ellipsometry data analysis: a tutorialweb.eecs.umich.edu/~fredty/wise2000/jellison.pdf ·...

Ellipsometry

Data Analysis: a Tutorial

G. E. Jellison, Jr.

Solid State Division

Oak Ridge National Laboratory

WISE 2000

University of Michigan

May 8-9, 2000

Motivation

The Opportunity:

Spectroscopic Ellipsometry (SE) is sensitive to

many parameters of interest to thin-film science,

such as

• Film thickness

• Interfaces

• Optical functions (n and k).

But

SE data is not meaningful by itself.

Therefore

One must model the near-surface region to get

useful information.

Outline

Data representations

Calculation of reflection from thin films

Modeling of optical functions (n and k)

Fitting ellipsometry data

Examples

Introduction

What we measure:

Stokes Vector of a light beam

�����

�

�

�����

�

�

−−−

=

�����

�

�

�����

�

�

=−

lcrc

oo

IIIIII

I

VUQI

4545

900S

Io ≥≥≥≥ (Q2 + U2 + V2 )1/2

To transform one Stokes vector to another:

Sout = M Sin

M = Mueller matrix (4X4 real)

Introduction

What we measure:

PSG PSD

Samples s

p pDetector

sourcelight

φ

Intensity of beam at the detector = SPSDT M SPSG

The matrix M includes:

Sample reflection characteristics

Intervening optics (windows and lenses)

Introduction

Isotropic Reflector:

Mueller Matrix:

�����

�

�

�����

�

�

−

−−

=

CSSC

NN

0000

001001

reflectorM

N = cos(2ψψψψ)

S = sin(2ψψψψ) sin(∆∆∆∆)

C = sin(2ψψψψ) cos(∆∆∆∆)

N2 + S2 + C2 = 1

Epin

Esin

Isotropic reflector

Epout

Esout

Introduction

What we calculate:

PSG PSD

Samples s

p pDetector

sourcelight

φ

��

�

�

��

�

�=��

�

����

�= i

sos

is

op

ip

os

ip

op

/ ÊÊ/ ÊÊ/ ÊÊ/ ÊÊ

sssp

pspp

rrrr

J

��

�

�

��

�

�=��

�

����

�= ∆

∆∆

1)tan()tan()tan(

1 sp

ps

isp

ips

i

sp

psss e

eer

ψψψ

ρρρ

.

There is no difference between Mueller and Jones IF

there is no depolarization!

Data Representations

Mueller-Jones matrices:

M = A ( J ⊗⊗⊗⊗ J* ) A-1,

�����

�

�

�����

�

�

−

−=

0001101001

1001

ii

A.

Assumes no depolarization!

NiSCe

EEEE

rr i

ins

outs

inp

outp

s

p

++==== ∆

1)tan(ψρ

Data Representations

Anisotropic samples:

�����

�

�

�����

�

�

−+−++−+++−+

+−+−−−−−++−−

=

1222

2111

21

21

11

ββξξββξξ

ζζαααζζα

CSSSSCCCSCN

SCN

psps

psps

spsppsspsp

spspps

M

psips

pspsps e

NiSC ∆=

++

= )tan(1

ψρ

spisp

spspsp e

NiSC ∆=

++

= )tan(1

ψρ

N2 + S2 + C2 + Sps2 + Cps

2 + Ssp2 + Csp

2 = 1

��

�

�

��

�

�=��

�

����

�= ∆

∆∆

1)tan()tan()tan(

1 sp

ps

isp

ips

i

sp

psss e

eer

ψψψ

ρρρ

J

Calculation of Reflection Coefficients

Single Interface: Fresnel Equations (1832):

)cos()cos()cos()cos(

1001

1001

φφφφ

ññññrpp +

−=

)cos()cos()cos()cos(

1100

1100

φφφφ

ññññrss +

−=

Snell’s Law (1621):

ξξξξ = ñ0 sin φφφφ0 = ñ1 sin φφφφ1.

