Session 3A
Non-idealities
Andrew Martin
UPenn, February 2014
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1
What could go wrong?
Assumptions
Each data point is from:
– ONE discrete
wavelength
– ONE discrete angle
– ONE discrete thickness
“Real” World
– Wavelength Spread
– Angle Spread
– Non-uniform film
thickness
Substrate is:
– Backside Reflections
– Infinitely thick
Film is:
– Smooth, Homogeneous
– Isotropic
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– Rough, Index Gradient
– Anisotropic Optical
Constants
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2
Wavelength Spread
A grating or prism is used to spread wavelengths out
and collect a small region on the detector.
– Discrete wavelength is actually a “spread” of
wavelengths.
Slit or
Detector
element
Grating
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3
Angle Spread
Focused-beam measurements will actually collect a
“range” of angles.
– Discrete angle actually a “spread” of angles
f
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Non-uniform Thickness
 Measurement spot can be large enough to see
variation in film thickness within the measured area
– Discrete thickness actually a “spread” of
thicknesses.
t
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Case: Thickness Nonuniformity
Experimental Data
80
 How are data affected
by measurements
over ’spread’ of
thicknesses.
Ideal Film
Same Film, 3% Thickness Non-Uniformity
 in degrees
60
Data from thicker
and thinner film go in
opposite directions –
34
average together.
32
40
20
0
0
300
Experimental
Data
600
900
1200
1500
1800
Wavelength (nm)
Ideal Data
With 3% non-uniformity
Varied Thickness Data
t
 in degrees
30
28
26
24
22
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Case: Thickness Nonuniformity
Experimental Data
80
Near a sharp feature,
the data are affected
by the thickness
variation.
Ideal Film
Same Film, 3% Thickness Non-Uniformity
 in degrees
60
40
20
Experimental Data
0
Ideal Data
0
With 3% non-uniformity
Varied Thickness Data
300
600
900
1200
Wavelength (nm)
1500
1800
Data from thicker and
thinner film go in same
directions – pulls
average down.
581
613
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644
675
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7
Affect of non-idealities on data
 Non-idealities can cause:
– Rounding/lowering of peaks and valleys in ψ and Δ
– Depolarization (A portion of the POLARIZED light
becomes randomly polarized upon reflection).
95%
5%
+
Non-uniform film
Unpolarized
Light
Substrate
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Thickness Uniformity
 Depolarization- hints towards non-ideality.
 The uniformity of a film is more important as
the film thickness increases.
Experimental Data
%Depolarization
80
200nm thick, 3% non-uniform
3 microns thick, 3% non-uniform
60
40
20
0
0
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300
600
900
1200
Wavelength (nm)
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1500
1800
9
Measuring Depolarization
 Possible only on Ellipsometers with COMPENSATOR
– VASE with AutoRetarder, M-2000, IR-VASE (rotating
compensator)
– All modern Woollam ellipsometers
 Acquiring data type “Standard” or “Isotropic”
measures: ψ, Δ
 Acquiring data type “Depolarization” measures:
ψ, Δ and % Depolarization.
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Measure Depolarization!
 Non-idealities can cause depolarization:
– Thickness non-uniformity
– Wavelength Spread
– Angle spread
Also:
– Backside substrate reflections
– Patterned films
Measure depolarization to help quantify
the non-ideality.
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How to deal with “non-ideal” world
1. Eliminate (or reduce) non-ideality.
 Spatially reduce collected “spread” of
wavelengths.
 Use collimated measurements
 Use smaller measurement spot or make
more uniform films.
2. Account for non-ideality in the
model.
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Model Options Dialog Box
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1
Example_1_Organic_on_Si_nonuniform.dat
– Determine Index, Thickness and Nonuniformity, based on SE + Depol data.
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2,3
Example_2_Dielectric_on_Si.dat
Example_3_SiO2_on_Si.dat
 Fit data and identify the likely
cause of depolarization for each
case.
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Coherent / Incoherent
 Incoherent beam summation can produce depolarization.
 Always must consider when working with transparent substrate
Reflected Beams
Rf
Input Light Beam
Tf • Rb • Tfr
(Tf • Rb • Tfr) • (Rfr • Rb)
(Tf • Rb • Tfr) • (Rfr • Rb)2
Multilayer Film Stack
(coherent propagation)
front side
Transparent Substrate
(incoherent propagation)
back side
(Tf • Tb) • (Rfr • Rb)3
(Tf • Tb) • (Rfr • Rb)2
(Tf • Tb) • (Rfr • Rb)
(Tf • Tb)
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Transmitted Beams
16
Incoherent Backside Reflections
 How to deal with backside reflections
– Roughen backside with sandblaster or sanding tool
– Index matching tape for glass substrate (translucent Scotch
tape)
– Focusing optics
– Wedge/thick substrate
– Account for it in software
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Summary
Non-idealities
Bandwidth
Nonuniform thickness
Angular spread
NEXT: Absorbing Films
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18
Absorbing Films
(l), D(l)
n(l), k(l)
AOI

