parameters, are also shown by solid lines in the same figures. The fitting parameters C, β, and γ are held fixed during the fitting procedure (see Table 5.4) and a satisfactory fitting was obtained, for the spectral range of 200-1400 nm. It is to be remarked that, the large value of the MSE as shown in Table 5.4 and the deviations of the fit from the experimental data below ~ 206 nm (see Figs. 5.13 and 5.14) are due to the limitation of the wavelength range of the optical constants for the substrate layer [140].
Figure 5.15 shows the optical constants n and κ of a 50 nm a-AlN thin film deposited on c-Si (111). The optical constants over the new wavelength range exhibit a strong dispersion and decrease monotonically with increasing wavelength. The values of refractive indices of a-AlN films were determined to be in the range 1.86-2.43 while the extinction coefficients were very small (< 2.5 × 10-4) at high wavelengths (> 400 nm) and increased smoothly with decreasing wavelength to reach 0.55 as a maximum value at 200 nm.
As mentioned in the introductory chapter the experimental and theoretical data on the optical constants of a-AlN thin films are surprisingly scarce. In addition, most of such studies are done on crystalline AlN films. As an example, Joo et al. [29] found that the refractive indices of c-AlN films were in the range of 2.1-3.4 at 200-800 nm and the 120 extinction coefficients were small (< 0.2) and nearly constant at high wavelength region
(see Fig.2.5).
90 Model Fit 80 Exp 70o o 70 Exp 70
60
50
(degrees) 40 Ψ
30
20
10 (a)
0 180 Model Fit 160 Exp 700 0 140 Exp 75
120
100
(degrees) 80 ∆
60
40
20 (b) 0 200 400 600 800 1000 1200 1400 Wavelenght (nm)
Figure 5.13 Measured and calculated Ψ and ∆ spectra of a-AlN thin film of 50 nm thickness sputtered onto c-Si (111). The dashed and dotted lines represent the measured data while the solid line is the model fit for angles of incidence of 70o and 75o.
121
90 Model Fit 80 Exp 70o o 70 Exp 70
60
50
(degrees) 40 Ψ
30
20
10 (a) 0 180 Model Fit 160 Exp 70o o 140 Exp 75
120
100 (degrees)
∆ 80
60
40
20 (b)
0 200 400 600 800 1000 1200 1400 Wavelength (nm)
Figure 5.14 Measured and calculated Ψ and ∆ spectra of a-AlN thin film of 100 nm thickness sputtered onto c-Si (111). The dashed and dotted lines represent the measured data while the solid line is the model fit for angles of incidence of 70o and 75o. 122
Table 5.4 Best-fit model structural parameters for a-AlN thin films sputtered onto c-Si (111) substrates determined by the VASE measurements in the wavelength range 200-1400 nm. a The parameter is not chosen as a fitting parameter and it has a fixed value during the fitting procedure while b represents the beginning of our wavelength range. The 90 % confidence limits are given by (±).
Sample no./ a b c Parameter
Film thickness (nm) 52.86 ± 9.02 × 10-2 85.5 ± 0.19 102.52 ± 0.28
a a a SiO2 thickness (nm) 2.0 2.0 2.0 A 1.85 ± 3.7 × 10-3 1.86 ± 3.13 × 10-3 1.94 ± 3.4 × 10-3 B 2.33 × 10-2 ± 4.7 × 10-4 2.2 × 10-2 ± 4.4 × 10-4 5 × 10-3 ± 2.8 × 10-4 C 0 a 0 a 0 a α 0.56 ± 3.7 × 10-2 0.52 ± 4.8 × 10-2 0.52 ± 2.24 × 10-2 β 1.5a 1.6 a 1.8 a γ (nm) 200 b 200 b 200 b MSE 2.89 3.21 4.03
2.5 0.6 n 2.4 κ κ 0.5 2.3
0.4 2.2
0.3 2.1 Extinction coefficient coefficient Extinction Index of refraction n of refraction Index 2.0 0.2
1.9 0.1
1.8 0.0 200 400 600 800 1000 1200 1400 WAvelength (nm)
Figure 5.15 Optical constants of a-AlN thin film of 50 nm sputtered onto c-Si (111) substrate obtained from a Cauchy-Urbach model (200 – 1400 nm).
123
From the extinction coefficient (κ) spectrum (Fig.5.15), we can obtain the absorption coefficient (α) via α = 4π κ/λ, where λ is the wavelength of the incident light. In Fig.5.16,
(E α n) 2 is plotted versus the photon energy, E. The (E α n) 2 law provides a good fit to the experimental data and by using a linear extrapolation technique we obtain the bandgap energy for a-AlN thin film of 50 nm thickness to be around 5.78 eV. By the same method, the bandgap energy for other a-AlN samples with different thicknesses
(Table 5.4) was found to be approximately 5.82 ± 0.05 eV (see Figs. 5.17 and 5.18). This value is in excellent agreement with previous results in the literature [146]. Hence the bandgap of a-AlN films is less than that in the c-AlN films as reported in Figs.5.16 - 5.18 and Table 5.5. This is attributed to the loss of long-range order on the electronic structure in the amorphous materials (see Chap.2, Sec.2)
3.0x1013 Experimental data of 50 nm a-AlN thin film Linear Fit 2.5x1013 ) -1 2.0x1013 (eV cm (eV 2 13
n) 1.5x10 α (E 1.0x1013
5.0x1012 E = 5.78 eV g
0.0 234567 Photon energy (eV)
Figure 5.16 Determination of the energy bandgap. The (E α n) 2 is Plotted vs. photon energy E for a-AlN thin film (50 nm) sputtered onto c-Si (111) substrate. 124
2.5x1013 Experimental data of 85 nm a-AlN thin film Linear Fit 2.0x1013 ) -1
1.5x1013 (eV cm (eV 2
n) α 13 (E 1.0x10
5.0x1012
Eg = 5.83 eV 0.0 234567 Photon energy (eV)
Figure 5.17 Determination of the energy bandgap. The (E α n) 2 is Plotted vs. photon energy E for a-AlN thin film (85 nm) sputtered onto c-Si (111) substrate.
2.5x1013 Experimental data of 100 nm a-AlN thin film 2.0x1013 Linear Fit ) -1 1.5x1013 (eV cm (eV 2
n) α 1.0x1013 (E
5.0x1012
Eg = 5.86 eV 0.0 234567 Photon energy (eV)
Figure 5.18 Determination of the energy bandgap. The (E α n) 2 is Plotted vs. photon energy E for a-AlN thin film (100 nm) sputtered onto c-Si (111) substrate. 125
Table 5.5 Optical constants of a-AlN thin films at bandgap energy.
(a) Experimental results, a-AlN thin films Sample Energy gap Index of Extinction No. (eV) refraction coefficient
a 5.78 2.35 0.298 b 5.83 2.36 0.315 c 5.86 2.28 0.306
(b) Literature data, c-AlN thin films Optical technique Deposition Energy gap Reference technique (eV)
RT method DCd-RMS1 5.91 [26] T method LA-CVD2 6.0 – 6.2 [147] Photoreflectance method PSMBE3 6.2 [72]
1 Direct current diode- reactive magnetron sputtering. 2 Laser assisted chemical vapor deposition. 3 Plasma source molecular beam epitaxy.
5.2.3 Polarized Optical Reflectivities and Brewster Angle
The spectral dependence of the polarized optical reflectivities, Rs and Rp, at seven angles of incidence θi for a 85 nm thick a-AlN film on a glass substrate are shown in Fig.5.19 (a) and (b), respectively. The Rs, p spectra were measured in the spectral range 300 - 1400 nm. The figure shows that the s-polarized reflectivity, Rs, increases with increasing wavelength for λ < 400 nm and decreases monotonically with increasing wavelength for
λ > 400 nm at all angles of incidence indicating that λ = 400 nm represents a “critical wavelength” in the spectral behavior of the s- polarized reflectivity. 126
0.6
0.5
0.4
0.3
s-Polarized Reflectance s-Polarized Reflectance 0.2
0.1 (a) 0.0 200 400 600 800 1000 1200 1400 Wavelength (nm)
0.3 20o 30o 40o 50o 0.2 60o 70o 80o p-Polarized Reflectance 0.1
(b) 0.0 200 400 600 800 1000 1200 1400 Wavelength (nm)
Figure 5.19 Spectroscopic reflection data for (a) s- polarized reflectance Rs (λ) (b) p-polarized reflectance Rp (λ) for the a-AlN thin film of 85 nm thickness sputtered onto a glass substrate.
127
Also, as can be seen from the figure, there is a “critical wavelength”, 400 nm, in the spectral dependence of the p-polarized (Rp) but it exhibits a different spectral behavior when the angle of incidence is grater than 60o.
The plots of Rs and Rp at two wavelengths, 400 and 1200 nm, versus angle of incidence θi are shown in Fig.5.20. As can be seen in the figure, the distinction between parallel and perpendicular components of the reflectance disappears when the angle of incidence goes to zero (normal incidence), due to the indistinguishability of the s- and p- directions at normal incidence. In general, when a p-polarized light wave is incident on the interface between two transparent media (i.e., κ1 = κ2 = 0) the reflected wave entirely
p disappears (R = 0) at a particular angle of incidence, called the Brewster angle θB, and the incident wave is totally transmitted into the second medium. The resulting reflected light is polarized perpendicular to the plane of incidence (mathematical proof was discussed in Chap.3, Sec.3.1).
Figure 5.20 shows that the reflectance Rp for p-polarized light at λ = 1200 nm reaches a minimum value of about zero at 61o, the Brewster angle. This value is marked by the dashed arrow in the figure. If we assume the first medium to be air (n1 = 1.0 + i
0.00) and the second medium to be a-AlN thin film (Fig.5.2) then by using the following equation:
n2 = n1 tan θ B = 1× tan 61 (5.8)
the index of refraction of the a-AlN thin film n2 = 1.81 at λ = 1200 nm. This value is comparable to the result obtained from the analysis of the measured ellipsometric spectra 128 of a-AlN films by the Cauchy-Urbach model (n = 1.86 + i 1.88 × 10-5) at the same wavelength as reported in Figs.5.8 – 5.10 and Fig.5.15.
0.7
λ = 400 nm (n = 1.99 + i 0.001) s -5 0.6 λ = 1200 nm (n = 1.86 + i 1.88 x 10 ) R
0.5 Rs
0.4
p R 0.3 Principal angle
0.2 p
Polarized Reflectances θ B R
0.1
0.0 20 30 40 50 60 70 80 Angle of Incidence θ (degrees) i
Figure 5.20 The polarized reflectances, Rp and Rs, for the p and s polarizations as functions of the angle of incidence θi (degrees). These curves are obtained assuming that medium 1 is air and medium 2 is a-AlN film with n = 1.99 + i 0.001 at λ = 400 nm (solid lines) and n = 1.86 + i 1.88 × 10-5 at λ = 1200 nm (dashed lines). The n values for a-AlN thin film are taken from Fig.5.8.
When the reflecting surface is not a dielectric material (i.e., κ2 ≠ 0), the situation becomes more complicated. The Fresnel’s reflection coefficients are complex numbers, so the
p reflectance does not go to zero at any θi .The angle for which R is a minimum is therefore called the principal angle of incidence, θp, or the pseudo-Brewster angle [134].
In Fig.5.20, at λ = 400 nm (solid lines; n = 1.99 + i 0.001), the principal angle is about
64o. Thus, the minimum value of Rp depends on the extinction coefficient κ. If the incident plane wave is s-polarized (instead of p-polarized), the corresponding polarized reflectance Rs increases monotonically from normal to grazing incidence, where it 129 approaches unity, without going through a minimum at any angle of incidence as shown in the plotted figure.
5.2.4 Polarized Optical Transmissivity and Absorptivity
The spectral dependence of the s- and p-polarized transmittances Ts and Tp at three angles of incidence, 50o, 60o and 70o, are shown in Figs. 5.21 and 5.22, respectively. We have chosen these angles to study the optical polarized transmittance behavior of a-AlN film
o s p up and below the Brewster angle, θB = 61 . The figure shows that T and T are also angle dependent with the value of the polarized transmittance increasing as the angle of incidence decreases. Also, the s-polarized transmittance shows a more systematic behavior than the p-polarized transmittance due to the fact that there are specific angles
(θB and θp) that change the optical behavior of the p-polarized transmittance. In addition, the dispersion curve of Ts increases with increasing wavelength for λ > 400 nm while Tp shows about constant a value in the same wavelength range. Moreover, the value of Tp is always larger than Ts indicating that the film is more transparent (80 % - 93 %) for the p- polarized state than for the s-polarized state in the visible and near-IR regions. This is further verified by the fact that the s-polarized reflectivity is greater than the p-polarized reflectivity (see Fig.5.19) while the s- and p- polarized absorptivities are about constant in the same wavelength range (the lower curves in Figs.5.21 and 5.22).
To fully understand the optical characteristics of a-AlN films we studied the optical polarized absorptivities As and Ap. The As and Ap were calculated using the law of conservation of energy, given by 130
R s, p (λ) + T s, p (λ) + As, p (λ) = 1, (5.9) where As, p is defined as the ratio of the polarized absorbed intensity to the incident intensity. The spectral dependence of As and Ap at three angles of incidence are shown in
Figs.5.21 and 5.22 (the lower curves).
0.7
0.6
0.5 T s 0.4
o 0.3 50 60o o 0.2 70
s-Polarized Optical Properties s 0.1 A
0.0 200 400 600 800 1000 1200 1400 Wavelength (nm)
Figure 5.21 Spectroscopic transmission and absorption data for s- polarized transmittance Ts (the upper curves) and absorptance As (the lower curves) for the a-AlN thin film of 85 nm thickness sputtered onto a glass substrate.
131
1.0 T p
0.8
0.6 50o 60o o 0.4 70 p-Polarized Optical Properties
0.2
A p 0.0 200 400 600 800 1000 1200 1400 Wavelength (nm)
Figure 5.22 Spectroscopic transmission and absorption data for p- polarized transmittance Tp (the upper curves) and absorptance Ap (the lower curves) for the a-AlN thin film of 85 nm thickness sputtered onto a glass substrate.
The above figures show that the polarized absorptivities are angle dependent with the value increasing as the angle of incidence increases. Besides, the a-AlN thin film shows a low polarized absorptivity, less than 15 %, in the visible and near infrared regions at the chosen angles.
As an intensive investigation to the optical behavior of sputtered a-AlN thin films onto glass substrates, we studied their s-and p-polarized optical properties at 60o, which is the closet angle to the critical angle (see Fig.5.20). In Figs.5.23 (a) and (b), we plotted the s-and p-polarized optical reflectivities Rs,p as functions of wavelength and film thickness.
