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 Motivation

Thin film optics

having to deal with multiple reflections and requiring refractive indices often only rely on optical measurements. Contents

• Basic Principles • Transfer Matrix Method • Multilayer Stack • Comparison to Experiment • Advanced Features • Surface Roughness • Inhomogeneous Layers • Effective Medium • User Interface • Outlook

3 Basic Principles – Transfer Matrix Method Basic Principles: Transfer Matrix Method Advanced Features User Interface

one wave with positive direction(E+) Superposition of electric field one wave with negative direction(E-)

Propagating through an interface:

+ + 1 , − = − , 1 𝑖𝑖 1, 𝑟𝑟𝑖𝑖 𝑗𝑗 𝐸𝐸𝑗𝑗 𝐸𝐸 𝑖𝑖 𝑗𝑗 𝑖𝑖 𝑡𝑡 𝑖𝑖 𝑗𝑗 𝑗𝑗 , =- , 𝐸𝐸= , 𝑟𝑟 , = 𝐸𝐸, , = ; 𝑁𝑁𝑖𝑖−𝑁𝑁𝑗𝑗 2𝑁𝑁𝑖𝑖 2𝑁𝑁𝑗𝑗 𝑖𝑖 𝑗𝑗 𝑗𝑗 𝑖𝑖 𝑁𝑁𝑖𝑖+𝑁𝑁𝑗𝑗 𝑡𝑡𝑖𝑖 𝑗𝑗 𝑁𝑁𝑖𝑖+𝑁𝑁𝑗𝑗 𝑡𝑡𝑗𝑗 𝑖𝑖 𝑁𝑁𝑖𝑖+𝑁𝑁𝑗𝑗 , are, 𝑟𝑟respectively,𝑟𝑟 the complex amplitude reflection and transmission Fresnel coefficients; is the complex refractive𝑟𝑟 𝑡𝑡 index of the layer 𝑁𝑁 Propagating within a layer:

+ 1 , + o E i = , = 2𝜋𝜋 o− E i 𝑖𝑖 −𝑖𝑖 𝑁𝑁𝑖𝑖𝑑𝑑 Propagation through 𝐸𝐸 − Φ − 𝜆𝜆 𝑖𝑖 Φ 𝑒𝑒 mediums at Where d is the𝐸𝐸 thickness of Φmedium, ω is the frequency of the propagating light and c is the speed of light normal incidence 4 Oblique incidence Basic Principles: Transfer Matrix Method Advanced Features User Interface

At interface Within the layer Medium i 𝞱𝞱𝑖𝑖 𝞱𝞱𝑖𝑖

Medium j P polarization:

, = 𝑗𝑗 + 𝞱𝞱 𝑁𝑁𝑗𝑗𝑐𝑐𝑐𝑐𝑐𝑐𝞱𝞱𝑖𝑖 − 𝑁𝑁𝑖𝑖𝑐𝑐𝑐𝑐𝑐𝑐𝞱𝞱𝑗𝑗 𝑟𝑟𝑖𝑖 𝑗𝑗 2 = 𝑁𝑁𝑗𝑗𝑐𝑐𝑐𝑐𝑐𝑐𝞱𝞱𝑖𝑖 𝑁𝑁𝑖𝑖𝑐𝑐𝑐𝑐𝑐𝑐𝞱𝞱𝑗𝑗 , + 𝑁𝑁𝑖𝑖𝑐𝑐𝑐𝑐𝑐𝑐𝞱𝞱𝑖𝑖 𝑖𝑖 𝑗𝑗 / 𝑡𝑡 𝑗𝑗 𝑖𝑖 𝑖𝑖 𝑗𝑗 Φ= S polarization:𝑁𝑁 𝑐𝑐𝑐𝑐𝑐𝑐𝞱𝞱 𝑁𝑁 𝑐𝑐𝑐𝑐𝑐𝑐𝞱𝞱 2𝜋𝜋 −𝑖𝑖 𝜆𝜆 𝑁𝑁𝑚𝑚𝑑𝑑 𝑐𝑐𝑐𝑐𝑐𝑐𝞱𝞱𝑗𝑗 𝑒𝑒 , = 𝑖𝑖 𝑖𝑖 + 𝑗𝑗 𝑗𝑗 𝑖𝑖 𝑗𝑗 𝑁𝑁 𝑐𝑐𝑐𝑐𝑐𝑐𝞱𝞱 − 𝑁𝑁 𝑐𝑐𝑐𝑐𝑐𝑐𝞱𝞱 𝑟𝑟 𝑖𝑖 2 𝑖𝑖 𝑗𝑗 𝑗𝑗 = 𝑁𝑁 𝑐𝑐𝑐𝑐𝑐𝑐𝞱𝞱 𝑁𝑁 𝑐𝑐𝑐𝑐𝑐𝑐𝞱𝞱 , + 𝑁𝑁𝑖𝑖𝑐𝑐𝑐𝑐𝑐𝑐𝞱𝞱𝑖𝑖 𝑡𝑡𝑖𝑖 𝑗𝑗 𝑁𝑁𝑖𝑖𝑐𝑐𝑐𝑐𝑐𝑐𝞱𝞱𝑖𝑖 𝑁𝑁𝑗𝑗𝑐𝑐𝑐𝑐𝑐𝑐𝞱𝞱𝑗𝑗 Fresnel coefficiencts for oblique incidence 5 Coherent Layers – Interference Effects Basic Principles: Multilayer Stack Advanced Features User Interface

