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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
æ
ç
ç
S = ç
ç
ç
è
æ
ö
ç
÷
Q ÷
ç I0
=
ç I
U ÷
ç 45
÷
ç I
V ÷ø
rc
è
I
o
I
o
− I
90
− I
− 45
− I
lc
ö
÷
÷
÷
÷
÷
ø
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:
light
source
PSG
p
PSD
p
φ
s
s
Detector
Sample
Intensity of beam at the detector = SPSDT M SPSG
The matrix M includes:
Sample reflection characteristics
Intervening optics (windows and lenses)
Introduction
Isotropic Reflector:
Epin
Epout
Esin
Esout
Isotropic reflector
Mueller Matrix:
M reflector
æ 1
ç
ç− N
=ç
0
çç
è 0
−N
1
0
0
0
C
0
−S
N = cos(2ψ
ψ)
S = sin(2ψ
ψ) sin(∆
∆)
C = sin(2ψ
ψ) cos(∆
∆)
N2 + S2 + C2 = 1
0ö
÷
0÷
S÷
÷÷
Cø
Introduction
What we calculate:
light
source
PSG
p
PSD
p
φ
s
Detector
s
Sample
æ rpp
J = çç
è rsp
æ ρ
= rss çç
è ρ sp
rps ö æ Ê op / Ê ip
÷÷ = ç o i
rss ø çè Ê p / Ê s
ρ ps ö æ tan(ψ )e i∆
÷÷ = ç
1 ø ç tan(ψ sp )e i∆
è
Ê so / Ê ip ö
÷
o
i ÷
Ês / Ê s ø
tan(ψ ps )e
sp
i∆ ps
1
There is no difference between Mueller and Jones IF
there is no depolarization!
ö
÷
÷.
ø
Data Representations
Mueller-Jones matrices:
M = A ( J ⊗ J* ) A-1,
æ1
ç
ç1
A=ç
0
ç
ç0
è
0
0
0
1
0
1
−i
i
1 ö
÷
− 1÷
0 ÷
÷.
0 ÷ø
Assumes no depolarization!
ρ=
rp
rs
=
in
E out
E
p
p
E
out
s
E
in
s
= tan(ψ ) e
i∆
C + iS
=
1+ N
Data Representations
Anisotropic samples:
1
æ
ç
ç − N − α sp
M=ç
C + ξ1
ç ps
ç− S +ξ
ps
2
è
ρps =
ρsp =
− N − α ps
1 − α sp − α ps
C sp + ζ 1
− C sp + ζ 1
− C ps + ξ 1
S ps + ξ 2
C + β1
− S + β2
Cps + iSps
1+ N
Csp + iSsp
1+ N
S sp + ζ 2 ö
÷
− S sp + ζ 2 ÷
S + β2 ÷
÷
C − β 1 ÷ø
i∆ps
= tan(ψ ps )e
i∆ sp
= tan(ψ sp )e
N2 + S2 + C2 + Sps2 + Cps2 + Ssp2 + Csp2 = 1
i∆
æ ρ ρps ö æ tan(ψ )ei∆ tan(ψ ps )e ps ö
÷
÷÷ = ç
J = rss çç
i
∆
sp
ç
÷
1
è ρsp 1 ø è tan(ψsp )e
ø
Calculation of Reflection Coefficients
Single Interface: Fresnel Equations (1832):
0
φ0
1
φ1
r pp =
ñ 1 cos( φ 0 ) − ñ 0 cos( φ 1 )
ñ 1 cos( φ 0 ) + ñ 0 cos( φ 1 )
rss =
ñ 0 cos( φ 0 ) − ñ 1 cos( φ 1 )
ñ 0 cos( φ 0 ) + ñ 1 cos( φ 1 )
Snell’s Law (1621):
ξ = ñ0 sin φ0 = ñ1 sin φ1.