The figures show that, the values of Rs,p decrease monotonically with increasing wavelength and increase with increasing film thickness for λ > 400 nm. As can be noticed in the figures, the values of Rs ranging from 30 % to 50 % are greater than Rp (4 ×10-3 – 132
2.15 ×10-2) indicating that the resulting reflected light is polarized perpendicular to the plane of incidence at 60o.
The s-and p-polarized optical transmissivities Ts,p as functions of wavelength and film thickness at 60o are shown in Figs.2.24 (a) and (b), respectively. From the figures it is evident that the measured Ts of a-AlN thin films are dependent upon the wavelength of the incident light as well as the film thickness while the measured Tp are relatively featureless for λ > 400. The value of Ts increases with increasing wavelength whereas it decreases when the thickness of the film increases. In addition, the range of Ts and Tp values was found to be about 45 % - 65 % and 90 % - 95 % , respectively at the chosen angle. Moreover, the figures show that the value of Tp is always larger than Ts indicating that the thin films are more transparent for the p-polarized state than for the s-polarized state. This is further verified by the fact that the s-polarized reflectivity is greater than the p-polarized reflectivity (see Figs.5.23 (a) and (b)).
To verify the above results, the s-and p-polarized optical absorptivities As,p of the studied films were calculated using equ.(5.6) at 60o. The As,p data are reported in
Figs.5.25 (a) and (b). In the figures, the polarized optical absorptivities were nearly constant and very small (< 8 × 10-2) at high wavelength ranges (> 400 nm), but exhibited various dispersion features at low wavelength ranges, which is again “similar” to the result obtained from the analysis of the extinction coefficient (Fig. 5.12). 133
0.5
0.4
0.3
0.2
s-polarized reflectivitys-polarized Sample #1 (25 nm) Sample #2 (50 nm) Sample #3 (85 nm) 0.1 Sample #4 (100 nm)
(a) 0.0 200 400 600 800 1000 1200 1400 Wavelength (nm)
0.024 0.022 Sample #1 (25 nm) 0.020 Sample #2 (50 nm) Sample #3 (85 nm) 0.018 Sample #4 (100 nm) 0.016 0.014 0.012 0.010
p-polarized reflectivityp-polarized 0.008 0.006 0.004 0.002 (b) 0.000 200 400 600 800 1000 1200 1400 Wavelength (nm)
Figure 5.23 Spectroscopic reflection data at 60o for (a) s-polarized reflectivity Rs and (b) p- polarized reflectivity Rp for the a-AlN thin films sputtered onto glass substrates. 134
0.7
0.6
0.5
0.4 Sample #1 (25 nm) Sample #2 (50 nm) 0.3 s-Polarized transmissivity Sample #3 (85 nm) Sample #4 (100 nm)
0.2
(a) 0.1 200 400 600 800 1000 1200 1400 Wavelength (nm)
1.00
0.95
0.90
0.85
0.80
0.75
0.70
0.65
p-polarized transmissivity Sample #1 (25 nm) 0.60 Sample #2 (50 nm) 0.55 Sample #3 (85 nm) Sample #4 (100 nm) 0.50 (b) 0.45 200 400 600 800 1000 1200 1400 Wavelength (nm)
Figure 5.24 Spectroscopic transmission data at 60o for (a) s-polarized transmittance Ts and (b) p- polarized transmittance Tp for the a-AlN thin films sputtered onto glass substrates.
135
0.55
0.50 Sample #1 (25 nm) 0.45 Sample #2 (50 nm) Sample #3 (85 nm) 0.40 Sample #4 (100 nm) 0.35
0.30
0.25
0.20 s-polarized absorptivity 0.15
0.10
0.05 (a) 0.00 200 400 600 800 1000 1200 1400 Wavelength (nm)
0.60 0.55 Sample #1 (25 nm) 0.50 Sample #1 (50 nm) 0.45 Sample #1 (85 nm) 0.40 Sample #1 (100 nm) 0.35 0.30 0.25
p-polarized absorptivity 0.20 0.15 0.10
0.05 (b) 0.00 200 400 600 800 1000 1200 1400 Wavelength (nm)
Figure 5.25 Spectroscopic absorption data at 60o for (a) s-polarized absorptivity As and (b) p- polarized absorptivity Ap for the a-AlN thin films sputtered onto glass substrates.
136
5.3 The Optical Characteristics of Sputtered Amorphous Indium Nitride
Thin Films
In order to fully understand the optical behavior of sputtered amorphous InN as single layers on different substrates, nine types of measurements were carried out in air at room temperature in the wavelength range 300 – 1400 nm. These types of measurements are summarized in Table 5.6.
Table 5.6 Optical data types measured on the sputtered amorphous InN thin films
Substrate type Data type Angle of incidence Description
c-Si (111) SE 70 o –75 o by 5 o Standard VASE: Ψ and ∆ Glass s-IR 75 o s- polarized intensity reflectance Glass p-IR 75 o p- polarized intensity reflectance Glass s-IT 75 o s- polarized intensity transmittance Glass p-IT 75 o p- polarized intensity transmittance Glass s-IR 20 o – 80 o by 10 o s- polarized intensity reflectance Glass p-IR 20 o – 80 o by 10 o p- polarized intensity reflectance Glass s-IT 10 o – 80 o by 10 o s- polarized intensity transmittance Glass p-IT 10 o – 80 o by 10 o p- polarized intensity transmittance
5.3.1 The Optical Constants
The set of samples (1 – 4) shown in Table 4.1 of Chap.4 were used to characterize amorphous indium nitride, a-InN, thin films. The model structure used to fit the SE data taken on all a-InN thin films is shown in Fig.5.26. Again the optical constants of the crystalline silicon substrate and silicon dioxide (SiO2) over layer were taken from the literature and were not allowed to vary during the fitting process [140]. Modeling of InN 137 thin films is quite challenging since the optical properties of these thin films are affected by the amount of disorder in the sample, the deposition rate, impurities, vacuum conditions, the lack of a suitable substrate and other factors [64, 70]. Furthermore, surface conditions and the formation of an oxide layer on the surface may strongly affect the ellipsometric measurements. To model the a-InN thin films, the Tauc-Lorentz model was used to obtain the optical constants. Recently Jellison and Modine [148] derived a
Tauc-Lorentz (TL) model for the optical functions of amorphous semiconductor materials. This formulation uses a combination of the Tauc joint density of states and the
Lorentz oscillator model [142] for the dielectric response for a collection of non- interacting atoms. If only a single transition is considered, then the imaginary part of the complex dielectric function ε2 is given by
2 ⎧ AΕ 0C(Ε − Ε g ) 1 * (Ε > Ε ) (5.10) ⎪ 2 2 2 2 2 g ε 2 (Ε) = ⎨(Ε − Ε 0 ) + C Ε Ε ⎪ ⎩0 (Ε ≤ Ε g )
As with Forouhi and Bloomer [142] the real part of the dielectric function ε1 is determined in a closed form from ε2 using Kramers-Kroning integration, given by
2 ∞ ε (E) = ε (∞) + P∫ ξε (ξ)(ξ 2 − E2 )−1dξ (5.11) 1 1 π Eg 2 where the P stands for the Cauchy principal part of the integral. The model dielectric function, Tauc-Lorentz, employs five fitting parameters: the optical bandgap energy Eg, the Lorentz resonant frequency Eo, the broadening parameter C, the prefactor A, which includes the optical transition matrix elements, and the non-dispersive term ε1 (∞). The three fitting parameters, Eg, Eo, and C are in units of energy while A and ε1 (∞) are dimensionless. ε1 (∞) represents the contribution of the optical transitions at higher 138 energies and appears as an additional fitting parameter [149]. It is usually assumed that
ε1 (∞) equals unity [36,142], but it can be greater than one if there is a significant optical transition at an energy greater than is sampled by the ellipsometer [150]. Also, the exact value of ε1 (∞) depends on the particular material. Klazes et al. [151] found the value of
ε1 (∞) for a-Si: H to be 2.28. Joseph et al. [152] found ε1 (∞) for a-TiO2 to equal 1.76, while Postava et al. [153] found the value for TiO2 to be 2.19. Our results, however, indicated that ε1 (∞) >1.
Although the TL expression (Eqs.5.10 and 5.11) is empirical, it does satisfy the major criteria for models of dielectric functions. The TL expression is consistent with known physical phenomena, within the limitations of the model. At large E, the ε2 (E) of the TL model → 0 as 1/E3. This is consistent with observed behavior in the x-ray and γ- ray regime, where it is known that the absorption coefficient is very small. Furthermore,
ε2 (E) = 0 below Eg. The only mechanisms that give a non-zero value of ε2 (E) below the band gap are the mechanisms that are explicitly ignored in the TL model, such as Urbach tail absorption and vibrational absorption in the infrared. Also, the TL expression is
Kramers-Kronig consistent, in that ε1 (E) is determined by Kramers-Kronig integration.
This implies that the integration of Eq.5.11 must converge, which requires that ε2 (E)
2 3 must approach 0 at least as fast as 1/E . As mentioned above, ε2 (E) → 0 as 1/E .
The above discussion therefore demonstrates that if ε2 is given by equation (5.10) then five parameters, i.e., A, B, C, Eg and ε1 (∞), are sufficient to fully describe the dependence of the optical constants on the incident energy E. The surface roughness, SR, layer was chosen as an over layer (see Fig.5.26) to obtain accurate a-InN thin film optical 139 constants and a good fitting to the experimental data. The SR was modeled using a
Bruggeman effective medium approximation [36] consisting of 50 % of the a-InN thin film and 50 % voids. The fitting was performed using a Levenberg – Marquardt algorithm, where five parameters were fitted: the film thickness, the surface roughness thickness, and three parameters, A, B, Eg, from the Tauc – Lorentz model. C, ε1 (∞), SiO2 thickness, and the Kramers – Kroing (KK) parameters layer [139] : Pole # 1 position
(eV), Pole # 2 position (eV), Pole # 1 magnitude and Pole # 2 magnitude parameters were set to a fixed value. The KK parameters are known as center energy (position) and strength (magnitude) of two zero width oscillators. The best-fit model parameters with a satisfactory MSE for the four samples (Table 4.1) are summarized in Table 5.7. The value of SR as shown in the Table confirms that there is no strong correlation between the roughness thickness and some other fitting parameters in the model. Also the SR yielded a significantly better MSE. Furthermore, the values for each fitting parameter for all samples are close, confirming the validity of the model used for a-InN. A good quality fitting with a satisfactory MSE was obtained for the spectral range of 300 – 1400 nm.
Air
Surface roughness
a-InN thin film
SiO2 layer
c-Si (111)
Figure 5.26 Schematic drawing of the optical model structure for the SE analysis of a-InN film. 140
Table 5.7 Best-fit model structural parameters for a-InN thin films sputtered onto c-Si (111) substrates determined by the VASE measurements in the wavelength range 300-1400 nm. a The parameter is not chosen as a fitting parameter and it has a fixed value during the fitting procedure. The 90 % confidence limits are given by (±).
Sample no. / 1 2 3 4 Parameter Film thickness 87.98 ± 0.13 157.23 ± 0.23 220.03 ± 0.49 257.18 ± 0.60 (nm) a a a a SiO2 thickness 2.2 2.0 2.0 2.0 (nm) SR thickness 3.72 ± 0.11 5.825 ± 0.19 5.88 ± 0.21 6.58 ± 0.23 (nm) Amp (eV) 59.94 ± 0.40 56.23 ± 0.53 59.54 ± 0.60 54.30 ± 0.59 En (eV) 4.49 ± 3.6 4.35 ± 5.2 4.40 ± 5.8 4.65 ± 6.98 × 10-2 × 10-2 × 10-2 × 10-2 C (eV) 10 a 10 a 10 a 10 a
Eg (eV) 1.65 ± 5.6 1.72 ± 5.9 1.68 ± 7.2 1.57 ± 8.4 × 10-3 × 10-3 × 10-3 × 10-3 a a a a ε1(∞) 2.4 2.6 2.4 2.65 Pole # 1 position 13 a 13 a 13 a 13 a (eV) Pole # 2 position 0.001 a 0.001 a 0.001 a 0.001 a (eV) Pole # 1 magnitude 8 a 8 a 8 a 8 a Pole # 2 magnitude 0.85 a 0.50 a 0.40 a 0.35 a MSE 1.43 2.01 2.47 3.41
Figures 5.27 – 5.30 show the measured and fitted spectra of the ellipsometric parameters,
Ψ and ∆, for four samples of a-InN thin films sputtered onto c-Si (111) substrates over
the spectral range 300 – 1400 nm. The Ψ and ∆ as a function of wavelength and at two angles of incidence, 70o (dashed lines) and 75 o (dotted lines), show steep peaks which are more pronounced in the spectrum of ∆ (λ) for λ > 600 nm. These peaks indicate the minimum value of the extinction coefficient κ in that region. In addition, we found that as 141 the thickness of the film increases the number of peaks increases with a constant maximum ∆ value of about 177o when λ > 600 as shown in Fig.5.31.
35 Model Fit Exp 70o 30 Exp 75o
25
20 (degrees)
Ψ 15
10
5 (a)
0 200 400 600 800 1000 1200 1400 Wavelength (nm)
180
160 Model Fit Exp 70o 140 Exp 75o
120
100
(degrees) 80 ∆ 60
40
20 (b)
0 200 400 600 800 1000 1200 1400 Wavelength (nm)
Figure 5.27 Ellipsometric data for parameters (a) Ψ (λ) and (b) ∆ (λ) of a-InN thin film (sample no.1). The dashed and doted lines represent the experimental data while the solid line is the model fit for angles of incidence, 70o and 75o.
142
40 Model Fit Exp 70o 35 Exp 70o 30
25
20 (degrees) Ψ 15
10
5 (a) 0 200 400 600 800 1000 1200 1400 Wavelength (nm)
180 Model Fit o 160 Exp 70 Exp 70o 140
120
100
(degrees) 80 ∆
60
40
20 (b) 0 200 400 600 800 1000 1200 1400 Wavelength (nm)
Figure 5.28 Ellipsometric data for parameters (a) Ψ (λ) and (b) ∆ (λ) of a-InN thin film (sample no.2). The dashed and doted lines represent the experimental data while the solid line is the model fit for angles of incidence, 70o and 75o.
143
35
30
25
20
(degrees) 15 Ψ Model Fit Exp 70o 10 Exp 75o
5 (a) 0 200 400 600 800 1000 1200 1400 Wavelength (nm)
180 Model Fit 160 Exp 70o o 140 Exp 75
120
100
(degrees) 80 ∆
60
40
20 (b) 0 200 400 600 800 1000 1200 1400 Wavelength (nm)
Figure 5.29 Ellipsometric data for parameters (a) Ψ (λ) and (b) ∆ (λ) of a-InN thin film (sample no.3). The dashed and doted lines represent the experimental data while the solid line is the model fit for angles of incidence, 70o and 75o.
