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Optical Properties of Thin Transparent Conducting
Oxide Films on Glass for Photovoltaic Applications
Mark Stockett
Professor John H Scofield, Advisor
April 15, 2006
Abstract
This project was motivated by my earlier work with John Scofield on thin-film
copper indium diselenideide (CIS) photovoltaic devices. We were attempting to
fabricate transparent conducting films of tantalum oxide (Ta2 O5 ) for use as a window layer in these devices. Exploring different deposition parameters and doping
techniques, many samples were produced that needed characterization to determine their quality and properties. Properties of interest include the film thickness,
the real and imaginary parts of the index of refraction, the bandgap energy, and the
carrier density in doped films. As a result of this project, we are now able to obtain
information on all of these quantities through the study of a single measurement,
namely the transmission spectrum. This document summarizes the fabrication
work done by me and previous students and details the method I developed for
using transmission measurements to characterize the samples.
1
Contents
1 Introduction
1.1 Solar Cell Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.2 CIS Device Components . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.3 Alternative Window Materials . . . . . . . . . . . . . . . . . . . . . . . . . .
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2 Fabrication
2.1 DC Magnetron Reactive Sputtering from Metallic Targets
2.2 Deposition Parameters . . . . . . . . . . . . . . . . . . . .
2.3 Reactive Doping . . . . . . . . . . . . . . . . . . . . . . . .
2.4 Target Doping . . . . . . . . . . . . . . . . . . . . . . . . .
2.5 Routine Characterization . . . . . . . . . . . . . . . . . . .
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3 Theory
3.1 Statement of Problem . . . . . . . . . . . . . . .
3.2 Maxwell’s Equations and Electromagnetic Waves
3.3 Reflection and Transmission . . . . . . . . . . . .
3.4 Propagation . . . . . . . . . . . . . . . . . . . . .
3.5 Multiple Reflections . . . . . . . . . . . . . . . . .
4 Measurement and Analysis
4.1 UV-Vis-NIR Spectrophotometer
4.2 Refractive Index of Substrate .
4.3 Interference Envelope . . . . . .
4.4 Refractive Index of Film . . . .
4.5 Film Thickness . . . . . . . . .
4.6 Absorption Coefficient . . . . .
4.7 Band Edge . . . . . . . . . . . .
4.8 Absorption in Doped Films . .
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5 Results and Discussion
5.1 Reconstructed Curves . . . . . . . . . . . . . . . . .
5.2 Refractive Indices . . . . . . . . . . . . . . . . . . .
5.3 Optical Versus Mechanical Thickness Measurements
5.4 Bandgap Energies . . . . . . . . . . . . . . . . . . .
5.5 Carrier Densities . . . . . . . . . . . . . . . . . . .
6 Conclusion
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29
2
List of Figures
1.1
1.2
1.3
2.1
2.2
2.3
2.4
3.1
3.2
3.3
4.1
4.2
4.3
4.4
4.5
The layers of a typical CIS thin film photovoltaic device[1]. The window layer (highlighted in
green) was the focus of my research. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
The p-n junction. Circled charges are fixed lattice ions. Non-circled charges are free carriers.
Both p- and n-type regions are charge neutral. An electron-hole pair created in either region
may be split by the electric field in the depletion region and lead to a photocurrent[2]. . . . .
Solar blackbody energy density spectrum B(λ) (smooth red curve). The vertical lines indicate
the wavelengths corresponding to the bandgap energies of Ta2 O5 , ZnO and CIS. Horizontal
arrows show transparancy ranges of Ta2 O5 and ZnO window layers on CIS absorber layer.
The Ta2 O5 transmits 11% more of the sun’s energy. . . . . . . . . . . . . . . . . . . . . . . .
Left: Top view of vacuum chamber. The positioner can be moved to locate the substrate over
either sputter gun. Oxygen is fed in through the front side of the chamber while argon is fed
up through the guns. Right: The geometry of the sputter gun (shown in cross-section in lower
portion) results in a nonuniform thickness profile (graph at top of frame). Lines show possible
paths for metal atoms ejected from the target ring (cathode). . . . . . . . . . . . . . . . . . .
Left: X-Ray diffraction (XRD) pattern for a polycrystalline Ta2 O5 film[3]. The location of
the peaks is related to the lattice parameters of the crystallites. Right: XRD pattern for
an amorphous Ta2 O5 film (sample WT-148). Note that both vertical scales are arbitrary.
The pattern for the amorphous film is likely on the order of the background noise of the
polycrystalline pattern. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Schematic diagram of four probe resistivity measurement[2]. This setup eliminates errors from
lead wire and contact resistances. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
The Hall effect causes a potential difference VH across a two-dimensional semiconductor related
to the carrier density. The applied magnetic field B is out of the page and the current I is
indicated by the arrow. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Interface between two media with different indices of refraction, showing incident, reflected
and transmitted EM waves and angles of incidence and refraction. The z-axis points into the
page . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Schematic diagram of a thin film (thickness d) between two infinite media, showing contribution to reflected and transmitted wavefronts from rays reflected within the film multiple
times. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
The physical situation for thin film measurements in the lab. The film (index n2 ) rests on top
of a thick but finite substrate (index n3 ) bounded on either side by air (index n1 ). . . . . . .
The “chopper” disk alternates the path of the measurement beam as it rotates. It has three
sections: transparent (1; white), mirror (2; grey) and matte (3; black). . . . . . . . . . . . . .
Simplified schematic of spectrophotometer. The chopper sends the measurement beam alternately through the sample and not through the sample. . . . . . . . . . . . . . . . . . . . . .
Example of raw data from spectrophotometer showing blank substrate (flat upper curve) and
ZnO coated sample (oscillating lower curve). Sample Zn-005. . . . . . . . . . . . . . . . . . .
Left: Example transmission spectrum of blank substrate. Right: Calculated refractive index
of blank substrate. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Simulated transmission curve (red) for a 2000 nm thick film, showing interference envelope
functions TM (Equation 4.4a) and Tm (Equation 4.4b) in blue. . . . . . . . . . . . . . . . . .
3
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4.6
Left: Comparison of interference envelopes built from first (red), second (blue) and third
(green) order interpolations. Dashed line is a measured transmission spectrum for sample
WT-152. Right: Example of a measured spectrum (green) with linear interference envelope
(blue). Sample Zn-005. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.7 Example of extracted refractive (yellow) index with Cauchy fit (red, Equation 4.8). The blue
dashed lines show the 3% relative uncertainty in n. Sample Zn-005. . . . . . . . . . . . . . . .
4.8 Graphical method for determining film thickness, interference order `/2 is plotted against n/λ.
The slope of the line is twice the film thickness. Sample Zn-005, with m0 = 3/2 and d = 617
nm. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.9 Left: Example absorption coefficients calculated from the refractive index n (yellow) and
the Cauchy fit to n (red). Blue dashed lines show uncertainty. Right: Example absorption
coefficient near band edge. The substrate itself begins absorbing below 250 nm, so data in
this region should be ignored. Note the difference in scales on the two plots. Sample Zn-005.
4.10 Left: Plot of α2 vs hν with fit to linear region. The intercept is the bandgap energy. Right:
Plot of ln α vs hν with fit to linear region. The slope is the inverse of the Urbach disorder
parameter. Sample Zn-005, with Eg = 3.28 eV and E0 = 75 meV. . . . . . . . . . . . . . . . .
4.11 Left: Example transmission spectra for a doped ZnO film (blue) showing infrared absorption
and uncoated glass microscope slide (red). Right: By fitting the infrared absorption coefficient
(blue) with a λ2 dependence (red), one can determine the ratio of the carrier density N to
the scattering time τ . Sample Zn-044 with N/τ = 7.1 × 1035 cm−3 s−1 as determined through
Hall effect measurements. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5.1 Example of reconstructed transmission curve. The blue curve is raw data and the red curve
is the reconstruction. Sample Zn-005. Compare Figure 4.3. . . . . . . . . . . . . . . . . . . .
5.2 Left: If n is taken as a constant, the reconstructed transmission curve (blue curve) fails to
capture the variation in the amplitude of the interference fringes present in the data (red
curve). Right: If n is allowed to vary, better agreement is achieved, particularly in the fringe
amplitude. Sample WT-148. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5.3 Close-up of Ta2 O5 transmission curve near band edge (lower curve). The proximity of the
Ta2 O5 bandgap to that of the substrate (upper curve) complicates analysis in this region.
Sample WT-152. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5.4 Example of reconstructed transmission curve for a doped film. The blue curve is raw data
and the red curve is the reconstruction. Sample Zn-044, with N/τ = 7.1 × 1035 cm−3 s−1 as
determined through optical measurements. Compare Figure 4.11. . . . . . . . . . . . . . . . .
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List of Tables
5.1
5.2
5.3
5.4
Real part of the index of refraction n as extracted with Equation 4.6 and literature values.
Uncertainties in parentheses. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Comparison of optical (dopt ) and mechanical (dmech ) thickness measurements for Ta2 O5 films.
Uncertainties in optical measurements in parentheses. . . . . . . . . . . . . . . . . . . . . . .
