Silicon Carbide – Materials, Processing and Applications in Electronic Devices

130
of diamond particles. The reason behind this is that the thermal conductivity of the
nominally pure aluminium matrix is influenced by take-up of some silicon from the silicon
carbide. It was shown in (Molina et al., 2008b) that the intrinsic value of the thermal
conductivity of pure aluminium in composites fabricated via gas-pressure infiltration might
be as low as 185 W/mK, because of the presence of silicon in solid solution in combination
with precipitated silicon phase in the matrix. Reactivity of liquid aluminium and SiC in the
“as-received” condition seems unavoidable in gas-pressure infiltration, since the time
elapsed during pressurization of the chamber and posterior solidification is of the order of
some minutes, depending on the special characteristics of the equipment at hand.
Decreasing as much as possible the infiltration temperature seems then to be a successful
way to avoid metal-ceramic reactivity in SiC-based systems.
4.3.2 Metal/SiC-graphite flakes composites
A very recent family of composite materials has been developed and patented at the
University of Alicante (Narciso et al., 2007; Prieto et al., 2008). The invention is concerned
with a composite material with high thermal performance and low cost which has a layered
structure achieved by proper combination of different components. The components of the
material are three: 1) a phase mainly formed by graphite flakes (phase A); 2) a second phase
(phase B) involving particles or fibers of a material which can act as a phase separator of
phase A (phase B is a ceramic material preferably selected from the group of SiC, BN, AlN,
TiB
2
, diamond and carbon fibers); and finally, 3) a third phase (phase C) formed by a
metallic alloy. The three present phases must have good thermal properties, although their
main function is different for each one: phase A (graphite flakes) is the principal responsible
of the properties of the final material, phase B acts as a separator of the layers of phase A
and phase C has to consolidate the preform.

(pp
m/K
)

60% graphite flakes + 40% SiC Al-12%Si 0.88
x
y
: 368
z: 65
z: 11
x
y
: 7.0
63% graphite flakes + 37% SiC Ag-3%Si 0.88
x
y
: 360
z: 64
z: 11
x
y
: 8.0
Table 4. Thermal properties of metal/SiC-graphite flakes composites. xy refers to the
graphene planes, while the direction perpendicular to it is denoted by z
Although the properties presented in Table 4 are very good, the authors of the patent (Narciso
et al., 2007) have already encountered even more promising values when special conditions of
infiltration are used. The thermal properties of these composites are currently being evaluated
by means of different modelling schemes, conveniently adapted to account for both the
anisotropic microstructure of the materials at the mesoscale and the anisotropy in the intrinsic
thermal properties of the graphite flakes. Modelling on this system arouses special interest

simple. One of the methods to get the threshold pressure of a given system is to infiltrate a

Silicon Carbide – Materials, Processing and Applications in Electronic Devices

132
preform at various applied pressures and measure, for a fixed time, the infiltrated height for
each pressure (Garcia-Cordovilla et al., 1999, Molina et al., 2004; Molina et al, 2008; Piñero et
al, 2008). Data are then analysed by means of Darcy’s law:

()
()
2
2
1
o
r
kt
hPP
V
μ
⋅
=−
⋅−
(11)
where
k is the permeability of the porous solid, t is the infiltration time and
μ
the viscosity of
the liquid metal.
P

lv
the surface tension of the molten metal at the infiltration
temperature. The value of
P
0
is clearly dependent on the wetting characteristics of the
system and, hence, may be strongly affected by the reactive phenomena occurring at the
interface between metal and substrate while the infiltration front moves over the substrate
surface. The study of this parameter becomes especially interesting for those systems where
infiltration front movement and reaction cannot be decoupled in time. A remarkable fact
that has to be taken in consideration is that if infiltration occurs too rapidly, reaction could
be prevented and the system may behave as non-reactive. A conclusive study regarding
these points was presented in (Molina et al., 2007b; Tian et al., 2005), which discusses results
for infiltration of pure Al and Al-12wt%Si into compacts of as-received and thermally
oxidized SiC particles. The main results of this study are summarized in Fig. 11a.

