Session 1A Lincoln 2004-09

J.A. Woollam Co., Inc.© Sept 20, 2004 All Rights Reserved

Short course onShort course on

Data Analysis for Data Analysis for Spectroscopic EllipsometrySpectroscopic EllipsometrySession 1A: Introduction to EllipsometrySession 1A: Introduction to Ellipsometry

Tom TiwaldTom TiwaldLincoln, NebraskaLincoln, Nebraska

September 20 September 20 –– 23, 200423, 2004

2Session 1A, Basics and TheoryJ.A. Woollam Co., Inc.© Sept 20, 2004 All Rights Reserved

Course OverviewCourse OverviewDay 1: Introduction to Ellipsometry

Basic theoryOptical ConstantsMeasure, Model, FitSubstrates

Day 2: Ellipsometry AnalysisTransparent LayersGlobal FittingUV absorbing filmsMultilayersThin Metal filmsEMA models

Day 3: Intermediate ModelingDispersion Model TheoryGeneralized Oscillator ModelAbsorbing Films

Day 4: Advanced Topics & ReviewNon-idealitiesGradingAnisotropyReview


Optical CharacterizationOptical CharacterizationIntensity-Based– Reflectance– Transmittance– Absorbance– Scatterometry

Polarization-Based– Polarimetry– Ellipsometry (reflected or transmitted)

Complete Description– Mueller-matrix

Measure how Light is affected by interaction

with materials


IntensityIntensity--based Measurementsbased Measurements

T = It / I1R = Ir / I1A = 1 - R - TAdvantages:– often directly of interest.– Simple System Design

Disadvantages:– Low information content– Not sensitive to ultra-thin films– Accuracy affected by scattering, pinholes,

misalignment, baseline or reference wafer.

I1 Incident Light Ir Reflected Light

reflects / transmitsat each interface

can be absorbedby material

It Transmitted Light


What is Ellipsometry?What is Ellipsometry?

plane of incidence

E

E

p-plane

p-plane

s-plane

s-plane

Measures the polarization change (Ψ and ∆) when light reflects from a surface.

s

pi

RR

eρ ~~

)tan( =Ψ= ∆

Substrate

film

Interference

tn,kCan determine optical constants and film thickness (n,k,t).


What can Ellipsometry measure?What can Ellipsometry measure?Ellipsometry Measures: Properties of Interest:

Film ThicknessRefractive Index

Surface RoughnessInterfacial Mixing

CompositionCrystallinityAnisotropyUniformity

Psi (Ψ)Delta (∆)


Data Analysis FlowchartData Analysis Flowchart


Session 1A OverviewSession 1A OverviewMeasurements– Light and Polarization– Ellipsometry– Instrumentation – Measurement considerations

Interaction between light and materials– Optical Constants– Fresnel Coefficients– Thin Films– Data Interpretation

Modeling & Fitting– Why Analyze Data?– Building a Model– Regression– Mean Squared Error– Best Fit?


A. Light and PolarizationA. Light and Polarization

MeasurementsA. Light and Polarization

– Electromagnetic Plane Waves– Superposition of Waves– Polarization (Linear, Circular, Elliptical)– Polarization Descriptions

B. EllipsometryC. InstrumentationD. Measurement considerations


• Light• Ellipsometry• Instrumentation• MeasurementsElectromagnetic Plane WaveElectromagnetic Plane Wave

From Maxwell’s equations we can describe a plane wave

Directionof propagation

X

Y

Z

Electric field, E(z,t)

Magnetic field, B(z,t)

⎟⎠⎞

⎜⎝⎛ +−−= ξ

λπ )v(2sin),( 0 tzEtzE

velocityvelocity

arbitrary phasearbitrary phase

wavelengthwavelength

amplitudeamplitude


• Light• Ellipsometry• Instrumentation• MeasurementsLightLight

)(240,1)(nm

heVEλ

υ ≅=Planck’s constantPlanck’s constant

Describe as plane wave or quantized particle (photon)

Frequency remains constant

nc

=vvelocity & wavelength vary in different materials

λυ v

=

n = 1 n = 2


Electromagnetic SpectrumElectromagnetic Spectrum

Figure taken from P.W. Atkins Physical Chemistry, 5th

Ed., (W.H. Freeman, New York,1994), p. 541.

106 1015109 1012 1018 1021Freq (Hz)

10-10 10110-6 10-2 103 106Energy (eV)

• Light• Ellipsometry• Instrumentation• Measurements


• Light• Ellipsometry• Instrumentation• MeasurementsCombining WavesCombining Waves

Coherent waves with same frequency and traveling in the same direction combine into a wave of that same frequency.

Superposition of 4 waves - same frequency

1000 1500 2000 2500 3000

Am

plitu

de

-3.0

-2.0

-1.0

0.0

1.0

2.0

3.0

4.0

5.0ETOT = E1 + E2 + E3 + E4E1E2E3E4


• Light• Ellipsometry• Instrumentation• MeasurementsConstructive InterferenceConstructive Interference

Constructive interference– Two or more waves with

phase difference ~0°– coherent interference

summation yields maximum-amplitude wave

E2 lags E1 by 15°

-2.5-2

-1.5-1

-0.50

0.51

1.52

2.5

0 5 10 15 20Radians

Am

plitu

de

E1E2E1+E2

E2 lags E1 by 0°

00.5

11.5

22.5

Am

plitu

de

E1E2E1+E2

-2.5-2

-1.5-1

-0.5

0 5 10 15 20Radians


• Light• Ellipsometry• Instrumentation• MeasurementsDestructive InterferenceDestructive Interference

Destructive interference– Two or more waves with

phase difference ~180°– coherent interference

summation yields ~zero-amplitude wave

E2 lags E1 by 175°

-1.5

-1

-0.5

0

0.5

1

1.5

0 5 10 15 20Radians

Am

plitu

de

E1E2E1+E2

E2 lags E1 by 180°

-1.5

-1

-0.5

0

0.5

1

1.5

0 5 10 15 20Radians

Am

plitu

de

E1E2E1+E2


• Light• Ellipsometry• Instrumentation• MeasurementsIntensity vs. PolarizationIntensity vs. Polarization

Intensity = “Size” of Electric field.

Polarization = “Shape” of Electric field travel.

X

Y

ELess

Intense

X

Y EMore

Intense

2EI ∝

Different Size (DifferentIntensity)

Same Shape! (same

Polarization)


• Light• Ellipsometry• Instrumentation• MeasurementsPolarizationPolarization

Polarization state defined by orientation & phase of E-field vector

Any state of polarization can be described with superposition of multiple plane waves – Minimum number: two with orthogonal E-fields.

2 orthogonal linearly-polarized components2 circularly-polarized (left & right) components

X

Y

Z

wave1

wave2


• Light• Ellipsometry• Instrumentation• MeasurementsLinearly Polarized LightLinearly Polarized Light

Orthogonal EX & EY propagating in same direction: waves are in phase with each otherResult : linearly polarized wave– the 'plane of vibration' depends on relative amplitudes of Ex & EY

X

wave1

wave2Y

E


• Light• Ellipsometry• Instrumentation• MeasurementsCircularly Polarized LightCircularly Polarized Light

Orthogonal EX & EY: 90° out-of-phase & equal in amplitude with each otherResult : circularly polarized wave

X

Y

Z

wave1

wave2

E

E (t)y

tX

Y

Net E-fieldE (t)x

tLooking Down Z-Axis


• Light• Ellipsometry• Instrumentation• MeasurementsElliptically Polarized LightElliptically Polarized Light

X

Y

Z

wave1

wave2

E

Orthogonal EX & EY: Arbitrary phase & amplitude with each otherResult : Elliptically polarized wave– linear and circular are subsets of elliptical polarization– Most general description of polarization state


• Light• Ellipsometry• Instrumentation• MeasurementsEllipse of PolarizationEllipse of Polarization

Locus of points traced out by E-field vector– Can be described as superposition of two complex numbers:

( ) { }( ) { }y

x

itiyoyyoy

itixoxxox

eeEtEE

eeEtEE

defining

δω

δω

δω

δω

Recos~Recos~

=+=

=+=

X

Y E

yx iyo

ixoyx eEeEEE δδ +=+ ~~


• Light• Ellipsometry• Instrumentation• MeasurementsJones VectorsJones Vectors

Describes polarized light.If Light is traveling in z-direction:

Examples:⎥⎦

⎤⎢⎣

⎡=⎥

⎦

⎤⎢⎣

⎡x

x

iy

ix

y

x

eEeE

EE

ϕ

ϕ

0

0

• E0x & E0y - amplitudes• ϕx & ϕy - phases.


• Light• Ellipsometry• Instrumentation• MeasurementsJones MatrixJones Matrix

Jones Matrix Examples:Describes change in polarization caused by optical element (or sample)

Horizontal Linear Polarizer ⎥

⎦

⎤⎢⎣

⎡0001

Vertical Linear Polarizer ⎥

⎦

⎤⎢⎣

⎡1000

Linear Polarizer at +45° ⎥

⎦

⎤⎢⎣

⎡1111

21

Linear Polarizer at -45° ⎥

⎦

⎤⎢⎣

⎡−

−1111

21

Quarter-wave plate, fast axis vertical

⎥⎦

⎤⎢⎣

⎡− i

ei

0014/π

Quarter-wave plate, fast axis horizontal

⎥⎦

⎤⎢⎣

⎡i

ei

0014/π

Homogeneous circular Polarizerright

⎥⎦

⎤⎢⎣

⎡− 11

21

ii

Homogeneous circular Polarizerleft

⎥⎦

⎤⎢⎣

⎡ −1

121

ii

Light inLight out

⎥⎦

⎤⎢⎣

⎡⎥⎦

⎤⎢⎣

⎡=⎥

⎦

⎤⎢⎣

⎡

−

−

−

−

iny

inx

yyyx

xyxx

outy

outx

EE

jjjj

EE

⎥⎦

⎤⎢⎣

⎡

yyyx

xyxx

jjjj


Ratio description of polarization ellipseRatio description of polarization ellipse

Ellipse of polarization can also be described as (complex) ratio of Ex/Ey

X

Y

NOTE: total E-field amplitude (light intensity) information is lost.

