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Analisi delle proprietà ottiche di un materiale
Mediante ellissometria spettroscopica ad angolo variabile:• in riflessione e trasmissione• dall’UV (300nm) al vicino infrarosso (1700nm)• angolo variabile• controllo in temperatura fino a 200°C
Fasi dell’esperienza:1. Studio della letteratura sulla tecnica e sul materiale scelto2. Acquisizione dati3. Modellizzazione ed analisi dei dati
Possibili materiali da analizzare:• Cristalli liquidi antiferroelettrici• Fluoruro di Lantanio• Sistemi ed elevata correlazione elettronica (perovskiti e manganiti)
L’ellissometria
• Polarized light is reflected at an oblique angle to a surface• The change to or from a generally elliptical polarization is measured.• From these measurements, the complex index of refraction and/or the
thickness of the material can be obtained.
Ratio of the complex Fresnel reflection coefficients for the p and s polarizations :
r
rp
s
It is often convenient to write it in the form
tan ei
L’ellissometria
Using Jones Matrix notation:– where and are complex Fresnel reflection coefficients.
ins
inp
s
p
outs
outp
E
E
r
r
E
E~0
0~
ins
inp
s
p
outs
outp
E
E
r
r
E
E~0
0~
spi
s
p
s
pins
outs
inp
outpi e
r
r
r
r
EE
EEe ~
~tan spi
s
p
s
pins
outs
inp
outpi e
r
r
r
r
EE
EEe ~
~tan
pr~
sr~
s
p
r
r)tan(
s
p
r
r)tan(
sp sp
tan ei
L’ellissometria
Ellipsometry measures the change in polarization of light reflected (transmitted) from sample.By determining complex ratio of output/input E-fields
ins
outs
inp
outpi
EE
EEe tan
ins
outs
inp
outpi
EE
EEe tan
Generalized EllipsometryMeasure diagonal and off-diagonal elements of the sample Jones Matrix
Muller Matrix Ellipsometry for depolarising samples
poutsout
rpp rsprps rss
pinsin
INOUTS
S
S
S
MMMM
MMMM
MMMM
MMMM
S
S
S
S
3
2
1
0
44434241
34333231
24232221
14131211
3
2
1
0
S = Stokes vector.
Measured data: Mij
)exp(tan ppppppss
ppiJ
j
j
)exp(tan pspspspp
psiJ
j
j
)exp(tan spspspss
spiJ
j
j
Caratteristiche dell’ellissometria
• Repeatable & accurate: – self-referencing (single-beam experiment) ellipsometry measures
ratio of orthogonal light components Ep/Es Thus, reduced problems with:
• Source Fluctuation• Light Beam Overlapping Small Sample
• Sensitive:– Phase term D is very sensitive to film thickness
Measure only two parameters
L’ellissometria, lo schema dell’apparato
Optical Fiber
Sample
PolarizerPhotoelasticModulator
Analyzer
Detector
Monochromator
Data Acquisitionand Computer
Xe lamp Shutter
Cosa ci può dire l’ellissometria
Geometrical properties:
• Layer thickness • Surface roughness• Interfacial roughness
Material Properties:
• Alloy ratio• Doping concentration• Microstructure• Depth profile
Optical Properties:
• Refractive index• Extinction coefficient• Anisotropy
Modellizzazione
• No direct access to optical and dielectric constants.• Modeling is required to determine sample’s properties from
measured data.• A model is an idealized mathematical representation of the
sample.• To construct a model, one has to assume each layer’s:
a. thickness
b. dielectric functions
c. composition• Remember: if the model is no good, then the interpretation of the
data isn’t good either.
Inversione dei dati
• Load Experimental Data.
• Build Model that represents the
sample.
• Generate data from model.
• Compare calculated and measured
curves.
• “Normal” Fit finds best match (lowest
MSE).
• Is this the correct answer?Check your model for reliability
Crosscheck, if possible, with one of the “direct” measurement technique: TEM, SEM, XAFS,....
Results
Fit
Optical Model
and Measurement
Experimental Data
Multilayers model
Generated Data
Generated Data
Exp. Data
Comparison
ne, no
thicknessroughnessuniformity
Mean Squared Error
We use the Mean Squared Error (MSEMSE) to quantify the difference between experimental and model-generated data.
A smaller MSE implies a better fit.MSE is weighted by the error bars of each measurement, so noisy data are weighted less.
N
i i
ii
i
ii
MNMNMSE
1
2
2
exp,
expmod2
exp,
expmod
2
1
2
1
Esempio: cristalli liquidi
Dispersion curves of 5CB for different temperatures are found to be well approximated by the 3-parameter Cauchy formula
Esempio: cristalli liquidi (VANs)
“Small angle” model for voltage under 6V (corresponds to the theoretical solution)
“Saturated” model for voltage over 6V.