0
200
400
600
800
1000
1200
1400
500 600 700 800 900
pressure (kPa)
h
2
(mm
2

). The straight lines are linear fittings of experimental data; (b) Threshold
pressure
P
0
versus γ
lv
 for the different systems in (a). The line corresponds to a fitting with
equation
P
0
= 0.603
γ
lv

+ 32.9 kPa

SiC as Base of Composite Materials for Thermal Management

133
The most important conclusion is that the contact angle derived from a fitting of the
experimental data (Fig. 11b) by means of Eq. (12) is the same for all cases studied. The
infiltration behaviour of the different systems, governed by a unique contact angle, indicates
that the metal/particle interface is in both cases the same. Instead of being a metal/SiC
contact, there exists an interlayer of silica between both. The very thin silica layer that covers
naturally the SiC particles seems to be thick enough to partly remain after reaction with the
metal during infiltration at these low temperatures and relatively rapid infiltration kinetics.
Another system with remarkable interest is Ag/SiC (Garcia-Cordovilla et al., 1999; Molina et
al., 2003b). Silver is a metal with high capacity for dissolution of oxygen in the molten state.
This oxygen can rapidly oxidize the SiC particles. This was observed to affect directly the
threshold pressure of the system by increasing its value. The apparent contact angle derived

==
⋅⋅

(13)
where
P, S and V
r
are saturation, applied pressure and volume fraction of reinforcement,
respectively;
A
v
is the particle specific surface area per unit volume of preform. The contact
angle can be easily derived by making use of the following relationship:

cos
ilv
W
γ
θ
=⋅ (14)
Recently, a new technique was proposed for the direct measurement of capillary forces during
the infiltration process of high-temperature melting non-wetting liquids into ceramic
preforms. In essence, the equipment is a high-temperature analogue of mercury porosimetry.
The device can track dynamically the volume of metal that is displaced during pressurization
and hence allows obtaining in a single experiment the entire drainage curve characterizing
capillarity in high-temperature infiltration of particles by molten metal (Bahraini et al., 2005;
Bahraini et al., 2008; Molina et al., 2007a; Molina et al., 2008d). The technique was validated in
an study of wetting of silicon carbide by pure aluminium and by aluminium-silicon eutectic
alloy using drainage curves obtained during gas pressure infiltration at 750ºC.


1
1.2
00.511.52
pressure (MPa)
saturation
SiC320/Hg
SiC320/Al
SiC320/Al12Si

Fig. 12. Drainage curves of SiC320 infiltrated with Hg, Al and Al-12%Si at 750ºC

0
0.2
0.4
0.6
0.8
1
1.2
024681012
pressure (MPa)
saturation
SiC1000/Al (0.13MPa/s)
SiC1000/Al (0.05MPa/s)

0
0.2
0.4
0.6
0.8
1

at two largely different infiltration kinetics. Fig. 14a resumes the thermal conductivities for
both series of composites together with modelling predictions using the DEM scheme.

150
160
170
180
190
200
210
220
230
240
250
0 0.2 0.4 0.6 0.8
fraction of coarse particles
thermal conductivity (W/mK)
SC
GPI - fast
GPI - slow
0
2
4
6
8
10
12
0.50.550.60.650.70.75
particle volume fraction
CTE (ppm/K)

2
K. For the slow GPI this
parameter has a value which is about the half, most probably due to the abundant reaction
product (Al
4
C
3
) at the interface (Weber et al., 2010).
The results of the CTE measurements are collected in Fig. 14b. The physical CTEs (measured
in a range of ±5ºC around the indicated temperature) are given for the SC and the fast GPI
samples only, yet for two temperatures of technical interest, i.e., 25ºC and 125ºC. The CTE
decreases in general with increasing SiC volume fraction and is typically 1–1.5 ppm/K
higher at 125ºC than at ambient temperature.
5.4 Effect of porosity
In a non-wetting system like Al/SiC infiltration of the metal into the open channels of the
preform does not take place at a single, well-defined pressure but, as already seen, it rather
takes place progressively with the applied pressure when this pressure exceeds a certain
threshold (threshold pressure). In order to obtain a hundred percent filling of the porous
space of the preform by the metal an infinitely large pressure, impossible to obtain in
laboratory, would be needed. For a given infiltration pressure, therefore, defects at the
contact area of particles will exist and porosity will hence be unavoidable.

