Angle resolved Mueller Polarimetry, Applications to periodic structures

Angle resolved Mueller Polarimetry, Applications to periodic structures

PhD DefenseClément Fallet

Under the supervision of Antonello de Martino

2

Motivations and introduction to polarization

Design and optimization of a Mueller microscope

Fourier space measurements : application to semiconductor metrology

Real space measurements : example of characterization of beetles

Conclusions and perspectives

Outline of the presentation

PhD Defense - Clément Fallet - October 18th

PhD Defense - Clément Fallet - October 18th 3

Various applications of polarization of light over the past decades.

A lot of studies, but mainly driven by classical ellipsometry spectral resolution (discrete angle, averaged over the illuminated region)

Spatial dependency of polarimetric properties is only qualitatively assessed

Motivations of the study


What we propose, discrete wavelength : Angular resolution (averaged over the field) Spatial resolution (averaged over the angles)

Possibility to use the same system for both measurements.

Evolution of a classical bright-field microscope ease of use

Motivations of the study


LET’S TALK ABOUT POLARIZATION

6

Introduction to polarization


𝑆𝑜𝑢𝑡

11 12 13 14

21 22 23 24

31 32 33 34

41 42 43 44

M M M MM M M MM M M MM M M M

M=𝑆𝑜𝑢𝑡=𝑀 .𝑆𝑖𝑛

7

A word about polarimeters


CCD

camera

PS

GP

SA

(Stokes Polarimeter)Mueller PolarimeterB = A.M.W

M = A-1.B.W-1

W = [S1, S2, S3’ S4]PSG Basis Stokes vectors

At = [S’1, S’2, S’3, S’4]PSA Basis Stokes vectors

A and W must be as close as possible to unitaryTheir condition numbers must be optimized(E.Compain 1999, S. Tyo 2000, M. Smith, 2002)

Calibration : eigenvalue method No instrument modelling(E.Compain, Appl. Opt 38, 3490 1999)


DESIGN & OPTIMIZATION OF A MUELLER MICROSCOPE


Specifications of the set-up

Complete Mueller polarimeter at discrete λ▪ Complete measurement of the Mueller Matrix

(4 by 4 matrix). First setup by S. Ben Hatit. 2 imaging modes

▪ Fourier Space we’re not imaging the sample itself but the back

focal plane of a high-aperture microscope objective

▪ Real space Design based on classical microscopy

10

Epi-Illumination scheme


CCD

Back focal plane

Sample

retractable lens

Beamsplitter

34

1 – Aperture diaphragm2 – Field diaphragm3 – PSG : Polarization State Generator4 – PSA : Polarization State Analyser5 – Aperture Mask

1 2

5

Source

Aperture image : angularly resolvedReal image : spatially resolved

Interferential filter

Strain-freeMicroscope objective

LColl L1 L2

Lim1

Lim2

Lim3


Illumination arm

Collectionlens

Aperturediaphragm

L1 L2

Back focal plane

Fielddiaphragm

Rays

em

ergi

ng

from

the

sour

ce


Detection arm

400nm pitch grating


Strain-free Nikon objectives▪ Specified for quantitative polarization▪ No polarimetric signature in real space

But small dichroism and birefringence when used in Fourier space calibration of the objective with well-characterized reference samples (c-Si, SiO2 on c-Si) (method explained in the manuscript)

Choice of the objectives


Aperture Vs Field

objective Full Field Maximum Aperture

5x 360µm 0-8°

20x 90µm 0-26°

50x 36µm 0-53°

100x 18µm 0-64°

with our current pinhole, the field (spot size) can be discreased down to 10µm Use of a pinhole with smaller diameter to achieve 5µm


Description of the measurements

𝑟 ∝ sin𝜃

c-Si wafer, 633nm

-0.2

0.2

-0.2

0.2

𝑟𝑝𝑟 𝑠

=tan (Ψ )𝑒𝑖 Δ

dichroism retardance


From (x,y) to (s,p)

(x,y) (s,p)Isotropic sample

psx

y

-0.2

0.2

-0.2

0.2

𝑀 (𝑥 , 𝑦 )=𝑅 (𝜑 ) .𝑀 (𝑠 ,𝑝 ) .𝑅 (−𝜑 )



APPLICATION TO OVERLAY CHARACTERIZATION IN THE SEMICONDUCTOR INDUSTRY


To keep increasing the power of microprocessors, we need to decrease the size of the transistors

Transistor fabrication = layer by layer With the decrease in size (currently

22nm), better metrology is required

Motivations


We engrave specially designed marks in the scribe lines

We measure :▪ The profile (critical dimension …) : ASML contract

Metrology requirements

▪ The overlay (shift between the 2 structures) : MuellerFourier contract with Horiba Jobin Yvon and CEA-LETI

CD


Overview of the metrology techniques

•Reflectometry, classical ellipsometry (q = 70°, f =0°, 0.75 – 6.3 eV)•Mueller matrix polarimetry (spectroscopic or angle-resolved)

State of the art AFM (gold standard for CD metrology) CD-SEM Optical techniques :


More about optical techniques Image Based overlay (IBO) :

▪ box in box or bar in bar marks imaged with a bright-field microscope.