1

0 φ0

φ1

Calculation of Reflection Coefficients

Two Interfaces: Airy Formula (1833):

)2exp(1)2exp(

,,2,,1

,,2,,1, ibrr

ibrrr

ssppsspp

ssppssppsspp −+

−+=

)cos(2

fff ñ

db φ

λπ

=

Calculation of Reflection Coefficients

Three or more interfaces: Abeles Matrices (1950):

Represent each layer by 2 Abeles matrices:

�����

�

�

�����

�

�−

=)cos()sin(

)cos(

)sin()cos(

)cos(

,

jjj

j

jj

jj

ppj

bbñ

i

bñ

ib

φ

φ

P

���

�

�

���

�

�

=)cos()sin()cos(

)cos()sin(

)cos(,

jjjj

jj

jj

ssj

bbiññ

bib

φφP

Matrix multiply:

ppsub

N

jppjpppp ,

1,,0 )( χχ ∏

=

= PM

∏=

=N

jsssubssjssss

1,,,0 )( χχ PM

Calculation of Reflection Coefficients

Three or more interfaces: Abeles Matrices:

Substrate and ambient characteristic matrices:

����

�

�

����

�

�

−=

0

0,0 )cos(1

)cos(1

21

ñ

ñpp φ

φ

χ ����

�

�

����

�

�

−=

)cos(11

)cos(11

21

0

0,0

φ

φχ

ñ

ñss

���

�

�

���

�

�=

01

0)cos(, sub

sub

ppsub ñφ

χ ���

�

�

���

�

�=

01

0)cos(

1, subsubsssub ñ φχ

Final Reflection coefficients:

pp

pppp M

Mr

,11

,21=

ss

ssss M

Mr

,11

,21=

Calculation of Reflection Coefficients

Anisotropic materials:

2 2X2 Abeles matrices become one 4X4 Berreman

matrix (1972)

Reason: s- and p- polarization states are no longer

eigenstates of the reflection.

Inhomogeneous layers:

If a layer has a depth-dependent refractive index, there

are two options:

1) Build up many very thin layers

2) Use interpolation approximations

Models for dielectric functions

Tabulated Data Sets:

1) Usually good for substrates

2) Not good for thin films

3) Even for substrates: problems

a) surface roughness

b) surface reconstruction

c) surface oxides

4) Most tabulated data sets do not include error

limits

Measured optical functions of silicon depend on the face!

[near 4.25 eV = 292 nm, εεεε2(100)<εεεε2(111)]

Models for dielectric functions

Lorentz Oscillator Model (1895):

� +−+==

j jjo

j

iA

ñλζλλ

λλλε 2

,2

22 1)()(

� Γ+−+==

j jjo

j

EiEEB

EñE 22,

2 1)()(ε

Sellmeier Equation:

� −+==

j jo

jAn 2

,2

22 1

λλλ

ε For Insulators

Cauchy (1830):

�+=j

jjB

Bn 20 λ

Drude (1890):

11)(

jj

j

iEEB

EΓ−

−= �ε For Metals

Models for dielectric functions

Amorphous Materials (Tauc-Lorentz):

EEE

EEEEA

EkEnE g

o

g )()(

)()()(2)( 2222

2

2

−ΘΓ+−

−==ε

ξξ

ξξεπ

εε dE

PEgR�∞

−+∞= 22

211

)(2)()(

Five parameters:

Eg Band gap

A Proportional to the matrix element

E0 Central transition energy

ΓΓΓΓ Broadening parameter

εεεε1(∞∞∞∞) Normally = 1

Forouhi and Bloomer

(Forouhi and Bloomer Phys Rev. B 34, 7018 (1986).)

Extinction coefficient:

CBEEEEA

Ek gFB +−

−= 2

2)()(

Refractive index: (a Hilbert transform)

�∞

∞− −∞−+∞= ξ

ξξ

πd

EkkPnEnFB

)()(1)()(

Problems: • kFB(E)>0 for E<Eg. Unphysical.

• kFB(E) →→→→ constant as E→→→→∞∞∞∞. Experiment states that

k(E) →→→→ 0 as 1/E3 as E→→→→∞∞∞∞.

• Time-reversal symmetry required [k(-E) = -k(E)].

• Hilbert Transform ≠≠≠≠ Kramers Kronig [k(∞∞∞∞) = 0].

Models for dielectric functions

Models for Crystalline materials:

Critical points, excitons, etc. in the optical spectra

make this a very difficult problem!

