Note: Descriptions are shown in the official language in which they were submitted.
20~6848
BACKGROUND OF THE INVENTION
Field of the Invention
This invention relates to optical interferometric methods
for determining the thickness of a thin layer or film at a large
number of sites. Recent developments in the field of semiconduc-
tor device fabrication have created a need for rapid thickness
mapping.
Description of tha Prior Art
The thickness of a thin transparent layer is commonly deter-
10 mined from its reflectance spectrum using a well understoodinterferometric principle which is fully described in standard
optics textbooks, such as PRINCIPLES OF OPTICS by M. Born and E.
20~6~48
Wolfe, published by PERGAMON PRESS (1970). There are many de-
vices and methods which employ this principle, a few of which are
described in U.S. Patents 3,099,579 by Spitzer et al., 4,254,337
by Yasujima et al., 4,645,349 by Tabata, 4,776,6~5 by van Pham
et al. and 4,787,749 by Ban et al. The relevant physics will be
briefly summarized in the following paragraphs.
Figure 1 presents a cross sectional view of a layered struc-
ture consisting of three dielectric materials. There is a thin
layer 101 having refractive index nL, in contact with a substrate
having refractive index nS on one side and a cover material
having refractive index nc on the other side. The cover material
is often air. The substrate can have the same refractive index
as the cover, but nL cannot equal either nC or ns. The wave-
length dependence of the refractive indices must be known to
interpret optical interference as thickness. The cover material
and the material forming the layer are essentially transparent to
light. The thickness of the layer is T.
Light ~of wavelength Lambda in vacuum) that is incident on
layer 101 from the side of the cover material will be partially
20 reflected by each of the interfaces between different media. For
transparent media in the absence of scattering mechanisms, the
Fresnel relations give the amplitude reflectivity and transmit-
tivity of an interface. The reflectivity and transmitivity are
coefficients that determine the field strength of the reflected
~5 and transmitted waves when they are multiplied by the field
strength of an incident wave. An interface between two media,
labeled j and k, will have reflectivity rjk when light is inci-
203~8
dent from medium j and transmitted into medium k; the order ofthe subscripts gives the direction of incidence on the interface.
Refer to Figure 1 and consider incident light 104. Reflec-
tions from the interfaces 100 and 102 result in an infinite
S number of emerging electromagnetic fields, each with a different
amplitude and phase. The net reflected light from the layer has
field strength equal to the sum of these fields. This infinite
sum may be evaluated to give the net amplitude reflectivity, r:
r = ( rCL + rLse~i ) I ( 1 + rCLrLSe ) (1)
where
= (2nLT/Lambda3 2~ cos(eL) (2)
and the subscripts C,L and S denote cover, layer and sub-
strate.
The total reflectance or intensity reflection factor, R, is
equal to the squared modulus of the reflectivity, ¦r¦2,
R = [ rCL2 + rLS2 + 2rcLrLscos(~) ] /
[ 1 + rCL2 rLS2 + 2rCLrLscos (~) ]
(3).
The first reflection, 106 in Figure 1, does not travel
20 through the thickness of layer 101, whereas all light reflected
from the interface 102 has travelled through the thickness of
layer lOl. This results in the phase difference ~ in equations
(1) and (2). The interference between light reflected from the
two interfaces 100 and 102 i5 thickness dependent. The cosine
25 function in equation ~2) varies from -1 to +1, which causes R to
20~6~48
vary over a range denoted Rmin to Rmax.
Figure 2 shows a reflectance spectrum for the layer 101 of
Figure 1, where oscillations are observed in reflectance, R, as
the wavelength of light varies. R depends upon the layer thick-
ness, T, the refractive indices and wavelength of the incidentlight. With knowledge of the refractive indices, T may be deter-
mined from an observed reflectance spectrum either by fitting the
spectrum to the functional form given by equation (3), or by
using successive maxima. Because it is accurate and non-
destructive, this is a commonly used technique for finding thethickness at a single site on a layer.
A suitable prior art apparatus for making spatially resolved
thickness determinations using reflective interference, and its
mode of operation will now be described. Figure 7 shows a sim-
plified schematic diagram of a scanning microreflectometer fromU.S. Patent 4,844,617 by Kelderman et al., adapted to make spec-
tral reflectance measurements at vi~ible and ultraviolet wave-
lengths. The instrument images a small area of a specimen and
then measures the spectral intensity distribution of the light in
20 the image. The thickness of a thin layer can be determined from
the spectrum. The imaged small area is varied in a scanning
pattern across the specimen. Because of the high spatial resolu-
tion of this type of instrument, it is well suited for spatially
resolved thickne~s mapping.
Referring to Figure 7, visible light source 706 provides
light which is collected by lens 712 and directed toward beam
splitter 722. Si~ilarly, a W light source provides light which
~6~48
is collected by lens 710 and directed toward beamsplitter 724.
The beam splitters 722 and 724 are positioned to reflect light
from these sources into the objective lens 726, which focuses the
light to the spot of illumination 750 on the specimen 728. The
specimen 728 is attached to the positioning stage 730, which
allows the spot 750 to be scanned and the focus to be adjusted.
Light reflected from the specimen 728 is collected by the objec-
tive lens 726 and a portion is focussed through the beamsplitters
722 and 724 onto a small aperture 704. Aperture 704 is imaged by
the objective lens 726 to a spot 3 microns by 3 microns in size
on the specimen 728. Light emerging from aperture 704 is imaged
with concave holographic grating 702 onto a multi-element detec-
tor array 700, which is positioned to receive light in the first
diffraction order, thereby measuring the reflectance spectrum at
750. Shutters 718 and 720 can be positioned to block the light
from either broadband light sources 706 or 708. This system is
also equipped with an autofocus light source 732, positioned in
the zero order diffraction direction, and control circuitry
external to the optical system (not shown) uses a signal produced
by 732 to adjust the stage 730 and maintain focus.
The instrument of Figure 7 can be used to determine the
thickness of a thin layer at any single site by measuring the
reflectance spectrum there. However, it is often necessary to
characterize the thickness of a thin layer over the entire sur-
~5 face of the specimen, which is called "thickness mapping". Forexample, the specimen could be a gallium arsenide wafer (GaAs)
20868~8
bearing an epitaxially grown layer of aluminum gallium arsenide
(AlGaAs). Figure 4A is a perspective representation of such a
specimen. For simplicity in the drawing, the layer 404 has been
drawn to have a flat interface with the substrate 406, but this
need not be the case in general. The variable thickness, T, in
Figure 4B, of the AlGaAs layer 404 is to be determined at many
sites. Figures 4A and 4C show that these sites are arranged in a
rectangular sampling grid 400 on the surface of specimen 490
having the layer 404 and substrate 406. The grid sites are
preferably closely spaced in the XY-plane. To accurately charac-
terize the surface, the sampling intervals Delta X and Delta Y
must be small compared to the rate of change of the thickness of
the layer. The determined thickness values are placed in an
array, with position in the array corresponding to position on
the grid 400. The resulting array of values is called the
"thickness map". It is desired to have about 65,000 thickness
determinations in a typical map over the area of a semiconductor
wafer.
Obviously, this thickness mapping task can be accomplished
by repeated application of the spectral reflectance technique at
many sites. This prior art method is briefly described with
re~erence to Figures 5, 6 and 7. Each site of sampling grid 400
is positioned under spot of illumination 750 of Figure 7 and the
reflectance spectrum is measured there, as described in U.S.
Patent 4,844,617. Figure 5 shows the arrangement of the grid
sites and the sequence in which they can be positioned under spot
of illumination 750. Figure 6 is a flowchart of the prior art
208~3~8
method for thickness mapping, which involves the steps of 1
selecting a site from the sampling grid, 2 positioning the se-
lected site under the spot of illumination, 3 illuminating the
site with broadband polychromatic light, 4 measuring the spectral
distribution of the light reflected from the site, and 5 analyz-
ing the reflectance spectrum to determine the thickness at the
site. Step 6 is repeating steps 1 to 5 until the thickness has
been determined at every site in the sampling grid 400. A com-
puter (not shown in Figure 7) is used to coordinate the operation
of the apparatus in steps 1, 2, 3, 4 and 6, and to carry out step
5 to determine thickness. The procedural details for carrying
out each of these steps are well known in the art.
Because of its spatial resolution and scanning ability, an
instrument such as that depicted in Figure 7 has been the most
widely used means of producing detailed thickness maps of thin
layers on semiconductor wafers. However, in the case of AlGaAs
on GaAs substrate, this instrument would be unable to function
because the specimen is not transparent at the wavelengths sup-
plied by the visible and W light sources. Infrared light must
be used. Even if an infrared light source and detector array
replaced the visible light counterparts in Figure 7, this or a
similar instrument would still have disadvantages, because of the
large number of spectra which must be sampled and analyzed. To
be practical, such a system must make many measurements quickly.
Sampling spectra quickly poses difficulties. In general, an
apparatus which samples spectra quickly is expensive, complicated
and can be inaccurate. Simpler, less expensive systems are too
2~868~8
slow to be useful in mass-production environments.
For example, the detector array 700 of the instrument in
Figure 7 samples a spectrum quickly by detecting many wavelengths
simultaneously. The array 700 will only sample a fixed set of
wavelengths, however, and an inadequate representation of the
spectrum may result. Also, photodetector arrays for use at the
infrared wavelengths in excess of 1 micron, which are required to
study many modern semiconductor materials, are difficult to
manufacture and have poor signal-to-noise ratios. An instrument
using such an array for infrared measurements will require an
extended signal averaging time, thereby reducing overall speed.
The price of detector arrays is high, and they require additional
signal reading and processing mechanisms which further add to the
co~t.
lS A single element detector is less expensive, is generally
more sensitive and has a higher frequency response. However, a
wavelength scanning monochromator is required to sequentially
scan through a wavelength range for each spectrum again and
again, whiCh ta~es a significant amount of time. High speed
scanning mirrors and gratings could be used to accelerate the
process, such as in the apparatus disclosed in U.S. Patent
4,254,337 to Yasujima et al, but these systems are excessively
delicate, prone to wobble during scans, and are expensive.
The process of acquiring a spectrum and making a thickness
determination at a site requires from one-half to several sec-
onds, using currently available technology of moderate cost.
