Crystal Structure Theory and Applications, 2012, 1, 114-120
http://dx.doi.org/10.4236/csta.2012.13021 Published Online December 2012 (http://www.SciRP.org/journal/csta)
Computational Study of Induction Heating Pr ocess in
Crystal Growth Systems—The Role of
Input Current Shape
Mohammad Hossein Tavakoli1*, Tayebe Nadery Mostagir2
1Physics Department, Bu-Ali Sina University, Hamedan, Iran
2Department of Physics, Kurdistan University, Sanandaj, Iran
Received October 28, 2012; revised November 30, 2012; accepted December 8, 2012
ABSTRACT
A set of 2D steady state finite element numerical simulations of electromagnetic fields and heating distribution for an
oxide Czochralski crystal growth system was carried out for different input current shapes (sine, square, triangle and
sawtooth waveforms) of the induction coil. Comparison between the results presented here demonstrates the importance
of input current shape on the electromagnetic field distribution, coil efficiency, and intensity and structure of generated
power in the growth setup.
Keywords: Computer Simulation; Induction Heating; Czochralski Method; Growth from Melt; Metals
1. Introduction
Radio frequency induction heating is frequently used in
crystal growth technology. The process principle consists
of applying an alternating current in a conductor or coil
called inductor (RF-coil) that generates an alternating
electromagnetic field in the space. The alternating elec-
tromagnetic field induces eddy currents in metal crucible
where the crystal material is placed and should be to melt.
These currents lead to Joulean heating (RI2) of the cruci-
ble in the form of temporal and spatial volumetric heat-
ing. Distribution and control of the induced power along
the crucible cross-section and length are quite important
which result in temperature difference and flow field in
the growth setup [1-3].
In order to produce the required heating pattern within
a metal crucible and afterheater it is necessary to accu-
rately model and predict the electromagnetic field and
the eddy currents distribution produced by the RF-coil
under different operating conditions such as geometry
and orientation of the metallic parts, cross section of the
coil turns, the crucible shape and position, and frequency
choice [4-7]. Selection of input current shape is another
critical issue, which is particularly important for certain
selective heating applications. In this article, we try to
investigate the effects of different input voltage shapes,
i.e., sine, square, triangle and sawtooth waveforms (Fig-
ure 1) on the strength and distribution of the electro-
magnetic fields and heat generation in a Czochralski se-
tup using the mathematical modeling and computer si-
mulation. It should be noted, however, that despite of the
differences in the patterns, each pattern is periodic. This
point is important for our analysis of the driving current
shape, i.e., they can be represented as closely as desired
by the combination of a sufficiently large number of si-
nusoidal patterns that form a harmonic series (Fourier
series). Every non-sinusoidal current pattern consists of a
fundamental and a complement of harmonics, which can
be considered as a superposition of sine pattern of a fun-
damental frequency ω and integer multiples of that fre-
quency [8].
2. Mathematical Model
2.1. Governing Equations
Since the real induction heating process is very complex,
we make some simplifying assumptions in our approach.
In our mathematical model used for numerical calcula-
tions, we make the following five assumptions: 1) the
heating system is rotationally symmetric about the z-axis,
so that all quantities are independent of the azimuthal
coordinate φ; 2) all materials are isotropic, non-magnetic
and have no net electric charge; 3) the displacement cur-
rent is neglected; 4) the distribution of driving electrical
current (also voltage) in the RF-coil is uniform; and 5)
*Corresponding author.
C
opyright © 2012 SciRes. CSTA

M. H. TAVAKOLI, T. N. MOSTAGIR 115
Figure 1. Four input current shapes (a) Sine; (b) Square; (c)
Triangle; and (d) Sawtooth wave for ms of the induction coil.
the driving and induced currents have only one angular
component (i.e., φ-direction). Under these assumptions,
the governing equations are [4];
0
11
ˆ
BB
B
J
rr rzr r



 





 (1)
where
11
ˆB
rr rzrr
B

  







(2)
and
coil coilcoil
crucible
1
driving and eddy currents in the coil
1eddy currents in the crucible
co B
de d
cr B
e
JJ Jrr t
J
Jrr t










 

(3)
in which
B

is the magnetic stream function defined by

t r

,,,, ,
BrzA rzt 


 where Aφ is the azimuthal com-
ponent of the vector potential, the cylindrical
coordinates, J the charge current density, σ the electrical
conductivity, 0