( ) ( ) ∆− Ψ== ii

yo

xo

y

x eeEE

EE yx tan~~

δδ

( ) yxyo

xo

EE δδ −=∆=Ψ ,tan


• Light• Ellipsometry• Instrumentation• MeasurementsEllipse ratio examplesEllipse ratio examples


Examples of (complex) Ex/Ey ratios

( ) ( ) ∆− Ψ== ii

yo

xo

y

x eeEE

EE yx tan~~

δδ

X

Y

(45°,0°)(45°,180°)

(30°,0°)(30°,180°)

(Ψ,∆)

(90°,0°)

(0°,0°)

Ψ=30°

X

Y

Ψ=45°, ∆=90°

X

Y

(∆=130°)

(∆=30°)(∆=90°)


• Light• Ellipsometry• Instrumentation• MeasurementsGeometric descriptionGeometric description

Ellipse of polarization can also be described geometrically (ellipticity & orientation of major axis)

X

Y

θ

γ

( )axismajoraxisminor

axismajorofnorientatiodefining

=

=

γ

θ

tan

θγ

θγ

2sin2tantan

,2cos2cos2cos

=∆

−=Ψ

E-field ratio parameters (Ψ & ∆) can be related to geometrical parameters (γ & θ) by

Example: θ=45°, γ=30°

Ψ=45°, ∆=60°

Summary: Light & PolarizationSummary: Light & Polarization• Light• Ellipsometry• Instrumentation• Measurements


Light can be described as a plane wave.Polarization state of light beam can be described as superposition of two orthogonal light beamsPolarization can be linear, circular or elliptical.The Polarization Ellipse can be described as:– Sum two orthogonal electric field vectors, or– Using Jones vectors– Ratio of two electric field vectors– Geometrically (orientation angle θ & ellipticity angle γ)


B. EllipsometryB. Ellipsometry

MeasurementsA. Light and PolarizationB. Ellipsometry

– Plane of Incidence– Definitions– Advantages

C. InstrumentationD. Measurement considerations


• Light• Ellipsometry• Instrumentation• MeasurementsEEpp & E& Ess components definedcomponents defined

Electric Field Vectors

Plane-of-Incidence– incident– reflected– transmitted

s-waves and p-waves– “senkrecht” and “parallel”

plane ofincidence

Ep

Es

Ep

EsEp

Es

EsEp

Transmitted light

Reflected lightIncident light


• Light• Ellipsometry• Instrumentation• MeasurementsPolarized ReflectionPolarized Reflection

P- and S- waves do not mix (isotropic-case)Material differentiates between p- and s- light

Angle of Incidence (°)0 20 40 60 80 100

Ref

lect

ion

0.0

0.2

0.4

0.6

0.8

1.0

RpRs

Measures change in polarizationMeasures change in polarization• Light• Ellipsometry• Instrumentation• Measurements


Ellipsometry measures the change in polarization of light reflected (transmitted) from sample.– By determining complex ratio of output/input E-fields

plane of incidence

E

E

p-plane

p-planes-plane

s-plane

1. Known input polarization

2. reflect off sample ...

3. Measure output polarization

( ) ins

outs

inp

outpi

EEEE

e =Ψ= ∆tanρ


• Light• Ellipsometry• Instrumentation• MeasurementsEllipsometry measures...Ellipsometry measures...

Using Jones Matrix notation:

– where and are complex Fresnel reflection coefficients (more later).

⎟⎟⎠

⎞⎜⎜⎝

⎛⎟⎟⎠

⎞⎜⎜⎝

⎛=⎟⎟

⎠

⎞⎜⎜⎝

⎛ins

inp

s

pouts

outp

EE

rr

EE

~00~

( ) ( )spi

s

p

s

pins

outs

inp

outpi e

rr

rr

EEEE

e δδρ −∆ ===Ψ= ~~

tan

pr~ sr~

s

p

rr

=Ψ)tan( sp δδ −=∆


• Light• Ellipsometry• Instrumentation• MeasurementsEllipsometry MeasurementsEllipsometry Measurements

Repeatable & accurate: – self-referencing (single-beam experiment) ellipsometry

measures ratio of orthogonal light components Ep/EsThus, reduced problems with:

fluctuation of source intensitylight beam overlapping sides of small samples

Sensitive:– Phase term ∆ is very sensitive to film thickness

(Example on next page)

Measure two parameters


• Light• Ellipsometry• Instrumentation• MeasurementsEllipsometry vs. reflectivityEllipsometry vs. reflectivity

Phase information gives Ellipsometry much higher sensitivity to very thin films.

Wavelength (nm)200 400 600 800 1000

∆ in

deg

rees

0

30

60

90

120

150

180

Gen E 75° (1)Gen E 75° (2)

Generated Data

200 400 600 800 1000

Ψ in

deg

rees

0

10

20

30

40

Gen E 75° (1)Gen E 75° (2)

∆

Ψ

1nm & 2nm oxide on Si


Generated Data

Wavelength (nm)200 400 600 800 1000

Ref

lect

ion

0.30

0.40

0.50

0.60

0.70

0.80

Gen sR 0° (1)Gen sR 0° (2)

R



Summary Summary -- EllipsometryEllipsometry

Measures the change in light polarization caused by interaction with the sample.

This change can be described as two terms:

Ψ and ∆.

Measurement can be highly accurate, repeatable and sensitive to thin films.


C. InstrumentationC. Instrumentation

MeasurementsA. Light and PolarizationB. EllipsometryC. Instrumentation

– Light source, Spectrometer, Monochromator, Detector

– Polarizer and Compensator– System Configurations– Case Example: Rotating Analyzer Ellipsometer

D. Measurement considerations


• Light• Ellipsometry• Instrumentation• MeasurementsEllipsometersEllipsometers

Every Ellipsometer contains the following componentsSE also needs wavelength selection.

PolarizationGenerator Analyzer

Sample

LightSource Detector


• Light• Ellipsometry• Instrumentation• MeasurementsEllipsometer TypesEllipsometer Types

Rotating Analyzer (RAE)

Rotating Polarizer (RPE)

Rotating Compensator (RCE)

Polarization Modulation (PME)

Null Ellipsometer

Lig

ht S

ourc

e Detector

P

P

P

P

P

A

A

A

A

A

S

S

S

S

S

C

M

C


• Light• Ellipsometry• Instrumentation• MeasurementsSE Principles of OperationSE Principles of Operation

Example: Rotating Analyzer Ellipsometer

sample

Polarizeranalyzer

detectorsource

( ) ( )spi

s

pins

outs

inp

outpi e

rr

EEEE

e δδρ −∆ ==Ψ= tan


Operation of Rotating AnalyzerOperation of Rotating Analyzer

Linearly Polarized

t

V(t)

Detector Detector converts lightconverts light

to voltageto voltage

ftttA πω 2)( ==

samplePolarizerRotatinganalyzer

Linearly Polarized Light:- 100% modulated sinusoidal signal at

2x analyzer rotational frequency- light totally extinguished when

analyzer is ‘crossed’ with linearly polarized beam – i.e. signal =0

- Maximum signal at ωt = θ, (θ + 2π)− γ = 0 when modulation = 100%

(θ, γ from geometric description of polarization ellipse)

θ

P

S

γ=0

θ


Operation of Rotating AnalyzerOperation of Rotating Analyzer• Light• Ellipsometry• Instrumentation• Measurements


Circularly Polarized Light:- constant DC signal as the circularly

polarized beam always has equal component through rotating analyzer, regardless of orientation

− θ is undefined− γ = 45° when modulation = 0


Circularly Polarized

t

V(t)

ftttA πω 2)( ==




P

S


Operation of Rotating AnalyzerOperation of Rotating Analyzer

Linearly Polarized



ftttA πω 2)( ==


For any Polarized Light:- Maximum signal at ωt = θ, (θ + 2π)− γ = proportional to %modulation


P

S

θγ

t

V(t)

θ

Modulation ↔ γ



• Light• Ellipsometry• Instrumentation• MeasurementsHow RAE measures How RAE measures ΨΨ and and ∆∆

t

DC

α cos(2ωt)

β sin(2ωt)

V(t)

t

V(t)

θ

Modulation ↔ γ

α and β are normalizedFourier coefficients

From Jones Matrix analysis of the RAE optical system:

V(t) = DC + α cos(2ωt) + β sin(2ωt)

From Jones Matrix analysis of the RAE optical system:

V(t) = DC + α cos(2ωt) + β sin(2ωt)

α = a

DC = tan2 Ψ - tan2 Ptan2 Ψ + tan2 P

β = b

DC = 2 tan Ψ cos ∆ tan P

tan2 Ψ + tan2 P

Sensitivity of RAE EllipsometerSensitivity of RAE Ellipsometer• Light• Ellipsometry• Instrumentation• Measurements


Invert equations to get:

β & α are sources of noise:– Noise minimized when ψ = P (i.e., when α = 0)– Noise minimized when ∆ = 90° (i.e., when β = α = 0)– Noise in Delta becomes large when

∆ = 0° or ∆ = 180°

2α-1β)cos( =∆)Ptan(

α1α1)ψtan(

−+

=


• Light• Ellipsometry• Instrumentation• MeasurementsAutoRetarderAutoRetarderTMTM

AutoRetarderTM changes polarization delivered to sample for optimum measurement condition.


• Light• Ellipsometry• Instrumentation• MeasurementsAutoRetarderTMAutoRetarderTM


• Light• Ellipsometry• Instrumentation• MeasurementsRotating CompensatorRotating Compensator

Accurate over complete range– Psi = 0 to 90, Delta = 0 to 360

Minimizes source and detector polarization sensitivity

No DC signal required to measure Psi and Delta– DC signal allows

depolarization to be measured

– Stokes matrix can be measured

Until recently, lack of rotatablespectroscopic compensators has kept this technology from widespread use!

Until recently, lack of rotatablespectroscopic compensators has kept this technology from widespread use!

SAMPLE

P

Ls

C A

D


Summary Summary –– InstrumentationInstrumentation

Ellipsometer consists of light source, detector, wavelength selector, polarization generator, and polarization analyzer.

Polarizers convert light to linear polarization.

Compensators convert light to circular polarization.

Ellipsometer selects a “known” polarization to send toward sample and detects reflected/transmitted polarization: the change is due to the sample.


