The ellipsometer
Introduction
An ellipsometer enables to measure the refractive index and the thickness of semi-transparent thin films. The
instrument relies on the fact that the reflection at a dielectric interface depends on the polarization of the light while the
transmission of light through a transparent layer changes the phase of the incoming wave depending on the refractive
index of the material. An ellipsometer can be used to measure layers as thin as 1 nm up to layers which are several
microns thick. Applications include the accurate thickness measurement of thin films, the identification of materials
and thin layers and the characterization of surfaces.
Principle of operation
The principle of operation of an ellipsometer is illustrated by the schematic drawing of the ellipsometer shown in the
figure below:
It consists of a laser (commonly a 632.8
Polarizer
Analyzer
nm helium/neon laser), a polarizer and
1 / 4 wave plat e
HeNe
Det ect or
a quarter wave plate which provide a
state of polarization which can be
t hin layer
Subst rat e
Fig.1: Schematic drawing of an ellipsometer
varied from linearly polarized light to
elliptically polarized light to circularly
polarized light by varying the angle of
the polarizer. The beam is reflected off
the layer of interest and then analyzed with the analyzer. The operator changes the angle of the polarizer and analyzer
until a minimal signal is detected. This minimum signal is detected if the light reflected by the sample is linearly
polarized, while the analyzer is set so that only light with a polarization which is perpendicular to the incoming
polarization is allowed to pass. The angle of the analyzer is therefore related to the direction of polarization of the
reflected light if the null condition is satisfied. In order to obtain linearly polarized light after reflection, the polarizer
must provide an optical retardation between the two incoming polarizations which exactly compensates for the optical
retardation caused by the polarization dependent reflections at each dielectric interface. Since the amplitude of both
polarizations was set to be equal, the ratio of the amplitudes after reflection equals the tangent of the angle of the
analyzer with respect to the normal.
Derivation of the ellipsometer equations and curves
The calculation of the expected angles of the polarizer and analyzer corresponding to the null condition starts with the
reflection coefficients at each of the dielectric interfaces for each polarization or:
r01,|| 
n 0 cos  0  n1 cos1
n 0 cos  0  n1 cos1
(1)

r01, 
n 0 cos 1  n1 cos 0
n 0 cos 1  n1 cos 0
(2)

r12,|| 
n1 cos 1  n 2 cos 2
n1 cos 1  n 2 cos 2
(3)

r01, 
n1 cos  2  n 2 cos1
n1 cos  2  n 2 cos1
(4)
where

(5)
n0 sin 0  n1 sin 1  n2 sin 2
where the subscripts, 0, 1 and 2 refer to air, the thin layer and the substrate and 0 is the angle of the transmitted wave
with respect to the normal to the interface as shown in the figure below:

R|| = r|| r||* is the reflectivity if the electric field is parrellel to
n0


the plane that transverse to the propagation direction and
parallel to the interface while RTM = rTM rTM* is the
reflectivity if the magnetic field is transverse to the
n1
propagation direction and parallel to the interface. These

reflectivities are angle dependent as well as being different for
each polarization. An example of the reflection as a function
Fig.2: Incident, reflected and transmitted waves at a
of the incident angle is shown in the figure below:
dielectric interface.
refl.xls - ellips3.gif
Fig.3: Reflectivity at an air-GaAs interface (ignoring the imaginary part of the refractive index) as a function of the
incident angle for both polarizations (TE and TM) of the incident wave. The reflectivity, RTM, goes to zero if the
incident angle equals tan-1(n1/n0)
The two dielectric interfaces yield a combined reflection coefficient which can be obtained using the Fabry-Perot
equations:
(ell8)
and
(ell9)
with
(ell10)
Combining the above equations yields the expression for rTM and rTE, or the reflection coefficient of the dielectric
stack for each polarization. Based on the above discussion the ratio of the two reflection coefficients can be split into an
amplitude and phase factor, thereby defining the ellipsometer parameters Y and D: (ell11)
The angles Y and D are related to the measured angles, P1, A1, P2 and A2 in the following way:
D = 3p/2 - 2 P1 = 5p/2 - 2 P2 = 2p - P1 - P2
Y = A1 = p - A2 = (A1 - A2 + p)/2
The minimal signal is obtained when both polarizations incident on the analyzer are in phase. This can be obtained for
two different positions of the polarizer, hence the two values P1 and P2. In principle one could measure either one. In
practice both values are measured to eliminate any possible misalignment of the instrument thereby yielding a more
accurate result. Y - D curves are typically used to visualize the ellipsometer parameters for different layer thickness and
refractive index. An example of such curves as obtained for silicon dioxide layers (n1 = 1.455) on silicon (n2 = 3.875 0.018 i ) using a helium-neon laser (l = 632.8 nm) is shown below.
ellips.xls - ellips4.gif
Fig.4: Y - D curves for silicon dioxide on silicon. The layer thickness increases counter clock wise from 0 (square
marker on the left) in steps of 10 nm (black diamonds) and in steps of 100 nm (squares). The incident angle of the
He/Ne laser beam (632.8 nm) is 70 degrees.
Since the silicon dioxide was assumed to be transparent, one finds the values for both Y and D
to be identical for layers which differ in thickness by l/(2 n1 cos
f1) = 284 nm. The corresponding curves for the measured values P1, A1, P2 and A2 are also shown in the figure below:
ellips5.gif
Fig.5: A1 - P1 and A2 - P2 curves for silicon dioxide on silicon. The thickness increases counter clock wise from 0 (at
the square marker on the left) for A1 versus P1 and counter clock wise from 0 (square marker on the right) for A2
versus P2
, both in steps of 10 nm (black diamonds) and in steps of 100 nm (squares). The incident angle of the He/Ne laser beam
(632.8 nm) is 70 degrees from the normal to the surface.
The table below lists the refractive index n* = n - i k for different materials as well as the minimum or maximum angle
of the analyzer and the half wavelength thickness when using a He/Ne laser and an incident angle of 70 degrees.
The refractive indices from this table can be used to generate the Y - D curves for any material combination. In addition
the minimum/maximum value of A1 can be used to help identify an unknown material, since it is directly related to the
refractive index which is unique for each material. However when using this technique one should realize that the
surface of the material should be polished and clean. A thin oxide layer (1 - 3 nm) which naturally grows on most
materials must therefore be removed before measuring A1. Since the cleaning procedures vary from material to
material, this technique is rather limited when trying to identify unknown materials. However it is a very sensitive and
therefore valuable technique to verify that one has a clean surface.
Table: containing refractive indices of various materials.
Problems:
Derive equations for r01,TE and r01,TM
Derive expressions for Y and D Prove that the reflectivity, RTM, goes to zero if the incident angle, f0, equals tan1(n1/n0) and only if the imaginary part of the refractive index is zero.
&copy Bart Van Zeghbroeck, 1997