Measured and literature values of energy bandgap Eg , and Urbach slope E0 for ZnO and
In2 O3 films. Uncertainties in parentheses. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Values of N/τ as extracted from optical data ((N/τ )opt ) and from Hall effect measurements
((N/τ )Hall ). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4
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1
1.1
Introduction
Solar Cell Background
Solar cells (photovoltaic devices) transform light from the sun into electrical work. Cells based on crystalline
silicon have dominated the industry since their development in 1954[2]. In recent years, polycrystalline
devices based on materials such as copper indium diselenide (CIS) have gained increased attention from the
scientific community. Unlike silicon, these materials may be made into light-weight thin films which are
flexible and do not degrade in direct sunlight[2]. Dr Scofield has at times collaborated with scientists at
the National Aeronautics and Spaces Administration’s (NASA) Glenn Research Center. NASA’s interest in
these devices stems from the significant cost of launching satellites equipped with heavy silicon modules.
The efficiency (the fraction of the sun’s energy converted to work) of CIS devices has been shown to be
comparable to that for silicon cells (10-15%)[4].
1.2
CIS Device Components
Figure 1.1 shows the layers of a typical CIS solar cell. As thin films, these materials do not have the
mechanical strength to be self-supporting, so they must be deposited on some sort of substrate. I sued
Corning 2947 plain soda glass microscope slides. On top of the glass substrate, a metallic molybdenum (Mo)
layer forms the back contact. Then follows the CIS itself, called the absorber layer, a cadmium sulfide (CdS)
buffer layer, and the zinc oxide (ZnO) window layer. ZnO is one of the most common widow layer materials
and has been studied extensively by Dr Scofield and his students, but other possibilities may be viable. The
top two layers are the aluminum-nickel (Al-Ni) contact grid and a magnesium fluoride (MgF2 ) anti-reflective
coating. The Mo back contact and the Al-Ni contact grid define the electrical circuit through which the
work done by the photovoltaic device is extracted.
Figure 1.1: The layers of a typical CIS thin film photovoltaic device[1]. The window layer (highlighted in
green) was the focus of my research.
Dr Scofield and his student have studied and fabricated many of these layers. My research focused on
the window layer.
A solar cell is essentially a p-n junction (see Figure 1.2 and [5]), where a “p-type” layer is joined to an
“n-type” layer. At the junction there is a “depletion region” where the oppositely charged carriers annihilate
each other and in equilibrium the now unscreened lattice ion cores establish an electric field. Absorption of a
photon in the (usually) p-type absorber layer creates an electron-hole pair, which are accelerated in opposite
5
Figure 1.2: The p-n junction. Circled charges are fixed lattice ions. Non-circled charges are free carriers.
Both p- and n-type regions are charge neutral. An electron-hole pair created in either region may be split
by the electric field in the depletion region and lead to a photocurrent[2].
directions by the depletion region electric field. If the absorber is p-type, the electron from the electronhole pair may diffuse into the depletion region and be accelerated into the n-type window layer, causing an
excess negative charge in this layer with an equal and opposite excess charge in the other material from the
hole that got left behind. This charge buildup can be collected through the grid and back contacts as a
“photocurrent”.
In an actual device, one would like the absorption event to occur spatially as close as possible to the
junction to maximize the probability that the excited carriers will be collected. This is achieved by putting
the window layer on top of the absorber layer so that solar photons passing completely through the window
layer will be absorbed as soon as they enter the absorber layer at the junction. This requires that the window
layer be highly transparent to photons which the absorber layer can absorb, ie those with energy greater
than the absorber layer’s bandgap energy.
Creating an effective p-n junction requires that the window layer be highly conducting, which is unfortunately at odds with the transparency requirement, as free charge carriers lead to absorption at infrared
wavelengths. Any window layer will also absorb photons with energy greater than its own bandgap energy,
so materials with wide bandgaps are needed for efficient window layers.
1.3
Alternative Window Materials
Between September 2004 and May 2005, I worked investigating tantalum oxide (Ta2 O5 ) as a possible window
layer for CIS solar cells. Ta2 O5 is frequently used for its high resistivity as a dielectric material in capacitors
and memory devices[6]. With a wide (4.2 eV[3]) bandgap, a window layer of Ta2 O5 would allow a greater
fraction of the sun’s blackbody spectrum to pass through to the absorber layer than would a ZnO (3.3 eV)
window (see Figure 1.3). The solar spectrum peaks in the visible but has substantial power in the ultraviolet.
Assuming perfect transmission, the wider bandgap of Ta2 O5 would lead to an 11% increase in cell efficiency.
Much of this increase would likely be wiped out by ultraviolet absorption in the Earth’s atmosphere, but for
space applications this advantage could be significant.
While I was fabricating and studying these films, we became interested in their optical properties and
thin-film interference effects. Since our motivation for studying this material was the possibility of increased
optical transparency, quantifying and understanding the optical properties of our samples was very important.
The optical analysis turned out to be complicated and tedious, so I developed a computerized analysis method
6
Figure 1.3: Solar blackbody energy density spectrum B(λ) (smooth red curve). The vertical lines indicate
the wavelengths corresponding to the bandgap energies of Ta2 O5 , ZnO and CIS. Horizontal arrows show
transparancy ranges of Ta2 O5 and ZnO window layers on CIS absorber layer. The Ta2 O5 transmits 11%
more of the sun’s energy.
which allowed us to extract all of the relevant parameters in a matter of minutes. This is the subject of
Section 4.
The most widely used window layer material for silicon solar cells is tin doped indium oxide (ITO). I
studied the optical properties of a few indium oxide (In2 O3 ) films, as well as several ZnO films, all deposited
by Dr Scofield and his students.
2
2.1
Fabrication
DC Magnetron Reactive Sputtering from Metallic Targets
Most of the samples I studied were fabricated using the Sloan sputter system located in Wright Laboratory
at Oberlin. The system consists of a vacuum chamber equipped with two DC magnetron sputter guns and a
moving substrate holder with a heating element. Thin metal-oxide films were deposited onto glass microscope
slides using high purity metallic targets (tantalum, zinc, indium etc.) in an argon/oxygen reactive mixture.
After the chamber pressure is reduced to 10−6 Torr using a diffusion pump, argon is fed into the base of the
gun, while oxygen and other gasses are fed directly into the vacuum chamber. The layout of the vacuum
chamber is shown in Figure 2.1 Left.
A large negative voltage is applied to an anode disk in the sputter gun, which is mounted in the base
plate of the vacuum chamber . This creates an electric field between the anode and the target, an electrically
grounded metallic ring 1/8 inch thick with a 3 inch inner diameter, which is a few millimeters larger than
the anode disk’s diameter. This field is strong enough to ionize the argon gas. The Ar ions are accelerated
into the target ring where they eject metal (in my case tantalum) atoms up towards the substrate, which is
held above the gun in the positioner (see Figure 2.1 Right). This is different from many DC sputter systems
which use flat targets and sputter downwards.
2.2
Deposition Parameters
The deposition process is a dynamic one, with many parameters affecting the resulting film. For making
metal-oxides, one of the most important variables is the ratio of argon (Ar) to oxygen (O2 ) in the reactive
7
Figure 2.1: Left: Top view of vacuum chamber. The positioner can be moved to locate the substrate over
either sputter gun. Oxygen is fed in through the front side of the chamber while argon is fed up through
the guns. Right: The geometry of the sputter gun (shown in cross-section in lower portion) results in a
nonuniform thickness profile (graph at top of frame). Lines show possible paths for metal atoms ejected
from the target ring (cathode).
gas mixture[3]. A higher oxygen fraction leads to higher oxygen incorporation in the film, ie a film that is
more metal-oxide than metal. Transparent Ta2 O5 films were made at a 5:1 Ar:O2 flow ratio.
The current delivered by the power supply primarily affects the sputter rate, with higher currents leading
to higher deposition rates[4]. Sputtering too fast can lead to oxygen-poor, opaque films. The sputter guns
are powered by a power supply configured to run in constant-current mode, but the plasma can become
unstable at high voltages (>400 V for our system). This presented a problem for depositing Ta2 O5 which
has the unusual property that increasing the oxygen flow rate leads to an increase in the sputter voltage for a
constant current. The solution to this problem came by adjusting yet another parameter, the total pressure.
By increasing both the O2 and Ar flow rates while maintaining their ratio, the overall chamber pressure can
be varied. At a 5:1 ratio, Ta2 O5 films were made on the Sloan system at 20 mTorr total pressure with 175
mA current and 375-400 V voltage. Deposition rates varied from 5-10 nm/min.
Other deposition parameters (substrate temperature, anode material and size, etc) were varied throughout
the research, but their effects were not systematically studied.
2.3
Reactive Doping
Transparent oxide films such as Ta2 O5 are typically highly insulating with resistivities on the order of 106
Ωm[7]. To be useful in photovoltaic devices, they must be doped to increase conduction.
We attempted to dope the Ta2 O5 films by substituting carbon or fluorine for oxygen atoms in the lattice.
This was done by adding gasses containing these elements to the reactive gas mixture. For carbon doping
the oxygen gas normally used was replaced with carbon dioxide (CO2 ), which was used in the same ratio as
the O2 [6]. In the case of fluorine doping, small amounts of carbon tetra-fluoride (CF4 ) were added to the
Ar-O2 mixture through a third independent gas line.
This approach is appealing as it allows the doping rate to be varied simply by changing the concentration
of the doping gas. Unfortunately we were not able to appreciably increase the conduction of our samples in
this way.
