Introduction to Ellipsometry - EPFL 2008

8/2/2019 Introduction to Ellipsometry - EPFL 2008

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EPFL October 7th 2008 1

Introduction to ellipsometry

J.Ph.PIEL (PhD)Application Lab ManagerSOPRALAB


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OUTLINE :

- Basic Theory

- GES5 description

- Data Analysis

- Conclusion


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I - BASIC THEORY

- Brief history of Ellipsometry

- Principle

- Physical meaning of and


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Phase measurement

For the 1st time Paul DRUDE use ellipsometry in 1888

2000


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Alexandre Rothen Published in 1945

a paper where the word Ellipsometer appears for the 1st time

Thickness sensitivity : 0.3 A


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Bref Historique

Schematic representation of the mounting from Alexandre Rothen(Rev. of scientifique instruments Feb 1945)


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Malus in 1808 : discovery of the polarisation of light by reflexion.

Fresnel in 1823 : Wave theory of the light.

Maxwell in 1873 : developpment of the Electrogmagnetic field theory.

Drude in 1888 : Use of the extreme sensitivity of the ellipsometry to detectultra thin layers (monolayer).

Abeles in 1947: Developpment of matrix formalism applied to thin layersstack.

Rothe in 1945 : Introduction of the word : Ellipsometer

Hauge, Azzam, Bashara in 1970 : description and developpement ofdifferent ellipsometer settings.

1980 : development of Personal Computer , automatisation of thetechnique. Industrial development of tools.

Few key people in ellipsometry area


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INTRODUCTION

ELLIPSOMETRY is a method based on measurement of the change of the

polarisation state of light after reflection at non normal incidence on the

surface to study

-The measurement gives two independent angles: and

- I t i s an absolute measurement: do not need any reference

- I t i s a non-direct technique: does not give directly the physicalparameters of the sample (thickness and index)- It is necessary to always use a model to describe the sample

SPECTROSCOPIC ELLIPSOMETRY (SE) gives more comprehensive resultssince it studies material on a wide spectral range


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PRINCIPLE OF SPECTROSCOPIC ELLIPSOMETRY

=

rp

rs = tan.e

j= f( ni, ki, Ti )

Substrate (ns, ks)

Thin Film 1 (n1, k1, T1)

Thin Film 2 (n2, k2, T2)Thin Film i (ni, ki, Ti)

Ambient (n0, k0)

ES

EiEP

rs

rp

Erlinear polarisation

EiEpEs

Er Ep.rp

Es. rs

elliptical polarisation

0

Ellipsometry measures the complex reflectance ratio :


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- After reflection on the sample, the extremity of the electric field vector describes anellipse

0

rp

rs

p

s

- This ellipse is characterised by

*the ellipticity Tan which is the ratio of the large axis to the small axis

*the angle of rotation between the main axis and the P axis:


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Extremity of the electric field vector descrives an ellipse


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Physical meaning of and .Those parameters give all relevant information about the polarization state of the light ata given wavelength.

- Tan gives exactly the angle of the first diagonal of the rectangle in which theellipse is enclosed.

- Cos gives roughly how fat is going to be the ellipse (shape).

Its mathematically linked to the ratio of short axis to long axis of the ellipse in its fundamentalframe.

However, and are not straightforwardly linked to the easiest geometrical features ofthe ellipse.

: angle of rotation of the long axis of the ellipseversus axis p

= CosTanTan )2()2( - 1 - 0.5 0.5 1

- 1

- 0.5

0.5

1

40 .


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- 1 - 0.5 0.5 1

- 1

- 0.5

0.5

10

- 1 - 0.5 0.5 1

- 1

- 0.5

0.5

120.

- 1 - 0.5 0.5 1

- 1

- 0.5

0.5

140.

- 1 - 0.5 0.5 1

- 1

- 0.5

0.5

1

60 .

- 1 - 0.5 0.5 1

- 1

- 0.5

0.5

1

80.

- 1 - 0.5 0.5 1

- 1

- 0.5

0.5

1

90.

Example of some different phase shift () for a given value. On those graphics, the long axis is the ellipse is represented and it has to be compared with the

diagonal of the rectangle.

Because is fixed, the ellipse is enclosed in the same rectangle for each graphic.

Modification of change both angle of inclination and ellipticity.