The time required to produce a detailed thickness map of some
208~8
~5,000 measurements is in excess of eight hours. This is too
slow for production-line wafer processing. Therefore, prior art
devices do not function well as high resolution thickness map-
pers.
It is frequently desired to combine several optical measure-
ment functions within a single instrument, in order to obtain
different types of spatially resolved information on the same
specimen. An example of this would be a combined thickness and
photoluminescence mapping device, which could determine layer
thickness and variations in material composition across the
layer. Most of the devices using spectral reflectance have been
designed with the single purpose of thickness determination, and
are therefore unable to provide additional types of spatially
resolved information. The instrument of Figure 7 has the poten-
tial to combine several functions by replacin~ or adding lightsources, but this is hindered by a disadvantage inherent in its
design. There are already two beamsplitters in the image path of
system, and to add more light sources, more beamsplitters and
shutters must be introduced. Additional beamsplitters in the
image path cause loss of light by reflections, which can lead to
poor signal-to-noise ratios in the observed data.
SUMMARY OF THE INYENTION
The disadvantages of the prior art techniques for mapping
the thickness of thin layers are primarily related to the exces-
sive time required to determine absolute layer thicknesses at a
2 ~ 4 ~
.arge number of sites. Detailed layer thickness maps are usefulduring the processing of semiconductor wafers in an industrial
production line environment. Accordingly, objects of the present
invention are:
a) to determine the thickness of thin layers, which are
transparent or partially transparent, at a large number of sites
with high spatial resolution;
b) to produce thickness maps in much less time than is
possible using prior art methods.
c) to produce thickness maps for layers of semiconductor
materials with small band gaps, which are transparent at long
infrared wavelengths;
d) to produce thickness maps in an automated manner;
e) to provide methods which compensate for the effects of
non-ideal (real-world) reflectance data during the production of
thickness maps.
An advantage of the present invention is that it is possible
to combine in a single apparatus the thickness mapping function
with several other spatially resolved semiconductor characteriza-
0 tion functions, such as:photoluminescence measurements,
reflectance measurements, and
optical beam induced current (OBIC) measurements,
without adversely affecting the performance of any optical func-
tion. As one preferred means of carrying out the methods of theinvention, a scanning optical measurement apparatus is described
.... .
20~3~8
below. This instrument makes spatially resolved thickness deter-
minations in addition to spatially and spectrally resolved photo-
luminescence measurements. Operating according to the methods of
the invention, this instrument is a useful semiconductor analysis
tool, because variations in both layer thickness and material
composition (from photoluminescence) for epitaxially grown layers
of semiconductor material can be studied. The integration of
the thickness and photoluminescence functions is economical,
since common optical elements are shared.
According to one aspect of the invention, the disclosed
instrument operates to carry out the novel method. However, the
methods of the invention may be employed by any apparatus which
can measure spatially resolved monochromatic reflectance at a
large number of sites on a thin layer, and then independently
determine the layer thickness at a small number of these sites.
This flexibility is an advantage, since an existing thickness
measuring apparatus could be easily adapted to map thickness
efficiently.
To achieve these objects and advantages, the present inven-
tion exploits the monochromatic interference fringes formed by
light reflecting from the thin layer to help determine absolute
layer thickness at many sites. These fringes are alternating
bands of maximum and minimum reflectance, caused by optical
interference. To our knowledge, the use of monochromatic inter-
ference fringes has been restricted to applications where theuniformity of optical surfaces or thin layer thickness is to be
tested, and their use to map the absolute thickness of a non-
4 ~
leformable structure is novel.
Briefly, a method is proposed for mapping the thickness of athin layer, which is essentially described by the steps of deter-
mining the reflectance of the layer under illumination with mono-
chromatic light at many sites, selecting a small set of siteswhich bracket each fringe extremum, calculating the layer thick-
ness at sites in the small set by well known means, determining
the half-order number of the interference from these thicknesses,
and then calculating the layer thickness at all remaining sites
from the value of the monochromatic reflectance and the deter-
mined interference orders. The method is preferably carried out
by a type of scanning optical measurement apparatus, which is
equipped with at least one broadband light source, the light of
which is directed along an optical path to a specimen. The opti-
cal path is shared by several light sources. The light source isselected, dependent upon the purpose to which the instrument is
to be applied. The broadband light directed toward the specimen
is focussed to a small spot of illumination by a microscope
objective lens, and collected in reflection. Collected light is
spectrally resolved by a monochromator with a diffraction grating
before impinging on a photodetector, so that either spectral or
monochromatic reflectance can be determined. A translating
positioning stage provides the means to scan the specimen. The
instrument is controlled by a computer, which serves to collect
and analyze data in order to map the thickness of a thin layer in
an automated fashion.
Other objects, features and advantages of the present inven-
20~48
tion will become clear from study of the accompanying descriptionand illustrative drawings.
13
20~68~8
BRIEF DESCRIPTION OF THE DRAWINGS
Figure 1 is a cross-section of a thin layer and the adjoin-
ing materials, showing typical rays of the beams reflected by the
layer interfaces.
Figure 2 is a plot of the reflectance, R, as a function of
the wavelength of light incident upon a thin layer.
Figure 3 is a plot of the reflectance, R, from a thin layer
as a function of layer thickness, T.
Figure 4A is a perspective ~iew of a specimen having a sub-
strate and a thin layer.
Figure 4B is a representative cross-sectional view of the
specimen from Figure 4A.
Figure 4C is a view of the sampling grid, with the sites for
thickness determination distributed across it.
Figure 5 is an illustration of the site-to-site processing
scheme for thickness mapping.
Figure 6 is a flowchart which shows the sequence of steps
involved in thickness mapping by a prior art method.
Figure 7 is a simplified schematic diagram of a prior art
scanning optical measurement apparatus.
Figure 8A is a diagram of thickness contour lines for the
thin layer of a specimen depicted in Figure 4A.
Figure 8B is a dot density representation of monochromatic
interference fringes which are ohserved in reflectance from the
thin layer of a specimen depicted in Figure 4A.
Figure 9A is a plot of the reflectance alonq the path P-Q in
Figure 8 for ideal data.
14
2086~4~
Figure 9B is a plot of the reflectance for non-ideal data.
Figure 9C is a plot of ideal reflectance data at a fringe
maximum.
Figure 9D is a plot of non-ideal reflectance data at a
fringe maximum.
Figure lO is a diagram which shows the extent of regions of
the layer having constant half-order number, H.
Figure 11 is a diagram which shows one of the regions of
Figure lO and the sites of the sampling grid that lie within it.
Figure 12 is a flowchart which shows the sequence of steps
for thickness mapping by the method of the invention, where the
regions of Figure lO are sequentially examined.
Figure 13 is a flowchart which shows the sequence of steps
for thickness mapping by the method of the invention, where rows
of sites on the sampling grid are sequentially examined.
Figure 14 is a flowchart which shows the sequence of steps
for thickness mapping by the method of the invention, where non-
ideal reflectance data must be corrected.
Figure 15A is a plot of a reflectance profile which shows
extrema and selected bracket sites.
Figure 15B is a plot of the reflectance values near a fringe
maximum in Figure 15A.
Figure l5C is a plot of the reflectance values near a minor
extremum in Figure 15A.
Figure 15D is a plot of the reflectance values near a fringe
minimum in Fi~ure 15A.
Figure 16A is a diagram showing poxtions of three adjacent
2Q86848
rows of grid sites.
Figure 16B is a diagram showing the sequence processing for
calculating thickness values across a row.
Figure 17 is a series of plots of a reflectance profile
during the various steps of processing.
Figure 18 is a diagram which shows the calculated changes in
half-order number for the typical row of reflectance data of
Figure 17.
Figure 19 is a diagram showing the arrangement of sites for
1~ a portion of the fringe map in the vicinity of a bracket site.
Figure 20A is a plot of a reflectance profile which shows
parameters for stretching reflectance data in the simplest case.
Figure 20B is a plot of a reflectance profile which shows
parameters for stretching reflectance data in a more complicated
case.
Figure 20C is a plot of a reflectance profile which shows
important parameters for stretching reflectance data at the end
of a row.
Figure 21 is a dot density representation of a monochromatic
fringe pattern showing a fringe discontinuity.
Figure 22 is a simplified schematic diagram of the scanning
photoluminescence and thickness mapping apparatus.
16
2086~8
REFERENCE NUMERALS IN DRAWINGS
100 cover/layer interface
101 layer
102 layer/substrate interface
104 incident light
106 light reflected from cover/layer interface 100
400 sampling grid
402 grid site
404 thin layer
406 substrate
408 interface between thin layer and substrate
410 surface of thin layer
490 specimen
700 detector array
702 concave holographic grating
704 aperture
706 broadband visible light source
708 broadband W light source
710 W collecting lens
712 visible collecting lens
718 visible light shutter
720 UV light shutter
722 beamsplitter
724 beamsplitter
- 25 726 objective lens
728 specimen
730 xyz positioninq stage 2 Q 8 6 ~ ~ 8
732 autofocus light source
750 site of illumination
700 sampling grid
5 702 site on grid
704 thin layer
706 substrate
802 thickness contour
804 thickness contour
10 806 thickness contour
810 dark fringe minimum
812 bright fringe maximum
814 dark fringe minimum
816 dark fringe minimum
15 818 bright fringe maximum
902 local maximum
920 fringe maximum
922 fringe minimum
924 fringe minimum
20 926 fringe maximum
928 local minimum
930 spike
932 spike
1001 region of constant half-order number
25 1002 region of constant half-order number
10812 contour line along bright fringe maximum 812
10818 contour line along bright fringe maximum 818
.500 bracket site 2 0 8 ~ 8 ~ ~
1501 fringe maximum
1502 bracket site
1503 fringe minimum
5 1504 bracket site
1505 fringe maximum
1506 local minimum
1507 fringe maximum
1508 bracket site
10 1509 fringe minimum
1510 bracket site
1511 fringe maximum
1512 bracket site
1513 fringe maximum
15 1516 bracket site
1517 minor extremum
1620 bracket site
1622 grid site
1624 grid site
20 1626 bracket site
1630 bracket site
1632 grid site
1634 grid site
1636 grid site
25 1691 grid site at fringe extremum
1702 spike
1703 spike
~710 minimum
1711 maximum 208~
1712 minimum
1713 maximum
5 1714 minimum
1715 maximum
1716 minimum
1717 minimum
1718 maximum
10 1719 maximum
1720 bracket site
1722 bracket site
1723 bracket site
1724 bracket site
15 1725 bracket site
1726 bracket site
1727 bracket site
1728 bracket site
1729 bracket site
20 1900 bracket site
1901 column bracket site
1902 fringe extremum site
1904 fringe extremum site
2200 collimated broadband light source
25 2202 white light bulb
2204 aperture
2206 collimating lens
208 laser light source 2 0 8 6 8 4 8
2210 laser light source
2211 movable selecting mirror
2212 mirror position
5 2213 mirror position
2214 mirror position
2216 beam emerging from broadband light source
2220 beam stop
2222 shield wall
10 2223 opening
2226 beam direction
2230 partially reflecting mirror
2232 beam direction
2234 objective lens
15 2236 spot of illumination
2238 beam direction
2239 beam direction
2240 monochromator
2242 diffraction grating
20 2244 beam emerging from monochromator
2246 photodetector
2248 A/~ converter
2250 monochromator stepping motor
2252 computer
25 2254 positioning stage controller
2256 XYZ positioning stage
2270 optical filters
208~848
JETAILED DESCRIPTION OF THE INVENTION
The present invention comprises methods for mapping the
thickness of thin layers which exploit measurements of monochro-
matic reflectance. This reduces the number of spectral reflect-
ance measurements which must be made.
