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

Light Materials (optical constants) Interaction between light and materials Ellipsometry Measurements Data Analysis

From Maxwell’s equations we can describe a plane wave π ⎛ 2λ ⎞ E(z,t) = E0 sin⎜ − (z − vt) + ξ ⎟ ⎝ ⎠ 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

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

Describes how Electric Field travels through space and time.

X wave1

Y E wave2 Z

Jones Vector Stokes Vector

Describe polarized light Describe any light beam with amplitude & phase. as vector of intensity

2 2 ϕ ⎡S ⎤ ⎡ E + E ⎤ i x 0 x0 y0 ⎡Ex ⎤ ⎡E0xe ⎤ ⎢ ⎥ ⎢ 2 2 ⎥ = S1 ⎢ Ex0 −Ey0 ⎥ ⎢ ⎥ ⎢ iϕy ⎥ ⎢ ⎥ = E E e ⎢ ⎥ ⎢ ⎥ ⎣ y ⎦ ⎣⎢ 0y ⎦⎥ S2 2Ex0Ey0 cosΔ ⎢ ⎥ ⎢ ⎥ ⎣S3 ⎦ ⎣⎢2Ex0Ey0 sinΔ⎦⎥

velocity & c wavelength vary v = in different n materials

n = 1 •n = 2

Frequency remains constant v υ = λ

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

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.”

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

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 (°)

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 θ θ ⎛ E ⎞ n cosθ − n cosθ ⎛ E ⎞ 2θn cos r = ⎜ r ⎟ = t i i t t = ⎜ t ⎟ = i i p ⎜ E ⎟ n cos + n cosθ p ⎜ E ⎟ n cos + n cosθ ⎝ i ⎠ p i t t i ⎝ i ⎠ p i t t i θ θ ⎛ E ⎞ n cosθ − n cosθ ⎛ E ⎞ 2θn cos r = ⎜ r ⎟ = i i t t t = ⎜ t ⎟ = i i s ⎜ E ⎟ n cos + n cosθ s ⎜ E ⎟ n cos + n cosθ ⎝ i ⎠s i i t t ⎝ i ⎠s i i t t

~ ~ ~ out in δ ψ r E E Eout p / Ein p p = p p = ei()p −δ s = tan( )eiΔ = ρ ~ ~ out ~ in rs Es Es Eout s / Ein s

E p-plane

s-plane p-plane E

plane of incidence s-plane

Change in polarization is related to sample properties.

~ n cosθ − n cosθ r r ~ t θi i t s,p p = tan()Ψ eiΔ = ρ r = n ~ p n cos + n cosθ i rs i t t i

nt n cosθ − n cosθ ts,p ~r = i i t t s n cos + n cosθ i i t t

n cosθε − n cosθ n cosθ − n cosθ ρ = ( t i i t )/( t i i t ) n cos + n cosθ n cos + n cosθ i t t i θi t t i

θ 2 For ni=1 ⎡ ρ ⎤ ~ 2 2 2 ⎛1− ⎞ = n = sin ()i ⋅ ⎢1+ tan i ⋅⎜ ()⎟ ⎥ Easy! ⎣⎢ ⎝1+ ρ ⎠ ⎦⎥

Multiple reflections inβ single film lead to an infinite series

-2i 2 -4iβ rtot = r01 + t01r12t10e + t01r12 r10t10e +...

−i2 r t r t t r r r t r01( p,s) +r12(p,s)t01(p,s)t10(p,s)e β 10 01 12 10 01 12 10 12 10 rtot( p,s) = −i2β No 1−r01(p,s)r12(p,s)e

N1 β FILM PHASEπ THICKNESS N2 t01t12 t01r12r10t12 t01r12r10r12r10t12 ⎛ dλ1 ⎞ = 2 ⎜ ⎟n1 cosθ1 ⎝ ⎠

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 ⎠

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

Example: Rotating Analyzer Ellipsometer

θ Rotating γ Polarizer sample analyzer A(t) = ωt = 2πft P

Detector converts light to voltage V(t)

Modulation ↔γ ~ ~ E out Ein p p = tan()Ψ eiΔ θ t ~out ~in Es Es

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

Rotating Analyzer with AutoRetarder

P AR S A Detector

r r r e e le e iz d p z r r ly la ta m a Light Source e a n Po R S A to u A

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!

Properties of Interest: Ellipsometry Measures: Film Thickness

Refractive Index Psi (Ψ) Surface Roughness Delta (Δ) Interfacial Mixing Composition

Crystallinity

Anisotropy

Uniformity

Ensure data contains information to solve all unknowns.

