THIN FILM ANALYSIS USING SPECTROMETRY AND ELLIPSOMETRY
I.
OBJECTIVE 1:
I.1.
Spectrophotometer
A spectrophotometer varies the wavelength of a collimated beam of light
over a certain range. The beam of light is transmitted through a sample to a
detector to measure percent transmission versus wavelength. To do this, first a baseline measurement must be made, which
includes transmitting the beam through the air (no sample) or through a
"dummy" sample to allow percent transmission to be determined for a
sample afterward. One use for the
spectrophotometer is to measure the absorption frequencies of solid, liquid,
or gas dielectric material. These
tests are valuable for determining the chemical makeup and properties of
materials.
If the wavelengths of the spectrophotometer are "monochromatic
enough", at each interval in the wavelength scan, then thin film studies
can be performed. The frequency
or spectral bandwidth is a measure of how monochromatic a light source is
[1]. The smaller the bandwidth,
the closer the light source is to perfect monochromatic light, and the larger
the coherence length. The
coherence length is a measure of the possible path difference two parts of the
same wave front can take and still interfere when recombined.
A wave front passing through a thin film is divided (reflected) at the
two interfaces (surface of the film). The
reflected wave fronts interfere, causing a variation of the intensity of
transmitted light as the wavelength varies.
The way the transmitted intensity varies with wavelength allows for the
determination of the thickness or the index of refraction of the film.
The Cary 5 Spectrophotometer, used in this study, uses a filter wheel
and then three slit devices to produce very monochromatic light from a
mercury lamp light source. After
the second slit, the wavelength has a spectral bandwidth of 2 nm (λ � 2
nm). The third slit is adjustable
and can reduce the bandwidth to a greater degree.
The Cary 5 has a total wavelength range of 175 to 3,300 nanometers,
through the use of two light sources and five unique filters in conjunction
with the three slit devices.
An example of a transmission test on a transparent thin film is shown
in Figure 1. Figure 1 shows that
above a wavelength of 400 nm, over 90% of the radiation is transmitted.
But there are sinusoidal variations.
The maximum and minimum values of these variations corresponds to
wavelengths given by the following equation:
1

where m must be an
integer, depending on the environment on each side of the film, maxima and
minima correspond of m being odd or even.
n is the index of refraction and d is the thickness of film.
φ' is the angle of the wave front in the film.
Equation 1 indicates that a maxima or minima occurs when the optical
path difference, caused by travel through the film, is equal to an integral
number of half wavelengths (λ/2).
FIGURE 1: Example of spectrophotometer trace of a transparent, thin film sample.
By determining the wavelengths at maxima and minima, the product of n*d
can be determined. However, neither n nor d can be determined independently.
This is a major drawback to this method because either n or d must be
known. In semiconductor work, as
in this work, it is n that is usually assumed from literature or other
experimentation. However, the
value of n can vary due to the nature of the manufacture of the film.
This not only can lead to significant errors in d but also neglects
vital information to be gleamed from the value of n.
(For example, a lower than expected value of n may indicate a porous
film.) Both n and d could be evaluated from several maxima (and/or
minima) if the dispersive equation for the index of refraction of the film is
known to be of a definite form. However,
dispersive equation may also vary with the quality of the film.
I.2
Adaption to Reflectometer:
In order to determine the n and d values for transparent films on top
of a reflecting (nontransmitting) surface, a special adaptive fixture was
designed and built for the Cary 5 Spectrophotometer, as shown in Figure 2.
The fixture reflects the beam of light on a front surface (probably
silver) mirror up to the sample. The
light strikes the sample at an incidence angle of 30^{o}.
Then the light is reflected on a second front surface mirror to the
detector. The fixture was
meticulously machined to assure precise angles.
The fixture was designed and produced to hold the sample parallel to
sample holder and the mirrors set, at high tolerance, 30^{o} to the
sample holder.
FIGURE 2:
Adaptive fixture to convert the Cary
5 Spectrophotometer into a reflectometer.
The concerns with this adaptation include:
the fact that the light beam diverges (as shown in Figure 3) and is not
designed to travel the extra distance this fixture imposes, and the fact that
reflection on the two mirrors alter the nature of light.
These concerns are alleviated by the fact that data taken in reference
to a primary baseline trace. To
do a baseline measurement at the proper wavelength range, the fixture is
placed on the sample holder and a "bare" substrate is placed on the
fixture. The reflected intensity
of subsequent sample traces is recorded as the percentage of the intensity
which reached the detector when the baseline was taken.
Therefore, abnormalities caused by the fixture will be factored out.