140
160
180
200
220
240
140 160 180 200 220 240
exp thermal conductivity (W/mK)

6. Conclusion
Several composite materials containing SiC as reinforcement, either single or combined with
other ceramics, have been presented as serious candidates to cover the specific demand of
heat dissipation for thermal management applications. Aside from the metal/SiC
composites with monomodal distribution of SiC particles, which nowadays define the state
of the art in materials for electronics, those derived from combinations of SiC with either SiC
of another largely different size (bimodal mixtures) or other ceramics (hybrid mixtures with
diamond or graphite flakes) present high values of thermal conductivity and coefficients of
thermal expansion extremely low such as to represent the future generation of heat sinks for
electronics. The use of these composites is mainly determined by the specific requirements
for every application, taking into account not only the thermal properties but also density,
isotropy or ease of machinability (when complex shapes are needed). The spectrum covered
by the SiC-based composites aims to offer specific solutions for the different problems of
heat dissipation encountered in the energy-related industries such as electronics or
aeronautics.
This contribution emphasizes the fact that the choice of a proper fabrication processing is as
important as a good selection of the constituents of the composite material. Being
aluminium a very used metal for the fabrication of SiC-based composites, processing by
liquid state routes must take into account the high reactivity between Al and SiC at the
temperature of molten aluminium. In these sense, squeeze casting, which operates allowing
very short contact times between metal and reinforcement, offers composites with the
highest values of thermal conductivity. Several specific conditions should be taken into
account in gas pressure infiltration to give appropriate materials with acceptable thermal
properties. In any case, porosity has to be avoided because dramatically decreases the
thermal conductivity of the materials. For this purpose, a certain minimum pressure that
ensures complete saturation is needed along with a certain pressurization rate in order to
force that infiltration and reactivity can be decoupled in time, since interfacial reaction can
hinder infiltration.
7. Acknowledgement
The author acknowledges all those who have actively participated to the research presented

Comprehensive Composite Materials, A. Kelly & C. Zweben (Eds.), 1-26, Elsevier
Science, ISBN 0-080437214 (Volume 3), Oxford UK, United Kingdom
Clyne, T.W (2000). Thermal and electrical conduction in MMCs, In:
Comprehensive Composite
Materials
, A. Kelly & C. Zweben (Eds.), 447-468, Elsevier Science, ISBN 0-080437214
(Volume 3), Oxford UK, United Kingdom
Garcia-Cordovilla, C.; Louis, E. & Narciso, J. (1999). Pressure infiltration of packed ceramic
particulates by liquid metals.
Acta Materialia, Vol.47, No.18 (August 1999), pp. 4461-
4479, ISSN 1359-6454
Molina, J.M.; Saravanan, R.A.; Arpon, R.; Narciso, J.; Garcia-Cordovilla, C. & Louis, E.
(2002). Pressure infiltration of liquid aluminium into packed SiC particulares with a
bimodal size distribution.
Acta Materialia, Vol.50, No.2, (September 2001), pp. 247-
257, ISSN 1359-6454

Molina, J.M.; Arpon, A.; Saravanan, R.A.; Garcia-Cordovilla, C.; Louis, E. & Narciso, J.
(2003). Thermal expansion coefficient and wear performance of aluminium/SiC
composites with bimodal particle distributions.
Materials Science and Technology,
Vol.19, (July 2002), pp. 491-496, ISSN 0861-9786
Molina, J.M.; Garcia-Cordovilla, C; Louis, E. & Narciso, J. (2003). Pressure infiltration of
silver into compacts of oxidized SiC.
Materials Science Forum, Vols.426-432, (July
2003), pp. 2181-2186, ISSN 0255-5476
Molina, J.M.; Arpon, R.; Saravanan, R.A.; Garcia-Cordovilla, C.; Louis, E. & Narciso, J.
(2004). Threshold pressure for infiltration and particle specific surface area of
particle compacts with bimodal size distributions.
Scripta Materialia, Vol.51, (June