Validity condition: n << 1 θ 𝑑𝑑 ∆𝜆𝜆𝐶𝐶𝐶𝐶퐶𝐶𝐶𝐶𝐶𝐶𝐶𝐶𝐶𝐶𝐶𝐶𝐶

Phase difference between transmission orders:

= cos 2𝜋𝜋 𝛿𝛿 2𝑛𝑛𝑛𝑛 𝜃𝜃 When = 𝜆𝜆 , there will be constructive interference in T δ 2𝑚𝑚𝜋𝜋 𝑚𝑚 ∈ ℕ 6 Incoherent Layers & Substrate Layers Basic Principles: Multilayer Stack Advanced Features User Interface

To removed coherency, calculate the Intensity instead of the Electric Field

= = | | = | | ∗ 2 𝑖𝑖𝑖𝑖𝑛𝑛�𝑑𝑑 −𝑖𝑖𝑖𝑖𝑛𝑛�𝑑𝑑 2 0 0 𝐼𝐼 𝐸𝐸 𝐸𝐸 𝐸𝐸phase𝑒𝑒 information𝑒𝑒 𝐸𝐸

>> 1 𝑡𝑡 ∆𝜆𝜆𝐶𝐶𝐶𝐶퐶𝐶𝐶𝐶𝐶𝐶𝐶𝐶𝐶𝐶𝐶𝐶𝐶

Phase relationships between interior reflections is destroyed – therefore there is no interference 7 Multilayer Stack Basic Principles: Multilayer Stack Advanced Features User Interface

Coherent Stack • Includes interference • Typical thickness 0 ~ 2000 nm

Incoherent Layer • Interference “turned off” • Typical thickness 1 mm

• 9 Total layers implemented in RefDex

• Combine coherent and incoherent layers in any order

• For R,T calculation, d, n and k must be known for all layers 8 Input Spectrum – R and T Basic Principles: Comparison to Experiment Advanced Features User Interface

Absorbing Region

• Reflection loses coherency peaks • Transmission drops to zero due to absorption

Transparent Region

• R and T both show coherency peaks • R does not drop to 0 due to reflection from glass substrate

9 Comparison to Experiment Basic Principles: Comparison to Experiment Advanced Features User Interface

An example: thin film on substrate

10 Problem of Uniqueness Basic Principles: Comparison to Experiment Advanced Features User Interface

, = 0 Choose the n, k values which minimise the 𝑐𝑐𝑐𝑐𝑐𝑐 𝑒𝑒𝑒𝑒𝑒𝑒 difference between our 𝑅𝑅 𝑛𝑛 λ ,𝑘𝑘 λ − 𝑅𝑅 λ = 0 model and experiment 𝑇𝑇𝑐𝑐𝑐𝑐𝑐𝑐 𝑛𝑛 λ 𝑘𝑘 λ − 𝑇𝑇𝑒𝑒𝑥𝑥𝑥𝑥 λ

, = , Adding these equations together we get a function + , 𝑐𝑐𝑐𝑐𝑐𝑐 𝑒𝑒𝑒𝑒𝑒𝑒 which takes n and k as input 𝐹𝐹 𝑛𝑛 𝑘𝑘 𝑅𝑅 𝑛𝑛 λ 𝑘𝑘 λ − 𝑅𝑅 λ 𝐹𝐹 𝑇𝑇𝑐𝑐𝑐𝑐𝑐𝑐 𝑛𝑛 λ 𝑘𝑘 λ − 𝑇𝑇𝑒𝑒𝑥𝑥𝑥𝑥 λ Problems arise because two , = , = 0 different n,k input pairs can both ′ equal zero! 𝐹𝐹 𝑛𝑛푛 𝑘𝑘� 𝐹𝐹 𝑛𝑛 ′ 𝑘𝑘𝑘

One Physically Many Unphysical Meaningful Solution Solutions 11 Problem of Uniqueness – Physical Picture Basic Principles: Comparison to Experiment Advanced Features User Interface

Spurius solution branches

Physically meaningful solution

Results need to be interpreted – More on this Later!

