Model-free extraction of refractive index from measured optical data
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Transcript of Model-free extraction of refractive index from measured optical data
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Model-free extraction of refractive index from measured optical data
A Tool for Refractive InDex Simulation
Martina Schmid, Guanchao Yin, Phillip Manley
Helmholtz-Zentrum Berlin, Nanooptical concepts for
photovoltaics
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Motivation
Thin film optics
having to deal with multiple reflections
and requiring refractive indices
often only rely on optical
measurements.
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Contents
โข Basic Principles
โข Transfer Matrix Method
โข Multilayer Stack
โข Comparison to Experiment
โข Advanced Features
โข Surface Roughness
โข Inhomogeneous Layers
โข Effective Medium
โข User Interface
โข Outlook
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Basic Principles โ Transfer Matrix Method
Superposition of electric fieldone wave with positive direction(E+)
one wave with negative direction(E-)
Propagating through an interface:
=-=, = , =;
are, respectively, the complex amplitude reflection and transmission Fresnel coefficients; is the complex refractive index of the layer
=
Propagating within a layer:
= ฮฆ=Where d is the thickness of medium, ฯ is the frequency of the propagating light and c is the speed of light
Propagation through mediums at normal incidence 4
Basic Principles: Transfer Matrix Method Advanced Features User Interface
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Oblique incidence
๐ฑ๐๐ฑ๐
๐ฑ ๐
P polarization:
๐ ๐ , ๐=๐ ๐ ๐๐๐ ๐ฑ๐โ๐ ๐๐๐๐ ๐ฑ ๐
๐ ๐๐๐๐ ๐ฑ๐+๐ ๐๐๐๐ ๐ฑ ๐
๐ก๐ , ๐=2๐ ๐๐๐๐ ๐ฑ๐
๐ ๐๐๐๐ ๐ฑ๐+๐ ๐๐๐๐ ๐ฑ ๐
S polarization:
๐ ๐ , ๐=๐ ๐๐๐๐ ๐ฑ ๐โ๐ ๐ ๐๐๐ ๐ฑ ๐
๐ ๐๐๐๐ ๐ฑ๐+๐ ๐๐๐๐ ๐ฑ ๐
๐ก๐ , ๐=2๐ ๐๐๐๐ ๐ฑ๐
๐ ๐๐๐๐ ๐ฑ ๐+๐ ๐๐๐๐ ๐ฑ ๐
Medium i
Medium j
At interface Within the layer
ฮฆ=
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Basic Principles: Transfer Matrix Method Advanced Features User Interface
Fresnel coefficiencts for oblique incidence
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Coherent Layers โ Interference Effects
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<< 1
๐ฟ=( 2๐๐ )2๐๐cos๐
nฮธ
Validity condition:
Phase difference between transmission orders:
When there will be constructive interference in T
Basic Principles: Multilayer Stack Advanced Features User Interface
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Incoherent Layers & Substrate Layers
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>> 1
Phase relationships between interior reflections is destroyed โ therefore there is no interference
To removed coherency, calculate the Intensity instead of the Electric Field
=
phase information
Basic Principles: Multilayer Stack Advanced Features User Interface
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Multilayer Stack
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Coherent Stackโข Includes interferenceโข Typical thickness 0 ~ 2000 nm
Incoherent Layerโข Interference โturned offโโข Typical thickness 1 mm
โข 9 Total layers implemented in RefDex
โข Combine coherent and incoherent layers in any order
โข For R,T calculation, d, n and k must be known for all layers
Basic Principles: Multilayer Stack Advanced Features User Interface
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Input Spectrum โ R and T
Absorbing Region
โข Reflection loses coherency peaks
โข Transmission drops to zero due to absorption
Transparent Region
โข R and T both show coherency peaks
โข R does not drop to 0 due to reflection from glass substrate
Basic Principles: Comparison to Experiment Advanced Features User Interface
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Comparison to Experiment
An example:
thin filmon substrate
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Basic Principles: Comparison to Experiment Advanced Features User Interface
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Problem of Uniqueness
Choose the n, k values which minimise the difference between our model and experiment
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๐น [๐ โฒ ,๐ โฒ ]=๐น [๐โฒ โฒ ,๐ โฒ โฒ ]=0
Adding these equations together we get a function which takes n and k as input
Problems arise because two different n,k input pairs can both equal zero!
One Physically Meaningful Solution
Many Unphysical Solutions
|๐ ๐๐๐ (๐ ( ฮป ) ,๐ ( ฮป ) )โ๐ ๐๐ฅ๐ ( ฮป )|=0
|๐ ๐๐๐ (๐ ( ฮป ) ,๐ ( ฮป ) )โ๐ ๐๐ฅ๐ ( ฮป )|=0
Basic Principles: Comparison to Experiment Advanced Features User Interface
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Problem of Uniqueness โ Physical Picture
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Results need to be interpreted โ More on this Later!
