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Ellipsometry Data Analysis: a Tutorial G. E. Jellison, Jr. Solid State Division Oak Ridge National Laboratory WISE 2000 University of Michigan May 8-9, 2000 Motivation The Opportunity: Spectroscopic Ellipsometry (SE) is sensitive to many parameters of interest to thin-film science, such as • Film thickness • Interfaces • Optical functions (n and k). But SE data is not meaningful by itself. Therefore One must model the near-surface region to get useful information. Outline Data representations Calculation of reflection from thin films Modeling of optical functions (n and k) Fitting ellipsometry data Examples Introduction What we measure: Stokes Vector of a light beam æ ç ç S = ç ç ç è æ ö ç ÷ Q ÷ ç I0 = ç I U ÷ ç 45 ÷ ç I V ÷ø rc è I o I o − I 90 − I − 45 − I lc ö ÷ ÷ ÷ ÷ ÷ ø Io ≥ (Q2 + U2 + V2 )1/2 To transform one Stokes vector to another: Sout = M Sin M = Mueller matrix (4X4 real) Introduction What we measure: light source PSG p PSD p φ s s Detector Sample Intensity of beam at the detector = SPSDT M SPSG The matrix M includes: Sample reflection characteristics Intervening optics (windows and lenses) Introduction Isotropic Reflector: Epin Epout Esin Esout Isotropic reflector Mueller Matrix: M reflector æ 1 ç ç− N =ç 0 çç è 0 −N 1 0 0 0 C 0 −S N = cos(2ψ ψ) S = sin(2ψ ψ) sin(∆ ∆) C = sin(2ψ ψ) cos(∆ ∆) N2 + S2 + C2 = 1 0ö ÷ 0÷ S÷ ÷÷ Cø Introduction What we calculate: light source PSG p PSD p φ s Detector s Sample æ rpp J = çç è rsp æ ρ = rss çç è ρ sp rps ö æ Ê op / Ê ip ÷÷ = ç o i rss ø çè Ê p / Ê s ρ ps ö æ tan(ψ )e i∆ ÷÷ = ç 1 ø ç tan(ψ sp )e i∆ è Ê so / Ê ip ö ÷ o i ÷ Ês / Ê s ø tan(ψ ps )e sp i∆ ps 1 There is no difference between Mueller and Jones IF there is no depolarization! ö ÷ ÷. ø Data Representations Mueller-Jones matrices: M = A ( J ⊗ J* ) A-1, æ1 ç ç1 A=ç 0 ç ç0 è 0 0 0 1 0 1 −i i 1 ö ÷ − 1÷ 0 ÷ ÷. 0 ÷ø Assumes no depolarization! ρ= rp rs = in E out E p p E out s E in s = tan(ψ ) e i∆ C + iS = 1+ N Data Representations Anisotropic samples: 1 æ ç ç − N − α sp M=ç C + ξ1 ç ps ç− S +ξ ps 2 è ρps = ρsp = − N − α ps 1 − α sp − α ps C sp + ζ 1 − C sp + ζ 1 − C ps + ξ 1 S ps + ξ 2 C + β1 − S + β2 Cps + iSps 1+ N Csp + iSsp 1+ N S sp + ζ 2 ö ÷ − S sp + ζ 2 ÷ S + β2 ÷ ÷ C − β 1 ÷ø i∆ps = tan(ψ ps )e i∆ sp = tan(ψ sp )e N2 + S2 + C2 + Sps2 + Cps2 + Ssp2 + Csp2 = 1 i∆ æ ρ ρps ö æ tan(ψ )ei∆ tan(ψ ps )e ps ö ÷ ÷÷ = ç J = rss çç i ∆ sp ç ÷ 1 è ρsp 1 ø è tan(ψsp )e ø Calculation of Reflection Coefficients Single Interface: Fresnel Equations (1832): 0 φ0 1 φ1 r pp = ñ 1 cos( φ 0 ) − ñ 0 cos( φ 1 ) ñ 1 cos( φ 0 ) + ñ 0 cos( φ 1 ) rss = ñ 0 cos( φ 0 ) − ñ 1 cos( φ 1 ) ñ 0 cos( φ 0 ) + ñ 1 cos( φ 