,rz


the magnetic permeability of free
space and t the time.
The energy dissipation rate in all metallic parts (coil,
crucible and afterheater) is computed as

2
,,
J
Przt


 (4)
Finally we average over one period to obtain the vo-
lumetric heat generation rate (i.e., the time averaged
quantity),
 
2π
0
,Pr,z,
2π
qrz



where

is the frequency of the electrical current in the
induction coil.
a) Sine Waveform
Assuming the driving current in the RF-coil as a sine
form 0sin
d
J
Jt


, we can consider a solution of the
form


sine ,sin ,cos
BSrzwt Crzwt
  (6)
where


,Srz is the in-phase component and


,Crz
is the out-of-phase component of the solution.
Now the coupled set of elliptic PDE’s for


,Srz
and


,Crz is:
00
sine
0
coil
ˆcrucible
0elsewhere
co
co
JC
r
SC
r
















 (7)
0
sine
0
coil
ˆcruci ble
0elsewhere
co
co
S
r
CS
r













 (8)
where is the linear operator defined in (2).
ˆ

After solving (7) and (8) for and

,Srz



,Crz,
the eddy currents distribution and the energy dissipation
rate can be computed via
 
sine ,cos ,sin
cos sin(9)
B
e
CS
J
SrzwtCrzwt
rt r
JwtJwt
  
 






and



2
2
2
200
2
2
22
2
sin coil
sin2crucible (1
,,
0)
co rr
co co
cr
J
JJ
SCSC
Przt
t
r
SCCS t
r


 
 
 




 











Consequently, the volumetric heat generation rate is


2
2
20
2
sine
2
22
2
coil
,
crucible
co r
co
cr
J
SC
r
qrz
SC
r















(11)
tdt (5)

M. H. TAVAKOLI, T. N. MOSTAGIR
116
b) Square Waveform
The square waveform of the driving current in the RF-
coil can be approximated by a sum of harmoni
Fourier series as
cs using
 
square 0
0
1
sin 21
4
π21
d
n
nt
J
JJF tn




 
 (12)
and then
square square
1
BBn
n


 

 
1
sin 21cos2
nn
n
SntC n








1t


(13)





square square
11
21 sin 21
cos 21
en
en
nn
n
n
J
JCn
r
Sn








t
t





(14)



0
square
21
4coil
21π
21
ˆcrucible
0elsewhere
(15)
co
co n
cr cr
nn
n
JC
nr
n
SC
r





















square
21 coil
21
ˆcrucible
0elsewhere
(16)
co co
n
cr cr
nn
nS
r
n
CS
r












 



square square
1
2
22
20
22
1
22
22
2
1
,,
21 4coil
221π
21 crucible (17)
2
n
n
co
nn
nco
cr
nn
n
qrz qrz
nJr
SC
rn
nSC
r














 















c) Triangle Waveform
The triangle waveform of the input current in the coi
is approximated as
l
 


1
triangle 0
022
1
1
8sin 21
π21
n
d
n
J
J
JFtn t
n








(18)
t

1
BBn
n

 
1
sin 21cos 21
nn
n
SntCn

  


triangle triangle

 


(19)

 

triangle triangle
1
1
21 sin 21
cos 21
een
n
n
n
n
JJ
nCn
r
Sn






 





t
t


(20)




1
0
22
triangle
18 21coil
21π
ˆ21 crucib le
0elsewhere
(21)
n
co
co n
cr cr
nn
Jn C
r
n
n
SC
r























triangle
21 coi l
21
ˆcruc ib le
0elsewhere
co
n
c
c
rcr
n
o
n
nS
r
n
CS
r


 








 (22)








triangle
22
2
1
1
0 triangle22
32
1
22
122
2
,
21
2
18 coi l
21π
21 crucible
2
co
n
n
nn n
nco
ncr
nn
qrz
n
r
Jr
qS C
n
nSC
r
















 















(23)
d) Sawtooth Waveform
The related equations of the sawtooth waveform of the
input current can be written similar to the squ
triangle waveforms. They are
are and

sawtooth 0
02
1
sin
π
d
n
Jnt
JJF tn





(24)
n


sawtooth sawtooth
B
Bn


11nn
sin cos
n
SntCnt


 (25)


sawtooth sawtooth
11
sin cos
e
en
nn
n
nn
J
JSntCnt
r





 
 (26)
0
sawtooth
coil
π
ˆcrucible
0elsewhere
co
co n
cr cr
nn
Jn C
nr
n
SC
r




















(27)