▪ Grating based Advanced Imaging Method (AIM) by KLA-TENCOR

▪ Limited by the aberrations and size of the marks ( 15x15 – 30x30 µm²)

Diffraction Based Overlay (DBO) : Collection of the light diffracted, scattered and reflected by the sample and analysis as a function of either the wavelength (spectroscopic) or the angle of incidence

▪ Empirical DBO : no modeling of the structure needed but at least 2 measurements of calibrated targets

▪ Model-Based DBO : overlay as a parameter of the fit. Only 1 measurement needed but model-dependent. Limited by the model and the size of the marks (30x60µm², ASML Yieldstar)


THE ITRS RoadMap

2011 1.6nm

2012 1.4nm


Properties of the Mueller matrix

4,32,1 jiMMMMMM

rightji

leftij

rightij

rightij

leftji

leftij

The Mueller matrix elements are sensitive to the profile structure and its asymmetry. For a structure presenting an asymmetry,

we have :

where left and right stand for the direction of the shift in the structure.

25

Simulation of the Mueller matrix of a superposition of 2 gratings with the same pitch but with a lateral shift

Simulation by Rigorous coupled wave analysis : All the electromagnetic quantities (E, H and ε,μ) are expanded in Fourier series. Simulations by T.Novikova and M.Foldyna

Simulations and RCWA



Simulations of structures of interest

Piece-wise layer dielectric function

Continuity of field assured by Lalanne / Li factorization rules

Propagation of S matrices

Based on our knowledge on Mueller matrix symmetries, we compute to define possible estimators

tMM

0 5 10 15 20 25 300

0.10.20.3

R² = 0.999998726171532

Overlay (nm)

Estim

ator

tMM


Description of the test samples Test samples designed and manufactured @ CEA-LETI

Nominal overlays (nm) : ±150, ± 100, ± 50, ± 40, ± 30, ± 20 ± 10, 0 Nominal CDs L1 and L2 also vary to extensively test the simulations 84 different grating combinations

50µm


Sample 1 : CD N1 150 N2 300

Normalized Mueller matrix measurement EstimatortMME

-0.2

0.2

-0.2

0.2


Scalar estimatorManually selected maskKept constant for all measurements of the same CD comination

Scalar estimator :

E = <E14>mask

E14


2 possibilities▪ 1 – Check the linearity of the estimator based

on the overlay actually present on the wafer. Gold standard established by Advanced Imaging Method (AIM)

▪ 2 – Measurement of the uncontrolled overlay (overlay in addition of the nominal overlay)

How to use our estimator?


VALIDATION OF THE LINEARITY OF ESTIMATOR E14


Sample 1 (N1 150 N2 300) : Linearity

-20 0 20 40 60 80 1000

2

4

6

8

10

12

f(x) = NaN x + NaNR² = 0 Estimator overlay Y

AIM overlay (nm)Gold standard

Valu

e of

the

estim

ator


Sample 1 : comparison with simulations

-60 -40 -20 0 20 40 60 80 100

-0.4

-0.3

-0.2

-0.1

0

0.1

0.2

0.3

0.4

R² = 0.993582309213263

R² = 0.999937738839223R² = 0.999976456213567

Max(E14(mask)) simu

Linear (Max(E14(mask)) simu)

Mean(E14(mask)) simu

Overlay (nm)

valu

e of

the

estim

ator


Sample 2 : CD N1 130 N2 300

-40.00 -20.00 0.00 20.00 40.00 60.00 80.00

-0.2

-0.15

-0.1

-0.05

0

0.05

0.1

0.15

0.2

0.25

0.3

f(x) = 0.00429680543212432 x + 0.00909874060495459R² = 0.966683795799765

mean(E14) Overlay Y

AIM overlay (nm)

estim

ator


Sample 2 : CD N1 130 N2 300

-60 -40 -20 0 20 40 60

-0.4

-0.3

-0.2

-0.1

0

0.1

0.2

0.3

f(x) = − 0.00549447722353445 x − 0.0300912498760537R² = 0.997823073653477

mean(E14) Overlay X

AIM overlay (nm)

estim

ator


Influence of the CD

-60 -40 -20 0 20 40 60

-0.14

-0.12

-0.1

-0.08

-0.06

-0.04

-0.02

0

0.02

0.04

f(x) = − 0.00141753887375114 x − 0.0664328571428571

f(x) = NaN x + NaNInfluence of the CD

overlay Y 200 200Linear (overlay Y 200 200)overlay Y 200 220

nominal overlay (nm)Specified value

valu

e of

the

estim

ator

-45nm -25nm


Estimator OK linear with overlay measured by AIM, which is considered as gold standard.

