# Slide 1

Semiconductor

Nov 2, 2013 (4 years and 8 months ago)

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Augustin

Cauchy

August 21, 1789

May 23, 1857

1810
-

Graduated in civil engineering and went to work as a
junior engineer where Napoleon planned to build a naval base

1812

(age 23) Lost interest in engineering, being more
attracted to abstract mathematics

Cauchy had many major accomplishments in both
mathematics and science in areas such as complex functions,
group theory, astronomy, hydrodynamics, and optics

Cauchy made 789 contributions to scientific journals

One of his most significant accomplishments involved
determining when an infinite series will converge on a
solution

In wave theory, he defined an empirical relationship between
the refractive index and wavelength of light for transparent
materials
--

Cauchy’s Dispersion Equation

Cauchy’s Dispersion Equation

Simple

Works well in the visible
spectrum (400→750nm) for
transparent material

n

refractive index

λ

wavelength (um)

A,B,C
-

coefficients that can be determined for a material by fitting the equation to
measured refractive indices at known wavelengths

SiO2:
A = 1.451, B = 317410, C = 0

An Application of the Cauchy Equation

The Cauchy Dispersion Equation is
used in semiconductor manufacturing
when monitoring film thickness

Films less than a few hundred
angstroms in thickness are required in
semiconductor manufacturing (1um =
10,000 angstroms)

A gate oxide on a transistor might be
between 50
-
100 Å and if off more than
a few angstroms the device may not
work correctly

Assumptions for Example

Initial medium is air (n
0

= 1)

Transparent film (k=0)

Normal incident light source

Measurement Sequence

Spectral
Reflectometry

Measurement

Measurement Data

As light strikes the surface of a film, it is
either transmitted or reflected

Light that is transmitted hits the bottom
surface and again is either transmitted or
reflected

The light reflected from the upper and
lower surfaces will interfere

The amplitude and periodicity of the
reflectance of a thin film is determined by
the film’s thickness and optical constants.

or destructively, due to the wavelike nature
of light and the phase relationship
determined by the difference in optical
path lengths of two reflections.

Fresnel Equations

(normal incidence)

Model and Fit

To obtain the best fit between the
theoretical and measured spectra, the
dispersion for the measured material is
needed

A material dispersion is typically
represented mathematically by an
approximation model that has a limited
number of parameters. One commonly
used model is the
Cauchy model
.

Best fit is determined through a regression
algorithm, varying the values of the
thickness and selected dispersion model
parameters in the equation until the best
correlation is obtained between theoretical
and measured spectra.

Recursive Fit

Thickness

The two parallel beams leaving the film at
A and C can be brought together by a
converging lens

The wavelength of light

n
in a medium of
refractive index n is given by

n

=

0

/n,
where

0

is the wavelength in air

The optical path difference (OPD) for
normal incidence is (AB+BC) times the
refractive index of the film.

(AB+BC) is approximately equal to twice
the thickness, so OPD = n(2t)

Reflections are in
-
phase and therefore
add constructively when the light path is
equal to one integral multiple of the
wavelength of light.

Reflectance of thin films will vary
periodically with 1/

n

References

http://utopia.cord.org/step_online/st1
-
4/st14eiii3.htm

http://en.wikipedia.org/wiki/Fresnel_equations

http://en.wikipedia.org/wiki/Thin
-
film_interference

http://en.wikipedia.org/wiki/Ellipsometry

http://en.wikibooks.org/wiki/Waves/Thin_Films

www.chem.agilent.com/Library/applications/uv90.pdf

http://www.jawoollam.com/resources.html