n
k
?
D
d
d
Known Substrate
Challenge:
More unknown
sample properties
than measured
values.
Issues: no unique answer
-Correlation between thickness and optical properties©2014 J.A. Woollam Co., Inc.
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19
Can Multiple Angle SE break the correlation?
40
Not necessarily!!
30
25
5.0
Exp E 45°
Exp E 55°
Exp E 65°
Exp E 75°
20
15
200
400
600
800
1000
30
25
2.5
20
1200
Wavelength (nm)
180
160
0.0
15
Exp <  1>-E 65°
Exp <  1>-E 75°
Exp <  2>-E 65°
Exp <  2>-E 75°
-2.5
140
120
-5.0
200
100
Exp E 45°
Exp E 55°
Exp E 65°
Exp E 75°
80
60
40
200
400
600
800
Wavelength (nm)
1000
400
600
800
1000
<  2>
<  1>
 in degrees
35
D in degrees
TiN film
Silicon
10
5
0
1200
Wavelength (nm)
1200
“Pseudo” optical constants remove the angle
dependence to compare information content
 Change in path length from multiple angles is minimal for
thin film - does not provide new information.
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Can Multiple Angle SE break the correlation?
40
35
30
MSE
25
20
15
10
5
0
0
10
20
30
40
50
60
TiN Thickness - nm
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21
Methods on Absorbing Films
 Reducing unknowns
– Opaque Layer
– Optical Constant
Parameterization
– Extrapolation from
Transparent Region


n
k
d
n
n
k
k
d
d
 Adding information
–
–
–
–
SE + Transmission
Interference Enhancement
Multi-Sample Analysis
In-Situ
DD
 D D
Above methods are often combined!
J. Hilfiker, “Survey of methods to characterize thin absorbing
films with Spectroscopic Ellipsometry” Thin Solid Films 516 (2008) 7979-7989.
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Opaque Layers
 Sample absorbs light in the
entire measured range.
n

k
D
 Typical for metal substrates,
thick metal films (>50-100nm)
 Model assumptions affect
accuracies of final optical
properties.
Method
Opaque
layers
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Direct fit
Advantages
Simplified data analysis –
ignore the film thickness.
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Disadvantages
Ignores surface films (oxide
and/or roughness).
Opaque layer may have different
“n,k” from thin layers.
23
Parameterization
(l) D(l)
p1 pp11pp3 p2 d
1
 Use dispersion
equations to describe
sample optical
properties.
 Often combined with
other techniques
Cauchy
GenOsc
Method
Advantages
Disadvantages
Optical
Constant
Parameterization
Reduce # of Fit Parameters
Smooth, continuous curves for n,k that are
often KK consistent
Work better for well-known and welldefined optical properties, such as a-Si or
DLC
By itself, results can
often be correlated
Need to choose best
of many different
options.
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Extrapolation from Transparent region
Transparent region
nt d

K=0
Absorption region
D
na

k
D
Pt by pt
Genosc
Cauchy
 Sample is transparent in some measured region
– Get thickness and n in the transparent region (k=0)
– Extend to absorbing region to get n, k and thickness
 Correct model should fit both transparent and absorbing
regions
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Extrapolation from Transparent region Cont.
Method
Advantages
Extrapolation High sensitivity to thickness
in transparent region.
from
Transparent
Pt by pt fit quickly and easily
Region
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Disadvantages
Requires film to be
transparent over some
measured wavelengths.
extends into absorbing
region.
Possible for NK to get lost in
absorbing region
Genosc models enhance KK
consistency.
Complicated
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SE + Intensity
n
k
d