The methods of the invention are described in herein. A
photoluminescence mapper to which has been added a polychromatic
light source is capable of carrying out the methods of the inven-
tion. This apparatus is also described herein. To facilitate
understanding of the present invention, the interference fringes
for a thin layer under monochromatic illumination will first be
discussed.
MONOCHROMATIC INTERFERENCE FRINGES
Consider the effect when surface 410 of specimen 490 shown
in Figure 4A is illuminated with monochromatic light, of wave-
length Lambda in vacuum. The variation of reflectance with
thickness is shown in Figure 3, as given by equation (3) for
representative refractive indices, and it is seen that there are
reflectance maxima and minima as thickness varies. A maximum and
an adjacent minimum are spaced by the fundamental thickness
increment, d, where
d = Lambda/4nL (4).
If the whole specimen were illuminated with a parallel beam
2~68`~8
-t- monochromatic light, the surface would appear as in Figure 8B,
where the dot density is varied to represent alternating dark and
bright bands of continuously varying intensity. Such bands are
called fringes of equal thickness. They are formed because sites
on the layer of equal thickness reflect the same intensity of
light, according to equation (3). Because the reflectance is
related to the thickness through the periodic cosine function,
any sites differing in thickness by an integral multiple of 2d
will also reflect the same intensity. In essence, this interfer-
ence pattern is a type of thickness contour map of the layer ofthe specimen. This can be seen by comparing Figure 8B with
Figure 8A, which represents thickness contour lines for the
specimen. The contour spacing in Figure 8A has been chosen to be
2d to emphasize the similarity between the contours and the
fringes.
Although the positions of the contour lines as defined above
and the reflectance interference fringes coincide, the absolute
value of thickness for the contour lines cannot be directly
obtained from the interference fringes because of ambiguity.
In order to demonstrate this ambiguity, equation (3), which
relates thickness to reflectance, will be solved for thickness by
using the definitions of ~ and d from equations (2) and (4).
The expression for the thickness T as a function of reflectance
of the specimen is found to be:
T = d ( 2~N + arccos( F(R) ) I ~ cos( eL ) (5)
where,
23
20~68~8
F(R) = ( R + R rCL2rLS2 ~ rCL ~ rLS
/ [ 2 rcL rLs (l-R) ]
and
N is an integer called the fringe order or interference
order.
The ambiguity arises because the fringe order, N, is un-
known.
For this reason, the monochromatic interference fringes are
commonly used only when absolute thickness information is not
needed, such as in a test of thickness uniformity.
However, if N can be determined, then the thickness of the
layer at a site may be calculated from the monochromatic reflect-
ance, R, using equation (5). Since all numerical implementations
of the arccosine function return values in the range of O to ~
radians only, it is more useful to consider the fringe half-order
number, H. The quantities H and N are related: the truncation of
the quotient H/2 is equal to N. The fringe half-order number for
a site having thickness T will be given by the truncation of the
quotient T/d. For light at normal incidence (eL=o)~ the follow-
ing relations replace equation (5):
H even, T = d t H ~ + arccos~ F~R) ) ] / ~ ~6a)
H odd, T = d [ ~H~ - arccos~ F(R) ) ] / ~ (6b).
APPARATUS
A preferred apparatus for carrying out the methods of the
24
20~43
invention will now be described. Refer to Figure 22. Light beam
2216 emitting from collimated broadband light source 2200, is
directed toward beam stop 2220. One of the known methods for
producing collimated broadband light source 2200 is illustrated
S in Figure 22. Incandescent white light bu]b 2202 directs light
through field limiting aperture 2204 and collimating lens 2206.
White light bulb 2202 has enhanced infrared emission, and typi-
cally emits light with wavelengths in the range from 400 nm
(visible) to 2600 nm (infrared). As will be apparent to those
skilled in the art, other suitable methods for producing colli-
mated broadband light may be utilized.
Movable selecting mirror 2211 in position 2212 reflects the
beam 2216 which emerges from collimated broadband source 2200
into beam direction 2226. (With movable selecting mirror 2211 in
either position 2213 or 2214, laser light sources 2208 and 2210
are selected. The laser light sources are used to stimulate
photoluminescence in specimens for photoluminescence mapping.)
With selecting mirror 2211 in position 2214, light beam 2226
passes through opening 2223 in shield wall 2222 and impinges on
partially reflecting mirror 2230. Shield wall 2222 blocks stray
light in the apparatus.
Partially reflecting mirror 2230 reflects a portion of beam
2226 along beam direction 2232 into objective lens 2234. Beam
2232 underfills objective lens 2234. Objective lens 2234 is a
high quality microscope objective with a numerical aperture
chosen for the purpose of the invention. A typical value of the
numerical aperture suitable for thickness mapping is 0.1.
2086848
Objective lens 2234 focuses beam 2232 to spot of illumina-
tion 2236 on specimen 490, which is mounted on XYZ positioning
stage 2256. Spot of illumination 2236 is the location on speci-
men 490 where measurements are made. Under the direction of the
apparatus operator or a computer program, computer 2252 directs
positioning stage controller 2254 to position stage 2256 to
illuminate the sites on specimen 490 at which measurements are to
be made. Positioning stage controller 2254 and its interface to
computer 2252 are not described, because such systems are well
known.
Refer to Figures 4 and 22. The representative specimen on
which thickness mapping is performed is shown generally at 490.
Specimen 490 comprises a thin layer 404 of one material overlay-
ing a substrate 406 of a material of different refractive index.
The cover material overlaying thin layer 404 in Figures 4 and 22
is air, but this is not a restriction on the invention; thin
layer 404 can be overlaid by other materials without impairing
the utility of the invention. For thickness mapping, the cover
material must be essentially transparent to the ranye of wave-
lengths used to make measurements, and have a refractive index
that is not equal to the refractive index of layer 404. Refer to
Figure 4A. For simplicity, interface 408 between layer 404 and
substrate 406 is shown as a plane parallel to the XY plane. In
practicing the invention, interface 408 need not be a plane.
Refer to Figure 22. Light reflected from spot of illumina-
tion 2236 is collected by objective lens 2234 and directed along
beam direction 2238 to partially reflecting mirror 2230. A
2086~8
portion of beam 2238 is transmitted through partially reflecting
mirror 2230 toward monochromator 2240 as light beam 2239. Light
beam 2239 entering monochromator 2240 is reflected onto grating
2242, where it is separated into its spectral co~ponents by
diffraction. Under the control of the apparatus operator or a
computer program, computer 2252 directs stepping motor 2250 to
select the angular position of grating 2242 and thereby the
wavelength of the light exiting from the monochromator as light
beam 2244. Such use of a computer-controlled stepping motor and
monochromator is well known in the art.
Single wavelength light beam 2244 impinges upon photodetec-
tor 2246. Photodetectors suitable for use with broadband light
source 2200 include: a normal InGaAs photodiode for detecting
infrared wavelengths from 900 nm to 1800 nm; an extended range
InGaAs photodiode for detecting wavelength in the range 1300 nm
to 2600 nm; and a silicon photodiode for detecting wavelengths in
the visible to near infrared range. The output of photodetector
2246 is an electrical signal proportional to the intensity of the
light falling on it. This signal is digitized by analog-to-
digital (AJD) converter 2248 to produce the detector output valuewhich is stored in computer 2252. A/D data acquisition systems
and interfaces are not described here because they are well known
in the art.
The apparatus operator executes the methods of the invention
by controlling computer 2252, which is pre-programmed to collect
and process data. The interface between the operator and computer
2252 is typically a keyboard. Software tools available to the
2~85~8
apparatus operator include routines which display the collected
data as digital images on computer monitors, gradient evaluating
routines for finding edges in digital images, and smoothing
filters for smoothing digital images. These tools, which are
useful with the method of the invention, are not described in
detail as they are known in the art.
DETERMINATION OF REFLECTANCE WITH APPARATUS
Calibration
The manner in which the detector output value is used to
determine the reflectivity at the spot of illumination 2236 on
specimen 490 will now be described. First, the apparatus cali-
bration procedure will be described.
Typically, the calibration specimen comprises only the
substrate 406 of specimen 490. Alternatively, a highly reflec-
tive front surface mirror i8 used as the calibration specimen.The reflectance of the calibration specimen is determined using a
reflectometer or calculated by substiting the refractive indices
of the materials comprising the calibration specimen into equa-
tion (3) of the BACKGROUND section. The calibration specimen is
mounted on positloning s~age 2256 and illuminated at spot of
illumination 2236 by broadband light source 2200. It is only
necessary to illuminate a single site on the calibration speci-
men. Under the control of the apparatus operator and computer
225Z, steppinq motor 22~0 causes monochromator 2240 to scan
28
2 ~ 8
through the range of wavelengths to be used for thickness map-
ping. Typically, for a specimen comprised of GaAs substrate and
AlGaAs layer, the wavelength range is 775 nm to 850 nm and the
resolution is 0.5 nm. The signal from photodetector 2246 for
each wavelength is digitized by A/D converter 2248 and the re-
sulting detector output value at each wavelength sent to computer
2252 where it is divided by the previously determined reflectance
of the calibration specimen to yield the adjusted detector output
value. The adjusted detector output values for each wavelength
are stored in the computer in a table called the "reference
spectrum".
