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Introduction to Ellipsometer |
4. Drude's Equations |
Ellipsometry was first used by Drude to measurement very thin
film in 1889. Drude's equation is a counterpart of Fresnel's
equation for the film structure. It is the basic of the
ellipsometry.
Drude's equations can be derived from Fresnel's equation with
combination of the interference between the layers. The reflected light is a superposition of beams
,
and
,
where the subscript 01 means light enter medium 1 from medium
0 and b is the phase delay the beam experiences during
propagating from the top surface of the film to the bottom
surface of the film. The subscripts s and p are ignored here for
both components, following this rule. From last section we know
that
and
.
These lead us to the Drude's equations:
For a stack of multiple layers, the Drude's equations can be
used recursively from the bottom to the top layer. Frensel's
reflection and transmission amplitude coefficient of each
surface are first calculated; an effective coefficient of the
bottom film is calculated by substituting the amplitude
coefficient into Drude's equations; then by using this effective
coefficient as an amplitude coefficient, an effective
coefficient of next-to-the-bottom layer is calculated by the
Drude's equations, until the top layer is reached.
For transparent films, will be periodic as the thickness of film
increases, causing the periodicity of vector (Y,D)
as functions of film thickness. This non-uniqueness is a main
limitation of ellipsometer.
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