Ellipsometry Data Analysis: a Tutorial
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1 llipsometry Data Analysis: a Tutorial G.. Jellison, Jr. Solid State Division Oak Ridge National Laboratory WIS University of Michigan May 8-9,
2 Motivation The Opportunity: Spectroscopic llipsometry (S is sensitive to many parameters of interest to thin-film science, such as Film thickness Interfaces Optical functions (n and k. But S data is not meaningful by itself. Therefore One must model the near-surface region to get useful information.
3 Outline Data representations Calculation of reflection from thin films Modeling of optical functions (n and k Fitting ellipsometry data xamples
4 Introduction What we measure: Stokes Vector of a light beam lc rc o o I I I I I I I V U Q I S I o (Q + U + V / To transform one Stokes vector to another: S out M S in M Mueller matrix (4X4 real
5 Introduction What we measure: light source PSG PSD p p s φ s Sample Detector Intensity of beam at the detector S T PSD M S PSG The matrix M includes: Sample reflection characteristics Intervening optics (windows and lenses
6 Introduction Isotropic Reflector: p in p out s in s out Isotropic reflector Mueller Matrix: M reflector N cos(ψ N S sin(ψ sin( C sin(ψ cos( N + S + C N C S S C
7 Introduction What we calculate: light source PSG PSD p p s φ s Sample Detector J o i o i rpp rps Ê p/ Ê p Ê s / Ê p o i o i rsp rss Ê p/ Ê s Ê s / Ê s ρ ρsp ρ i tan( ψ e i tan( ψ sp e ps rss sp tan( ψ ps e i ps. There is no difference between Mueller and Jones IF there is no depolarization!
8 Data Representations Mueller-Jones matrices: M A ( J J * A -, i i A. Assumes no depolarization! N is C e r r i in s out s in p out p s p + + tan(ψ ρ
9 Anisotropic samples: Data Representations M N α sp C ps + ξ S ps + ξ N α C S ps α sp ps α + ξ + ξ ps ps C sp C S sp + ζ + ζ C + β + β S sp + ζ S sp + ζ S + β C β ρ ρ ps sp Cps + isps tan( ψps e + N Csp + issp tan( ψ sp e + N i ps i sp N + S + C + S ps + C ps + S sp + C sp J ρ ρsp i ρ tan( ψ e i tan( ψsp e ps rss sp tan( ψ e ps i ps
10 Calculation of Reflection Coefficients Single Interface: Fresnel quations (83: φ φ r ñ cos( φ pp ñ cos( φ + ñ ñ cos( cos( φ φ r ñ cos( φ ss ñ cos( φ + ñ ñ cos( φ cos( φ Snell s Law (6: ξ ñ sin φ ñ sin φ.
11 Calculation of Reflection Coefficients Two Interfaces: Airy Formula (833: r pp, ss r, pp, ss + r +, pp, ss r, pp, ss r, pp, ss exp( ib exp( ib πd b λ f ñ f cos( φ f
12 Calculation of Reflection Coefficients Three or more interfaces: Abeles Matrices (95: Represent each layer by Abeles matrices: P, pp cos( b ñ i sin( b cos( φ cos( φ i sin( b ñ cos( b P, ss iñ cos( b cos( φ sin( b i sin( b ñ cos( φ cos( b Matrix multiply: M N pp χ, pp( P, pp χsub, pp M ss χ N, ss P, ss ( χ sub, ss
13 Calculation of Reflection Coefficients Three or more interfaces: Abeles Matrices: Substrate and ambient characteristic matrices: χ cos( ñ φ φ, pp cos( ñ χ, ss ñ ñ cos( φ cos( φ χ cos( φ ñ sub, pp sub sub χ sub ss ñsub cos( φ, sub Final Reflection coefficients: r pp M M, pp, pp r ss M M, ss, ss
14 Calculation of Reflection Coefficients Anisotropic materials: X Abeles matrices become one 4X4 Berreman matrix (97 Reason: s- and p- polarization states are no longer eigenstates of the reflection. Inhomogeneous layers: If a layer has a depth-dependent refractive index, there are two options: Build up many very thin layers Use interpolation approximations