Scripta
Materialia
, Vol.59, (March 2008), pp. 243-246, ISSN 1359-6462
Molina, J.M.; Bahraini, M.; Weber, L & Mortensen, A. (2008). Direct measurement of
drainage curves in infiltration of SiC particle preforms: influence of interfacial
reactivity.
Journal of Materials Science, Vol.43, No.15 (April 2008), pp. 5061-5067,
ISSN 0022-2461
Molina, J.M.; Prieto, R.; Narciso, J. & Louis, E. (2009). The effect of porosity on the thermal
conductivity of Al-12wt%Si/SiC composites.
Scripta Materialia, Vol.60, (December
2008), pp. 582-585, ISSN 1359-6462
Molina, J.M.; Narciso, J. & Louis, E. (2010). On the triple line in infiltration of liquid metals
into porous preforms.
Scripta Materialia, Vol.62, (March 2010), pp. 961-965, ISSN
1359-6462
Narciso, J.; Weber, L.; Molina, J.M.; Mortensen, A. & Louis, E. (2006). Reactivity and thermal
behaviour of Cu-Si/SiC composites: effects of SiC oxidation.
Materials Science and
Technology
, Vol.22, No.12, (February 2006), pp. 1464-1468, ISSN 0861-9786
Narciso, J.; Prieto, R.; Molina, J.M. & Louis, E. (2007). Production of composite materials
with high thermal conductivity. Spanish patent (P002700804 2007), European
Application Patent (EP2130932-A1 2009), US Application Patent (US 20100143690-
A1 2010)
Piñero, E.; Molina, J.M.; Narciso, J. & Louis, E. (2008). Liquid metal infiltration into particle
compacts chemically and morphologically heterogeneous.
Materials Science &
Engineering A
, Vol.495, (November 2007), pp. 288-291, ISSN 0921-5093

SiC Single Crystal
Lina Ning and Xiaobo Hu
JiaXing University & Shandong University
China
1. Introduction
Sublimation method was used to grow bulk SiC by J.A. Lely for the first time in 1955 (Lely,
1955). It was improved then by Tairov and Tsvetkov and became the most mature method
for bulk SiC growth. In this chapter, we will introduce the growth of hexagonal SiC.
Although the bulk growth method is well known and used widely, there are still plenty of
details which are different and unique for different groups.
The growth of 4H-SiC is not as stable as that of 6H-SiC. That is to say the growth of 4H-SiC
needs a harsh growth conditions. In order to grow high quality 4H polytype, the polytype
transition of 4H-SiC single crystals had been studied.
Although single crystals of SiC are commercially available, owing to the specific structures
of SiC, there are still some structural defects, such as micropipes, mis-orientations,
dislocations, stacking faults, basal plane dislocations, particle inclusions, precipitates and so
on, which hinder its applications. So in this chapter we also introduce the recent progress in
research of structural defects in 6H-SiC single crystals. Three kinds of typical structural
defects in 6H-SiC single crystals were investigated. First, we describe the strain field of a
micropipe by the theory of screw dislocation. Stress birefringence images from micropipes
with different Burgers vectors have been simulated. The results are compared with
polarized optical microscopic observations. Second, elementary screw dislocations were
observed by back-reflection synchrotron radiation topography (BRSRT). Based on the
reflection geometry, the image of an elementary screw dislocation was simulated.
Elementary screw dislocation is a pure screw dislocation with Burger vector lc. Finally,
Basal plane bending was detected by high resolution X-ray diffractometry (HRXRD) and
transmission synchrotron white-beam x-ray topography (SWBXT).
The observation and investigation of the structural defects helped us to understand their
formation mechanisms. This makes it possible for us to further decrease or eventually
eliminate them.

region, screw dislocation mechanism controls the growth process. Fig. 2 is the morphology
of the area out of the facet. In this region, rough surface growth mechanism dominants. In
Fig. 2, there are also two different morphologies. In area A, the growth steps are very fine. In
area B, the surface is smooth. There is a slit between the two areas, which is not a scratch
caused by machining or annealing after growth. Normally the slit is along the <11-20>
direction and extends several or dozens of milli-meters on the as-grown surface.

Bulk Growth and Characterization of SiC Single Crystal

143

Fig. 2. Micrograph of the as-grown surface showing the existence of 4H-SiC, 15R-SiC at two
sides of the slit Fig. 3. Schematic diagram of one-dimensional Raman scanning route across the slit
In order to identify the polytype structures in the two areas with different morphologies,
Raman spectroscopy were used. One-dimensional Raman scanning was done cross the slit
in a range of 100 μm, as shown in Fig. 3.The dashed line represents the scanning path, and
the real line is the actual position of slit which is along the <11-20> direction.
The intensity ratio of folded transverse acoustic (FTA) mode of 15R-SiC (Raman shift at
172.3 cm
-1
) (Wang et al., 2004) and 4H-SiC (Raman shift at 204.99 cm
-1
) (Wang et al., 2004)
was introduced. In Fig. 4, the horizontal coordinate is along the dashed line in Fig. 3, and the
longitudinal coordinate is the intensity ratio. According to the intensity ratio, the scanning
scope can be divided into three regions. In region A, the intensity ratio is much greater than
AB