Reflectance at a Single Site
Refer to Figure 22 and Figure 4A. The method for determin-
ing the reflectance at a single site on the specimen will now be
described. Light from source 2200 incident at spot of illumina-
tion 2236 on specimen 490 is partially reflected from surface 410
of thin layer 404 and from interface 408 between layer 404 and
substrate 406. The light reflected from the two surfaces recom-
bines and is directed towards the monochromator as described
under APPARATUS. The apparat~s operator selects the wavelength
to be used for the reflectance measurement. ~he detector output
value at the selected wavelength which emerges from the A/D
converter is sent to the computer where it is divided by the
adjusted detector output value at the same wavelength (obtained
from the reference spectrum) to yield the measure of the reflect-
ance at the spot of illumination 2236 of specimen 490. The
29
2~&6~48
reflectance at the spot of illumination 2236 is stored in the
computer.
Reflectance at Grid Sites - Fringe Mapping
The method of the invention for obtaining reflectances at
many sites on a specimen will now be described. This process is
called "fringe mapping".
The apparatus operator selects the wavelength of the light
to be used for the reflectance measurements so that two or three
interference fringes will be present across the width of the
typical wafer. The selection of the light wavelength is based on
the experience of the operator, or when required, the operator
will conduct trials to determine a suitable light wavelength.
For a specific specimen, increasing the light wavelength will de-
crease the number of fringes across the specimen and decreasing
the light wavelength will increase the number of fringes across
the specimen.
~ efer to Figure 4. Measurements of reflectance on a speci-
men are made at sites defined by a rectangular sampling grid
superposed on the specimen. The number and spacing of the grid
lines is selected by the apparatus operator. Typically, there
would be 200 to 300 grid lines across a 5 cm wafer. For clarity,
the sampling grid 400 shown in Figure 4C has fewer grid lines
than would be used in practice. In Figure 4A, grid 400 is shown
superposed on specimen 4so. Typical sampling site 402 is shown
2~858~8
ln Figures 4A, 4B, and 4C. Refer to Figure 4C. The X- and Y-
directions of Figure 4C correspond to the X- and Y-directions of
positioning stage 2256. The position of a grid site can be
specified by X and Y grid coordinates.
Refer to Figure 22. After the apparatus operator selects
the fringe mapping parameters to be used, computer 2252 begins to
collect the monochromatic reflectance data. Under the control of
computer 2252 stage 2256 is positioned so that one of the grid
sites at which reflectance is to be measured falls under spot of
illumination 2236. The reflectance at each grid site is deter-
mined in the manner described above and stored in an array called
the "fringe map". The storage position in the fringe map corre-
sponds to the grid coordinates of the grid site at which the
reflectance is measured.
CHARACTERISTICS OF THE FRINGE MAP
Ideal Data
The fringe map is a discrete representation of monochromatic
interference fringes, and is therefore a digital image.
Refer to Figures 4 and 8B. Figure 8B is a representative
plot of the fringe map obtained by the methods of the invention
for thin layer 404 of specimen 490 of Figure 4A.
Refe~ to Figures 9A and 8B. Figure 9A is a representative
plot of the reflectance values along a typical line P-Q sho~n in
Fi~ure 8B, for ideal reflectance data. The reflectance curve in
Figure 9A is smoothly varying. Rmin and Rmax are the minimum and
2~863~
maximum possible reflectances for specimen 490, and are calculat-
ed using equation (3) of the BACKGROUND section. Extrema, such
as 812, 810, 814 and 816 in Figure 9A, with reflectances very
close to Rmin or Rmax are labelled fringe extrema. Other extre-
ma, such as 902 in Figure 9A, are labelled local extrema. Thereflectance data are discrete, but closely spaced. A larger
scale plot of the data in the vicinity of fringe maximum 812 is
shown in Figure 9C.
The difference in layer thickness between a fringe maximum
and an adjacent fringe minimum is one quarter wave, d, which is
defined by equation (4) of MONOCHROMATIC INTERFERENCE FRINGES.
Non-Ideal Data
Refer to Figures 9B and 9D, which show representative plots
of non-ideal reflectance data. The high frequency features of
Figure 9B are random measurement noise. Reflectance values in
the vicinity of fringe maximum 920 of Figure 9B are plotted in
Fi~ure 9D, to show the effect on a larger scale. Typically, the
maximum reflectance value at 920 is not as close to the value
Rmax as it is for ideal data. Large narrow s~ikes, which are
usually downwardly directed (typically 930 and 932 of Figure 9B)
are also present. Spikes are due to imperfections in the layer
or dust particles on the surface. The methods of the invention
require smoothly varying reflectance data, and it is therefore
necessary to remove noise and spikes from non-ideal data before
processing. This is accomplished in the manner described in the
DATA CONDITIONING section below.
20~8'18
Other non-ideal features which may be found in fringe maps
are invalid regions, fringe discontinuities and contrast varia-
tion. Each of these will be briefly described.
Invalid regions of the fringe map include areas outside the
boundary of the specimen, and near the outer edges of some speci-
mens. For example, the edges of a semiconductor wafer bearing
an epitaxial layer are known to have highly irregular reflectivi-
ty, due to unusual growth conditions for the layer. Invalid
regions are excluded from processing.
A typical fringe discontinuity is shown in Figure 21. Such
discontinuities are the result of abrupt layer thickness changes
in the specimen. Regions of the fringe map separated by fringe
discontinuities are processed separately.
Typical contrast variation is shown in Figure 9B. The re-
flectances at fringe maxima are not necessarily constant; like-
wise reflectances at fringe minima are not necessarily constant.
As a consequence, the fringes show contrast variation across the
specimen. For example, difference in reflectance between fringe
minima 922 and fringe maximum 920 is not the same as the differ-
ence in reflectance between fringe maximum 926 and fringe minimum924 in Figure 9B. The method of the invention compensates for
fringe contrast variation.
The treatment of the non-ideal features of the fringe map is
described in detail in the section NON-IDEAL DATA PROCESSING
METHoD-
2 Q ~
DATA CONDITIONING
The fringe map is a digital image, and well-known digital
image processing techniques are utilized to condition the data
for further processing. Such techniques are described in DIGITAL
IMAGE PROCESSING by Kenneth R. Castleman published by Prentice-
Hall (New Jersey. 1979) and other publications.
The first step in data conditioning is smoothing. Based on
experience, the apparatus operator chooses the smoothing filter
to be used. Typical smoothing filters include: 5-point averaging
window applied along each row of the fringe map; and convolution
of the fringe map with square averaging matrices, typically 3x3
or 5x5 in dimension. The smoothing step removes most of the
random noise features. Smoothing in this manner does not com-
pletely remove large narrow spikes~ The residual spikes in the
smoothed data are broader and smaller in amplitude than the
spikes in the original data.
Residual spikes are next removed from the fringe map. The
apparatus operator chooses one of the known edge-finding or
gradient filters stored in computer 2252 to locate residual
spikes. The reflectance values at the residual spikes are re-
moved from the fringe map and replaced with values interpolated
from the surrounding values in the fringe map.
The data condi~ioning procedures described above prepare
non-ideal data for further processing.
In some cases, the conditioned fringe map closely approxi-
mates the fringe map for ideal data, and the conditioned data are
34
20~58~
:hen processed by the same methods that are used with ldeal data
to obtain the thickness map.
LOCATION OF FRINGE EXTREMA IN THE FRINGE MAP
Based on experience the apparatus operator selects the
spatial density of grid sites so that the fringe extrema can be
accurately located.
A preferred method for locating fringe extrema in data which
are smoothly varying will now be described.
Rmax and Rmin are calculated.
Based either on experience or reflectance data statistics,
the apparatus operator specifies values for fa~tors K1 and K2.
The value. of Kl is typically O.O1; the range of values for K2 is
typically O.Ol-0.05. The operator also specifies an integer
value for the number K3, which is greater than 3 (typically 5).
Parameters used in the determination of fringe extrema are
calculated according to the following relations:
Tolerance TOL = Kl~ Rmax - Rmin )
Significance Factor ~R = K2( Rmax - Rmin )
Upper Threshold Rhi = Rmax - TOL
Lower Threshold Rlo = Rmin + TOL
A row in the fringe map is selected for processing.
The column grid coordinate of the first site to be consid-
ered in the row is stored in the computer variable START. The
column grid coordinate of the final site to be considered in the
row is stored in the computer variable STOP. The computer varia-
20g6g'1~
ble EXTREME is initially set equal to START. R(EXTREME) is thereflectance value at grid coordinate EXTREME. Value (EXTREME+1)
is initially stored in computer variable C.
A straight line is fitted to the first K3 points of the
selected row, and the slope of the line is determined. The re-
flectances at column grid coordinates are sequentially examined
in ascending order commencing with the site at column C. The
reflectance at column grid coordinate C is R(C). If the slope of
the line is positive, the maximum seeking procedure described
below is followed, otherwise the minimum seeking procedure, also
described below, is followed.
The maximum seeking procedure will now be described. The follow-
ing calculations and decisions are made at each stage:
DIFF = R(EXTREME) - R(C) is calculated;
if DIFF < 0, EXTREME is set equal to C;
if DIFF > ~R,
EXTREME is flagged as a maximum and stored;
C is incremented by one to become equal to C~1;
if C > STOP, the process of locating extrema ends;
if EXTREME is flagged as a maximum and stored,
the minimum seeking procedure is commenced;
otherwise, the maximum seeking procedure is continued with
the incremented value in C.
36
208~48
The minimum seeking procedure will now be described. The follow-
ing calculations and decisions are made at each stage:
DIFF = R(EXTREME) - R(C) is calculated;
if DIFF > O, EXTREME is set equal to C;
if DIFF < ~R,
EXTREME is flagged as a minimum and stored;
C is incremented by one to become equal to C+1;
if C > STOP, the process of locating extrema ends;
if EXTREME is flagged as a minimum and stored,
the maximum seeking procedure is commenced;
otherwise, the minimum seeking procedure is continued with
the incremented value in C.
The maximum and minimum seeking procedures are applied
alternately as described until the end of the data (column STOP)
is reached. The column grid coordinates at each extremum are
stored for later use.