D. Measurement ConsiderationsD. Measurement Considerations

MeasurementsA. Light and PolarizationB. EllipsometryC. InstrumentationD. Measurement considerations

– Calibration– Choosing Angle of Incidence– Choosing Wavelength Range


• Light• Ellipsometry• Instrumentation• MeasurementsCalibration Calibration

Optical element orientations relative to plane of incidence –e.g. polarizer, analyzer, &

compensator azimuths

Optical element non-idealities (often done at factory set-up & testing)

Electronic signal delays

plane of incidence

calibration ...

Determining:

• Ps (polarizer Azimuth wrt to plane of incidence)

• As (analyzer Azimuth wrt to plane of incidence)

• Cs (compensator Azimuth wrt to plane of incidence)

• etc…

Calibration accurately determines:


• Light• Ellipsometry• Instrumentation• MeasurementsCalibrationCalibration

Ellipsometry is self referencing; therefore calibration does NOT depend on a specific “known” reference sampleOften a variety of sample types can be used for calibration.– Some are better than others.

Follow recommendations of instrument manufacturers.


• Light• Ellipsometry• Instrumentation• MeasurementsData Acquisition parametersData Acquisition parameters

Choose– What Spectral Range ?

Wavelengths of interest? Region of material transparency?Film Thickness?

– How Many Wavelengths ?How thick is the film?Are there sharp features in data?

– What angles and how many angles?What is the Substrate and Film Materials? Single or Multilayers?


• Light• Ellipsometry• Instrumentation• MeasurementsHow Many Wavelengths?How Many Wavelengths?

Resolve oscillations and features in data.

0 si_jaw 1 mm1 sio2_jaw 25000 Å

0 si_jaw 1 mm1 sio2_jaw 250 Å

Experimental Data

Wavelength (nm)0 300 600 900 1200 1500 1800

Ψ in

deg

rees

0

20

40

60

80

100Exp E 65°Exp E 75°

Experimental Data

Wavelength (nm)0 300 600 900 1200 1500 1800

Ψ in

deg

rees

0

10

20

30

40

50

60

Exp E 65°Exp E 75°

Data every 2nmData every 25nm


• Light• Ellipsometry• Instrumentation• MeasurementsFilm ThicknessFilm Thickness

1micron oxide on silicon

Wavelength (nm)200 400 600 800 1000

∆in

deg

rees

50

100

150

200

250

300

350

50nm oxide on silicon

200 400 600 800 1000

∆in

deg

rees

60

90

120

150

180

210

240

270 200nm oxide on silicon

200 400 600 800 1000

∆in

deg

rees

0

100

200

300

400

5 micron oxide on silicon

Wavelength (nm)200 400 600 800 1000

∆in

deg

rees

50

100

150

200

250

300


• Light• Ellipsometry• Instrumentation• MeasurementsWavelength Units?Wavelength Units?

Data collected in nanometers does NOT resolve short wavelength features as well.

,1240

nmeVE

λ=

Data collected every 12.4 nm (82 total wavelengths)

Wavelength (nm)200 400 600 800 1000 1200 1400

Ψ in

deg

rees

0

20

40

60

80

100

Exp E 75°

Photon Energy (eV)1.0 2.0 3.0 4.0 5.0

Ψ in

deg

rees

0

20

40

60

80

100

Exp E 75°

Wavelength (nm) Photon Energy (eV)


• Light• Ellipsometry• Instrumentation• MeasurementsWavelength Units? Wavelength Units?

Data collected in photon energy spread equally over interference structure.

Data collected every 0.05 eV (82 wavelengths)

Photon Energy (eV)1.0 2.0 3.0 4.0 5.0

Ψ in

deg

rees

0

20

40

60

80

100

Exp E 75°

Wavelength (nm)200 400 600 800 1000 1200 1400

Ψ in

deg

rees

0

20

40

60

80

100

Exp E 75°

Wavelength (nm) Photon Energy (eV)

Wavelength & Photon EnergyWavelength & Photon Energy• Light• Ellipsometry• Instrumentation• Measurements


Thick, transparent film– Rapidly changing features in Ψ & ∆ at short wavelengths.

Best to measure in terms of eV.

Thick, UV absorbing film– Data features flat at short wavelengths.

Best to measure in terms of nm.

Very Thick transparent films (>3 microns)– Difficult to resolve short wavelength features

Measure only long wavelengths in terms of nm(e.g. 700 – 1700 nm by 2 nm)

Recommended Wavelength StepsRecommended Wavelength Steps• Light• Ellipsometry• Instrumentation• Measurements


Film Thickness Steps (eV) Steps (nm)0 to 200 nm 0.1 eV 20 nm

200 nm to 500 nm 0.05 eV 10 nm

500 nm to 800 nm 0.025 eV 5 nm, well resolved for wavelengths >300 nm

800 nm to 1.2 µm 0.02 eV 5 nm, well resolved for wavelengths >800 nm

1.2 µm to 2 µm 0.01eV 2 nm, well resolved for wavelengths >400 nm.

3 µm 2 nm, well resolved for wavelengths >500 nm


• Light• Ellipsometry• Instrumentation• MeasurementsHow many angles?How many angles?

Multiple angles increase confidence, but not always necessary

Multiple angles best for:– Multiple layers– Absorbing films– Anisotropic films– Graded films


• Light• Ellipsometry• Instrumentation• MeasurementsBenefiting from multiple anglesBenefiting from multiple angles

Fit Uniqueness Simulations: Thick Film Index and Thickness

05

101520253035404550

1.45 1.46 1.47 1.48 1.49 1.50 1.51 1.52 1.53 1.54 1.55Film Index

MSE

Single angle: close to Brewster angle is best– e.g. compare 45, 60 and 75.

Multiple angles: improve sensitivity

Two widely spaced angles adequate– e.g. compare

45-75 to 70-75

45°60°75°70°, 75°45°, 75°45°, 60°, 75°45,50,55,60,65,70,75°


• Light• Ellipsometry• Instrumentation• MeasurementsWhy Multiple Angles HelpWhy Multiple Angles Help

Single angle: index sensitivity only in the peak maximums:

Multiple angles: the frequency shifts for both angle and index

60° Only, Data Mis-fit with Index fixed at 1.45

Wavelength (nm)200 300 400 500 600 700 800

Ψ in

deg

rees

20

30

40

50

60

70

80

Model Fit Exp E 60°

45° & 75°, Data Mis-fit with Index fixed at 1.45

Wavelength (nm)200 300 400 500 600 700 800

Ψ in

deg

rees

0

20

40

60

80

100Model Fit Exp E 45°Exp E 75°


• Light• Ellipsometry• Instrumentation• MeasurementsAngle of Incidence SettingsAngle of Incidence Settings

Why more than one?– Some in “good” region– Averaging

When determining OC’s, use two or three angles– One near Brewster Angle– One above Brewster Angle– One below Brewster Angle

When OC’s are known– one angle usually sufficient– more if time permits

Brewster's Angle

50°

55°

60°

65°

70°

75°

80°

85°

1 1.5 2 2.5 3 3.5 4 4.5 5

Refractive IndexA

ngle

Ge*

*approx. value of N@ λ=650nm

GaAs*

Si*InP*

ITO*Al2O3

*

ZnSe*

SiC*

TiO2*

Si3N4Ta2O5

*

SiO2*

H2O*


• Light• Ellipsometry• Instrumentation• MeasurementsTypical AnglesTypical Angles

Angle of Incidence

(spot-length)/ (beam-dia.)

25° 1.135° 1.245° 1.455° 1.765° 2.475° 3.980° 5.885° 11.5

Spot size vs. angleTypical Angle Combinations:» Thin films on Si: 65°, 75°» Thick films on Si: 60°, 75°

or 55, 65°, 75°» n-matched films on glass: 55°, 56.5°, 58°» Other films on glass: 55°, 65°, 75°» Films on metals: 70°, 80°» Anisotropic & Graded films: 55°, 65°, 75°

or 45°, 60°, 75°

Typical Angle Combinations:» Thin films on Si: 65°, 75°» Thick films on Si: 60°, 75°

or 55, 65°, 75°» n-matched films on glass: 55°, 56.5°, 58°» Other films on glass: 55°, 65°, 75°» Films on metals: 70°, 80°» Anisotropic & Graded films: 55°, 65°, 75°

or 45°, 60°, 75°


Summary Summary –– Measurement ConsiderationsMeasurement Considerations

Ellipsometer is calibrated for accurate measurement of

ellipsometry parameters: Ψ and ∆, without “known” sample.

– Component performance and orientation to plane of incidence.

Range and number of wavelengths depends on application.

Angles to match materials under study

– Above, below, and near Brewster condition.


Measurements SummaryMeasurements SummaryC. Instrumentation– Light source, Spectrometer,

Monochromator, Detector– Polarizer and Compensator– System Configurations– Case Example: Rotating Analyzer

Ellipsometer

D. Measurement considerations– Calibration– Choosing Angle of Incidence– Choosing Wavelength Range

A. Light and Polarization– Electromagnetic Plane Waves– Superposition of Waves– Polarization (Linear, Circular,

Elliptical)– Polarization Descriptions

B. Ellipsometry– Plane of Incidence– Definitions– Advantages


Interaction between Light & MaterialsInteraction between Light & Materials

E. Optical Constants– Phase velocity, Absorption, Reflection, Refraction– Transparent Materials– Absorbing Materials: mechanisms for absorption

F. Fresnel Coefficients & Brewster AnglesG. Thin Films

– Interference– Single and Multilayer Films

H. Data Interpretation– Substrates– Films (Thickness and Index of Refraction)– Envelopes


E. Optical ConstantsE. Optical Constants

Interaction between light and materialsE. Optical Constants

– Phase velocity, Absorption, Reflection, Refraction– Transparent Materials

– Absorbing Materials: mechanisms for absorptionF. Fresnel Coefficients & Brewster AnglesG. Thin FilmsH. Data Interpretation


• Optical Constants• Fresnel Coefficients• Thin Films• Data InterpretationComplex Refractive IndexComplex Refractive Index

Materials are described by two numbers: n and kn = “refractive index”– phase velocity = c/n

– direction of propagation (refraction)

k = “extinction coefficient”– Loss of wave energy to the material

Together called “Complex Refractive Index”:

ñ(λ) = n(λ) – ik(λ)Both n and k vary with wavelength.