12 Determination of optical constants in multiple-layer configuration Basic Principles: Comparison to Experiment Advanced Features User Interface

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 Surface Roughness – Effect on R and T Basic Principles Advanced Features: Surface Roughness User Interface

Absorbing Region

• Reflection Strongly Reduced • Transmission Slightly Reduced

Transparent Region

• R and T reduced prefferentially at coherency peaks

14 Modified Transfer Matrix Method – Scalar Scattering Theory Basic Principles Advanced Features: Surface Roughness User Interface

Scalar Scattering Theory Rough Interface

= 2( / ) Medium a , , ′ 2 2 𝑖𝑖 𝑗𝑗 𝑖𝑖 𝑗𝑗 𝑖𝑖 𝑟𝑟 = 𝑟𝑟 𝑒𝑒𝑒𝑒𝑒𝑒 −2( 2𝜋𝜋𝜋𝜋/ 𝜆𝜆) 𝑛𝑛 Modified , , ′ 2 2 𝑟𝑟𝑗𝑗 𝑖𝑖 𝑟𝑟𝑗𝑗 𝑖𝑖𝑒𝑒𝑒𝑒𝑒𝑒 − 2𝜋𝜋𝜋𝜋 𝜆𝜆 𝑛𝑛𝑗𝑗 Fresnel , = , 2 ( ) /2 coefficients ′ 2𝜋𝜋𝜋𝜋 2 𝑡𝑡𝑖𝑖 𝑗𝑗 𝑡𝑡𝑖𝑖 𝑗𝑗𝑒𝑒𝑒𝑒𝑒𝑒 − 𝑛𝑛𝑖𝑖 − 𝑛𝑛𝑗𝑗 Medium b 𝜆𝜆 , = , 2 ( ) /2 ′ 2𝜋𝜋𝜋𝜋 2 𝑡𝑡𝑗𝑗 𝑖𝑖 𝑡𝑡𝑗𝑗 𝑖𝑖𝑒𝑒𝑒𝑒𝑒𝑒 − 𝑛𝑛𝑗𝑗 − 𝑛𝑛𝑖𝑖 𝜆𝜆 • is the interface roughness • Gives us the loss of specular beam intensity𝜎𝜎 due to interface roughness

15 Modified Transfer Matrix Method - Examples Basic Principles Advanced Features: Surface Roughness User Interface

Determination of G. Yin et al.,The effect of surface optical constants roughness on the determination of optical constants of CuInSe2 and CuGaSe2 thin films, J. Appl. Phys., 133, 213510 (2013)

σ = 9nm σ = 20nm

16 Inhomogeneous Layers – Effect on R and T Basic Principles Advanced Features: Inhomogeneous Layers User Interface

Absorbing Region

• Small reduction in R and T

Transparent Region

• Coherency reduced for both R and T • Transmission strongly reduced

17 Inhomogeneous Layers – Coherent / Incoherent Decomposition Basic Principles Advanced Features: Inhomogeneous Layers User Interface

a)

b) e)

c)

d) a) 2D slice through the 3D inhomogeneous film b) Overlay a rectangular grid c) The resulting discretised representation of the film d) 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 18 Inhomogeneous Layers – Coherent / Incoherent Decomposition Basic Principles Advanced Features: Inhomogeneous Layers User Interface

, = + (Same equations for T not shown here) 𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐 𝑐𝑐 𝐶𝐶 𝐼𝐼 𝐼𝐼 Standard𝑅𝑅 Calculation𝑛𝑛 𝑘𝑘 𝑤𝑤 𝑅𝑅 𝑤𝑤 𝑅𝑅

Replace propagation operator inside inhomogeneous layer with:

, ( ) 0 , = , 0 = , , = 𝑀𝑀 𝑛𝑛 𝑚𝑚 𝑛𝑛 𝑚𝑚 𝑛𝑛 𝑚𝑚−1 𝑛𝑛 0 ¬ 0 , = �𝑖𝑖 𝑚𝑚=1 �𝑚𝑚 � 𝑚𝑚 𝑚𝑚−1 �0 𝑛𝑛 𝑚𝑚 𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒 𝑚𝑚 ≠ 𝑷𝑷 ∏ 𝑷𝑷 𝑫𝑫 𝑷𝑷 𝑛𝑛 𝑚𝑚 � 0 =𝑛𝑛 𝑚𝑚or 𝑜𝑜𝑜𝑜𝑜𝑜0 = ∗ ∗ 𝑛𝑛 𝑛𝑛𝑖𝑖 𝑛𝑛 𝑛𝑛𝑣𝑣 • Void scattering as from a rough surface. (Slide 13) • Requires statistical knowledge of 3D void distribution as input

