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THORIUM-BASED MIRRORS IN THE EXTREME ULTRAVIOLET by Nicole Farnsworth
THORIUM-BASED MIRRORS IN THE EXTREME ULTRAVIOLET
by
Nicole Farnsworth
Submitted to Brigham Young University in partial fulllment
of graduation requirements for University Honors
Department of Physics and Astronomy
March 2005
R. Steven Turley, Advisor
Eric N. Jellen, Honors Representative Copyright c 2005 Nicole Farnsworth
All Rights Reserved ACKNOWLEDGMENTS
I would like to thank my advisor, Dr. R. Steven Turley for the countless hours he
has spent with me on this project. I would also like to thank Dr. David D. Allred
for his assistance and care. I am grateful to all of the BYU Thin Films Research
Group for their valuable assistance on this project. This project was funded in part
by the BYU Physics Department and by the National Science Foundation. I would
also like to thank my parents and my sisters for their love and support through my
undergraduate experience. Contents
1
Introduction
1
1.1
Interest in the Extreme Ultraviolet and Thorium-based Mirrors
. . .
1
1.2
Project Background . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2
1.3
Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3
1.3.1
Optical Constants . . . . . . . . . . . . . . . . . . . . . . . . .
3
1.3.2
Experimentally determining optical constants
. . . . . . . . .
8
1.3.3
Roughness . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
13
1.3.4
Accepted methods of accounting for roughness . . . . . . . . .
14
1.3.5
Project Focus . . . . . . . . . . . . . . . . . . . . . . . . . . .
16
1.3.6
Film Deposition . . . . . . . . . . . . . . . . . . . . . . . . . .
17
1.3.7
Characterization theory
. . . . . . . . . . . . . . . . . . . . .
17
2
Roughness
20
2.1
Characterization of Roughness . . . . . . . . . . . . . . . . . . . . . .
22
2.2
Problems with Characterization . . . . . . . . . . . . . . . . . . . . .
23
2.3
Accounting for Roughness . . . . . . . . . . . . . . . . . . . . . . . .
29
3
Finding the Optical Constants of Thorium Oxide
34
3.1
Reectance and Transmittance Measurements . . . . . . . . . . . . .
34
iv 3.2
Reectance and Transmittance Data
. . . . . . . . . . . . . . . . . .
42
3.3
Data Fitting . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
48
4
Conclusions
61
A MATFIT Source Code
63
A.1 FitFrame.class . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
63
A.2 ErrorDialog.class . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
80
A.3 LightDialog.class . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
85
A.4 OsetDialog.class . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
89
A.5 nkLookup.class . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
94
A.6 rindex.class . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
99
A.7 EditLayerDialog.class . . . . . . . . . . . . . . . . . . . . . . . . . . .
101
A.8 Mirror.class . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
116
A.9 Film.class . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
118
A.10 index.class . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
119
A.11 FitParameter.class
. . . . . . . . . . . . . . . . . . . . . . . . . . . .
120
A.12 MirrorFunc.class
. . . . . . . . . . . . . . . . . . . . . . . . . . . . .
121
A.13 layer.class . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
128
A.14 re.class . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
129
A.15 ReFit.class . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
140
A.16 Complex.class . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
148
A.17 Matrix.class . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
152
B AFM Tip Source Code
155
B.1 gaussianRandomness.m . . . . . . . . . . . . . . . . . . . . . . . . . .
155
B.2 uniformRandomness.m . . . . . . . . . . . . . . . . . . . . . . . . . .
156 B.3 SeveralRuns2.m . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
157
B.4 SemiCorrelated.m . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
159
B.5 SemiCorrelatedSeveralRuns.m . . . . . . . . . . . . . . . . . . . . . .
160
C Representative Fits
163 List of Tables
3.1
Grating, lter, and order sorters used to acheive spectral purity for the
given wavelength range.
. . . . . . . . . . . . . . . . . . . . . . . . .