• Optical Constants• Fresnel Coefficients• Thin Films• Data InterpretationLight in a materialLight in a material

Velocity and wavelength vary in different materials

Frequency of wave remains constant

Energy of photon

nc

=v

λυ v

=

n = 1 n = 2

)(240,1)(nm

heVEλ

υ ≅=

speed of light

speed of light

Planck’s constantPlanck’s constant


• Optical Constants• Fresnel Coefficients• Thin Films• Data InterpretationAbsorption CoefficientAbsorption Coefficient

Absorption Coefficient, α

zoz eII α−=)(

Io

I(z)

zDp λλπλα )(4)( k

=


• Optical Constants• Fresnel Coefficients• Thin Films• Data InterpretationCharacteristic DepthCharacteristic Depth

Characteristic Depth, Light penetration depth.

– 1 * Dp , I/Io » 0.37– 2 * Dp , I/Io » 0.15– 3 * Dp , I/Io » 0.05– 4 * Dp , I/Io » 0.02

Significance: Can light penetrate a film?– If thickness > 4 Dp, film is opaque (ignore underlying materials).

wave has 1/e ofenergy remaining

(36.7%).

wave has 1/e ofenergy remaining

(36.7%).kDp π

λα 41

==


• Optical Constants• Fresnel Coefficients• Thin Films• Data InterpretationCharacteristic DepthCharacteristic Depth

Examples

Mat’l λ k Dp

Si wafer 632.8nm .016 ~3.1 µ Si wafer 301nm 4.09 58Å

W 632.8nm 2.63 191Å

Al 632.8nm 6.92 73Å


• Optical Constants• Fresnel Coefficients• Thin Films• Data InterpretationSi Characteristic DepthSi Characteristic Depth

Silicon Characteristic Depth

1

10

100

1000

10000

100000

200 300 400 500 600 700 800 900wavelength (nm)

Dp

(nm

)

0

1

2

3

4

5

6

Extinction C

oefficient, k

Dp (nm)Extinction Coefficient k


• Optical Constants• Fresnel Coefficients• Thin Films• Data InterpretationSnellSnell’’s Laws Law

Reflection

Refraction (Snell’s Law)– for dielectrics, i.e., k = 0

– in general

φ i = φ r

index, Ñ2velocity, v

index, Ñ1velocity, c

φ1φ1

φ2

n1 sin φ1 = n2 sin φ2

Ñ1 sin φ1 = Ñ 2 sin φ2


• Optical Constants• Fresnel Coefficients• Thin Films• Data InterpretationDielectric FunctionDielectric Function

Dielectric function describes material’s response to electro-magnetic radiationRefractive index describes EM radiation’s interaction with materials – is the complex square root of the dielectric function

21~ εεε iikn +==−


• Optical Constants• Fresnel Coefficients• Thin Films• Data InterpretationDielectric Dielectric ⇔⇔ index conversionindex conversion

ε1 + i ε2 ←→ n - i kconversions

Frequency (energy)

ε1n

ε2k

21~ εεε iikn +==−

221 kn −=ε

nk22 −=ε

Phase velocity=vphase= c/n


• Optical Constants• Fresnel Coefficients• Thin Films• Data InterpretationTransparent RegionTransparent Region

Index decreases for longer wavelengths.Higher index for stronger UV absorption.

1

2

3

4

0 500 1000 1500 2000Wavelength in nm

Inde

x, n

SiO2Si3N4Si


• Optical Constants• Fresnel Coefficients• Thin Films• Data InterpretationDielectricDielectric

Electric field causes oscillating electron charge cloud.

Electrons move easily, atomic core too heavy to move.

Displaced charge = Electric Dipoles.

Photon In



Atoms “borrow” energy from electric field. Then give it back with NO LOSS. DIPOLE RADIATION!

New Photon Out !

- Same Frequency- Same Energy = hν

Oscillating Dipole



Oscillations take time– Re-emitted wave is “delayed”.

Wave has been slowed– Index, n = (speed in void)/(speed in material)

Reference Wave

In Phase Out of Phase

Dielectrics: Refractive IndexDielectrics: Refractive Index • Optical Constants• Fresnel Coefficients• Thin Films• Data Interpretation


Oscillating Charge Cloud = Mechanical Oscillator.Mechanical Oscillators:– Mass on Spring.– Simple oscillator

Mechanical Oscillators have:– RESONANT FREQUENCY

– Oscillators can be driven at frequencies Below, At, or Abovethe Resonant Frequency. 3 different cases.



RESONANCE:Below, At, or Above Resonant Frequency

DISPERSION = Index Change with λDielectric Optical Constants: Resonance

Wavelength (nm)200 400 600 800 1000 1200 1400

Inde

x of

refra

ctio

n 'n

'

Extinction C

oefficient 'k'

1.70

1.80

1.90

2.00

2.10

2.20

2.30

2.40

0.00

0.10

0.20

0.30

0.40

0.50

0.60

nk

Below

Above

Resonance



Case 1: Below ResonanceDipole response increases as frequency approaches resonance– Energy added with each

cycle, Energy more efficiently coupled to oscillators

Result: Longer delay in re-admitted wave →Index increases with frequency (decreases with longer λ’s).

Dielectric Optical Constants

Wavelength (nm)200 400 600 800 1000 1200 1400

Inde

x of

refra

ctio

n 'n

'

Extinction C

oefficient 'k'

1.70

1.80

1.90

2.00

2.10

2.20

2.30

2.40

0.00

0.10

0.20

0.30

0.40

0.50

0.60

nk

“NORMAL DISPERSION”



Case 2: ResonanceMaximum amplitude of Electron cloud displacement

Absorption = maximum transfer of energy to other processes (for example: thermal energy (heat))

Dielectric Optical Constants: At Resonance

Wavelength (nm)200 300 400 500 600

Inde

x of

refra

ctio

n 'n

'

Extinction C

oefficient 'k'

1.70

1.80

1.90

2.00

2.10

2.20

2.30

2.40

0.00

0.10

0.20

0.30

0.40

0.50

0.60

nk

Dipole Oscillators response is no longer in phase with driving fields



Case 3: Above ResonanceDriving frequency to fast, oscillators can no longer “keep-up”

Dipoles related to this resonance can no longer respond driving frequency

– they contribute less & less to dielectric response as λ decreases

At higher frequencies (shorter λ’s):

– Dielectric response, velocity & index become are affected by more tightly-bound electron orbitals that resonate at even higher frequencies

Dielectric Optical Constants: At Resonance

Wavelength (nm)200 300 400 500 600

Inde

x of

refra

ctio

n 'n

'

1.70

1.80

1.90

2.00

2.10

2.20

2.30

2.40

0.00

0.10

0.20

0.30

0.40

0.50

0.60

nk

AbsorptionDecreasing

Extinction C

oefficient 'k'


• Optical Constants• Fresnel Coefficients• Thin Films• Data InterpretationKramersKramers--Kronig RelationKronig Relation

Frequency (energy)

nk

ε1ε2

Theory describing relationship between absorption processes & normal/anomalous dispersionsReal and imaginary parts related to each other


• Optical Constants• Fresnel Coefficients• Thin Films• Data InterpretationAbsorbing RegionsAbsorbing Regions

EM fields lose energy to a material process– Electronic Transition– Molecular Vibration– Lattice Vibration– Free-carrier

ε

Photon Energy (eV)0.03 0.1 0.3 1 3 10

ε 1 2

-50

-38

-25

-13

0

13

25

0

10

20

30

40

50

60

ε1ε2

LatticeVibrations

ElectronicTransitions

UVIR Visible

Transparent

Rutile TiO2

UV to IR


• Optical Constants• Fresnel Coefficients• Thin Films• Data InterpretationElectronic TransitionsElectronic Transitions

Electrons in energy bands.

Gap between valence and conduction band.

If photon has energy larger than gap, electron may excite to higher state.



Energy bands can be complicated with different acceptable transitions (direct or indirect).

S. Adachi, Optical properties of crystalline and amorphous semiconductors,Kluwer Academic Publishers, Boston (1999).



Absorption associated with each transition.Band-to-band transitions produce absorption bandsCritical point energies (Ecp) = photon energies where there is a high probability of band-to-band transitions

Photon Energy (eV)1.0 2.0 3.0 4.0 5.0 6.0 7.0

Imag

(Die

lect

ricC

onst

ant),

ε2

0

10

20

30

40

50

SiliconGaAs

Ecp

Ecp

Ecp

Ecp

Band-gap energy (Eg): minimum energy where transitions can occur

~Eg(Si & GaAs)


• Optical Constants• Fresnel Coefficients• Thin Films• Data InterpretationAmorphous or CrystallineAmorphous or Crystalline

Electronic transition critical points broaden in amorphous materials


• Optical Constants• Fresnel Coefficients• Thin Films• Data InterpretationMolecular VibrationMolecular Vibration

Electric Field with slow frequency (IR) can vibrate molecules (stretch or bend)

CH2 stretch(symmetric)

CH2 stretch(asymmetric)

N-Hstretch

C=Ostretch


• Optical Constants• Fresnel Coefficients• Thin Films• Data InterpretationLattice VibrationsLattice Vibrations

Electric Field causes lattice to vibrate– Longitudinal (LO) or Transverse (TO) Optical

+ + + +- - - -+ + + +- - - -+ + + ++ + + +- - - -- - - -

E kE k

M. Schubert, 2002Wave Number (cm-1)

600 900 1200 1500 1800 2100∆

in d

egre

es

0

60

120

180

Wave Number (cm-1)600 900 1200 1500 1800 2100

∆in

deg

rees

0

60

120

180

0

60

120

180

TO+LA

2TOTO+LO

T.E. Tiwald et al., Thin Solid Films, 313-314 (1998) 661.

TO


• Optical Constants• Fresnel Coefficients• Thin Films• Data InterpretationFreeFree--carrierscarriers

Metals have more electrons than needed for bonding (covalent bonds = sharing electrons).Electric field causes “free” electrons to move! When electrons collide with atomic cores in lattice, EM Wave energy lost as heat.