8
2.4
Target Doping
The more traditional method of doping metal-oxide semiconductors is to add metallic impurities[8]. Previous
students accomplished this by adding small amounts of aluminum (Al) to their zinc and indium targets. Many
of these films conducted, and I studied the effect of this form of doping on the optical properties of the films.
2.5
Routine Characterization
After depositing a film, several tests are usually performed to characterize its properties. Thickness measurements were performed using a Dektak IIA stylus profilometer. The sputtering geometry used results in
a nonuniform thickness profile (see Figure 2.1 Right).
Structural properties were assessed using a Philips X’Pert X-ray diffractometer. This system measures
the intensity of X-rays reflected off of a thin film as a function of the angle of incidence θ at a given wavelength
λx . In a crystalline or polycrystalline sample, one expects to see sharp peaks in this spectrum at angles
given by the Bragg condition,
2a sin θ = mλx
m = 1, 2, 3, ...,
(2.1)
where a is the spacing of crystal planes perpendicular to the plane of incidence of the X-rays[5]. See, for
example, the polycrystalline pattern in Figure 2.2 Left. This film was fabricated by another group using
the same sputtering method, but was subjected to a post-deposition annealing process which led to the
formation of crystal structure[3]. The absence of sharp peaks in the spectra of the Ta2 O5 films I studied
(see Figure 2.2 Right) suggests that they were amorphous, which is in line with observations of non-annealed
films made by[3].
Resistivities were measured using a four probe method and two lock-in amplifiers. This method (shown
schematically in Figure 2.3) ensures that the lead wire and probe contact resistances are not included in the
measured sample resistance. The current I is provided by the reference channel of the first lock-in and is
measured as the voltage drop across a 10 Ω resistor. The ballast resistance RB is chosen to be large enough
so that I is unaffected by the sample, lead and contact resistances. The second lock-in measures the voltage
drop ∆V across a second pair of probes. If the film thickness d is much less than the probe spacing, then
Figure 2.2: Left: X-Ray diffraction (XRD) pattern for a polycrystalline Ta2 O5 film[3]. The location of the
peaks is related to the lattice parameters of the crystallites. Right: XRD pattern for an amorphous Ta2 O5
film (sample WT-148). Note that both vertical scales are arbitrary. The pattern for the amorphous film is
likely on the order of the background noise of the polycrystalline pattern.
9
Figure 2.3: Schematic diagram of four probe resistivity measurement[2]. This setup eliminates errors from
lead wire and contact resistances.
the resistivity ρ is given by
ρ=d
π ∆V
,
log 2 I
(2.2)
where log is the natural logarithm[9]. This technique provides precise measurements for small resistivities,
but for the insulating films I studied it is less illuminating.
Another important electrical property of semiconductors is the carrier density. This is typically measured
using the Hall effect, whereby a magnetic fields establishes a potential gradient across a two-dimensional
sample which is perpendicular to the direction of the field (see Figure 2.4). The size of this potential difference
(called the Hall voltage VH ) is a measure of the carrier density N . If a current I is applied to a sample of
thickness d and a magnetic field B is directed into it, the carrier density is given by
N=
BI
,
edVH
(2.3)
where e is the carrier charge[10]. The sign of VH determines the charge of the carriers. If both positively and
negatively charged carriers are present, the N in Equation 2.3 is the difference in the positive and negative
carrier densities, with the sign determining the charge of the dominant carrier.
Optical transmission measurements were taken using a UV-Vis-NIR spectrophotometer (see Section 4).
Figure 2.4: The Hall effect causes a potential difference VH across a two-dimensional semiconductor related
to the carrier density. The applied magnetic field B is out of the page and the current I is indicated by the
arrow.
10
3
3.1
Theory
Statement of Problem
One of the most valuable pieces of information used to characterize our samples is the optical transmission
spectrum. This measurement gives the ratio of the intensity of light transmitted through the sample to the
incident intensity as a function of wavelength. The purpose of this section is to derive an equation for the
transmission spectrum in terms of the optical properties of the material.
The sample consists of a thin dielectric film with an unknown permeability µ and permittivity deposited
on a thick but finite substrate. Both media are considered to be infinite slabs of uniform thickness with
plane-parralell sides with air on either side. Maxwell’s equations, with appropriate boundary conditions, can
be applied to find the coefficients of reflection and transmission at each interface (air/film, film/substrate
and substrate/air)and then find the net transmission and reflection of the composite system.
3.2
Maxwell’s Equations and Electromagnetic Waves
Maxwell’s equations (in CGS units) for linear dielectric media are[11]
~ = ∇ · E
~ = 4πρ,
∇·D
(3.1a)
~ = µ∇ · H
~ = 0,
∇·B
(3.1b)
~
~ = − µ ∂H ,
∇×E
c ∂t
(3.1c)
and
~
~ + ∂E .
~ = 4πσ E
(3.1d)
∇×H
c
c ∂t
~ r, t), the electric field E(~
~ r, t), the magnetic flux density B(~
~ r, t)
They relate the electric displacement D(~
10
~
and the magnetic field H(~r, t) where the speed of light c = 3.0 × 10 cm/s. For media with zero conductivity
or free charge density (σ = ρ = 0), the wave equations
~−
∇2 E
~
µ ∂ 2 E
=0
2
2
c ∂t
(3.2a)
~ −
∇2 H
~
µ ∂ 2 H
=0
2
c ∂t2
(3.2b)
and
can be derived.
√
√
The phase velocity v = c/ µ = c/n, where n = µ is called the index of refraction of the medium. For
√
most materials, the permeability µ in CGS units is practically unity and can be ignored, leaving n = .
~ = E~0 ei(~k·~r−ωt) where the
The plane wave solution to Equation 3.2a is the real part of the exponential E
wavevector |~k| = 2πn/λ points in the direction of propagation. Substituting this solution into Equation 3.1c
and integrating with respect to time gives
~ =
H
√
~ = n(ŝ × E),
~
(ŝ × E)
(3.3)
where ŝ is a unit vector in the direction of propagation[12]. For non-conducting media, , n and k are all
purely real.
11
3.3
Reflection and Transmission
~ A ei(k·r−ωt) is incident
When an electromagnetic wave with an electric field vector given by the real part of E
~ A (which may be complex if
on the boundary between two homogeneous dielectric media, it’s amplitude E
k
iφ
there is a phase offset given by e ) may be resolved into a component EA polarized parallel to the plane of
⊥
incidence (defined by the dashed lines in Figure 3.1) and one perpendicular to it EA
. Referencing Figure 3.1
~
i(kR ·~
r −ωt)
~
we see a reflected wave with an electric field vector given by the real part of ER e
and a transmitted
i(k~T ·~
r −ωt)
~
wave with an electric field vector given by the real part of ET e
, the amplitudes of both of which
k
k
(which may be complex) can also be resolved into components parallel to the plane of incidence (ER , ET )
⊥
and perpendicular to the plane of incidence (ER
, ET⊥ ). One can also find the parallel and perpendicular
components of the magnetic field vectors associated with each of these electric field vectors using Equation
3.3.
Figure 3.1: Interface between two media with different indices of refraction, showing incident, reflected and
transmitted EM waves and angles of incidence and refraction. The z-axis points into the page
If a coordinate system is established with the plane of incidence in the xz plane and the interface between
the two media in the yz plane, the total components of the electric and magnetic field vectors in the x and y
directions can be found by adding the parallel and perpendicular components of the incident, reflected and
transmitted waves. In the first medium
k
k
E1x = (EA + ER ) cos φ1 ,
⊥
⊥
E1y = EA
+ ER
(3.4a)
and
⊥
⊥
H1x = n1 (−HA
+ HR
) cos φ1 ,
k
k
H1y = n1 (HA − HR ).
(3.4b)
In the second medium
k
E2x = ET cos φ2 ,
E2y = ET⊥
(3.5a)
and
H2x = −n2 HT⊥ cos φ2 ,
k
H2y = n2 HT .
(3.5b)
Our definition of the plane of incidence fixes Ez = Hz = 0. The boundary conditions on Maxwell’s
equations require that both components of the the electric and magnetic field vectors must be continuous
if there is no surface charge at the interface[12]. One may then solve for the amplitudes of the transmitted
and reflected waves in terms of that of the incident wave[11]:
12
k
=
n1 cos φ2 − n2 cos φ1
≡ rk ,
n1 cos φ2 + n2 cos φ1
(3.6a)
=
2n1 cos φ1
≡ tk ,
n1 cos φ2 + n2 cos φ1
(3.6b)
⊥
ER
n1 cos φ1 − n2 cos φ2
=
≡ r⊥ ,
⊥
n
EA
1 cos φ1 + n2 cos φ2
(3.6c)
ET⊥
2n1 cos φ1
=
≡ t⊥ .