D
(l), D(l)
%R(l)
I
n(l), k(l)
Known Substrate
Parameterization
Including intensity
Uniqueness fit
Method
d
%T(l)
Advantages
SE +
Extra information from
Transmission Intensity breaks correlation.
Disadvantages
 Not for absorbing substrates.
 Need accurate Intensity
measurements.
B. Johs et al. ,Opt. Int. Coat. Tech. Digest. 15 (1992) 433.
B. Johs et al. Thin Solid Films. 253 (1994) 25.
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Interference Enhancement
(l), D(l) X 2 Angles
n k d1 d2
1D12D2
n(l), k(l)
Thick Transparent Film
d2
d1
Known Substrate
Method
Advantages
Interference
Enhancement
 Extra info. from multi-angle data.
Effective “substrate” features enhance
sensitivity to correct film thickness/n,k.
Great method for absorbing substrates.
Disadvantages
Requires extra, thick
dielectric layer.
Increases complexity
Demonstrated on a-C:H films
W.A. McGahan, B. Johs, J.A. Woollam, Thin Solid Films 234 (1993) 443-446.
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Multi-Sample and in-situ Analysis
n k d1 d2
1 D1 2 D2
(l), D(l)
(l), D(l)
n(l), k(l)
(l), D(l)
n(l), k(l)
(l), D(l)
d
n(l), k(l)
Parameterization
Multi-sample analysis
Method
Advantages
Multi-Sample More information about same
material.
and in-situ
Easy to achieve from map of
Analysis
non-uniform sample.
Known Substrate
Disadvantages
Requires consistent optical
constants
In-situ requires ellipsometer
integration into system.
C.M. Herzinger et al. “Determination of AlAs optical constants by variable angle spectroscopic
ellipsometry and a multisample analysis” J. Appl. Phys. 77 (9) 1995, p. 4677-4687.
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SE + Intensity
 Increase measured
information.
 Transmission requires
transparent
substrate
 Must accurately
measure intensity
(l), D(l)
%R(l)
d
n(l), k(l)
Known Transparent
Substrate
%T(l)
B. Johs et al. ,Opt. Int. Coat.
Tech. Digest. 15 (1992) 433.
B. Johs et al. Thin Solid Films.
253 (1994) 25.
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4
Example_4_Cr_on_glass_SE.dat
Example_4_Cr_on_glass_T.dat
 Use Glass_genosc.mat for substrate
 Append Transmission data to Fit
Thickness, n and k simultaneously.
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Cr on Glass – SE Only
 Thickness fixed at 12, 18, and 24nm.
12
200
Model Fit
Exp E 45°
Exp E 60°
Exp E 75°
100
50
9
1
D in degrees
150
6
Fixed at 12 nm
Fixed at 18 nm
Fixed at 24 nm
3
0
30
21
20
18
15
2
 in degrees
0
10
12
0
300
600
900
1200
Wavelength (nm)
MSE = 0.8, 0.5, 0.9
for 12, 18 and 24nm
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1500
1800
9
6
300
600
900
1200
1500
1800
Wavelength (nm)
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32
 Use “fit” n,k to
simulate
transmission and
reflection
intensity.
Transmission, 0°
 Compare to
measured values,
only match the
18nm data.
P-Reflection, 45°
SE + Intensity – Cr on Glass
0.50
0.40
0.30
Experimental
Model-12nm
Model-18nm
Model-24nm
0.20
0.10
0.24
0.21
0.18
0.15
0.12
0.09
300
600
900
1200
1500
1800
Wavelength (nm)
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SE + Intensity
10
Normalized Relative MSE
9
8
SE Only
SE + T
SE + R
7
6
5
4
3
2
1
0
0
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5
10
15
20
Cr Thickness - nm
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25
30
35
34
Interference Enhancement
 Thick dielectric
below absorbing
layer enhances
change in path
length to provide
new information
from multiple
angles.
 Increases sample
complexity.
(l), D(l) X 2 Angles
n(l), k(l)
Thick Transparent Film
d
d
Known Transparent
Substrate
Demonstrated on a-C:H films
Applied to Data Storage Industry for thin magnetic layers
W.A. McGahan, B. Johs, J.A. Woollam,
Thin Solid Films 234 (1993) 443-446.
M.T. Kief, G. Al-Jumaily, and G.S. Mowry, IEEE Transactions on
Magnetics, 33 (1997) 2926.
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Multiple Angles (new info?)
 Check <n,k> to see if new angles help.
Experimental Data
Experimental Data
50
8
Exp E 60°
Exp E 70°
Exp E 80°
30
20
0
200
4
2
10
400
600
800
Wavelength (nm)
Experimental
Data
1000
0
200
1200
400
600
800
1000
1200
Wavelength (nm)Data
Experimental
4.0
300
Exp E 60°
Exp E 70°
Exp E 80°
3.0
2.0
<k>
200
D in degrees
Exp E 60°
Exp E 70°
Exp E 80°
6
<n>
 in degrees
40
100
1.0
0.0
Exp E 60°
Exp E 70°
Exp E 80°
-1.0
0
-2.0
-100
200
400
600
800
Wavelength (nm)
1000
-3.0
200
1200
400
600
800
Wavelength (nm)
1000
1200
Thin TiN / SiO2 / Si substrate
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Thin TiN on SiO2 / Silicon
 Fit n, k, t1, and t2 unique result!
25
20
MSE
15
10
Uniqueness Simulation shows a minimum
in the MSE for a “unique” thickness
5
0
0
5
10
15
20
25
30
35
TiN thickness in nm
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37
5
Example_5_Pt_on_SiO2_on_Si.dat