• Optical Constants• Fresnel Coefficients• Thin Films• Data InterpretationFreeFree--carrierscarriers

Nickel

Wavelength (nm)0 300 600 900 1200 1500 1800

Inde

x of

refra

ctio

n, n

Extinction C

oefficient, k

1.0

1.5

2.0

2.5

3.0

3.5

4.0

1

2

3

4

5

6

7

8

nkStrongly Absorb

low frequency EM wavesk increases in IR spectral range

Aluminum

Wavelength (nm)0 300 600 900 1200 1500 1800

Inde

x of

refra

ctio

n, n

Extinction C

oefficient, k

0.0

0.5

1.0

1.5

2.0

2.5

3.0

0

3

6

9

12

15

18

nk


• Optical Constants• Fresnel Coefficients• Thin Films• Data InterpretationDielectricsDielectrics

Low index Low reflectanceSmall dispersion in visible and infrared

Generated Reflectance

Wavelength (nm)300 600 900 1200 1500 1800

Ref

lect

ion

0.02

0.04

0.06

0.08

0.10

0.12

0.14

Si3N4SiO2

Index of Refraction, N

Wavelength (nm)300 600 900 1200 1500 1800

Inde

x of

refra

ctio

n 'n '

1.4

1.6

1.8

2.0

2.2

SiO2Si3N4


• Optical Constants• Fresnel Coefficients• Thin Films• Data InterpretationSemiconductorsSemiconductors

Higher index and absorption above bandgap

Silicon

Wavelength (nm)200 400 600 800 1000 1200

Ref

lect

ion

0.30

0.40

0.50

0.60

0.70

0.80

Reflection

Silicon

Wavelength (nm)200 400 600 800 1000 1200

Inde

x of

refra

ctio

n, n

Extinction C

oefficient, k1.0

2.0

3.0

4.0

5.0

6.0

7.0

0.0

1.0

2.0

3.0

4.0

5.0

6.0

NK


Ensemble model of dielectric functionEnsemble model of dielectric function

Dielectric function can be modeled as an sum or ensemble of various kinds of functions, or “oscillators”:

Each oscillator type has its own distinct shape, & describes a different portion of the dielectric spectral function

Each oscillator type has its own distinct shape, & describes a different portion of the dielectric spectral function

( ) ∑+=n

nntypeoffset AmpOsc ,...),,(~ γωεωε

Osctype= Lorentz, Gaussian, Drude, Pole, etc.

Photon Energy (eV)0.0 1.0 2.0 3.0 4.0 5.0

Imag

(Die

lect

ric C

onst

ant),

ε 2

0.0

0.5

1.0

1.5

2.0

2.5

ito pbpdrudegaussiantauc-lorentz

Drude

Gaussian

Tauc-Lorentz

• Optical Constants• Fresnel Coefficients• Thin Films• Data Interpretation


Optical Constant SummaryOptical Constant Summary

Two values describe the materialn – ik or ε1 + iε2

Transparent region: index decreases toward longer wavelengths (normal dispersion)Absorbing region: KK consistency creates anomalous dispersionDifferent absorbing mechanisms described with different mathematical functions.


F. Fresnel F. Fresnel CoefCoef. & Brewster Angles. & Brewster Angles

Interaction between light and materialsE. Optical Constants

F. Fresnel Coefficients & Brewster Angles– Fresnel Coefficients defined– Measurable Properties– Ellipsometry & Fresnel Coefficients– Brewster & Principle Angles defined

G. Thin FilmsH. Data Interpretation


• Optical Constants• Fresnel Coefficients• Thin Films• Data InterpretationFresnel CoefficientsFresnel Coefficients

Solutions to boundary value problem at interfaces:Describe reflection and transmission at interface:Depend on angle and polarization direction (p or s)

ttii

ii

si

ts nn

nEEt

θθθcoscos

cos2

0

0

+=⎟⎟

⎠

⎞⎜⎜⎝

⎛=

ttii

ttii

si

rs nn

nnEEr

θθθθ

coscoscoscos

0

0

+−

=⎟⎟⎠

⎞⎜⎜⎝

⎛=

itti

tiit

pi

rp nn

nnEEr

θθθθ

coscoscoscos

0

0

+−

=⎟⎟⎠

⎞⎜⎜⎝

⎛=

itti

ii

pi

tp nn

nEEt

θθθcoscos

cos2

0

0

+=⎟⎟

⎠

⎞⎜⎜⎝

⎛=


• Optical Constants• Fresnel Coefficients• Thin Films• Data InterpretationFresnel CoefficientsFresnel Coefficients

Fresnel Coefficients are complex numbers ( ) ( ) si

ssss errirr δ~~Im~Re~ =⋅+=

( ) ( ) pipppp errirr δ~~Im~Re~ =⋅+=

( ) ( ) sissss ettitt δ~~Im~Re~ =⋅+=

( ) ( ) pipppp ettitt δ~~Im~Re~ =⋅+=


• Optical Constants• Fresnel Coefficients• Thin Films• Data InterpretationMeasurable PropertiesMeasurable Properties

Relate to Fresnel coefficients:– Reflectance and Transmittance (intensities)

– Reflected and Transmitted Ellipsometry

2

pp rR = 2

pp tT =2

ss rR = 2ss tT =

s

pi

rr

e =∆)ψtan(s

pi

tt

e =∆)ψtan(

Reflectance / TransmittanceReflectance / Transmittance• Optical Constants• Fresnel Coefficients• Thin Films• Data Interpretation


Reflectance / Transmittance– ratio of intensity– square of Amplitude– for a single interface

Reflections caused by:– DIFFERENCES in Refractive Index: – Air to glass, Glass to Silicon, etc.

I0 IR

IT

Transmission = I /IReflection = I /I

T 0

R 0

R n nn n

= −+

( )( )

i t2

i t2 For Uncoated Substrate

At Normal Incidence.


Ellipsometry & the Fresnel coefficientsEllipsometry & the Fresnel coefficients• Optical Constants• Fresnel Coefficients• Thin Films• Data Interpretation

Ellipsometry measures

In other words, Ellipsometry measures the ratio of the Fresnel Reflection (transmission*) coefficients

∆− Ψ==== ii

s

p

s

pincidents

refls

incidentp

reflp ee

rr

rr

EEEE

sp )tan(~~

~~

)( δδρ

s

pincidents

refls

incidentp

reflp

rr

EEEE

~~

≡

* Replace rp & rs with tp & ts


• Optical Constants• Fresnel Coefficients• Thin Films• Data InterpretationEllipsometry measures...Ellipsometry measures...