144
30 Model Fit Exp 70o o 25 Exp 75
20
15 (degrees) Ψ
10
5 (a) 0 200 400 600 800 1000 1200 1400 Wavelength (nm)
180 Model Fit o 160 Exp 70 Exp 75o 140
120
100
(degrees) 80 ∆
60
40
20 (b) 0 200 400 600 800 1000 1200 1400 Wavelength (nm)
Figure 5.30 Ellipsometric data for parameters (a) Ψ (λ) and (b) ∆ (λ) of a-InN thin film (sample no.4). The dashed and doted lines represent the experimental data while the solid line is the model fit for angles of incidence, 70o and 75o.
145
200 a-InN thin films Model Fit 180 Sample no.1 Sample no.2 160 Sample no.3 140 Sample no.4
120
100
(degrees)
∆ 80
60
40
20
0 200 400 600 800 1000 1200 1400 Wavelength (nm)
Figure 5.31 The spectra of the ellipsometric parameter ∆ for a-InN thin films as a function of wavelength at 70o. The lines (dash, dot, dash dot dot, and short dash dot) represent the experimental data of the samples 1, 2, 3, and 4, respectively while the solid line is the model fit.
Also, the oscillatory behavior of the spectra of Ψ and ∆ come from interference effects and is more evident in the thicker samples (see Fig.5.31). As a snapshot of this behavior, we plotted the relative phase change, ∆, as a function of film thickness at 70o and wavelength 490 nm (see Fig.5.32). From the figure, it is very obvious the relative phase change is not a periodic function of the film thickness. Therefore, the damping of the ∆ oscillation is attributed to the increase in the absorption of the probing light with increase in the a-InN film thickness and hence low skin depth (d = α-1). Moreover, it is noticed that the relative phase change and the relative amplitude change, Ψ, show angle dependence. By changing the angle of incidence you are also changing the path length of the probe beam as it travels through a film. Clearly, the fit shown in Figs.5.27 – 5.30 is a 146 good one indicating that the Tauc-Lorentz model works very well and can be used to calculate the spectroscopic optical constants of the a- InN thin films.
100
λ = 490 nm 90 o θ = 70
80
70
60 (degrees) ∆ 50
40
30
80 100 120 140 160 180 200 220 240 260 280 Film thickness (nm)
Figure 5.32 The relative phase change ∆ for a-InN thin films as a function of film thickness at 70o and λ = 490 nm.
Figures 5.33 – 5.36 show the optical constants, n and κ, for a-InN samples. The refractive index, n, of the a-InN samples in this study varied from 2.05 to 2.45 in the range of 300 –
1400 nm, with the maximum occurring at about 500 nm. The extinction coefficient, κ, varies smoothly from 0.0 to 0.59. Above the wavelength of about 750 nm, κ is zero indicating that the films are transparent in this region. Because of the limited experimental and theoretical data on the optical constants of a-InN, we compared the general behavior of the optical constants of our thin films to that of c-InN films [64, 100, and 114] and observed the same behavior with different values of the optical constants 147
(Chap.2, Sec.5.2, Figs.2.13 and 2.15). Again we attribute the difference to the loss of long-range order on the electronic structure in the amorphous materials. In addition, the apparent thickness dependence of the optical constants in Fig.5.37 and Table 5.8 (< 0.1 for n and < 0.07 for κ) is due to the different values of the thicknesses of the surface roughness over layer. The Table shows that the thicknesses of the thin films by SE are in substantial agreement with those obtained by the crystal oscillator inside the vacuum chamber during the sputtering process and the surface profilometer.
2.4 n 0.6 κ 2.3 0.5 κ 2.2
0.4 2.1
2.0 0.3 Extinction coefficient coefficient Extinction Index of refraction n Index 1.9 0.2 1.8 0.1 1.7
1.6 0.0 200 400 600 800 1000 1200 1400 Wavelength (nm)
Figure 5.33 Index of refraction and the extinction coefficient obtained from the Tauc-Lorentz model for 90 nm a-InN thin film sputtered onto c-Si (111) substrate.
148
2.4 0.6 n 2.3 κ 0.5 κ 2.2 0.4 2.1
0.3 2.0 Extinction coefficient coefficient Extinction Index of refraction n Index 0.2 1.9
0.1 1.8
1.7 0.0 200 400 600 800 1000 1200 1400 Wavelength (nm)
Figure 5.34 Index of refraction and the extinction coefficient obtained from the Tauc-Lorentz model for 160 nm a-InN thin film sputtered onto c-Si (111) substrate.
2.4 0.6 n 2.3
κ κ 0.5 2.2 0.4 2.1
0.3 2.0 Extinction coefficient coefficient Extinction Index of refraction n Index 0.2 1.9
0.1 1.8
1.7 0.0 200 400 600 800 1000 1200 1400 Wavelength (nm)
Figure 5.35 Index of refraction and the extinction coefficient obtained from the Tauc-Lorentz model for 222 nm a-InN thin film sputtered onto c-Si (111) substrate.
149
2.5 0.6 n 2.4 κ 0.5 κ 2.3
0.4 2.2
0.3 2.1 Index of refraction n of refraction Index Extinction coefficient coefficient Extinction 0.2 2.0
1.9 0.1
1.8 0.0 200 400 600 800 1000 1200 1400 Wavelength (nm)
Figure 5.36 Index of refraction and the extinction coefficient obtained from the Tauc-Lorentz model for 260 nm a-InN thin film sputtered onto c-Si (111) substrate.
2.50 0.65 2.45 0.60 2.40 a-InN thin films 2.35 )
0.55 κ 2.30 2.25 0.50 2.20 0.45 2.15 2.10 0.40 2.05 0.35 2.00 1.95 0.30 1.90 n (sample no. 2) 0.25 Extinction Coefficient ( Coefficient Extinction
Index of refraction (n) of refraction Index 1.85 n (sample no. 3) 1.80 n (sample no. 4) 0.20 1.75 0.15 1.70 κ (sample no.2) 1.65 κ (sample no.3) 0.10 1.60 κ (sample no.4) 0.05 1.55 1.50 0.00 200 400 600 800 1000 1200 1400 Wavelength (nm)
Figure 5.37 Optical constants obtained from the Tauc-Lorentz model for a-InN thin films sputtered onto c-Si (111) substrates. The lines (solid, dash, and dash dot dot) represent the optical constants for samples 2, 3, and 4 (Table 5.7), respectively. 150
Table 5.8 Thicknesses of a-InN samples measured by using different techniques and their optical constants at two different wavelengths, 500 nm and 1200 nm. The 90 % confidence limits are given by (±).
Nominal Film Film Roughness Index of Extinction Sample film thickness by thickness by by refraction coefficient No. thickness Profilometer SE BEMAL at at (nm) (nm) (nm) (nm) 500 1200 nm 500 1200 nm
1 90 87.3 87.98 ± 0.13 3.72 ± 0.11 2.38 1.97 0.20 0 2 154 157.8 157.23 ± 0.23 5.83 ± 0.19 2.36 2.06 0.17 0 3 218 226 220.03 ± 0.49 5.88 ± 0.21 2.38 2.08 0.19 0 4 255 264.3 257.18 ± 0.60 6.58 ± 0.23 2.44 2.19 0.23 0
5.3.2 Optical Bandgap Energy
Recently there has been an exciting conflict on the optical bandgap of InN. This conflict was raised only just when a few groups showed by photoluminescence (PL) measurement that the band gap energy of InN is between 0.65 and 0.90 eV, [154, 155] much smaller than 1.9 eV,[88] the most commonly accepted value. The absorption edge or optical band gap energy has been primarily determined by optical absorption or transmission measurements. Osamura et al. performed optical absorption measurements near the fundamental absorption edge for polycrystalline thin films of the alloy of InN and GaN grown by electron beam plasma technique over the entire composition region [156]. They determined the band gap energy of InN to be 1.95 eV at room temperature and to 2.11 eV at 78 K. Some other groups also reported similar values of band gap energy, around 2 eV, measured by optical absorption in the polycrystalline InN film [157, 158]. Later, Tansley and Foley measured a fundamental absorption edge at 1.89 ± 0.01 eV in high purity InN films grown by reactive sputtering [88]. The measured band gap, i.e., 1.9 eV, has been 151 the most commonly accepted and referred band gap value of InN. There are also some other reports which support this value for the band gap of InN measured by optical absorption even for single crystal InN film [159].
In 2001, Yodo et al. [160] observed strong band edge luminescence at 8.5–200 K from 200 to 880 nm thick InN film grown on Si (001) and Si (111) substrates by electron cyclotron resonance-assisted molecular beam epitaxy (ECR-MBE). The InN film on the
Si (001) substrate exhibited strong band edge PL emission at 1.814 eV at 8.5 K, while those on Si (111) showed other strong band edge PL emission at 1.880, 2.081, and 2.156 eV at 8.5 K and 100 K, respectively.
Evidence of a narrower band gap for InN was reported in 2001. Inushima et al. insisted that the fundamental absorption edge of MBE grown InN layer lies around 1.1 eV, which is much lower than the previously reported values [161]. Davydov et al. reported a band gap value of 0.9 eV for high quality MBE grown InN, studied by means of optical absorption, PL, and photoluminescence excitation spectroscopy (PLE) [162].
They further studied in detail with different high quality hexagonal InN films grown by different epitaxy methods. Analysis of optical absorption, PL, PLE, and photoreflectivity data obtained on single crystalline hexagonal InN film leads to the conclusion that the true band gap of InN is around 0.7 eV [163]. Several other groups also have reported that the band gap of crystalline InN is less than 1 eV [154, 155]. Wu et al. showed good coincidence of data for optical absorption edge, PL peak energy, and photomodulated reflectance for MBE grown epitaxial layers with a carrier concentration of order of 1018 cm-3 and a mobility higher than 1000 cm2/Vs, which showed an energy gap for InN 152 between 0.7 and 0.8 eV [154]. Matsuoka et al. observed at room temperature strong PL at
0.76 eV as well as a clear absorption at 0.7–1.0 eV for the MOVPE grown InN film
[155].
After the above argument, the question that comes to the mind is that which value is the true bandgap of crystalline InN, ~ 2 or ~ 0.7 eV? And why does one material have two values for its bandgap? In fact there is no absolute answer, but from the standpoint that the larger band gap is correct, the emission at around 0.7 eV (1771.4 Ǻ) is interpreted as a deep level emission value. From the viewpoint of narrow band gap, the larger band gap cited in the literature may be due to the formation of oxynitrides, which have much larger band gaps than that of crystalline InN. As can be seen in Fig.5.38, energy gap data less than 1 eV were obtained for single crystalline InN film with a relatively low carrier concentration, while the larger values were mostly for polycrystalline InN film. It should also be pointed out that the band gap obtained from epitaxial films shows a remarkable dependence on carrier concentration [88], which is different from the larger one obtained from polycrystalline films. Polycrystalline films show a similar band gap (~ 2 eV) in spite of the wide range variation of carrier concentration 1016–1021 cm-3. Thus, these two values, 2 and 0.7 eV seem to be from different materials. As Motlan reported, oxygen
incorporation is one of the causes for the increase in band gap [164] Therefore, the larger
values may be related to oxygen incorporation into grown InN because polycrystalline
films can contain a high density of oxygen atoms at their grain boundaries. Davydov et al
[163] showed that the sample with band gap in the region of 1.8–2.1 eV contained up to
20% of oxygen, much higher than for samples with narrow band gap. It can be assumed 153 that oxygen is responsible for a high concentration of defects. In this case an increase of the band gap in wide gap samples can be caused by formation of oxynitrides, which have much larger band gap than that of InN film. 2.65
Poly-crystal (InNO) Motlan et al. [164] 2.2 Poly-crystal Tansley and Foley [88] 1.77
1 Bandgap energy (eV) Single-crystal Davydov et al. [163] 0.65
1x1016 1x1020 1x1021
Carrier concentration (1/cm)
Figure 5.38 Bandgap energy for InN films as a function of carrier concentration.
As discussed in Chap. 2.6, in a case of defect-free crystalline semiconductor, a gap in the distribution of electronic states occurs, the conduction band and valence band
distributions of states terminating abruptly at their respective band edges (see Fig.2.18).
In contrast, in an amorphous semiconductor, the distributions of conduction band and
valence band states do not terminate abruptly at the band edges. Instead, distributions of
tail states invade into the gap region (Fig.2.18), these states arising as a consequence of
the disorder which characterizes these semiconductors. In addition to tail states, there are
also states deep within the gap region, which arise from structural defects, such as
vacancies [165]. Thus, in an amorphous semiconductor, there is no true gap (i.e., quasi- 154 gap) [106] in the distribution of electronic states. Despite this fact, for the purposes of material characterization it has proven instructive to devise empirical measures for the badgap of an amorphous semiconductor. Measures of the gap which relate to measurements of the optical absorption coefficient spectrum, α (E), are the most common. Unfortunately, there is no pronounced feature of α (E) for an amorphous semiconductor which can be directly related to an optical badgap. While α (E) associated with a defect-free crystalline semiconductor terminates abruptly at the fundamental gap, in an amorphous semiconductor a tail in the absorption spectrum intrudes into the gap region [166]. This tail in the optical absorption coefficient spectrum, arising as a consequence of the tail and gap states [167], makes the optical badgap of an amorphous semiconductor difficult to define experimentally.
For many years, the Tauc model [168] has served as the standard empirical model whereby the optical badgap of an amorphous semiconductor may be determined. In this model, it is assumed that the disorder characteristic of amorphous semiconductors relaxes the momentum conservation rules. Assuming square root distributions of conduction band and valence band states, and assuming that the momentum matrix element is independent of ħω, Tauc et al. [168] concluded that an extrapolation of the essentially linear functional dependence of α(hω)hω , observed in amorphous semiconductors at a
sufficiently large ħω, allows an empirical optical gap to be defined. Cody et al. [169]
suggested that it is the dipole matrix element instead of the momentum matrix element
which is independent of ħω. Thus, they assert one should instead extrapolate
α(hω) / hω to define the optical gap. What is more, they pointed out that with the tail in 155 the optical absorption coefficient spectrum it is always possible to plot any root of α (ħω) as a function of ħω and obtain an apparent linear variation with ħω over some range of energies. The energy range over which this apparent linear variation is observed is also the subject of some controversy. However, to date there is no report on the optical bandgap value for amorphous InN, or on a method to investigate it. The determination of the bandgap energy of a-InN thin film presents a challenge and may add another dimension to the Eg conflict.