1 ) Snell’s Law (1621): ξ = ñ0 sin φ0 = ñ1 sin φ1. Calculation of Reflection Coefficients Two Interfaces: Airy Formula (1833): rpp ,ss = b= r1, pp ,ss + r2, pp ,ss exp(−2ib) 1 + r1, pp ,ss r2, pp ,ss exp( −2ib) 2πd f λ ñ f cos(φ f ) Calculation of Reflection Coefficients Three or more interfaces: Abeles Matrices (1950): Represent each layer by 2 Abeles matrices: P j , pp cos(φ j ) æ ö ç −i cos(b j ) sin(b j ) ÷ ñj ç ÷ =ç ÷ ñj çi ÷ sin(b j ) cos(b j ) ç cos(φ ) ÷ j è ø P j , ss æ ç cos( b j ) =ç ç iñ cos( φ ) sin( b ) j j è j i sin( b j ) ö ÷ ñ j cos( φ j ) ÷ cos( b j ) ÷ø Matrix multiply: N M pp = χ 0, pp (∏ Pj , pp ) χ sub, pp j =1 N M ss = χ 0,ss ( ∏ P j ,ss ) χ sub ,ss j =1 Calculation of Reflection Coefficients Three or more interfaces: Abeles Matrices: Substrate and ambient characteristic matrices: χ 0 , pp æ ç 1 1ç = 2 ç−1 ç è χ sub, pp æ cos( φ ) ö ç1 ÷ 1 ñ0 ÷ χ ç 0 , ss = cos( φ ) ÷ 2 ç1 ç ñ0 ÷ø è 1 æ ö æ cos(φ sub ) ö 0÷ 0 ç ç ÷ χ = ç ñsub sub,ss = ç ñsub cos(φ sub ) ÷÷ ÷ ç ç ÷ 1 0 1 0 è ø è ø Final Reflection coefficients: rpp = 1 ö ÷ ñ0 cos( φ ) ÷ −1 ÷ ñ0 cos( φ ) ÷ø M 21 , pp M 11 , pp rss = M 21 , ss M 11 , ss Calculation of Reflection Coefficients Anisotropic materials: 2 2X2 Abeles matrices become one 4X4 Berreman matrix (1972) 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: 1) Build up many very thin layers 2) Use interpolation approximations Models for dielectric functions Tabulated Data Sets: 1) Usually good for substrates 2) Not good for thin films 3) Even for substrates: problems 4) a) surface roughness b) surface reconstruction c) surface oxides Most tabulated data sets do not include error limits Measured optical functions of silicon depend on the face! [near 4.25 eV = 292 nm, ε2(100)<εε2(111)] Models for dielectric functions Lorentz Oscillator Model (1895): ε (λ ) = ñ(λ ) 2 = 1 + å j ε ( E ) = ñ( E ) 2 = 1 + å j A j λ2 λ2 − λo2 , j + iζ j λ Bj E o2, j − E 2 + iΓ j E Sellmeier Equation: ε = n2 = 1+ å j A j λ2 λ2 − λ2o , j For Insulators Cauchy (1830): n = B0 + å j Bj λ2 j Drude (1890): ε (E) = 1 − å j Bj E 1 E − iΓj For Metals Models for dielectric functions Amorphous Materials (Tauc-Lorentz): ε 2 ( E ) = 2n ( E ) k ( E ) = A( E − E g ) 2 Θ( E − E g ) ( E 2 − Eo2 ) 2 + Γ 2 E ∞ ξε (ξ ) 2 ε 1 ( E ) = ε 1 (∞ ) + P ò 2 2 2 dξ π Rg ξ − E Five parameters: Eg Band gap A Proportional to the matrix element E0 Central transition energy Γ Broadening parameter ε1(∞ ∞) Normally = 1 Forouhi and Bloomer (Forouhi and Bloomer Phys Rev. B 34, 7018 (1986).) Extinction coefficient: k FB ( E ) = Refractive index: A( E − E g ) 2 E 2 − BE + C (a Hilbert transform) ∞ 1 k (ξ ) − k ( ∞ ) dξ n FB ( E ) = n ( ∞ ) + P ò π −∞ ξ − E Problems: • kFB(E)>0 for E<Eg. Unphysical. • kFB(E) → constant as E→ →∞. Experiment states that →∞. k(E) → 0 as 1/E3 as E→ • Time-reversal symmetry required [k(-E) = -k(E)]. • Hilbert Transform ≠ Kramers Kronig [k(∞ ∞) = 0]. 