Thin Pt film on thick thermal oxide.
Assume fixed nk for Pt and SiO2.
Fit thickness only.
Allow Pt nk to fit directly.
Check uniqueness.
Fit to GenOsc (Extra Credit).
Interference Enhancement
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38
Example 5 Result

Simultaneously fit Pt and SiO2
thickness, and Pt nk
Generated and Experimental
45
3
2
1
0
 in degrees
40
35
Model Fit
Exp E 45.2°
Exp E 60.3°
Exp E 75.5°
30
25
pt
sio2_jaw
intr_jaw
si_jaw
15.374 nm
1709.768 nm
1.000 nm
1 mm
20
15
400
600
800
Wavelength (nm)
1000
1200
Optical Constants
4.5
Generated and Experimental
D in degrees
160
140
120
Model Fit
Exp E 45.2°
Exp E 60.3°
Exp E 75.5°
100
80
60
400
600
800
Wavelength (nm)
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1000
1200
4.0
3.5
Pt - Direct nk fit, n
Pt - GenOsc nk fit,
n
Pt - Direct nk fit, k
Pt - GenOsc nk fit,
k
7.0
6.0
3.0
5.0
2.5
4.0
2.0
1.5
400
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600
800
Wavelength (nm)
1000
Extinction Coefficient 'k'
Index of Refraction 'n'
180
8.0
3.0
1200
39
Multilayer films
 Need to determine many unknowns!
– Includes thickness, n and k for each layer.
 SE data feature: Oscillation patterns will be
superimposed!
80
200 nm
500 nm
1 mm
Multilayer
Oxide Only
Nitride Only
Substrate
60
 in degrees
2 nitride
1 oxide
0 substrate
40
20
0
0
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300
600
900
1200
Wavelength (nm)
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1500
1800
40
Multi-Layer Strategies
 Preferred Method #1
– Measure n,k from single-layers: use dispersion models.
– Fit thickness only.
– If poor fit, add dispersion parms. for least stable film.
Film A
Film B
Film C
Substrate
Substrate
Film B
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Film C
Film A
Substrate
Substrate
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41
Multi-Layer Strategies
 Preferred Method #2
– Measure n,k for each “new” layer with previous layers
fixed: use dispersion models.
– If poor fit, add thickness and then dispersion
parameters for previous layers.
Film C
Film B
Film B
Film A
Film A
Film A
Substrate
Substrate
Substrate
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42
Multilayer Fit Strategy
Strategy #1:
– Measure “n, k” from single layer samples.
– Then, fit thicknesses only for multi-layers.
Strategy #2:
– Fix “n, k” for known layers.
– Use dispersion or alloy models for unknown layers.
Additional Strategy:
– If same coating is repeated in stack, use same
(coupled) optical constants for each.
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43
Repeated Layer Structures in WVASE
 WVASE32 allows user to very simply model
repeated structures…“Superlattices”.
– Thickness and optical constants
automatically coupled in repeated layers.
4
3
2
1
0
gaas-ox
(gaas) Coupled to #0
algaas x=0.350
algaas x=0.200
gaas
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15 Å
200 Å
3000 Å
2000 Å
1 mm
5
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44
6
Example_6_a-Si_on_SiO2_on_Si.dat
 Use tabulated n and k for Si and SiO2
 Use Gen-Osc for the a-Si layer.
– (a-Si.mat as reference)
 Fit thickness of layers
 Fit thickness and Gen-Osc Parameter of
layers.
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45
Example 6: a-Si / SiO2 / Silicon
 Fit Thickness and All Oscillator
Parameters
3
2
1
0
(sio2) Coupled to #1
genosc
sio2
si_jaw
300
40
Model Fit
Exp E 65°
Exp E 70°
Exp E 75°
200
D in degrees
30
 in degrees
2.301 nm
374.234 nm
320.232 nm
1 mm
Generated and Experimental
Generated and Experimental
20
Model Fit
Exp E 65°
Exp E 70°
Exp E 75°
100
0
10
0
300
MSE = 2.17
400
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500
600
Wavelength (nm)
700
800
900
-100
300
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400
500
600
Wavelength (nm)
700
800
900
46
7
Example_7_TiO2 on SiO2 on Si.dat
 Nominal structure 30 nm TiO2 over
240 nm SiO2 on Silicon.
 Use reference n and k for Si
substrate and SiO2.
 Use Cauchy equation for index of
TiO2
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47
Example 7: TiO2/SiO2 on Si
2 cauchy
1 sio2_jaw
0 si_jaw
MSE=2.776
Thick.1
Thick.2
An.2
Bn.2
Cn.2
29.991 nm
237.105 nm
1 mm
Generated and Experimental
100
180
80
150
D in degrees
 in degrees
Generated and Experimental
60
40
Model Fit
Exp E 60°
Exp E 65°
Exp E 70°
20
0
300
600
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900
