Dielectric function ™ Optical vibrations (phonons) ™ Free electrons (plasma) ™ Electronic transitions (valence Æ conduction band) ™ Dielectric function and refractive index are generally complex: ε = ε ' + iε '' ; ; ε ' = n 2 -n2 ; ε '' = 2n n Phys 774: r r r r R I r R I absorption coefficient α =2kn n - extinction coefficient Ellipsometry 0 I I

Fall 2007

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Dielectric function contributions Spectroscopic Measurements in IR ε()ω =+ 1χωPh () + χω FC () +χω E () =ε∞ χω + Ph () χω + FC ()

⎛⎞S 2 Fourier Transform Infra-Red χω()= ⎜⎟j ph ∑ 22 Spectroscopy j ⎜⎟ω −+ωωγi ⎝⎠TOj j

-1 2 2 ELLIPSOMETRY 30 – 10000 cm Ω p 2 ne χωFC ()= 2 Ω=p * −+ω iωγ ε 0m

2 • Technique that determines the change in polarization state P of light reflected from a sample. () j χωE = ∑ 22 • Typically used for characterizing thin films, strongly j ω 0 j −+ωωγi j correlated materials. • Used in a wide range of industrial applications. • Non contact, non destructive method. 3 4 Principles and goals of Ellipsometry Ellipsometry and Grating Spectrometers

r Disadvantages: ρ = p Cannot use lenses due to r Depolarization effects s PMT or Photodiode 0φ variable Normalization is difficult due to complicated response of the grating

Ellipsometry:

Linear polarized light is incident on the sample under an angle

φ0 and polarization of the reflected light is measured. 5 6

Ellipsometry and Grating Spectrometers Spectroscopic Ellipsometer Disadvantages here ? SOPRA GESP5 Resolution decreases if we PMT or cannot illuminate all groves • SOPRA variable angle spectroscopic ellipsometer GESP5 Photodiode of the grating

7 8 FT-IR Ellipsometry Advantages: 1. Multiplexing Types of Ellipsometer 2. Throughput (no slits needed)

• Ellipsometers may be single wavelength or spectroscopic • They may operate as rotating element or as nulling ellipsometers • They may be single point or imaging

IR-bolometer 9 10

Plane of Incidence Fresnel’s Equations

ncos12φ − ncosφ iyδ • Plane of incidence of light is the plane containing the normal to the • Fresnel’s equations rryy==⋅e n12 cosφ + n cosφ sample surface and the incident direction of light. give the complex φ ncosφ − ncosφ • The s-plane is perpendicular to the plane of incidence. amplitude of the rr==⋅21 eiδ x xxncosφ + ncosφ • The p-plane is parallel to the plane of incidence. reflected light 21 y φ φ IIy − x x Polarization≡= P IIy + x ϕ Jones Formalism Optical elements (polarizers, mirrors, etc) Jones ⎡⎤cosα EElinear = 0 ⎢⎥ sinα ⎡⎤EEJJ ⎡ ⎤ ⎣⎦ Jones ⎡⎤JJ11 12 x ⎡⎤11 12 0x J = ⎢⎥⎢⎥=⋅⎢⎥ ⎢ ⎥ EEyyJJ 0 Jones ⎡⎤1 ⎣⎦JJ21 22 ⎣⎦⎣⎦21 22 ⎣ ⎦ EEcircular = 0 ⎢⎥ 11 ⎣⎦i 12 Ellipsometric Equations What does an ellipsometer measure? • The polarization change of light reflected from a sample in 22⎡⎤1− ρ εφφφ=+⋅⋅sin00 sin tan 0⎢⎥ terms of two parameters ∆ and Ψ ⎣⎦1+ ρ r • These values are related to the ratio of the Fresnel reflection ρ ==p tan Ψ⋅ ei∆ coefficients for p- and s- polarized light rs r φ variable, tanΨ=p , ∆=δδ − 0 r p s r s p =Ψ⋅tan ei∆ ε is a function of frequency (energy) rs in real experiment we need to connect ρ with the polarizer angles of the instrument • This ratio is complex using, e.g., Jones matrices •tanΨ measures the ratio of the modulus of the amplitude reflection ratio Pseudo-dielectric function, which coincides with • The phase difference between p- and s-polarised reflected “dielectric function” for uniform and isotropic bulk materials light is given by ∆ Anisotropy can be measured directly by changing φ 0 13 14

What can ellipsometry tell us about Select the model samples? and Optical Functions • Taken on there own values of ∆ and Ψ tell us little about a sample. The optical model will generally consist of • We take the measured values of ∆ and Ψ, typically • The thickness of a film or layer as a function of wavelength and angle of incidence • The complex refractive index (Variable Angle Spectroscopic Ellipsometry). • An optical model is then built using as much N = nr –ini information about the sample as possible Sometimes, the complex dielectric constant ε = εr –iεi is given, where What cannot be directly determined N2 = ε from ellipsometry measurement? • For a Mixture of materials use the Effective Medium Approximation Multilayer sample structure (it is not a forensic tool) 15 16 Extracting data in ellipsometry Regression • Acquire data of spectral range and angle of incidence required • Regression algorithms are used to vary unknown parameters

measure and minimize the difference between model and experimental • Build an optical model that describes cos ∆ an d t anΨ data. sample structure. • The Levenberg-Marquardt algorithm is the most widely used. • Generate theoretical data from optical construct optical model. model The algorithm seeks to minimise the function • Compare experimental and theoretical

data compare data 1 2 2 and model χ = ∑{}()()tanψ measured − tanψ model + cos∆ measured − cos∆ model • Vary model parameters, such as film 2N − M N thickness, until a “best fit” to

Global minimum. experimental data is obtained. p rint mo del dat a where N is the number of data points at the different measured wavelengths and M the number of model parameters.