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 (E) = εo + å j Aj e iφ j E − E j + iΓ j Can end up with MANY terms Models for dielectric functions Effective Medium theories: ε j −εh < ε > −ε h =åfj < ε > +γεh ε j + γεh j Choice of host material: 1) Lorentz-Lorentz: εh = 1 2) Maxwell-Garnett: εh = ε1 3) Bruggeman γ εh = <εε> Depolarization factor ~2. Fitting Models to Data Figure of Merit: Experimental quantities: ρexp(λ λ) Calculated quantities: λ, z) ρcalc(λ, z: Vector of parameters to be fit (1 to m) film thicknesses, constituent fractions, parameters of optical function models, etc. Minimize 2 N [ρ ( λ ) ρ ( λ , )] − z 1 j calc j exp χ2 = å N − m − 1 j =1 σ (λ j ) 2 σ(λ λ) = random and systematic error Fitting Models to Data Calculation Procedure: 1) 2) 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. χ2>>1), go back to 1). 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 (2nd ed., Cambridge, 1992), Ch. 15, pg. 650. Fitting Models to Data Levenberg-Marquardt algorithm: Curvature matrix: 1 ∂2χ 2 α kl = = 2 ∂ z k ∂z l ∂ρ calc ( λ j , z ) ∂ρ calc ( λ j , z ) 1 å 2 ∂z k ∂z l j =1 σ ( λ j ) N Inverse of curvature matrix: ε = α-1 Error in parameter zj = εjj. Cross correlation coefficients: proportional to the elements of ε. Fitting Models to Data Meaning of fitted parameters and errors: Assume air/SiO2/Si structure Parameterize SiO2 using Sellmeier dispersion (λ λo=93 nm): ε = n2 = 1+ å j A j λ2 λ2 − λ2o , j Two fit parameters: df, Af Af • • • • • •• • • • • •• •• •• ••• ••• • • • • σΑ • •••••• •••••••• •• • • • • •• • •• •• •• •• •• • •• • • • • • σd df Fitting Models to Data An Example: a-SixNy:H on silicon: 3 SiN / c-Si 2 Re(ρpp) 1 0 -1 2 χ =0.96 -2 T-L -3 Lorentz χ =3.84 -4 F&B 2 2 χ =47.8 3 Im(ρpp) 2 1 0 -1 -2 1 2 3 Energy (eV) 4 5 Fitting Models to Data An Example: a-SixNy:H on silicon: Model: 1) 2) air surface roughness Bruggeman EMA (50% air, 50% a-SiN) 3) a-SiN (3 models) Lorentz Forouhi and Bloomer Tauc-Lorentz 4) interface Bruggeman EMA (50% Si, 50% a-SiN) 5) silicon Fitting parameters: d2, d3, d4, A, Eo, Γ, ε(∞ ∞) and Eg (F&B and T-L) Fitting Models to Data An Example: a-SixNy:H on silicon: Lorentz F&B T-L 2.1±0.3 4.9±0.7 1.9±0.3 Film thickness (nm) 197.8±0.7 195.6±1.1 198.2±0.4 Interface thick (nm) 0.6±0.3 -0.6±0.6 -0.1±0.2 A 201.9±4.6 4.56±1.9 78.4±12.7 Εo (eV) 9.26±0.05 74.4±30.5 8.93±0.47 Γ (eV) 0.01±0.02 0.74±23.5 1.82±0.81 Eg (eV) ---- 2.85±0.48 4.35±0.09 1.00±0.02 0.93±0.51 1.38±0.26 3.64 47.5 0.92 