Ellipsometry drawings by expert

1. Ellipsometry
Matt Brown
Alicia Allbaugh
Electrodynamics II Project
10 April 2001

2. Ellipsometry
A method
of probing
surfaces
with light.

3. Introduction
 History
 Methodology
 Theory
 Types of Ellipsometry
 Applications
 Summary

4. History
 Fresnel derived his equations which
determine the Reflection/Transmission
coefficients in early 19th century.
Ellipsometry used soon thereafter.
 Last homework assignment
Electrodynamics I.
 Ellipsometry became important in
1960’s with the advent of smaller
computers.

5. Methodology
 Polarized light is reflected at an oblique angle
to a surface
 The change to or from a generally elliptical
polarization is measured.
 From these measurements, the complex
index of refraction and/or the thickness of the
material can be obtained.

6. Theory
 Determine r = Rp/Rs (complex)
 Find r indirectly by measuring the shape of
the ellipse
 Determine how e varies as a function of
depth, and thickness L of transition layer.
Note: We will focus on the case of very thin films.
In this case, only the imaginary part of r matters.

7. Maxwell’s equations for a wave incident
On a discontinuous surface. (Gaussian Units)
y
z
x H
z
H
y
E
t
c 







e
z
x
y H
x
H
z
E
t
c 







e
x
y
z H
y
H
x
E
t
c 







e
z
y
x E
y
E
z
H
t
c 







1
x
z
y E
z
E
x
H
t
c 







1
y
x
z E
x
E
y
H
t
c 







1
Boundary Conditions
2
1 x
x E
E  2
1 y
y E
E  2
2
1
1 z
z E
E e
e 
2
1 x
x H
H  2
1 y
y H
H  2
1 z
z H
H 
1
2 x
z
y

8. Derivation of Drude Equation
Fundamentals of Derivation
 Concept: Integrate a Maxwell Equation along z
over transition region of depth L. Result will be a
new Boundary Condition.
 Fundamental Approximations:
 a.
 b. We assume certain field components ,
which vary slowly along z, are constant.
X
Y
Z
I R
T
L

Y
1

L

Incident
beam

x
H

x
H

p
H

p
H
Example: Since Hx+= Hx-, and
/L<<1, Hx1~Hx2.

9. z
y
x E
y
E
z
H
t
c 







1
Integrate along z over L

 




L
y
L
x dz
E
z
dz
H
t
c 0
0
1
X
Y
Z
I R
T
L

Derivation of Drude Equation
Assumption that is uniform
With respect to y
z
E
z
y
x E
y
E
z
H
t
c 







1
0
Incident
beam

x
H

x
H

p
H

p
H

10. Derivation of Drude Equation
1
2
0
y
y
y
L
E
E
E
z
dz 




1
2 y
y
x E
E
H
t
c
L




Assumption that varies little:
Since , = constant.
x
H
x
L
x H
t
c
L
dz
H
t
c 





0
1

  x
x H
H
x
y
y H
t
c
L
E
E



 1
2
and
Substituting
Rearrangement yields
Incident
beam

x
H

x
H

p
H

p
H
2
1 x
x H
H 

11. x
z
y E
z
E
x
H
t
c 







1
2
2
1
1 z
z E
E e
e 
Integrate
)
(
1
1
2
0
0
x
x
L
z
L
y E
E
dz
E
x
dz
H
t
c









y
H z
z E
e
and vary
little over L


 







L
z
z
z
L
z
z
z
L
z dz
E
x
dz
E
x
dz
E
x 0
0
0
1
e
e
e
e
2
2
2
2
1 z
y
x
x E
x
q
H
t
c
L
E
E






 e 

L
dz
q
0
1
e

y
H

y
H


  p
p E
D e

p
D

z
D

z
D
X
Y
Z
I R
T
L
Y

where
;