Consistency between X and Y overlays. The slope highly depends on the CD of

the gratings. Value of the experimental estimator

smaller than predicted by simulations.

Conclusion


MEASUREMENTS OF THE UNCONTROLLED OVERLAY


We distinguish the nominal overlay (specified) and real overlay

The nominal overlay is a controlled bias, intentionally introduced.

Only the uncontrolled overlay is relevant

Definitions

𝑜𝑣𝑟𝑒𝑎𝑙=𝑜𝑣𝑛𝑜𝑚+𝑜𝑣𝑢𝑛𝑐


I

Linear fit on the measurements

Method 1

𝐸14=𝑆∗𝑜𝑣𝑛𝑜𝑚+𝑜𝑓𝑓𝑠𝑒𝑡

𝑜𝑣𝑢𝑛𝑐=− 𝑜𝑓𝑓𝑠𝑒𝑡𝑆

Given by linear regression -50 -40 -30 -20 -10 0 10 20 30 40 50

-0.16-0.14-0.12

-0.1-0.08-0.06-0.04-0.02

-2.77555756156289E-170.020.04

f(x) = − 0.0014175389 x − 0.0664328571

Overlay Y

nominal overlay (nm)

Estim

ator


(H)

Method 2


Verification of H

Method 1

Method 2

AIM overlay (nm)

Module 10 N1 170 N2 300, overlay Y

Method 2 is validated for high nominal overlays


Correlation between AIM et Mueller

-100 -50 0 50 100 150

-100

-50

0

50

100

150

200

f(x) = 1.04558846108261 x + 1.52931313231716R² = 0.967448625485969

Correlation AIM - Mueller Overlay Y

AIM overlay (nm)

Mue

ller o

verla

y (n

m)


Correlation between AIM et Mueller

-80.00 -60.00 -40.00 -20.00 0.00 20.00 40.00 60.00

-80

-60

-40

-20

0

20

40

60

f(x) = 0.94503672902141 x + 0.708960842034468R² = 0.97077398340729

Correlation AIM - Mueller Overlay X

AIM overlay (nm)

Mue

ller o

verla

y (n

m)


Map of the overlay on a field

Map of the uncontrolled overlay (all measurement in nm)


TMU : total measurement uncertainty

A few quality estimators


Total measurement uncertainty (TMU) for commercial instruments

▪ AIM : TMU ~ 2nm (2008)

▪ Yieldstar : TMU = 0,2nm (2011)

▪ Nanometrics : TMU ~ 0,4nm (2010)

Comparisons with existing apparatus


Characterization of the overlay with a (fast), non-destructive technique. No modelling required but 2 very-well characterized structures for calibration

Uncertainty relatively small ~ 2nm Measurements in 20 x 20µm² boxes

Conclusions


Very good linearity of the scalar estimator respect to the overlay defect (R² between 0,94 and 0,99)

However, experimental values of the estimators are lower than what simulation predicted.

Estimators are very sensitive to the chosen mask

Conclusions (2)


Possibility to go down to 5 x 5µm² boxes with the correct pinhole

Automatic selection of the mask Increase the repeatability of the

measurements to decrease Tool Induced Shift and its variability to decrease total uncertainty

Integrate CD measurement through fitting of the Mueller matrix to approach Ausschnitt’s MOXIE (Metrology Of eXtremely Irrational Exuberance)

Perspectives


MEASUREMENTSON

BEETLES


A twisted multilayer structure : Bouligand structures

Each layer consists of a chitin structure with uniaxial anysotropy

Organization of the cuticle

L. Besseau and M.-M. Giraud-Guille, J. Mol. Biol., no. 251, pp. 197–202, 1995.

10µm


Fit of spectroscopic Mueller ellipsometry Optical model of the cuticle (K. Järrendahl)

Spatial homogeneity is assumed; but need of a more complex model to take into account the spatial variations

Modeling of the structure

Image from K. Järrendahl.


Compare the results obtained on same species with different characterization methods

Characterize the spatial variations of the polarimetric response to improve the model

Purpose of this study


Variable Angle Spectroscopic ellipsometer RC2

Angular range 20°-70° 2θ configuration Average on the field Spectral resolution Only the specular

reflection

Angle resolved Mueller polarimeter

All incidence at a time Average on the angle

Spatial resolution All the light emitted at a

certain angle (reflection + scattering)

Comparisons of the results


Cetonia aurata

Cetonia aurata 5x imageImaged area 360µm

20x imageImaged area 90µm


Cetonia aurata

M14

20X

[ 1 0 0 − 10 0 0 00 0 0 0

−1 0 0 1

]

[1 0 0 00 𝑎 0 00 0 −𝑎 00 0 0 𝑏

]


Chrysina argenteola

20x imageImaged area 90µm


Chrysina argenteola

M14

20X


Difficult to accurately compare the results obtained with different techniques.