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Fit the data Regression Analysis Advantages of Ellipsometry • The data are then compared with the data generated from the theoretical model. • It measures the ratio of two values so is highly accurate and reproducible, does not need a • Unknown parameters in the optical model, such as reference sample, and is not so susceptible to light film thickness or optical constants, are varied to try source fluctuation to produce a “best fit” to experimental data. • Since it measures phase, it is highly sensitive to • Regression algorithms, such as Levenberg-Marquadt, the presence of ultra-thin films (down to sub- are used to vary unknown parameters and minimize monolayer coverage). the difference between experimental and model- generated data. • It provides two pieces of data at each wavelength. More film properties can be determined. • Physical parameters are obtained once a good fit is achieved.

19 20 What is far-IR Ellipsometry? Some applications of Ellipsometry: • Data storage: – Characterization of thin films used the manufacturing of storage media • Display technology: – Characterisation of materials used in flat panel displays • Optical coatings: – Study of coatings on filters, beam splitters and other optical components. • Biology / Chemistry: – Study of organic semiconductors, and tissue samples. • In-situ measurement: – Allows real time monitoring of etching and thin film deposition Far-Infrared Ellipsometry is a technique which allows one to measure very accurately and with high reproducibility the complex dielectric function ε(ω) = ε(ω)1 + i ε(ω)2 of oxide thin films and single processes. crystals. It measures the change in polarization of Infrared light upon non-normal reflection on the surface of a sample to be studied. To extend the Ellipsometry technique to the Far-Infrared part of the electromagnetic spectrum, we are going to carry out these experiments at Brookhaven National Laboratory, 21 National Synchrotron Light Source. Synchrotron light provides three orders of magnitude more brilliant 22 light in the Far-Infrared as compared to conventionally available light sources, like mercury arc lamps.

Example of far-IR spectroscopic Ellipsometry: Another example: Sensitivity to thin film Hardening of the soft modes in SrTiO3 thin films

Ellipsometry of SrTiO3 thin films

Bare Si

Bare Si

Si + 1nm oxide

A.A. Sirenko, et al., Nature 404, 373 (2000) 23 24 Problems with transparent Problems with transparent samples samples • The error in ∆ is proportional to cosec ∆ • For transparent samples, ∆ ~ 1800 • This can be overcome by placing a retarder in front of the light source to introduce a phase shift into the light. • Alternatively, a rotating compensator ellipsometer could be used

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Problems with transparent Hafnium oxide on silicon samples

Poor fit

27 28 Hafnium oxide on silicon Partially oxidised sacrificial a-Si on SiC

Excellent fit with parameterisation

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Partially oxidised sacrificial a-Si on 4H-SiC Ellipsometry of conventionally grown oxide on 4H-SiC • Model (SiO2 database – aSi Forouhi) ε unoxidised, etched i 10 ε oxidised, etched i ε unoxidised, etched r ε oxidised, etched r 8

6 SiO2 (1.41176 µm) ε

4

aSi + SiO2 with density gradient (27.92 nm) 2

aSi (64.7 nm) 0 12345 Photon Energy eV SiC substrate 31 32 Very thin films Grazing Incidence X-ray Reflectometry

• For films of thickness about 10 nm or less, • X-rays used in XRR have wavelengths of the there exists a correlation between thickness order of 1Å. Since the refractive index of and refractive index materials at these short wavelengths is slightly less than unity, x-rays are totally externally reflected at • Complementary techniques must be used to angles of incidence typically up to a few tenths of measure some parameters, usually a degree. thickness. • As the incidence angle increases above the • GIXR employed critical angle for total reflection, x-rays penetrate the layers of the material.

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Grazing Incidence X-ray Reflectometry vs. Ellipsometry Ellipsometry of conventionally • At the interface between each layer, some of the radiation is reflected to the grown oxide on 4H-SiC layer above, while some is transmitted

(refracted) to the layer below. 0.50 Best fit using non-oxidised substrate • The wave interference produces a series 0.9 of fringes in the reflectivity curve. 0.45

• Layer densities, thickness and interface 0.40 tan (ψ) data 0.8 tan (ψ) best fit

roughness can be determined from the ) cos (∆) data cos ( ψ 0.35 cos (∆) best fit ∆

reflectivity curve. tan ( 0.7 )

0.30

0.6 0.25

0.20 0.5 0.2 0.3 0.4 0.5 0.6 0.7 0.8 Wavelength (µm)

35 36 Ellipsometry of conventionally Another example: grown oxide on 4H-SiC Comparison of XTEM and

0.50 Best fit using etched thermal oxide as substrate Spectroscopic Ellipsometry 0.9 0.45

0.40 tan (ψ) data 0.8 tan (ψ) best fit ) cos (∆) data cos ( ψ 0.35 cos (∆) best fit ∆

tan ( 0.7 )

0.30

0.6 0.25

0.20 0.5 0.2 0.3 0.4 0.5 0.6 0.7 0.8 Wavelength (µm)

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