However, one factor unaccounted for is the fact that there will be a
native oxide on the substrate used for the baseline.
However, with silicon for example, the native oxide (SiO_{2})
will be so thin that it will not cause gross variations.
I.3
Sample Preparation:
The author was fortunate to be granted access to the
cleanroom, to be gifted with seven 3" diameter silicon wafers, and
most notably, to receive the assistance of David Price.
All eight wafers were given an RCA clean to remove ionic and organic
contaminants as well as the native oxides.
(The description of the RCA clean procedure is included in the
Appendix.)
Five of the wafers were wet oxidized.
This involves placing the cleaned wafers into a furnace.
The furnace seals the wafers in a fused silica chamber heated by
elements in its wall. Oxygen and
hydrogen are introduced into the chamber at high pressure, and release heat
while turning into steam. The
steam diffuses into the silicon at a much faster rate than oxygen does, due to
ability of hydrogen to diffuse into silicon rapidly.
Therefore, wet oxidation (with steam) is used when thick oxides are
required. However, the oxide
produced is generally more porous and of lesser quality than oxide produced by
dry oxidation (no steam). Figure
4 shows the rates versus temperature for wet and dry oxidation.
Hydrogen and oxygen are introduced in a ratio of 2 : 1 and produce the
following chemical reaction with the silicon:
Si + H_{2}O � SiO_{2} + 2H_{2 }
The parameters for the oxidation were designed to produce an oxide
thickness of one micron. The parameters include:
Ramp up: 20 minutes
Preanneal: 20
minutes
Oxidation: 17
minutes (Steam at 1,000 ^{o}C)
Postanneal: 40 minutes
Ramp Down: 40 minutes
Silicon nitride, which is typically used as a high temperature mask,
was deposited on one wafer (after RCA clean).
The nitride was deposited by chemical vapor deposition at a temperature
of 800 ^{o}C and a NH_{3} flowrate of 120 liters/minutes.
The nitride was deposited for five minutes.
After the RCA clean, wax was applied to the five wafers which were
oxidized and to the wafer which was deposited with nitride.
The concept was that the wax prevents oxidation or nitride deposit.
Therefore, after these two procedures, the wax was removed (by hot
trichloroethylene) to reveal "bare" silicon surfaces.
Then a machine called a Dektak, which is basically a profilometer, uses
a mechanical stylus to slide along the "bare" silicon surface into
the surface covered with the oxide or the nitride.
The machine measures the vertical displacement of the stylus when this
occurs. This measurement is inaccurate, but it ballparks the
thickness, which is useful because the ellipsometer returns several possible
thickness values, equally and significantly separated.
The above procedure has been described because during assistance in the
cleanroom, the wax was applied in typical manner, which was inappropriate for
this work. A thin "X" was made with the wax over the
entire wafer, as shown in Figure 5. At
first it was thought that this unfortunate action was terminal because there
was no way to position the wafer on the fixture and totally avoid radiation of
the "X" region by the spectrophotometer beam. However, not wanting to abuse the generosity of the cleanroom
personnel, it was considered that any light incident on the "X"
region the same as the baseline measurement made on a "bare" silicon
wafer. Therefore, the affect of
this region will only be seen in the maximum and minimum amplitude values, but
will not affect the wavelength values for the maxima and minima, most
important for Objective 1. This
concern will be discussed further in Objective 3.
I.4
Experimental Data Curves:
The wafer which was not coated was placed on the fixture and a baseline
trace was made, ranging from 200 to 1,000 nm.
This is shown in Figure 6. Then
the oxidized wafer was traced. The
trace is shown in Figure 7 and the test of the nitridecoated wafer is shown
in Figure 8. Before delving into calculations and manipulations of the
data, a "big picture" view of the data can be obtained by
theoretical curves for the same physical conditions.
FIGURE 6:
A baseline
spectrophotometer trace. A
"bare" silicon wafer was placed on the reflectometry adaptor and this
reference trace was made.
FIGURE 7:
Spectrophotometry trace of
reflection from an oxidized silicon wafer.
FIGURE 8:
Spectrophotometry trace of
reflection from a nitride coated silicon wafer.
I.5
Theoretical Amplitude Curves:
"Mathcad" software [2], was used to plot the theoretical
curves. The reflected amplitudes
of radiation with electric field oscillations perpendicular and parallel to
the incidence plane (R_{s} and R_{p}) were calculated via a
formula which appeared as follows:
2

where r'_{s}
is the reflection (of perpendicularly polarized light) from the air to
dielectric interface, r''_{s} is the reflection from the
dielectricmetal interface, and x_{s} is the phase change caused by
the optical path length travelled into and out of the film plus the phase
change caused by the reflection from the metal surface δ_{s}:
3

Note r'_{s}
and r'_{p}, r''_{s} and r''_{p}, δ_{s}
and δ_{p}, and R_{s} and R_{p}, are different for
a nonzero angle of incidence. Therefore,
another major difference between this type of reflection measurements and
typical transmission spectrophotometry is the difference in the traces of
polarization components.