15 Models for dielectric functions Tabulated Data Sets: Usually good for substrates Not good for thin films 3 ven for substrates: problems a surface roughness b surface reconstruction c surface oxides 4 Most tabulated data sets do not include error limits Measured optical functions of silicon depend on the face! [near 4.5 ev 9 nm, ε (<ε (]
16 Models for dielectric functions Lorentz Oscillator Model (895: + + o i A ñ λ ζ λ λ λ λ λ ε, ( ( Γ + + o i B ñ, ( ε ( Sellmeier quation: + o A n, λ λ λ ε For Insulators Cauchy (83: + B B n λ Drude (89: ( i B Γ ε For Metals
17 Models for dielectric functions Amorphous Materials (Tauc-Lorentz: A k n g o g ( ( ( ( ( ( Θ + Γ ε ξ ξ ξ ξε π ε ε d P R g + ( ( ( Five parameters: g Band gap A Proportional to the matrix element Central transition energy Γ Broadening parameter ε ( Normally
18 Forouhi and Bloomer (Forouhi and Bloomer Phys Rev. B 34, 78 (986. xtinction coefficient: k FB ( A( g B + C Refractive index: (a Hilbert transform k( ξ k( n FB ( n( + P dξ π ξ Problems: k FB (> for < g. Unphysical. k FB ( constant as. xperiment states that k( as / 3 as. Time-reversal symmetry required [k(- -k(]. Hilbert Transform Kramers Kronig [k( ].
19 Models for dielectric functions Models for Crystalline materials: Critical points, excitons, etc. in the optical spectra make this a very difficult problem! Collections of Lorentz oscillators: ε ( ε o + A e i φ + iγ Can end up with MANY terms
20 Models for dielectric functions ffective Medium theories: < ε > ε h < ε > + γε h ε εh f ε + γε h Choice of host material: Lorentz-Lorentz: ε h Maxwell-Garnett: ε h ε 3 Bruggeman ε h <ε> γ Depolarization factor ~.
21 Figure of Merit: Fitting Models to Data xperimental quantities: Calculated quantities: ρ exp (λ ρ calc (λ, z z: Vector of parameters to be fit ( to m film thicknesses, constituent fractions, parameters of optical function models, etc. Minimize χ N N m [ ρ exp ( λ ρ σ ( λ calc ( λ, z] σ(λ random and systematic error
22 Calculation Procedure: Fitting Models to Data Assume a model. a. Number of layers b. Layer type (isotropic, anisotropic, graded Determine or parameterize the optical functions of each layer 3 Select reasonable starting parameters. 4 Fit the data, using a suitable algorithm and Figure of Merit 5 Determine correlated errors in z and cross correlation coefficients If the Figure of Merit indicates a bad fit (e.g. χ >>, go back to.
23 Fitting Models to Data fitting of parameters is not the end-all of parameter estimation. To be genuinely useful, a fitting procedure should provide (i parameters, (ii error estimates on the parameters, and (iii a statistical measure of goodness-of-fit. When the third item suggests that the model is an unlikely match to the data, then items (i and (ii are probably worthless. Unfortunately, many practitioners of parameter estimation never proceed beyond item (i. They deem a fit acceptable if a graph of data and model looks good. This approach is known as chi-by-eye. Luckily, its practitioners get what they deserve. Press, Teukolsky, Vettering, and Flannery, Numerical Recipes ( nd ed., Cambridge, 99, Ch. 5, pg. 65.
24 Fitting Models to Data Levenberg-Marquardt algorithm: Curvature matrix: α kl z N χ ρ calc ( λ, z ρ calc ( λ, zk zl σ ( λ zk zl Inverse of curvature matrix: ε α - rror in parameter z ε. Cross correlation coefficients: proportional to the elements of ε.
