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1. Introduction & Theory 1. Introduction & Theory Neha Singh October 2010 Course Overview Day 1: Day 2: Introduction and Theory Genosc Layer Transparent Films Absorbing Films Microstructure – EMA If time permits: – Surface roughness Non-idealities – Grading (Simple and Ultra thin films function-based ITO) Uniqueness test – Thickness non-uniformity UV Absorption Review – Point-by-point fit Actual Samples © 2010, All Rights Reserved 2 Introduction & Theory Light Materials (optical constants) Interaction between light and materials Ellipsometry Measurements Data Analysis © 2010, All Rights Reserved 3 Light Electromagnetic Plane Wave From Maxwell’s equations we can describe a plane wave ⎛ 2π ⎞ E(z,t) = E0 sin⎜ − (z − vt) + ξ ⎟ ⎝ λ ⎠ Amplitude Amplitude arbitraryarbitrary phase phase X Wavelength Wavelength VelocityVelocity λ Electric field E(z,t) Y Z Direction Magnetic field, B(z,t) of propagation © 2010, All Rights Reserved 4 Intensity and Polarization Intensity = “Size” of Electric field. I ∝ E 2 Polarization = “Shape” of Electric field travel. Different Size Y •Y E More Intense Less (Intensity) Intense E Same Shape! X (Polarization) •X © 2010, All Rights Reserved 5 What is Polarization? Describes how Electric Field travels through space and time. X wave1 Y E wave2 Z © 2010, All Rights Reserved 6 Describing Polarized Light Jones Vector Stokes Vector Describe polarized light Describe any light beam with amplitude & phase. as vector of intensity ⎡S ⎤ ⎡ E2 + E2 ⎤ iϕx 0 x0 y0 ⎡Ex ⎤ ⎡E0xe ⎤ ⎢ ⎥ ⎢ 2 2 ⎥ = S1 ⎢ Ex0 −Ey0 ⎥ ⎢ ⎥ ⎢ iϕy ⎥ ⎢ ⎥ = E E e ⎢ ⎥ ⎢ ⎥ ⎣ y ⎦ ⎣⎢ 0y ⎦⎥ S2 2Ex0Ey0 cosΔ ⎢ ⎥ ⎢ ⎥ ⎣S3 ⎦ ⎣⎢2Ex0Ey0 sinΔ⎦⎥ © 2010, All Rights Reserved 7 Light-Material Interaction velocity & c wavelength vary v = in different n materials n = 1 •n = 2 Frequency remains constant v υ = λ © 2010, All Rights Reserved What are Optical Constants n , k Describe how materials and light interact. Complex Refractive Index : ñ= n+ ik – Describes how material changes the light wave. Alternatively … Complex Dielectric Function: ε = ε1+ iε2 – Describes how light changes the material. ε = ñ2 © 2010, All Rights Reserved 9 Complex Refractive Index ñ(λ) = n(λ) + ik(λ) n and k vary with wavelength n = “Refractive Index” – phase velocity = c/n – direction of propagation (refraction angle) k = “Extinction Coefficient” – Loss of wave energy to the material. Intensity is “Extinguished.” Io −α z I (z) = Ioe I(z) 4πk(λ) α(λ) = λ Dp z © 2010, All Rights Reserved Light at interface Interaction Reflections caused by index difference: Normal incidence: Oblique incidence 2 ()nn12− Reflection φ = φ R = i r 2 Refraction ()nn12+ Ñ 1 sin φ1 = Ñ 2 sin φ2 index, Ñ1 velocity, c φ1 φ1 index, Ñ2 velocity, v φ2 © 2010, All Rights Reserved 11 Polarized light at interface Electric field either parallel Ep Ep (p) or perpendicular (s) to plane of incidence. Es Es Material differentiates Ep between p- and s- light plane of incidence Es 1.0 Rp 0.8 Rs 0.6 0.4 Reflection 0.2 0.0 0 20 40 60 80 100 Angle of Incidence (°) © 2010, All Rights Reserved 12 Fresnel Coefficients Maxwell’s Equations at boundary conditions. Describes reflection and transmission rs,p n of polarized light (p or s). i Depends on angle, n(λ),k(λ) n t ts,p ⎛ E ⎞ n cosθ − n cosθ ⎛ E ⎞ 2n cosθ r = ⎜ r ⎟ = t i i t t = ⎜ t ⎟ = i i p ⎜ E ⎟ n cosθ + n cosθ p ⎜ E ⎟ n cosθ + n cosθ ⎝ i ⎠ p i t t i ⎝ i ⎠ p i t t i ⎛ E ⎞ n cosθ − n cosθ ⎛ E ⎞ 2n cosθ r = ⎜ r ⎟ = i i t t t = ⎜ t ⎟ = i i s ⎜ E ⎟ n cosθ + n cosθ s ⎜ E ⎟ n cosθ + n cosθ ⎝ i ⎠s i i t t ⎝ i ⎠s i i t t © 2010, All Rights Reserved What does Ellipsometry measure? Measures change in polarization of reflected light. ~ ~ out ~ in r E E Eout p / Ein p p = p p = ei()δ p −δ s = tan(ψ )eiΔ = ρ ~ ~ out ~ in rs Es Es Eout s / Ein s E p-plane s-plane p-plane E plane of incidence s-plane © 2010, All Rights Reserved 14 Optical constants from Ellipsometry Change in polarization is related to sample properties. ~ n cosθ − n cosθ r r ~ t i i t s,p p = tan Ψ eiΔ = ρ r = n ~ () p n cosθ + n cosθ i rs i t t i nt n cosθ − n cosθ ts,p ~r = i i t t s n cosθ + n cosθ i i t t n cosθ − n cosθ n cosθ − n cosθ ρ = ( t i i t )/( t i i t ) ni cosθt + nt cosθi ni cosθt + nt cosθi 2 For ni=1 ⎡ ⎤ ~ 2 2 2 ⎛1− ρ ⎞ ε = n = sin ()θi ⋅ ⎢1+ tan ()θi ⋅⎜ ⎟ ⎥ Easy! ⎣⎢ ⎝1+ ρ ⎠ ⎦⎥ © 2010, All Rights Reserved 15 Thin Film on Substrate Multiple reflections in single film lead to an infinite series -2iβ 2 -4iβ rtot = r01 + t01r12t10e + t01r12 r10t10e +... −i2β r t r t t r r r t r01( p,s) +r12(p,s)t01(p,s)t10(p,s)e 10 01 12 10 01 12 10 12 10 rtot( p,s) = −i2β No 1−r01(p,s)r12(p,s)e N1 FILM PHASE THICKNESS N2 t01t12 t01r12r10t12 t01r12r10r12r10t12 ⎛ d1 ⎞ β = 2π ⎜ ⎟n1 cosθ1 ⎝ λ ⎠ © 2010, All Rights Reserved Jones and Mueller Matrix Describe how sample changes polarization of light Light in Light out ⎛ So _ out ⎞ ⎡m11 m12 m13 m14 ⎤ ⎛ So _in ⎞ ⎜ ⎟ ⎢ ⎥ ⎜ ⎟ ⎡E ⎤ ⎡ j j ⎤⎡E ⎤ ⎜ S1_ out ⎟ m m m m ⎜ S1_ in ⎟ x−out = 11 12 x−in = ⎢ 21 22 23 24 ⎥ ⋅ ⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎜ S ⎟ ⎜ S ⎟ E j j E 2_ out ⎢m31 m32 m33 m34 ⎥ 2_in ⎣ y−out ⎦ ⎣ 21 22 ⎦⎣ y−in ⎦ ⎜ ⎟ ⎢ ⎥ ⎜ ⎟ ⎜ S ⎟ ⎜ S ⎟ ⎝ 3_ out ⎠ ⎣m41 m42 m43 m44 ⎦ ⎝ 3_in ⎠ © 2010, All Rights Reserved 17 What is Ellipsometry? Measures change in polarization of reflected light. Ψ, Δ = function of ( n, k, d, λ, θ,…) E p-plane s-plane E p-plane plane of incidence s-plane © 2010, All Rights Reserved 18 Ellipsometer Components Measurement Example: Rotating Analyzer Ellipsometer S θ Rotating γ Polarizer sample analyzer A(t) = ωt = 2πft P Detector converts light to voltage V(t) Modulation ↔γ ~out ~in Ep Ep = tan()Ψ eiΔ θ t ~out ~in Es Es © 2010, All Rights Reserved Rotating Analyzer Limitations: Δ ‘handedness’ is not determined. Uncertainty in Δ near 0°, 180°. ⎡ 1 − N 0 0⎤ ⎢ ⎥ ⎢− N 1 0 0⎥ ⎢ 0 0 C S ⎥ ⎢ ⎥ ⎣ 0 0 − S C⎦ N = cos(2Ψ) C = sin()()2Ψ sin Δ S = sin()()2Ψ cos Δ No measure of S © 2010, All Rights Reserved 20 Rotating AnalyzerVASE with AutoRetarder ® Ellipsometer Light Source P Polarizer AR AutoRetarder © 2010, All Rights Reserved S Sample A Analyzer Detector 21 AutoRetarderTM