The thickness and index of refraction of the dielectric films were
determined, for these theoretical plots, from ellipsometry measurements (to be
discussed in the next chapter). The
cleanroom ellipsometer uses a beam laser with a wavelength of 623.8 nm.
Therefore, the value of n determined by ellipsometry is only good for
that wavelength. Since the values
of n as a function of λ are required across the range of λ used in
theoretical calculations, the Cauchy dispersion equation was assumed:
4

where A, B, and C are constants. These constants were approximately determined, by using the index value from ellipsometry (n_{632.8}) and the indices at various wavelengths given in Table I for vitreous silica [3]. (The same was done for the nitride so that only the value of A differs from nitride and silica input.)
TABLE I: Index of refraction values for vitreous silica. This data was used, along with ellipsometry data, to generate constants for the Cauchy dispersion equation. (A = 1.416; B = 4,000; C = 1.533 * 10^{7}) Table taken from N. P. Bansal and R. H. Doremus, Handbook of Glass Properties, Academic Press, New York, 1986.
Other data required for generation of theoretical curves is the complex
index of refraction for silicon (= ns + ks*i).
Both the real and imaginary (ns and ks) parts of the index vary greatly
over the wavelength range used, as illustrated by the plot of ns and ks versus
λ shown in Figure 9. The
data used for complex index was obtained from Aspnes and Theeten [4,5] and is
shown in Table II.
The mistake was made of entering complex index values at intervals of
only 1 nm and the software limited the quantity of inputs.
Therefore, theoretical calculations are only made for λ = 200 to
799 nm by 1 and not λ from 200 to 1,000 nm as in the experimental work.
However, the 200 to 799 nm range is broad enough to yield understanding
of the theory. Lastly, the
complex index of refraction is slightly dependent on angle of incidence, but
test calculation showed variations of only tenths of a percent, thus no
attempt is made to correct for the different angles of incidence in this
study.
The Mathcad work is shown here.
Figures 10a and 11 show three theoretical curves
plotted on the Yaxis: R_{s}/R_{bs}
is the ratio of the amplitude of the perpendicular reflected component from
the film covered substrate over the amplitude of perpendicular reflected
component which would be reflected from the bare substrate.
Therefore, R_{bs} is a theoretical baseline for the
perpendicular component. Similarly,
R_{p}/R_{bp} is plotted for the parallel components.
Finally,
The first thing that one notices is that the sinusoidal nature of the
theoretical curves for the two polarization components matches the sinusoidal
nature of the experimental curve. But
when the two component curves are averaged to determine the total reflected
amplitude ratio, the curve cannot be considered to be sinusoidal.
It would seem that the experimental curve would be predicted by the
averaged theoretical curve, because no polarizer was purposely placed in the
system. However, this is clearly
not the case. Therefore, the Cary
5's optical elements (slits, diffraction gratings, etc.) must cause
significant polarization of the light beam.
It is noticed that the maxima and minima of the experimental curve
(Figure 7) match the maxima and minima of the amplitude curve for the parallel
polarization component exactly. This
indicates first that accurate n*d values will be determinable from the
experimental data. Secondly, it
indicates that the perpendicular polarization component has been significantly
reduced in the optical system. This
is easily proven by placing a polarizer in the system and performing baseline
traces (this test has not been performed at this time).
The second difference is seen as the sharp reduction in oscillation
amplitude for λ's below about 370 nm in the theoretical curve.
The experimental curve (Figure 7) retains significant oscillation
amplitudes in that range. The
reason for this is easily seen by comparing the experimental baseline (Figure
6) to the theoretical baseline (Figure 10b).
The experimental reflection off silicon decreases sharply for λ's
below 370 nm, while the theoretical reflection increases greatly below 370 nm.
Ignoring the native oxide, the reflection off the silicon is only
dependent upon the complex index of refraction of silicon.
This indicates how dramatically the complex index can changes in this
wavelength range, especially if the silicon is doped differently.
The silicon used by Aspnes and Theeten [4] had a resistivity of 26
Ω cm. The resistivity of the
wafers used in this study was not determined, but could be determined in the
future.
Something that is not as easily seen is that the peaks of the
theoretical curve exceed unity by a very small percentage.