19 Inhomogeneous Layers – Modelling Distribution of Voids Basic Principles Advanced Features: Inhomogeneous Layers User Interface

• Measurement of real 2D surface used to generate 3D distribution

• From 3D distribution we obtain inputs for the RefDex calculation

20 Inhomogeneous Layers – Recalculating n and k Basic Principles Advanced Features: Inhomogeneous Layers User Interface

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, 21 (submitted) Effective Medium Approximation - Background Basic Principles Advanced Features: Effective Medium User Interface

= + Volume Fraction Approximation • Direct mixing of the two materials via 𝑛𝑛𝑒𝑒𝑒𝑒𝑒𝑒 𝑤𝑤ℎ𝑛𝑛ℎ 𝑤𝑤𝑖𝑖𝑛𝑛𝑖𝑖 the volume fraction = + • Does not consider polarisation effects arrising due to mixing 𝑘𝑘𝑒𝑒𝑒𝑒𝑒𝑒 𝑤𝑤ℎ𝑘𝑘ℎ 𝑤𝑤𝑖𝑖𝑘𝑘𝑖𝑖

Maxwell Garnett Approximation = • Based on elementary electrostatics + 2 + 2 • Assumes spatially separated 𝜀𝜀𝑒𝑒𝑒𝑒𝑒𝑒 − 𝜀𝜀ℎ 𝜀𝜀𝑖𝑖 − 𝜀𝜀ℎ 𝑤𝑤𝑖𝑖 polarisable particles 𝜀𝜀𝑒𝑒𝑒𝑒𝑒𝑒 𝜀𝜀ℎ 𝜀𝜀𝑖𝑖 𝜀𝜀ℎ

Bruggeman Approximation = • Assumes two kinds of spherical particles randomly arranged. ℎ+ 2 𝑒𝑒𝑒𝑒𝑒𝑒 𝑖𝑖+ 2 𝑒𝑒𝑒𝑒𝑒𝑒 𝜀𝜀 − 𝜀𝜀 𝜀𝜀 − 𝜀𝜀 • Spatial separation between 𝑤𝑤ℎ −𝑤𝑤𝑖𝑖 𝜀𝜀ℎ 𝜀𝜀𝑒𝑒𝑒𝑒𝑒𝑒 𝜀𝜀𝑖𝑖 𝜀𝜀𝑒𝑒𝑒𝑒𝑒𝑒 particles should be small (i.e. is large) 𝑤𝑤𝑖𝑖 22 ELLIPSOMETRY MODE

= 𝒓𝒓𝒑𝒑 𝒊𝒊∆ 𝐭𝐭𝐭𝐭𝐭𝐭𝐭𝐭 𝒆𝒆 𝒓𝒓𝒔𝒔 Type equation here.

• Ellipsometric parameters Ψ and Δ simulated by RefDex = • Useful for highly absorbing substrates 𝒓𝒓𝒑𝒑 𝒊𝒊∆ • Currently incompatable with roughness and 𝐭𝐭𝐭𝐭𝐭𝐭𝐭𝐭 𝒆𝒆 inhomogeneity advanced features 𝒓𝒓𝒔𝒔 23 n k Data from Ellipsometry – Example of Mo film

24 Main Interface Basic Principles Advanced Features User Interface

25 Advanced options Basic Principles Advanced Features User Interface

26 Data Extraction Process Basic Principles Advanced Features User Interface

• Interactive fitting process

• Place nodes which are automatically connected by a smooth function

• User selects physically meaningful solutions from multiply degenerate solution space

27 Summary and Outlook

RefDex • calculates T, R (n,k) for a multilayer stack = → extracts n,k from T, R 𝒓𝒓𝒑𝒑 𝒊𝒊∆ 𝐭𝐭𝐭𝐭𝐭𝐭𝐭𝐭 𝒆𝒆 • considers surface roughness 𝒓𝒓𝒔𝒔 Type equation here. • applies to inhomogeneous layers • has also basic features for ellipsometry . . . • is freely available from

http://www.helmholtz-berlin.de/forschung/oe/enma/nanooptix/index_en.html

Impulse and Networking Fond: VH-NG-928 28