12. Similarly, we now find new B.C. for and
x
H y
H
x
y
y H
t
c
L
E
E



 1
2
2
2
2
2
1 z
y
x
x E
x
q
H
t
c
L
E
E






 e 2
2
1
1 z
z E
E e
e 
2
1 z
z H
H 
2
2
2
1 y
z
x
x E
t
c
p
H
x
L
H
H






 2
2
1 x
y
y E
t
c
p
H
H






L
dz
q
0
1
e
New complete Boundary Conditions
X
Y
Z
I R
T
L
Y

Where


L
dz
p
0
e

13. We now solve Maxwell’s equations with
these new Boundary Conditions
x
y
y H
t
c
l
E
E



 1
2
)
(
0
t
r
k
i
e
E
E 



0
ˆ
)
( 


 n
T
R
Einc
E
k
H 
 ˆ
e
y
y
y
y
inc
y T
E
R
E
E 

 2
,
1 ,
)
]
[
1
( 2
2
,
c
L
Cos
i
E
R
E y
y
y
inc e
 Y



X
Y
Z
I R
T
L
Y

2
p
H
Boundary
Condition
Relate
H and E
Form of E field (to
satisfy Maxwell eq.)
Continuity

14. Again solve Maxwell’s equations
with these new Boundary Conditions
2
2
2
2
1 z
y
x
x E
x
q
H
t
c
l
E
E






 e
)
(
0
t
r
k
i
e
E
E 



0
ˆ
)
( 





 n
T
k
R
k
E
k t
r
i
E
k
H 
 ˆ
e
p
p
p
inc
p T
R
E
E 

 p2
,
1 E
))
]
[
(
]
[
(
]
[
)
( 2
2
2
2
, q
Sin
l
c
i
Cos
E
Cos
R
E p
p
p
inc e
e

 Y


Y


X
Y
Z
I R
T
L
y

Boundary Condition
Relate
H and E
Form of E field (to
satisfy Maxwell eq.)
Continuity
Note on notation:
Subscript p refers to
component parallel to
incident plane (x-z plane),
and subscript s refers to
perpendicular (same as y)
component.

15. )
]
[
1
( 2
2
,
c
L
Cos
i
E
R
E y
y
y
inc e
 Y



))
]
[
(
]
[
(
]
[
)
( 2
2
2
2
, q
Sin
L
c
i
Cos
E
Cos
R
E p
p
p
inc e
e

 Y


Y


))
]
[
(
]
[
(
]
[
)
( 2
2
2
2
1
,
c
p
Sin
L
i
Cos
E
Cos
R
E y
y
y
inc 
Y

Y

 e

e

e
)
]
[
(
)
( 2
2
1
,
c
p
Cos
i
E
R
E p
p
p
inc Y


 
e
e
This results in 4 relations between , , and .
inc
E 2
E
R

16. 2
1
2
2
1
2
1
2
1
2
)
]
[
2
(
]
[
]
[
(
2
]
[
]
[
)
]
[
2
(
]
[
]
[
(
2
]
[
]
[
e
e

e



e
e

e
e

e



e
e

Sin
q
L
Cos
pCos
i
Cos
Cos
Sin
q
L
Cos
pCos
i
Cos
Cos
E
R
p
p


Y

Y



Y

Y


Algebraically eliminate transmission terms.
Example: Parallel components


L
dz
q
0
1
e 

L
dz
p
0
e
Notice that if we assume p and q terms to be
Proportional to L, the imaginary parts of top and
Bottom are proportional to

L
where

17. )
]
[
]
[
]
[
]
[
]
[
2
1
(
]
[
]
[
]
[
]
[
2
1
2
2
2
2
2
2
1
1
2
1
2
Y

Y


Y


Y

Y


Cos
Cos
Sin
q
L
pCos
Cos
i
Cos
Cos
Cos
Cos
E
R
p
p
e

e
e
e
e


e
e

e
e

)
]
[
]
[
]
[
2
1
(
]
[
]
[
]
[
]
[
2
1
2
2
2
2
1
2
1
Y



Y

Y


Cos
Cos
p
L
Cos
i
Cos
Cos
Cos
Cos
E
R
s
s
e

e
e


e
e

e
e

Approximation for when L<< such that terms
in second order of L/ can be neglected.