1200
Wavelength (nm)
237.105±0.101
29.991±0.114
2.0502±0.00239
0.028775±0.00131
0.0036188±0.000253
1500
120
90
Model Fit
Exp E 60°
Exp E 65°
Exp E 70°
60
30
1800
0
300
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600
900
1200
Wavelength (nm)
1500
1800
48
8
Example_8_SOI.dat
 Crystalline Si over SiO2 on Si
(thickness unknown).
 Start by using reference n and k for
all layers.
 Don’t forget surface oxide!!
 Bonus: Try Global fitting thickness.
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49
Example 8: SOI
3
2
1
0
(sio2) Coupled to #1
(si_jaw) Coupled to #0
sio2
si_jaw
2.682 nm
131.568 nm
393.074 nm
1 mm
Generated and Experimental
Generated and Experimental
60
300
Model Fit
Exp E 45°
Exp E 65°
Model Fit
Exp E 45°
Exp E 65°
200
40
D in degrees
 in degrees
50
30
20
100
0
10
0
0
300
©2014 J.A. Woollam Co., Inc.
600
900
Wavelength (nm)
1200
1500
1800
-100
0
www.jawoollam.com
300
600
900
Wavelength (nm)
1200
1500
1800
50
Summary
Non-idealities
Bandwidth
Nonuniform thickness
Angular spread
Absorbing Films
Opaque Films
SE+Transmission
Interference Enhancement
Multilayer Films
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51
Extra Slides
 The following slides provide additional
details pertaining Generalized
Ellipsometry and Mueller Matrix data.
©2014 J.A. Woollam Co., Inc.
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52
More Multilayer Examples
 Example_9A_ARC on Si.dat
 Example_9B_Photoresist on Si.dat
 Example_9C_Photoresist on ARC on
Si.dat
– Model single layers first, ARC and
Photoresist.
– Then fit thickness only in multilayer
sample.
– Try fitting thickness and oscillator
parameters (Extra Credit).
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53
Example 9A: ARC on Silicon
AR Coating
Generated and Experimental
100
Model Fit
Exp E 60°
Exp E 75°
1 ar coating-cauchy
0 si_jaw
1601.411 Å
1 mm
 in degrees
80
60
40
20
0
0
300
600
ARC Optical Constants
2.00
900
Wavelength (nm)
1200
1500
1800
Generated and Experimental
0.40
0.30
1.80
0.20
1.70
0.10
1.60
1.50
0
300
©2014 J.A. Woollam Co., Inc.
600
900
Wavelength (nm)
1200
1500
Model Fit
Exp E 60°
Exp E 75°
150
D in degrees
n
k
1.90
Extinction Coefficient ' k'
Index of Refraction ' n'
180
0.00
1800
www.jawoollam.com
120
90
60
30
0
0
300
600
900
Wavelength (nm)
1200
1500
1800
54
Example 9B: Photoresist on Si
PhotoResist
Generated and Experimental
100
Model Fit
Exp E 60°
Exp E 75°
1 photoresist-cauchy
0 si_jaw
6062.039 Å
1 mm
 in degrees
80
60
40
20
0
0
300
600
photoresist-cauchy Optical Constants
0.15
0.12
0.09
1.80
0.06
1.70
1.60
0
0.03
300
©2014 J.A. Woollam Co., Inc.
600
900
Wavelength (nm)
1200
1500
1800
Generated and Experimental
1500
150
D in degrees
n
k
1.90
1200
180
Extinction Coefficient ' k'
Index of Refraction ' n'
2.00
900
Wavelength (nm)
0.00
1800
www.jawoollam.com
120
Model Fit
Exp E 60°
Exp E 75°
90
60
30
0
0
300
600
900
Wavelength (nm)
1200
1500
1800
55
Example 9C : Photoresist on ARC
 Use Results from single layers.
 Build and fit 2 layer model. Fit both layer
thicknesses.
2 photoresist
6129.155 Å
3110.700 Å
1 mm
1 ar coating
0 si_jaw
Generated and Experimental
Generated and Experimental
100
Model Fit
Exp E 60°
Exp E 75°
60
40
20
0
0
Model Fit
Exp E 60°
Exp E 75°
150
D in degrees
 in degrees
80
180
120
90
60
30
300
600
900
Wavelength (nm)
1200
1500
1800
0
0
300
600
900
Wavelength (nm)
1200
1500
1800
 ARC film was twice the nominal value of 1600 Å!
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56
Example 4C : Photoresist on ARC
 Resist and AR indices:
– Very similar in visible and infrared.
– VERY different in UV.
 UV data makes analysis possible!
Optical Constants
Optical Constants
0.40
ar coating
photoresist
1.90
Extinction Coefficient ' k'
Index of Refraction ' n'
2.00
1.80
1.70
1.60
1.50
0
300
©2014 J.A. Woollam Co., Inc.
600
900
Wavelength (nm)
1200
1500
1800
ar coating
photoresist
0.30
0.20
0.10
0.00
0
www.jawoollam.com
300
600
900
Wavelength (nm)
1200
1500
1800
57
Traditional Ellipsometry
 Measures change in polarization from complex
ratio of output/input Electric Fields
  tan eiD 
in
E out
E
p
p
Esout Esin
1. Known input polarization
E
p-plane
s-plane
3. Measure output polarization
p-plane
E
s-plane
plane of incidence
2. reflect off sample ...
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58
Based on Jones Matrix Notation…
For Isotropic
sample:
~
rp
– where
~
rp
 E out


