Transcript Ellipsometry - Kansas State University

Ellipsometry
Matt Brown
Alicia Allbaugh
Electrodynamics II Project
10 April 2001
Ellipsometry
A method
of probing
surfaces
with light.
Introduction






History
Methodology
Theory
Types of Ellipsometry
Applications
Summary
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.
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.
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.
z
1
y
x
2
Maxwell’s equations for a wave incident
On a discontinuous surface. (Gaussian Units)
e 


Ex 
Hz  Hy
c t
y
z
1 


H x  E y  Ez
c t
z
y
e 


Hx  Hz
z
x
1 


Hy 
Ez  Ex
c t
x
z


Ez  H y  H x
c t
x
y
1 


Hz 
Ex  E y
c t
y
x
c t
Ey 
e 
Boundary Conditions
Ex1  Ex 2
Ey1  Ey 2
e1Ez1  e 2 Ez 2
H x1  H x 2
H y1  H y 2
H z1  H z 2
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. 
L
I
 R
L
Z
Y
X Y
 1
b. We assume certain field components ,
which vary slowly along z, are constant.

Incident
beam
H p
Example: Since Hx+= Hx-, and
/L<<1, Hx1~Hx2.
H x
H x
H p
T
Derivation of Drude Equation
1 


H x  E y  Ez
c t
z
y
Incident
beam
H p
H x
Assumption that E is uniform
z
With respect to y
H x
H p
0
1 


H x  E y  Ez
c t
z
y
I 
Integrate along z over L
1 

H x dz   E y dz

c t 0
z
0
L
L
R
L
Z
Y
T
X
Derivation of Drude Equation
Inc ident
Assumption that H x varies little:
beam
Since H  H , H  H = constant.
x
x
x1
x2
1 
L 
H x  dz 
Hx
c t
c t
0
L
and

0 dz z E y  E y 2  E y1
L
H x
H x
H p
Substituting
L 
H x  E y 2  E y1
c t
Rearrangement yields
Ey2
H p
L 
 E y1 
Hx
c t
1 


Hy 
Ez  Ex ;
c t
x
z
e1Ez1  e 2 Ez 2
1 

H y dz   Ez dz  ( Ex 2  Ex1 )

c t 0
x
0
Z
Y
T
Y
X
L
H y
Dz 
Dp   e  E p 
Dp
H y and e z Ez vary
H y
little over L

 ez

1
0 x Ez dz  0 x e z Ez dz  x e z Ez 0 e z dz
L
R
L
Integrate
L

I
L
L
L 

E x1  E x 2 
H y 2  qe 2
Ez 2
c t
x
Dz 
L
1
where q   dz
e
0
Similarly, we now find new B.C. for
H x and H y
New complete Boundary Conditions
L

E x1  E x 2 
H y 2  qe 2 E z 2
c t
x

p 
H x1  H x 2  L H z 2 
Ey2
x
c t
L 
E y 2  E y1 
Hx
c t
p 
H y1  H y 2 
Ex 2
c t
Where
L
1
q   dz
e
0

I
L
p   dze
0
e1Ez1  e 2 Ez 2
H z1  H z 2
R
L
Z
Y
X
T
Y
We now solve Maxwell’s equations with
these new Boundary Conditions
Boundary
Condition
Relate
H and E
Ey2
H  e kˆ  E
Form of E field (to
satisfy Maxwell eq.)
Continuity
l 
 E y1 
Hx
c t

I
i ( k r t )
E  E0e
( Einc  R  T )  nˆ  0
R
L
Z
Y
X
Ey1  Einc, y  Ry , Ey 2  Ty
L
Einc, y  R y  E y 2 (1  iCos[Y ] e 2 )
c
Y
T
H p2
Again solve Maxwell’s equations
with these new Boundary Conditions
Boundary Condition
Relate
H and E
Note on notation:
l 