⊥
n1 cos φ1 + n2 cos φ2
EA
(3.6d)
ER
k
EA
k
ET
k
EA
and
The above relations define the Fresnel reflection and transmission coefficients r⊥ , rk , t⊥ and tk . Note that
the derivation does not depend on whether the refractive indices n1 and n2 are real or complex. For normal
incidence (when the incident wave is normal to the interface), φ1 = φ2 = 0 and the distinction between
parallel and perpendicular components disappears. The Fresnel coefficients for normal incidence upon an
interface between any two media i and j are given by
rij =
ni − nj
ni + nj
tij =
2ni
.
ni + nj
(3.7)
The Fresnel transmission coefficient gives the ratio of the amplitude of the transmitted wave to that of
the incident wave. Similarly, the reflection coefficient gives the ratio of the amplitude of the reflected wave
to that of the incident wave. In the experimental setting, the quantities of interest are the ratios of the
intensities of the transmitted and reflected waves to that of the incident wave. The intensity of an EM wave
is proportional to the square of its amplitude, so the ratios of the intensities of the reflected and transmitted
waves to that of the incident wave (called simply the reflection and transmission) are are given by
Rij = |rij |2 =
(ni − nj )2
(ni + nj )2
Tij =
nj
nj
4n2i
.
|tij |2 =
ni
ni (ni + nj )2
(3.8)
The factor of nj /ni in the transmission reflects the fact that the energy carried by an EM wave is
proportional to its phase velocity as given by the index of refraction. To deal with multiple interfaces, the
Fresnel coefficients should be combined in the appropriate way and then the result squared as above to
obtain the reflection or transmission.
3.4
Propagation
Since light is an electromagnetic wave, it’s propagation is profoundly affected by the electrical properties
of the medium. In a metal or semiconductor where there may be some electrical conductivity (σ 6= 0), the
wave equation for the electric field vector becomes[12]
~ =
∇2 E
~
~
4πµσ ∂ E
µ ∂ 2 E
+ 2
.
2
2
c ∂t
c
∂t
(3.9)
~
The ∂∂tE term implies that the wave is damped and will be attenuated as it propagates. If time dependence
of the form e−iωt is assumed, one finds
~ + k̃ 2 E
~ = 0.
∇2 E
(3.10)
This expression is equivalent to Equation 3.2a, but the wavevector |k̃| = 2π
λ ñ is now complex. The
complex index of refraction ñ = n − iκ where κ is called the extinction coefficient and n, κ are real. For our
13
purposes it will be useful to write κ = −αλ/4π where α ≥ 0 is called the absorption coefficient. One may
α
α
then write |k̃| = 2π
λ n + i 2 = |k| + i 2 . The plane wave solution to Equation 3.10 for propagation in the x
direction is
~
E(x,
t) = Re[E~0 ei(kx−ωt) e−αx/2 .]
3.5
(3.11)
Multiple Reflections
Consider the configuration in Figure 3.2 where a single film (thickness d) is “sandwiched” between two media
which are assumed to be non-absorbing. A normally incident wave propagating in the x direction with an
~ A ei(kx−ωt) will be split into transmitted and reflected waves with amplitudes given
electric vector given by E
~ T = t12 E
~ A and E
~ R = r12 E
~ A . If the film is absorbing, its index of refraction
by the Fresnel coefficients as E
is complex with n2 = n − iκ. If the first interface is taken to be at x = 0 and the second at x = d, then
~ A e−i(ωt−φ/2) e−αd/2 where
using Equation 3.11 the amplitude at the second interface is found to be t12 E
φ/2 = 2π
λ nd is the phase acquired traversing the film and α is the same as above. The phase and attenuation
1
factors can be combined by defining the complex phase β ≡ 2π
λ n2 d = 2 (φ + iαd). The amplitude at the
~ A e−iωt eiβ . For the rest of this discussion, we will ignore the time
second interface can then be re-written as E
part since the quantity of interest is simply the ratio of the transmitted intensity to the incident intensity.
Figure 3.2: Schematic diagram of a thin film (thickness d) between two infinite media, showing contribution
to reflected and transmitted wavefronts from rays reflected within the film multiple times.
To calculate the intensity of light transmitted through the film, the contribution from rays reflected
multiple times must be calculated[11]. The ratio of the amplitude of a wave passing directly through the film
to the amplitude of the incident wave is equal to the product of the Fresnel transmission coefficients at the two
interfaces and the attenuation term: t12 t23 eiβ . For a wave transmitted through the first interface, reflected
off the second and then again off the first and then transmitted through the second interface, this ratio would
further be multiplied by the Fresnel reflection coefficients of the two interfaces yielding t12 t23 r23 r21 e3iβ , where
the r21 subscript denotes reflection into medium 2 and the factor of 3 in the exponential denotes the fact
that the light passes through the film three times. Light can bounce back and forth in the film any number
of times, so the ratio t123 of the total transmitted amplitude to that of the incident wave is given by
2 2 5iβ
t123 = t12 t23 eiβ + t12 t23 r23 r21 e3iβ + t12 t23 r23
r21 e + ... = t12 t23 eiβ
∞
X
`=0
This sum is a simple geometric series and can be written as
14
[r21 r23 e2iβ ]` .
(3.12)
t123 =
t12 t23 eiβ
.
1 − r21 r23 e2iβ
(3.13)
A similar calculation gives the ratio r321 of the reflected amplitude of a wave incident from the substrate
side to that of the incident wave:
r321 =
r32 + r21 e2iβ
.
1 + r32 r21 e2iβ
(3.14)
Equation 3.13 gives the ratio of the amplitude of the transmitted wave to that of the incident wave. The
transmission (the ratio of the transmitted intensity to the incident intensity) T123 can be calculated from
∗
Equation 3.8. Accordingly, T123 = (n3 /n1 )t∗123 t123 . Noting that |eiβ |2 = e−iβ eiβ = e−i(φ−iαd)/2 ei(φ+iαd)/2 =
e−αd , one finds
T123 =
n3
|t12 t23 |2 e−αd
.
n1 1 − 2r21 r23 e−αd cos φ + |r21 r23 |2 e−2αd
(3.15)
∗
The ratio of the intensity of reflected light to incident light is given by R321 = r321
r321 , or
R321 =
|r32 |2 + 2r32 r21 e−αd cos φ + |r21 |2 e−2αd
.
1 + 2r32 r21 e−αd cos φ + |2r32 r21 |2 e−2αd
(3.16)
The transmission in Equation 3.15 does not accurately portray the physical situation under which thin
film spectra are obtained in the laboratory. Figure 3.3 shows the configuration correctly, with the thin film
on top of a thick (but finite) substrate, usually a glass microscope slide. For a thick dielectric slab, Equation
3.15 breaks down because the multiple reflections in the slab do not interfere as they do in the film.
Figure 3.3: The physical situation for thin film measurements in the lab. The film (index n2 ) rests on top
of a thick but finite substrate (index n3 ) bounded on either side by air (index n1 ).
The above analysis assumes that the light is monochromatic and coherent and can thus be expressed as
a simple plane wave. For optical transmission measurements, the wavelength of light is on the order of the
film thickness, which itself is small compared to the other dimensions of the problem. Over such short length
scales the light is approximately coherent. The substrate thickness is several hundreds of thousands of times
greater than that of the film. Over such long length scales, the finite bandwidth of the spectrometer comes
into play and the light can not be considered monochromatic, a prerequisite for interference[13]. Interference
effects can be averaged out (just as they are by the spectrometer) by integrating |t123 |2 (Equation 3.12) over
the effective bandwidth ∆λ [14]. One finds
15
Z
∞
X
|t12 t23 eiβ
[r21 r23 e2iβ ]` |2
dλ
=
∆λ
Z
∞ X
∞
X
dλ
.
∆λ
∆λ
∆λ
`=0
`=0 `0 =0
(3.17)
In a thick sample, the real part of β must be much greater than the imaginary part for the reflected light
to be detectable at all. This means that the factors of e2iβ` in Equation 3.17 oscillate rapidly with λ and
vanish when averaged over ∆λ. The only terms in the sum that remain are then the terms where ` = `0 and
the real parts of β cancel. One then has
T =
T = |t12 t23 |2 e−αd
∞
X
|t12 t23 |2 e−αd
[|r21 r23 |2 e−2αd ]` =
`=0
0
∗ ∗ ` 2i(β`−β
(r21 r23 )` (r21
r23 ) e
|t12 t23 |2 e−αd
.
1 − |r21 r23 |2 e−2αd
∗ 0
`)
(3.18)
Equation 3.18 gives the ratio of the transmitted intensity to the incident intensity for a thick slab between
two infinite media. To account for the presence of the film the transmission and reflection coefficients for
the first interface are replaced by those for the film from Equations 3.13 and 3.14 (this is why the reflected
intensity was calculated from the substrate side). The result is given by
T =
(1 − R13 )T123
,
1 − R13 R321
(3.19)
where R13 = |r13 |2 is the reflection from the glass/air interface. Substituting Equations 3.15 and 3.16 and
simplifying yields
T =
n3 (1 − |r13 |2 )|t12 t23 |2 e−αd
.
1 − |r13 r32 |2 + 2r21 r32 (1 − |r13 |2 )e−αd cos φ + |r21 |2 (|r32 |2 − |r13 |2 )e−2αd
(3.20)
One can now finally substitute the Fresnel coefficients from Equation 3.7 with n1 = 1 for air, n2 = n and
the substrate index n3 = s. Also plugging in the above expression for φ, Equation 3.21 is obtained, which is
the same expression quoted by Swanepoel[15].
T =
(n +
1)3 (n
+
s2 )
−
2(n2
−
1)(n2
16n2 se−αd
.