39
vii List of Figures
1.1
CXRO computed reectance as a function of wavelength at 15 . Al-
though gold, nickel, and indium are commonly used reector materials
in this region, thorium based mirrors may be more reective at wave-
lengths longer than 160
A. . . . . . . . . . . . . . . . . . . . . . . . .
3
1.2
A thin Gaussian pillbox. . . . . . . . . . . . . . . . . . . . . . . . . .
6
1.3
A thin Amperian loop. . . . . . . . . . . . . . . . . . . . . . . . . . .
6
1.4
The geometry used to calculate theoretical reectance and transmittance.
7
1.5
a) a monochromator b) reection measurement c) transmission mea-
surement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
9
1.6
Large scale roughness results in diuse scattering, decreasing the re-
ectivity of the surface at the specular angle. . . . . . . . . . . . . . .
13
1.7
Interdiusion of layers. In an ideal multilayer, composition changes
innitely fast from one material to another. In real multilayers, inter-
faces have a nite width. left: an innitely sharp boundary right: a
diuse boundary
. . . . . . . . . . . . . . . . . . . . . . . . . . . . .
14
1.8
The gradient in density at an interface brought about by surface rough-
ness can be approximated as a series of steps in density from vacuum
to the material. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
16
viii 1.9
A schematic of the nite dierences approximation. A gradient in the
index of refraction between points a and b is approximated as a nite
number of layers whose indices of refraction vary linearly from n1 to n2. 17
1.10 The photoelectric eect . . . . . . . . . . . . . . . . . . . . . . . . . .
18
1.11 AFM tip . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
18
2.1
Reectance data for a single layer of thorium on a Silicon substrate at
150
A. Along the x-axis is plotted angle, and along the y-axis is plotted
reectance. The t assumes smooth boundaries and abrupt interfaces.
The inset shows the poorness of the t at middle angles (10 30 ). .
21
2.2
Atomic force microscopy proles of thorium at two length scales: 1000x1000
A
and 10,000x10,000
A. At the 1000x1000
A length scale, RMS roughness
equals 36
A. At the 1000x1000
A length scale, RMS roughness equals
43
A. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
22
2.3
Power spectral density plot of sputtered thorium. The highest density
of roughness occurs in the range from 400 to 800
A, small-scale roughness. 23
2.4
X-ray photoelectron spectroscopy depth prole of thorium. The surface
of the sample has oxidized nearly linearly for about 50
A. The thorium-
silicon edge gives as an idea of the resolution of this technique because
we assume that this boundary is sharp. . . . . . . . . . . . . . . . . .
24
2.5
An AFM tip that is on the order of the size of the roughness being
measured cannot characterize roughness accurately. . . . . . . . . . .
24
2.6
A rough surface approximated by Gaussian random numbers (red line).
The blue line is the surface as measured by the AFM tip. The RMS
roughness of the surface is 10.3
A while the RMS roughness detected
by the tip was 2.3
A. . . . . . . . . . . . . . . . . . . . . . . . . . . .
25 2.7
The power spectral density of the surface (red) and the surface detected
by the tip (blue). For this surface, the tip not only fails to detect
high frequency roughness on the surface, it actually detects more low
frequency roughness than is actually present. . . . . . . . . . . . . . .
26
2.8
A rough surface approximated by correlated Gaussian random numbers
(red line). The blue line is the surface as measured by the AFM tip.
The RMS roughness of the surface is 14.9
A while the RMS roughness
detected by the tip was 13.3
A. . . . . . . . . . . . . . . . . . . . . .
27
2.9
The power spectral density of the surface (red) and the surface detected
by the tip (blue). For this surface, the PSD of the real surface and the
surface detected by the tip match almost identically. . . . . . . . . . .
28
2.10 The power spectral density of the surface (red) and the surface detected
by the tip blue. As the length scale of correlation gets longer, the PSD
measured by the tip gets closer to the actual PSD of the surface. . . .
29
2.11 Fit of thorium reection data after correction using the Debye-Waller
factor. The factor improved the t only at low angles. . . . . . . . . .
30
2.12 Fit of thorium reection data after correction using the Nevot-Croce
factor.