2.4±0.3 4.7±0.6 1.8±0.2 Film thickness (nm) 198.6±0.4 195.3±0.9 198.1±0.3 A 202.5±1.4 5.03±0.35 97.7±3.2 Εo (eV) 9.26±0.02 70.7±2.2 9.61±0.0.03 Γ (eV) 0.01±0.01 40.1±10.8 3.07±0.33 Eg (eV) ---- 2.97±0.27 4.44±0.04 χ2 3.75 48.0 0.96 Roughness thick (nm) ε(∞) χ2 Roughness thick (nm) Fitting Models to Data An Example: a-SixNy:H on silicon: 0.2 exp-calc Re(ρpp) 0.1 0.0 T-L Lorentz F&B -0.1 -0.2 0.2 Im(ρpp) 0.1 0.0 -0.1 -0.2 1 2 3 Energy (eV) 4 5 Optical Functions from Ellipsometry refractive index (n) Optical Functions from Parameterization: T-L Lorentz F&B 2.1 2.0 1.9 extinction coefficient (k) 1.8 0.015 0.010 0.005 0.000 1 2 3 4 5 Energy (eV) Error limits: Use the submatrix αs from the associated fitted parameters. Optical Functions from Ellipsometry Newton-Raphson algorithm: Solve: λ, φ, nf, kf, …)) - Re(ρ ρexp(λ Re(ρ ρcalc(λ λ)) = 0 Im(ρ ρcalc(λ λ, φ, nf, kf, …)) - Im(ρ ρexp(λ λ)) = 0 Jacobian: æ ∂ ρ re ç J = ç ∂n çç ∂ ρ im è ∂n nnew = nold + δn; Propagate errors! ∂ ρ re ∂k ∂ ρ im ∂k ö ÷ ÷ ÷÷ ø where δn = -J-1 ρ Optical Functions from Ellipsometry Optical functions of semiconductors: Dielectric function from air/substrate system: ε = ε 1 + i ε 2 = sin 2 (φ ){1 + [ 1− ρ 2 ] tan 2 (φ )} 1+ ρ Only valid if there is no overlayer (almost never true) If there is a thin film, Drude showed: 2 4π d n f − 1 ∆ = ∆ o (ns , k s ) + K λ n 2f Optical Functions from Ellipsometry Optical functions of semiconductors: Pseudo-dielectric functions of silicon with 0, 0.8, and 2.0 nm SiO2 overlayers. 8 0.0 0.8 2.0 <n> 6 4 2 0 6 <k> 4 2 0 10 10 10 10 6 < α > (1/cm ) 5 4 3 1 2 3 Energy (eV) 4 5 Optical Functions from Ellipsometry Optical functions of semiconductors: Error limits of the dielectric function of silicon: 0.05 δn δk Error in n and k 0.04 0.03 0.02 0.01 0.00 1 2 3 Energy (eV) 4 5 Optical Functions from Ellipsometry Optical functions of thin films: Sm all grain poly silicon Re(ρ) 1 0 -1 Im(ρ) 0.5 0.0 -0.5 1 2 3 Energy (eV) 4 5 Optical Functions from Ellipsometry Optical functions of thin films: Method of analysis: A. Restrict analysis region to interference oscillations. Parameterize the optical functions of the film. B. 1 air 2 surface roughness (BEMA) 3 T-L model for film 4 Lorentz model for a-SiO2 5 c-Si Fit data to determine thicknesses and Lorentz model parameters of a-SiO2. C. Calculate optical functions of thin film using Newton-Raphson. Optical Functions from Ellipsometry Optical functions of thin films: 8 n 6 4 2 6 k 4 2 0 10 10 10 10 6 α (1/cm) c-Si lg p-Si sg p-Si a-Si 5 4 3 1 2 3 Energy (eV) 4 5 Parting Thoughts SE 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, SE gives very accurate thicknesses and values of the complex dielectric function.