3
2 −2αd
− s2 )e−αd cos ( 4π
λ nd) + (n − 1) (n − s )e
(3.21)
The use of a purely real index of refraction n2 = n may seem inappropriate as Equation 3.20 was derived
with the assumption that n2 was complex. A more rigorous expression would use n2 = n − iκ. Over most
of the spectrum, however, κ is nonzero but is several orders of magnitude smaller than the real part n. The
effect of the imaginary part enters through the exponential terms. In regions of strong absorption, however,
this approximation breaks down and different expressions must be used.
The phenomenon of thin film interference is buried in the φ dependence of Equation 3.20. The transmission oscillates between extreme values when φ takes on values which are integer multiples of π. To determine
the nature of these extrema, simply take the second derivative of Equation 3.21 and determine its sign when
φ = 2mπ, or when
2nd = mλ
m = 1/2, 1, 3/2, 2, 5/2, ...
(3.22)
This is the classic expression for thin film interference. The nature of the extremes turns out to depend
on the sign of (n − s). If n < s, the interference order m = 1, 2, 3, ... for transmission maxima and m =
1/2, 3/2, 5/2, ... for minima. If n > s, the opposite is true. Note that this equation gives information on
the product nd based on the measured wavelengths of the interference fringes and cannot give either n or
d independently. It is only through use of the full transmission equation that n may be extracted without
knowledge of d.
16
4
4.1
Measurement and Analysis
UV-Vis-NIR Spectrophotometer
I made transmission measurements using the Varian Cary 5e UV-Vis-Nir spectrophotometer located in the
Chemistry department. It is configured to split the measurement beam in a 3-phase “chopper” cycle. The
chopper itself is a disk (see Figure 4.1) which spins at 30 Hz and is divided into transparent, mirror and
matte sections.
Figure 4.1: The “chopper” disk alternates the path of the measurement beam as it rotates. It has three
sections: transparent (1; white), mirror (2; grey) and matte (3; black).
The chopper is placed between the light source and the sample chamber (see Figure 4.2) and as it spins,
the beam from the source is incident on each of the three section in turn. Each cycle of the chopper disk
then has three “phases”. In the transparent phase (the phase during which light is passing through the
transparent section of the disk) the light beam from the source follows a path taking it through the sample
and to the detector on the other side of the sample chamber. During the mirror phase the beam is reflected
off the mirror portion of the disk onto a path taking it through the sample chamber but not through the
sample. The intensity of light measured by the detector during this phase is equal to the intensity of light
incident on the surface of the sample. The spectrometer software divides the transparent phase intensity by
the mirror phase (incident) intensity to obtain the transmission.
Figure 4.2: Simplified schematic of spectrophotometer. The chopper sends the measurement beam alternately
through the sample and not through the sample.
During the third phase the beam from the source is absorbed by the matte black portion of the disk
and no light from the source is incident on the detector. This is appropriately called the “no-measurement”
phase. In his phase a stepper motor rotates a diffraction grating to change the wavelength of the beam. This
no-measurement phase wavelength stepping eliminates errors associated with the motion of the grating and
increases resolution.
17
Before every measurement, a baseline scan with no sample in the beam line is performed to calibrate
the system. This eliminates errors associated with differences between the two beam paths. A scan of an
uncoated microscope slide is performed next. Once these measurements are made the samples of interest
(the coated slides) can be scanned. All measurements are made from 200 - 2000 nm. The spectral bandwidth
of the beam (full width at half height) was 2 nm and the scanning rate was 600 nm/min. An example of the
resulting raw data for a ZnO sample are presented in Figure 4.3.
The uncertainty in the transmission measurements is estimated by looking at the spectrum of a blank
substrate. Over small wavelength ranges the transmission should be approximately constant. By observing
the statistical noise in the data, I found the error to be less than 0.1%.
Note that by using the classic interference condition, Equation 3.22, for the 1 mm thick glass substrate,
the wavelength spacing between adjacent maxima can be shown to be less than 0.1 nm, which can not be
resolved at this bandwidth. This justifies our use of Equation 5.2.
Figure 4.3: Example of raw data from spectrophotometer showing blank substrate (flat upper curve) and
ZnO coated sample (oscillating lower curve). Sample Zn-005.
4.2
Refractive Index of Substrate
Before the optical properties of the thin film can be determined, those of the thick substrate on which it
rests must be found. The absorption coefficient is assumed to be zero, though absorbing substrates can
in principle be treated as in [16]. The thickness of the substrate does not enter into Equation 3.21, so all
that remains is to calculate the index of refraction s. One begins by measuring the transmission of a blank
substrate (see Figures 4.3 and 4.4 Left). The transmission for this simple case can be found using Equation
3.18 and is given by
2s
.
(4.1)
s2 + 1
Inverting this expression allows us to determine s as a function of wavelength from the measured transmission Ts . Note from Figure 4.3 that there is a strong absorption feature in the short wavelength region of
the measured spectrum. This is caused by a sudden increase in the absorption coefficient of the substrate,
not the real part of the refractive index. To account for this effect, s is taken to be constant for wavelengths
below 500 nm, although s probably increases by about a factor of 2 in this region. The increase in the
imaginary part, by comparison, is likely a factor of 100. The result is shown in Figure 4.4. The substrate’s
refractive index is seen to vary by less than 5% across the measurement region. Though s could be assumed
to be a constant across the whole spectrum, allowing it to vary gives more accurate results.
Ts =
18
Figure 4.4: Left: Example transmission spectrum of blank substrate. Right: Calculated refractive index of
blank substrate.
From Equation 4.1 and the rules of error analysis[17] one can see that the relative uncertainty in s is the
same as the relative uncertainty in Ts , or about 0.1%.
4.3
Interference Envelope
Equation 3.21 may be more succinctly written as
T =
Ax
,
B − Cx cos φ + Dx2
(4.2)
where
A ≡ 16n2 s,
(4.3a)
B ≡ (n + 1)3 (n + s2 ),
(4.3b)
C ≡ 2(n2 − 1)(n2 − s2 ),
(4.3c)
D ≡ (n − 1)3 (n − s2 ),
(4.3d)
x ≡ e−αd ,
(4.3e)
4π
n.
λ
(4.3f)
and
φ≡
Using this form, the interference envelope consisting of the functions that pass through the extreme points
in the spectrum (cos φ = ±1) are defined by
TM ≡
Ax
B − Cx + Dx2
and
(4.4a)
Ax
.
(4.4b)
B + Cx + Dx2
The upper bounding function TM passes through the maxima of the spectrum and Tm through the
minima, as in Figure 4.5, where the envelope for a simulated transmission curve is shown.
In practice these functions are built by creating interpolating functions between the extrema in the
transmission data. Figure 4.6 Left shows a comparison between first, second and third order interpolations
Tm ≡
19
Figure 4.5: Simulated transmission curve (red) for a 2000 nm thick film, showing interference envelope
functions TM (Equation 4.4a) and Tm (Equation 4.4b) in blue.
of an example data set. The higher order interpolations tend to give spurious results especially if the film
is thin and few interference fringes are visible. A first order (linear) interpolation, while crude, is the most
likely to capture the smoothly varying interference envelope (see Figure 4.6 Right).
Figure 4.6: Left: Comparison of interference envelopes built from first (red), second (blue) and third (green)
order interpolations. Dashed line is a measured transmission spectrum for sample WT-152. Right: Example
of a measured spectrum (green) with linear interference envelope (blue). Sample Zn-005.
4.4
Refractive Index of Film
Following Swanepoel[15], note that
1
2C
1
−
=
.
Tm
TM
A
The quantity on the right hand side is independent of the thickness of the film. Solving for n yields
(4.5)
n = [N + (N 2 − s2 )1/2 ]1/2 ,
(4.6)
TM − Tm
s2 + 1
+
.
T M Tm
2
(4.7)
where
N = 2s
20
Using the previously calculated values for s(λ) and the linear interference envelope described in the
previous section, the index of refraction of the film can be extracted without knowledge of the film thickness.
As n depends on s1/2 , the relative uncertainty in s (about 0.1%) leads to a relative uncertainty in n of
less than 0.05%[17]. This is small compared to the error arising from our use of interpolated interference
envelopes. According to Swanepoel, a 1% error in the envelopes leads to a 1-3% error in n. I conservatively
estimate the error in n to be about 3%.
For many of the films studied, the index of refraction was found to fit well to a second order Cauchy
dispersion relation[18] of the form
n2
n1
+ 4.
(4.8)
λ2
λ
This empirical fit was derived by Cauchy based on a theory of light propagation later proven to be false.
Nevertheless, it is often used by optical spectroscopists for its simplicity[18]. An example of an extracted
n(λ) and it’s accompanying Cauchy fit are shown in Figure 4.7.
n(λ) ≈ n0 +
Figure 4.7: Example of extracted refractive (yellow) index with Cauchy fit (red, Equation 4.8). The blue
dashed lines show the 3% relative uncertainty in n. Sample Zn-005.
4.5
Film Thickness
Now that the index of refraction is known, the classic thin film interference equation (Equation 3.22) may be
used to determine the thickness of the film. This is done using a simple graphical method. First, Equation
3.22 is rewritten as
2nd = (m0 − `/2)λ,
(4.9)
where ` = 0, 1, 2, ... and m0 is the unknown order of the first interference fringe in the measured spectrum
(ie the extreme that occurs at the longest wavelength). Then a plot is made of the straight line
`/2 = 2d(n/λ) − m0 .