Based on the extinction coefficient κ spectrum, we can obtain the absorption
coefficient α by α = (4π κ) / λ, where λ is the wavelength of the incident light. In this
analysis, we determine the optical bandgap using a modified Tauc extrapolation, i.e., by
extrapolating the essentially linear functional dependence of Eα(E)n(E) , observed at sufficiently large E, to the horizontal axis. We modulated the linear functional dependence of Tauc extrapolation by incorporating the index of refraction n (E) into our
Tauc extrapolation. Figs.5.39 – 5.42 show the square root law Eα(E)n(E) for samples
1, 2, 3, and 4 as a function of photon energy, respectively. By using a linear extrapolation
of Eα(E)n(E) the absorption edge (or Tauc optical bandgap, EgTauc) was found to be
about 1.67 eV for the sample 1 (90 nm), 1.75 eV for the sample 2 (160 nm), 1.73 eV for
the sample 3 (220 nm), and 1.56 eV for the sample 4 (260 nm). These values are nearly
equal to the values of the optical bandgap energy obtained as fitting parameters in the
Tauc-Lorentz model (Table 5.7). 156
1600 Experimental data of 90 nm a-InN film 1400 Linear fit
1/2 1200 ) -1 1000 (eV cm (eV 1/2 800 n) α
(E 600 E = 1.67 eV g Tauc 400
200
0 1.01.52.02.53.03.54.04.5 Photon energy (eV)
Figure 5.39 Determination of the Tauc optical bandgap energy. Plot of (E α (E) n (E)) 1/2 as a function of photon energy E for 90 nm a-InN thin film sputtered onto a c-Si (111).
1600 Experimental data of 160 nm 1400 a-InN film Linear Fit 1200 1/2 ) -1 1000
(eV cm (eV 800 1/2
n) α 600 (E E = 1.75 eV 400 g Tauc
200
0 1.01.52.02.53.03.54.04.5 Photon energy (eV)
Figure 5.40 Determination of the Tauc optical bandgap energy. Plot of (E α (E) n (E)) 1/2 as a function of photon energy E for 160 nm a-InN thin film sputtered onto a c-Si (111).
157
1600
Experimental data of 220 nm 1400 a-InN film Linear Fit 1200 1/2 ) -1 1000
(eV cm (eV 800 1/2
n)
α E = 1.73 eV 600 g Tauc (E
400
200
0 1.01.52.02.53.03.54.04.5 Photon energy (eV)
Figure 5.41 Determination of the Tauc optical bandgap energy. Plot of (E α (E) n (E)) 1/2 as a function of photon energy E for 220 nm a-InN thin film sputtered onto a c-Si (111).
1600 Experimental data of 260 nm a-InN film 1400 Linear Fit
1/2 1200 ) -1 1000 (eV cm (eV 1/2 800 n) α
(E E = 1.56 eV 600 g Tauc
400
200
0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 Photon energy (eV)
Figure 5.42 Determination of the Tauc optical bandgap energy. Plot of (E α (E) n (E)) 1/2 as a function of photon energy E for 260 nm a-InN thin film sputtered onto a c-Si (111).
158
Figures 5.43 (a) and (b) show the determination of the optical bandgap energy of sputtered 160 nm a-InN thin film onto c-Si (111) and glass substrates by using spectroscopic ellipsometry and spectrophotometric (SP) methods, respectively. The SP method depends on the transmitted intensity of the monochromated light through the film. The absorption coefficient α using SP method is calculated according to the Beer-
Lambert law (Chap.3, Sec.2). As shown in the inset figure, the values of α by means of
SE and SP methods are almost equal either from the beginning of the absorption edge or at the end of the desired photon energy range. It is also shown that the value of the optical bandgap via the SP method (1.74 eV) is much closer not only to that by the analysis of
SE data method through the quadratic law (1.75 eV), but also to the values of Eg as a fitting parameter in the chosen model (Table 5.7). The modified Cody extrapolation used in our analysis for the same sample is shown in Fig.5.44. The corresponding Cody optical gap, EgCody, is found to be 1.72 eV which is “exactly” equal to the fitting value: 1.72 ± 5.9
× 10-3 eV. It should also be pointed out that the square root law provides a good fit to the
experimental a-InN thin films data (Figs.5.39 – 5.44) as well as a-Si thin films [24, 170].
As verification we can prove mathematically the correct power of α (E) as a
function of E. The average values of the optical bandgap energy as fitting parameters in
the Tauc – Lorentz model, Eg average, are equal to 1.66 ± 0.055 eV. Now if we accept these
values as a “fact” according to what we obtained for the spectral range of 300 – 1400 nm
(such as a high quality fitting and a satisfactory MSE, see Figs.5.29-5.30) we can simply find the correct root of α (E) as shown in Fig.5.45. The figure shows that the slope of the 159 relationship between ln (E α n) and ln (E – 1.69) is equal to 0.49. This value is in excellent agreement with Tauc et al. [168] and Cody et al. [169] results.
0.5
2.5x105 (a) ) -1 2.0x105
0.4 (cm α 1/2 ) 5 -1 1.5x10
0.3 1.0x105 (eV nm (eV Absorption Coefficient Coefficient Absorption
1/2
4 n) 0.2 5.0x10 α (E 0.0 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 Photon Energy (eV) 0.1 E = 1.75 eV g Tauc Experimental data of 160 nm a-InN film on c-Si (111) by SE Linear Fit 0.0 1.01.52.02.53.03.54.04.5 Photon Energy (eV)
(b) 1000 2.5x105 ) -1
(cm 5 α 2.0x10
1/2 800 ) 5 -1 1.5x10
1.0x105 (eV cm (eV 600 Absorption Coefficient 1/2 )
α 5.0x104 (E
400 1.52.02.53.03.54.0 Photon Energy (eV) E = 1.74 eV g Tauc Experimental data of 160 nm a-InN film on glass by spectrophotometric 200 Linear Fit
1.52.02.53.03.54.0 Photon Energy (eV)
Figure 5.43 Determination of the energy bandgap using (a) Spectroscopic Ellipsometry and (b) Spectrophotometric methods. (Eα(E)n(E)) 1/2 is plotted vs. the photon energy E for a-InN thin film of 160 nm thickness sputtered onto c-Si (111) substrate. The absorption coefficient as a function of wavelength is shown in the inset. 160
140 Experimental data for sample no.2 (160 nm) a-InN film on c-Si (111)
-1/2 120 Linear Fit
100 (cm eV) (cm 1/2 80 )(E)/E) n E = 1.72 eV 60 gCody (E)( α ( 40
20
0 0.8 1.0 1.2 1.4 1.6 1.8 2.0 2.2 Photon energy (eV)
Figure 5.44 Determination of the Cody optical bandgap energy. Plot of the quadratic law (α(E)n(E)/E) 1/2 as a function of photon energy for 160 nm a-InN thin film sputtered onto a c-Si (111).
m 1 (α n E) = A (E-E ) g m ln (α n E) = ln A + ln (E-1.66) m = 0.49 ( The slope) 0
-1 ln (E-1.66) ln
-2
-3 -14 -12 -10 -8 -6 -4 -2 ln (α n E)
Figure 5.45 Why for a-InN thin films the square-root dependence be used?
161
Figure 5.46 shows photoluminescence PL spectra at T< 325 K of sample 4 (260 nm).
Only one strong band edge PL emission was observed at 1.6 eV. Also, T< 325 K PL spectra of sample 4 exhibited only one weak band edge PL emission at 1.63 eV. The strong PL emission value is in excellent agreement with those values obtained either as a fitting parameter in the Tauc-Lorentz model (Table 5.7) or by the analysis of the absorption coefficient through the quadratic law (Eg Tauc) as shown in Fig.5.42.
In summary, the bandgap of a-InN thin films (1.68 ± 0.071 eV) is less than the
most acceptable value in c-InN films (1.9 eV) and also it is much higher than the recently
controversial value (~ 0.75 eV). This is because disorder is known to influence the optical
bandgap of amorphous semiconductors [171].
InN/Si (111) 1.6 eV sample no.4 (260 nm) T < 325 K
1.63 eV PL intensity (arb. (arb. units) intensity PL
740 760 780 800 820 840 860 Wavelength (nm)
Figure 5.46 Photoluminescence (PL) spectra at 20 K of sample no.4 (260 nm).
162
5.3.3 Polarized Optical Reflectivities and Brewster Angle
The spectral dependence of the polarized optical reflectivities Rs and Rp at different
angles of incidence θi for a 160 nm thick film are shown in Figs.5.47 (a) and (b),
respectively. There exist oscillations in the 300 – 750 nm ranges and there are no
oscillations in the 750 – 1400 nm ranges. This points out that the film is practically
absorbing in the oscillation region, as is indicated by the sharp peaks especially at θi <
60o. Also, as shown in the figure the spectral dependence of s-polarized reflectivity Rs (λ)
increases with the angle of incidence whereas the spectral dependence of p-polarized
reflectivity Rp (λ) does not. This means that there are some particular angles in the optical
behavior of the Rp (λ) (see Fig.5.48).
The plots of Rs and Rp at two wavelengths, 350 and 1200 nm, versus angle of incidence θi are shown in Fig.5.48. As can be seen in the figure, the distinction between parallel and perpendicular components disappears when the angle of incidence goes to zero (normal incidence). Also, the figure shows that the reflectance for p-polarized light
Rp at λ = 1200 nm reaches a minimum value of zero at the Brewster angle of 64o. This
value is marked by the dashed arrow in the figure.
If we assume the first medium to be air (n1 = 1) and the second medium to be a-
InN thin film then the index of refraction of the thin film (n2) can be calculated as
following:
n2 = n1 × tan θ B = 1× tan 64 = 2.05, λ = 1200 nm (5.12) 163
This value is similar to the result obtained by the analysis of the measured ellipsometric spectra of a-InN thin films through the Tauc-Lorentz model (n2 = 2.06 + i 0.00) at the
same wavelength as shown in Table 5.8 and Fig.5.37.
0.5
0.4
0.3 s-Polarized reflectivity 0.2 20o 60o 30o 70o 40o 80o 0.1 50o (a)
200 400 600 800 1000 1200 1400 Wavelength (nm)
0.22 20o o 0.20 30 40o 0.18 50o o 0.16 60 70o 0.14 0.12 0.10
p-Polarized reflectivity p-Polarized 0.08 0.06 0.04
0.02 (b) 0.00 200 400 600 800 1000 1200 1400 Wavelength (nm)
Figure 5.47 Spectroscopic reflection data for (a) s- polarized reflectance and (a) p- polarized reflectance for the a-InN thin film of 160 nm thickness sputtered onto a glass substrate.
164
0.6
λ = 350 nm (n = 2.26 + i 0.44) 0.5 λ = 1200 nm (n = 2.06 + i 0.0) Rs 0.4 Rs 0.3
Rp 0.2 Principal angle = 67o Polarized Reflectances 0.1 θ = 64o p B R
0.0 20 30 40 50 60 70 80 Angle of Incidence θ (degrees) ι
Figure 5.48 The polarized reflectances, Rp and Rs, for the p- and s-polarizations as functions of the angle of incidence θi (degrees). These curves are obtained assuming that medium 1 is air and medium 2 is a-InN film with n = 2.26 + i 0.44 at λ = 350 nm (solid lines) and n = 2.06 + i 0.00 at λ = 1200 nm (dashed lines). The n values for a-InN thin film are taken from Fig.5.37.
The angle for which Rp is a minimum is known as the principal angle of incidence or
pseudo-Brewster angle. In Fig.5.48, at λ = 350 nm (solid lines), the principal angle is
about 67o. Thus, the minimum value of Rp depends on the value of the extinction coefficient κ. If the incident plane wave is s-polarized (instead of p-polarized), the corresponding polarized reflectance Rs increases monotonically without going through a
minimum at any angle of incidence as shown in figure. 165
5.3.4 Polarized Optical Transmissivities and Absorptivities
The spectral dependence of the polarized transmittances Ts (λ) and Tp (λ) at four angles of
incidence namely, 60o, 65o, 70o and 75o, for a 160 nm thick film are shown in Figs.5.49
(a) and (b), respectively. The figure shows that Ts (λ) and Tp (λ) are also angle dependent with the value of the polarized transmittance increasing as the angle of incidence decreases. As can be noticed in the figure, Ts and Tp show systematic behavior (as the angle of incidence increases Ts and Tp decreases). Moreover, the figure shows that the
value of Tp is always larger than Ts indicating that the film is more transparent (75% -
84%) for the p-polarized state than (20% - 45%) for the s-polarized state. This is further verified by the fact that the s-polarized reflectivity is always greater than the p-polarized reflectivity (see Figs.5.47 and 5.48) whereas the s- and p-polarized absorptivities are about constant in the same wavelength range (Fig.5.50).
0.5
0.4
0.3
0.2 60o s-Polarized Transmissivity 65o 0.1 70o 75o 0.0 (a)
200 400 600 800 1000 1200 1400 Wavelength (nm)
Figure 5.49 (a) Spectroscopic transmission data for s- polarized intensity transmittance Ts for the a-InN thin film of 160 nm thickness sputtered onto a glass substrate. 166
1.0
0.8
0.6
0.4 600 650 0
p-Polarized Transmissivity p-Polarized 70 0.2 0 75
0.0 (b)
200 400 600 800 1000 1200 1400 Wavelength (nm)
Figure 5.49 (b) Spectroscopic transmission data for p-polarized intensity transmittance Tp for the a-InN thin film of 160 nm thickness sputtered onto a glass substrate.
In order to totally understand the optical behavior of our amorphous InN samples we
calculated the optical polarized absorptivities As and Ap. The As and Ap were calculated
using the law of conservation of energy (equ.5.9). These quantities, however, become
significant when the material of a thin film is an absorber (i.e., metal, semimetal, and
semiconductor with a narrow bandgap). The spectral dependence of As and Ap at different
angles of incidence are shown in Figs.5.50 (a) and (b). We find that there is higher
absorptivity in the visible region than in the near infrared region. Also, the figure shows
the spectral dependence of the polarized absorptivities to be strongly dependent on
wavelength for λ < 725 nm. This wavelength, ~ 725 nm, therefore indicates that the
absorption edge for a-InN film is about 1.70 eV. By comparing this result with the
previous results about the optical bandgap energy (Chap.5.3.2), our values are similar to 167 those results from the analysis of the absorption coefficient α as a function of energy.
Besides, As shows more angle dependence than Ap for λ > 725 nm. What is more, the a-
InN thin film shows a low polarized absorptivity, less than 25 %, in the near infrared
region at the chosen angles.
Table 5.9 summarizes the relevant features of the various methods for the
determination of the optical bandgap energy Eg of a-InN thin films. The Table shows the
Eg results obtained for thin films of 160 nm thick sputtered onto different substrates. Two
features of interest and importance are shown by these results. (1) The values of Eg obtained by different methods are approximately the same with average values equal to
1.726 ± 0.017 eV. (2) The substrate type does not effect on the obtained value of Eg.