p
 out   
E 
 s  0
0  E inp 
 in 
~
rs  Es 
~
rs
and
are complex Fresnel
reflection coefficients.
  tan eiD 
tan() 
©2014 J.A. Woollam Co., Inc.
~
rp rp i  p  s 
~  e
in
Es
rs rs
in
E out
E
p
p
Esout
rp
rs
D   p  s
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59
Stokes vector
 Describe any light beam as a vector
of different polarized intensities
2
2

Eox  Eoy 
S0 
  S1   Eox2  Eoy2 
S  
 S 2  2 Eox Eoy cos  

  
 S3   2 Eox Eoy sin  
Total Intensity
Horizontal – Vertical
+45° – (-45°)
Right Hand – Left Hand
–(  is the phase difference between Eox and Eoy):
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60
Mueller Matrix
 Describes polarization change by mapping
Stokes Vectors
– Can describe Depolarization and Anisotropy
Light in
Light out
 S o _ out   m11

 
 S1_ out  m21
S
  m
 2 _ out   31
S
 m
 3 _ out   41
©2014 J.A. Woollam Co., Inc.
m12
m13
m22
m23
m32
m33
m42
m43
www.jawoollam.com
m14   S o _ in 



m24   S1_ in 

m34  S 2 _ in 

 
m44   S3 _ in 
61
Mueller Matrix of sample
 For isotropic sample:
 1
 N
M sample  
 0

 0
N
0
1
0
0
C
0
S
0
0 
S

C
N  cos 2 
C  sin 2 sin D 
S  sin 2  cos D 
P  N 2  C2  S 2
%depolarization  1  P 100%
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62
Generalized Ellipsometry...
 For Anisotropic Samples…
 pout  rpp rsp   pin 
 
 s   r
r
 out   ps ss   sin 
Generalized
Ellipsometry allows
cross-terms  0
 3 Ellipsometry Ratios instead of 1…
–
r pp
i
D
tan()  e 
AnE:
rss
tan( ps )  e
– Aps:
iD ps
– Asp:tan(sp )  e
iD sp
©2014 J.A. Woollam Co., Inc.

r ps
r pp
rsp

rss
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63
Anisotropic Mueller Matrix
 When sample is anisotropic and depolarizing.
 mm11 mm12
 s1 
mm
s 
mm
21
22
2
  