Once the extrema in the row have been located, all extrema
having reflectance greater than Rhi are classified as fringe
maxima; extrema having reflectance less than Rlo are classified
as fringe minima. All others are classified as local extrema.
The extrema classifications are stored along with the grid coor-
dinates for future processing.
37
2 Q ~ 8
For an example of the process, refer to Figure 15A, which
shows a plot of reflectance data along a typical row. The
extrema which must be found by the algorithm are visible to the
reader in Figure 15A, and the following paragraphs summarize what
would occur during the processing of such data.
The fitting of a straight line to the reflectance values at
the left end of the plot of Figure 15A yields a positive slope;
thus, the trend is increasing, and a maximum is sought.
Figure 15B is a plot of the reflectance values for the
columns in the vicinity of maximum 1501. Maximum 1501 is located
along the row by sequentially examining each of the reflectance
values in columns C = 1, 2, 3,..., J-2, J-1, and so on, while
always storing the position of the largest value found in varia-
ble EXTREME. The reflectance values are generally increasing
until column J. Since it is the largest value found so far, J
is stored in variable EXTREME. The value in column J+1 is less
than that in column J, as are the values in column J+2, and so
on. Eventually, in column J+6, the reflectance has decreased
from the value in column J by ~R. Thus, maximum 1501 is assigned
to the location of column J, and column index J is stored for
future use. The difference in position between the actual re-
flectance extremum and the nearest grid site corresponding to
column J is insignificant. After finding a maximum, a minimum is
sought.
Refer to Figures 15A and 15C. Minimum 1503 is located along
the row by sequentially examining the reflectance value from
column J+7, J+g, and so on, always storing the column position of
2~68~8
the smallest value found in variable EXTREME. In Figure 15C, the
reflectance values in the vicinity of 1517 of Figure 15A are
plotted, and they are generally decreasing until column K. Since
it is the smallest value found so far, K is stored in EXTREME.
The value in column K+1 is greater than that in column K, as are
the ~alues in columns K+2, through K+10, but thereafter they tend
to become less again. Since none of the reflectance values in
columns K+1 through K+12 exceed the value in column K by more
than ~R, column K is not considered to be the minimum. Thus,
insignificant extrema are ignored at this stage. Refer to Fig-
ures 15A and 15D. The reflectance values in the vicinity of 1503
are plotted. The process continues looking for a minimum, stor-
ing the position of the smallest reflectance as the values in
columns K+12 through L are examined. Eventually, in column L+9,
the reflectance has increased by ~R from its value in column L.
Therefore, minimum 1503 is located in column L. Once a minimum
is found, a maximum is sought in a similar manner to before.
Referring to Figure 15A, fringe maxima 1501, 1505, 1507,
1511, and 1513 are the extrema with maximum reflectance greater
~han Rhi, and fringe minima 1503 and 1509 are the extrema with
minimum reflectance less than Rlo. Refer again to Figures 15B
and 15D. The threshold values are shown to identify fringe
extrema.
Extrema are located and classified for any row of the fringe
map in a similar manner. Likewise, the procedure i5 used on any
portion of a row, by taking the starting and ending positions as
required.
39
2~8~4~
BRACKET SITES
To reduce the number of independent thickness measurements
that must be made while carrying out the method of the invention,
a subset of the total set of grid sites comprising the fringe map
is chosen for thickness determination. The members of the chosen
subset are called bracket sites. The rules for selecting the
locations of bracket sites will be presented for each embodiment
of the invention.
THICKNESS DETERMINATION AT A SINGLE GRID SITE
Refer to Figures 4 and 22. The method for determining the
; thickness of the thin layer 404 at a specific site on specimen
490 will now be described. Positioning stage 2256 is moved to so
that spot of illumination 2236 falls on the grid site at which
the thickness is to be determined. Under the control of computer
2252, the reflectance is measured at each of the calibration
wavelengths to yield the discrete reflectance spectrum at the
site. The discrete reflectance spectrum is stored in a discrete
reflectance table in computer 2252. The thickness of layer 404
at the specific grid site on specimen 490 is determined using
computer 2252 to mathematically fit the discrete reflectance
spectrum to its known functlonal form in the prior art manner
referred to in BACKGROVND. The thickness at the grid site is
20~48
ored in the computer along with the coordinates of the grid
site.
HALF-ORDER NUMBER DETERMINATION AT A SITE OF KNOWN THICKNESS
The half-order number at a site in the fringe map having
known thickness is determined in the following manner. The
fundamental thickness increment, d, is calculated by using equa-
tion (4). The known thickness is divided by d; the decimal
fractional part of the quotient is discarded; the resulting
integer is the half-order number at the site.
THICKNESS MAPPING
The methods of the invention for converting the reflectance
data stored in the fringe map to the thicknesses at the corre-
sponding grid sites will now be described. Three methods will be
described: (1) the TWO-DIMENSIONAL REGION PROCESSING METHOD; (2)
the ROW-BY-ROW PROCESSIN~ METHOD; and (3) the NON-IDEAL DATA
PROCESSING METHOD. Methods (l) and (2) are used when the re-
flectance data are ideal, that is, with uniform fringe contrast
and without discontinuities or noise. Methods (1) and (2) are
used to process a rectan~ular hlock of data that falls within the
specimen boundaries. Methods (1) and (2) can readily be extended
to process any area contained within the boundaries of a specimen
that is made up of rectangles. Method (3) processes data that
41
2~68~
cannot be processed by Methods (1) and (2).
TWO-DIMENSIONAL REGION PROCESSING METHOD
The method of the invention for processing two-dimensional
regions will be explained with reference to Figures 4A, 8B, lO,
and 11. The steps in the method are also presented in the flow-
chart of Figure 12. Specimen 490 of Figure 4A is representative
of the rectangular area of a specimen wafer for which the thick-
nesses at the grid sites are to be determined by the method of
the invention.
The apparatus operator initiates the thickness mapping
process by initializing parameters and issuing the commands
necessary for computer 2252 to collect and process the data.
Step 1. Mapping the Monochromatic Interference Fringes.
The fringe map for the rectangular area of representative
specimen 490 is obtained in the manner described above in the
section Reflectance at Grid Sites - Fringe Mapping.
Step 2. Determining Threshold Values.
Rmax and Rmin for specimen 490 are calculated and threshold
values Rhi and Rlo are determined, as previously described above
in the section LOCATION OF FRINGE EXTREMA IN THE FRINGE MAP.
Rmid = (Rlo+Rhi)/2 is also calculated.
Step 3. Locating the Fringe Extrema~
The grid coordinates of the fringe extrema are determined
42
2~S~8~8
ar each row of the fringe map, in the manner described in the
section LOCATION OF FRINGE EXTREMA IN THE FRINGE MAP.
Step 4. Partitioning into Regions With Fringe Extrema
The next step is to partition the fringe map into regions,
each region being the set of grid coordinates of contiguous sites
that have the same half-order number. Areas between adjacent
fringe extrema are suitable regions. Well known image segmenta-
tion techniques are used to partition the data into the regions
delineated by the fringe extrema, and each region is given a
unique identification number. Image segmentation is described in
Chapter 15 of DIGITAL IMAGE PROCESSING by Castleman and Chapters
7&8 of DIGITAL IMAGE PROCESSING by R.C. Gonzalez & P. Wintz
published by Addison-Wesley (Mass. 1987) and other publications.
Amongst the known image segmentation techniques are region grow-
ing and line segment extraction. The grid coordinates of the~ites within each region and the corresponding region identifica-
tion number are stored together in computer 2252 for processing
in the next step.
Figures 10 and 11 will be used to illustrate the result of
the image segmentation process. Figure 10 is a representation of
the regions of constant half-order number obtained by applying
image segmentation techniques to the fringe map of Figure 8B.
Each grid coordinate is uniquely assigned to a region. 1001 and
1002 of Figure 10 are typical regions. Figure 11 is an enlarge-
~5 ment o~ region 1001 of Figure 10, in which the dots are the posi-
43
2~85~`~8
ions of grid sites belonging to region 1001. In practicing theinvention, the sites would be more densely packed. The density
of sites shown within region lOO1 has been reduced for the sake
of clarity; this reduction does not affect the method of the
invention.
Step 5. Selecting a Region.
A region corresponding to one of the identification numbers
is selected for thickness determination at the sites within the
region.
For illustrative purposes, consider region lOO1 of Figure 11
to be the selected region.
Step 6. Selecting a Bracket Site Within the Selected Region.
A bracket site for the selected region is located in the
following manner. The reflectance at each of the grid sites
belonging to the selected region is compared with Rmid, and the
grid coordinates of the site having reflectance closest in value
to Rmid are stored. The storPd coordinates define the position
of the bracket site in the selected region.
For illustrative purposes, consider "S" of Figure 11 to be
the bracket site in region lOQ1.
Step 7. Determining the Half-Order Number for Selected Region.
The thickness of the thin layer at the brac~et site in the
selected region is determined by the method described above, in
the section THICKNESS DETERMINATION AT A SINGLE GRID SITE. The
thickness is stored in an array called the "thickness map." The
44
2~5848
thickness map has the same grid coordinate system as the fringe
map. The half-order number for the selected region is determined
by determining the half-order number at the bracket site in the
selected region as previously described in the section HALF-
S ORDER NUM~ER DETERMINATION AT A SITE OF KNOWN THICKNESS.
Step 8. Calculating Thickness Values for Selected Region.
The thickness values at the remaining sites in the selected
region are calculated as follows. A site in the selected region
at which the thickness has not been determined is selected. The
thickness at this site is calculated by substituting the reflect-
ance value for the site retrieved from the fringe map and the
half-order number for the selected region into equation (6a) when
the half-order number is even, and into equation (6b) when it is
odd; the calculated site thickness is stored at the appropriate
location for the site in the thickness map. Equations (6a) and
(6b~ are given in the section MONOCHROMATIC INTERFERENCE FRINGES.
The thicknesses at all sites in the selected region are deter-
mined in a similar manner.
Step 9. Repeating For Remaining Regions.
The steps 5-8 are repeated until the thickness values have
been determined for all regions of the fringe map.
The thickness map is obtained when all regions of the fringe
map have been processed.
2~o~
ROW-BY-ROW PROCESSING METHOD
The row-by-row processing method will now be described. The
row-by-row method is simpler to program for computer implementa-
tion because advanced computer memory structures and algorithms
to segment the fringe map are not required. Figure 13 is the
flowchart for the row-by-row thickness mapping process.