(4.10)
The value of m0 is chosen to be the nearest integer or half-integer to yield a good fit to the data, bearing in
mind that if n > s then an integer-valued m0 implies that the the first interference extremum is a maximum,
with a half-integer implying a minimum. The slope of the line is 2d where d is the film thickness. An
example of this method is shown in Figure 4.8. Sometimes the points corresponding to higher order extrema
will deviate significantly from the linear relation in Equation 4.10. This is due to absorption effects as the
21
imaginary part of the refractive index becomes non-negligable near the band-edge. These points should be
ignored when determining the thickness. Uncertainty in the slope of the line (and hence the thickness) is
determined through standard weighted lest-squares methods[17].
If the extracted n(λ) is fit well by Equation 4.8 than using this fit in Equation 4.10 can reduce dispersion
and lead to more accurate thickness measurements.
Figure 4.8: Graphical method for determining film thickness, interference order `/2 is plotted against n/λ.
The slope of the line is twice the film thickness. Sample Zn-005, with m0 = 3/2 and d = 617 nm.
4.6
Absorption Coefficient
If one now adds the reciprocals of Equations 4.4a and 4.4b, one can solve for x = e−αd [15] and find
x=
F − [F 2 − (n2 − 1)3 (n2 − s4 )]1/2
,
(n − 1)3 (n − s2 )
(4.11)
where
F = 4n2 s
T M + Tm
.
TM Tm
(4.12)
The uncertainty in α is found through error propagation and for the sample in Figure 4.9 Right varies
from 3-4×10−5 nm−1 across the measurement region. This uncertainty is most sensitive too the error in
Tm , which is largest in the long-wavelength regime where there are insufficient data to construct accurate
interpolations.
4.7
Band Edge
Equation 4.11 is only valid when our assumption n κ holds. At wavelengths corresponding to photon
energies near the bandgap energy of the film, this assumption fails as α increases by orders of magnitude.
This strong absorption washes out interference effects and, since x is very small, Equation 4.2 can be rewritten as T0 ≈ Ax/B. The values of A and B however are not the ones given in Equation 4.3. The
exponential nature of this relation, however, allows a simple fit to the the transmission data near the band
edge. Such a fit is shown in Figure 4.9 Right.
The value of the absorption coefficient near the band edge provides key information about the film’s band
structure. The energy gap Eg can be found from the well known relation[19]
22
Figure 4.9: Left: Example absorption coefficients calculated from the refractive index n (yellow) and the
Cauchy fit to n (red). Blue dashed lines show uncertainty. Right: Example absorption coefficient near band
edge. The substrate itself begins absorbing below 250 nm, so data in this region should be ignored. Note
the difference in scales on the two plots. Sample Zn-005.
α ∝ (E − Eg )1/2 ,
(4.13)
where E = hν = hc/λ is the photon energy (Planck’s constant h = 4.136×10−15 eV s). This expression
assumes a direct bandgap and parabolic valence and conduction bands. Ta2 O5 , ZnO and In2 O3 all have
direct bandgaps[3][18][20]. A linear fit to a plot of α2 vs E has an E-intercept at Eg , as in Figure 4.10 Left.
The absorption coefficient can also be fit to the Urbach relation α ∝ eE/E0 , where E0 is called the Urbach
slope and characterizes the width of exponential band-tailing into the gap and is frequently used as a measure
of disorder and impurities in thin films[19]. The Urbach slope is found from a linear fit to a plot of log α vs
photon energy, as in Figure 4.10 Right.
It should be noted from Figure 4.3 that the substrate itself may have strong absorption features. If the
bandgap energy of the film is close to or greater than that of the substrate, the film’s band edge may become
convoluted with or hidden by substrate absorption. In cases such as these, as in Ta2 O5 with Eg = 4.2 eV
right at the bandgap of the glass substrates used, the analysis in this section should be applied with caution.
Figure 4.10: Left: Plot of α2 vs hν with fit to linear region. The intercept is the bandgap energy. Right:
Plot of ln α vs hν with fit to linear region. The slope is the inverse of the Urbach disorder parameter. Sample
Zn-005, with Eg = 3.28 eV and E0 = 75 meV.
23
Once α has been determined near the band edge, it can be joined to the interference-zone absorption to
obtain α(λ) across the whole spectrum. Mathematically,
α(λ) = αgap u(λc − λ) + αint u(λ − λc ),
(4.14)
where u(λ) is the unit step function and λc is the wavelength where the band edge drop meets the interferencezone transmission.
4.8
Absorption in Doped Films
The doping processes discussed in Section 2 can have a dramatic effect on the transmission spectrum, as
in Figure 4.11 Left, where I have graphed the transmission spectrum of a doped ZnO film in blue and
that of an uncoated substrate in red. The increased absorption in the infrared region of the spectrum is
due to excitations of donor-level electrons to the conduction band at these energies. A nearly-free-electron
model of doped semiconductors gives the following relation between the absorption coefficient α and the
room-temperatue carrier density N associated with the doping[21]:
N 2
e2
λ ,
(4.15)
4π 2 m∗ 0 nc3 τ
where e is the electron charge, m∗ is the effective mass of conduction band carriers, 0 is the vacuum
permittivity, n is the index of refraction and c is the speed of light. The scattering time τ is related to the
mobility µ = eτ /m∗ . By fitting the infrared absorption coefficient with a λ2 dependence, one can extract
the parameter N/τ (see Figure 4.11 Right). To fully determine the carrier density and mobility, a 4-probe
resistivity measurement can be made and the relation 1/ρ = σ = N eµ employed.
The complete absorption coefficient is now built from its value in the three regions of the spectrum, the
band edge, the interference zone, and the infrared “carrier zone”:
α=
α(λ) = αgap u(λc − λ) + αint u(λ − λc )u(λd − λ) + αcar u(λ − λd ),
(4.16)
where αcar is the carrier zone absorption and λd is the critical wavelength where doping effects become
noticeable.
Figure 4.11: Left: Example transmission spectra for a doped ZnO film (blue) showing infrared absorption
and uncoated glass microscope slide (red). Right: By fitting the infrared absorption coefficient (blue) with a
λ2 dependence (red), one can determine the ratio of the carrier density N to the scattering time τ . Sample
Zn-044 with N/τ = 7.1 × 1035 cm−3 s−1 as determined through Hall effect measurements.
24
5
5.1
Results and Discussion
Reconstructed Curves
Once s(λ), n(λ), d, and α(λ) have been found according the above analysis, the full transmission curve can
be reconstructed using Equation 3.21. This is a useful check to ensure the extracted optical properties are
accurate. Figure 5.1 is a typical example. It also shows one of the limitations of this method, which is
that in the long wavelength region there are insufficient data (transmission extrema) to construct accurate
transmission envelopes. This results in uncertainty in all properties derived from these envelopes. It is
sometimes possible to improve the reconstruction by extrapolating n from the visible region using Equation
4.8, or some other dispersion rule. For very thin (d < 200 nm) films with only 2-3 measurable fringes, even
this approach is difficult.
Figure 5.1: Example of reconstructed transmission curve. The blue curve is raw data and the red curve is
the reconstruction. Sample Zn-005. Compare Figure 4.3.
5.2
Refractive Indices
Refractive indices for materials are often quoted as a single number which is independent of wavelength.
Taking note of Figure 4.7, this may be seen to be a reasonable approximation. The values of n calculated
for transparent films usually vary by only 5-10% across the entire measurement region. For calculating the
thickness from Equation 3.22, approximating n as a constant results in an error of 5-10%, which for our
deposition process is less than the variation in the thickness across the substrate. The same is true for many
other properties of the material that depend on n.
For reconstructing the transmission curve, however, this approximation falters. The term in the transmission equation that determines the amplitude of the interference fringes (Equation 4.3c) depends on n4 ,
meaning that even a 5% error in n will yield a 20% error in the fringe amplitude. This effect can be seen in
Figure 5.2, where the location of the maxima is accurate but the amplitude is distorted. Allowing n to vary
with wavelength allows us to capture the complete transmission curve more accurately. Transmission near
the bandgap (where this effect is most pronounced) is of special importance for photovoltaic applications,
making the constant n approximation unjustifiable.
Physically, the real part n of the refractive index must vary with wavelength to satisfy causality in the
universe. The imaginary part κ can be seen to vary with wavelength from the absorption features at the band
edge an in the infrared. Because n and κ are the real and imaginary parts of a single complex parameter,
they must satisfy the Kramers-Kronig relationships[21],
25
1
n(ω) = 1 +
π
and
κ(ω) = −
1
π
Z
Z
∞
−∞
∞
−∞
κ(ω 0 )
dω 0
ω0 − ω
n(ω 0 ) − 1 0
dω ,
ω0 − ω
(5.1a)
(5.1b)
where ω = 2πν = 2πc/λ is the radial frequency. So any variation in κ at any wavelength implies that n will
also have to vary with λ. These relationships can be used, in principle, to determine the imaginary part of
the refractive index from the real part and vice- versa, but this would require information over a broader
spectral range[21].
Figure 5.2: Left: If n is taken as a constant, the reconstructed transmission curve (blue curve) fails to
capture the variation in the amplitude of the interference fringes present in the data (red curve). Right: If
n is allowed to vary, better agreement is achieved, particularly in the fringe amplitude. Sample WT-148.