Fig. 4. The intensity ratio of FTA mode along the dashed line in Fig. 3.
A
B
C
D

Bulk Growth and Characterization of SiC Single Crystal

145
From Fig. 5c and 5d, 15R and 4H polytypes appear at the same time. The characteristic
peak of 15R-SiC dominates at point C. Both the characteristic peaks of 15R and 4H-SiC
are weak and the intensity of the background signal is strong at point D. That is to say,
the phonon state density is irregular in this area. In other words, the Si-C di-atom
stacking near slit is not completely disorder but contains short range order of 4H and
15R-SiC.

Fig. 5. The Raman spectra at different points of one-dimensional Raman scanning, (a)
region A, 15R-SiC; (b) region B, 4H-SiC; (c) point C 15R- and 4H-SiC; (d) point D 4H- and
15R-SiC.
2.3 Summary
In summary, the polytype transition is a process in which the stacking structure changes
from long range order to short range order and then back to long range regular. The
transition region in our observation is in a range of about 2-3μm. The slit is just the sign of
the polytype transition.
3. Characterizations
There are some structure defects in SiC single crystals which hinder its applications. For

micropipes (Giocondi et al., 1997; Liu et al., 2005). It has been proposed that the large steps
interact with unit screw dislocations and the heterogeneous phase to form the micropipe.
3.1.1 Experimental observations
Fig. 6a and 6c show the stress birefringence images of two typical micropipes with different
diameters respectively. They look like butterflies with four bright wings, and have a dark
core in the center. The wings’ length varies with the diameter of micropipe. The wings’
brightness was also not uniform from the center to the outside. The longer the distance from
the center, the weaker is the brightness of the wings. Fig. 6b and 6d exhibit the bright field
images of the two micripipes respectively. The open cores were visible distinctly in the
centers of micropipes especially for the one with large diameter. When we rotated the
sample, the birefringence pattern rotated simultaneously (as shown in Fig. 7).
3.1.2 Theoretical explanations
In this chapter we describe the strain field of a micropipe by the theory of screw dislocation.
According to Frank’s theory, micropipes are dislocations with large Burgers vectors. Therefore
the strain field caused by the micropipes could be described by dislocation theory and stress
birefringence image from a micropipe could be simulated. The existence of micropipes
changes the crystal from uniaxial to biaxial crystal in the neighbour area of a micropipe.
So the interference intensity near a micropipe should be written as follows:

22
0
0
( / )sin (2 2 ) when
0 when
Ar r r
I
rr
θα

−>

, the contour equation of the intensity curve can be obtained

222 2
0
sin (2 2 )
I
rf m
I
θαμ
=−

(4)

Bulk Growth and Characterization of SiC Single Crystal

147
2
d
b
a
-50 0 50
um
-25 0 25um

Silicon Carbide – Materials, Processing and Applications in Electronic Devices

148

Fig. 7. Microscopic images of micropipe observed by a polarizing optical microscope with
different α. The red and white arrows represent the directions of the polarizer and the <2-1-
10> direction, respectively. (a) α =0. (b) α =
π/4.
From Eq. 4, we can get interference intensity contours with different I
0
/I, as shown in Fig. 6.
The smaller the intensity of incident beam, the longer is the birefringence pattern’s radius.
In Fig. 6 the diameters of MP
1
and MP
2
are about 4.0μm and 0.66μm. According to Eq. 1 and
Eq. 4, the lengths of the brightest wings (in case of I=I
0
) for the two micropipes could be
calculated (Table 1). The observed and calculated values for these wings are nearly the same
with two different micropipes. The insets in the upper rights of Fig. 6a and 6c give the

Observed Calculated
MP1
4.00μm 22.2μm 21μm
MP2
0.66μm 8.3μm 9μm
Table 1. The observed and calculated wing lengths of MP
1
and MP
2
in case of I=I
0
a b

Bulk Growth and Characterization of SiC Single Crystal

149
3.1.3 Evolution of micropipes during growth
Fig. 8 shows the microscopic images for the same area but in different growth stages of the
same crystal. The micropipes’ distribution proves that Fig. 8a and 8b, 8c and 8d are the same
areas in different growth stages respectively. The diameter of the micropipe is increasing
with the crystal growth which can be proved by the size of bright wings in the birefringence
images. It means that the stress can be released through micropipes during cryatal growth.
According to statistic of the densities of micropie along crystal growth, the micropipe
density decreased with growth process for almost all samples, except one. That is to say, in
case of optimizing growth conditions, micropipe density can be decreased and crystal
quality can be improved.