The apparatus operator initiates the thickness mapping
process by initializing parameters and issuing the commands
necessary for computer 2252 to collect and process the data.
Step 1. Mapping the Monochromatic Interference Fringes.
The fringe map for the rectangular area of representative
specimen 490 is obtained in the manner described above in the
section Reflectance at Grid Sites - Fringe Mapping.
Step 2. Determining Threshold Values.
Rmax and Rmin for specimen 490 are calculated and threshold
values Rhi and Rlo are determined, as previously described above
in the section LOCATION OF FRINGE EXTREMA IN THE FRINGE MAP.
Rmid = (Rlo+Rhi)/2 is also calculated.
Step 3. Selectinq a Row of the Fringe Map.
A row of the frin~e map is selected for processing. Typi-
cally, the first row selected is the row closest to the middle
(centre) of the frin~e map. Let the row number of the first
46
20~ 4~
,lected row be m.
Step 4. Locating Fringe Extrema Along Selected Row.
The fringe extrema in the selected row are located, as
previously described in the section LOCATION OF FRINGE EXTREMA IN
THE FRINGE MAP.
Step 5. Selecting Bracket Sites Along Selected Row.
A set of bracket sites for the selected row is chosen in the
following manner.
The located fringe extrema for the selected row partition
the row into intervals. The first interval is between the first
column of the selected row and the first located fringe extremum.
Each pair of adjacent fringe extrema define an interval in the
row. The last interval is between the last located fringe extre-
mum and the last column of the selected row.
In each interval, the site having reflectance closest to
~mid in value is chosen as a bracket site for the selected row.
The grid coordinates of the bracket sites are stored in computer
2252 for future use.
For purposes of illustration, refer to Figure 15A~ The
application of the above bracket site selection procedure locates
the bracket sites 1500, 1502, 1504, 1508, 1510, lS12, 1514 and
1516.
Step 6. Determining Half-Order Number at Bracket Sites.
The half-order number is determined at each bracket site in
47
20868~
he selected row in following manner.
There are two cases to be considered: case 1) when the
selected row is the first row to be processed; case 2~ when the
first row is other than the first selected row.
Case 1). Half-order numbers are independently determined at each
bracket site. To determine the half-order number independently
at a particular bracket site:
the reflectance value, R, at the bracket site is retrieved
from the fringe map and the quantity TEST = ¦R-Rmid¦ is calculat-
ed;
if TEST < TOL, where TOL is defined in the section LOCATING
FRINGE EXTREMA IN THE FRINGE MAP, then the thickness, T, is
determined at the bracket site as described in the section THICK~
NESS DETERMINATION AT A SINGLE GRID SITE,
otherwise (TEST > TOL), a more suitable site for half-order
number determination is sought along the column of the bracket
site as follows:
fringe extrema are located along the column which contains
the bracket site in the selected row; this is done in a similar
manner to that described in LOCATION OF FRINGE E~TREMA IN THE
FRINGE MAP;
the interval between fringe extrema ~or between a fringe
extremum and the edge of the fringe map) along the column which
contains the bracket site in the selected row is identified;
2S the site in the identified interval which has reflectance
most nearly equal to Rmid is located (this is the same procedure
as choosing a bracket site along the column);
48
2~8~
the thickness T at the located site is determined in the
manner described in the section THICKNESS DETERMINATION AT A
SINGLE GRID SITE;
The half-order number, H, is determined from the thickness T
in the manner described in the section HALF-ORDER NUMBER DETERMI-
NATION AT A SITE OF KNOWN THICKNESS;
the half-order number at the bracket site is equal to H.
For purposes of illustration, refer to Figure 19. Site 1900
of Figure 19 is a bracket site at column C of the selected row N,
where the half-order number is to be determined independently.
TEST is calculated and found to be ~reater than TOL; therefore
another site in column C is sought. Typically, 1902 and 1904 are
a fringe extrema, located in column C by the methods described
previously in the section LOCATING FRINGE EXTREMA IN THE FRINGE
MAP. In the example of Figure 19, 1901 is the site between
fringe extrema 1902 and 1904 which has reflectance closest to
Rmid. The thickness and half-order number are determined at 1901
as previously described. The half-order number at 1900 is set
equal to that at 1901.
Case 2). There is a previously processed adjacent row to the
selected row. The half-order number at a particular bracket site
in the selected row can often be quickly ascertained from values
calculated for the previously processed adjacent row, as will now
be described:
if any of the nearest R4 neighbours (in the adjacent row) to
the bracket site are at fringe extrema, then an independent
49
2Q86~
determination of the half-order number for the bracket site is
made, as described in case l); the value of K4 is specified by
the operator of the apparatus (typically K4=3);
otherwise, the half-order number at the bracket site is
equal to the half-order number at the site with the same column
number in the previously processed adjacent row.
For illustrative purposes, refer to Figure 16A, which shows
the arrangement of grid sites along a portion of several rows.
Row N is the currently selected row, and row N-1 is the previous-
ly selected row. 1620 and 1630 are bracket sites in row N, wherethe half-order number is required. The thickness, and therefore
half-order number, has already been determined at all of the
sites in row N-l. For this example, K4 equals 3. If none of the
sites 1622, 1624 or 1626 are located at fringe extrema, the
half-order number at bracket site 1620 is equal to that at adja-
cent site 1624; otherwise, an independent determination of
half-order ~umber is required. Similarly, the half-order number
at bracket site 1630 is set equal to the half-order number at
site 1634, provided that none of the sites 1632, 1634 or 1636 are
at a fringe extremum.
Step 7. Calculating Thickness Values for Selected Row.
Thickness values are calculated at sites across the selected
row of the fringe map as follows.
The first bracket site in the selected row is chosen as
starting point. ~ is set equal to the half-order number at the
2~ 48
tarting bracket site. Then H is used in the appropriate equa-
tion (6a) or (6b) to calculate thickness for each site along the
row to the right until a fringe extremum is reached. Each calcu-
lated thickness value is stored in its place in the equivalent
row of the thickness map. At the fringe extremum, H is set equal
to the half-order number at the next bracket site along the row
to the right of the fringe extremum. This value of H is then
used in equation (6a) or (6b) until the next fringe extremum is
reached. The calculations continue in this manner until the end
of the row is reached. Then, the portion of the row to the left
of the starting bracket site is processed in a similar fashion in
a leftward direction, until the end of the row is reached. As a
result of this, the thickness values (and half-order numbers) at
all sites along the row are determined.
For purposes of illustration, refer to Figure 16B, which
shows the arrangement of sites along a portion of a typical
selected row of the fringe map. 1620 and 1630 bracket the site
1691, which has been previously identified as the site of a
fringe extremum. In this example 1620 is selected as the start-
ing point for the row.
The determined half-order number for bracket site 1620 i5
used in the appropriate equation (6a) or (6b) as previously
described to calculate the thickness from reflectance values at
each site sequentially to the right of 1620, and the thickness
values are stored in the thickness map. At fringe extremum 1691
the half-order number from the next bracket site alon~ the row to
the right, 163~, is used and the calculations using the appropri-
51
2 Q 8 ~
te equation (6a) or (6b) continue for sites to the right.Processing continues in this manner, until the end of the row.
Upon reaching the right end of the row, the portion of the row to
the left of starting bracket site 1620 (which has not been proc-
essed) is processed to the left in the same manner as describedimmediately above.
Step 8. Repeating for Remaining Rows.
If there are more rows remaining to be processed in the
fringe map, then an unprocessed row is selected adjacent to a
previously processed row (step 3). After the first selected row,
m, rows are selected in sequence m+1, m+2, ..., until the highest
row; and then, the rows are selected in the sequence m-1, m-2,...
1, to complete the processing of the fringe map. Steps 4 through
7 are repeated for each selected row.
It will be apparent that the procedure just described is
also applicable to processing fringe map data on a column-by-
column basis. The thickness map can be verified after row-by-row
processing by column-by-column processing.
NON-IDEAL DATA PROCESSING METHOD
The method of the invention for treating non-ideal data will
now be described. Typically non-ideal data contains noise,
spikes and contrast variations. See the section ~on-Ideal Data
of CHARACTERISTICS OF THE FRINGE MAP for details. The method
2Q~'a~
removes noise and spikes from the data, and compensates for
contrast variations. The steps of the method, which uses the
techniques of row-by-row processing, will now be described. The
sequence of steps for producing a thickness map from non-ideal
data is shown as a flowchart in Figure 14.
The apparatus operator initiates the thickness mapping
process by initializing parameters and issuing the commands
necessary for computer 2252 to collect and process the data.
Step 1. Mapping the Monochromatic Interference Fringes.
The fringe map for a rectangular area is obtained in the
manner described above in the section Reflectance at Grid Sites -
Fringe Mapping. The area of the fringe map may extend beyond the
boundaries of the specimen to be processed.
Refer to Figure 17. Figure 17A shows a typical row of non-
ideal reflectance data. Figures 17B-E illustrate how the row of
Figure 17A i~ affected by applying the steps of the non-ideal
processing method.
Step 2. Identifying Invalid Areas.
The normal range of reflectance values for a particular type
of ~pe~imen i5 known. The reflectances at sites in the fringe
map are compared to the normal range. Sites at which the re-
flectance is outside the normal range are assigned a flag value,
typically -1. Areas of the fringe map having the flag value are
considered to be invalid, and are ignored during processing.
2Q~848
Aese invalid areas are generally located outside the boundaries
of the specimen.
Based upon experience, the apparatus operator identifies any
additional invalid areas, such as the edge regions of the speci-
men, which are to be excluded from processing.
Refer to Figure 17A, which is the typical row of reflectance
data extending across the specimen. The ends of the row corre-
spond to the edge regions of the specimen, and have been marked
"INVALID". The invalid areas are not displayed in Figures 17B
through 17E.
Step 3. Partitioning Fringe Map into Discontinuity-Free Sub-
Areas
Gradient (edge finding) methods are used to search for and
locate discontinuities. If discontinuities are present, they
partition the fringe map into sub-areas. Each sub-area is free
of discontinuities. Each of these sub-areas is processed indi-
vidually. If the fringe map is entirely free of discontinui-
ties, which is frequently the case, then the "sub-area" consists
of the entire fringe map, less invalid areas. Figure 21 is a
representation of a fringe discontinuity which divides the fringe
map into two sub-areas. There may be many sub-areas in a specif-
ic specimen, and they are identified and stored in a similar
manner to that used to identify and store regions in the section
TWO-DIMENSIONAL REGION PROCESSING METHOD.
