Table of Contents -
Study Aids -
Ellipsometer data table -
In this section:
- Principle of operation
- Derivation of ellipsometer equations and curves
- Table of ellipsometer data for different materials
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.
The principle of operation of an ellipsometer is illustrated by the
schematic drawing of the ellipsometer shown in the figure below:
Fig.1: Schematic drawing of an ellipsometer
It consists of a laser (commonly a 632.8 nm helium/neon laser),
a polarizer and a quarter wave plate which provide a state of polarization
which can be varied from linearly polarized light to elliptically
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.
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:
where the subscripts, 0, 1 and 2 refer to air, the thin layer and the
substrate and f0 is the angle of
the transmitted wave with respect to
the normal to the interface as shown in the figure below:
Fig.2: Incident, reflected and transmitted waves at a dielectric
RTE = rTE
rTE* is the reflectivity if the
field is transverse to the propagation direction and parallel to the
RTM = rTM
rTM* is the
reflectivity if the magnetic field is
transverse to the 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 of
the incident angle is shown in the figure below:
refl.xls - ellips3.gif
Fig.3: Reflectivity at an air-GaAs interface (ignoring the
imaginary part of the refractive index) as a
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:
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
D are related to the measured angles,
P2 and A2 in the following way:
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
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
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
to be identical for layers which differ
in thickness by
l/(2 n1 cos
f1) = 284 nm. The
for the measured values P1,
A1, P2 and A2
are also shown in the
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.
table below lists the refractive index n* =
n - i k for different
materials as well as the minimum or maximum angle of the
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
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.
- Derive equations for r01,TE and
- Derive expressions for Y and
- Prove that the reflectivity, RTM,
goes to zero if the incident angle,
tan-1(n1/n0) and only if
the imaginary part of the refractive index is zero.
© Bart Van Zeghbroeck, 1997