( ) ( )spi

s

pins

outs

inp

outpi e

rr

EEEE

e δδρ −∆ ==Ψ= ~~

~~~~

tan

The normalized polarization transfer function described by from sample’s Jones Matrix :

⎟⎟⎠

⎞⎜⎜⎝

⎛⎟⎟

⎠

⎞

⎜⎜

⎝

⎛=⎟⎟

⎠

⎞⎜⎜⎝

⎛ins

inp

s

p

outs

outp

EE

rr

EE

10

0~~

* The off-diagonal components equal zero for isotropic samples and certain anisotropic samples where the optical axis is oriented along axes of symmetry

**For transmission ellipsometry, replace rp & rswith tp & ts.


Behavior of rBehavior of rpp &r&rs s : Brewster Angle: Brewster Angle• Optical Constants• Fresnel Coefficients• Thin Films• Data Interpretation

for dielectrics, i.e., k = 0rs negative and non-zerorp passes through zeroℜp goes to zero (φb)The Brewster Angle (φb)– Reflected light is s-polarized

tan φB =

n2

n1


Principle Angle (for metals)Principle Angle (for metals)• Optical Constants• Fresnel Coefficients• Thin Films• Data Interpretation

if k is non-zero, rp and rs are complex– cannot plot rp and rs vs. AOI– still can plot ℜ– ℜp has a minimum,

not zero

– called the “principle angle”


BrewsterBrewster’’s s (or Principle) angle for various materials(or Principle) angle for various materials

• Optical Constants• Fresnel Coefficients• Thin Films• Data Interpretation

… at λ = 650nm

Brewster's Angle

50°

55°

60°

65°

70°

75°

80°

85°

1 1.5 2 2.5 3 3.5 4 4.5 5

Refractive Index

Angl

e

Ge*

*approx. value of N@ λ=650nm

GaAs*

Si*InP*

ITO*Al2O3

*

ZnSe*

SiC*

TiO2*

Si3N4Ta2O5

*

SiO2*

H2O*

Glasses, polymers & other dielectrics

Semiconductors

metals


Fresnel Coefficients SummaryFresnel Coefficients Summary

Fresnel Coefficients– Complex functions that describe reflection &

transmission at interfaces

Measurable Properties of Fresnel Coefficients– Transmittance, Reflectance & Ellipsometry

Ellipsometry & Fresnel Coefficients– Ellipsometry measures the complex ratio rp/rs (or tp/ts)

Brewster & Principle Angles– Angle at which rp = 0 or rp at minimum


F. Thin FilmsF. Thin Films

Interaction between light and materialsE. Optical ConstantsF. Fresnel Coefficients & Brewster Angles G. Thin Films

– Interference– Oscillation Periods– Single and Multilayer Films

H. Data Interpretation


• Optical Constants• Fresnel Coefficients• Thin Films• Data InterpretationThin FilmsThin Films

If transparent, causes interference oscillations

If absorbing, film looks like substrate, no interference oscillations (optically thick)

Wavelength (nm)120 150 180 210 240 270 300

∆in

deg

rees

-50

0

50

100

150

200

250

300


• Optical Constants• Fresnel Coefficients• Thin Films• Data InterpretationThin Film InterferenceThin Film Interference

Each reflected wave will have a different phase and amplitude.

N0

N1

N2


• Optical Constants• Fresnel Coefficients• Thin Films• Data InterpretationFresnel Coefficients & filmsFresnel Coefficients & films

Multiple reflections in a thin film lead to an infinite series for transmitted and reflected light

...ee -4i1010

21201

-2i10120101 +++= ββ trrttrtrrtot

β

β

2),(12),(01

2),(10),(01),(12),(01

),( 1 ispsp

ispspspsp

sptot errettrr

r −

−

−

+=

N2

N1

No

t01t12 t01r12r10t12 t01r12r10r12r10t12

t01r12r10r12t10r10 t01r12t10

FILM PHASE THICKNESS

111 cos2 θ

λπβ nd

⎟⎠⎞

⎜⎝⎛=


• Optical Constants• Fresnel Coefficients• Thin Films• Data InterpretationSurface & Interface beamsSurface & Interface beams

rsurf (Surface beam) r int (Interface beam)

N2

N1

No

t01t12 t01r12r10t12 t01r12r10r12r10t12

t01r12r10r12t10r10 t01r12t10

=totr ...ee -4i1010

21201

-2i101201 ++ ββ trrttrt01r +

01r ( )β

β

21210

221012

11

i

i

errerr−

−

−−

+=totr

– All Fresnel quantities are complex


• Optical Constants• Fresnel Coefficients• Thin Films• Data InterpretationMultilayersMultilayers

Summation of multiple reflections quickly becomes complicated for multiple layersScattering matrix method is used instead:

– Matrix describes each INTERFACE

– Matrix describes each LAYER

( ) ⎥⎦

⎤⎢⎣

⎡=

11

1ab

ababab r

rtI

⎥⎦

⎤⎢⎣

⎡=

− β

β

i

i

ee

L0

0

23212101 ILILIS =


Thin Films SummaryThin Films Summary

Optical constants determine how light interacts with film.

Fresnel reflection coefficients calculate the reflected and refracted light along different polarization directions

Theory allows calculation of substrates, single-layer films, and multi-layers.


G. Data InterpretationG. Data Interpretation

Interaction between light and materialsE. Optical ConstantsF. Fresnel Coefficients & Brewster Angles G. Thin FilmsH. Data Interpretation

– Substrates– Films (Thickness and Index of Refraction)– Envelopes


• Optical Constants• Fresnel Coefficients• Thin Films• Data InterpretationMotivationMotivation

Data Interpretation:– Estimate sample based on “raw” data.– Reduce analysis time and improve ability to identify

correct models.Experimental Data

Wavelength (nm)300 600 900 1200 1500 1800

Ψin

deg

rees

0

5

10

15

20

25

30

Exp E 65°Exp E 75°

Data interpretation not quantitative

We assume ideal samples and

measurements

Data interpretation not quantitative

We assume ideal samples and

measurements


• Optical Constants• Fresnel Coefficients• Thin Films• Data InterpretationTools: Generating DataTools: Generating Data

Demonstrate data caused by model variations:1. Build Model in WVASE.2. Choose ‘Range’ from

Generate Data Window.3. Press ‘Generate Data’

Select:Angle, Wavelength and Data Type

Select:Angle, Wavelength and Data Type


• Optical Constants• Fresnel Coefficients• Thin Films• Data InterpretationRole of Optical ConstantsRole of Optical Constants

Index difference leads to reflection– Large ∆N Large Reflection– Small ∆N Small Reflection

Material 1 Material 2 ReflectanceAir (N=1) SiO2 (N=1.5) ~3.3%

Air (N=1) Si (N=3.5) ~31%

a-Si (N=3.4) Si (N=3.5) ~0.02%


• Optical Constants• Fresnel Coefficients• Thin Films• Data InterpretationBare SubstratesBare Substrates

What should we expect?– Large N gives Large Reflection

– At 0°, Psi and Delta are 45° and 180°, respectively.

– Psi will decrease as angle is increased until it goes through a minimum (at Brewster angle) and then rises.

– Delta will proceed from 180° to 0°, crossing 90° at the Brewster angle.


• Optical Constants• Fresnel Coefficients• Thin Films• Data InterpretationBare Substrate: DielectricsBare Substrate: Dielectrics

Low index Low reflectanceSmall dispersion in visible and near infrared

Generated Reflectance

Wavelength (nm)300 600 900 1200 1500 1800

Ref

lect

ion

0.02

0.04

0.06

0.08

0.10

0.12

0.14

Si3N4SiO2

Index of Refraction, N

Wavelength (nm)300 600 900 1200 1500 1800

Inde

x of

refra

ctio

n 'n '

1.4

1.6

1.8

2.0

2.2

SiO2Si3N4


• Optical Constants• Fresnel Coefficients• Thin Films• Data InterpretationDielectrics: EllipsometryDielectrics: Ellipsometry

Data depend on angle of incidence.


Ψin

deg

rees

0

10

20

30

40

50

SiO2Si3N4


∆in

deg

rees

0

50

100

150

200

SiO2Si3N4

Spectroscopic Data for SiO 2

Wavelength (nm)300 640 980 1320 1660 2000

Ψin

deg

rees

0

5

10

15

20

25

45°

50°

55°

60°

65°

70°

Substrates: SemiconductorsSubstrates: Semiconductors• Optical Constants• Fresnel Coefficients• Thin Films• Data Interpretation


Higher index and absorption above bandgap

Silicon

Wavelength (nm)200 400 600 800 1000 1200

Ref

lect

ion

0.30

0.40

0.50

0.60

0.70

0.80

Reflection

Silicon

Wavelength (nm)200 400 600 800 1000 1200

Inde

x of

refra

ctio

n, n

Extinction C

oefficient, k1.0

2.0

3.0

4.0

5.0

6.0

7.0

0.0

1.0

2.0

3.0

4.0

5.0

6.0

NK

Semiconductors: EllipsometrySemiconductors: Ellipsometry• Optical Constants• Fresnel Coefficients• Thin Films• Data Interpretation


Psi minimum depends on value of ‘k’.Delta away from 0° or 180° Silicon

Wavelength (nm)200 400 600 800 1000 1200

Ψin

deg

rees

0

10

20

30

40

50

55°65°75°

Wavelength (nm)200 400 600 800 1000 1200

∆in

deg

rees

0

50

100

150

200

55°65°75°

Silicon


Ψin

deg

rees

0

10

20

30

40

50

Wvl=270nmWvl=670nm


∆in

deg

rees

0

50

100

150

200

Wvl=270nmWvl=670nm


• Optical Constants• Fresnel Coefficients• Thin Films• Data InterpretationBare Substrates: MetalsBare Substrates: Metals

Significant ‘n’ and ‘k’ at all wavelengths.

Aluminum

Wavelength (nm)0 300 600 900 1200 1500 1800

Ref

lect

ion

0.86

0.88

0.90

0.92

0.94

0.96

0.98Aluminum

Wavelength (nm)0 300 600 900 1200 1500 1800

Inde

x of

refra

ctio

n, n

Extinction C

oefficient, k0.0

0.5

1.0

1.5

2.0

2.5

3.0

0

3

6

9

12

15

18

NK


• Optical Constants• Fresnel Coefficients• Thin Films• Data InterpretationMetals: EllipsometryMetals: Ellipsometry

Psi near 45°, Delta away from 180° or 0°.Aluminum

Wavelength (nm)200 400 600 800 1000 1200

Psi

in d

egre

es

36

38

40

42

44

46

65°75°85°

Wavelength (nm)200 400 600 800 1000 1200

Del

ta in

deg

rees

0

30

60

90

120

150

180

65°75°85°

Aluminum

0 20 40 60 80 100

Psi

in d

egre

es

40.0

41.0

42.0

43.0

44.0

45.0

Exp E 350nmExp E 950nm


Del

ta in

deg

rees

0

50

100

150

200

Exp E 350nmExp E 950nm


• Optical Constants• Fresnel Coefficients• Thin Films• Data InterpretationThin film interferenceThin film interference

rtot(p,s) depends on waves reflected from each interface have a different phase & amplitude.rtot(p,s) is a function of e-i2β & will oscillate period of 2ββ a function of δ1 & 1/λ.

111 cos2 θ

λπβ nd

⎟⎠⎞

⎜⎝⎛=

β

β

2),(12),(01

2),(10),(01),(12),(01

),( 1 ispsp

ispspspsp

sptot errettrr

r −

−

−

+=

FILM PHASE THICKNESS N2

N1

No

t01t12 t01r12r10t12 t01r12r10r12r10t12

t01r12r10r12t10r10 t01r12t10


• Optical Constants• Fresnel Coefficients• Thin Films• Data InterpretationSingleSingle--wavelengthwavelength

Data circle as film thickness increases (approximately every 1/2λ).