mm31 mm32
 s3 

 
 s 4  in mm41 mm42
mm13
mm23
mm33
mm43
mm14   s1 
mm24   s 2 

mm34   s3 
  
mm44   s 4  out
Anisotropy in off-diagonal blocks
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64
MM elements measured?
Figure from:
Spectroscopic Ellipsometry: Principles and
Applications by Hiroyuki Fujiwara (2007)
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65
When is g-SE needed?
 Anisotropic materials
“may” cause crosspolarization between pand s-.
 Off-diagonal elements only
non-zero when optical axis
misaligned with
ellipsometer.
Figure from:
Spectroscopic Ellipsometry:
Principles and Applications
by Hiroyuki Fujiwara (2003)
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66
Euler Angles
 The Euler Angles (f, ,  ) define the orientation
of SAMPLE DIELECTRIC TENSOR relative to the
LABORATORY Frame of reference.
Laboratory frame of reference
is defined by:
sample surface (x-y plane)
plane of incidence (x-z plane)
Plane of
incidence
x
Shown:
(f, ,  ) = (0°, 0°, 0°)
©2014 J.A. Woollam Co., Inc.
y
z
sample
surface
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22 direction
11 direction
33 direction
67
Euler Angles (f, ,  )
 3 Euler Angle
steps:
2)  : rotate
y’,z’ about
sample
x’
surface
x
y
1) f : rotate x,y
about z
x
y
sample
surface
22 direction
33 direction
11 direction
z

z
”
y
’
22
y
”
direction
z
f
11 direction
33
direction
z
z
’
3)  : rotate x”,y”
about z”
sample
surface
x
’
y
’
x
’
x
y
22 direction
33
direction
11 direction
z

y
’
’
’
©2014 J.A. Woollam Co., Inc.
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y
”
z
’
’
’
x
”
x’
’
’
68
Anisotropy - challenge
 MANY UNKNOWNS
nx(l), ny(l), nz(l)
Euler Angles: f, , 
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69
Anisotropic Substrates
Anisotropic effects from substrate can be minimized by removing
backside reflections.
Hard coating
Hard coating
polycarbonate
polycarbonate
20
20
 in degrees
 in degrees
Exp Data, 65°
16
12
16
12
8
8
Model Fit
Exp Data, 65°
4
300
70
500
©2014 J.A. Woollam Co., Inc.
700
Wavelength (nm)
900
1100
4
300
www.jawoollam.com
500
700
Wavelength (nm)
900
1100
70
Handling Substrates
1.
2.
3.
4.
Roughen backside (lightly with sandpaper)
Align with “specific” direction.
Measure Pseudo-N
Save Optical Constants as reference for
future measurements with same alignment.
OR

Complete Characterization of substrate
analyzing g-SE or MM data.
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71
G-VASE data acquisition strategies