0.8
o 0.7 60 65o o 0.6 70 75o 0.5
0.4
0.3 s-Polarized Absorptivity 0.2
0.1 (a) 0.0 200 400 600 800 1000 1200 1400 Wavelength (nm)
Figure 5.50 (a) Sectroscopic absorption data for s-polarized intensity absorptance As for the a- InN thin film of 160 nm thickness sputtered onto a glass substrate. 168
1.0
60o 65o 0.8 70o 75o
0.6
0.4 p-Polarized Absorptivity
0.2
(b) 0.0 200 400 600 800 1000 1200 1400 Wavelength (nm)
Figure 5.50 (p) Sectroscopic absorption data for p-polarized intensity absorptance Ap for the a- InN thin film of 160 nm thickness sputtered onto a glass substrate.
Table 5.9. Summary of methods of determining bandgap energy of sputtered 160 nm a-InN thin film on two types of substrates. ______
Method Bandgap value (eV) Remarks ______Fitting parameter 1.72 ± 5.9× 10-3 Deposited on c-Si (111) Modified SE Tauc extrapolation 1.75 Deposited on c-Si (111) Modified SP Tauc extrapolation 1.74 Deposited on glass Modified Cody extrapolation 1.72 Deposited on c-Si (111) Polarized absorptivity 1.70 Deposited on glass ------Average value 1.726 ± 0.017
169
Figure 5.51 shows the p-polarized transmissivity T p and absorptivity A p as functions of photon energy E at 75o for samples no.3 (220 nm) and no.4 (260 nm). The value of T p increases monotonically with decreasing photon energy to reach about 85 % as a maximum value. While the value of A p increases monotonically with increasing photon
energy to reach about 90 % in the near-infrared region. Thus, the optical behavior of T p and A p is similar to “a picture and its image in a plane mirror at value 45 “. This is
because the value of p-polarized reflectivity is less than 10 % over the same photon
energy range. Further, there exist no significant absorption until E > 1.5 eV (absorption
edge). This result is in a good agreement with the obtained value of the bandgap energy
for the same thin films using different techniques (Chap.5.3.2).
The effect of absorption with increasing film optical thickness n × h at λ = 480 nm
(~ 2.6 eV) reduces the maximum value of the s- and p- polarized transmissivity as shown in Fig.5.52. At this wavelength the absorption becomes large with A s,p > 55 % (Figs.5.50
and 5.51) and κ ~ 0.26 (Fig.5.37). As we expected, the polarized transmissivity of the a-
InN thin films as well as the polarized reflectivity are not periodic functions of the film
optical thickness. 170
1.0 1.0
0.9 0.9
0.8 0.8
0.7 0.7
0.6 0.6 T p Sample # 3 (220 nm) 0.5 p 0.5
T Sample # 4 (260 nm) p 0.4 A Sample # 3 (220 nm) 0.4 p A Sample # 4 (260 nm) p-Polarized absorptivity p-Polarized transmissivity 0.3 o 0.3 θ = 75 ι 0.2 0.2
0.1 0.1
0.0 0.0 0.51.01.52.02.53.03.54.04.5 Photon energy (eV)
Figure 5.51 The p-polarized transmissivity and absorptivity of a-InN thin films as functions of photon energy at 75o.
0.16 Ts 0.7 p 0.14 T o 0.6 θ = 75 ι 0.12 λ = 480 nm 0.5 0.10 0.4 0.08
0.3 0.06
0.2 p-Polarized transmissivity s-Polarized transmissivity 0.04
0.02 0.1
0.00 0.0 200 300 400 500 600 700 Optical thickness nh (nm)
Figure 5.52 The s- and p-polarized transmissivity of a-InN thin film as functions of its optical thickness n × h at 75o and λ = 480 nm. The index of refraction value is taken from Fig.5.37 at the same wavelength. 171
5.4 The Optical Characteristics of Sputtered Amorphous Gallium Nitride
Thin Films
The fact that gallium nitride (GaN) has a wide and direct band gap (~ 3.4 eV) has led researchers to consider it as a candidate material for short wavelength optical devices, especially for a laser diode. Due to this interest, much work has recently been reported on the various physical properties of crystalline GaN. However, there has been little work on the optical properties of amorphous GaN [146].
In this work a comparative study on a-GaN thin films was done to obtain a complete optical behavior of group III-V nitrides in the visible to near infrared regions.
The a-GaN samples for SE measurements were prepared by a reactive RF magnetron sputtering method under the same deposition conditions as a-(Al, In) N samples (Table
4.1) with deposition rate 0.4 – 0.7 Å/sec. The optical model used to fit the SE data taken on the films is shown in Fig.5.53. The optical constants of a-GaN film were parameterized using the Tauc-Lorentz model (Sec.5.3.1). The addition of the surface roughness SR layers (< 6 nm) on a-GaN thin films yielded a high quality fitting and a minimum mean square error. The SR was modeled using a Bruggeman EMA consisting of 50 % of the a-GaN film and 50 % voids. The fitting was performed using a Levenberg
– Marquardt algorithm (equ.5.7).
The measured SE data, ∆ (λ) and Ψ (λ), of the a-(Al, Ga, In) N films sputtered onto c-Si (111) substrates at 70o are shown in Fig.5.54. The solid lines represent the fitted
∆ and Ψ spectra simulated with the best-fit Cauchy-Urbach and Tauc-Lorentz models for a-AlN and a-(Ga, In) N, respectively. In the figure the best-fit calculated data are 172 indistinguishable from the measured data, indicating that the models are unique over a wide wavelength range (300 – 1400 nm) with MSE of 0.79 (a-AlN), 1.05 (a-GaN) and
1.2 (a-InN). Also, it is to be noted that for λ > 700 nm the spectral dependence of Ψ for all three films is almost the same indicating that the extinction coefficients have minimum values in that region. Further, there are very sharp peaks in the dispersion curves of ∆ (λ) and Ψ (λ). These peaks can be attributed to the effect of constructive (in phase) and destructive (out of phase) interferences.
The optical constants, n (λ) and κ (λ), are shown in Figs.5.55 and 5.56. In figure
Fig.5.55 the refractive indices of a-(Al, Ga) N films exhibit strong dispersion and decrease monotonically with increasing wavelength while n (λ) for a-InN rolls over at about 500 nm. The behavior of n (λ) for a-InN was discussed in detail in the previous section. Refractive indices of a-(Al, Ga, In) N films were found to be in the range 1.82 –
2.11, 2.03 – 2.46, and 2.05 – 2.45, respectively. As obvious in these values, the maximum and minimum points of n (λ) for a-(Ga, In) N films are much closer while nInN > nGaN >
nAlN for λ > 400 nm.
Air
Surface roughness
a-GaN film
SiO2 layer
c-Si (111)
Figure 5.53 Schematic drawing of the optical model for the SE analysis of a-GaN thin film. 173
180 Model Fit (70o) Exp 50 ∆ (a-AlN film (85 nm)) 160 Exp ∆ (a-GaN film (85 nm)) Exp 140 ∆ (a-InN film (90 nm)) Exp 40 Ψ (a-AlN film (85 nm)) Exp 120 Ψ (a-GaN film (85 nm)) Exp Ψ (a-InN film (90 nm)) 100 30 (degrees)
Ψ (degrees) 80 ∆ 20 60
40 10 20
0 0 200 400 600 800 1000 1200 1400 Wavelength (nm)
Figure 5.54 Measured and model fit ellipsometric parameters, ∆ (λ) and Ψ (λ), for a-(Al, Ga, In) N thin films sputtered onto c-Si (111) substrates at 70o.
2.5
a-AlN thin film 2.4 a-GaN thin film a-InN thin film 2.3
2.2
2.1 Index of refractionIndex n
2.0
1.9
200 400 600 800 1000 1200 1400 Wavelength (nm)
Figure 5.55 Indices of refraction of a-AlN and a-(Ga, In) N thin films obtained from the Cauchy- Urbach and Tauc-Lorentz models, respectively 174
0.010 0.7 a-AlN thin film a-InN thin film
) 0.008 0.6 κ ) κ 0.5 0.006 0.4
0.004 0.3 Extinction Coefficient ( Coefficient Extinction 0.2
0.002 ( Coefficient Extinction 0.1
0.000 0.0 200 400 600 800 1000 1200 1400 200 400 600 800 1000 1200 1400 Wavelength (nm) Wavelength (nm)
0.25 a-GaN thin film
0.20 ) κ
0.15
0.10 Extinction Coefficinet ( Coefficinet Extinction 0.05
0.00 200 400 600 800 1000 1200 1400 Wavelength (nm)
Figure 5.56 Extinction coefficients of a-AlN and a-(Ga, In) N thin films obtained from the Cauchy-Urbach and Tauc-Lorentz models, respectively.
In figure 5.56 the extinction coefficients for all three film materials vary smoothly and found to be in the range 1.5 × 10-5 – 9 × 10-3, 5 × 10-4– 0.25, and 0.0 – 0.59 over 300 –
1400 nm. These values indicate that the studied films are transparent in this region.
From the analysis of the absorption coefficient α as a function of photon energy E
(in the range 0.93 – 4.11 eV), we can figure out the optical bandgap energy of an a-GaN thin film. In figure 5.57, (α n E) 2 is plotted versus E. The (α n E) 2 law provides a good fit
to the experimental data as well as in a-AlN thin film case which is very clear in the
figure. By using a linear extrapolation technique we obtained the bandgap energy for a-
GaN thin film to be around 3.44 eV. While the bandgap of a-GaN as a fitting parameter 175 in the Tauc-Lorentz model is equal to 3.2 ± 8.2 × 10-3 eV, where the ± signs represent the
90 % confidence limits. By the same method, the bandgap energies for other a-GaN samples with different thicknesses were found to be approximately 3.38 ± 0.05 eV,
respectively. These values are in a good concord with previous results in the literature
[51].
1.4x1012 12 4x10 Experimental data E a-AlN = 5.83 eV g of a-AlN film 12 Experimental data 1.2x10 of a-GaN film 3x1012 Linear Fit 1.0x1012 for a-AlN
2 ) for a-GaN
-1
11 2 8.0x10 ) -1
2x1012 11 (eV cm (eV
2 6.0x10 (eV cm (eV n) 2 α n) 11 (E 4.0x10 α 12 (E 1x10 a-GaN Eg = 3.44 eV 2.0x1011
0 0.0 1234567 Photon energy (eV)
Figure 5.57 Determination of the energy bandgaps for 85 nm a-(Al, Ga) N thin films sputtered onto c-Si (111) substrates.
The plots of s and p components of the polarized reflectivity, Rs, p, for a 150 nm thick a-
GaN film on a glass substrate versus angle of incidence θi at two wavelengths, 350 and
1200 nm, is shown in Fig.5.58. The figure shows that the reflectance Rp at λ = 1200 nm
reaches a minimum value of about zero at 63o (the Brewster angle). This value is marked
by the dashed arrow in the figure. If we assume the first medium to be air (n1 = 1.0 + i
0.00) and the second medium to be an a-GaN film then by using “tan θB = n film / n air” 176 the index of refraction of the a-GaN film n film = 1.96 at λ = 1200 nm. This value is
comparable to the result obtained from the analysis of the measured ellipsometric spectra
of a-GaN film by the Tauc-Lorentz model (n = 2.03+ i 0.0006) at the same wavelength
(see Fig.5.55). While at λ = 440 nm (solid lines; n = 2.34 + i 0.12), the principal angle θp is about 66o. Thus, the values of the Brewster and principal angles for a-GaN thin film are
in an intermediate position between those in a-AlN and a-InN thin films. In other words:
a-AlN a-GaN a-InN a-AlN a-GaN a-InN (θB ) < (θB ) < (θB ) and (θp ) < (θp ) < (θp ) which are similar to the refractive indices at the same wavelength (Fig.5.55). If the incident plane wave is s- polarized (instead of p-polarized), the corresponding polarized reflectance Rs increases
monotonically from normal to grazing incidence without going through a minimum at
any angle of incidence.
0.6 a-GaN thin film Rs λ = 350 nm (n = 2.34 + i 0.12) 0.5 λ = 1200 nm (n = 2.03 + i 0.0006)
0.4 Rs
p 0.3 R Principal angle 0.2 = 66o o Polarized Reflectances Polarized θ = 63 B p 0.1 R
0.0 20 30 40 50 60 70 80 Angle of Incidence θ (degrees) i
Figure 5.58 The polarized reflectances for the s-and p- polarizations as functions of the angle of incidence θi for a-GaN thin film sputtered onto glass substrate.
177
Figures 5.59 and 5.60 shows the spectral dependence of the s-and p-polarized transmissivity Ts, p of 150 nm a-GaN thin film on glass substrate at different angles of
incidence. Again, we have chosen these angles to study the optical polarized
transmittance behavior of the film up and below the Brewster and principal angles
(Fig.5.58). To completely understand the optical characteristics of these films we studied
the optical polarized absorptivities As, p. The spectral dependence of As, p at the same
angles of incidence are shown in the lower curves of the figures. We found that the a-
GaN thin films show a highly p-polarized transmissivity (85 % – 91 %) which is larger
than that in the case of s-polarized transmissivity (35 % –60 %). This is because of Rs >
Rp as obvious in Fig.5.58. While the optical polarized absorptivities of these films show low absorptivity, (< 18 %), when λ > 400 (3.1 eV), similar to the result obtained from the analysis of the extinction coefficient (see figure 5.57). 178
0.8 50o 0.8 60o 0.7 o 0.7 70
0.6 0.6
0.5 0.5
s 0.4 T 0.4
0.3 0.3 s-Polarized Absorptivity s-Polarized Transmissivity
0.2 0.2
0.1 As 0.1
0.0 0.0 200 400 600 800 1000 1200 1400 Wavelength (nm)
Figure 5.59 The spectral dependence of the s- polarized transmissivity (the upper curves) and absorptivity (the lower curves) data at different angles of incident for a-GaN thin film of 150 nm thickness sputtered onto glass substrates.
1.0 1.0
0.9 p 0.9 T 0.8 0.8
0.7 0.7
0.6 50o 0.6 60o
0.5 0.5 70o
0.4 0.4 p-Polarized Absorptivity
p-Polarized Transmissivity 0.3 0.3
0.2 0.2
0.1 Ap 0.1 0.0 0.0 200 400 600 800 1000 1200 1400 Wavelength (nm)
Figure 5.60 The spectral dependence of the p-polarized transmissivity (the upper curves) and absorptivity (the lower curves) data at different angles of incident for a-GaN thin film of 150 nm thickness sputtered onto glass substrates.