Imaginary(rho)-6 -4 -2 0 2 4 6

Rea

l(rho

)

-2

0

2

4

6

8

10

12

Full Thickness Cycle

φλ

φ 220

21 sin~~2 nn

D−

= nmD 226=φ

50, 276, 502nm…

SiO2 on Si

Substrate


• Optical Constants• Fresnel Coefficients• Thin Films• Data InterpretationSpectroscopic EllipsometrySpectroscopic Ellipsometry

Vary Wavelength– Probe different n,k– Thickness constant

Wavelength (nm)400 600 800 1000 1200 1400

Ψin

deg

rees

0

20

40

60

80

100

50 nm276 nm502 nm

Data match at 500 nm


• Optical Constants• Fresnel Coefficients• Thin Films• Data InterpretationTransparent FilmsTransparent Films

5 micron Oxide

Wavelength (nm)0 300 600 900 1200 1500 1800

Ψin

deg

rees

0

20

40

60

80

100

Exp E 75°

100nm Oxide

Wavelength (nm)0 300 600 900 1200 1500 1800

Ψin

deg

rees

0

20

40

60

80

100

Exp E 75°Thicker films produce more interference oscillations.

500nm

Wavelength (nm)0 300 600 900 1200 1500 1800

Ψin

deg

rees

0

20

40

60

80

100

Exp E 75°


• Optical Constants• Fresnel Coefficients• Thin Films• Data InterpretationThickness EffectsThickness Effects

As thickness increases:*Interference shifts toward red*Shorter interference period (peaks closer together)

Wavelength (nm)0 300 600 900 1200 1500 1800

Ψin

deg

rees

0

20

40

60

80

100

T


Period of interference oscillationsPeriod of interference oscillations• Optical Constants• Fresnel Coefficients• Thin Films• Data Interpretation

Film Phase delay has (approx.) constant period T when Ψ & ∆ are plotted in 1/λ scale

Generated Data

Photon Energy (eV)0.0 1.0 2.0 3.0 4.0 5.0 6.0 7.0

∆ in

deg

rees

50

100

150

200

250

300

Gen E 68°T

Generated Data

Wavelength (nm)0 300 600 900 1200 1500 1800

∆ in

deg

rees

50

100

150

200

250

300

Gen E 68°

using

eV scale (α 1/λ)

λ scale

111 cos2 θ

λπβ nd

⎟⎠⎞

⎜⎝⎛=

( ) ( )eVdn

T111 cos2

1240θ

=

)nm(,coscos 11

01 d

nθθ =


Period of interference oscillationsPeriod of interference oscillations• Optical Constants• Fresnel Coefficients• Thin Films• Data Interpretation

Film Phase delay has (approx.) constant period T when Ψ & ∆ are plotted in 1/λ scale

111 cos2 θ

λπβ nd

⎟⎠⎞

⎜⎝⎛=

Generated Data

Photon Energy (eV)0.0 1.0 2.0 3.0 4.0 5.0 6.0 7.0

∆ in

deg

rees

50

100

150

200

250

300

Gen E 68°T

Generated Data

Wavelength (nm)0 300 600 900 1200 1500 1800

∆ in

deg

rees

50

100

150

200

250

300

Gen E 68°

( ) ( )eVnd

T11 cos2

1240θ

= eV scale (α 1/λ)

λ scale

)nm(,coscos 11

01 d

nθθ =

using


• Optical Constants• Fresnel Coefficients• Thin Films• Data InterpretationEstimating thickness from # Estimating thickness from # oscosc’’ss

# periods 400 < λ < 1000 nm, P = 1.86eV (2•nd•cosθ1)Generated Data n=1.6, t=500nm

Wavelength (nm)400 500 600 700 800 900 1000

Ψ in

deg

rees

0

20

40

60

80

100

Gen E 68°

P ≈ 2,t=500nm

n=1.6

0 1 2 3 4 5 6 7 8 9 100

500

1000

1500

2000

2500

3000

3500

Oscillations from 400 < wvl < 1000nm

film

thic

knes

s (nm

)

N=3.2

N=2.2

N=1.6

N=1.2

68° angle of incidence

Generated Data, n=2.2, t=1200nm

Wavelength (nm)400 500 600 700 800 900 1000

Ψ in

deg

rees

10

20

30

40

50

Gen E 68°P ≈ 7,

t=1200nmn=2.2



A chart for thicker films:# periods 400 < λ < 1000nm P = 1.86eV (2•nd•cosθ1)

0 5 10 15 20 250

1000

2000

3000

4000

5000

6000

7000

8000

9000

Oscillations from 400 < wvl < 1000nm

film

thic

knes

s (nm

)

N=3.2

N=2.2

N=1.6

N=1.2

68° angle of incidence

Generated Data, n=1.6, t=5000 nm

Wavelength (nm)400 500 600 700 800 900 1000

Ψ in

deg

rees

0

20

40

60

80

100

Gen E 68°P ≈ 20, t=1200nm n=2.2



Table: P = 1.86eV (2•nd•cosθ1) (400 < λ < 1000nm)(copy inside front cover)

Thickness Estimate (nm) from # ψ/∆ Oscillation periods for 400nm < λ < 1000nm

#osc n = 1.2 n = 1.6 n = 2.2 n = 3.20.5 190 120 80 501 390 250 170 1102 780 500 330 2203 1160 750 500 3304 1550 1000 660 4305 1940 1240 830 5406 2330 1490 1000 6507 2720 1740 1160 7608 3100 1990 1330 8709 3490 2240 1490 980

10 3880 2490 1660 109011 4300 2700 1800 120012 4700 3000 2000 130013 5000 3200 2200 140014 5400 3500 2300 150015 5800 3700 2500 160020 7800 5000 3300 220025 9700 6200 4200 270030 11600 7500 5000 330035 13600 8700 5800 380040 15500 10000 6600 430045 17500 11200 7500 4900


• Optical Constants• Fresnel Coefficients• Thin Films• Data InterpretationEnvelopesEnvelopes

Psi and Reflection oscillations restricted to envelope, related to ∆n.

Vary Oxide Thickness on Silicon

Wavelength (nm)0 300 600 900 1200 1500 1800

Ref

lect

ion

0

0.2

0.4

0.6

0.8

R (0)R (1000)R (2000)R (3000)R (4000)


• Optical Constants• Fresnel Coefficients• Thin Films• Data InterpretationExample: SiO2 on GlassExample: SiO2 on Glass

Index match gives small interference, except in ∆.

Optical Constants

Wavelength (nm)200 400 600 800 1000

Inde

x, n

1.44

1.46

1.48

1.50

1.52

1.54

1.56

7059SiO2


• Optical Constants• Fresnel Coefficients• Thin Films• Data InterpretationReflectance: SiOReflectance: SiO22 on Glasson Glass

Reflectance is enveloped by RU and RL.

Reflectance Curves

Wavelength (nm)200 400 600 800 1000

Ref

lect

ion

0.025

0.030

0.035

0.040

0.045

0.050500nm SiO2 / GlassBare GlassBare SiO2

2

11

+−

=s

sU N

NR

Upper and Lower Envelopes:

2

2

2

fs

fsL NN

NNR

+−

=


• Optical Constants• Fresnel Coefficients• Thin Films• Data InterpretationEllipsometry:SiOEllipsometry:SiO22 on Glasson Glass

Psi also enveloped, depends on angle.

Ellipsometry (500nm SiO2 on Glass)

Wavelength (nm)200 400 600 800 1000

Ψin

deg

rees

0

3

6

9

12

15

18

SiO2/Glass, 55°SiO2/Glass, 65°Glass, 55°Glass, 65°SiO2, 55°SiO2, 65°


• Optical Constants• Fresnel Coefficients• Thin Films• Data InterpretationEllipsometry:SiOEllipsometry:SiO22 on Glasson Glass

Coated vs. Uncoated Glass (Delta)

Wavelength (nm)200 400 600 800 1000

∆in

deg

rees

0

50

100

150

200

Bare Glass 55°Bare Glass 65°500nm SiO2 55°500nm SiO2 65°


• Optical Constants• Fresnel Coefficients• Thin Films• Data InterpretationSurface & Interface beamsSurface & Interface beams

rsurf (Surface beam) r int (Interface beam)

N2

N1

No

t01t12 t01r12r10t12 t01r12r10r12r10t12

t01r12r10r12t10r10 t01r12t10

=totr ...ee -4i1010

21201

-2i101201 ++ ββ trrttrt01r +

01r ( )β

β

21210

221012

11

i

i

errerr−

−

−−

+=totr

– All Fresnel quantities are complex


• Optical Constants• Fresnel Coefficients• Thin Films• Data InterpretationFilm Index EffectsFilm Index Effects

Index difference influences peak-to-peak strength of oscillations

Wavelength (nm)300 600 900 1200 1500 1800

Ψin

deg

rees

0

20

40

60

80

100Nf


• Optical Constants• Fresnel Coefficients• Thin Films• Data InterpretationHow to use EnvelopesHow to use Envelopes

1) Change film index to match envelope Be careful because multiple solutions may exist.

2) Change thickness to match oscillation period


• Optical Constants• Fresnel Coefficients• Thin Films• Data InterpretationAbsorbing FilmsAbsorbing Films

Oscillations are ‘damped’ by absorption

No oscillations, ellipsometry can’t measure thickness.

Simulated SiGe / Si Wavelength (nm)0 300 600 900 1200 1500 1800

Ψin

deg

rees

0

10

20

30

40

Wavelength (nm)0 300 600 900 1200 1500 1800

∆in

deg

rees

-100

0

100

200

300


• Optical Constants• Fresnel Coefficients• Thin Films• Data InterpretationMultilayersMultilayers

Both oscillation patterns are superimposed!

Multilayer Example

Wavelength (nm)0 300 600 900 1200 1500 1800

Ψ in

deg

rees

0

20

40

60

80

n=4 1 mmn=3 1000 nmn=1.5 200 nm


• Optical Constants• Fresnel Coefficients• Thin Films• Data InterpretationTHIN FilmsTHIN Films

“Very Thin” ⇒ thickness < 100 Å

Determine thickness with known OC’s.

Difficult to determine OC’s.


• Optical Constants• Fresnel Coefficients• Thin Films• Data InterpretationDetermining ThicknessDetermining Thickness

High sensitivity from phase change (Delta)

20, 40, 65,80, 100 Å


• Optical Constants• Fresnel Coefficients• Thin Films• Data InterpretationDetermining Determining OCOC’’ss

Don’t determine OC’sif film is too thin.

Choose similar material to get “equivalent thickness”


Data Interpretation SummaryData Interpretation SummaryUse “raw” data to interpret sample.

Watch how data vary with angle and wavelength.

Oscillations give info about thickness and ∆n

No oscillations for absorbing films

Cannot independently measure n & thickness if films are too thin


SummarySummary

Interaction between Light & MaterialsE. Optical Constants

– Phase velocity, Absorption, Reflection, Refraction– Transparent Materials

– Absorbing Materials: mechanisms for absorption

F. Fresnel Coefficients & Brewster AnglesG. Thin Films

– Interference– Single and Multilayer Films

H. Data Interpretation– Substrates– Films (Thickness and Index of Refraction)– Envelopes


Modeling and FittingModeling and Fitting

Modeling and Fitting– Why Analyze Data?– Building a Model– Regression– Mean Squared Error– Best Fit?


Why Analyze Data?Why Analyze Data?

What Ellipsometry Measures: What we are Interested in:

Psi (Ψ)Delta (∆)

Film ThicknessRefractive Index

Surface Roughness Interfacial Regions

CompositionCrystallinityAnisotropyUniformityDesired information must be extracted

Through a model-based analysis using equations to describe interaction of

light and materials


Direct SolutionDirect Solution

For bulk material with no overlayers, a direct solution exists to determine optical constants from ellipsometry data.

( ) ( ) ( )⎥⎥⎦

⎤

⎢⎢⎣

⎡⎟⎟⎠

⎞⎜⎜⎝

⎛+−

⋅+⋅=+==+=2

2222

21 11tan1sin~

ρρφφεεε kinni


Regression AnalysisRegression Analysis

For most samples, the equations are transcendental – therefore a direct inversion of equations are not

possible

Use regression analysis:– Calculate response expected from model (generated

data)– Compare to measurement (experimental data)