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II - GES5 DESCRIPTION

- Physical description

- Jones Formalism

- Mathematical treatment of the signal

- Example


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Goniometer

Analyser Arm

Polariser Arm

Sample

Mapping

rho/theta

Microspots


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SPECTROSCOPIC ELLIPSOMETER

Scanninig

channelC.C.D.

channel

Xe Lamp

PA

Optical Fiber

Goniometer

PhotoMultiplier

Tube


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Spectrometer

Spectrograph

Schematics of dispersion elements

Grating : rotating

Prism : rotating

NIR detector

PMT

Entrance Fiber

Grating : fixed

Multichannel

Detector

Entrance Fiber


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GENERAL GES5 DESCRIPTION:

- Xenon Lamp:75 W, Short arc, High brilliance- MgF2 Rotating Polarizer MgF2 : 6 Hz- Adjustable Analyzer

- Goniometer from 7 up to 90- Microspots : spot size : 400 m

- High resolution way :

-Spectral range : 190 nm up to 2000 nm- resolution < 0.5 nm- double monochromator: prism + grating- Photon counting PMT

-High speed way :- Spectral range : 190 nm up to 1700 nm- Measurement duration : few seconds- Fixed grating spectrograph

- CCD (1024x64 pixels)and OMA NIR (256 pixels) detectors


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Jones Formalism

BUT : Jones formalism can only workif there is no depolarisation effects induced by the material


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Optical system with no depolarisation effectsis characterized by this following Jones Matrix :

2X2 complex Matrix


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If the material is isotrope :

Ellipsometric Angles and measured simultaneously


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P =R(-).Px.R()

Jones Matrix for a Linear Polarizer : Px = 00 01

P =

cossin

sincos

Rotation Matix :

General expression for a polarizerwhere the main axis is oriented with an angle


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I = Edp . Edp* + Eds . Eds*

I (t) = I0 . ( 1 + Cos 2 (t) + Sin 2 (t) )

A : Angle between Analyser and plane of Incidence.(t) : Angle between Polariseur and plane of Incidence.


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Edp 1 0 Cos A Sin A rp 0 Cos P - Sin P 1 0 Ep

Eds 0 0 -Sin A Cos A 0 rs Sin P Cos P 0 0 Es=

Detector Analyser Rotation Sample Rotation Polariser Lamp

A : Angle between Analyser axis and Plane of Incidence.P(t) : Angle between Polarizer axis and Plane of Incidence.

I = Edp . Edp*

+ Eds . Eds*

I = I0 . ( 1 + Cos 2 P(t) + Sin 2 P(t) )


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S1 S4S2 S3 S1

Time

Intensity

S I P dP10

4

= ( )

S I P dP2

4

2

=

( )

S I P dP32

34

= ( )

S I P dP43

4

= ( )

[S1 - S2 -S3 + S4 ]

2 I0 =[S1 + S2 - S3 - S4 ]

2 I0 =[S1 + S2 + S3 + S4 ]

0 =

HADAMART TRANSFORM


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CALCULATION OF THE ELLIPSOMETRIC PARAMETERS

Cos 2 ATan 2 + Tan 2 A

0 =

Tan 2 - Tan 2 ATan 2 + Tan 2 A

= 2 Cos . Tan . Tan ATan 2 + Tan 2 A

=

1 + 1 -

Tan = Tan A .

1 - 2

Cos =


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DUV UV VISIBLE NIR

ELLIPSOMETRIC MEASUREMENT


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RESULTS ANALYSIS

Ti , ni , ki

Experimental Measurement

=Model Simulation ?

No

Yes

Cos

Tan

Experimental

Measurement

Physical ModelEstimated sample structure

- Film Stack and structure- Material n, k, dispersion- Composition Fraction of Mixture

REAL SAMPLE STRUCTURE

Non direct technique.

Need the use of modelsto interpret the measurementsand to get physical parameters of thelayers


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III - DATA ANALYSIS

- Which physical parameters can we get ?

- Sensisitivy of the technique

- Description of the main models used

- How to describe optical properties of thematerials.


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Spectroscopic ellipsometry sensitiviy

Phase variation : isextremelly sensitive to ultra

thin layers

Angle of incidence: 75

S t i lli t iti i


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0nm

10 A

Spectroscopic ellipsometry sensitiviy

Angle of incidence: 75

Sensitivity could reach

0.01 A

or 1 picometer


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Description of the main models used


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2 media model : Substrat alone

1

0Ambiant : air

Substratns = s - i.ks

rp = (ns.cos0 n0.coss)/(ns.cos0 + ns.cos1)

and

rs= (n0.cos0 - ns.coss)/(ns.cos0 + ns.cos1)

with = r

p/ r

s= Tan.exp(i.)