179
5.5 Optical Characteristics of Nitride Bilayer and Multilayer Thin Film
Systems
This section deals with the fabrication and study of optical characteristics of
insulator/semiconductor nitride multilayer thin film systems. The polarized optical
properties of two sets of AlN / InN multilayer thin film systems have been investigated in
the visible and near infrared regions. The first set consisted of five samples that were
composed of 53 nm InN and 64 nm AlN layered distinctly on top of each other with identical period, d′ (mh) = 117 × m nm; m = 1,2,...,5. Thus, the total physical thickness
was 585 nm. The second set also consisted of three samples that were composed of 294
nm InN and 244 nm AlN layered distinctly on top of each other with identical period,
d′ (mh) = 538 × m nm; m = 1,2,3. Thus the total physical thickness was 1.6 µm. In these
systems, semiconductor thin film InN was sputtered first onto a quartz substrate while
AlN thin film, an insulator, was deposited as a second layer to build one bilayer structure
in the periodic multilayer system (Chap.4, Sec.1.2). In order to design a Broadband
Antireflection Coating (BBARC), the system was carefully built according to the
refractive index and thickness of each single layer.
5.5.1 Bilayer Antireflection Coatings
Reflective and antireflective (AR) optical coatings have long been developed for a variety
of applications in all aspects of use; for optical and electro-optical systems in
telecommunications, medicine, military products and consumer products. AR coatings
have been widely used in many applications including glass lenses, eyeglasses, lasers,
mirrors, solar cells, IR diodes, narrow band-pass filters, and flat panel displays. In 180 addition, highly reflecting dielectric mirrors have been developed to be used in gas lasers and in Fabry-Perot interferometers. Numerous studies have been undertaken on this subject [1, 172-176]. Among the existing methods to produce such coatings, the most used are evaporation in high vacuum, reactive sputtering and electron beam evaporation.
All these methods are capable of producing films of uniform thickness and good optical properties. However, magnetron sputtering techniques provide good uniformity for smaller area coatings [174 – 176].
The antireflection coating for its operation depends more or less on the complete cancellation of the reflected light and enhancement of the total transmitted intensity over selected wavelength intervals. This result is achieved by the addition of one or more surface layers that produce reflections that interfere destructively with those originating from the substrate. To obtain antireflection for a single wavelength, single layer coating solves the problem [1, 177]. When light reflects from an interface between two media there may also be a phase shift. If light travels from a lower index of refraction to a higher index of refraction, the reflected light will experience a 180o or λ/2 phase shift. If the light travels from a higher index material to a lower index material, however, there is no phase change [122, 126, and 178].
Figure 5.61 shows light entering an AlN (or InN) film with index of refraction of
1.93 (or 2.35) at 500 nm (see Figs.5.8 and 5.37) from air (no = 1). Some of the light is
transmitted (top of figure) and some is reflected from the interface. Since it is reflected
from a higher index of refraction, the wave undergoes a 180o phase shift. The light that is
transmitted through the film enters a quartz substrate with an index of refraction of 1.54 181 at 500 nm [179], where a portion of the light is transmitted and a fraction is reflected.
This time there is no change of phase because the reflection is from a lower index of refraction. Hence, in order to have destructive interference, the path in the film must be
λ/2, 3λ/2, 5λ/2,…or, the mathematical condition for antireflection is then given by
⎛ 1 ⎞ 2hn film = ⎜m − ⎟λ m = 1, 2, 3, …(5.13) ⎝ 2 ⎠ The minimum film thickness occurs when m = 1, or
λ h = (5.14) 4n film
Thus, to ensure that the relative phase shift is 180o so that the beams cancel, the optical
thickness of the film should be made one quarter wavelength. In other words, a simple
antireflection coating should consist of a single layer of optical thickness one quarter of a
wavelength.
λ/2 Phase no Phase change change
Transmitted wave
Air AlN film Quartz Reflected from first surface
Reflected from second surface
Figure 5.61 Phase shifts in an AlN film. The reflected waves are shown separately for clarity.Reflection from a higher index of refraction produces a change of phase. Reflection from a lower index of refraction does not cause a change of phase in the reflected wave. 182
The disadvantage of the single layer coating is the limited number of adjustable parameters [1]. In practice, the refractive index is not a parameter that can be varied at will. Materials suitable for use as thin films are limited in number and the designer has to use what is available. A more satisfying approach, therefore, is to use more layers
(specifying obtainable refractive indices for all layers at the start) and to achieve zero reflectivity by varying the thickness. Then, besides, there is the limitation that the single layer coating can give zero reflectance at one wavelength only and low reflectance over a narrow region. A wider region of high performance requires that the number of layers must be increased in order to design a broadband antireflection coating.
More complex coating arrangements consisting of two or more layers have been developed to address the need for more broadband AR coatings, with considerable success. J. Mouchart [172, 173] proposed general conditions linking the thicknesses and indices of refraction of two and three nonabsorbent layer coatings to obtain an antireflection filter for a given wavelength. Figure 5.62 shows the conditions for the existence of solutions for no = 1 and ns = 1.52 in the case of two layers with indices of
refraction n1 and n2. n1 is the index of the second layer while n2 is the index of the first
deposited layer on a substrate as shown in the inset figure. The hatched parts of the figure
correspond to possible solutions.
One material that has received little attention as multilayer or even double layer
antireflection coatings, but which may offer many advantages in the optical coating field,
is the nitride material.
183
n1 ns = n2 no n2 on substrate
no
n1 n1 ns = n2 no n2
ns
n1
nons = n2
Figure 5.62 Diagram indicating values for indices n1 and n2 (hatched zone) to obtain an antireflection coating comprising two layers on a substrate with index ns = 1.52 and no = 1 [172].
Since we know exactly the values of indices of refraction of nitride materials over a wide
wavelength range (Chap.5.2 and 5.3), it is therefore very easy to apply Mouchart
conditions to build a double layer (or one bilayer) antireflection coating of nitride
materials. Because the index of refraction of InN is always greater than AlN and quartz
(see Fig.5.55), we first sputtered InN onto the substrate as an inner layer and AlN was
sputtered as an outer layer to form one bilayer structure (Chap.4.2). The schematic
drawing of a bilayer film system of AlN / InN/ Quartz is shown in Fig.5.63. This system was optimized at 500 nm (cyan color) where the indices of refraction of InN and AlN are
2.35 and 1.94, respectively. Further, the thicknesses of each layer in the bilayer system
were chosen according to the destructive interference condition (equ.5.14). Both layers
have an optical thickness of a quarter wave at the wavelength of interest. The thickness
and deposition conditions of each single layer in the bilayer and multilayer systems were
discussed in Chap.4.1.2 (Tables 4.2 and 4.3). 184
First
Second
Third
Air (no = 1) Phase chang AlN film λ/4 (n1 = 1.94) Phase chang
InN film λ/4 (n2 = 2.35)
Quartz No (ns = 1.54) Phase
Figure 5.63 Schematic drawing of quarter / quarter antireflection coating of AlN / InN / Quartz bilayer system.
The studied bilayer system therefore represents a double layer antireflection coating for two reasons: firstly, the point of intersection for AlN and InN indices of refraction lies within the hatched area of the figure 5.62. Secondly, the indices of refraction for quarter / quarter coating of AlN (1.94) and InN (2.35) for one wavelength satisfies the following formula for exact zero reflectance [180]:
n2n (1.94)2 ×1.54 n = 1 s = = 1.04 0 2 2 (5.15) n2 (2.35) where no is the refractive index of air (approximated as 1.0) and ns is the refractive index of the quartz substrate.
The s- and p-polarized reflectivities of sputtered bilayer system of AlN (64 nm, outer layer) and InN (53 nm, inner layer) onto quartz substrate over the spectral range
300 – 1400 nm are shown in Figs.5.64 (a) and (b), respectively. The system was optimized for one wavelength (500 nm) at three angles of incidence, 50o, 60o and 70o. 185
These angles were chosen to study the optical behavior and also to maximize the
o sensitivity near the Brewster angles of the quartz substrate (θB = 57 ) and of the (Al, In) N
films (Chap.5, Secs.2.3 and 3.3). As can be noticed in the figures, the spectral
dependence of s-polarized reflectivity Rs (λ) shows systematic behavior (as the angle of
incidence increases Rs increases) whereas the spectral dependence of p-polarized reflectivity Rp (λ) does not. Again, this can be attributed to the effect of the Brewster
angle on the optical behavior of Rp. It should be pointed out that the s-polarized
reflectivity is always greater than the p-polarized reflectivity at any angle of incidence.
Additionally, both of these reflectivities reach maximum values only in the visible range
and decrease in the near infrared region.
The designed bilayer system (Fig.5.63) provides extremely low reflectance over
an intermediate range of wavelength (380 – 460 nm) as shown in Fig.5.65. The p-
polarized reflectance is < 2 % while the s-polarized reflectance is < 4 % at 50o. 186
0.7 One bilayer: 117 nm x 1
0.6
0.5
0.4
0.3 50o s-Polarized reflectivity 60o 0.2 70o
0.1 (a)
0.0 200 400 600 800 1000 1200 1400 Wavelength (nm)
0.16 One bilayer: 117 nm x 1 50o 0.14 60o 70o 0.12
0.10
0.08
0.06 p-Polarized reflectivity p-Polarized
0.04
0.02 (b)
0.00 200 400 600 800 1000 1200 1400 Wavelength (nm)
Figure 5.64 The spectral dependence of the (a) s-polarized reflectivity and (b) p-polarized reflectivity at 50o, 60o , and 70o of one bilayer system of 53 nm InN film coated with 64 nm AlN film (the period of the structure d′ = 117 × 1 nm).
187
0.14 Relectance of intermediate band AR coating for visible 0.12 Optimized for 500 nm, 50o incidence
0.10
0.08
0.06 plarized reflectivityplarized Rs 0.04 Rp 0.02
0.00 380 400 420 440 460 Wavelength (nm)
Figure 5.65 Antireflection coating with two layers of the same optical thickness (one bilayer system of period d′ = 117 × 1 nm). The coating optimized for 500 nm at 50o incidence.
The spectral dependence of the polarized transmittances Tp (λ) and Ts (λ) of a bilayer system of periodic structure 117 nm at different angles of incidence are shown in
Figs.5.66 (a) and (b), respectively. The figures illustrate that Ts is more angle-dependent
and it shows more systematic behavior (as the angle of incidence increases Ts decreases)
than the transmissivity in the p-direction. This is further verified by the fact that the
spectral dependence of the s-polarized reflectivity does not have any special angle while
the p-polarized reflectivity does (see Chap.5.2.3 and 5.3.3).
The most obvious feature of Fig.5.66 is that the polarized optical transmissivities increase toward longer wavelengths from 10% to 77% and from 3% to 43% at 300 ≤ λ ≤
1400 nm in the case of Tp (λ) and Ts (λ), respectively. This indicates that the bilayer system of AlN / InN / quartz is more transparent for the p-polarized state than for the s- 188 polarized state. This is additionally confirmed by the fact that the s-polarized reflectivity is greater than the p-polarized reflectivity over the considered wavelength range (see
Fig.5.64).
0.8 One bilayer: 117 nm x 1 0.7
0.6
0.5
0.4
o 0.3 50 o p-Poloarized transmissivity p-Poloarized 60 o 0.2 70
0.1 (a)
0.0 200 400 600 800 1000 1200 1400 Wavelength (nm)
0.45 One bilayer: 117 nm x 1 0.40
0.35
0.30
0.25
0.20 50o s-Polarized transmissivity 0.15 60o 0.10 70o
0.05 (b)
0.00 200 400 600 800 1000 1200 1400 WAvelength (nm)
Figure 5.66 The spectral dependence of the (a) p-polarized transmissivity and (b) s-polarized transmissivity at 50o, 60o , and 70o of one bilayer system of 53 nm InN film coated with 64 nm AlN film (the period of the structure d′ = 117 × 1 nm). 189
5.5.2 Multilayer Antireflection Coatings
To obtain antireflection for a single wavelength or even for a narrow wavelength range, double layer (or one bilayer) coating solves the problem as indicated in the previous section (see Fig.5.65). However, in the design of a broadband antireflection (BBAR) coating four optical factors have to be handling. The factors which influence the results
[181]: 1) the refractive indices of the material used; 2) the overall thickness of the coating; 3) the number of layers in the design, and 4) the bandwidth over which the reflection is to be reduced. The bandwidth, defined as the ratio of the longest wavelength to the shortest, is usually specified, leaving the materials, overall thickness, and number of layers as variables. The choice of materials is limited by the spectral range of interest; we have confined the materials used here to be (Al, In) N with indices of 2.35 and 1.94 at
500 nm (reference or design wavelength). The schematic drawing of a multilayer of alternate high (H)-and low (L)-index nitride films (HL periods); all one quarter wavelength thick is shown in Fig.5.67.
AlN film Low index
InN film High index . . . Multilayer structure AlN film Low index
InN film High index
AlN film Low index One bilayer structure
InN film High index
Quartz substrate
Figure 5.67 Schematic drawing of a periodic AlN / InN / Quartz multilayer system with H, L quarter waves of indices 2.35, 1.94 and design wavelength 500 nm. 190
The spectral dependence of the s- and p polarized reflectivities of a coating of design:
Air | HLHL | Quartz (2π system with d′ = 117 × 2 nm)
at different angles of incidence are shown in Figs.5.68 (a) and (b). The total thickness of a four-layer design of antireflection coating for the visible and near infrared regions is
234 nm, and the total thickness of the low-index and the high-index layers are 128 and
106 nm, respectively. Fig.5.68 (a) shows that, the coating design reduces Rs to 0% at 50o and less than 4% at 60o from 400 to 430 nm at the design wavelength. As well, the design
provides low Rs (< 20 %) over a broad range of telecommunication wavelengths through
the near infrared (NIR) region.
The two bilayer (2π) or four-layer design provides extremely low Rp at two
separate wavelength bands anywhere within the visible and NIR spectral at 50o incidence
as shown in Fig.5.68 (b). Rp is < 4% over the first band (350 – 430 nm) while it is < 2%
over the second band (770 – 1050 nm). In the world of AR coatings such design at the
reference wavelength 500 nm and 50o incidence is known as Dual Band AR coating [182,
183]. On the other hand, the 2π design provides very low Rp over a broad range of
wavelengths anywhere within the NIR spectra at 60o. At this angle, Rp is < 1.5% in the
spectral range from 750 to 980 nm. Therefore, the performance of a four-layer design of
antireflection coating for the visible and NIR regions is very sensitive to an angle of
incidence.