SAMPLE UNKNOWNS

DATA CONTENT Over-Determined

DATA CONTENT SAMPLE UNKNOWNS

DATA CONTENT

SAMPLE UNKNOWNS Under-Determined

2λ 2λ DATA UNKNOWNS Ag Experimental Data 8 1.8 Exp -E 70° Exp -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 ε Wavelength (nm) Single Interface Inversion equation: φ

⎡ φ ρ 2 ⎤ ~ ~ 2 2 2 ⎛1− ⎞ iΔ rp = n = sin ()⋅ ⎢1+ tan ⋅⎜ () ⎟ ⎥ where ρ = tan(Ψ)e = ⎜1+ ρ ⎟ ~ ⎣⎢ ⎝ ⎠ ⎦⎥ rs

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)

Collect Ψ,Δ versus angle versus wavelength

– 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

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)

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)

•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

Mean Squared Error (MSE) used to quantify the difference between experimental and model- generated data.

σ 2 2 1 N ⎡⎛ Ψ mod − Ψexp ⎞ ⎛ Δmod − Δexp ⎞ ⎤ MSE ⎢⎜ i i ⎟ ⎜ i i ⎟ ⎥ = ∑ exp + exp 2N − M ⎢⎜ ⎟ ⎜ σ ⎟ ⎥ i=1 ⎣⎝ Ψ,i ⎠ ⎝ Δ,i ⎠ ⎦

A smaller MSE implies a better fit. There is no target (best) MSE value.

MSE starting thickness (guess)

Local Minima

BEST Thickness FIT

* W.H. Press et al., Numerical Recipes in C, Cambridge, UK: Cambridge University Press, 1988.

Compare experimental and generated data How low is MSE? Can it be reduced further by increasing model complexity? Are fit parameters physical? – Normal dispersion, K-K consistency Check other mathematical “goodness of fit” indicators – Correlation matrix – Uniqueness Test – Error bars

Investigate further to ensure unique. – Adjust fit values by 10-20%, do they return to same result? – Can one of the correlated parameters be fixed at nominal value and still get good MSE?

Find the simplest optical model that fits Experimental Data.

Verify uniqueness of the model.

Optical 'constants' for materials are not always constant, and quality of fit can only be as good as the optical constants assumed in the model.

Model: Layered structure including current results and fit parameters.

Material File: Optical constants or dispersion parameters for layer

Environment: Everything shown on the screen. – Can be sensitive to software version.

Demo1_SiO2 on Si – Use published tabulated values for n,k.

CTRL-D: Defaults CTRL-R: Range-Select Data CTRL-G: Generate CTRL-F: Normal Fit CTRL-T: Toggle Through Graphs SHIFT-CTRL-T: Toggle graphs in reverse order.

1. Hiroyuki Fujiwara, Spectroscopic Ellipsometry: Principles and Applications, John Wiley & Sons, 2007.

2. Handbook of Ellipsometry, Tompkins and Irene, eds., William Andrew Publishing, NY, 2005.

3. H. G. Tompkins, and W.A.McGahan, Spectroscopic Ellipsometry and Reflectometry, John Wiley & Sons, New York, 1999.

4. H. G. Tompkins, A User’s Guide to Ellipsometry, Academic Press, San Diego, 1993.

5. R.M.A. Azzam, and N.M.Bashara, Ellipsometry and Polarized Light, North Holland Press, Amsterdam 1977, Second edition, 1987.

ICSE Conference Proceedings:

1. Thin Solid Films Vol. 455-456, (2004) M. Fried, K. Hingerl, and J. Humlicek, Editors, Elsevier Science.

2. Thin Solid Films Vol. 313-314 (1998) R.W.Collins, D.E.Aspnes, and E.A. Irene, Editors, Elsevier Science.

3. Spectroscopic Ellipsometry, A.C.Boccara, C.Pickering, J.Rivory, eds, Elsevier Publishing, Amsterdam, 1993.

The following slides provide additional details pertaining to this section. Specifically, they cover instrumentation in more detail.

Wavelengths (Range and Number)? – Wavelengths of interest? – Where is film transparent? – Film Thickness? – Sharp features in data? Angles? – What are Substrate and Films? – Single or Multilayers? – Complex materials?