18. )
]
[
]
[
(
]
[
]
[ 2
2
2
1
2
2
1
2
1
2
2 
e

e
e
e
e
e

e Cos
Sin
Cos
Cos 


Y

2
1
2
[
[ e
e
e

e 


Y

 

 Cos
Cos
]
[
]
[ 2
1 Y
 Sin
Sin e

e
Using Snell’s Law,
We get
Set polarization at 45 degrees. Then
s
p
s
s
p
p
R
R
R
E
E
R

Again, keeping only terms to first order in L/, and using binomial expansion,


e

e


e
e
e
e




)
]
[
]
[
]
[
]
[
4
1
(
]
[
]
[
2
2
2
1
2
2
1
1
2
Cos
Sin
Sin
Cos
i
Cos
Cos
R
R
s
p



Y

Y











L
dz
q
L
p
0
2
1
2
1
2
1
)
)(
(
)
(
e
e
e
e
e
e
e
e
e

where

19. 

e

e


e
e
e
e




)
]
[
]
[
]
[
]
[
4
1
(
]
[
]
[
2
2
2
1
2
2
1
1
2
Cos
Sin
Sin
Cos
i
Cos
Cos
R
R
s
p



Y

Y











L
dz
q
L
p
0
2
1
2
1
2
1
)
)(
(
)
(
e
e
e
e
e
e
e
e
e








L
s
p
dz
R
R
0
2
1
2
1
2
1 )
)(
(
]
Im[
e
e
e
e
e
e
e
e
e


r
For thin films, we often take to be the dielectric constant
Of air, to be that of our substrate, and to be constant
in the film. Then
Recall that at Brewster’s angle Ep is minimized
So near Brewster’s Angle, we get
1
e
2
e e
L
R
R
s
p
e
e
e
e
e
e
e
e
e


r
)
)(
(
]
Im[ 2
1
2
1
2
1 





This is the
Drude
Equation.

20. Types of Ellipsometry
 Null Ellipsometry
 Photometric Ellipsometry
 Phase Modulated Ellipsometer
 Spectroscopic Ellipsometry

21. We choose
our polarizer
orientation
such that the
relative phase
shift from
Reflection is
just cancelled
by the phase
shift from the
retarder.
We know that the relative phase
shifts have cancelled if we can null
the signal with the analyzer
Null Ellipsometry







s
p
r
r
Im
seek
We r

22. Example Setup
Phase modulated ellipsometer







s
p
R
R
Im
seek
We r

23. How to get r,an example.
Phase Modulated Ellipsometry

24. )
ˆ
2
1
ˆ
2
1
(
0 p
s
E
E 


The polarizer polarizes light to
45 degrees from the incident plane.
How to get r,an example.
Phase Modulated Ellipsometry

25. )
ˆ
2
1
ˆ
2
1
(
0 p
s
E
E 


)
ˆ
]
[
exp[
ˆ
(
2
0
0
0
p
t
Sin
i
s
E
E 




The birefringment modulator
introduces a time varying phase shift.
extrema.
the
determines
unchanged.
is
on
polarizati
,
0
]
[
at
that
Note
0
0

 
t
Sin
How to get r,an example.
Phase Modulated Ellipsometry

26. For a continuous
interface,
)
ˆ
]
[
exp[
ˆ
(
2
0
0
0
p
t
Sin
i
s
E
E 




]])
[
exp[
ˆ
]
exp[
(
2
0
0
0
t
Sin
i
i
r
s
i
r
E
E p
p
s
s 


 



Upon reflection both the parallel
and perpendicular components are
changed in phase and amplitude.
.
s
p 
 
How to get r,an example.
Phase Modulated Ellipsometry
s
p 
 
For a discontinuous
interface,

27. ]])
[
exp[
ˆ
]
exp[
(
2
0
0
0
t
Sin
i
i
r
s
i
r
E
E p
p
s
s 


 



]])
[
exp[
ˆ
]
exp[
(
2
0
0
0
t
Sin
i
i
r
s
i
r
E
E p
p
s
s 


 


How to get r,an example.
Phase Modulated Ellipsometry

28. Photomultiplier Tube measures intensity.
Lockin Amplifier














 ]]
[
[
2
1
)
( 0
0
2
2
2
2
2
t
Sin
Cos
r
r
r
r
r
r
E
I
p
s
p
s
p
s 


s
p 
 


Where
   
 
   
 
...
]
3
[
2
]
[
2
]
[
...
]
2
[
2
]
[
]]
[
[
0
3
0
0
1
0
0
2
0
0
0
0










t
Sin
J
t
Sin
J
Sin
Cos
J
J
Cos
t
Sin
Cos
o









How to get r,an example.
Phase Modulated Ellipsometry
Note: The J’’s are the Bessel Functions

29. Lockin Amplifier
  small.
is
since
,
2
1
2
2
s
p
s
p
s
p
s
p
r
r
r
r
r
r
r
r
a 














765
.
0
:
J
of
zero
a
to
set
to
modulator
nt
birefringe
orient the
We
0
0
0

   
  