191
0.6 Two bilayer: 117 nm x 2
0.5
0.4
0.3 50o
s-Polarized reflectivity o 0.2 60 70o
0.1 (a) 0.0 200 400 600 800 1000 1200 1400 Wavelength (nm)
0.18 Two bilayers: 117 nm x 2 0.16 50o o 0.14 60 70o 0.12
0.10
0.08
p-Polarized reflectivity 0.06
0.04
0.02 (b) 0.00 200 400 600 800 1000 1200 1400 Wavelength (nm)
Figure 5.68 The spectral dependence of the (a) s-polarized reflectivity and (b) p-polarized reflectivity at 50o, 60o , and 70o of a two bilayer system of 53 nm InN film coated with 64 nm AlN film (the period of the structure d′ = 117 × 2 nm).
192
Figures 5.69 (a) and (b) show comparisons between the s- and p-polarized reflectivities of
one bilayer ( d′ = 117 nm) and two bilayer ( d′ = 234 nm) systems of AlN / InN / quartz
at 60o incidence, respectively. It is to be noticed that when the period ( d′ ) of these
systems increases the values of Rs and Rp drastically decrease, more obvious when λ >
600 nm. This result is in excellent agreement with the antireflection coating theory:
“when the antireflection function is broadened to cover a range of wavelengths, the
number of layers must also be increased” [172]. Also, the spectral dependence of the
polarized reflectivity of a two bilayer system shows oscillatory behavior which is not
observed in that of a one bilayer system. This can be attributed to the interference effect.
The most observable aspect of Figs.5.69 (a) and (b) or even Figs.5.68 (a) and (b)
is that the value of the polarized reflectivity over the visible range is higher than that in
the near infrared region at any angle of incidence. This is more noticeable in the p-
direction than the s-direction of the polarized reflectivity. For a discussion and
explanation of this point, we studied the optical behavior of the s- and p-polarized
transmissivities over the wavelength range 300 – 1400 nm. The spectral dependence of Tp
(λ) and Ts (λ) of a two bilayer system of periodicity 117 nm at different angles of
incidence are shown in Figs.5.70 (a) and (b), respectively. The figures reveal low transmissivity in the visible range at any angle of incidence. Outside this range it shows a significant increase reaching a plateau value in the near infrared region. The two bilayer system therefore showed considerable reduction of the polarized transmissivities within the visible range than in the near infrared region (see Fig.5.70 and Table 5.10). Table
5.10 also shows a comparison between the s- and p-polarized transmissivities of one and
193 two bilayer systems through the visible and near infrared regions at two angle of incidence, 50o and 60o.
0.6
0.5
0.4
0.3 s-Polarized reflectivity 0.2 One bilayer system: 117 x 1 nm Two bilayers system: 117 x 2 nm 0.1 o θ = 60 (a) 0.0 200 400 600 800 1000 1200 1400 Wavelength (nm)
0.12 One bilayer system: 117 x 1 nm Two bilayers system: 117 x 2 nm 0.10 o θ = 60
0.08
0.06
p-Polarized reflectivity p-Polarized 0.04
0.02 (b) 0.00 200 400 600 800 1000 1200 1400 Wavelength (nm)
Figure 5.69 Comparisons between the (a) s-and (b) p-polarized reflectivities of one bilayer ( d′ = 117 nm) and two bilayer ( d′ = 234 nm) systems of AlN / InN / quartz at 60o incidence, respectively.
194
0.7
Two bilayers: 117 nm x 2 0.6
0.5
0.4
o 0.3 50 60o o p-Polarized transmissivity 0.2 70
0.1 (a)
0.0 200 400 600 800 1000 1200 1400 Wavelength (nm)
0.4 Two bilayers: 117 nm x 2
50o o 0.3 60 70o
0.2 s-Polarized transmissivity s-Polarized
0.1
(b)
0.0 200 400 600 800 1000 1200 1400 Wavelength (nm)
Figure 5.70 The spectral dependence of the (a) p-polarized transmissivity and (b) s-polarized transmissivity at 50o, 60o , and 70o of two bilayer system of 53 nm InN film coated with 64 nm AlN film (the period of the structure d′ = 117 × 2 nm).
195
Table 5.10 The s- and p-polarized transmissivities of one and two bilayer systems through the visible and near infrared regions at two angles of incidence, 50o and 60o.
Visible range NIR range Property One bilayer Two bilayer One bilayer Two bilayer
o θi = 50
Tp 10 % - 55 % 0 % - 30 % 55 % - 70 % 30 % - 59 %
Ts 6 % - 33 % 0 % - 16 % 33 % - 43 % 16 % - 42 %
o θi = 60
Tp 10 % - 61% 0 % - 32 % 61 % - 77 % 32 % - 62 %
Ts 5 % - 25 % 0 % - 15 % 25 % - 33% 16 % - 34 %
As pointed out in the above discussion, the number of layers must be increased in order to obtain a broadband antireflection coating. Therefore, more complex coating arrangements consisting of three, four, and five bilayer systems of AlN / InN / quartz were developed to address the need for more broadband AR coatings, with considerable success (see Chap.4.1.2). In these systems, the only change was the number of layers (or the total optical thickness) whereas the identity period d′ of each bilayer system (or the optical thickness of each layer in the bilayer system) and the coating indices arrangement of HLHL… on the quartz substrate (Fig.5.67) were fixed. The complete designs for all configurations on quartz at 500 nm as a reference wavelength are then:
Air | H L H L H L | Quartz (3π system with d′ = 117 × 3 nm)
Air | H L H L H L H L | Quartz (4π system with d′ = 117 × 4 nm)
196
Air | H L H L H L H L H L | Quartz (5π system with d′ = 117 × 3 nm)
In these designs the optical thickness of each single layer with high or low index of
refraction is equal to the one quarter of wavelength (λ = 125 nm). Thus, the above
configurations with a quarter-wave optical thickness for the wavelength for which the
antireflection condition is desired (equ.5.14) are known as (quarter-quarter) q – wave coatings, where q = 3, 4, 5.
The spectral dependence of the s- and p- polarized optical reflectivities for three,
four, and five bilayer systems at different angles of incidence with identical period 117
nm are displayed in Figs.5.71 − 5.73 (a) and (b), respectively. As observed from the (a)
curves in the figures, the s- reflectance of all the (quarter-quarter) q – wave coatings of
AlN / InN / quartz is < 2 % and < 5 % over a narrow range of 380 – 420 nm, at the design wavelength of 500 nm, 50o and 60o incidence. Also, it is seen that the s- reflectance for
all the coatings first decreases rapidly and then increases again after passing a minimum
to reach a maximum value within the visible range. This behavior is consistent with the
general behavior of the s- reflectance in the case of one and two bilayer systems of
identical period of 117 nm (see Fig.5.69).
These multilayer designs provide very low p-reflectivity over a broad range of
wavelengths anywhere within the visible and NIR spectra as shown in Figs.5.71- 5.73
(b). The p-reflectivity values depend on the exact wavelength range and the angle of
incidence. For the visible region would be Rp < 8.5 % average and < 4.8 % average while
for the NIR region would be Rp < 7 % average and < 3 % average at 50o and 60o, respectively.
197
Three bilayer: 117 nm x 3 0.6 50o 60o o 0.5 70
0.4
0.3 s-Polarized reflectivity 0.2
0.1 (a)
0.0 200 400 600 800 1000 1200 1400 Wavelength (nm)
0.12 Three bilayer: 117 nm x 3
0.10 50o 60o 0.08 70o
0.06
p-Polarized reflectivity 0.04
0.02 (b) 0.00 200 400 600 800 1000 1200 1400 Wavelength (nm)
Figure 5.71 The spectral dependence of the (a) s-polarized reflectivity and (b) p-polarized reflectivity at 50o, 60o , and 70o of a three bilayer system of 53 nm InN film coated with 64 nm AlN film (the period of the structure d′ = 117 × 3 nm).
198
0.6
50o 0.5 60o 70o 0.4
0.3
s-Polarized reflectivity s-Polarized 0.2
Four bilayers: 117 nm x 4 0.1 (a) 0.0 200 400 600 800 1000 1200 1400 Wavelength (nm)
0.135 Four bilayers: 117 nm x 4 0.120
o 0.105 50 60o 0.090 70o
0.075
0.060
p-Polarized reflectivity 0.045
0.030
0.015 (b) 0.000 200 400 600 800 1000 1200 1400 Wavelength (nm)
Figure 5.72 The spectral dependence of the (a) s-polarized reflectivity and (b) p-polarized reflectivity at 50o, 60o , and 70o of a four bilayer system of 53 nm InN film coated with 64 nm AlN film (the period of the structure d′ = 117 × 4 nm).
199
0.50 Five bilayers: 117 nm x 5 0.45
0.40
0.35
0.30
0.25
o 0.20 50 o s-Polarized reflectivity s-Polarized 60 0.15 70o 0.10
0.05 (a)
0.00 200 400 600 800 1000 1200 1400 Wavelength (nm)
0.120 Five bilayers: 117 nm x 5 0.105 50o 0.090 60o 70o 0.075
0.060
0.045 p-Polarized reflectivity p-Polarized
0.030
0.015 (b)
0.000 200 400 600 800 1000 1200 1400 Wavelength (nm)
Figure 5.73 The spectral dependence of the (a) s-polarized reflectivity and (b) p-polarized reflectivity at 50o, 60o , and 70o of a five bilayer system of 53 nm InN film coated with 64 nm AlN film (the period of the structure d′ = 117 × 5 nm).
200
The comparison of the p-reflectivity spectra of three and five bilayer systems with that of the one bilayer system over the NIR region at 60o is shown in Fig.5.74. It is clear from the graph that as the number of layers (or bilayer) increases the reflectivity completely decreases to minimum values (~ 0 %) over a wide range of wavelengths. Hence, the
(quarter-quarter) q – wave coatings of AlN / InN / quartz optimized for 500 nm could be a good candidate for antireflection coating applications.
0.09
1π (d` = 117 x 1 nm) 0.08 3π (d` = 117 x 3 nm) 5π (d` = 117 x 5 nm) 0.07 o θ = 60 ι 0.06
0.05
0.04
p-Polarized eflectivity p-Polarized 0.03
0.02
0.01
700 800 900 1000 1100 1200 1300 1400 Wavelength (nm)
Figure 5.74 Comparison of the p-reflectivity spectra of three and five bilayer systems with one bilayer system over the NIR region at 60o.
Furthermore, two main features in the spectral dependence of the s- and p- reflectivities of these designs are: (1) both of them show relatively maximum value within the visible range at any angle of incidence and (2) the number of minimum peaks increases with increasing the number of layers (or the total physical thickness). This behavior was expected because of the interference effect.
201
s p The average polarized reflectivity (Rave = (R + R ) / 2)) of the multilayer systems with identical period of 117 nm versus the total bilayer thickness (h total = 117 × m; m = 1, 2
…5) at wavelength of 850 and 60o incidence is shown in Fig.5.75. At higher bilayer thickness, Rave values decrease and it also shows a somewhat damping oscillation. The
Rave therefore is not a complete periodic function of the total bilayer thickness. The reason for this may be due to two optical phenomena. The first is the interference phenomenon: as the number of layers increases the number of the reflected beams from each interface which are out of phase increasing. Secondly, the absorption phenomenon: the absorption of the probing light increases with increasing number of layers of AlN /
InN / quartz multilayer systems. However, quantitative justification of the second phenomenon not only depends on the absorption coefficient of each single layer in the bilayer system but also on the general shape of the absorption coefficient of each bilayer structure in the multilayer system. The absorption coefficient of AlN and InN as single layers is lower for the NIR region than for the visible region (strong absorption occurred when λ < 350 nm, Chap.5.2.1 and 5.3.1). This is understandable since this range represents a high-energy region of the visible spectrum. In this region intense absorption occurs from electronic transitions between the valance and conduction bands.
202
0.32
0.30 λ = 850 nm θ = 60o 0.28
0.26
0.24 ave
R 0.22
0.20
0.18
0.16
0.14 (a) 0.12 100 200 300 400 500 600 Total bilayer thickness (nm)
q Figure 5.75 The average polarized reflectivity Rave of (quarter-quarter) – wave coatings of AlN / InN / quartz as function of their total bilayer thickness at λ = 850 nm and 60o incidence.
Figures 5.76 − 5.78 (a) and (b) show the spectral dependence of the p- and s- polarized
transmissivities of multilayer systems of three, four, and five bilayers with identical
period of 117 nm, respectively. It is noted that the transmissivities of these systems
increase in a monotonic way as the wavelength increases regardless of the angle of
incidence. The increase in transmissivity either in s- or p- direction is more pronounced in the NIR region than the visible region. Also, it does not show any plateau value in that region and hence the transmissivity of these systems increases in the direction of the longest wavelength (infrared region) as expected. This result is due to the low values of the reflectivity through the NIR region (see Fig. 5.74).
203
0.5 Three bilayers: 117 nm x 3 50o o 0.4 60 70o
0.3
0.2 p-Polarized transmissivity p-Polarized
0.1
(a) 0.0 200 400 600 800 1000 1200 1400 Wavelength (nm)
0.35 Three bilayers: 117 nm x 3
0.30 50o 60o 0.25 70o
0.20
0.15 s-Polarized transmissivity 0.10
0.05 (b)
0.00 200 400 600 800 1000 1200 1400 Wavelength (nm)
Figure 5.76 The spectral dependence of the (a) p-polarized transmissivity and (b) s-polarized transmissivity at 50o, 60o , and 70o of a three bilayer system of 53 nm InN film coated with 64 nm AlN film (the period of the structure d′ = 117 × 3 nm).
204
0.4 Four bilayers: 117 nm x 4 50o 60o o 0.3 70
0.2 p-Polarized transmissivity p-Polarized
0.1
(a)
0.0 200 400 600 800 1000 1200 1400 Wavelength (nm)
0.30 Four bilayers: 117 nm x 4
0.25 50o 60o 0.20 70o
0.15
0.10 s-Polarized transmissivity s-Polarized
0.05 (b)
0.00 200 400 600 800 1000 1200 1400 Wavelength (nm)
Figure 5.77 The spectral dependence of the (a) p-polarized transmissivity and (b) s-polarized transmissivity at 50o, 60o , and 70o of a four bilayer system of 53 nm InN film coated with 64 nm AlN film (the period of the structure d′ = 117 × 4 nm).
205
0.40 Five bilayers: 117 nm x 5 0.35 50o 0.30 60o 70o 0.25
0.20
0.15 p-Polarized transmissivity 0.10
0.05 (a)
0.00 200 400 600 800 1000 1200 1400 Wavelength (nm)
0.20 Five bilayers: 117 nm x 5 50o 60o 0.15 70o
0.10 s-Polarized transmissivity
0.05
(b)
0.00 200 400 600 800 1000 1200 1400 Wavelength (nm)
Figure 5.78 The spectral dependence of the (a) p-polarized transmissivity and (b) s- polarized transmissivity at 50o, 60o , and 70o of a five bilayer system of 53 nm InN film coated with 64 nm AlN film (the period of the structure d′ = 117 × 5 nm).