Session 1A, Basics and TheoryJ.A. Woollam Co., Inc.© Sept 20, 2004 All Rights Reserved

Ellipsometry Flow ChartEllipsometry Flow Chart


Constructing a ModelConstructing a Model

Propose a layered structure– Thickness of each layer– OC’s of each layer

n,k (film 2)

n,k (film 1)

n,k (substrate)

t1

t2

0 si 1 mm1 sio2 1000 Å2 amorphous si 500 Å


Model schematic & Matrix Model schematic & Matrix eqseqs..Scattering matrix :

– I Matrix describes each INTERFACE

– L Matrix describes each LAYER

n,k (film 2)

n,k (film 1)

n,k (substrate)

t1

t2

Model schematic

( ) ⎥⎦

⎤⎢⎣

⎡=

11

1ab

ababab r

rtI

23212101 ILILIS =Model Matrix equation

⎥⎦

⎤⎢⎣

⎡=

− β

β

i

i

ee

L0

0


assumptions of modelassumptions of modelAssumption of parallel layers Difficulty with scattering (diffuse) surfaces– When features (surface & interface roughness, grain sizes, etc.) are

similar dimension as the wavelength, then light will scatter (diffuse).

Best results from specular (mirror-like) surfaces – Inhomogeneities must be approximately 1/10 or smaller than the

wavelength of light


Constructing a ModelConstructing a Model

Representing Optical Functions

Tabulated list (no variation)Often used as a starting point (seed values)

– Mixtures (minimum variation)EMA

– Alloy Files (known variation)Compound Semiconductors

– Dispersion Models (maximum variation)

CauchyOscillator Models

0 si 1 mm1 sio2 1000 Å2 amorphous si 500 Å

0 si 1 mm1 sio2_cauchy 1000 Å2 ema a-si/15% si/7% void 500 Å

0 si 1 mm1 sio2_cauchy 1000 Å2 polysi_c x=0.100 500 Å

0 si 1 mm1 sio2_cauchy 1000 Å2 a-si_genosc 500 Å


Regression: Mean Squared ErrorRegression: Mean Squared Error

WVASE32 uses the Mean Squared Error (MSEMSE) to quantify the difference between experimental and model-generated data.

A smaller MSE implies a better fit.MSE is weighted by the error bars of each measurement, so noisy data are weighted less.

∑= ∆Ψ −

=⎥⎥

⎦

⎤

⎢⎢

⎣

⎡

⎟⎟⎠

⎞⎜⎜⎝

⎛ ∆−∆+⎟

⎟⎠

⎞⎜⎜⎝

⎛ Ψ−Ψ−

=N

i i

ii

i

ii

MNMNMSE

1

2

2

exp,

expmod2

exp,

expmod

21

21 χ

σσ


Fit ExampleFit Example

Software adjusts “fit”parameters to find best match between model and experimental curves.MSE (mean square error) is used to quantify difference.

Wavelength (nm)200 400 600 800 1000

Ψin

deg

rees

0

20

40

60

80

100

Thickness

MSE


Minimizing the MSEMinimizing the MSEThe Marquardt-Levenberg* algorithm is a commonly-used regression routine (there are others)Good starting values are important

MSE

Thickness

startingthickness(guess)

BESTFIT

Local Minima

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


Fitting the DataFitting the Data

Getting starting values close

– Generate from model to compare with Experimental Data

– Adjust “fit” parameters to get better starting values.

Generated and Experimental

Wavelength (nm)300 600 900 1200 1500 1800

Ψ in

deg

rees

0

20

40

60

80

Model Fit Exp E 70°Exp E 75°

Generated and Experimental

Wavelength (nm)300 600 900 1200 1500 1800

Ψ in

deg

rees

0

20

40

60

80

Model Fit Exp E 70°Exp E 75°


Starting values can be criticalStarting values can be critical

Wavelength (nm)200 400 600 800 1000 1200

Ψ in

deg

rees

0

20

40

60

80

100


400nm SiN / 3000nm SiO2

200 400 600 800 1000 1200

Ψ in

deg

rees

0

20

40

60

80

100


One solution – Global fit:Searches a ‘grid’ of starting points for the best match.

94.990.285.480.775.971.266.561.757.052.247.542.738.033.228.523.719.014.29.54.70.0

MSE

Global Minimum“Valley” of local

minima


Quality of Results!Quality of Results!

Once fit is complete, resulting fit parameters must be evaluated for sensitivity and possible correlation.– Compare experimental data with generated data

– How low is MSE? Can it be reduced further by increasing model complexity?

– Are fit parameters physical?

– Check other mathematical “goodness of fit” indicatorsi.e.

Correlation matrix

90% confidence limits …


Correlation MatrixCorrelation Matrix

Example of Correlation Matrix– matrix of two-parameter correlation coefficients calculated

from the covariance matrix

Correlated Parameters: Cn.2 and Bn.2

Any off diagonal elements near +/-1 are correlated.


Correlation MatrixCorrelation Matrix

Watch for strong correlation: greater than ±0.92

Investigate further to insure unique fit.

If parameters appear correlated.

– Reset fit and turn off 1 correlated parameter – can you get same MSE?

– Adjust fit values by 10-20%. Do they return to the same final values that produce lowest MSE?


FOM (90% Confidence Limit x MSE)FOM (90% Confidence Limit x MSE)

Indicates parameter sensitivity and correlation±values =

(90% confidence limit x MSE)

MSE MSE

Fit Parameter Fit Parameter


Are the Results Physical?Are the Results Physical?

n must decrease with

increasing λ if k = 0 !!!– i.e. No absorption --> normal dispersion

Absorption = anomalous dispersion– KK consistent

k can’t be negativeFor metals, generally, both n and k increase with wavelength


General RulesGeneral Rules

Find the simplest optical model that adequately fits the experimental data (this can be subjective; herein lies the ‘art’ of ellipsometric data analysis)

Verify the uniqueness of the model fit

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

Session 1A, Basics and TheoryJ.A. Woollam Co., Inc.© Sept 20, 2004 All Rights Reserved

Modeling SummaryModeling Summary


Session 1A SummarySession 1A SummaryMeasurements– Light and Polarization– Ellipsometry– Instrumentation – Measurement considerations

Interaction between light and materials– Optical Constants– Fresnel Coefficients & Brewster angles– Thin Films– Data Interpretation

Model and Fit– Building a Model– Regression– Mean Squared Error– Best Fit?


References: EllipsometryReferences: Ellipsometry

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

2. R.M.A.Azzam, Selected Papers on Ellipsometry, SPIE Milestone Series MS 27, 1991.

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

6. Spectroscopic Ellipsometry, R.W.Collins, D.E.Aspnes, and E.A. Irene, Editors, Elsevier Science S.A.,1998 , Lausanne, Switzerland; also appears as Vol. 313-314 Thin Solid Films, Numbers 1-2, 1998.

7. R.H. Muller, “Principles of Ellipsometry” Adv. Electrochem. Eng., 9, 167-226 (1973).

8. R.H. Muller, "Ellipsometry as an in Situ Probe for the Study of Electrode Processes," in Techniques for Characterization of Electrodes and Electrochemical Processes, Ravi Varmaand J.R. Selman, Ed. (Wiley & Sons, New York, 1991), pp. 31-125.

9. Hecht, Optics, 2nd Edition, (Addison-Wesley, 1987).


Further Reading: Optics & MaterialsFurther Reading: Optics & Materials

1. Wooten, Frederick, Optical Properties of Solids, Academic Press, 1972.

2. Hecht, Eugene, and Alfred Zajac, Optics, 2nd Edition, Addison-Wesley Publishing Company., 1987.

3. Moller, K.D., Optics, University Science Books, 1988.

4. Rancourt, James D., Optical Thin Films User’s Handbook, McGraw-Hill Publishing Company, 1987.

5. Streetman, Ben G., Solid State Electronic Devices, Prentice-Hall, Inc., 1980.

6. Pankove, Jacques I., Optical Processes in Semiconductors, Dover Publications, 1971.

7. Fowles, Grant R., Introduction to Modern Optics, Dover Publications, 1968.

8. Adachi, Sadao, Optical Properties of Crystalline and AmporphousSemiconductors, Kluwer Academic Publishers, Boston, 1999.


Supplemental: Instrument componentsSupplemental: Instrument components

Light SourcesDetectorsSelecting WavelengthsPolarizerRetarders & Compensators


Light SourcesLight SourcesLaser– Single wavelength (high intensity and well collimated)

Lamps– Broad spectral coverage

DeuteriumXenonTungstenGlobar

– Ideally produce UNPOLARIZED LIGHTcombination of many random polarizations


• Light• Ellipsometry• Instrumentation• MeasurementsDetectorsDetectors

Measure the light intensity

Types:

– Photodiodes

– Photomultipliers

– Diode Arrays

– CCD Arrays

2EI ∝


• Light• Ellipsometry• Instrumentation• MeasurementsWavelength RangeWavelength Range

Combine components for wider range.

9VUV

Spectral Range in eV (not to scale)

DUV UV VIS NIR6.5 3.25 1.25 0.73 0.5 0.09 0.03

D2

Light Sources:

Detectors:

Xe Arc

QTH

SiC Globar

Si

InGaAs

DTGS

UV enhanced PMT

IR


• Light• Ellipsometry• Instrumentation• MeasurementsSelecting WavelengthsSelecting Wavelengths

Separate wavelengths from broad spectrum source

Czerny-Turner Design

Gra

ting

Light in


• Light• Ellipsometry• Instrumentation• MeasurementsPolarizersPolarizers

Pass Linearly Polarized Light– Optical axis determines direction of polarization allowed to pass.– Extinction ratio measures ratio of light that passes parallel and

perpendicular to polarizer: Typically 106 or better

X

Y

Z

E

EPolarizer

Axis


• Light• Ellipsometry• Instrumentation• MeasurementsTypes of Types of PolarizersPolarizers

Different physical mechanisms used to reject one of the orthogonal light components:– Birefringence

Superior performance

– DichroismSuperior Angle acceptability and flexible (shape)

– Reflection


• Light• Ellipsometry• Instrumentation• MeasurementsRetardationRetardation

Anisotropy delays (retards) one wave component relative to the orthogonal waveFast Axis has lower index of refraction.

δ


• Light• Ellipsometry• Instrumentation• MeasurementsCompensatorsCompensators

An optical retarder with exactly 90° retardation (1/4 wave)Difficult to maintain perfect 90°, as the retardation is a strong function of wavelength and alignment

X

Y

E

EPolarizerPolarizer

AxisAxis

Z

CompensatorCompensatorAxisAxis

E


Supplement Material:Supplement Material:Understanding Psi EnvelopesUnderstanding Psi Envelopes

Envelope depends on Angle.

Film Index1.0 2.0 3.0 4.0 5.0 6.0

Ψin

deg

rees

0

20

40

60

80

10060°

70°75°80°

Ns=4

Documents

Session 1A Lincoln 2004-09