Step 4. Selecting a Discontinuity-Free Sub-Area.
54
2n~ g
one of the discontinuity-free sub-areas is selected for
processing.
Refer again to Figure 17A, in which a discontinuity is
evident in the reflectance data. The data to the left of the
discontinuity belongs to one discontinuity-free sub-area, and the
data to the right belongs to another sub-area. For illustrative
purposes, the section to the left of the discontinuity is in the
selected sub-area for further processing, and only this section
is shown in Figures 17B-E.
Step 5. Smoothing Reflectance Data.
The data in the selected sub-area of the fringe map is
smoothed in the manner previously described in the section DATA
CONDITIONING.
Figure 17B shows how the reflectance data along the typical
row in the selected sub-area appears after smoothing.
Step 6. Determining Threshold Values.
Rmax and Rmin for the specimen being processed are calculat-
ed and threshold values Rhi and Rlo are determined, as previously
described above in the section LOCATION OF FRINGE EXTREMA IN THE
FRINGE MAP. Rmid = (Rlo+Rhi)/2 is also calculated.
The horizontal dashed Iines in Figure 17 are the theoretical
minimum and maximum reflectance, Rmin and Rmax, of the specimen.
~g>~)~48
Step 7. Selecting a Row.
A row in the selected sub-area is selected for processing.
Typically, the first row selected is the row closest to the
middle (centre) of the fringe map. Let the row number of the
first selected row be m.
Step 8. Locating Extrema Along Selected Row.
Extrema are located along the selected row and classified as
previously described, in the section LOCATION OF FRINGE EXTREMA
IN THE FRINGE MAP; grid coordinates of ALL located extrema are
stored for future reference. In Figure 17B, 1710, 1711, 1712,
1713, 1714, 1715, 1716, 1717, 1718 and 1719 are extrema in the
typical selected row. The located extrema are initially classi-
fied as "fringe" or "local'l type, according to the criteria of
the section LOCATION OF FRINGE EXTREMA IN THE FRINGE MAP. In
Figure 17C, extrema marked F are classified as fringe extrema,
and extrema marked L are classified as local extrema.
Step 9. Selecting Bracket Sites Along Selected Row.
From the selected row, a set of sites which bracket the
located fringe extrema in the row is determined as described in
the section ROW-BY-ROW PROCESSING METHOD. In Figure 17C, bracket
sites 1720, 1711, 1722, 1723, 1724, 1725, 1726, 1727, 1728, and
1729 are located. Note that the local extremum 1711 is also
selected as a bracket site by this method.
The data values in the row are scanned to determine all of
56
2 ~ 4 ~
the locations where the Rmid threshold is crossed. The grid
coordinates of the site having reflectance closest to Rmid at
each crossing are temporarily stored. The locations of the
crossing sites are compared to the locations of the previously
determined bracket sites, and those crossing sites which are not
already bracket sites are selected as additional bracket sites.
The correspondin~ crossing site grid coordinates are stored with
the previously determined bracket site coordinates in the select-
ed row for future use. As a result, all peaks and valleys of the
reflectance along the row which extend across the Rmid threshold
are bracketed, reqardless of "fringe" or "local" classification.
In Figure 17C, 1723 and 1728 are the additional bracket
sites located by the Rmid threshold crossing technique. Extrema
1713 and 1717, which initially are classified as local extrema
are now bracketed. The additional bracket sites allow the clas-
sifications of the extrema to be verified, as discussed below.
Step 10. Determining Half-Order Number at Bracket Sites.
The half-order numbers for each of the bracket sites in the
selected row are determined in the same manner as described in
the ROW-BY-ROW PROCESSING METHOD.
Step 11. Rationalizing Extrema Classifications.
The half-order numbers of the bracket sites in the selected
row are used to check the correctness of the initial extrema
classifications. If the initial classification is correct, the
20g~ 4~
xtremum is considered verified. If the initial classification
of the extremum is incorrect, the classification of the extremum
is corrected. This procedure, which is called rationalization,
will now be described.
The change in half-order number, Delta H, between each pair
of adjacent bracket sites in the selected row is evaluated. The
values for Delta H are integers, most often o or 1, but sometimes
2, 3 or higher. Each pair of adjacent bracket sites is selected
in turn, and the number and type of the extrema between them are
determined. The following cases must be considered for each pair
of bracket sites.
Delta H = 0:
if the number of fringe extrema is equal to 0 or 1,
the extrema within the pair of selected
bracket sites are correctly classified;
if the number of fringe extrema is not equal to 0 or 1,
all fringe extrema within the pair of selected
bracket sites are reclassified as local extrema.
Delta H = 1:
if the number of fringe extrema is equal to 1,
the extremum within the pair of selected
bracket sites is correctly classified;
if the number of fringe extrema does not equal 1,
then if there is no relativ~ phase chanqe between
reflections at the two interfaces for the layer,
which occurs when nC<nL<nS, then
if the larger half-order number for the
58
208~4~
pair of bracket sites is even
the local maximum within the
bracket sites which has reflectance
closest to Rmax is reclassified to
be a fringe maximum;
if the larger half-order number for the
pair of bracket sites is odd
the local minimum within the
bracket sites which has reflectance
closest to Rmin is reclassified to
be a fringe minimum;
or if there is a relative phase change of ~ radians
between reflections at the two interfaces for the
layer, then
if the larger half-order number for the
pair of bracket sites is odd
the local maximum within the
bracket sites which has reflectance
closest to Rmax is reclassified to
2~ be a fringe maximum;
if the larger half-order number for the
pair of bracket sites is e~en
the local minimum within the
bracket sites which has reflectance
closest to Rmin is reclassified to
be a fringe minimum;
59
2 ~ 8
elta H = 2:
if the number of fringe extrema is equal to 2,
then the extremum within the pair of selected
bracket sites is correctly classified;
if the number of fringe extrema equals o, then
the local maximum closest to Rmax and local minimum
closest to Rmin are reclassified to be fringe extrema;
if there is one fringe maximum, then
the local minimum closest to Rmin is reclassified to be
a fringe minimum;
if there is one fringe minimum, then
the local maximum closest to Rmax is reclassified to be
a fringe maximum.
The process for reclassifying extrema can be extended for cases
when the value of Delta H is greater than 2 by applying the
following rules.
1) Delta H has magnitude unity or zero for bracket sites
around an interval containing a single fringe extremum;
2) Delta H has magnitude zero for bracket sites around an
interval containing only local extrema.
3) Delta H equals the total number of fringe extrema in the
interval between two bracket sites;
4) for even values of Delta H, the number of fringe maxima
must equal the number of fringe minima between two bracket sites;
5) for odd values of Delta H, fringe maxima are one less or
one greater in number than fringe minima in the interval between
the bracket sites; when there is no relative phase change be-
20~5348
tween reflections from the cover/layer interface and from the
layer/substrate interface (nC<nL<nS), then there are more fringe
maxima between two bracket sites when the larger H is even.
Conversely, there are more fringe minima between two bracket
sites when the larger H is odd. For cases when there is a rela-
tive phase shift of ~ radians between the reflections at the two
different interfaces, then these stated rules for odd Delta H are
reversed.
An example to illustrate the rationalization process in part
follows. Consider the case where the interval between two brack-
eting sites has a single fringe maximum and Delta H equals 2;
there is a misclassified local minimum in the region. The local
minimum is reclassified to fringe minimum, even if it does not
satisfy the Rlo threshold. As another example, if the refractive
indices for the specimen are such that nC<nL<nS and an interval
i8 bounded by two sites with half-order numbers of 137 and 142,
then there must be 3 fringe maxima and 2 fringe minima between
the sites. If the extrema classifications do not support this,
then extrema are reclassified accordingly.
The correction of misclassifications may lead to more fringe
extrema for the row. Each fringe extremum along the row is
checked to determine whether or not it is bracketed. All addi-
tional bracket sites that are needed to bracket the fringe extre-
ma are selected and their half-order numbers are determined as
before. The rationalization process continues until the row is
fully rationalized, that is, the extrema classifications obey the
~Q~53~8
above stated rules.
For a specific example, refer to Figure 18, which is a
composite of Eigures 17B and 17C on a larger scale. Typical
changes in half-order number, Delta H, between each pair of
adjacent bracketing sites, are listed below the data in Figure
18. Delta H has unit or zero magnitude around fringe extrema
such as 1715, 1716, and 1718, in accordance with rule 1). Delta
H around local extrema such as 1717 is zero, in accordance with
rule 2).
Around local extremum 1713, the change in half-order number
between the bracket sites 1722 and 1723 is found to be non-zero.
Therefore, 1713 is misclassified. The other extrema in the row
depicted in Figure 18 are correctly classified.
Refer to Figure 17D, where 1713 is reclassified to be a
fringe extremum (marked F).
Step 12. Stretching Reflectance Data.
The data values in the selected row are stretched so that
the fringe minima have reflectance Rmin and the fringe maxima
have reflectance Rmax. Intermediate data values are stretched
proportionally between Rmin and Rmax.
Refer to Figure 17E, which illustrates the effect of
stretching the reflectance data of Figure 17D. The method for
stretching data will now be described.
Stretching transformations are the relations which transform
observed data into a form from which thickness values can be
62
2Q~6~48
alculated using equation (6) of MONOCHROMATIC INTERFERENCE
FRINGES. The stretching transformations depend upon the local
characteristics of the reflectance profile for the selected row.
There are three types of row "pieces", each of which requires a
specific stretching procedure. The row is stretched piecewise
until the whole row is stretched~ The different types of pieces
and the corresponding stretching procedures will now be de-
scribed.