Table 5.1 summarizes the typical values of n extracted using my method. In total I analyzed the spectra
for 4 Ta2 O5 films, 15 ZnO films (about half of which were doped) and 2 In2 O3 films (both of which were
doped). For Ta2 O5 and ZnO, a range of values is given to show the variation of n with wavelength. Bulk
Ta2 O5 is usually taken to have n ≈ 2.2, with thin films of good transparency showing n ≈ 2.1[3]. The films
I studied showed slightly smaller n values. The observed indices for the ZnO films was in good agreement
with accepted values[18]. Too few In2 O3 films were studied to get a handle on the statistics, but there was
rough agreement with accepted values[22].
Material
Ta2 O5
ZnO
In2 O3
Measured n
1.96(5)-2.04(4)
1.87(6)-1.95(7)
2.0(5)
Literature n
2.09-2.13[3]
1.93[18]
1.95[22]
Table 5.1: Real part of the index of refraction n as extracted with Equation 4.6 and literature values.
Uncertainties in parentheses.
5.3
Optical Versus Mechanical Thickness Measurements
To assess the accuracy of this method, the thicknesses determined through the optical measurement, dopt ,
were compared to those made using a Dektak stylus profilometer, dmech . The results for the Ta2 O5 films
26
studied are presented in Table 5.2. Given the surface irregularity inherent to our sputtering system (Figure
2.1 Right), the numbers show good agreement, supporting the validity of this method.
Film
WT-148
WT-150
WT-151
WT-152
dopt (nm)
588(2)
351(3)
353(3)
306(3)
dmech (nm)
535
300
400
325
% Difference
9%
14%
12%
6%
Table 5.2: Comparison of optical (dopt ) and mechanical (dmech ) thickness measurements for Ta2 O5 films.
Uncertainties in optical measurements in parentheses.
5.4
Bandgap Energies
For the ZnO and In2 O3 studied, the bandgap energies were found using Equation 4.13. Average values are
reported in Table 5.3. The bandgap for ZnO was observed to be slightly smaller than the literature value[18].
Though only two In2 O3 films were analyzed, the bandgap energies were in excellent agreement with accepted
values[20].
Average values of the Urbach slope parameter E0 are also included in Table 5.3. This energy characterizes
the width of exponential band tailing (“fuzziness”) associated with sample impurities[19]. Note the wider
band tail observed in In2 O3 .
Material
ZnO
In2 O3
Measured Eg (eV)
3.27(3)
3.63(2)
Literature Eg (eV)
3.35[18]
3.6[20]
E0 (meV)
69(8)
570(70)
Table 5.3: Measured and literature values of energy bandgap Eg , and Urbach slope E0 for ZnO and In2 O3
films. Uncertainties in parentheses.
This analysis was also applied to the Ta2 O5 films. The measured bandgaps agreed well with the accepted
4.2 eV value[3], but the proximity of the film’s band edge to that of the substrates casts doubt on these
measurements (see Figure 5.3).
Figure 5.3: Close-up of Ta2 O5 transmission curve near band edge (lower curve). The proximity of the Ta2 O5
bandgap to that of the substrate (upper curve) complicates analysis in this region. Sample WT-152.
27
5.5
Carrier Densities
A number of complications arise when one tries to calculate carrier densities of doped films using this method
and Equation 4.15. My method of extracting the real part of the index of refraction relies on measuring the
amplitude of interference fringes, which are wiped out in the infrared by carrier absorption (see Figure 4.11
Left). The index of refraction must then be extrapolated from the interference zone. As n is nearly constant
away from the band edge, this is at most a 10% error.
I tried several methods of obtaining numerical values for the absorption coefficient in the carrier zone,
including solving Equation 4.2 for x. In the end, a simple exponential dependence was used. A more general
expression than Equation 3.21 may be needed to obtain accurate information about α. That being said, the
values of α used fit well to a λ2 dependence (Figure 4.11 Right) and the reconstructed transmission curves
agreed well with the data (Figure 5.4). This qualitative agreement suggests that quantitative results are not
unattainable.
Figure 5.4: Example of reconstructed transmission curve for a doped film. The blue curve is raw data and
the red curve is the reconstruction. Sample Zn-044, with N/τ = 7.1 × 1035 cm−3 s−1 as determined through
optical measurements. Compare Figure 4.11.
After fitting the α values to Equation 4.15, I extracted the values of N/τ in Table 5.4. I compared these to
those obtained earlier by John Scofield from Hall effect measurements. For the ZnO films (m∗ = 0.25me [18])
I studied, the results were somewhat inconsistent. Agreement between the two measurements ranged from
being within a factor of 2 to a factor of 50, though almost all were within an order of magnitude.
A further complication arises when impurities in the sample are considered. The Hall measurements
determine the difference between the positively and negatively charged carrier densities (see Equation 2.3. If
both types of carriers are present in significant numbers, as is likely the case with these films, the measurement
Film
Zn-025
Zn-037
Zn-041
Zn-043
Zn-044
(N/τ )opt (1035 cm−3 s−1 )
2.5
2.9
3.6
0.5
2.1
(N/τ )Hall (1035 cm−3 s−1 )
160
18
4.8
3.5
7.1
Table 5.4: Values of N/τ as extracted from optical data ((N/τ )opt ) and from Hall effect measurements
((N/τ )Hall ).
28
will underestimate N and overestimate τ by as much as a factor of 2. This does not explain the discrepancy
between the Hall and optical measurements, as the optically obtained values were always smaller. The
impurity density contributes to the absorption coefficient additively, but with a different effective mass and
scattering time[21]. The effect of impurities on absorption thus depends on the mobility of the carriers
associated with them.
6
Conclusion
This project has shown the power of Maxwell’s equations in understanding the optical and electrical properties of thin films. Performing a thorough analysis of the transmission spectrum using Equation 3.21 allows
us to deduce many of the parameters important to photovoltaic applications. Armed with these techniques,
we can more deterministically tailor the performance of a complete cell (Figure 1.1). By systematically
determining the effects of the deposition parameters on the optical properties, one could use the same equations to simulate the transmission of a film before it is made. Trade-offs such as conductivity versus infrared
absorption could be computationally optimized without the need to deposit dozens of films by trial and error.
Linking deposition parameters and optical properties (now readily available using this analysis) is then an
important next step for solar cell research at Oberlin.
It is likely that improved accuracy could be achieved by incorporating the full complex refractive index
in the analysis, especially in the infrared where the absorption is more gradual than near the band edge.
Combined with Hall effect measurements, this would enable complete and accurate characterization of the
n- and p-type carrier densities. These are critical factors for photovoltaic applications (see Figure 1.2), but
more work is needed to understand the role of impurities and their effect on the transmission. Pinning down
the discrepancy between the Hall effect and optical measurements is the logical starting point.
We have developed this analysis as a tool for quickly determining the properties of the samples we
generate to guide our research in an informed, orderly fashion.
Acknowledgements
I’d like to thank my advisor, John Scofield, for his years of support and guidance, and the rest of the
Physics and Astronomy community.
References
[1] J H Scofield et el. Sputtered molybdenum bilayer back contact for copper indium diselenide-based
polycrystalline thin film solar cells. Thin Solid Films, 260:26, 1995.
[2] J Gaumer. Fabrication and characterization of zno thin films for cis solar cell window layers. Oberlin
College Honors Thesis, 2002.
[3] J M Ngaruiya et al. Preparation and characterization of tantalum oxide films produced by reactive dc
magnetron sputtering. Physics Status Solidi, 198:99, 2003.
[4] T J Gillespie et al. Reactive magnetron sputtering of transparent and conductive zinc oxide films
deposited at high rates onto cis/cigs photovoltaic devices. Proceedings of the 26th IEEE Photovoltaic
Specialists Conference, 1997.
29
[5] C Kittel. Introduction to solid state physics. Wiley, 2005.
[6] K Chu and J P Chang. Material and electrical characterization of carbon-doped tantalum oxide films
for embedded dynamic random access memory applications. Journal of Applied Physics, 91:308, 2002.
[7] E Krikorian and R J Sneed. Deposition of tantalum, tantalum oxide, and tantalum nitride with controlled electrical characteristics. Journal of Applied Physics, 37:3674, 1966.
[8] D A Sager et al. Diffusion of neodymium into sputtered films of tantalum pentoxide. Journal of the
American Ceramics Society, 85:2581, 2002.
[9] L Maissel et al. Handbook of thin film technology. McGraw-Hill, 1970.
[10] B G Streetmnan. Solid state electronic devices. Prentice Hall, 1990.
[11] O Heavens. Optical properties of thin solid films. Dover, 1965.
[12] M Born and E Wolf. Principles of Optics. Pergamon, 1959.
[13] F Manoocheri S Nevas and E Ikonen. Determination of thin-film parameters from high accuracy measurements of spectral regular transmittance. Metrologia, 40:S200, 2003.
[14] F Manoocheri A Haapalinna and E Ikonen. High-accuracy measurements of specular spectral reflectance
and transmittance. Anal Chim Acta, 380:317, 1999.
[15] R Swanepoel. Determination of the thickness and optical constants of amorphous silicon. J Phys E,
16:1214, 1983.
[16] J Cisneros. Optical characterization of dielectric and semiconductor thin films by use of transmission
data. Applied Optics, 37:5262, 1998.
[17] J Taylor. An introduction to error analysis. University Science Books, 1997.