Collections of Lorentz oscillators:

� Γ+−+=

j jj

ij

o iEEeA

Ejφ

εε )(

Can end up with MANY terms

Models for dielectric functions

Effective Medium theories:

� +−

=+><−><

j hj

hjj

h

h fγεεεε

γεεεε

Choice of host material:

1) Lorentz-Lorentz: εεεεh = 1

2) Maxwell-Garnett: εεεεh = εεεε1

3) Bruggeman εεεεh = <εεεε>

γγγγ Depolarization factor ~2.

Fitting Models to Data

Figure of Merit:

Experimental quantities: ρρρρexp(λλλλ)

Calculated quantities: ρρρρcalc(λ,λ,λ,λ, z)

z: Vector of parameters to be fit (1 to m)

film thicknesses,

constituent fractions,

parameters of optical function models, etc.

Minimize

�=

−−−

=N

j j

jcalcj

mN 12

2exp2

)()],()([

11

λσλρλρ

χz

σσσσ(λλλλ) = random and systematic error

Fitting Models to Data

Calculation Procedure:

1) Assume a model.

a. Number of layers

b. Layer type (isotropic, anisotropic, graded)

2) Determine or parameterize the optical functions of

each layer

3) Select reasonable starting parameters.

4) Fit the data, using a suitable algorithm and Figure

of Merit

5) Determine correlated errors in z and cross

correlation coefficients

If the Figure of Merit indicates a bad fit (e.g. χχχχ2>>1), go

back to 1).

Fitting Models to Data

“…fitting of parameters is not the end-all of parameter

estimation. To be genuinely useful, a fitting procedure

should provide (i) parameters, (ii) error estimates on

the parameters, and (iii) a statistical measure of

goodness-of-fit. When the third item suggests that the

model is an unlikely match to the data, then items (i)

and (ii) are probably worthless. Unfortunately, many

practitioners of parameter estimation never proceed

beyond item (i). They deem a fit acceptable if a graph

of data and model “looks good.” This approach is

known as chi-by-eye. Luckily, its practitioners get what

they deserve.”

Press, Teukolsky, Vettering, and Flannery, Numerical

Recipes (2nd ed., Cambridge, 1992), Ch. 15, pg. 650.

Fitting Models to Data

Levenberg-Marquardt algorithm:

Curvature matrix:

�= ∂

∂∂

∂=

∂∂∂=

N

j l

jcalc

k

jcalc

jlkkl zzzz 1

2

22 ),(),()(

121 zz λρλρ

λσχα

Inverse of curvature matrix: εεεε = αααα-1

Error in parameter zj = εεεεjj.

Cross correlation coefficients: proportional to the

elements of εεεε.

Fitting Models to Data

Meaning of fitted parameters and errors:

Assume air/SiO2/Si structure

Parameterize SiO2 using Sellmeier dispersion

(λλλλo=93 nm):

� −+==

j jo

jAn 2

,2

22 1

λλλ

ε

Two fit parameters: df, Af

A

d

f

f

σd

σΑ ••••

• ••

••

••

••

•

••

••

••

•••

•

••

•

•

••

•••

••

•• •

••

••• •

••

••

••

•••• •

••

••

•••

• •

••

•

•

•

Fitting Models to Data

An Example: a-SixNy:H on silicon:

1 2 3 4 5

-2

-1

0

1

2

3

Energy (eV)

Im(ρ

pp)

-4

-3

-2

-1

0

1

2

3SiN / c-Si

T-L χ2=0.96

Lorentz χ2=3.84

F&B χ2=47.8

Re(

ρ pp)

Fitting Models to Data

An Example: a-SixNy:H on silicon:

Model: 1) air

2) surface roughness

Bruggeman EMA (50% air, 50% a-SiN)

3) a-SiN (3 models)

Lorentz

Forouhi and Bloomer

Tauc-Lorentz

4) interface

Bruggeman EMA (50% Si, 50% a-SiN)

5) silicon

Fitting parameters: d2, d3, d4, A, Eo, ΓΓΓΓ, εεεε(∞∞∞∞) and

Eg (F&B and T-L)

Fitting Models to Data

An Example: a-SixNy:H on silicon:

Lorentz F&B T-L Roughness thick (nm) 2.1±0.3 4.9±0.7 1.9±0.3 Film thickness (nm) 197.8±0.7 195.6±1.1 198.2±0.4 Interface thick (nm) 0.6±0.3 -0.6±0.6 -0.1±0.2 A 201.9±4.6 4.56±1.9 78.4±12.7 Εo (eV) 9.26±0.05 74.4±30.5 8.93±0.47

