Udo_ME2006.ppt, © Udo v. Toussaint, 11. July 2006 1 Bayesian Analysis of Ellipsometry Measurements...

udo_ME2006.ppt, © Udo v. Toussaint, 11. July 20061

Bayesian Analysis of Ellipsometry Measurements

Udo v. Toussaint and Thomas Schwarz-Selinger

• Ellipsometry

• Example

• Bayesian Analysis

• Results and Conclusion


immer noch komplexer Teilchenzoo!

Detection of the change of polarization of linearly polarized light due to the reflection at the sample surface

shutter

Principles of Ellipsometry

here: single wavelength rotating analyzer ellipsometer


Camera

Laser

polarizer

analyzer

controller ellipsometer

powersupplies

highpressure

lampmonochromator

sample stage

Jobin Yvon PZ 2000 Ellipsometer

wavelength: 632 nm (400 -800 nm)

spot size: 10 x 30 m (10 m)

motorized xyz sample stage

positioning accuracy: 30 m

sample thickness: 2 mm

measurement range: Å - 30 m

measurement accuracy: > 0. 1 Å (1 nm)



Principles of Ellipsometry: Reflection of light I

solid sample: Fresnel equations

.cosˆcosˆ

cosˆ2

,cosˆcosˆ

cosˆ2

,cosˆcosˆ

cosˆcosˆ

,cosˆcosˆ

cosˆcosˆ

2211

11

012,

1221

11

012,

2211

2211

012,

2112

2112

012,

nnn

E

Et

nnn

EE

t

nnnn

EE

r

nnnn

EE

r

s

ts

p

tp

s

rs

p

rp

inn ˆcomplex index of refraction:n

ncphasevdefined by:

: extinction coefficient


Principles of Ellipsometry: Reflection of light II

multilayer system:

123*

123123222312

2312123 ,cosˆ

2,

)2iexp(1)2iexp(

rrRdnrrrr

r

Snell’s law: 2211 cosˆcosˆ nn



.tans

p ier

r

Measured data:

detection of the change of polarization of linearly polarized light due to the reflection at the sample surface

in fact we measure and (ellipsometric angles):

each measurement delivers only 2 pieces of information

but depends on: incident angle ni and i and di of each medium is

p

r

r

for a single measurement result is ambiguous if neither ni nor i is known!


immer noch komplexer Teilchenzoo!


the plane:,

Use of empirical models:


Duoplasmatron (Ivan Bizyukov): a-C:H flux probe (bombardment by 1 keV D+)

But sometimes…


Duoplasmatron (Ivan Bizyukov): a-C:H flux probe (bombardment by 1 keV D+)

But sometimes…

?

?


Duoplasmatron (Ivan Bizyukov): a-C:H flux probe (erosion by 1 keV D+)

But sometimes…

measurement

model for the plasma deposited a-C:H film


Surface reconstruction from interference images

Interference images from ellipsometry:

2 data values (angles) per measurement point


Bayesian Model

Likelihood: Gaussian likelihood

2

2

2

2

2

,,exp

2

,,exp

1,,,|,

dngdnf

ZIdnp

Ill-posed problem: no. of parameters larger than no. of data

Use prior-information: optical properties vary on a different length scale

Two-scale approach: Nested grids for d and n,

Prior: Bounded, flat :

MinMax xx

Ixp

1

|

dnx

,,


Bayesian Model• Posterior: Bayes theorem

Ip

IdnpIdnpIndp

|,

,,,|,|,,,,,,|

******

• Model specifications: 4 layers, 6 unknowns (in 2 layers)

domain size inner grid: 3x3 -5x5

• Optimization with respect to the parameters: Results were disappointing

• Why?



Ambigous solutions possible Important: Stay on correct branch of solution

Virtually indistinguishable

solutions: identical 2



Interference images from ellipsometry



And what about the edge?


Conclusions & Outlook

• Ellipsometry is a great non-perturbing surface analytical tool - but

• ML - evaluation of data may not be straightforward or even misleading

• Prior information is essential

• Derived parameter estimation algorithm works reliable

Outlook:

• Model comparison for number of layers• Improved consideration of correlations

Conclusions:


experimental setup ICP


20 30 40 50

1,6

1,8

2,0

2,2

2,4 U

SB=-200 V

USB

=- 30 V U

SB=floating

USB

=floating, a-C:D

Bre

chu

ng

sin

de

x n

p

H/(H+C) (at %)

film properties like- hydrogen content- density- refractive index

are closely correlated

T.Schwarz-Selinger, A. von Keudell, W.Jacob, J.Appl. Phys. 86, 3988 (1999)

quantification of ellipsometry data


tender spot in general

detection of atomic hydrogen in the plasma environment

needed: hydrogen sensor

12 scans

and measuring the erosion depth with ex-situ-ellipsometry

is complicted in the plasma environment

erosion of a dense a-C:H-film at 650 K

T. Schwarz-Selinger, W. Jacob, A. von Keudell, JVST A. 18 (3), 995 (2000)


Profilometry versus Ellipsometry/Reflectometry

Profilometry: mechanical contact with the sample topography

Ellipsometry/Reflectometry: optical response of the sample

thickness x refractive index

general take home message:

Udo_ME2006.ppt, © Udo v. Toussaint, 11. July 2006 1 Bayesian Analysis of Ellipsometry Measurements...

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