Film Steps Steps (nm) Experimental Data Thickness (eV) 100 Exp E 65° < 200 nm 0.1 eV 20 nm Data every 2nm Exp E 75° 80 200 - 500 nm 0.05 eV 10 nm 60 500 nm - 1 40

0.025 eV 5 nm in degrees

μm Ψ 20 1 -3 μm 0.01eV 2 nm 0 0 300 600 900 1200 1500 1800 2 nm, Wavelength (nm) >3 μm Long wavelengths 2.5 μm Oxide

One angle is often 80° GaAs* Ge* * Si* 75° InP sufficient, but more ZnSe* SiC* 70° Si3N4 angles helps with * Ta2O5 Angle 65° ITO* * confidence. Al2O3 60° * SiO2 * TiO2 * When choosing 55° * approx. value of H2O N@ λ=650nm 50° multiple angles, 1 1.5 2 2.5 3 3.5 4 4.5 5 best to have one Refractive Index

above, one below, Generated for n=3.5, k=0 and one near 1.0 Rp 0.8 Brewster Angle Rs 0.6

0.4 Reflection

0.2

Typical Angle Combinations: Angle of (spot-length)/ » Thin films on Si: 65°, 75° Incidence (beam-dia.) » Thick films on Si: 60°, 75° 25° 1.1 35° 1.2 or 55, 65°, 75° 45° 1.4 » n-matched films on glass: 55°, 56.5°, 58° 55° 1.7 » Other films on glass: 50°, 60°, 70° 65° 2.4 75° 3.9 » Films on metals: 65°, 75° 80° 5.8 » Anisotropic & Graded films: 55°, 65°, 75° 85° 11.5 or 45°, 60°, 75° Spot size vs. angle

P ola G riza ion Light ene tio rizat r n ola er Detector Source ator P lyz Ana

Sample

SE also needs wavelength selection.

Polarizer: only allows linear polarization to exit. Compensator: retard orthogonal electric fields by 90°.

X Polarizer E Axis Y Compensator E Axis

Linearly Polarized

Rotating S Polarizer sample analyzer A(t) = ωt = 2πft γ θ

Detector P converts light to voltage V(t)

Modulation ↔γ

θ t

V(t) V(t) α cos(2ωt) DC Modulation ↔γ β sin(2ωt)

θ t t

FromFrom Jones Jones Matrix Matrix analysis analysis of of the the RAE RAE optical optical system: system: V(t)V(t) = = DC DC + + α αcos(2cos(2ωωt)t)+ + β βsin(2sin(2ωωt)t)

a tan2 Ψ - tan2 P α = DC = 2 2 α and β are normalized tan Ψ + tan P Fourier coefficients b 2 tan Ψ cos Δ tan P β = = DC tan2 Ψ + tan2 P

Polarizers can not distinguish unpolarized polarizer and circular polarizations.

polarizer

Compensators can measure unpolarized from compensator circular polarizations.

compensator

AutoRetarderTM changes polarization delivered to sample for optimum measurement condition.

© 2010, All Rights Reserved RCE P o lar te L Gen iza ta s er tion n S a S tio A) α tor ta a S (P te riz (P I RCE = dc + α 2 cos(2Cs )+ β 2 sin(2Cs )+ α 4 cos(4Cs)+ β 4 sin(4Cs ) P S ola er G) P lyz D na α A P ⋅C P C dc = 1+ = 1+ sin()()2Ψ cos Δ A 2 2 ()() 2 = −P ⋅ S = −P sin 2Ψ sin Δ β 2 = 0 P ⋅C P P ⋅ N P SAMPLE = − = − sin()()2Ψ cos Δ β = = cos()2Ψ 4 2 2 4 2 2

For isotropic (depolarizing) sample: To calculate Ψ & Δ ⎡ 1 − N m 0 0 ⎤ ⎡ 1 − PN 0 0 ⎤ ⎢− N 1 0 0 ⎥ ⎢− PN P 0 0 ⎥ -α /α yields tan(Δ) ⎢ m ⎥ ⎢ ⎥ 2 4 M sample = = ⎢ 0 0 Cm S m ⎥ ⎢ 0 0 PC PS ⎥ ⎢ ⎥ ⎢ ⎥ -β4, α2, or α4 yields Ψ ⎣ 0 0 − S m C m ⎦ ⎣ 0 0 − PS PC ⎦ -dc required to calculate depolarization N = cos()2Ψ C = sin()()2Ψ sin Δ ()() S = sin 2Ψ cos Δ P = N 2 + C 2 + S 2 ≤ 1 %depolarization = (1− P)×100%