Principles
The method of measurement of thin
film deposition or etch rate is based on the interference principle of light
waves. Interference is the phenomenon that can be observed when two or more
light beams from the same source, reach the same point at the same time but
they have travel different optical paths. The existence of this
phenomenon is due to the coherence property of waves. It is easily observed
when a monochromatic light source is used. This happens because two
monochromatic light beams that reach the same point at the same time, have a
coherence coefficient equal to 1. The physical result of waves
overlapping is the observation of dark and bright regions that are called
fringes. Bright regions are observed when a number of light waves interact
resulting to a light wave of maximum intensity. On the other hand, dark
regions are observed when a number of light waves interact, resulting to a
light wave of minimum, very often null, intensity.
The observation of minima and maxima
of light intensity, when a light beam is reflected on a thin film surface,
corresponds to conditions of minimum and maximum coherence of light waves
that are reflected on the thin film-air
interface (0-1), of the light waves that are reflected on the thin
film-substrate interface (1-2) and finally of the waves that are reflected on
substrate - substrate holder interface (2-3). The variation of the thin film
thickness, either increase (deposition) or decrease (etching), results
to a continuous change of the optical path of the light wave that
transmits the 0-1 interface or/and the 1-2 interface. These light waves that
are finally reflected either on the 1-2 or on
the 2-3 interface, interfere with the light beam that is initially reflected
on the 0-1 interface, leading to the observation of fringes.
The mathematical description of the
interference phenomenon that can lead to the calculation of the deposition
and the etch rate is based on the Freshnel
formulas. According to this method the electric field and consequently the
intensity of a light wave of general polarization that is reflected on a
planar surface can be analyzed to two components - one parallel (from now one
denoted as p) and one vertical (denoted as s) to the surface. The ratio of
the reflected and the incident electric field
components are called Freshnel complex-amplitude
reflection coefficients and can be related to the angles of light incidence
and the dielectric indexes of the materials. For the system that is presented
on the above figure that includes three interfaces (0-1, 1-2 and 2-3) and
four materials (air, thin film, substrate and substrate holder) there are
three parallel and and three vertical reflection
coefficients that corresponds to each of the three interfaces (0-1, 1-2 and
2-3) These coefficients can be written as:
In the above relations the unknown
angles φ1 and φ2 has been expressed as a function of the angle of
incidence φo and the refraction indexes no,
n1, n2 according to Snell' s law
This allows the expression of the Freshnel coefficients as a function of dielectric
constants and the angle of incidence φο which are all known. Moreover, the resultant
reflected wave in medium 0, which is the measured quantity, will be the sum
of the wave that is initially reflected on 0-1 boundary and of the planar
waves that after multiplex reflections in all
of the interfaces they finally reach again the 0-1 interface. The total
reflectance R of the parallel and the vertical components of the incident
light wave can be expressed as geometric series of the Freshnel
coefficients by using the relations:
where
In the last two formulas, L is the
wavelength of the incident beam in Å, d1 the film thickness and d2 the
thickness of the substrate. In fact β1 and β2 express the phase
change that the multiply-reflected wave experiences inside the film (β1)
or inside the substrate (β2). They are called phase angles and are
the most important parameters in the method.
Finally, the intensity of the
reflected light wave that result from the interference of the multiple
reflected light waves can be expressed in terms of the total p and s
reflectance and the phase angles β1 and β2,
as:
In the figures a, b is presented the variation of the
reflected intensity of the light beam in terms of thin film thickness as
calculated from the above formula. The theoretical calculations have been
performed for a system with three interfaces and the materials: air (0),
microcrystalline silicon thin film (1, n1=3.65), glass
Corning 7059 (2, substrate) and stainless steel (3, substrate holder). In
figure a, one can observe a periodic variation of the reflected intensity
with film thickness, the thickness that corresponds to one period being about
895 Å. The interference of
the light waves result to the appearance of two maxima and one minimum during this period. The first maximum results from
the interference of light waves that are reflected from the 0-1 and the 1-2
interface and take place when β1 becomes equal to:
while the second peak result
from the interference of light waves that are reflected from the 0-1 and the
2-3 interface and take place when β1 becomes equal to:
It has to be mentioned that
these values of β1 stand only for the specific system
(three interfaces and n1=3.65).
Figure b presents a case
that is quite closer to conditions of the deposition of a thin film from
plasma i.e the thickness of a thin film increases
and simultaneously the refractive index (real part of n1)
and the absorption coefficient (imaginary part of n1) of
the film increases. This scheme result to a variation of the reflected
intensity that is not periodic and to a continuous dumping of the reflected
intensity. The film thickness that corresponds to the appearance of two
successive maxima is reduced due to the increase of the refractive index,
while the second peak, which has been attributed to the interference of light
waves that are reflected from the 0-1 and the 2-3 interfaces, is step by step
disappeared, due to the increase of the film absorption.
Figures c, d present the variation of the reflected
intensity with the film thickness for the simpler case of two interfaces (0-1
and 1-2). In figure c, are presented results that correspond to a film
refractive index of 3.65. In this case the variation of the reflected
intensity is periodic and the film thickness that corresponds to the
appearance of two successive maxima or minima is 895 Å i.e equal to the
thickness presented in figure a. This is not a surprising result because the appearance
of the maxima and minima are based on the value of β1 that in turn does not depends on the
existence of two or three interfaces but only on the value of the film
thickness. On the other hand, in this simple approach the second peak that
corresponds to the interference of light waves that are reflected from the
air-film and the substrate-substrate holder interfaces is not observed as the
2-3 interface has been ignored.
Finally, figure d presents a case where the refractive
index of the thin film has a quite low value (n1=1.5). In this situation, the
thickness that corresponds to the appearance of two minima is high (2800 Å)
and the intensity of the reflected wave is low, more than an order of
magnitude lower compared to the case where n1=3.65. In both cases (n1=3.65
and n1=1.5) the same value of the incident light wave has been used (Iin=2 x 10-5
mW).