But still, common features arise Only a preliminary work, a lot remains to

be done. To our knowledge, nobody has ever

published spatially resolved Mueller matrices for beetles

Conclusions


CONCLUSIONS & PERSPECTIVES


PERSPECTIVES


Understand the relationship between helicoidal structures and circular dichroism

Mimic the cuticle of beetles

Chiral structures

From G. z. Radnoczi et al. ,Physica status solidi. A. Applied

research, vol. 202, no. 7, pp. R76–R78.

Mueller Matrix @ 633nm M14

-0.2

0.2

-0.2

0.2


Periodic structuresSol-gel deposited silica spheres

Real image with 100x

M12

M34

Angle resolved MM

Hexagonal symmetry visible in both the structure and the Mueller matrix


CONCLUSIONS


Optimization of a Mueller microscope▪ Better illumation scheme Modified Köhler▪ Good calibration of the objective without any

prior modelling but only a (Ψ,Δ) matrix assumption

Measurements in both real and reciprocal space, different kind of applications presented

Conclusions


In Fourier space▪ Characterization of the overlay with a (fast),

non-destructive technique. No modelling required but 2 very-well characterized structures

▪ Uncertainty relatively small ~ 2nm In real space

▪ Accurate spatial characterization of entomological structures

▪ Major step for the study of the auto-organized structures

Conclusions


Acknowledgements Financial support of the French National Research Agency (ANR) through the joint project MuellerFourier with CEA-LETI and Horiba Jobin-Yvon. Hans Arwin, Kenneth Järrendahl and Roger Magnusson at LiU. Special thanks to Tatiana Novikova and Bicher Haj Ibrahim for their help and support.


Thank you



Calibration of the objective

Assumptions : ▪ Objective can be described by a (Ψ,Δ)

matrix.▪ The MM in forward and backward directions

are equal = Mobj

By measuring an isotropic sample (eg. c-Si wafer), we can calibrate the objective

iobj

oobj TMTM 1 i

objo

obj TMTM 1 iobj

oobj TMTM 1


Calibration of the objective

Mmeas = Mobj * McSi * Mobj

(Ψ,Δ) matrices commute Mmeas = Mobj² * McSi

Δmeas = 2 Δobj + ΔcSi

tanΨmeas = tanΨobj² * tanΨcSi


Results on objective calibration

Objective calibrated with cSi @633nmDifference between calibration with cSi and SiO2Difference between calibration @532nm and 633nm


Calibration of reflectivity

M11 is not calibrated in the ECM.

B = τ A’.M.W’.Isource with τ, total transmission of the device

M = 1/c .A’-1.B.W’-1 with c= τ.Isource

By measuring well-known samples, we can calibrate the factor c.


AIM marks clockwiseanticlockwise

y marksOverlay specified along x

x marksoverlay specified along Y

10

10

10

AIM marks 30x30 µ2

20

20

5

x grating

y grating

Level 1 Level 2

Details of the mark


Best results so far, N1 300 N2 180

-60 -40 -20 0 20 40 60

-0.3

-0.2

-0.1

0

0.1

0.2

0.3

f(x) = − 0.00573610047238372 x − 0.00474275000000001R² = 0.989273527602751

OVY

-60 -40 -20 0 20 40 60

-0.4

-0.3

-0.2

-0.1

0

0.1

0.2

0.3

0.4

f(x) = − 0.00691868272727273 x − 0.017844R² = 0.994721736748336

OVX

Value of E14 versus nominal overlay in nm for overlay along x and y axis

Main features :- the uncontrolled overlay is

close to 0.- Highest slope in the

measured samples

Is there a correlation between the slope and the uncontrolled overlay?


Intrensinc properties of the MM

A Stokes non-diagonalizable Mueller matrix (NSD MM) : theory

Image and equation from Ossikovski et al, Opt. Lett. 34, 974-976 (2009)


Intrensic properties of the MM


Beetles, natural occurrence of NSD MM

The MM can be regarded as the weighted average of 3 components

1000010000100001

1000010000100001

1001000000001001

ndM

LCP Mirror HWP

From Ossikovski et al., Opt. Lett. 34, 2426-2428 (2009)


Sum decomposition of the MM


DOP ellipse


Calibration of Bouligand structures

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Angle resolved Mueller Polarimetry, Applications to periodic structures

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