Calculation of Reflection Coefficients
Two Interfaces: Airy Formula (1833):
rpp ,ss =
b=
r1, pp ,ss + r2, pp ,ss exp(−2ib)
1 + r1, pp ,ss r2, pp ,ss exp( −2ib)
2πd f
λ
ñ f cos(φ f )
Calculation of Reflection Coefficients
Three or more interfaces: Abeles Matrices (1950):
Represent each layer by 2 Abeles matrices:
P j , pp
cos(φ j )
æ
ö
ç
−i
cos(b j )
sin(b j ) ÷
ñj
ç
÷
=ç
÷
ñj
çi
÷
sin(b j )
cos(b j )
ç cos(φ )
÷
j
è
ø
P j , ss
æ
ç
cos( b j )
=ç
ç iñ cos( φ ) sin( b )
j
j
è j
i
sin( b j ) ö
÷
ñ j cos( φ j ) ÷
cos( b j ) ÷ø
Matrix multiply:
N
M pp = χ 0, pp (∏ Pj , pp ) χ sub, pp
j =1
N
M ss = χ 0,ss ( ∏ P j ,ss ) χ sub ,ss
j =1
Calculation of Reflection Coefficients
Three or more interfaces: Abeles Matrices:
Substrate and ambient characteristic matrices:
χ 0 , pp
æ
ç 1
1ç
=
2 ç−1
ç
è
χ sub, pp
æ
cos( φ ) ö
ç1
÷
1
ñ0 ÷ χ
ç
0 , ss =
cos( φ ) ÷
2 ç1
ç
ñ0 ÷ø
è
1
æ
ö
æ cos(φ sub )
ö
0÷
0
ç
ç
÷ χ
= ç ñsub
sub,ss = ç ñsub cos(φ sub )
÷÷
÷
ç
ç
÷
1
0
1
0
è
ø
è
ø
Final Reflection coefficients:
rpp =
1
ö
÷
ñ0 cos( φ ) ÷
−1 ÷
ñ0 cos( φ ) ÷ø
M 21 , pp
M 11 , pp
rss =
M 21 , ss
M 11 , ss
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
4)
a)
surface roughness
b)
surface reconstruction
c)
surface oxides
Most tabulated data sets do not include error
limits
Measured optical functions of silicon depend on the face!
[near 4.25 eV = 292 nm, ε2(100)<εε2(111)]
Models for dielectric functions
Lorentz Oscillator Model (1895):
ε (λ ) = ñ(λ ) 2 = 1 + å
j
ε ( E ) = ñ( E ) 2 = 1 + å
j
A j λ2
λ2 − λo2 , j + iζ j λ
Bj
E o2, j − E 2 + iΓ j E
Sellmeier Equation:
ε = n2 = 1+ å
j
A j λ2
λ2 − λ2o , j
For Insulators
Cauchy (1830):
n = B0 +
å
j
Bj
λ2 j
Drude (1890):
ε (E) = 1 − å
j
Bj
E
1
E − iΓj
For Metals
Models for dielectric functions
Amorphous Materials (Tauc-Lorentz):
ε 2 ( E ) = 2n ( E ) k ( E ) =
A( E − E g ) 2
Θ( E − E g )
( E 2 − Eo2 ) 2 + Γ 2
E
∞
ξε (ξ )
2
ε 1 ( E ) = ε 1 (∞ ) + P ò 2 2 2 dξ
π Rg ξ − E
Five parameters:
Eg
Band gap
A
Proportional to the matrix element
E0
Central transition energy
Γ
Broadening parameter
ε1(∞
∞)
Normally = 1
Forouhi and Bloomer
(Forouhi and Bloomer Phys Rev. B 34, 7018 (1986).)
Extinction coefficient:
k FB ( E ) =
Refractive index:
A( E − E g ) 2
E 2 − BE + C
(a Hilbert transform)
∞
1
k (ξ ) − k ( ∞ )
dξ
n FB ( E ) = n ( ∞ ) + P ò
π −∞ ξ − E
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:
ε (E) = εo + å
j
Aj e
iφ j
E − E j + iΓ j
Can end up with MANY terms
Models for dielectric functions
Effective Medium theories:
ε j −εh
< ε > −ε h
=åfj
< ε > +γεh
ε j + γεh
j
Choice of host material:
1)
Lorentz-Lorentz: εh = 1
2)
Maxwell-Garnett: εh = ε1
3)
Bruggeman
γ
εh = <εε>
Depolarization factor ~2.
Fitting Models to Data
Figure of Merit:
Experimental quantities:
ρexp(λ
λ)
Calculated quantities:
λ, z)
ρcalc(λ,
z: Vector of parameters to be fit (1 to m)
film thicknesses,
constituent fractions,
parameters of optical function models, etc.
Minimize
2
N [ρ
(
λ
)
ρ
(
λ
,
)]
−
z
1
j
calc
j
exp
χ2 =
å
N − m − 1 j =1
σ (λ j ) 2
σ(λ
λ) = random and systematic error
Fitting Models to Data
Calculation Procedure:
1)
2)
Assume a model.
a.
Number of layers
b.
Layer type (isotropic, anisotropic, graded)
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. χ2>>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:
1 ∂2χ 2
α kl =
=
2 ∂ z k ∂z l
∂ρ calc ( λ j , z ) ∂ρ calc ( λ j , z )
1
å
2
∂z k
∂z l
j =1 σ ( λ j )
N
Inverse of curvature matrix: ε = α-1
Error in parameter zj = εjj.