electric fields must probe both e & o
Simultaneous analysis of multiple sample
rotational orientations and incident angles
For reflection-VASE, best if sample is “optically
thick” (eliminates backside effects)
In - p la n e s a m p le
r o ta tio n
T r a n s m is s io n
E llip s o m e t r y
In p u t L ig h t
M u ltip le a n g le s o f
in c id e n c e : n e g a tiv e & p o s it iv e
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R e f le c t io n
E llip s o m e t r y
72
ANISOTROPY
 What is Anisotropy?
 Building Anisotropic Models
 Uniaxial Thin Films?
 Extra slides:
– Generalized SE and MM
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73
Anisotropy
 Optically anisotropic material
– A material that exhibits different optical properties
depending on the polarization direction of light
beam propagating through the material
Beam C
Beam B
z
Beam A
y
x
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Anisotropic Materials
 Anisotropic crystalline materials
–
–
–
–
Tetragonal, Hexagonal, Rhombohedral, …
Crystalline organic chains
Liquid crystals
Sugars (chiral)
 Materials strained during processing
– PET sheets, spin-on films
 Materials with preferred orientation growth
– Certain columnar growth films
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75
Origins of Anisotropy
 Mechanical depiction of Anisotropy, where the springs have
different stiffness depending on direction
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76
Electron Cloud Model
 Spherical -- light encounters
same electron density
regardless of direction
Electron
cloud
 Oxygen Molecule has
different electron density
with direction BUT: rotates
at high speed and random
ordering of large number of
molecules
Incident light
Incident light
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77
Anisotropic Material Requires
 Direction-dependent distribution of
atomic or molecular properties
 Long-range alignment and
confinement.
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Anisotropic Optical Phenomena
 Linear Birefringence
– no  ne
 Linear Dicroism
– ko  ke
 Circular Birefringencen k
L, L
– nL  nR
no, ko
ne, ke
nR, kR
 Circular Dicroism
– kL  kR
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Angle of Refraction
 Birefringence  2 refraction angles
– Probe beam splits upon entering sample
p- & s- E-field
components
Refracted beam splits into
two beams
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80
Types of Anisotropy
Figure from:
Spectroscopic Ellipsometry:
Principles and Applications
by Hiroyuki Fujiwara (2003)
©2014 J.A. Woollam Co., Inc.
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81
Modeling Anisotropy
 Right click on layer and choose
‘Add Uniaxial Anisotropy’
0 Uniaxial
-1 Nz (cauchy)/100% (dNz)
-2 dNz
-3 cauchy
1 mm
0 nm
0 nm
0 nm
or
‘Add Biaxial Anisotropy’
0 Biaxial
-1 Ny (cauchy)/50% (dNxy)
-2 Nx (cauchy)/-50% (dNxy)
-3 dNxy
-4 Nz (cauchy)/100% (dNz)
-5 dNz
-6 cauchy
©2014 J.A. Woollam Co., Inc.
1 mm
0 nm
0 nm
0 nm
0 nm
0 nm
0 nm
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82
Biaxial.mat layer
Describe optical
constants in 3
orthogonal directions
To adjust optical
constants, ‘couple’ to
dummy layers below.
©2014 J.A. Woollam Co., Inc.
3
2
1
0
www.jawoollam.com
biaxial
cauchy2
cauchy
si_jaw
637.22 nm
0 nm
0 nm
1 mm
83
Polymer on Si
 Isotropic Cauchy Fit
MSE = 173.5
1 cauchy
0 si_jaw
Generated and Experimental
Generated and Experimental
100
300
60
40
200
D in degrees
Model Fit
Exp E 55°
Exp E 65°
Exp E 75°
80
 in degrees
602.4 nm
1 mm
100
0
Model Fit
Exp E 55°
Exp E 65°
Exp E 75°
-100
20
-200
0
400
600
800
©2014 J.A. Woollam Co., Inc.
1000
1200
Wavelength (nm)
1400
1600
1800
-300
400
www.jawoollam.com
600
800
1000
1200
Wavelength (nm)
1400
1600
1800
84
Polymer on Si
 Uniaxial Anisotropy Fit.
4
3
2
1
0
MSE < 21
Uniaxial
Nz (cauchy)/100% (dNz)
dNz
cauchy
si_jaw
Generated and Experimental
Generated and Experimental
300
100
200
Model Fit
Exp E 55°
Exp E 65°
Exp E 75°
60
D in degrees
 in degrees
80
40
100
Model Fit
Exp E 55°
Exp E 65°
Exp E 75°
0
-100
20
0
400
637.65 nm
0 nm
0 nm
0 nm
1 mm
600
800
©2014 J.A. Woollam Co., Inc.
1000
1200
Wavelength (nm)
1400
1600
1800
-200
400
www.jawoollam.com
600
800
1000
1200
Wavelength (nm)
1400
1600
1800
85
Polymer Result
biaxial Optical Constants
Index of refraction ' n'
1.80
nx
nz
1.75
1.70
1.65
1.60
1.55
400
©2014 J.A. Woollam Co., Inc.
600
800
1000
1200
Wavelength (nm)
www.jawoollam.com
1400
1600
1800
86
DEMONSTRATION
 Example 4-Polymer on Si
Fit with Anisotropic Model.
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87
Uniqueness of Anisotropy
Test for “uniqueness” of model from Fit
Window.
70
60
50
MSE
40
30
20
10
0
-0.2
-0.15
-0.1
-0.05
0
Index Difference (A-parm)
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88
How thin can we measure?
 Uniqueness Fit for 22nm Thin Film
25
20
MSE
15
10
5
0
-0.04
-0.02
0
0.02
0.04
Index Difference (Nz-Nx)
©2014 J.A. Woollam Co., Inc.
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89
Example
 Example 5- low k on Si.dat”
Fit data with “uniaxial” anisotropy.
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90
Low-K, Result
Uniaxial
Nz (cauchy)/100% (dNz)
dNz
cauchy
si_jaw
1179.971 nm
0.000 nm
0.000 nm
0.000 nm
1 mm
MSE=15
1.80
Generated and Experimental
100
Model Fit
Exp E 60°
Exp E 75°
 in degrees
80
60
40
20
0
300
nx
nz
1.70
1.65
1.60
1.55
1.50
300
600
900
1200
Wavelength (nm)
1500
600
1800
Generated and Experimental
900
1200
Wavelength (nm)
1500
1800
Generated and Experimental
300
80
Model Fit
Exp E 60°
Exp E 75°
60
%Depolarization
200
D in degrees
Uniaxial Optical Constants
1.75
Index of refraction ' n'
4
3
2
1
0
100
0
Model Fit
Exp dpolE 60°
Exp dpolE 75°
40
20
0
-100
300
91
600
©2014 J.A. Woollam Co., Inc.
900
1200
Wavelength (nm)
1500
1800
-20
300
600
900
1200
Wavelength (nm)
www.jawoollam.com
1500
1800
91