And the measurement maximums from the experimental curve are less than
one by a very small percentage. This
indicates two things: 1. the
light is highly polarized, if it were not the experiment curve would not be as
close to one because the parallel polarization component reaches a minimum at
nearly the same λ where the pcomponent is maximum, and 2. the system is
accurate (errors with this method are insignificant).
Before comparing the experimental and theoretical curves for the
nitride layer, it will be noted that the ellipsometric data for the nitride
revealed a film thickness of only 78.2 nm.
This means that there will be very few maxima and minima occurring over
the wavelength range at which n*d values can be determined.
In comparing the experimental curve (Figure 8) with the theoretical
curve (Figure 11), it can be seen that the two peaks in experimental curve
correspond to the two peaks of the theoretical amplitude of the perpendicular
polarization component (this will be discussed later).
The larger peak (occurring at 612 nm in the experimental curve)
corresponds closely to the theoretical maximum of the scomponent.
But the experimental peak is sharper and much higher (over 2 where the
theoretical peak is only slightly above 1).
The experimental peak at the smaller wavelengths occurs at a larger
wavelength than the theoretical, it is also sharper, but in this case it is
smaller than the theoretical. There
appears to be a third peak at, about λ = 360 nm, in the theoretical
curve. But comparison of the
shape of this peak with the theoretical baseline plot in Figure 10b, indicates
that this is simply an influence from the varying index of silicon data used.
Why is there such differences between the theoretical and experimental
for the nitride film, when the comparison for the silica was so close?
And, if the light beam is heavily polarized in the parallel
direction, as was concluded from the previous comparison, why does the
experimental curve match the perpendicular component in this case?
The answer is that the sample actually contains a double film on the
silicon. Since the nitride layer
is so thin and since nitride has a much higher index than silicon dioxide, the
native silicon dioxide layer has significant influence in this case.
Before answering the second question on polarization, it will be noted
that the theoretical amplitude curves for the two polarization components are
nearly π (180^{o}) out of phase with each other.
This will occur when δ_{s} is very close in value to
δ_{p}, which is true for the majority of the wavelength range
used, and the incident angle is within a certain range (0 < φ <
θ_{p}). To
demonstrate that this is valid consider the following case: Assume that δ_{s}  δ_{p} = 0.
At the angles used in this study, light waves normal to the plane of
incidence will experience a phase change of π when reflected at a
boundary where the incident medium is less dense than transmitting medium and
the normal waves will experience no phase change when reflected in a denser,
incident medium. The exact
opposite occurs for waves parallel to the plane of incidence.
For parallel waves, there is no phase change from reflection where the
incident medium is less dense and a π phase difference with a denser
incident medium. Since the parallel and perpendicular waves are compete
opposites, it is easy to see that reflection at a metal with one thin film
(where δ_{s}  δ_{p} = 0) will cause the maximum for
one component to occur at the same wavelength of a minimum for the other
component, and so on. Therefore,
it is fortunate that the Cary 5 inadvertently polarizes the measurement beam,
otherwise the data would have been useless.
If another film is added to create a double film stack on silicon (and
another interface), the p and s components would reverse.
In other words, the peaks would still be out of phase by π, but
each component would have changed phase by π.
Therefore, Figure 11 indicates that the perpendicular component matches
the experiment, because Figure 11 was generated for one film.
But it is actually the parallel component that matches.
I.6
Experimental Data:
The Cary 5's program allows for accurate determination of X and Y
coordinates from a trace. Therefore, these coordinates were determined for the plots in
Figures 7 and 8, for the silica and nitride films. The coordinates yield the wavelength values where a maxima or
a minima occurs (Xaxis) and Yaxis values, the ratio of reflected to baseline
amplitudes. Since, this system
will not be able to determine n and d independently, a film index determined
by ellipsometry was used initially. This
value is n_{632.8} = 1.426 for silicon dioxide and n_{632.8} =
1.997 for silicon nitride. This
is the index of refraction at the wavelength of the laser used with the
ellipsometer. This n_{632.8}
is used to obtain a value for d using a maxima or minima nearest to λ =
632.8. Then this value of d is
used to determine the value of n_{λ} at all the other maxima and
minima.
It was noted earlier that the experiment data for the silica film case
closely matched the theoretical curve for the parallel polarization component.
Equation 3 is the main equation used:
(3) 
where δ_{p}
is the phase change a wave front experiences when reflected at the
silicasilicon interface. It is
easier to calculate δ_{p}, by calculating Δ and δ_{s}:
6

and
7

8

where ns_{λ}
and ks_{λ} are the real and imaginary parts of the complex index
of refraction for silicon, as they vary with λ.