Physically meaningful solution
Spurius solution branches
Basic Principles: Comparison to Experiment Advanced Features User Interface
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Determination of optical constants in multiple-layer configuration
Take the configuration of CIGSe/TCO/glass substrate as an example:
G. Yin et al., Influence of substrate and its temperature on the optical constants of CuIn1-xGaxSe2 thin films, accepted for Journal of Physics D: Applied Physics 13
Basic Principles: Comparison to Experiment Advanced Features User Interface
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Surface Roughness โ Effect on R and T
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Absorbing Region
โข Reflection Strongly Reduced
โข Transmission Slightly Reduced
Transparent Region
โข R and T reduced prefferentially at coherency peaks
Basic Principles Advanced Features: Surface Roughness User Interface
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Modified Transfer Matrix Method โ Scalar Scattering Theory
Rough InterfaceScalar Scattering Theory
๐ ๐ , ๐โฒ =๐ ๐ , ๐๐๐ฅ๐ [โ2(2๐๐ /๐)2๐๐2 ]
๐ ๐ ,๐โฒ =๐ ๐ ,๐๐๐ฅ๐ [โ2(2๐๐ /๐)2๐ ๐
2 ]
๐ก ๐ ,๐โฒ =๐ก ๐ , ๐๐๐ฅ๐ [โ( 2๐๐๐ )
2
(๐ ๐โ๐๐)2/2 ]
๐ก๐ , ๐โฒ =๐ก ๐ , ๐๐๐ฅ๐[โ( 2๐๐๐ )
2
(๐๐โ๐ ๐)2/2 ]
Medium a
Medium b
Modified
Fresnel coefficients
โข is the interface roughnessโข Gives us the loss of specular beam
intensity due to interface roughness
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Basic Principles Advanced Features: Surface Roughness User Interface
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Modified Transfer Matrix Method - Examples
Determination of optical constants
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G. Yin et al.,The effect of surface roughness on the determination of optical constants of CuInSe2 and CuGaSe2 thin films, J. Appl. Phys., 133, 213510 (2013)
Basic Principles Advanced Features: Surface Roughness User Interface
ฯ = 9nm ฯ = 20nm
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Inhomogeneous Layers โ Effect on R and T
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Absorbing Region
โข Small reduction in R and T
Transparent Region
โข Coherency reduced for both R and T
โข Transmission strongly reduced
Basic Principles Advanced Features: Inhomogeneous Layers User Interface
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Inhomogeneous Layers โ Coherent / Incoherent Decomposition
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a) 2D slice through the 3D inhomogeneous filmb) Overlay a rectangular gridc) The resulting discretised representation of the filmd) Layers containing voids can be modelled incoherently allowing the use of
average layer thicknesses e) This reduces the number of transfer matrix calculations to 4
a)
b)
c)
d)
e)
Basic Principles Advanced Features: Inhomogeneous Layers User Interface
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Inhomogeneous Layers โ Coherent / Incoherent Decomposition
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Standard Calculation
Replace propagation operator inside inhomogeneous layer with:
, ๐ (๐ )={๐ (0 ) ,๐=๐๐ฃ๐๐ ,๐โ 0ยฌ๐ (0 ) ,๐=๐๐๐
or
(Same equations for T not shown here)
โข Void scattering as from a rough surface. (Slide 13) โข Requires statistical knowledge of 3D void distribution as input
Basic Principles Advanced Features: Inhomogeneous Layers User Interface
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Inhomogeneous Layers โ Modelling Distribution of Voids
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โข Measurement of real 2D surface used to generate 3D distribution
โข From 3D distribution we obtain inputs for the RefDex calculation
Basic Principles Advanced Features: Inhomogeneous Layers User Interface
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Inhomogeneous Layers โ Recalculating n and k
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n k data from an inhomogeneous CISe2 film is in good agreement to the n k data from a homogeneous film using the inhomogeneous layer feature.
P. Manley et al.,A method for calculating the complex refractive index of inhomogeneous thin films, (submitted)
Basic Principles Advanced Features: Inhomogeneous Layers User Interface
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Effective Medium Approximation - Background
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Volume Fraction Approximationโข Direct mixing of the two materials via
the volume fractionโข Does not consider polarisation
effects arrising due to mixing
( ๐๐๐๐โ๐h๐๐๐๐ +2๐h )=๐ค๐( ๐๐โ๐h๐๐+2 ๐h )Maxwell Garnett Approximationโข Based on elementary electrostaticsโข Assumes spatially separated
polarisable particles
Bruggeman Approximationโข Assumes two kinds of spherical
particles randomly arranged.โข Spatial separation between
particles should be small (i.e. is large)
๐คh ( ๐hโ๐๐๐๐๐h+2 ๐๐๐๐ )=โ๐ค๐( ๐๐โ๐๐๐๐๐๐+2๐๐๐๐ )
Basic Principles Advanced Features: Effective Medium User Interface
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ELLIPSOMETRY MODE
๐๐
๐ ๐=๐ญ๐๐ง๐ฟ ๐๐โ
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๐๐
๐ ๐=๐ญ๐๐ง๐ฟ ๐๐โ
โข Ellipsometric parameters ฮจ and ฮ simulated by RefDexโข Useful for highly absorbing substratesโข Currently incompatable with roughness and
inhomogeneity advanced features
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n k Data from Ellipsometry โ Example of Mo film
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Main Interface
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Basic Principles Advanced Features User Interface
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Advanced options
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Basic Principles Advanced Features User Interface
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Data Extraction Process
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โข Interactive fitting process
โข Place nodes which are automatically connected by a smooth function
โข User selects physically meaningful solutions from multiply degenerate solution space
Basic Principles Advanced Features User Interface
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Summary and Outlook
๐๐
๐ ๐=๐ญ๐๐ง๐ฟ ๐๐โ
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RefDexโข calculates T, R (n,k) for a multilayer stack
โ extracts n,k from T, R
โข considers surface roughness
โข applies to inhomogeneous layers
โข has also basic features for ellipsometry
. . .
โข is freely available from
http://
www.helmholtz-berlin.de/forschung/oe/enma/nanooptix/index_en.html
Impulse and Networking Fond: VH-NG-928