[18] U Ozgur et al. A comprehensive review of zno materials and devices. Journal of Applied Physics,
98:041301–1, 2005.
[19] A Aqili and A Maqsood. Determination of thickness, refractive index, and thickness irregularity for
semiconductor thin films from transmission spectra. Applied Optics, 41:219, 2002.
[20] H Jia et al. Efficient field emission from single crystalline indium oxide pyramids. Applied Physics
Letters, 82:4146, 2003.
[21] M Fox. Optical Properties of Solids. Oxford University Press, 2001.
[22] S Laux et al. Room temperature deposition of indium tin oxide thin films with plasma ion-assisted
evaporation. Thin Solid Films, 335:1, 1998.
30
Appendix A.nb
Clear@slidedata, slide100, t, sD;
slidedata = Import@"filename.csv"D;
êê Imports raw data on blank slide
slide100 = slidedata.881, 0<, 80, .01<<;
êê Converts data from 0 - 100 % to 0 - 1
t = Interpolation@Join@Drop@slide100, -300D, Transpose@
Join@8Array@Identity, 300, 200D<, 8Table@Last@slide100@@-300DDD, 8i, 300<D<DDDD;
êê Makes Ts a constant below 500 nm
s@x_D = 1 ê t@xD + H1 ê t@xD^ 2 - 1L ^ H1 ê 2L;
êê Determines s@lD
Clear@data, data100, intdataD;
data = Import@"êVolumesêMARKASAURUSêOpticsêSpectraêzn037.csv"D;
êê Imports raw data on film + slide
data100 = data.881, 0<, 80, .01<<;
êê Converts data from 0 - 100 % to 0 - 1
intdata = Interpolation@data100, InterpolationOrder Ø 1D;
Clear@max, min, TM, TmD;
max = 881135, .830656<, 8660, .8829886<, 8473, .8679934<<;
min = 88868, .788655<, 8543, .8069522<<;
êê These are example extreme data found using a peak - fitting program
TM = Interpolation@max, InterpolationOrder Ø 1D;
Tm = Interpolation@min, InterpolationOrder Ø 1D;
êê Builds the linear interference envelope
1
Appendix A.nb
2
Clear@n, cauchyD;
i
j1
2 s@xD H-Tm@xD + TM@xDL
j
n@x_D = . j
H1 + s@xD2 L + ÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅ
ÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅ +
j
j ÅÅÅÅ
2
Tm@xD TM@xD
k
z
1 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
2 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
s@xD H-Tm@xD +%%%%%%%%%%%%%%%%
TM@xDL 2% y
z
$%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
-s@xD2 + J ÅÅÅÅ H1
+ s@xD2 L + ÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅ
ÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅ%%%%%%%%%
ÅÅÅ N z
z
z;
2
Tm@xD TM@xD
{
êê Finds n@lD
Plot@8n@xD<, 8x, 200, 2000<, PlotStyle Ø [email protected]<D
êê Plots n@lD to determine limits for Cauchy fit
cauchy@x_D = Fit@Transpose@8Array@Identity, points, startD, Array@n, points, startD<D,
81, x ^ -2, x ^ -4<, xD
êê Finds Cauchy fit
Plot@8n@xD, cauchy@xD<, 8x, 200, 2000<, PlotStyle Ø [email protected], [email protected]<D
êê Plots both to determine fit success
Clear@all, linelist, ell, d, b, hD;
all = Reverse@Sort@Join@max, minDDD
linelist = 8<;
Do@linelist = Append@linelist, Join@8n@all@@iDD@@1DDD ê all@@iDD@@1DD<, 8Hi - 1L ê 2<DD,
8i, Length@allD<D
êê Creates the list of points used for graphical thickness measurement
ell@x_D = ReplaceAll@h * x - b ê 2, FindFit@linelist, h * x - b ê 2, 8b, h<, xDD
êê The parameter b is varied by hand
Show@ListPlot@Drop@linelist, -1D, PlotStyle Ø [email protected]<D,
Plot@ell@xD, 8x, 0, .005<, PlotStyle Ø [email protected]<DD
êê Makes the graph to determine fit accuracy
d = ReplaceAll@h ê 2, FindFit@linelist, h * x - b ê 2, h, xDD
êê Finds thickness from slope of fit line
Appendix A.nb
3
Clear@a, an, acD;
4 n s Hdown+upL
16 n s Hdown+upL
ÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅ
ÅÅÅÅÅÅÅÅ
ÅÅÅÅÅÅÅÅÅÅÅÅÅÅ -$%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
-H-1+n2 L3 Hn2
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
-s4 L+ ÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅ
%%%%%%%%%%%%%%%%
ÅÅÅÅÅÅÅÅ
ÅÅÅÅ
%%%%%%%
ÅÅÅÅ
down
up
2
2%%%%%%%%
LogA ÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅ
ÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅ
ÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅ E
H-1+nL3 Hn-s2 L
a@dee_, n_, s_, up_, down_D = - ÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅ
ÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅ
ÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅ
ÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅ ;
dee
2
4
2
down
2
up
êê Finds the absorption coefficient a@lD
an@x_D = a@d, n@xD, s@xD, TM@xD, Tm@xDD;
ac@x_D = a@d, cauchy@xD, s@xD, TM@xD, Tm@xDD;
Clear@T, Tn, TaD;
T@n_, s_, dee_, a_, x_D = H16 ‰-a *dee n2 sL í
4 * dee * n p
J‰-2 a *dee H-1 + nL3 Hn - s2 L + H1 + nL3 Hn + s2 L + 2 ‰-a* dee H-1 + n2 L Hn2 - s2 L CosA ÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅ EN;
x
Tn@x_D = T@n@xD, s@xD, d, an@xD, xD;
Tc@x_D = T@n@xD, s@xD, d, ac@xD, xD;
êê Builds the reconstructed transmission curves
Plot@8intdata@xD, Tn@xD, Tc@xD<,
8x, 200, 2000<, PlotStyle Ø [email protected], [email protected], [email protected]<D
êê Plots the reconstruction
Appendix A.nb
Clear@ao, cut, atotn, atotc, ad, fit, Ncm, cut, dopeD;
ao@x_D = -H1 ê dL Log@intdata@xD ê Ht@473DLD;
ad@x_D = -H1 ê dL Log@intdata@xD ê Ht@1200DLD;
êê Exponential absorption features near band edge and in infrared
fit@x_D = ReplaceAll@s * Hx - pL ^ 2 + b,
FindFit@Transpose@Join@8Array@Identity, points, startD<, 8Array@ad, points, startD<DD,
s * Hx - pL ^ 2 + b, 8s, b, p<, xDD
êê Fits infrared absorption to l2 dependence to find carrier density
Plot@8fit@xD, ad@xD<, 8x, 500, 2000<,
PlotStyle Ø [email protected], [email protected]<, PlotRange Ø 88500, 2000<, 8-.0001, .0005<<D
êê Plots infrared absorption and fit to assess fit accuracy
cut = 473;
dope = 1200;
atotn@x_D = ao@xD * UnitStep@cut - xD +
an@xD * UnitStep@x - cutD * UnitStep@dope - xD + Hfit@xDL * UnitStep@x - dopeD;
atotc@x_D = ao@xD * UnitStep@cut - xD +
ac@xD * UnitStep@x - cutD * UnitStep@dope - xD + ad@xD * UnitStep@x - dopeD;
êê Builds full absorption coefficient
Clear@Ttotn, TtotcD;
Ttotn@x_D = T@n@xD, s@xD, d, atotn@xD, xD;
Ttotc@x_D = T@n@xD, s@xD, d, atotc@xD, xD;
êê Builds full reconstructions
Plot@8intdata@xD, Ttotn@xD, Ttotc@xD<,
8x, 300, 2000<, PlotStyle Ø [email protected], [email protected], [email protected]<D
êê Plots reconstructions
4
Appendix A.nb
Clear@h, c, sub, e, low, high, asq, afitD
h = 4.14 * 10 ^ -15;
c = 3 * 10 ^ 17;
sub = h * c ê e;
e@x_D = h * c ê x
Plot@8ao@subD^ 2<, 8e, 3, 4.5<,
PlotStyle Ø [email protected], [email protected]<, PlotRange Ø 883, 4.5<, 80, .0004<<D;
asq@x_D = ao@xD ^ 2;
afit@e_D = ReplaceAll@m He - aL, FindFit@Transpose@
Join@8Array@e, points, startD<, 8Array@asq, points, startD<DD, m He - aL, 8m, a<, eDD
Plot@8ao@subD^ 2, afit@eD<, 8e, 3, 4.5<, PlotStyle Ø [email protected], [email protected]<D;
êê This linear fitting routine finds the bandgap energy
Clear@high, low, alog, aurbD
Plot@8Log@ao@subDD<, 8e, 3, 4<, PlotStyle Ø [email protected], [email protected]<D
alog@x_D = Log@ao@xDD;
aurb@e_D = ReplaceAll@e ê u + b, FindFit@Transpose@
Join@8Array@e, points, startD<, 8Array@alog, points, startD<DD, e ê u + b, 8u, b<, eDD
Plot@8Log@ao@subDD, aurb@eD<, 8e, 1, 6.2<,
PlotStyle Ø [email protected], [email protected]<, PlotRange Ø 882.8, 4.7<, 8-8, -4<<D
êê This linear fitting routine finds the Urbach disorder parameter
5