Ellipsometry

(1)
Ellipsometry
(how to measure the dielectric constant ~
directly and very accurately)
Setup:
ellipt.
pol. light
E
lin. pol. light
E s
A
nˆ
E p
P
monochromator or
interferometer
analyzer
detector: measures I ~ E
2
polarizer
light
source
ip
E E e , E e
p
is
s
coordinate system defined by plane of incidence of light
2
E
p
tan
2
i
E
tan e
, : ellipsometric angles
s
p
~
~
, ,
s
1 i 2
measure the change of the polarization state of light upon non-normal incidence
reflection on surface of a sample
obtain the complex dielectric function ~
Reflection of polarized light
ikr i
t
plane wave : E E0e
k E
B
k
Dispersion relation
n
k
c
n
magn. permeability 1 (non-magn. medium)
= dielectric function

z
k
(2)
'
n,
n
''
k
nˆ
k
boundary condition at z = 0
the same spatial and temporal evolution of the el. field on both sides
equal phase factors
n
k
r
z
0
k
r
z
0
k
r
z
0
(projection of k , k and k on x-y plane)
all the wave vectors must lie in the same plane
k sin
ksin
k sin
since k k
and
sin
k
n
sin
k n
Snell' s law
dynamical aspects from boundary conditions for Maxwell equations
(see previous lectures)
tang. comp. of E and H are continuous
normal components of D and B are continuous
E E
0 0
E
0 nˆ
0
k E0
k E0
k E
0 nˆ
0
E0
E0
E
0 nˆ
0
(1) D
(2) B
k
E0
k E0
k E
0
nˆ
0 (4) Ht
n
(3) E
t
n
D E
B H
n ˆ
unit vector,
ˆ1
nˆ
nˆ
1

(3)
s-component: perpendicular (senkrecht) to plane of incidence
B
E
'
n
n
B
E
nˆ
’'
E
B
E
s
nˆ
(1),(2) are satisfied
(3) E E
E
0
s s s
(4) n E E
cos
nE
cos0
s s s
with ncos n cos n n sin n
n
sin
2 2 2 2 2 2 2 2
Snell's law
Es
2n
cos
E
2 2 2
s n cos
n
n
sin
2 2 2
E n cos
n
n
sin
s
E
2 2 2
s n cos
n
n
sin
Fresnel equations for s -pol. light

(4)
p-component: parallel to plane of incidence
E
p
B
‘
n
n
E
p
B
’'
nˆ
B
E p
(1)
(2)
cos
E p E
p
cos
E
n E p E p nE
p 0
p
0
(E
t
)
(B )
t
E
E
p
p
n
2
2n
ncos
cos
n
n
2
n
2
sin
E
E
p
p
n
n
2
2
cos
n
cos
n
n
n
2
2
n
n
2
2
sin
sin
2
2
Fresnel equations for
p -pol. light

(5)
Brewster-angle:
2 2 2
for ncos
B
n n n
sin B
0
n
Ep
0
at this angle the refl. wave has no p-component
purely s-polarized one can make so - called Brewster -polarizers)
n
B
arctan
n
n 1.5 (glass) B
56.3
n
the larger n the closer is to 90
B
total internal reflection:
n
sin
for n n
n sin
90
if
B
n
arcsin
n
refracted wave propagates along the interface
no energy transport across interface; total internal reflection.

(6)
Glas / amb.
r s , r p 1
n 1.5
n
~ r
s,
p
r
s,
p
e
i s,
p
Es
rs
E
s
r s
r
p
E
E
p
p
r p
0
45° 90°
B
s
p
reflected wave
0
Reflectance
1
E
R
2
2
E
R s
R s
R P

define complex reflection coeff.
2 2 2
rp
n sin tan n
n
sin
r 2 2 2
s n sin tan n
n
sin
(7)
take
n 1, n
2
2
2 1
2 2
sin tan sin
1
It is important to note that
and
are complex quantities
i
1 2
Rp
e
1i
2
R e
Rp
i( ps)
e
R
s
s
i
p
i
s
relative attenuation
relative phase shift
R
R
p
s
p
tan
s
and
ellipsometric angles
1
2
sin
~
2
2cos
,
2
2
2
cos 2
sin 2sin
1
2
1
sin2cos
2
2 2
2sin
2sin
sin tan
2
1
sin2cos
,
tan
2
How do we measure and :
ellipt.
pol. light
E
E p
E s
P
FTIR
nˆ
A
analyzer
Ee
E
Ee
i
p
p
i
s
s

(8)
We want to know
I
E
2
det
as a function of
A and P
det ~
Jones matrix formalism :
Polarizer
1 0
change of coordinate system
cos
P sin P 1 0 cos
P sin P
0 0
sin
P cos P
0 0
sin P cos P
sample
r
~
p 0
0
~ rs
cos
A sin A1 0 cos
A sin A
r
~
p 0
cos
P sin P 1 0 cos
P sin P
E
det
sin
A cos A
0 0
sin
A cos A
0
r~
s
sin
P cos P
0 0
sin P cos P
analyzer
sample
polarizer
E
det
~ r
p
cos P cos A ~ rs
sin P sin A Ex
0
I
det
~
~ 1
E
2
det
~ 2 2 2
cos P sin P
~ 2 2 2
cos P sin P
cos 2A
Re
~ sin2P
~ 2 2 2
cos P sin P
sin 2A
I
I 0
modulation of intensity as a function of
Analyzer angle get , as second
Fourier coefficient
~
tan cos
i sin
~ 2 2
tan ;
Re
tan
tan
cos
2
2
2
cos
2
2
~
cos P sin P
P
P sin
2
2
tan cos
P
2 2
1tan
sin P1
2·A
1
tan
1
tan P
analogous :
cos
1
2

(9)
modulation of I
det
A
, tan
,cos
. Only cos is measured, not .
2nd Fourier
coefficients
ambiguity with respect to shift by can be determined with retarder unit, i.e.
introduce in addition to the sample a unit which leads to a known plane shift.
Examples for applications
1. Au + Semiconductors critical points in band structure
2. Oxidation of a semiconductor surface thin film characterization
3. Electrochemistry and biological systems. Radiation damage in solids
4. High-T c -Superconductors