Since δ_{p} is dependent on n, n is determined for each
λ by iteration.
For the silica film, using n_{632.8}, a d value of 1082
nanometers was determined. Also
the λ values and the calculated n values, for the silicon dioxide film
are listed in Table III. The
calculated index values in Table III were plotted in Figure 14 and a curve was
fit to the data assuming the form of Equation 4, the Cauchy dispersion
equation. The maximum deviation
of the experimental points from the fit curve was 0.6% and the standard
deviation of the point from the curve was 0.3%.
Therefore, this technique can determine film thicknesses to plus or
minus half a percent.
FIGURE 14: Index of Refraction versus Wavelength.
For the silicon nitride film the second dielectric layer will be
ignored at this time. Only two
maxima were used (the minimum appears to be ambiguous).
With the nitride, δ_{s} was used in Equation 5 instead of
δ_{p}, as discussed before.
δ_{s} is determined directly from Equation 7.
The largest maximum occurs at λ = 612, so using n = 1.998
(corresponding to n_{632.8} = 1.997) a nitride thickness of d = 76.6
nanometers was arrived at. The
value of n_{254} was then determined, for the other maxima at λ =
254 nm, to be n_{254} = 2.052.
TABLE III
Experimental and Calculated Data for the Silica Film
^{ }
^{*}Yaxis
values are referenced to reflection from "bare" silicon.
^{**}Index
for the silica film is calculated using the λ values at listed &
using an ellipsometer value (n_{632.8}=1.426) to set d=1082 nm.
λ
[nm] 
Reflection
Ratio^{*} 
Maxima or Minima 
Calculated Index (n)^{**} 
961 
0.9836 
MAX 
1.422 
826 
0.2464 
MIN 
1.426 
723 
0.9885 
MAX 
1.426 
643 
0.2191 
MIN 
1.426 
580 
0.9880 
MAX 
1.428 
529 
0.2764 
MIN 
1.431 
485 
0.9839 
MAX 
1.429 
448 
0.3349 
MIN 
1.426 
418 
0.9841 
MAX 
1.429 
392 
0.4498 
MIN 
1.430 
368 
0.9867 
MAX 
1.437 
348 
0.5055 
MIN 
1.443 
330 
0.9879 
MAX 
1.449 
313 
0.5123 
MIN 
1.452 
298 
0.9983 
MAX 
1.457 
285 
0.6428 
MIN 
1.461 
274 
0.9904 
MAX 
1.465 
262 
0.6548 
MIN 
1.457 
252 
0.9941 
MAX 
1.458 
243 
0.6118 
MIN 
1.463 
234 
1.0019 
MAX 
1.465 
227 
0.6435 
MIN 
1.472 
220 
1.0009 
MAX 
1.474 
214 
0.6774 
MIN 
1.481 
207 
1.0007 
MAX 
1.479 
202 
0.7261 
MIN 
1.488 
TABLE IV
Experimental
and Calculated Data for the Silicon Nitride Film
^{*}Yaxis
values are the ratio of reflection from the sample to reflection from a
"bare" silicon wafer used as a reference.
^{**}Index
for the silica film is calculated using the λ values at listed and using an
ellipsometer value (n_{632.8}=1.997), which allowed determination of d
= 76.6 nm.
λ
[nm] 
Reflection
Ratio^{*} 
Maxima or Minima 
Calculated Index (n)^{**} 
612 
2.09 
MAX 
1.997 
254 
0.88 
MAX 
2.052 
This index value
is well within the range of possible indices due to dispersion.
Therefore, it is again concluded that this technique can determine film
thicknesses accurately.
I.7
Suggested Improvements:
So far the system has proved to be ruggedly accurate for the
determination of the product of n*d. However, it is desirable to determine n and d independently.
It is obviously redundant to use the ellipsometry to determine a value of
n in practice, as it was determined in this study.
Furthermore, using a textbook value for n can yield significant errors in
d. This can be seen by comparing
the experimental and textbook values of n for silica shown in Tables III and I
respectively. There is over a two
percent difference in n from experimental and textbook.
The suggested method for measuring n and d is to start with a test beam
which is not significantly polarized and make two traces, one for ppolarized
then turn the polarizer 90^{o} for spolarized.
This requires four scans altogether (two baselines and two traces).
This would essential be a spectrophotomic ellipsometer and thus would be
more accurate, versatile, and capable than a single λ ellipsometer.
The use of ellipsometry is will be described further in the next chapter.
Go to Abstract & Intro. Go to Objective #2 Go to Objective #3 Go to Objective #4
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