Ex1  Ex 2 
H y 2  qe 2 Ez 2
c t
x
Subscript p refers to
component parallel to
incident plane (x-z plane),
and subscript s refers to
perpendicular (same as y)
component.
H  e kˆ  E
Form of E field (to
satisfy Maxwell eq.)
i ( k r t )
E  E0e
L
Z
Continuity
(ki  E  kr  R  kt  T )  nˆ  0
E p1  Einc, p  Rp
 R
I
Ep2  Tp
Y
X
y T
i
( Einc, p  R p )Cos[ ]  E2 p (Cos[Y ] 
( e 2 l  Sin[Y ]2 e 2 q))
c
This results in 4 relations between
Einc , R , and E2 .
L
Einc, y  R y  E y 2 (1  iCos[Y ] e 2 )
c
i
( Einc, p  R p )Cos[ ]  E p 2 (Cos[Y ] 
( e 2 L  Sin[Y ]2 e 2 q))
c
p
( Einc, y  Ry ) e1 Cos[ ]  E2 y ( e 2 Cos[Y ]  i ( L e 2 Sin[Y ]2  ))
c
p
( Einc, p  R p ) e1  E p 2 ( e 2  iCos[Y ] )
c
Algebraically eliminate transmission terms.
Example: Parallel components
Rp
Ep

Cos[ ] e 2  Cos[Y ] e1  i
2
( pCos[ ]Cos[Y ]  ( L  qe 2Sin[ ]2 ) e1e 2

2
Cos[ ] e1  Cos[Y ] e 2  i
( pCos[ ]Cos[Y ]  ( L  qe 2Sin[ ]2 ) e1e 2

L
where
1
L
q   dz
p   dze
0
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

Approximation for when L<< such that terms
in second order of L/ can be neglected.
Rp
Ep

Cos[ ] e 2  Cos[Y ] e1
Cos[ ] e 2  Cos[Y ] e1
e1  pCos[Y ]  Le 2  qe 2 2 Sin[Y ]2
(1  2iCos[ ]
)
2
2
e 2Cos[ ]  e1Cos[Y ]
Rs Cos[ ] e1  Cos[Y ] e 2

Es Cos[ ] e1  Cos[Y ] e 2
Le 2  p
(1  2iCos[ ]
)
2
2
e 2Cos[ ]  e1Cos[Y ]
R p Es R p
Set polarization at 45 degrees. Then

E p Rs Rs
Using Snell’s Law,
e1 Sin[ ]  e 2 Sin[Y]
We get
eCos[ 2  e Cos[Y  e1  e 2
e1  e 2
e 2Cos[ ]  e1Cos[Y] 
(e1Sin[ ]2  e 2Cos[ ]2 )
e2
2
2
Again, keeping only terms to first order in L/, and using binomial expansion,
Cos[  Y]
4 e 2 e1
Cos[ ]Sin[ ]2

(1  i
)
2
2
Rs
Cos[  Y]
 e1  e 2 e1Sin[ ]  e 2Cos[ ]
Rp
L
where
  p  L(e1  e 2 )  qe1e 2  
0
(e  e1 )(e  e 2 )
e
dz
Cos[  Y]
4 e 2 e1
Cos[ ]Sin[ ]2

(1  i
)
2
2
Rs
Cos[  Y]
 e1  e 2 e1Sin[ ]  e 2Cos[ ]
Rp
L
  p  L(e1  e 2 )  qe1e 2  
(e  e1 )(e  e 2 )
dz
e
Recall that at Brewster’s angle Ep is minimized
So near Brewster’s Angle, we get
0
 e1  e 2
r  Im[ ] 
Rs
 e1  e 2
Rp
This is the
L

0
(e  e1 )(e  e 2 )
e
dz
Drude
Equation.
For thin films, we often take e 1 to be the dielectric constant
Of air, e 2 to be that of our substrate, and e to be constant
in the film. Then
R p  e1  e 2 (e  e1 )(e  e 2 )
r  Im[ ] 
L
Rs
 e1  e 2
e
Types of Ellipsometry

Null Ellipsometry

Photometric Ellipsometry


Phase Modulated Ellipsometer
Spectroscopic Ellipsometry
Null Ellipsometry
We choose
our polarizer
orientation
such that the
relative phase
shift from
Reflection is
just cancelled
by the phase
shift from the
retarder.
 rp 
We seek r  Im 
 rs 
We know that the relative phase
shifts have cancelled if we can null
the signal with the analyzer
Example Setup
Phase modulated ellipsometer
 Rp 
We seek r  Im

 Rs 
How to get r,an example.
Phase Modulated Ellipsometry
How to get r,an example.
Phase Modulated Ellipsometry
The polarizer polarizes light to
45 degrees from the incident plane.