206
s p The average polarized transmissivity (Tave = (T + T ) / 2)) of the multilayer systems with
identical period of 117 nm versus wavelength at 60o and 70o incidence are shown in
Figs.5.79 (a) and (b), respectively. It is shown that for any bilayer structure there is a
o o slight difference between the values of Tave at 60 and 70 . The reason for this is that the
s-direction, which shows more angle dependence than the p-direction provides low
transmissivity whereas the p-direction makes available high transmissivity anywhere
within the NIR spectra. It is also very clear that when the number of bilayer increases the
transmissivity goes rapidly to zero over a broad range of wavelengths within the visible
region and then drastically increases toward the longest wavelengths.
0.6 1π 2π 0.5 3π 4π 5π 0.4 o θ = 60
0.3 ave T
0.2
0.1 (a)
0.0 200 400 600 800 1000 1200 1400 Wavelength (nm)
q Figure 5.79 (a) The average polarized transmissivity Tave of (quarter-quarter) – wave coatings of AlN / InN / quartz as a function of wavelength at 60o incidence.
207
0.5 1π 2π 3π 4π 0.4 5π o θ = 70 0.3
ave T
0.2
0.1 (b)
0.0 200 400 600 800 1000 1200 1400 Wavelength (nm)
q Figure 5.79 (b) The average polarized transmissivity Tave of (quarter-quarter) – wave coatings of AlN / InN / quartz as a function of wavelength at 70o incidence.
Figure 5.80 shows the dependence of the Tave on the total bilayer thickness of the multilayer systems with identical period of 117 nm at different wavelengths and 60o incidence. It is noteworthy that Tave values monotonically decrease to 0 % with increasing total thickness of the multilayer system at shorter wavelengths while they exhibit a kind off damping oscillation at the longest wavelength. Again a good explanation for that requires much optical information than the reflectivity such as the effective or pseudo optical constants,
208
0.5 λ = 360 nm λ = 500 nm λ = 850 nm 0.4 o θ = 60
0.3 ave T
0.2
0.1
0.0 100 200 300 400 500 600 Total bilayer thickness (nm)
q Figure 5.80 The average polarized transmissivity Tave of (quarter-quarter) – wave coatings of AlN / InN / quartz as function of their total bilayer thickness at λ = 360 , 500, and 850 nm and 60o incidence.
The second design of AlN / InN / quartz multilayer system is also optimized at 500 nm,
50o - 70o incidence. In this system the periodicity was 538 nm (294 nm InN, inner layer +
244 nm AlN, outer layer). Therefore, the only difference between this design and the
pervious one is just the value of the identical period i.e., the sputtered thicknesses of AlN
and InN in the second design are larger than their values in the first design.
The spectral dependence of the s- and p- polarized optical reflectivities for one, two, and three bilayer systems at different angles of incidence with identical period 538 nm are depicted in Figs.5.81 − 5.83 (a) and (b), respectively. If reflectance spectra of one bilayer with d′ = 538 nm and total thickness of 538 nm (1π538, 538) are compared with that
of a five bilayer system with d′ = 117 nm and total thickness of 585 nm (5π117, 538,
Fig.5.73), it is seen that the reflectance spectra of both show generally the same optical
209
behavior. The first superscript number represents the identity period while the second
represents the total physical thickness of the structure. Despite the fact that they have the
same behavior, the minimum peaks in the 1π538, 538 system are sharper than the 5π117, 538 system. The sharp peaks confirm the fact that the overall thickness of the coating plays an important factor in the design of a BBAR coating. The second multilayer AR coating designed at a reference wavelength of 500 nm for visible and NIR regions provides very low p-reflectance (< 8 %) in the visible region and almost zero over a broad range of wavelengths everywhere in the NIR range as shown in Fig.5.84. The figure clearly shows a reduction in reflectivity with increasing number of layers or bilayers over a wide band of wavelengths. The reduction for bilayer systems with periodicity of 538 nm is similar to that for with 117 nm periodicity. However, in the second design with high periodicity the performance is even better and the curves are e flatter and comparatively closer to the horizontal axis.
210
0.55 One bilayer: 538 x 1 0.50 0.45 0.40 0.35 0.30 0.25
s-Polarized reflectivitys-Polarized 0.20 50o 0.15 60o 0.10 70o 0.05 (a) 0.00 200 400 600 800 1000 1200 1400 Wavelength (nm)
0.10
0.09 One bilayer: 538 x 1
0.08 o 0.07 50 60o 0.06 70o 0.05
0.04 p-Polarized reflectivity p-Polarized 0.03
0.02
0.01 (b) 0.00 200 400 600 800 1000 1200 1400 Wavelength (nm)
Figure 5.81 The spectral dependence of the (a) s-polarized reflectivity and (b) p-polarized reflectivity at 50o, 60o , and 70o of a one bilayer system of 294 nm InN film coated with 244 nm AlN film (the period of the structure d′ = 538 × 1 nm).
211
0.45
0.40 Two bilayers: 538 nm x 2
0.35
0.30
0.25
0.20
s-Polarized reflectivity s-Polarized 0.15
0.10 50o 60o 0.05 o 70 (a) 0.00 200 400 600 800 1000 1200 1400 Wavelength (nm)
0.10
0.09 Two bilayers: 538 nm x 2
0.08 50o o 0.07 60 70o 0.06
0.05
0.04 p-Polarized reflectivity p-Polarized 0.03
0.02
0.01 (b) 0.00 200 400 600 800 1000 1200 1400 Wavelength (nm)
Figure 5.82 The spectral dependence of the (a) s-polarized reflectivity and (b) p-polarized reflectivity at 50o, 60o , and 70o of a two bilayer system of 294 nm InN film coated with 244 nm AlN film (the period of the structure d′ = 538 × 2 nm).
212
0.55
0.50 Three bilayers: 538 nm x 3 0.45
0.40
0.35
0.30
0.25
0.20 s-Polarized reflectivity s-Polarized 0.15 50o 0.10 60o 0.05 70o (a) 0.00 200 400 600 800 1000 1200 1400 Wavelength (nm)
0.12 Three bilayers: 538 nm x 3
0.10 50o 60o o 0.08 70
0.06
p-Polarized reflectivity 0.04
0.02 (b) 0.00 200 400 600 800 1000 1200 1400 Wavelength (nm)
Figure 5.83 The spectral dependence of the (a) s-polarized reflectivity and (b) p-polarized reflectivity at 50o, 60o , and 70o of a three bilayer system of 294 nm InN film coated with 244 nm AlN film (the period of the structure d′ = 538 × 3 nm).
213
0.08 Multilayer system of period = 538 nm 0.07 One bilayer 0.06 Two bilayers Three bilayers
0.05 o θ = 60
0.04
0.03 p-Polarized reflectivity p-Polarized
0.02
0.01
0.00 800 900 1000 1100 1200 1300 1400 Wavelength (nm)
Figure 5.84 Combined p- reflectance profiles for one, two, and three bilayer systems of InN / AlN / quartz of identical period of 538 nm at 60o incidence.
Figure 5.85 (a) and (b) show the spectral dependence of the average polarized transmissivity of one (1π538, 538) and two bilayer (2π538, 1) systems of the first superscript number represents the identity period while the second represents the total physical thickness of the structure period of 538 nm at different angles of incidence, respectively.
As can be seen the transmissivity spectra of the thick layered systems compared with the thin layered once show almost zero value when λ < 600 nm and then increase rapidly toward the longest wavelengths. This behavior is similar to that generated using the first multilayer system of the first superscript number represents the identity period while the second represents the total physical thickness of the structure period 117 nm. Even as the transmissivity values of 1π538, 538 are higher than the 5π117, 585 at any angle of incidence.
214
The discrepancy in the transmissivity values may be due to the interfacial roughness effect [61, 62].
0.35 One bilayer: 538 x 1 nm 50o 0.30 o 60 70o 0.25
0.20 ave T 0.15
0.10
0.05 (a) 0.00 200 400 600 800 1000 1200 1400 Wavelength (nm)
0.12 Three bilayers: 538 nm x 3
0.10 o 50 60o 0.08 70o
ave 0.06 T
0.04
0.02 (b) 0.00 200 400 600 800 1000 1200 1400 Wavelength (nm)
Figure 5.85 The spectral dependence of the average polarized transmissivity Tave of (a) one bilayer system and (b) two bilayers system of identical period of 538 nm at 50o, 60o , and 70o incidence.
215
5.5.3 Applications of Nitride Multilayer Systems: Broadband NIR-AR Coating
As mentioned in the above discussion the optical transmissivity and reflectivity of the
nitride multilayer systems are not sufficient to decide their applications. This is because
the nitride materials are not dielectrics. The bandgap of AlN and InN is about 5.8 and 1.7
eV. Also, these materials as single layers show high absorption coefficients within the
UV and visible regions and almost zero values over the NIR region (Chap.5.2.2 and
5.3.2). Therefore, the study of the absorptivity of AlN / InN / quartz multilayer systems with different identity period is an important step, in order to obtain enough optical information about these systems.
The optical polarized absorptivities As,p were calculated using the law of
conservation of energy (equ.5.9). The spectral dependence of the p-polarized optical
properties, transmissivity, reflectivity, and absorptivity of 1π117,117, 4π117,468, and 1π538, 538
systems at 60o incidence are showed in Figs.5.86 – 5.88, respectively. We find that the
systems show higher absorptivity in the visible region (30 % - 98 %) than in the near infrared region (20 % - 80 %). Also, the figures show that the absorptivity increases with
increasing the total physical thickness or the identity period of the multilayer system over
the visible and NIR spectra.
A snapshot of the NIR region reveals that the transmissivity tends to increase
smoothly while the absorptivity has a tendency to decrease with increasing the
wavelength. This means that the nitride bilayer and multilayer systems can be used as a
longwave-pass filter. However, since these systems provide extremely high absorptivity
anywhere within the visible spectra they could be used as solar cells with high efficiency.
216
0.8 0.9
0.7 0.8
0.6 Transmissivity 0.7 Reflectivity 0.5 p Apsorptivity 0.6 R p
o A & 0.4 θ = 60 p 117, 117 0.5 T 1π 0.3 0.4 0.2 0.3 0.1 0.2 0.0 200 400 600 800 1000 1200 1400 Wavelength (nm)
Figure 5.86 The spectral dependence of the polarized optical properties of a one bilayer system of identical period 117 nm and total physical thickness 117 nm at 60o incidence.
0.5
1.0
0.4
0.9
0.3 p Reflectivity Transmissivity 0.8 p A
& R p Absorptivity T 0.2 o θ = 60 0.7 4π 117, 468 0.1
0.6
0.0 200 400 600 800 1000 1200 1400 Wavelength (nm)
Figure 5.87 The spectral dependence of the polarized optical properties of a four bilayers system of identical period 117 nm and total physical thickness 468 nm at 60o incidence.
217
0.40 1.00
0.35 0.95
0.90 0.30 Transmissivity 0.85 0.25 Reflectivity p p
R Absoprtivity 0.80 A
o & 0.20
p θ = 60
T 0.75 538, 538 0.15 1π 0.70 0.10 0.65 0.05 0.60 0.00 200 400 600 800 1000 1200 1400 Wavelength (nm)
Figure 5.88 The spectral dependence of the polarized optical properties of a one bilayer system of identical period 538 nm and total physical thickness 538 nm at 60o incidence.
218
CHAPTER SIX
SUMMARY AND CONCLUSIONS
III-V nitride and AlN / InN multilayer thin films systems with two identical periods were
prepared by RF magnetron sputtering onto c-Si, (111), glass, and quartz substrates. The
amorphous structure of (Al, Ga, In) N as single layers was confirmed by the lack of any
detectable nitride peak in the X-ray diffraction spectra, while the polycrystalline structure
of multilayer systems showed a maximum diffraction peak at 31.2o. The value of the peak
increased as the number of layers increased. Additionally, the number of peaks increased
when the total time deposition increased.
The optical characteristics of a-(Al, Ga, In) N thin films were investigated using
the variable angle spectroscopic ellipsometry of the rotating analyzer type at room temperature in the spectral range 300 – 1400 nm. This study showed that the Cauchy-
Urbach and the Tauc-Lorentz models could represent adequately a-AlN and a-(Ga, In) N thin films, respectively over a wide spectral range. From the data analysis of the extinction coefficient κ, the bandgap of a-AlN film was determined to be 5.82 ± 0.05 eV.
In addition, the Tauc-Lorentz model provided indirect information about the energy band
gap. The bandgap of a-GaN was determined by an interpolation method and it showed a
good agreement with the Eg values as fitting parameters in Tauc-Lorentz model (3.2 ± 8 ×
-3 10 eV). The Eg of a-InN film which given much attention during this work was found to be 1.68 ± 0.071 eV. These values were verified using different methods such as
219
spectrophotometric, photoluminescence, and polarized absorptivity. Also, the deduced
bandgap energy values of a-InN showed an excellent agreement with the Eg as fitting
parameters in Tauc-Lorentz model (1.66 ± 0.055). The addition of the surface roughness
layer (< 6 nm) on the a-(Ga, In) N films yielded a significantly better MSE and therefore
a high quality fitting.
Analysis of the angle dependence of the p-polarized reflectivity Rp showed that
the minimum value of Rp depended on the extinction coefficient value. In addition, the
Brewster and principal angles of these thin films were deduced according to the
minimum value of Rp. Measurement of the polarized optical properties revealed a high
transmissivity (70 % – 95 %) and very low absorptivity (< 18 %) for all three thin films
in the visible and near infrared regions.
AlN / InN / quartz multilayer antireflection coating systems in the visible and near
infrared regions were designed at 500 nm and 50o – 70o incidence. The systems were
warily built according to the refractive index and thickness of each single layer. They
were starting from one bilayer up to five bilayer of alternate high (H)-and low (L)-index
configuration with two identical periods. The performance of each successive
configuration was optimized in order to demonstrate efficient use of multiple layers for
antireflection applications. The analysis of these designs revealed that by increasing
number of layers in a careful manner with appropriate thickness or identical period, the
reflectivity can be decreased to very low magnitude over a broad range of wavelengths
anywhere within the visible and near infrared spectra. Furthermore, the performance of
the multilayer designs of AR for the visible and NIR regions was very sensitive to an
220 angle of incidence. The NIR region revealed that the transmissivity tended to increase smoothly while the absorptivity showed a tendency to decrease with increasing the wavelength. Therefore, the nitride bilayer and multilayer systems could be used as a longwave-pass filter. However, since these systems provided extremely high absorptivity anywhere within the visible spectra they could be used as solar cells with high efficiency.
221
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