AutoRetarderTM changes polarization delivered to sample for optimum measurement condition. ⎡ 1 − N 0 0⎤ ⎢ ⎥ ⎢− N 1 0 0⎥ ⎢ 0 0 C S ⎥ ⎢ ⎥ ⎣ 0 0 − S C⎦ Use it! © 2010, All Rights Reserved 22 Analysis What can SE determine? Properties of Interest: Ellipsometry Measures: Film Thickness Refractive Index Psi (Ψ) Surface Roughness Delta (Δ) Interfacial Mixing Composition Crystallinity Anisotropy Uniformity © 2010, All Rights Reserved 23 Data Content versus Sample Unknowns Ensure data contains information to solve all unknowns. SAMPLE UNKNOWNS DATA CONTENT Over-Determined DATA CONTENT SAMPLE UNKNOWNS DATA CONTENT SAMPLE UNKNOWNS Under-Determined © 2010, All Rights Reserved 24 Direct Solution Single reflection can be directly ‘inverted’ to get n,k. 2λ 2λ DATA UNKNOWNS Ag Experimental Data 8 1.8 Exp <k>-E 70° Exp <n>-E 70° 1.5 6 1.2 < n > k 4 0.9 > Ψ,Δ < φ 0.6 2 0.3 n,k 0 0.0 200 400 600 800 1000 Wavelength (nm) Single Interface Inversion equation: ⎡ 2 ⎤ ~ ~ 2 2 2 ⎛1− ρ ⎞ iΔ rp ε = n = sin ()φ ⋅ ⎢1+ tan ()φ ⋅⎜ ⎟ ⎥ where ρ = tan(Ψ)e = ⎜1+ ρ ⎟ ~ ⎣⎢ ⎝ ⎠ ⎦⎥ rs © 2010, All Rights Reserved 25 Regression Analysis Interference n,k film t Substrate For most samples - “inverse” problem Result is known instead of Cause. Ellipsometry Sample Structure Measurement Experimental Data 90 180 2 srough 20.00 Å 80 160 Exp -E 73° Ψ Δ 70 Exp Δ-E 73° indegrees 140 60 1 film 1500.00 Å 120 50 in degrees 100 40 0 silicon 1 mm Ψ 30 80 Yes 20 60 200 400 600 800 1000 Wavelength (nm) © 2010, All Rights Reserved Data Analysis Flowchart © 2010, All Rights Reserved 27 1. Measure Sample Collect Ψ,Δ versus angle versus wavelength © 2010, All Rights Reserved Data Acquisition Example: VASE® data acquisition parameters: © 2010, All Rights Reserved Ellipsometry Measurements Repeatable & accurate: – Self-referencing, measures ratio of Ep/Es Thus, reduced problems with: • fluctuation of source intensity • light beam spilling over small samples Measure two parameters Psi and Delta – increased sensitivity to multiple film parameters © 2010, All Rights Reserved 2. Build Model Propose a layered structure Describes Thickness and Optical Constants of all layers t2 n,k (film 2) t1 n,k (film 1) n,k (substrate) © 2010, All Rights Reserved 31 3. Generated data Generated and Experimental Calculate response from 80 Model Fit Exp E 70° model 60 Exp E 75° 40 in degrees Ψ 20 Compare to Experimental 0 data. 300 600 900 1200 1500 1800 Wavelength (nm) Generated and Experimental 80 60 Adjust unknown (fit) parameters to get close to 40 in degrees Model Fit Ψ Exp E 70° solution. 20 Exp E 75° 0 300 600 900 1200 1500 1800 Wavelength (nm) © 2010, All Rights Reserved 32 4. Data Fit •100 Software adjusts “fit” •80 •60 parameters to find ••40 in degrees Ψ • best match •20 •0 between model •200 •400 •600 •800 •1000 and experiment. •Wavelength (nm) MSE is Difference. MSE Thickness © 2010, All Rights Reserved Mean Squared Error Mean Squared Error (MSE) used to quantify the difference between experimental and model- generated data.
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