Fresnel equation and Snell-Descartes law:

ns = sin0.(1 + ((1-)/(1+))2.Tan20)1/2Direct inversion of the ellipsometric parameters to get substrate indices


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EPFL October 7th 2008 37Silicium Angle of incidence 75

2 media model : Substrat alone


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3 media model : one layeron known substrate


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Native oxide (SiO2) on SiliconSicr

SiO2 30.8

3 media model : one layer on known substrate


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SicrSiO2 1200 SiO2 layer on silicon



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Thick SiO2 layer on Silicon SicrSiO2 1,9838m


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Rp et Rs are periodic functionSame periods Idem for Tan and Cos

If a =75Period=/(nb2-0.93)0.5

For SiO2 filmnb =1.5 Period = /2.3

At =450 nm, period = 200 nm

Sensitivity to :In this case variation de 360 for a period = 200 nm

sensitivity given by instrument: 5.10-2

Thickness sensitivity : 0.01 A


M l il k


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Multilayer stack

Multilayer stack


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Interface relations : FresnelPropagation inside the layer :Interferences

-

Multilayer stack


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How to describe optical properties of thematerials

Index library Effective medium mixing laws Dispersion law Harmonic oscillators laws

Drude laws

Index Library


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Index Library

3 main type of materials:

Dielectric : transparent in the visible range but absorbing in the UV and haveabsorbing band in the IR.Transparent materials : Oxides (SiO2, TiO2) Fluorides (MgF2)

Optical filters, Anti reflective coatings, dielectric mirror (lasers).

Semi-conductors dispersives laws extremelly rich in the visible range linkedto the band structures. Could be metallic in the IR.Silicon : Si; Germanium : Ge; Gallium arsenide : GaAs;

Gallium nitride: GaN;

Carbon Silicon SiC (blue diode);

Metal highly absorbing in the visible.Infrared mirrors :Au, Al, Cu

Magnetic Materials : Co, Ni, Fe, Gd

Handbook of Optical Constants from Palick

SOPRA library Direct measurements on bulk materials


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Effective medium model


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Popular effective medium model

I t f d S f h


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Interface and Surface roughnesstreated by Effective medium Model


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- Model is convenient for physical or mechanical mixing.Ex : Porous material with inclusion of void.

- Model is not convenient for chemical mixingEx : Inclusion of atom in elementary cellsor variable atomic concentrationEx : Si(1-x)Gex.

- Model is not applicable when the size of inhomogeneitiesexceed few hundredth (1/100) of the wavelenght of the beam.

Limits of the effective medium model


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Dispersion Law

Sellmeier law :

42

CBAn ++= 53

FEDk ++=

Cauchy law :

k =0 for transparent materials

)(

)1(1)(

2

22

B

Ar

+=

53)(

EDCi

++=

1

3

32

1

4

2

2

3

2

2

2

1

0 ).(.)()(..

)(

++=++=

BB

BnkA

A

A

A

AnSiNx example


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4 absorption peaks

SiO2 in the Infrared range2222

0

2

3

2222

0

2

2

0

22

.)(

..

.)(

)(..

+

=+

=L

A

L

LAir

Harmonic Oscillators

Absorption band in the measured spectral rangeLorentz oscillators

A: Intensity.

L0: Middle wavelength.

: Oscillator width.



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100 nm of an absorbing layer on Silicon substrate

Spectroscopic Ellipsometry data Indices of the layer


Drude Law


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Wavelength ( m)

2 4 6 8 10 12 14 16

n, k

0.0

0.5

1.0

1.5

2.0

2.5

3.0

3.5

4.0

Measure and Drude law

fit on doped silicon

undoped silicon

( )

1

2

2 2= p

( )

2

2

2 2=

p

( )

Indices are fitted

using Drude law :

p : Plama frequency : diffusion frequency

Semi conductors indices are sensitive to the doping level in the IR range

Drude Law

N() doped Silicon N()

k() doped Siliconk() undoped Silicon


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=e

m*

Spectroscopic ellipsometry fit gives :

Plasma frequency p and Diffusion Frequency

Material Conductivity

Carrier Density

Carrier Mobility

= 0

2p

N m e

p

=

* 0

2

2

For the sample correspondingto the previous measurement :

N =1.6 1019 at./cm3

= 104 cm2 V-1 s-1

= 264 -1 cm-1

Drude Law


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Conclusion


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EPFL O t b 7th 2008 57

The measurement gives two independent angles: and

-Absolute measurement: do not need any reference.

-Extremely sensitive to very thin layers (less than a monolayer).-fast : get the full spectrum (190 nm up 1700 nm) in few seconds

-Non-direct technique: does not give directly the physical

parameters of the sample (thickness and index)-Need to use a modelto describe the sample.

Determination of : but also :

Thickness PorosityRefractive index : n Resistivity

Extinction coefficient : k

Molecular bounds

Introduction to Ellipsometry - EPFL 2008

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