M. H. TAVAKOLI, T. N. MOSTAGIR 117
sawtooth
coil
ˆcrucible
0elsewhere
co co
n
cr cr
nn
nS
r
n
CS
r











 (28)




sawtooth
22
22
0
22
1
sawtooth
22
122
2
1
,
coil
2π
crucib le
2
co
nn
nco
n
ncr
nn
n
qrz
nJr
SC
rn
q
nSC
r
























(29)
2.2. The Calculation Conditions
The driving current density in the induction coil is calcu-
lated by

0coil coil
2π,
co
J
VRN   where is
total voltage of the coil, is the m
coil radius and N is the numof coil turns. e bound-
ary conditions are
coil
V
ean value of the
Th
the
coil
R
ber
0
B

; both in the far field


,rz and at thmetry (r
em for our
parameters
tem is shown in Figure und
sing
s been
method.
r distribution in the crucible, afterheater and
R
e axis of sym= 0).
ployedValues of electrical conductivity
calculations are presented in [9], operating
are listed in Table 1 and the geometry of the growth
heating sys 2. The famental
partial equations require u a numerical discretization
method to solve them. Calculation of the equations with
boundary conditions ha made by 2D finite element
The two-dimensional computational domain with the
finite element triangle mesh is shown in Figure 3. In the
space close to and in the metal parts (i.e. crucible, after-
heater and RF-coil) the mesh is denser because of the
high gradients of the electromagnetic fields. After solv-
ing the set of equations, we can obtain the electromag-
netic field structure in the system as well as the volumet-
ric powe
F-coil.
Table 1. Operating parameters used for calculations.
Description (units) Symbol Value
Crucible inner radius (mm)
Crucible wall thickness (mm)
Crucible inner height (mm)
Baffle inner radius (mm)
B
Heig
rc
lc
hc
rb
50
2
100
35
ottom heater height (mm)
ht of the thick bottom (mm)
Coil width (mm)
Coil wall thickness (mm)
D)
T
Cur
hbh
htb
l
50
10
13
Radius of the round bottom corner (mm)
Coil inner radius (mm)
rcb
rco
10
78
Height of coil turns (mm)
istance between coil turns (mm
otal voltage of the RF-coil (v)
rent frequency of RF-coil (kHz)
co
lco
hco
dco
Vcoil
f
1.5
20
3
200
10
Figure 2. Sketch of an oxide Czochralski growth heating.
Figure 3. The finite element mesh structure of the calcula-
tion domain.
3. Results and Discussion
We explain the results of electromagnetic field and heat-
ing pattern in an oxide CZ setup including a cylindrical
metal crucible, active afterheater and RF-coil correspond-
ing to a real growth situation with differeapes of driv- nt sh
ing current in the RF-coil and with unique amplitude and
frequency.
3.1. Electromagnetic Fields
Figures 4-7 show the distribution of in-phase component
and out-of-phase component of the magnetic stream
function (ψB) for the cases of sine, square, triangle, and
sawtooth waveforms, respectively, in the growth setup.

M. H. TAVAKOLI, T. N. MOSTAGIR
118
Figure 4. Components of the magnetic stream function (ψB)
calculated for the case of sine waveform. The left hand side
shows the in-phase component (S) and the right hand side
shows the out-of-phase component (C) in the setup.
Figure 5. Components of the magnetic stream function (ψB)
calculated for the case of square waveform. The left hand
side shows the in-phase component (S) and the right hand
side shows the out-of-phase component (C) in the heating
setup.
The maximum of in-phase component


max
S
while t
is located
at the lowest and top edges of the RF-coil he mi-
nimum


min
S is located on the middouter surle of the -
face of crucible and afterheater wall, for the square and
triangle waveforms. But for the sine and sawtooth wave-
forms, it is vice versa, that is, the positions of the