11
11 0);2
( dddd
SinCAd Bottom part:
Central part:
Top part:
21;2/ dddd
1);22
( 22
22 dddd
SinCAd
)( dCSinBdAd
d – cell gap fraction
10 dCondition: 121 ddError bars: ± 1°-2°
Esempio: silicio poroso infiltrato con CL
-4 cauchy 0 nm-3 delta_n 0 nm-2 ema void/20% si/0% (cauchy) 0 nm-1 sum_nk (ema)/100% (delta_n) 0 nm0 si 1 mm1 biaxial 0 nm2 graded (biaxial) 485.21 nm
Effective Medium Approximation (EMA) layer and a Graded anisotropic Layer
Ordinary and extraordinary refractive indices as
a function of and depth can be immediately
calculated from the fitted data resulting from the
described model. Effective no and ne values for
the whole samples, obtained by the simple
following formula,
d
eoeo dzznd
n0
,, )(1
5CB
Esempio: silicio poroso infiltrato con CL
Wavelength (nm)300 600 900 1200 1500 1800
in
deg
rees
0
20
40
60
80
100Model Fit Exp E 60°, PSExp E 60°, PS+ E7 for t=65°C
PS sample infiltrated with 5CB (19%Si, 52% 5CB)
Infiltration of a nematic increases anisotropy of samples in the infrared and decreases in the visible.
For temperature above TC LC escapes from the pores partially!
PS sample infiltrated with E7 (30%Si, 13% E7)
Esempio: Thue-Morse quasi-crystals
Multilayer structures can be organized in a quasi-crystal structure like the Thue-Morse.
A→AB
B→BA
S0= A
S1= AB
S2= ABBA
S3= ABBABAAB
S4= ABBABAABBAABABBASN → 2N layers
Esempio: Thue-Morse quasi-crystals
http://www.cs.uwaterloo.ca/~shallit/
Jeffrey O. ShallitProfessorSchool of Computer Science University of Waterloo Waterloo, Ontario N2L 3G1 Canada
Esempio: Thue-Morse quasi-crystals
“Photonic band gaps analysis of Thue-Morse multilayers made of porous silicon”Optics Express, Vol. 14 , pp. 6264-6272 (2006).
d)
e)
The photonic bandgap properties of the Thue-Morse multilayers have been theoretically investigated by means of the transfer matrix method and the integrated density of states.
Experimental (solid curves) and calculate (dashed curves) reflectivity for(a) S3 T-M structure, (b) S4 T-M structure and (c) S5 T-M structure. (d) S6 T-M structure: 64 PSi layers(e) S7 T-M structure: 128 PSi layers
Esempio: Fluoruro di Lantanio
LAYER 1 = rugosità del materiale ~9nm; LAYER 2 = LaF3 biassiale di 350μm.
Lo strato biassiale è stato modellizzato come composto da due materiali con due diversi indice di rifrazione, che sono l’ordinario e lo straordinario. Ciascuno di questi è stato descritto delle equazioni di dispersione di Cauchy.
2
BAn
Generated and Experimental
Wavelength (nm)800 1000 1200 1400 1600
in
deg
rees
40
50
60
70
80
90
Model Fit Exp AnEt -0°Exp Apst -0°Exp Aspt -0°Exp AnEt -0°Exp Apst -0°Exp Aspt -0°
Generated and Experimental
Wavelength (nm)800 1000 1200 1400 1600
in
deg
rees
-100
0
100
200
300
Model Fit Exp AnEt -0°Exp Apst -0°Exp Aspt -0°Exp AnEt -0°Exp Apst -0°Exp Aspt -0°
25 30 35 40 45 50 55 60 65 70
0.00704
0.00706
0.00708
0.00710
0.00712
0.00714
Temperature (°C)
Bir
efr
ing
en
ce
976 nm 1550 nm
Angular Dependence of and @976nm
Angle of Incidence (°)48 51 54 57 60 63 66
in
deg
rees
in degrees
0
30
60
90
120
150
180
0
2
4
6
8
10
12
Model FitExp Exp
Angular Dependence of and @1550nm
Angle of Incidence (°)48 51 54 57 60 63 66
in
deg
rees
in degrees
0
30
60
90
120
150
180
0
2
4
6
8
10
12
Model FitExp Exp
Esempio: Fluoruro di Lantanio
In futuro, l’ellissometria nel Terahertz
Bal. Ph-diodes
Si
Si
Ti:Salaser
800 mW @ 80 MHz
10 fs100 nm BW
800 mW @ 80 MHz
10 fs100 nm BW