1
1
E  E0 (
sˆ 
pˆ )
2
2
How to get r,an example.
Phase Modulated Ellipsometry
The birefringment modulator
introduces a time varying phase shift.

1
1
E  E0 (
sˆ 
pˆ )
2
2
 E0
E
( sˆ  exp[i 0 Sin[ 0t ] pˆ )
2
Notethatat Sin[0t ]  0,
polarization is unchanged.
 0 determinestheextrema.
How to get r,an example.
Phase Modulated Ellipsometry
Upon reflection both the parallel
and perpendicular components are
changed in phase and amplitude.
 E0
E
( sˆ  exp[i 0 Sin[ 0t ] pˆ )
2
For a discontinuous
interface,  p   s .
For a continuous
interface,  p   s
 E0
E
(rs exp[i s ]sˆ  rp exp[i p  i 0 Sin[0t ]])
2
How to get r,an example.
Phase Modulated Ellipsometry
E0
E
(rs exp[ i s ]sˆ  rp exp[ i p  i 0 Sin[ 0t ]])
2
 E0
E
(rs exp[i s ]sˆ  rp exp[i p  i 0 Sin[0t ]])
2
How to get r,an example.
Phase Modulated Ellipsometry
Photomultiplier Tube measures intensity.
2


2rs rp
2
2 
I  E  (rs  rp ) 1  2 2 Cos[   0 Sin[ 0t ]]
 r r

s
p


Lockin Am plifier
Where   p   s
Cos[   0 Sin[0t ]]  Cos[]J 0  0   2 J 2  0 Cos[20 ]  ...
 Sin[]2 J1  0 Sin[0t ]  2 J 3  0 Sin[3ot ]  ...
Note: The J’’s are the Bessel Functions
a
Lockin Am plifier
2rp  rs 
 rp 
1   
 rs 
2
2
rp
rs
, since
rp
rs
is small.
We orient t hebirefringent
modulat ort o set  0 t o a
zero of J 0 :  0  0.765
1  2aJ1  0 Sin[]Sin[ 0t ]

I  rs  r 




2
aJ

Cos
[

]
Cos
[
2

t
]

...
2
0
0



2
2
p

At the Brewster Angle,
Where   p   s
rp
 rp 
a Sin[]  2  Sin[]  2 Im[ ]  2 r
rs
 rs 
1  4 J1  0 rSin[ 0t ]

I  rs  r 

 2aJ2  0 Cos[]Cos[2 0t ]  ...

2
2
p

How to get r,an example.
Phase Modulated Ellipsometry
1  4 J1  0 rSin[ 0t ]

I  rs  r 

 2aJ2  0 Cos[]Cos[2 0t ]  ...

2
2
p

V0  2J1  0 r
V20   J 2  0 Cos[]
We find the Brewster angle by adjusting until
Which is where     2 .
V20  0,
Now we can use a calibration procedure to
Find the proportionality of V0 to r
Applications

Determining
the thickness
of a thin film

Focus of this
presentation
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
Ellipsometry


Ellipsometry can measure the oxide depth.
Intensity doesn’t vary much with film depth
but  does.
Other Methods

Reflectometry

Microscopic Interferometry

Mirau Interferometry
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.
Ellipsometry


Ellipsometry can measure the oxide depth.
Intensity doesn’t vary much with film depth
but  does.
Reflectometry
Reflectometry

Can be more accurate for thin metal films.
Microscopic Interferometry


Uses only
interference
fringes.
Only useful for
thick films and/or
droplets

Thickness h>/4
Mirau Interferometry


Accuracies to 0.1nm
x is less than
present ellipsometry


At normal incidence.
Kai Zhang is
constructing one for
use at KSU.
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.
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
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 firstorder wetting transition: Experimental results from contact angle
measurements.” Physical Review E 63(3).