Cross correlation coefficients: proportional to the
elements of ε.
Fitting Models to Data
Meaning of fitted parameters and errors:
Assume air/SiO2/Si structure
Parameterize SiO2 using Sellmeier dispersion
(λ
λo=93 nm):
ε = n2 = 1+ å
j
A j λ2
λ2 − λ2o , j
Two fit parameters: df, Af
Af
•
• • • • •• •
•
• • ••
•• •• ••• ••• • • •
•
σΑ • •••••• •••••••• •• •
• •
• •• • •• •• •• ••
•• • •• •
• • •
•
σd
df
Fitting Models to Data
An Example: a-SixNy:H on silicon:
3
SiN / c-Si
2
Re(ρpp)
1
0
-1
2
χ =0.96
-2
T-L
-3
Lorentz χ =3.84
-4
F&B
2
2
χ =47.8
3
Im(ρpp)
2
1
0
-1
-2
1
2
3
Energy (eV)
4
5
Fitting Models to Data
An Example: a-SixNy:H on silicon:
Model: 1)
2)
air
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, Γ, ε(∞
∞) and
Eg (F&B and T-L)
Fitting Models to Data
An Example: a-SixNy:H on silicon:
Lorentz
F&B
T-L
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
1.00±0.02
0.93±0.51
1.38±0.26
3.64
47.5
0.92
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
Roughness thick (nm)
ε(∞)
χ2
Roughness thick (nm)
Fitting Models to Data
An Example: a-SixNy:H on silicon:
0.2
exp-calc
Re(ρpp)
0.1
0.0
T-L
Lorentz
F&B
-0.1
-0.2
0.2
Im(ρpp)
0.1
0.0
-0.1
-0.2
1
2
3
Energy (eV)
4
5
Optical Functions from Ellipsometry
refractive index (n)
Optical Functions from Parameterization:
T-L
Lorentz
F&B
2.1
2.0
1.9
extinction coefficient (k)
1.8
0.015
0.010
0.005
0.000
1
2
3
4
5
Energy (eV)
Error limits:
Use the submatrix αs from the associated fitted
parameters.
Optical Functions from Ellipsometry
Newton-Raphson algorithm:
Solve:
λ, φ, nf, kf, …)) - Re(ρ
ρexp(λ
Re(ρ
ρcalc(λ
λ)) = 0
Im(ρ
ρcalc(λ
λ, φ, nf, kf, …)) - Im(ρ
ρexp(λ
λ)) = 0
Jacobian:
æ ∂ ρ re
ç
J = ç ∂n
çç ∂ ρ im
è ∂n
nnew = nold + δn;
Propagate errors!
∂ ρ re
∂k
∂ ρ im
∂k
ö
÷
÷
÷÷
ø
where δn = -J-1 ρ
Optical Functions from Ellipsometry
Optical functions of semiconductors:
Dielectric function from air/substrate system:
ε = ε 1 + i ε 2 = sin 2 (φ ){1 + [
1− ρ 2
] tan 2 (φ )}
1+ ρ
Only valid if there is no overlayer (almost never true)
If there is a thin film, Drude showed:
2
4π d n f − 1
∆ = ∆ o (ns , k s ) + K
λ
n 2f
Optical Functions from Ellipsometry
Optical functions of semiconductors:
Pseudo-dielectric functions of silicon with 0, 0.8, and 2.0
nm SiO2 overlayers.
8
0.0
0.8
2.0
<n>
6
4
2
0
6
<k>
4
2
0
10
10
10
10
6
< α > (1/cm )
5
4
3
1
2
3
Energy (eV)
4
5
Optical Functions from Ellipsometry
Optical functions of semiconductors:
Error limits of the dielectric function of silicon:
0.05
δn
δk
Error in n and k
0.04
0.03
0.02
0.01
0.00
1
2
3
Energy (eV)
4
5
Optical Functions from Ellipsometry
Optical functions of thin films:
Sm all grain poly silicon
Re(ρ)
1
0
-1
Im(ρ)
0.5
0.0
-0.5
1
2
3
Energy (eV)
4
5
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.
B.
1
air
2
surface roughness (BEMA)
3
T-L model for film
4
Lorentz model for a-SiO2
5
c-Si
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:
8
n
6
4
2
6
k
4
2
0
10
10
10
10
6
α (1/cm)
c-Si
lg p-Si
sg p-Si
a-Si
5
4
3
1
2
3
Energy (eV)
4
5
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.