Fig. 9. Micropipes distribution along growth direction
3.1.4 Summary
In summary, the birefringence images of micropipes in 6H-SiC single crystals were observed
by a polarization optical microscope. Based on the Frank’s theory, the micropipes were
treated as screw dislocations with huge Burgers vectors. Due to the strain caused by a
micropipe, the SiC crystal in the neighbour area around a micropipe transferred from
uniaxial into biaxial crystal. In the meanwhile, the refractive index ellipsoid was described
by three principal refractive indexes and birefringence occurred. Based on above
consideration, the birefringence images of micropipes with different Burgers vectors were
simulated. The results agreed well with the observations. It confirms indirectly the Frank’s
theory that a micropipe is actually the super-screw dislocation with huge Burgers vector.
With crystal growth, the micropipe density has a tendency of decrease and the diameter of
micropipes can be enlarged for the stress release.
3.2 Elementary screw dislocations
Although the elementary screw dislocations did not seriously deteriorate the performance of
device as micropipes, they prevent the realization of high-efficiency, reliable electronic
devices (Lendenmann, 2001; Malhan et al., 2003; Neudeck et al., 1998).
In order to assess the density of elementary screw dislocation, back-reflection synchrotron
radiation topograph of (0001) wafer was taken. In experimental setup, symmetric diffraction
geometry with 000
30 reflection was used. The angle between incident beam and sample
surface was 83°. In this case, the selected X-ray wavelength is 0.1nm. The distance between
sample and film was 20cm. The illumination area is 1mm x 1mm.

Bulk Growth and Characterization of SiC Single Crystal

151
Fig. 10 shows the back-reflection synchrotron radiation topograph of SiC (0001) wafer with
000
30 reflection. From the topograph, many white dots with the same diameters of 26-28μm

2221/2
2221/2
2221/2
(,,) /( 4 )
(,,) /( 4 )
(,,) 2 /( 4 )
x
y
z
nxyz byrb r
nxyz bxrb r
nxyz r b r
π
π
ππ
=+
=− +
=+

(6)
Where r=(x
2
+y
2
)
1/2
. For the numerical simulation of the screw dislocation, the distorted
lattice was divided into a set of small cubes of constant. All the cubes were assumed to have
the same integrated diffraction density I
0

The HRXRD was conducted by XPERT-PRO diffractometer. The incident beam was adjusted
by the beam conditioner of four Ge(022) collimated crystals so that the X-ray beam on the
sample was accurately the Cu-Kα
1
(λ=1.54056Å) radiation. The symmetrical diffraction
geometry has been used in the measurement. The slits for the incident beam and detector
are 0.5mm × 4mm and two degree respectively. The tube voltage and tube current are 40kV
and 40mA respectively.
Several samples were investigated by transmission synchrotron topography performed at
the 4W1A beam line of Beijing Synchrotron Radiation Laboratory. In our experiment, the
electron energy of storage ring was 2.2GeV and the beam current was 80-100mA.
Transmission synchrotron topographs were taken by transmission Laue geometry. The

Bulk Growth and Characterization of SiC Single Crystal

153
samples were orientated with the (0001) surface perpendicular to the incident beam and the
topographs were recorded by Fuji film.
3.3.1 Experimental results-HRXRD
Fig. 12 shows the plot of the relative ω
006
rocking curve peak position as a function of the
beam position across the diameters of a 2 inch wafer. The three groups of data in Fig. 12
were got from three different diameters of the same wafer. In case of precisely orientated
crystal without any basal plane bending, ω is equal to Bragg angle. But the HRXRD test here
is not the case. When the incident beam is located on the left of the center, the ω is larger
than Bragg angle and the deviation of ω from Bragg angle is proportional to the distance
between detected point and the center of the wafer. The farther the distance of the detected
point from the center, the larger the deviation of surface from the basal plane. When the
incident beam is located on the right of the center, the result is reversal, i.e. the ω is smaller