1) The first type of piece to be considered is that which is
between a fringe minimum and an adjacent fringe maximum (or
between a fringe maximum and an adjacent fringe minimum). Refer
to Figure 20A, which shows the reflectance profile along a row
between a fringe minimum, W, and a fringe maximum, Z. The trans-
formation for stretching the piece from W to Z will now be de-
scribed. The quantities used to calculate the stretching trans-
formation are shown in Figure 20A. ~1 and ~2 are the increments
between the reflectance at W and Rmin and reflectance at Z and
Rmax respectively:
RW + ~1 = Rmin, and
Rz + ~2 = Rmax. (7)
Let P be a site along the selected row in the W-Z interval
having stored reflectance R (in the fringe map). P is located a
distance u from the observed minimum, W, and a distance v from
the observed maximum, Z. A preferred transformation for the
2~ reflectance R in the W-~ interval, is:
R' = R + ~
63
2~ ~ ~ 3 ~
( v ~l + u ~2 ) / ( u + v ) (8).
~1 and ~2 may differ significantly, and they can be positive or
negative quantities. Within the row being processed, all such
pieces between a fringe minimum and an adjacent fringe maximum
are found. For each piece, ~l and ~2 are evaluated according to
equation (7); the data within the piece are then stretched ac-
cording to transformation (8). u and v are calculated for each
site in the piece, and ~ is added to the reflectance value R in
the fringe map to yield R'. The new value of reflectance R', is
stored in the fringe map in place of the value R. The procedure
is obviously extendable to the case where the fringe maximum is
to the left of the fringe minimum.
2) The second type of piece to be considered is that which
is between two fringe extrema of the same type, and with type 1
pieces on either side. The results of stretching the pieces of
type 1 in the row are used in stretching pieces of type 2. Refer
to Figure 20B, which presents a representative case of a type 2
piece, where there are two fringe maxima, E and F, with no inter-
vening fringe minimum. The point Q is the most significant local
minimum between the fringe maxima. Q is called the principle
local extremum and has reflectance RLE. Q divides the piece be-
tween the fringe maxima into the intervals of width A and B.
Reflectance values in the outside regions, C and D, have been
transformed when type l pieces were stretched. ~Cl and ~C2 for
region C and ~D1 and ~D2 for region D, have been determined. The
amount by which Q is stretched will now be determined using the
64
'~Q~34~
results from previously stretched type 1 pieces C and D. Two
quantities R'C and R'D are first calculated. R'C is equal to
the transformed reflectance of site, QC~ nearest Q in pi.ece C
which has reflectance equal to RLE. Similarly, R'D is equal to
the transformed reflectance of site, QD~ nearest Q in piece D
which has reflectance equal to RLE. The relations used are:
R'C = RLE + SC
R'D = RLE + SD
where the Sc and SD are defined for the pieces C and D according
to the equation (8). The transformed reflectance value for the
principle local extremum, Q, will be:
R Q = (B R'-c + A R'D) / (A + B)
and the increment for Q is:
SQ ( B ~C + A ~D ) / ( A + B ).
The reflectances at all sites in the intervals A and B are then
stretched individually, using the same functional form as given
by equation (8). For a site PA in interval A with reflectance R,
R' = R + ( vA SC2 + uA SQ )/( uA + vA ).
As before, uA and vA are distances from a site in interval A to
each bou~dary for the interval. Reflectance values in interval B
are transformed similarly.
It is readily apparent that the case where the piece to be
stretche~ falls between two fri.nge minima can be treated in a
2Q~48
anner analogous to that described for a piece falling between
two fringe maxima.
3) The third type of piece to be considered is that which
falls between the end of a row and a type 1) piece. Refer to
Figure 20C, which depicts the reflectance profile along one end
of a typical row of the fringe map. J is at the end of the
selected row. In this situation, the interval N to G is
stretched with equation (8) as previously described for the type
1) pieces. O represents any point in the G to J interval with
reflectance R. The point P in the interval N to G has reflect-
ance R. The transformed reflectance at O is set equal to the
transformed reflectance at P. This gives the same amount of
stretching in the interval G-J as an adjacent interval N-G.
After the piecewise stretching of the portion of the row
within the selected sub-area, the data are suitable for thickness
calculations.
Step 13. Calculating Thic~ness Values for Selected Row.
Thickness values are calculated at sites across the selected
row within the selected sub-area of the fringe map, as previously
described under ROW-BY-ROW PROCESSING METHOD.
Step 14. Repeating for Remaining Rows.
Steps 7-13 are repeated for all remaining rows within the
selected sub-area, in a similar manner to that descri~ed under
ROW-BY-ROW PROCESSING METHOD.
66
208~48
Step 15. Repeating for Remaining Sub-Areas
Steps 4-14 are repeated for all remaining sub-areas, thus
completing the thickness map for the specimen.
USES FOR THE THICKNESS MAP
A typical use for the thickness map is to display the map as
a colour coded video image on the computer 2252, for human in-
spection. Statistics for the layer thickness, such as mean value
and standard deviation, are also calculated from the values in
the thickness map. These statistics can be displayed by computer
2252.
In practicing the invention, thickness maps are obtained for
thin layers of material grown by molecular beam epitaxy (MBE) on
semiconductor wafers, prior to the production of devices on the
wafers. Thickness maps allow the useful area to ~e determined on
each wafer, and various processing parameters are set according
to the thickness values in the thickness maps. Also, detailed
analysis of the thickness variations on the wafers has been made
possi~le because of the introduction of the m~thods of the inven-
tion. This has lead to improvements in the MBE growth techniques
used.
CONCLUSION, RI~MIFICATIONS AND SCOPE
The methods and apparatus of the invention significantly
reduce the number of spectral measurement operations which must
67
2086848
be performed to map layer thickness, compared to the number
required with prior art methods. Large speed improvements over
prior art methods which make independent absolute thickness
determinations at each site are achieved by using the information
inherent in monochromatic interference fringes.
Because the methods of the present invention use high speed
computer processing of data, which is available at relatively low
cost, and a~e not dependent on fast measurement hardware, which
is expensive, the methods of the present invention are more
economical for thickness mapping than the prior art methods.
Improvements in computer processing speed will further enhance
the advantages of the method of the invention over the prior art
methods.
Three methods of the invention have been described under the
separate headings l) TWO-DIMENSIONAL REGION PROCESSING METHOD, 2)
ROW-BY-ROW PROCESSING METHOD and 3) NON-IDEAL DATA PROCESSING
METHOD; other methods which combine steps from these three
methods are possible. For example, the following sequence of
steps could be used to obtain a thickness map:
the monochromatic fringes are mapped (common to 1,2,3);
fringe map data are smoothed (from 3);
the extrema are located, stored, and classified in each row
of the fringe map (from 1);
row-by-row rationalization and stretching procedures are
applied to each row of the fringe map (from 3);
the fringe extrema are used to partition the fringe map into
two dimensional regions of constant half-order number (from 1);
2~8~'~4~
region-by-region processing is used to calculate the thick-
ness values (from 1).
As well, one of the three main methods of the invention can
be used to verify the thickness map generated by one of the other
methods. This can be done at the discretion of the apparatus
operator. Thickness maps generated by row-by-row processing can
also be verified by column-by-column processing.
Other variations in the mode of operation of the apparatus
used to map thickness will now be described. If the reflectance
data are sufficiently well behaved, thicknesses may be calculated
along a row while the next data row is being collected by the
apparatus. This would yield an improvement in speed.
The data collection and processing phases can be separated,
and carried out with different computers. Processing the data on
a computer other than apparatus-controlling computer 2252 in-
creases the rate at which specimens are processed, because the
apparatus of Figure 22 is free to collect data without having to
pause to analyze the data. In this mode of operation, the appa-
ratus obtains a fringe map, and determines the thickness of the
layer at all required sites in advance of fringe map analysis.
This is possible when the specimens are known to have highly
consistent thickness profiles. Epitaxial layers on semiconductor
wafers, whi~h usually have monotonic thickness profiles from
wafer centre to edge, tend to produce "bullseye" (concentric
rings) fringe patterns. For this type of frin~e pattern, it is
known that the only thickness measurements from spectral reflect-
ance that are required are across the middle row of the fringe
6g
2Q~6~48
map. Therefore, the fringe map can be obtained, and the thick-
ness determined at selected sites spaced across the middle row of
the fringe map. The data can then be processed on another com-
puter. When the fringe pattern is not of consistent form from
specimen to specimen, the apparatus operator must select the
sites where thickness measurement are made using spectral re-
flectance.
The specimen studied by the thickness mapping instrument
need not be a semiconductor wafer bearing a thin layer. The
methods of the present invention are equally applicable to other
materials, and even to thin films which are not supported by a
substrate material, such as a thin polymer film web. It will be
clear to those skilled in the art that the positioning stage
could be adapted to bear such self-supporting thin dielectric
layers for use with the present invention.
It will also be evident to those skilled in the art that
thickness maps can be made when the material comprising the layer
is partially absorbing to the light provided by the broadband
source. This involves modifying equation (3) for absorbing
(complex refractive index) materials; an analysis of this prob-
lem is found in the previously mentioned textbook by Born & Wolf.
This modified form of the equation is implicitly dependent upon
layer thickness, but is solvable by known numerical means (see
for example U.S. Patent 4787749 by Ban et al.).
Other apparatus can be used to map thickness in accordance
- with the methods of the present invention. For example, the
described apparatus can be made confocal, giving slightly better
2 0 ~ 8
spatial resolution in the plane of the layer and much better
resolution in depth. An autofocussing mechanism could also be
installed, to help assure high quality of data.
In the description of the apparatus of the invention, a
S broadband source which supplies visible and infrared wavelengths
is described. However, it will be seen by those skilled in the
art that there is no reason why the method cannot be employed
with an apparatus containing broadband W light sources, if the
specimen requires such wavelengths.
Another possible embodiment of the present invention com-
prises a computerized ellipsometer which is adapted to map thick-
nesses using the method of the invention. Those skilled in the
art will be familiar with the techniques of ellipsometry, and it
will be evident that an ellipsometer is capable of generating the
required monochromatic reflectance data. In this case, the
device would not employ the polarizer and the analyzer during the
mapping of the monochromatic fringes. The monochromatic fringes
would be observed at an oblique angle to the specimen. During
the analysis of the fringe map, the absolute layer thickness at
selected bracket sites would then be found from an ellipsometric
~easurement, which would also provide the refractive index and
absorption of the layer. An ellipsometer can be used with all of
the methods of the invention; the only change required is that
the s~ep of determining the thickness from spectral reflectance
measurement be replaced by the equivalent step of measuring the
thickness using ellipsometry.
208~8
The scope of the invention should be determined by the ap-
pended claims and their legal equivalents, and not snlely by the
examples given.
72