Γ (eV) 0.01±0.02 0.74±23.5 1.82±0.81

Eg (eV) ---- 2.85±0.48 4.35±0.09 ε(∞) 1.00±0.02 0.93±0.51 1.38±0.26

χ2 3.64 47.5 0.92

Roughness thick (nm) 2.4±0.3 4.7±0.6 1.8±0.2 Film thickness (nm) 198.6±0.4 195.3±0.9 198.1±0.3 A 202.5±1.4 5.03±0.35 97.7±3.2

Εo (eV) 9.26±0.02 70.7±2.2 9.61±0.0.03

Γ (eV) 0.01±0.01 40.1±10.8 3.07±0.33

Eg (eV) ---- 2.97±0.27 4.44±0.04

χ2 3.75 48.0 0.96

Fitting Models to Data

An Example: a-SixNy:H on silicon:

1 2 3 4 5-0.2

-0.1

0.0

0.1

0.2

Energy (eV)

Im(ρ

pp)

-0.2

-0.1

0.0

0.1

0.2exp-calc

T-L Lorentz F&B

Re(

ρ pp)

Optical Functions from Ellipsometry

Optical Functions from Parameterization:

1 2 3 4 5

0.000

0.005

0.010

0.015

Energy (eV)

extin

ctio

n co

effic

ient

(k)

1.8

1.9

2.0

2.1 T-L Lorentz F&B

refra

ctiv

e in

dex

(n)

Error limits:

Use the submatrix ααααs from the associated fitted

parameters.

Optical Functions from Ellipsometry

Newton-Raphson algorithm:

Solve:

Re(ρρρρcalc(λλλλ, φφφφ, nf, kf, …)) - Re(ρρρρexp(λλλλ)) = 0

Im(ρρρρcalc(λλλλ, φφφφ, nf, kf, …)) - Im(ρρρρexp(λλλλ)) = 0

Jacobian:

����

�

�

����

�

�

∂∂

∂∂

∂∂

∂∂

=

kn

knimim

rere

ρρ

ρρ

J

nnew = nold + δn; where δn = -J-1 ρ

Propagate errors!

Optical Functions from Ellipsometry

Optical functions of semiconductors:

Dielectric function from air/substrate system:

)}(tan]11[1){(sin 222

21 φρρφεεε

+−+=+= i

Only valid if there is no overlayer (almost never true)

If there is a thin film, Drude showed:

2

2 14),(f

fsso n

ndKkn−

+∆=∆λπ

Optical Functions from Ellipsometry

Optical functions of semiconductors:

Pseudo-dielectric functions of silicon with 0, 0.8, and 2.0

nm SiO2 overlayers.

1 2 3 4 5103

104

105

106<α> (1/cm)

Energy (eV)

0

2

4

6

<k>

0

2

4

6

8

0.0 0.8 2.0

<n>

Optical Functions from Ellipsometry

Optical functions of semiconductors:

Error limits of the dielectric function of silicon:

1 2 3 4 50.00

0.01

0.02

0.03

0.04

0.05

δn δk

Energy (eV)

Erro

r in

n an

d k

Optical Functions from Ellipsometry

Optical functions of thin films:

1 2 3 4 5

-0.5

0.0

0.5

Im(ρ

)

Energy (eV)

-1

0

1

Small grain poly silicon

Re(

ρ)

Optical Functions from Ellipsometry

Optical functions of thin films:

Method of analysis:

A. Restrict analysis region to interference oscillations.

Parameterize the optical functions of the film.

1 air

2 surface roughness (BEMA)

3 T-L model for film

4 Lorentz model for a-SiO2

5 c-Si

B. Fit data to determine thicknesses and Lorentz

model parameters of a-SiO2.

C. Calculate optical functions of thin film using

Newton-Raphson.

Optical Functions from Ellipsometry

Optical functions of thin films:

1 2 3 4 5103

104

105

106α (1/cm)

Energy (eV)

0

2

4

6

k

2

4

6

8

c-Si lg p-Si sg p-Si a-Si

n

Parting Thoughts

SE is a powerful technique, but modeling is critical.

Modeling should use an error-based figure of merit

Does the model fit the data?

Calculate correlated errors and cross-correlation

coefficients.

When used properly, SE gives very accurate thicknesses

and values of the complex dielectric function.