25 Fitting Models to Data Meaning of fitted parameters and errors: Assume air/sio /Si structure Parameterize SiO using Sellmeier dispersion (λ o 93 nm: ε n A λ + λ λ o, Two fit parameters: d f, A f A f σ Α σ d d f
26 Fitting Models to Data An xample: a-si x N y :H on silicon: 3 SiN / c-si Re(ρ pp T-L χ.96 Lorentz χ 3.84 F&B χ Im(ρ pp nergy (ev
27 Fitting Models to Data An xample: a-si x N y :H on silicon: Model: air surface roughness Bruggeman MA (5% air, 5% a-sin 3 a-sin (3 models Lorentz Forouhi and Bloomer Tauc-Lorentz 4 interface Bruggeman MA (5% Si, 5% a-sin 5 silicon Fitting parameters: d, d 3, d 4, A, o, Γ, ε( and g (F&B and T-L
28 Fitting Models to Data An xample: a-si x N y :H on silicon: Lorentz F&B T-L Roughness thick (nm.±.3 4.9±.7.9±.3 Film thickness (nm 97.8± ±. 98.±.4 Interface thick (nm.6±.3 -.6±.6 -.±. A.9± ± ±.7 Ε o (ev 9.6± ± ±.47 Γ (ev.±..74±3.5.8±.8 g (ev ± ±.9 ε(.±..93±.5.38±.6 χ Roughness thick (nm.4±.3 4.7±.6.8±. Film thickness (nm 98.6± ±.9 98.±.3 A.5±.4 5.3± ±3. Ε o (ev 9.6±. 7.7±. 9.6±..3 Γ (ev.±. 4.±.8 3.7±.33 g (ev ± ±.4 χ
29 Fitting Models to Data An xample: a-si x N y :H on silicon:. exp-calc. Re(ρ pp T-L Lorentz F&B. Im(ρ pp nergy (ev
30 Optical Functions from llipsometry Optical Functions from Parameterization: refractive index (n. T-L Lorentz F&B..9.8 extinction coefficient (k nergy (ev rror limits: Use the submatrix α s from the associated fitted parameters.
31 Optical Functions from llipsometry Newton-Raphson algorithm: Solve: Re(ρ calc (λ, φ, n f, k f, - Re(ρ exp (λ Im(ρ calc (λ, φ, n f, k f, - Im(ρ exp (λ Jacobian: J ρ n ρ n re im ρ k ρ k re im n new n old + δn; where δn -J - ρ Propagate errors!
32 Optical Functions from llipsometry Optical functions of semiconductors: Dielectric function from air/substrate system: } ( tan ] [ { ( sin φ ρ ρ φ ε ε ε i Only valid if there is no overlayer (almost never true If there is a thin film, Drude showed: 4, ( f f s s o n n d K k n + λ π
33 Optical Functions from llipsometry Optical functions of semiconductors: Pseudo-dielectric functions of silicon with,.8, and. nm SiO overlayers <n> <k> 6 <α> (/cm nergy (ev
34 Optical Functions from llipsometry Optical functions of semiconductors: rror limits of the dielectric function of silicon:.5.4 δn δk rror in n and k nergy (ev
35 Optical Functions from llipsometry Optical functions of thin films: Small grain poly silicon Re(ρ -.5 Im(ρ nergy (ev
36 Optical Functions from llipsometry Optical functions of thin films: Method of analysis: A. Restrict analysis region to interference oscillations. Parameterize the optical functions of the film. air surface roughness (BMA 3 T-L model for film 4 Lorentz model for a-sio 5 c-si B. Fit data to determine thicknesses and Lorentz model parameters of a-sio. C. Calculate optical functions of thin film using Newton-Raphson.
37 Optical Functions from llipsometry Optical functions of thin films: 8 6 n k 6 α (/cm nergy (ev c-si lg p-si sg p-si a-si
38 Parting Thoughts S is a powerful technique, but modeling is critical. Modeling should use an error-based figure of merit Does the model fit the data? Calculate correlated errors and cross-correlation coefficients. When used properly, S gives very accurate thicknesses and values of the complex dielectric function.
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