max
S
and

C are replaced. For the out-of-phase compo-
ent (C
max
), the minnimum is located on th
f sine,
ase o
compo
e outer surfaces of
the induction coil turns for the cases osquare and
triangle waveforms while the maximum is placed there
for the cf sawtooth waveform. The distribution of
C-component has a linear gradient in the space between
the coil and the crucible and afterheater wall for all cases.
The crucible and afterheater wall squeezes this -
Figure 6. Components of the magnetic stream function (ψB)
calculated for the case of triangle waveform. The left hand
side shows the in-phase component (S) and the right hand
side shows the out-of-phase component (C) in the heating
setup.
Figure 7. Components of the magnetic stream function (ψB)
calculated for the case of sawtooth waveform. The left hand
side shows the in-phase component (S) and the right hand
side shows the out-of-phase component (C) in the heating
setup.
nent to the area between the crucible and afterheater, and
the RF-coil. Some interesting advantages are:
 The gradient of the S-component is too high in the
rly visible for all cases (edge effect). For the
area close to the maximum and minimum points,
which is not true for other parts of the system;
 Deformation and distortion of the S-component in the
area close to the extreme edges of the RF-coil are
particula
crucible and afterheater, spatial distribution of the S-
component is along and parallel to their sidewall;
 The strongest electromagnetic fields belong to the
square waveform while the weakest fields are pro-
duced by the sawtooth waveform.

M. H. TAVAKOLI, T. N. MOSTAGIR 119
3.2. Heat Generation
The volumetric heat generation rate (q) in the crucible
and afterheater has been shown for all cases in Figure 8.
The power intensity is at its maximum value at the mid-
dle portion of the outer surface of the crucible sidewall,
which arises from the skin effect.
The most important features are:
 The heating structure of the crucible and afterheater is
expect for their intensity. The
is produced by square, sine,
the same for all cases
most powerful energy
triangle and sawtooth waveform, respectively, Figure
9. This feature is predictable from the related elec-
tromagnetic fields distribution;
(a) (b) (c) (d
Figure 8. Volumetric power distribution (q) in the crucib
and afterheater computed for (a) sine; (b) square; (c) trian-
gle; and (d) sawtooth waveform of the driving current (for a
better demonstration the crucible and afterheater sidewal
and bottom are separately magnified).
)
le
l
Figure 9. Profiles of the heat generated along the outer sur-
face of the crucible and afterheater side wall calculated for
(a) sine; (b) square; (c) triangle; and (d) sawtooth waveform
of the input current.
 The spatial distribution of heat generation in the in-
duction coil is mostly uniform with local “hot spots”
(highly heated areas) at the lowest and upper edges,
which is shown in Figure 10. The skin effect and pro-
ximity effect are responsible for appearance of these
undesirable overheating because the induced eddy cur-
rents are concentrated on the top and lowest corners
of the RF-coil [10,11];
 It is worth to note that despite of different total power
generation, the coil efficiency (i.e., the part of the en-
ergy delivered to the coil that is transferred to the
workpiece) does not change and is approximately the
same for all cases (Table 2).
4. Conclusions
erical calculations was performed.
To study the dependence of electromagnetic distribution
and heating pattern on the input current shape (sine,
square, triangle and sawtooth waveforms) of the induction
coil, a set of 2D num
(a) (b) (c) (d)
Figure 10. Volumetric power distribution (q) in the induc-
tion coil calculated for (a) Sine; (b) Square; (c) Triangle;
and (d) Sawtooth waveform of the input current.
Table 2. Detail information about the heat generated in the
CZ coil (Heating efficiency is the part of the energy deliv-
ered to the coil that is transferred to the crucible and after-
heater).
Waveform Crucible and
afterheater (kW)
Induction coil
(kW)
Heating
efficiency (%)
Sine 15 1.4 91.5
Square 26 2.4 91.6
Triangle10 0.9 91.5
Sawtooth1.9 0.2 92.1

M. H. TAVAKOLI, T. N. MOSTAGIR
120
From the computational results described above, we can
conclude:
 The spatial structure of electromagnetic fields and
generated heat is a complex function of several pa-
rameters such as setup geometry and driving current
shape;
 The electromagnetic fields distribution within the cru-
cible and afterheater as well as the RF-coil is not uni-
form. This electromagnetic fields nonuniformity causes
a nonuniform heating pattern in the crucible and af-
terheater, which in turn leads to a nonuniform tem-
perature profile in the growth system.
A square input current results in a high intense heating
of the crucible and afterheater while a sawtooth wave-
form leads to a low heating intensity in that part of the
system. Different amount of produced energy in the setu
is due to differences in the intensity and distribution of
the electromagnetic fields. Understanding the physics of
these n-
No. 3-4, 1989, pp. 792-826.
doi:10.1 (8
p
properties is important during designing of an i
duction system for certain crystal growth applications.
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