...
]
2
[
]
[
2
]
[
]
[
2
1
0
0
2
0
0
1
2
2
t
Cos
Cos
aJ
t
Sin
Sin
aJ
r
r
I p
s





s
p 
 


Where
r
2
]
Im[
2
]
[
2
]
[ 












s
p
s
p
r
r
Sin
r
r
a Sin
   
  










...
]
2
[
]
[
2
]
[
4
1
0
0
2
0
0
1
2
2
t
Cos
Cos
aJ
t
Sin
J
r
r
I p
s



r


At the Brewster Angle,

30.  r

 0
1
2
0
J
V 

  ]
[
0
2
2 0


 Cos
J
V 

   
  










...
]
2
[
]
[
2
]
[
4
1
0
0
2
0
0
1
2
2
t
Cos
Cos
aJ
t
Sin
J
r
r
I p
s



r


.
2




Now we can use a calibration procedure to
Find the proportionality of r
 to
0
V
How to get r,an example.
Phase Modulated Ellipsometry
We find the Brewster angle by adjusting until
Which is where
,
0
0
2 

V

31. Applications
 Determining
the thickness
of a thin film
 Focus of this
presentation

32. Applications - Continued
 Research
 Thin films, surface structures
 Emphasis on accuracy and precision
 Spectroscopic
 Analyze multiple layers
 Determine optical constant dispersion relationship
 Degree of crystallinity of annealed amorphous silicon
 Semiconductor applications
 Solid surfaces
 Industrial applications in fabrication
 Emphasis on reliability, speed and maintenance
 Usually employs multiple methods

33. Ellipsometry
 Ellipsometry can measure the oxide depth.
 Intensity doesn’t vary much with film depth
but  does.

34. Other Methods
 Reflectometry
 Microscopic Interferometry
 Mirau Interferometry

35. Reflectometry
 Reflectometry
 Intensity of reflected to incident (square of
reflectance coefficients).
 Usually find relative reflectance.
 Taken at normal incidence.
 Relatively unaffected by a thin dielectric
film.
 Therefore not used for these types of thin films.

36. Ellipsometry
 Ellipsometry can measure the oxide depth.
 Intensity doesn’t vary much with film depth
but  does.

37. Reflectometry

38. Reflectometry
 Can be more accurate for thin metal films.

39. Microscopic Interferometry
 Uses only
interference
fringes.
 Only useful for
thick films and/or
droplets
 Thickness h>/4

40. Mirau Interferometry
 Accuracies to 0.1nm
 x is less than
present ellipsometry
 At normal incidence.
 Kai Zhang is
constructing one for
use at KSU.

41. Ellipsometry
 Allows us to probe the surface structure of
materials.
 Makes use of Maxwell’s equations to
interpret data.
 Drude Approximation
 Is often relatively insensitive to calibration
uncertainties.

42. Ellipsometry
 Accuracies to the Angstrom
 Can be used in-situ (as a film grows)
 Typically used in thin film applications
 For more information and also this
presentation see our website:
html://www.phys.ksu.edu/~allbaugh/ellipsometry

43. Bibliography
1. Bhushan, B., Wyant, J. C., Koliopoulos, C. L. (1985). “Measurement of
surace topography of magnetic tapes by Mirau interferometry.” Applied
Optics 24(10): 1489-1497.
2. Drude, P. (1902). The Theory of Optics. New York, Dover Publications, Inc.,
p. 287-292.
3. Riedling, K. (1988). Ellipsometry for Industrial Applications. New York,
Springer-Verlag Wein, p.1-21.
4. Smith, D. S. (1996). An Ellipsometric Study of Critical Adsorption in Binary
Liquid Mixtures. Department of Physics. Manhattan, Kansas State University:
276, p. 18-27.
5. Tompkins, H. G. (1993). A User's Guide to Ellipsometry. New York,
Academic Press, Inc.
6. Tompkins, H. G., McGahan, W. A. (1999). Spectroscopic Ellipsometry and
Reflectometry: A User's Guide. New Your, John Wiles & Sons, Inc.
7. Wang, J. Y., Betalu, S., Law, B. M. (2001). “Line tension approaching a first-
order wetting transition: Experimental results from contact angle
measurements.” Physical Review E 63(3).